f ′ g (X s )dX s + 1 2 R L a t (X)f ′′ (da) ✭✹✮ ❉❡ ♥♦♠❜r❡✉① ❛✉t❡✉rs s❡ s♦♥t é❣❛•❡♠❡♥t ✐♥t➥❡ssés à •❛ ♣♦ss✐❜✐•✐té ❞❡ r❡♠♣•❛❝❡r •❛ ❝♦♥❞✐t✐♦♥ f ∈ C 2 ♣❛r ✉♥❡ ❝♦♥❞✐t✐♦♥ ❞✉ t②♣❡ f ∈ W p,1 ✱ ✉♥ ❡s♣❛❝❡ ❞❡ ❙♦❜♦•❡✈✳ P❛r ❡①❡♠♣•❡ ♣♦✉r •❡ ❝❛s ♣❛rt✐❝✉•✐❡r ❞✉ ♠♦✉✈❡♠❡♥t ❜r♦✇♥✐❡♥ ❡♥ ❞✐♠❡♥s✐♦♥ 1✱ ❬✶✸( ❡①♣♦s❡ •❡s ♣r♦♣r✐étés ❞❡ s❡♠✐♠❛rt✐♥❣❛•❡ ❞❡ x → L x t (X) à t ✜①é ❡t ét❛❜•✐t •❡ rés✉•t❛t s✉✐✈❛♥t✿ ❚❤é♦rè♠❡ ✵✳✶✳✷✳✸ ✭❋♦r♠✉•❡ ❞❡ ❇♦✉•❡❛✉✲(♦r✮✳ ❙♦✐t f ✉♥❡ ❢♦♥❝t✐♦♥ ❞ér✐✈❛❜•❡ s✉r R✱ ❞❡ ❞ér✐✈é❡ •♦❝❛•❡♠❡♥t ❜♦r♥é❡ ❡t W ✉♥ ♠♦✉✈❡♠❡♥t ❜r♦✇♥✐❡♥ ❞❡ ❞✐♠❡♥s✐♦♥ 1✳ ❖♥ ❛✿ f (W t ) = f (0) + t 0 f ′ (W s )dW s - 1 2 • R f ′ (x)d x L x t (X) ✭✺✮ ❈❡tt❡ ❝♦♥str✉❝t✐♦♥ ❢❛✐s❛♥t ✐♥t❡r✈❡♥✐r •❡ t❡♠♣s •♦❝❛• ❞✉ ♠♦✉✈❡♠❡♥t ❜r♦✇♥✐❡♥✱ ❡••❡ ❡st s♣é❝✐❢q✉❡ à •❛ ❞✐♠❡♥s✐♦♥ 1✳ ❈❡♣❡♥❞❛♥t✱ ❬✷✵( ♣r♦♣♦s❡ ✉♥❡ ❣é♥ér❛•✐s❛t✐♦♥ s♦✉s ✉♥❡ ❤②♣♦t❤ès❡ ❡♥❝♦r❡ ♣•✉s ❢❛✐❜•❡ q✉✐✱ q✉❛♥t à ❡••❡✱ s✬ét❡♥❞ ❛✉ ❝❛s N = d✿ ❚❤é♦rè♠❡ ✵✳✶✳✷✳✹ ✭❋♦r♠✉•❡ ❞❡ ❋ö••♠❡r✲Pr♦tt❡r✲❙❤✐r②❛❡✈✮✳ ❙♦✐❡♥t f ❞ér✐✈❛❜•❡ t❡••❡ q✉❡ Df ∈ L 2 loc R N , R N ❡t W ✉♥ ♠♦✉✈❡♠❡♥t ❜r♦✇♥✐❡♥ ❡♥ ❞✐♠❡♥s✐♦♥ N ✳ ❖♥ ❛✿ f (W t ) = f (0) + N i=1 ∂ i f (W s )dW s + 1 2 • [f ′ (W ), W ] t ✭✻✮ ♦ù •❛ ❝♦✈❛r✐❛t✐♦♥ q✉❛❞r❛t✐q✉❡ [f ′ (W ), W ] ❡st ❞é✜♥✐❡ ❝♦♠♠❡ ✉♥❡ •✐♠✐t❡ ❞❡ s♦♠♠❡s ❞❡ |✐❡♠❛♥♥✿ [f ′ (W ), W ] t = N i=1 lim n→∞ n k=1 f ′ (W k n ) -f ′ (W k-1 n ) • W k n -W k-
♦rè♠❡ ✵✳✶✳✷✳✺✳ ❙♦✐❡♥t W ✉♥ ♠♦✉✈❡♠❡♥t ❜r♦✇♥✐❡♥ à ✈❛•❡✉rs ❞❛♥s R d ❡t f ∈ W 2,1 R d ✳ ❆•♦rs ✐• ❡①✐st❡ ✉♥❡ ✉♥✐q✉❡ ❞é❝♦♠♣♦s✐t✐♦♥✿ f (W t ) = f (0) + M t (f ) + N t (f ) ✭✽✮ ♦ù M (f ) ❡st ✉♥❡ ♠❛rt✐♥❣❛•❡ •♦❝❛•❡ ❞❡ ❝❛rré ✐♥té❣r❛❜•❡ ❡t N (f ) ❡st ✉♥ ♣r♦❝❡ss✉s ❝♦♥t✐♥✉✱ ❛❞❞✐t✐❢✱ ❞✬é♥❡r❣✐❡ ♥✉••❡✳ ✵✳✶✳✸ ▲❡ ❝❛s ❞❡s ❊❉❙ ■• ❡①✐st❡ é❣❛•❡♠❡♥t ❞❡ ♥♦♠❜r❡✉① rés✉•t❛ts ♣❡r♠❡tt❛♥t ❞✬é❝r✐r❡ ✉♥❡ ❢♦r♠✉•❡ ❞✬■tô ♣♦✉r f ∈ W p,1 ♣♦✉r ❞❡s ♣r♦❝❡ss✉s ♣•✉s ❣é♥ér❛✉① q✉❡ •❡ s❡✉• ♠♦✉✈❡♠❡♥t ❜r♦✇♥✐❡♥✳ ❊♥ ❣é♥ér❛• ✐• ❡st ♥é❝❡ss❛✐r❡ ❞❡ ❢❛✐r❡ ✉♥❡ ❤②♣♦t❤ès❡ ❞✬❡••✐♣t✐❝✐té s✉r •❡ ♣r♦❝❡ss✉s X✳ P♦✉r •❡ rés✉•t❛t s✉✐✈❛♥t✱ ♦♥ s❡ ❞♦♥♥❡ ❞❡✉① ❢♦♥❝t✐♦♥s b ❡t σ ♣♦✉r •❡sq✉❡••❡s ✐• ❡①✐st❡ ✉♥❡ ❝♦♥st❛♥t❡ ❈ t❡••❡ q✉❡ ♣♦✉r t♦✉s x, y ∈ R N ✿ sup t≥0 |b(t, x) -b(t, y)| ≤ C •|x -y| ✭✾✮ sup t≥0 |σ(t, x) -σ(t, y)| ≤ C •|x -y| ✭✶✵✮ ❆•♦rs✱ ✐• ❡①✐st❡ ✉♥❡ ✉♥✐q✉❡ s♦•✉t✐♦♥ ❢♦rt❡ à •✬éq✉❛t✐♦♥ ❞✐✛✬❡r❡♥t✐❡••❡ st♦❝❤❛st✐q✉❡✿ X t = X 0 + t 0 b(s, X s )ds + t 0 σ(s, X s )dW s ✭✶✶✮ ❖♥ ♥♦t❡ ❡♥✜♥ A •❡ ❣é♥ér❛t❡✉r ❞✉ ♣r♦❝❡ss✉s❀ s✐ a = σ • t σ✿ A = b i • ∂ i + 1 2 • a ij • ∂ ij ✭✶✷✮ ❆✈❡❝ ❝❡s ♥♦t❛t✐♦♥s✱ ❬✸✽( ét❛❜•✐t✿ ❚❤é♦rè♠❡ ✵✳✶✳✸✳✶✳ ❙✉♣♣♦s♦♥s q✉❡ b ❡t σ s♦✐❡♥t ❜♦r♥és ♣r❡sq✉❡ sûr❡♠❡♥t ❡t q✉❡ A s♦✐t ✉♥✐❢♦r♠é♠❡♥t ❡••✐♣t✐q✉❡✱ ❝✬❡st à ❞✐r❡ q✉✬✐• ❡①✐st❡ ✉♥❡ ❝♦♥st❛♥t❡ C t❡••❡ q✉❡ ♣♦✉r t♦✉t ξ ∈ R N ✿ (A • ξ, ξ) = i,j a ij ξ i ξ j ≥ C • ξ 2 ✭✶✸✮ ❆•♦rs✱ ♣♦✉r t♦✉t f ∈ W 2,2 ✱ ♦♥ ♣❡✉t é❝r✐r❡ •❛ ❢♦r♠✉•❡ ❞✬■tô ♣♦✉r f (X t ) ❝♦♠♠❡ ❞❛♥s •❡ ❝❛s C 2 ✳ Σ ij (X) = (∇X i , ∇X j ) H ✭✶✽✮ ❡t ♦♥ s✉♣♣♦s❡ q✉❡ X ❡st ♥♦♥✲❞é❣é♥éré❡✱ ❝✬❡st à ❞✐r❡ q✉❡✿ γ(X) := Σ(X) -1 ∈ 1<p<∞ L p ✭✶✾✮ ✵✳✷✳ ❆PP❖|❚❙ ❉❯ ❈❆▲❈❯▲ ❉❊ ▼❆▲▲■❆❱■◆ ①✈✐✐ ❆•♦rs✿ E [∂ i f (X) • Y ] = E [f (X) • l i (Y, X)] ✭✷✵✮ ♦ù l i (Y, X) = δ Y • N i=1 γ ij (X) • X j ✭✷✶✮ ❈❡ rés✉•t❛t ♣❡r♠❡t ❞❡ ❞é✜♥✐r f ′ (X) ∈ D ′ ♣❛r ❞✉❛•✐té ♣♦✉r X ∈ D✱ ✈❛r✐❛❜•❡ ❛•é❛t♦✐r❡ ♥♦♥✲❞é❣é♥éré❡ s✉r •✬❡s♣❛❝❡ ❞❡ ❲✐❡♥❡r ❡t f ✉♥❡ ❢♦♥❝t✐♦♥ t❡••❡ q✉❡ f ′ ♥✬❡st ❞é✜♥✐❡ q✉✬❛✉ s❡♥s ❞❡s ❞✐str✐❜✉t✐♦♥s✳ P•✉s ♣ré❝✐sé♠❡♥t✱ ♣❛r ✉♥❡ ré❝✉rr❡♥❝❡ s✐♠✲ ♣•❡ ❢♦♥❞é❡ s✉r •❡ t❤é♦rè♠❡ ♣ré❝é❞❡♥t ❡t •❡ ❢❛✐t q✉❡ •✬❡♥s❡♠❜•❡ ❞❡s ❞✐str✐❜✉t✐♦♥s t❡♠♣éré❡s s✬♦❜t✐❡♥t✱ ♣♦✉r t♦✉t 1 < p < ∞✱ ❝♦♠♠❡✿ S ′ = k∈N Id + X 2 -∆ -k (L p ) ✭✷✷✮ ✐• ❛ été ét❛❜•✐ ❞❛♥s ❬✽✶( q✉❡ •✬♦♥ ♣❡✉t ❞é✜♥✐r T • X ∈ D ′ ♣♦✉r t♦✉t X ∈ D ♥♦♥✲ ❞é❣é♥éré❡ ❡t T ∈ S ′ ✳ ❈❡•❛ ❛ ♣❡r♠✐s ❞✬♦❜t❡♥✐r •✬❡①t❡♥s✐♦♥ s✉✐✈❛♥t❡ ❞❡ •❛ ❢♦r♠✉•❡ ❞✬■tô✱ ✈♦✐r ❬✼✽(✿ ❚❤é♦rè♠❡ ✵✳✷✳✷✳✷✳ ❙♦✐❡♥t b ❡t σ ❞❡✉① ♣r♦❝❡ss✉s st♦❝❤❛st✐q✉❡s ❛❞❛♣tés à •❛ ✜•tr❛✲ t✐♦♥ ❞✉ ♠♦✉✈❡♠❡♥t ❜r♦✇♥✐❡♥ W ❀ ♦♥ s✉♣♣♦s❡✱ ♣♦✉r t♦✉s p ❡t k✿ t 0 b s p D p,k ds < ∞ ✭✷✸✮ t 0 σ s p D p,k ds < ∞ ✭✷✹✮ ❡t ♦♥ ✐♥tr♦❞✉✐t✿ X t = X 0 + t 0 b s ds + t 0 σ s dW s ✭✷✺✮ ❙♦✐t Σ s •❛ ♠❛tr✐❝❡ ❞❡ ▼❛••✐❛✈✐♥ ❞❡ X s ❀ ♦♥ s✉♣♣♦s❡ ❡♥ ♦✉tr❡✿ t 0 Σ -1 s p L p ds < ∞ ✭✷✻✮ ♦✉✱ ❝❡ q✉✐ r❡✈✐❡♥t ❛✉ ♠ê♠❡✿ t 0 1 det Σ s p L p ds < ∞ ✭✷✼✮ ①✈✐✐✐ ■◆❚|❖❉❯❈❚■❖◆ ❆•♦rs ♣♦✉r t♦✉t ǫ > 0 ♦♥ ❛✿ T (t) • X t -T (ǫ) • X ǫ = M (ǫ) t + i t ǫ b i (u) • (∂ i T u ) • X u du + 1 2 i,j t ǫ a ij (u) • (∂ ij T u ) • X u du + t ǫ T (du) • X u ✭✷✽✮ ❛✈❡❝✱ ❡♥ ✉t✐•✐s❛♥t •❛ ♥♦t❛t✐♦♥ ❞❡s ✐♥té❣r❛•❡s st♦❝❤❛st✐q✉❡s ♣♦✉r •❛ ❞✐✈❡r❣❡♥❝❡✿ M (ǫ) t = j t ǫ i [b i (u) • (∂ i T u ) • X u ] dW j u ✭✷✾✮ ▲❛ ♣r❡✉✈❡ ❞❡ ❬✼✽( ❡st ❢♦♥❞é❡ s✉r •❛ st✉❝t✉r❡ ❞✬❡s♣❛❝❡ ❞❡ ❋ré❝❤❡t ♥✉❝•é❛✐r❡ ❞❡ S ′ ✳ ❊♥ ❡✛❡t✱ ♦♥ ♣❡✉t ❛•♦rs ❛♣♣•✐q✉❡r •❡ t❤é♦rè♠❡ ❞❡ •r♦t❤❡♥❞✐❡❝❦ ♣♦✉r ♦❜t❡♥✐r ✉♥❡ r❡♣rés❡♥t❛t✐♦♥ ❞✉ ❝❤❡♠✐♥ T t ❞✉ t②♣❡✿ T t = ∞ n=1 λ n • V n (t) • T n ✭✸✵✮ ♦ù (λ n ) ∈ l 1 ✱ (V n ) ❡st ✉♥❡ ❢❛♠✐••❡ ❜♦r♥é❡ ❞❡ ❢♦♥❝t✐♦♥s à ✈❛r✐❛t✐♦♥s ❜♦r♥é❡s s✉r R ❡t (T n ) ❡st ✉♥❡ ❢❛♠✐••❡ ✉♥✐❢♦r♠é♠❡♥t ❝♦♥t✐♥✉❡ ❞❡ S ′ ✳ ❖♥ ✈♦✐t ❛•♦rs q✉❡ •✬♦♥ ♣❡✉t ❝❤♦✐s✐r k t❡• q✉❡ φ t := (Id + X 2 -δ) -k T t s♦✐t ✉♥❡ ❢♦♥❝t✐♦♥ C 2 ❡♥ ❡s♣❛❝❡ ❞❡ ❢❛ç♦♥ ✐♥❞é♣❡♥❞❛♥t❡ ❞❡ t❀ ❞❡ ♣•✉s✱ •❛ ❢♦♥❝t✐♦♥✿ (t, x) → φ t (x) = Id + X 2 -δ -k T t (x) ✭✸✶✮ ❡st ❛•♦rs à ✈❛r✐❛t✐♦♥ ❜♦r♥é❡ ❡♥ t❡♠♣s ✭❡t C 2 ❡♥ ❡s♣❛❝❡✮✳ ▲❡ rés✉•t❛t ❡st ✜♥❛•❡♠❡♥t ♦❜t❡♥✉ ❡♥ ❛♣♣•✐q✉❛♥t •❛ ❢♦r♠✉•❡ ❞✬■tô ❝•❛ss✐q✉❡ à •❛ ❢♦♥❝t✐♦♥ φ ❡t ❡♥ ✧✐♥
♦rè♠❡ ✵✳✸✳✶✳✶✳ ❙♦✐t F ∈ D ∞ -,1+δ ❛✈❡❝ δ > 0 ✉♥❡ ✈❛r✐❛❜•❡ ❛•é❛t♦✐r❡ ♥♦♥✲ ❞é❣é♥éré❡ à ✈❛•❡✉rs ❞❛♥s R N ✳ ❖♥ s✉♣♣♦s❡ q✉❡ X ❛❞♠❡t ✉♥❡ ❞❡♥s✐té ❜♦r♥é❡ p X ✳ ❆•♦rs ♣♦✉r t♦✉t δ ′ < δ ❛♥❞ 1 < p < p ′ < ∞ ♦♥ ♣❡✉t tr♦✉✈❡r q, r > 1 ❡t θ > 0 t❡•s q✉✬♦♥ ❛✐t •✬❡st✐♠❛t✐♦♥ s✉✐✈❛♥t❡ ♣♦✉r t♦✉s φ ∈ S(R N )✿ φ • F D p,-δ ≤ C • 1 (detΣ) 2(k+1) q L q + 1 detΣ q L q 2m • F θ D r,1+δ ′ • p F ∞ • φ S p ′ ,-δ ′ ✭✸✻✮ ①① ■◆❚|❖❉❯❈❚■❖◆ ■❝✐ C ❡st ✉♥❡ ❝♦♥st❛♥t❡ ✉♥✐✈❡rs❡••❡ ❞é♣❡♥❞❛♥t ❞❡ p, δ, δ ′ , q, r, θ✳✳✳ ♠❛✐s ♣❛s ❞❡ X ♦✉ ❞❡ φ✳ ❊♥ ♣❛rt✐❝✉•✐❡r ♣♦✉r t♦✉t T ∈ S p ′ ,-δ ′ ✱ T • F ♣❡✉t êtr❡ ❞é✜♥✐ ❞❛♥s D p,-δ ❡t ♦♥ ❛ ❡♥❝♦r❡ •❡ ❝♦♥trô•❡ ❝✐✲❞❡ss✉s✳ P♦✉r ♣r♦✉✈❡r ❝❡ rés✉•t❛t✱ ♦♥ s✬❛♣♣✉✐❡ ❡♥ ♣❛rt✐❝✉•✐❡r s✉r tr♦✐s •❡♠♠❡s✳ ▲❡ ♣r❡✲ ♠✐❡r ♣❡r♠❡t ❞❡ ❝♦♥trô•❡r •❡s ♥♦r♠❡s ❞❡ ❙♦❜♦•❡✈ ❞❡ •✬✐♥✈❡rs❡ ❞❡ •❛ ♠❛tr✐❝❡ ❞❡ ▼❛••✐✲ ❛✈✐♥ ❞✬✉♥❡ ✈❛r✐❛❜•❡ ❛•é❛t♦✐r❡ s✉✣s❡♠♠❡♥t ré❣✉•✐èr❡✿ ▲❡♠♠❡ ✵✳✸✳✶✳✶✳ P♦✉r ✉♥ ❝❡rt❛✐♥ s > 1✱ ❝♦♥s✐❞ér♦♥s ✉♥❡ ✈❛r✐❛❜•❡ ❛•é❛t♦✐r❡ ♥♦♥ ❞é❣é♥éré❡ F ∈ D ∞-,s (R N ) ❡t s❛ ♠❛tr✐❝❡ ❞❡ ▼❛••✐❛✈✐♥ Σ✳ ❆•♦rs ♣♦✉r t♦✉t 1 ≤ s ′ < s✱ 1 det Σ ∈ D ∞ -,s ′ -1 (R) ✭✸✼✮ ❉❡ ♣•✉s✱ ♣♦✉r t♦✉t p ′ > p ✐• ❡①✐st❡ ✉♥ q > 1 ❡t ✉♥ α > 0 t❡•s q✉✬♦♥ ❛✐t •❡ ❝♦♥trô•❡ s✉✐✈❛♥t✿ 1 detΣ D p,s ′ -1 ≤ C • 1 (detΣ) 2(k+1) p L p ′ + 1 detΣ p L p ′ • F α Dq,s ✭✸✽✮ ♦ù k = [s] ❡t C ❡st ✉♥❡ ❝♦♥st❛♥t❡ ✉♥✐✈❡rs❡••❡ ♥❡ ❞é♣❡♥❞❛♥t q✉❡ ❞❡ p, p ′ , s, s ′ , q✳
▲❡ ❞❡✉①✐è♠❡ •❡♠♠❡ ♣❡r♠❡t ❞❡ ❝♦♥trô•❡r •❡s ♦♣ér❛t❡✉rs ❞✬✐♥té❣r❛t✐♦♥ ♣❛r ♣❛r✲ t✐❡s✿ ▲❡♠♠❡ ✵✳✸✳✶✳✷✳ P♦✉r t♦✉s p > 1 ❡t s > 0✱ ♣♦✉r t♦✉s p ′ > p ❡t s ′ > s✱

✐• ❡①✐st❡ q, r > 1✱ θ > 0 ❡t ✉♥❡ ❝♦♥st❛♥t❡ ✉♥✐✈❡rs❡••❡ ♥❡ ❞é♣❡♥❞❛♥t q✉❡ ❞❡s ♣❛r❛♠ètr❡s ❝✐✲❞❡ss✉s t❡•s q✉❡ •✬♦♥ ❛✐t •❡ ❝♦♥trô•❡ ❝✐✲❞❡ss♦✉s ♣♦✉r t♦✉t❡ ✈❛r✐❛❜•❡ ❛•é❛t♦✐r❡ G ∈ D ∞ -,s+1 ❡t t♦✉t❡ ✈❛r✐❛❜•❡ ❛•é❛t♦✐r❡ ♥♦♥✲❞é❣é♥éré❡ F ∈ D ∞ -,s ′ +2 ✿ l i (G, F ) Dp,s ≤ C • G D p ′ ,s+1 • 1 (detΣ) 2(k+1) q L q + 1 detΣ q L q • F θ D r,s ′ +2 ✭✸✾✮ ♦ù k = [s]✳ ❊♥ ♣r❛t✐q✉❡✱ ♥♦✉s ❡♥ ✉t✐•✐s❡r♦♥s •❛ ❣é♥ér❛•✐s❛t✐♦♥ s✉✐✈❛♥t❡✳ ❙♦✐t m i •✬♦♣ér❛t❡✉r ❞❡ ♠✉•t✐♣•✐❝❛t✐♦♥ ♣❛r •❡ ♠♦♥ô♠❡ X i ✳ ❈♦♥s✐❞ér♦♥s ✉♥❡ ❢❛♠✐••❡ ❞✬♦♣ér❛t❡✉rs µ j ♣♦✉r j = 1, . . . , m t❡•s q✉❡ ❝❤❛q✉❡ µ i s♦✐t ♦✉ ❜✐❡♥ •✬✉♥ ❞❡s l i ✱ ♦✉ ❜✐❡♥ •✬✉♥ ❞❡s m i ✱ i = 1, . . . , N ✳ ❖♥ ❞é✜♥✐t✿ λ(G, F ) = µ m (G, µ m-1 (G, µ m-2 (G, . . . , F ) . . . )) ✭✹✵✮ ♣♦✉r ✉♥❡ ✈❛r✐❛❜•❡ ❛•é❛t♦✐r❡ s✉✣s❡♠♠❡♥t ré❣✉•✐èr❡ G ❡t ✉♥❡ ✈❛r✐❛❜•❡ ❛❵•é❛t♦✐r❡ s✉✣s❡♠♠❡♥t ré❣✉•✐èr❡ ❡t ♥♦ ❞é❣é♥éré❡ F ✳ ❖♥ ♣❡✉t ❛•♦rs é♥♦♥❝❡r •❡✿ ✵✳✸✳ ❆PP❖|❚❙ ❉❊ ❈❊ ❚|❆❱❆■▲ ①①✐ ❈♦r♦••❛✐r❡ ✵✳✸✳✶✳✶✳ ❙♦✐❡♥t p ′ > p > 1 ❡t s ′ > s > 0✳ ❖♥ ❝♦♥s✐❞èr❡ λ ❝♦♠♠❡ ❝✐✲❞❡ss✉s✱ ❡♥ s✉♣♣♦s❛♥t q✉✬❛✉ ♣•✉s n ❞❡s µ j s♦✐t •✬✉♥ ❞❡s l i ✳ ❆•♦rs • ❡①✐st❡ q, r > 1✱ θ > 0 ❡t ✉♥❡ ❝♦♥st❛♥t❡ ✉♥✐✈❡rs❡••❡ C t❡•s q✉✬♦♥ ❛✐t •❡ ❝♦♥trô•❡ ❝✐✲❞❡ss♦✉s ♣♦✉r G ❛ss❡③ ré❣✉•✐èr❡ ❡t F ❛ss❡③ ré❣✉•✐èr❡ ❡t ♥♦♥✲❞é❣é♥éré❡✿ λ(G, F ) Dp,s ≤ C • G D p ′ ,s+m • 1 (detΣ) 2(k+1) q L q + 1 detΣ q L q m • F θ D r,s ′ +m+1 ✭✹✶✮ ♦✉ù k = [s] ❡t σ ❡st •❛ ♠❛tr✐❝❡ ❞❡ ▼❛••✐❛✈✐♥ ❞❡ F ✳ ❊♥✜♥✱ ♦♥ ✉t✐•✐s❡ •❡ •❡♠♠❡ ❞✬✐♥t❡r♣♦•❛t✐♦♥ s✉✐✈❛♥t✿ ❈♦r♦••❛✐r❡ ✵✳✸✳✶✳✷✳ ❙✉♣♣♦s♦♥s q✉❡ F ∈ D ∞ -,1 ❛✐t ✉♥❡ ❞❡♥s✐té ❜♦r♥é❡ p F ✳ ❆•♦rs ♣♦✉r t♦✉s 0 ≤ ρ < ρ ′ ≤ 1 ❡t 1 < p < p ′ < ∞ ✐• ❡①✐st❡ ✉♥❡ ❝♦♥st❛♥t❡ ✉♥✐✈❡rs❡••❡ C(ρ, ρ ′ , p, p ′ ) t❡••❡ q✉❡ ♣♦✉r t♦✉t φ ∈ S ♦♥ ❛✐t •❡ ❝♦♥trô•❡ ❝✐✲❞❡ss♦✉s✿ φ • F Dp,ρ ≤ C • p F ∞ • (1 + ∇F L q ) • φ S p ′ ,ρ ′ ✭✹✷✮ ❆✉ ♣❛r❛❣r❛♣❤❡ ✸✳✷✱ ♦♥ ❡①♣•✐q✉❡ ❝♦♠♠❡♥t ❝♦♠❜✐♥❡r ❝❡s •❡♠♠❡s ♣♦✉r ♦❜t❡♥✐r •❡ rés✉•t❛t ♣r✐♥❝✐♣❛•✳ ▲✬ ✐❞é❡ ❡st ❞✬ é❝r✐r❡ φ = K -l K l φ ♣♦✉r ✉♥ l ❡♥t✐❡r ❝❤♦✐s✐ ♣r♦❝❤❡ ❞❡ s ❞❛♥s ✉♥ s❡♥s q✉❡ •✬♦♥ ♣ré❝✐s❡✳ ▲❡s ❞❡✉① ♣r❡♠✐❡rs •❡♠♠❡s ♣❡r♠❡tt❡♥t ❛•♦rs ❞❡ ❢❛✐r❡ ✉♥ ♣r❡♠✐❡r ❝♦♥trô•❡ ❡♥ ❢❛✐s❛♥t [s] ✐♥té❣r❛t✐♦♥s ♣❛r ♣❛rt✐❡s ❡t •❡ ❞❡r♥✐❡r •❡♠♠❡ ♣❡r♠❡t ❞❡ r❛✣♥❡r ❛✜♥ ❞✬♦❜t❡♥✐r •❛ ♣❛rt✐❡ ❢r❛❝t✐♦♥♥❛✐r❡ ❞✉ ❝♦♥trô•❡✳ ❖♥ ❞♦♥♥❡ é❣❛•❡♠❡♥t ❞❡s rés✉•t❛ts s✉r •✬❡①✐st❡♥❝❡ ❡t •❛ ré❣✉•❛r✐té ❞❡ •❛ ❞❡♥s✐té p X ❛✜♥ ❞❡ s✐♠♣•✐✜❡r ❝❡ rés✉•t❛t✳ ❖♥ ♦❜t✐❡♥t ❡♥ ♣❛rt✐❝✉•✐❡r✿ ▲❡♠♠❡ ✵✳✸✳✶✳✸✳ ❙♦✐t δ > 1✳ ❙♦✐t ❛•♦rs X ∈ D ∞ -,1+δ ✉♥❡ ✈❛r✐❛❜•❡ ❛•é❛t♦✐r❡ ♥♦♥✲ ❞é❣é♥éré❡✳ ❆•♦rs X ❛❞♠❡t ✉♥❡ ❞❡♥s✐té ❝♦♥t✐♥✉❡ ❡t ❜♦r♥é❡❀ ♦♥ ❛ ♣•✉s ♣ré❝✐sé♠❡♥t X ∈ S p,δ ✳ ❖♥ ❛ ❞♦♥❝ •❛ ❣éér❛•✐s❛t✐♦♥ s✉✐✈❛♥t❡ ❞✉ t❤é♦rè♠❡ ✵✳✸✳✶✳✶✿ ❚❤é♦rè♠❡ ✵✳✸✳✶✳✷✳ ❙♦✐t F ∈ D ∞ -,1+δ ❛✈❡❝ δ > 1 ✉♥❡ ✈❛r✐❛❜•❡ ❛•é❛t♦✐r❡ ♥♦♥✲ ❞é❣é♥éré❡ à ✈❛•❡✉rs ❞❛♥s R N ✳ ❆•♦rs ♣♦✉r t♦✉t δ ′ < δ ❛♥❞ 1 < p < p ′ < ∞ ♦♥ ♣❡✉t tr♦✉✈❡r q, r > 1 ❡t θ > 0 t❡•s q✉✬♦♥ ❛✐t •✬❡st✐♠❛t✐♦♥ s✉✐✈❛♥t❡ ♣♦✉r t♦✉s φ ∈ S(R N )✿ φ • F D p,-δ ≤ C • 1 (detΣ) 2(k+1) q L q + 1 detΣ q L q 2m • F θ D r,1+δ ′ • p F ∞ • φ S p ′ ,-δ ′ ✭✹✸✮ ■❝✐ C ❡st ✉♥❡ ❝♦♥st❛♥t❡ ✉♥✐✈❡rs❡••❡ ❞é♣❡♥❞❛♥t ❞❡ p, δ, δ ′ , q, r, θ✳✳✳ ♠❛✐s ♣❛s ❞❡ X ♦✉ ❞❡ φ✳ ❊♥ ♣❛rt✐❝✉•✐❡r ♣♦✉r t♦✉t T ∈ S p ′ ,-δ ′ ✱ T • F ♣❡✉t êtr❡ ❞é✜♥✐ ❞❛♥s D p,-δ ❡t ♦♥ ❛ ❡♥❝♦r❡ •❡ ❝♦♥trô•❡ ❝✐✲❞❡ss✉s✳ ❖♥ ét❛❜•✐t é❣❛•❡♠❡♥t ✉♥❡ ❣é♥ér❛•✐s❛t✐♦♥ ❞✉ t❤é♦rè♠❡ ✵✳✸✳✶✳✶ ♣♦✉r •❡s ❡s♣❛❝❡s S ∞,s ✿ ①①✐✐ ■◆❚|❖❉❯❈❚■❖◆ ❚❤é♦rè♠❡ ✵✳✸✳✶✳✸✳ ❙♦✐t F ∈ D ∞ -,1+δ ✱ ♣♦✉r ✉♥ ❝❡rt❛✐♥ δ > 0 ✉♥❡ ✈❛r✐❛❜•❡ ❛•é❛✲ t♦✐r❡ ♥♦♥✲❞é❣é♥éré❡ à ✈❛•❡✉rs ❞❛♥s R N ✳ ❆•♦rs ♣♦✉r t♦✉s δ ′ < δ ❡t 1 < p < ∞ ✐• ❡①✐st❡ ✉♥❡ ❝♦♥st❛♥t❡ ✉♥✐✈❡rs❡••❡ C ❡t ❞❡s ré❡•s q, r > 1 ❡t θ > 0 t❡•s q✉✬♦♥ ❛✐t •❡ ❝♦♥trô•❡ s✉✐✈❛♥t ♣♦✉r t♦✉s •❡s φ ∈ S(R N )✿ φ • F D p,-δ ≤ C • 1 (detΣ) 2(k+1) q L q + 1 detΣ q L q 2m • F θ D r,1+δ ′ • φ S ∞,-δ ′ ✭✹✹✮ ❊♥ ♣❛rt✐❝✉•✐❡r ♣♦✉r t♦✉t T ∈ S p ′ ,-δ ′ ✱ T • F ♣❡✉t êtr❡ ❞é✜♥✐❡ ❞❛♥s D p,-δ ❡t ♦♥ ❛ ❡♥❝♦r❡ •❡ ❝♦♥t✂ r♦•❡ ❝✐✲❞❡ss✉s✳ ✵✳✸✳✷ ❯♥❡ ❢♦r♠✉•❡ ❞✬■tô ❢❛✐❜•❡ ▲❡ t❤é♦rè♠❡ ✷✳✸✳✷✳✷✱ q✉❡ ♥♦✉s r❡♣r♦❞✉✐s♦♥s ❝✐✲❞❡ss♦✉s✱ ❞♦♥♥❡ •❛ str✉❝t✉r❡ ❞❡s ❝❤❡♠✐♥s à ✈❛r✐❛t✐♦♥s ❜♦r♥é❡s à ✈❛•❡✉rs ❞❛♥s •❛ ❝❛ss❡ ❞❡ ❙❝❤✇❛rt③✿ ❚❤é♦rè♠❡ ✵✳✸✳✷✳✶✳ ❙♦✐t T ∈ BV ([0, 1], S ′ )✳ ♦✉r t♦✉t p ∈ [1, ∞] ✐• ❡①✐st❡ s ∈ R t❡• q✉❡ T ∈ BV ([0, 1], S p,s )✳ ▲❡s t❤é♦rè♠❡s ✵✳✸✳✶✳✷ ✭♦✉ ✵✳✸✳✶✳✸✮ ❡t ✵✳✸✳✷✳✶ rés♦•✈❡♥t •❡s ♣r✐♥❝✐♣❛•❡s ❞✐✣❝✉•tés ❞❡ •❛ ♣r❡✉✈❡ ✭❛✉ ❝❤❛♣✐tr❡ ✹✮ ❞❡ •❛ ❢♦r♠✉•❡ ❞✬■tô ❝✐✲❞❡ss♦✉s✱ q✉✐ ❡st •❡ ❝❡♥tr❡ ❞❡ ❝❡tt❡ t❤ès❡✿ ❚❤é♦rè♠❡ ✵✳✸✳✷✳✷✳ ❙♦✐t W •❡ ♠♦✉✈❡♠❡♥t ❜r♦✇♥✐❡♥ ❝❛♥♦♥✐q✉❡ ❞❡ ❞✐♠❡♥s✐♦♥ d s✉r •✬❡s♣❛❝❡ ❞❡ ❲✐❡♥❡r✳ ❙♦✐t X •❡ ♣r♦❝❡ss✉s ❞✬■tô ❞❡ ❡♥ ❞✐♠❡♥s✐♦♥ N t❡• q✉❡✿ dX t = b t dt + σ t dW t ✭✹✺✮ X 0 = x ✭✹✻✮ ♦ù b ❡st ✉♥ ♣r♦❝❡ss✉s ❛❞❛♣té ❞❡ ❞✐♠❡♥s✐♦♥ N ❡t σ ❡st ✉♥ ♣r♦❝❡ss✉s ♠❛tr✐❝✐❡• ❛❞❛♣té ❞❡ ❞✐♠❡♥s✐♦♥ N ⊗ d✳ ◆♦✉s ♥♦t♦♥s a = σ t σ✳ ❙♦✐t ❛✉ss✐ t ∈ [0, 1] → T (t) ∈ S ′ (R N ) ✉♥ ❝❤❡♠✐♥ à ✈❛r✐❛t✐♦♥s ❜♦r♥é❡s❀ ✐• ❡①✐st❡ ❛•♦rs ✉♥ s > 0 t❡• q✉❡✿ T ∈ BV (S p,-s ) ✭✹✼✮ ❖♥ ❢❛✐t é❣❛•❡♠❡♥t •❡s ❤②♣♦t❤ès❡s s✉✐✈❛♥t❡s✿ t 0 b u r D r,s ′ +3 du < ∞ ✭✹✽✮ t 0 σ u r D r,s ′ +3 du < ∞ ✭✹✾✮ sup ǫ≤u≤t 1 Σ(u) L r < ∞ ✭✺✵✮ ✵✳✸✳ ❆PP❖|❚❙ ❉❊ ❈❊ ❚|❆❱❆■▲ ①①✐✐✐ ♣♦✉r t♦✉t 0 < ǫ < 1✱ ♦ù Σ(u) ❡st •❛ ♠❛tr✐❝❡ ❞❡ ▼❛••✐❛✈✐♥ ❞❡ X u ✳ ❖♥ ❛ ❛•♦rs✿ T (t) • X t -T (ǫ) • X ǫ = M (ǫ) t + i t ǫ b i (u) • (∂ i T u ) • X u du + 1 2 i,j t ǫ a ij (u) • (∂ ij T u ) • X u du + t ǫ T (du) • X u ✭✺✶✮ ♦ù M (ǫ) t ǫ≤t≤1
❡st ✉♥❡ ♠❛rt✐♥❣❛•❡ ❢❛✐❜•❡ s✉r •✬❡s♣❛❝❡ ❞❡ ❲✐❡♥❡r ❡t ❛✉ s❡♥s ❞❡s ❞✐✈❡r❣❡♥❝❡s ♦♥ ❛✿

M (ǫ) t = j t ǫ i [b i (u) • (∂ i T u ) • X u ] dW j u ✭✺✷✮ ▲✬✐❞é❡ ❞❡ •❛ ♣r❡✉✈❡ ❡st q✉❡✱ ❣râ❝❡ ❛✉ t❤órè♠❡ ✵✳✸✳✷✳✶✱ •✬♦♥ ♣❡✉t tr♦✉✈❡r ✉♥ ❡♥t✐❡r l ✐♥❞é♣❡♥❞❛♥t ❞❡ t t❡•s q✉❡ •❡s K -l T t s♦✐❡♥t ❞❡s ❢♦♥❝t✐♦♥ C 2 ✳ ❆•♦rs ♦♥ é❝r✐t T t = K l K -l T t ✱ ♦♥ ❛♣♣•✐q✉❡ •❛ ❢♦r♠✉•❡ ❞✬■tô ❝•❛ss✐q✉❡ ❛✉① ❢♦♥❝t✐♦♥s K -l T t ❡t ♦♥ ❛♣♣•✐q✉❡ •❡ t❤é♦rè♠❡ ✵✳✸✳✶✳✷ ✭♦✉ ✵✳✸✳✶✳✸✮ ♣♦✉r r❡•❡✈❡r •❛ ❢♦r♠✉•❡ ❞✬■tô ❝•❛ss✐q✉❡ ❡♥ ♥♦tr❡ ❡①t❡♥s✐♦♥✳ ❈♦♠♠❡ ❛♥♥♦♥❝é ♥♦tr❡ ❢♦r♠✉•❡ ❡st ❜❡❛✉❝♦✉♣ ♣•✉s ❢❛❝✐•❡ à ❛♣♣•
X t D p,s ′ ≤ A exp(B( b C 0,s + σ C 0,s ) • t) ✭✺✹✮ ❙✉♣♣♦s♦♥s ♠❛✐♥t❡♥❛♥t q✉❡ σ ∈ C 0,s ♣♦✉r ✉♥ ❝❡rt❛✐♥ s ≥ 2 ❡t q✉❡ b ∈ C 0,1 ✳ ❆•♦rs ♥♦tr❡ ❊❉❙ ❛ ✉♥❡ ✉♥✐q✉❡ s♦•✉t✐♦♥✳ ❖♥ ♥♦t❡ [s] = k✳ P♦✉r i ≤ j ≤ d✱ ♦♥ ✐♥tr♦❞✉✐t •❡ ❝❤❛♠♣ ❞❡ ✈❡❝t❡✉rs✿ V j = σ j ∂ x j ✳ ❙♦✐t L 0 •✬❡♥s❡♠❜•❡ ❞❡ t♦✉s •❡s V j ✳ ❖♥ ♥♦t❡ [•, •] •❡ ❝♦♠♠✉t❛t❡✉r ❞❡ ❞❡✉① ❝❤❛♠♣s ❞❡ ✈❡❝t❡✉rs✳ ❆•♦rs✱ ♦♥ ❞é✜♥✐t ♣❛r ré❝✉rr❡♥❝❡ L i ❝♦♠♠❡ •✬❛•❣è❜r❡ ❞❡ ▲✐❡ ❡♥❣❡♥❞ré❡ ♣❛r •❡s [V j , Z]✱ ♦ù Z ∈ L i-1 ✳ ❖♥ ❛ •❡✿ ❚❤é♦rè♠❡ ✵✳✸✳✸✳✷✳ ❙✉♣♣♦s♦♥s q✉❡ ♣♦✉r ✉♥ ❝❡rt❛✐♥ ❡♥t✐❡r n ≤ k -2 ✐• ❡①✐st❡ c > 0 t❡• q✉❡ ♣♦✉r t♦✉t ξ ∈ S N -1 ♦♥ ❛✐t✿ n i=0 Z∈L k ξ, Z (0, x) > c ✭✺✺✮ ❙✉♣♣♦s♦♥s ❛✉ss✐ q✉❡ σ ∈ C (β,n+2) ♣♦✉r ✉♥ ❝❡rt❛✐♥ β > 0 ✭❝✬❡st à ❞✐r❡ q✉❡ •✬♦♥ s✉♣✲ ♣♦s❡ ✉♥❡ ré❣✉•❛r✐té ❤♦•❞❡r✐❡♥♥❡ ❡♥ t❡♠♣s✮✳ ❆•♦rs✱ s✐ Σ t ❡st •❛ ♠❛tr✐❝❡ ❞❡ ▼❛••✐❛✈✐♥ ❞❡ X t ✱ ♣♦✉r t♦✉t t > 0 ♦♥ ❛ ❞❡s ❝♦♥trô•❡s ❞✉ t②♣❡ s✉✐✈❛♥t✿ 1 Σ t L p ≤ C • 1 t ν • e Kt ✭✺✻✮
■❝✐ C✱ ν ❡t K s♦♥t ❞❡s ❝♦♥st❛♥t❡s q✉✐ ♥❡ ❞é♣❡♥❞❡♥t q✉❡ ❞❡ t✳ ❊♥ ♣❛rt✐❝✉•✐❡r✿ ❚❤é♦rè♠❡ ✵✳✸✳✹✳✶✳ ❙♦✐❡♥t p, q, r > 1 t❡•s q✉❡ 1/p + 1/q = 1/r✳ ❙✉♣♣♦s♦♥s q✉❡

T ǫ 1 Σ t p L p dt < ∞ ✭✺✼✮ ❊♥ ❝♦♠❜✐♥❛♥t ❝❡s ❞❡✉① t❤é♦rè♠❡s✱ ♦♥ ♦❜t✐❡♥t✿ ❚❤é♦rè♠❡ ✵✳✸✳✸✳✸✳ ❙✉♣♣♦s♦♥s q✉❡ •❡s ❝♦❡✣❝✐❡♥ts ❞❡ ♥♦tr❡ ❊❉❙ C (0,s) ❢♦r s♦♠❡ s > 2 ❛♥❞ •❡t k = [s]✳ ❙✉♣♣♦s♦♥s q✉❡ •❛ ❝♦♥❞✐t✐♦♥ ❞❡ ❍♦r♠❛♥❞❡r ❞✉ t❤é♦rè♠❡ ♣ré❝é❞❡♥t s♦✐t ✈ér✐✜é❡ ♣♦✉r ✉♥ ❝❡rt❛✐♥ n ≤ k -2 ❡t q✉❡ σ ∈ C (β,n+2) ✳ ❆•♦rs s✐ T ❡st ✉♥ ❝❤❡♠✐♥ à ✈❛r✐❛t✐♦♥s ❜♦r♥é❡s ❞❛♥s S p,-(s-1) ♦♥ ❛ •❛ ❢♦r♠✉•❡ ❞✬■tô ❝✐✲❞❡ss♦✉s✿ T t • X t -T ǫ • X ǫ = M ǫ t + i t ǫ b i (u, X u ) • (∂ i T u ) • X u du + 1 2 i,j t ǫ a ij (u, X u ) • (∂ ij T u ) • X u du + t ǫ T (du) • X u ✭✺✽✮ ✵✳✸✳ ❆PP❖|❚❙ ❉❊ ❈❊ ❚|❆❱❆■▲ ①①✈ ❡t ❛✉ s❡♥s ❞❡s ❞✐✈❡r❣❡♥❝❡s✿ M (ǫ) t = j t ǫ i [b i (u, X u ) • (∂ i T u ) • X u ] dW j u ✭✺✾✮ P♦✉r •❡s ❛♣♣•✐❝❛t✐♦♥s ❛✉ ❝❤❛♣✐tr❡ ✺✱ ♥♦✉s ✉t✐•✐s❡r♦♥s •✬❡①t❡♥s✐♦♥ s✉✐✈❛♥t❡✿ ❚❤é♦rè♠❡ ✵✳✸✳✸✳✹✳ ❙♦✐❡♥t X ❡t T ❝♦♠♠❡ ❞❛♥s •❡ t❤é♦rè♠❡ ♣ré❝é❞❡♥t✳ ❈♦♥✲ s✐❞❡r♦♥s ❞❡✉① ❢♦♥❝t✐♦♥s b Y , σ Y ∈ C (0,s) ❡t s♦✐t Y •✬✉♥✐q✉❡ s♦•✉t✐♦♥ ❢♦rt❡ ❞❡ •✬❊❉❙✿ dY t = b(t, Y t )dt + σ(t, Y t )dW t ✭✻✵✮ Y 0 = y ✭✻✶✮ ❙♦✐t ❡♥✜♥ f ∈ BV (C s )✳ ❆•♦rs ♦♥ ♣❡✉t é❝r✐r❡ ✉♥❡ ❢♦r♠✉•❡ ❞✬■tô ♣♦✉r✿ f (t, Y t ) • T t • X t ✭✻✷✮ ✵✳✸✳✹ ❆✉tr❡s ❡①t❡♥s✐♦♥s ◆♦✉s ❛✈♦♥s é❣❛•❡♠❡♥t
•✬♦♥ ❛✐t✿ • f ∈ L p (C 1 b ) • f ∈ D p,1 (C 0 b ) • X ∈ D q,1 (R N ) ❆•♦rs✱ ♦♥ ❛✱ f (X) ∈ D r,1 ❡t •❛ rè❣•❡ ❞❡s ❝❤❛î♥❡s s✉✐✈❛♥t❡✿ ∇(f (X)) = N k=1 ∂ k f (X)∇X k + (∇f )(X) ✭✻✸✮ ①①✈✐ ■◆❚|❖❉❯❈❚■❖◆ ❉❡ ♣•✉s✱ ♣♦✉r t♦✉t φ ∈ D p * ,1 ♦♥ ❛✿ E [∂ k f (X) • φ] = E [f (X) • l k (X, φ)] -E [φ • T k (f, X)(X)] ✭✻✹✮ ❊♥ s✉✐✈❛♥t •❛ ♠ê♠❡ ❞é♠❛r❝❤❡ q✉❡ ❞❛♥s •❡ ❞é❜✉t ❞✉ ❝❤❛♣✐tr❡ ✸✱ ♦♥ ❡♥ ❞é❞✉✐t ❡♥ ♣❛rt✐❝✉•✐❡r •❡✿ ❚❤é♦rè♠❡ ✵✳✸✳✹✳✷✳ P♦✉r ✉♥ δ > 0✱ s♦✐t X ∈ D ∞ -,1+δ ✉♥❡ ✈❛r✐❛❜•❡ ❛•é❛t♦✐r❡ ♥♦♥✲❞é❣é♥éré❡ à ✈❛•❡✉rs ❞❛♥s R N ✳ ❆•♦rs✱ ♣♦✉r t♦✉s δ ′ < δ ❡t 1 < p < ∞ ✐• ❡①✐st❡ ✉♥❡ ❝♦♥st❛♥t❡ ✉♥✐✈❡rs❡••❡ C ❡t ❞❡s ré❡•s p 1 , p 2 , p 3 > p ❡t θ > 0 t❡•s q✉❡ •✬♦♥ ❛✐t •❡ ❝♦♥trô•❡ s✉✐✈❛♥t ♣♦✉r t♦✉t φ ∈ S(D)✿ φ • X D p,-δ ≤ C • 1 (detΣ) 2(k+1) p 1 L p 1 + 1 detΣ p 1 L p 1 θ • X θ D p 2 ,1+δ ′ × [δ] k=0 φ D p 3 ,k (S ∞,-(δ ′ -k) ) + φ D p 3 ,k (W α,-(δ ′ -k-N α ) ) ✭✻✺✮ ❊♥ ♣❛rt✐❝✉•✐❡r T ∈ D r,k (S p ′ ,-δ ′ )✱ T • X ♣❡✉t êtr❡ ❞é✜♥✐❡ ❞❛♥s D p,-δ ❡t ❞❛♥s ❝❡ ❝❛s ♦♥ ❛ ❡♥❝♦r❡ •❡ ❝♦♥trô•❡ ❝✐✲❞❡ss✉s✳ ❈❡•❛ ♥♦✉s ❛ ♣❡r♠✐s ❞❡ ♣r♦✉✈❡r ✉♥❡ ✈❡rs✐♦♥ ❢❛✐❜•❡ ❞❡ •❛ ❢♦r♠✉•❡ ❞✬■tô✲❲❡♥t③❡••✳ ❊♥ ❡✛❡t✱ ❝♦♥s✐❞ér♦♥s ❞❡✉① ♣r♦❝❡ss✉s✿ dX t = b t dt + σ t dW t ✭✻✻✮ ■❝✐ X ✈ér✐✜❡ •❡s ♠ê♠❡s ❤②♣♦t❤ès❡s q✉❡ ❞❛♥s •❡ t❤é♦rè♠❡ ✵✳✸✳✷✳✷✳ ❖♥ s✬✐♥tér❡ss❡ ❛✉① T t • X t ♦ù T ❡st ✉♥ ✢✉① à ✈❛•❡✉rs ❞❛♥s S ′ ✳ P•✉s ♣ré❝✐sé♠❡♥t✱ ♦♥ ✐♥tr♦❞✉✐t ❞❡s ♣r♦❝❡ss✉s✿ D t , V t ∈ k<[s] D ∞ -,k S ∞,-(s-k) ∩ D ∞ -,k W α,-(s-k-N α ) ✭✻✼✮ ■❝✐✱ ♦♥ s✉♣♣s❡ q✉❡ α ❡st t❡• q✉❡ {s} > N α ❀ s✐ s ❡st ❡♥t✐❡r ♦♥ s✉♣♣♦s❡ s❡✉•❡♠❡♥t✿ D t , V t ∈ k<[s] D ∞ -,k S ∞,-(s-k) ✭✻✽✮ ❖♥ s✉♣♣♦s❡ q✉❡ ❝❡s ♣r♦❝❡ss✉s s♦♥t ❛❞❛♣tés ❞❛♥s •❛ ✜•tr❛t✐♦♥ ❞❡ W ✳ ❖♥ ♣❡✉t ❛•♦rs ❞é✜♥✐r ✉♥❡ s❡♠✐♠❛rt✐♥❣❛•❡ à ✈❛•❡✉rs ❞❛♥s S ∞,-s ♣❛r ✉♥❡ ✈❛•❡✉r ✐♥✐t✐❛•❡ T 0 ∈ S ∞,-s ❡t •✬éq✉❛t✐♦♥✿ dT t = D t dt + V t dW t ✭✻✾✮ ✵✳✸✳ ❆PP❖|❚❙ ❉❊ ❈❊ ❚|❆❱❆■▲ ①①✈✐✐ ❈✬❡st à ❞✐r❡ q✉❡ ♦✉r t♦✉t φ ∈ S✿ T t , φ = T 0 , φ + t 0 D s , φ ds + t 0 V s , φ dW s ✭✼✵✮ q✉✐ ❡st ✉♥❡ s❡♠✐♠❛rt✐♥❣❛•❡ ré❡••❡✳ ❖♥ ♣r♦✉✈❡ ❛•♦rs✿ ❚❤é♦rè♠❡ ✵✳✸✳✹✳✸✳ ❙♦✉s •❡s ❤②♣♦t❤ès❡s ❝✐✲❞❡ss✉s✱ ♣♦✉r t♦✉s p ′ > p ❡t s ′ > s✱ ♣♦✉r t♦✉t ǫ > 0✱ ♦♥ ❛ •❛ ❢♦r♠✉•❡ s✉✐✈❛♥t❡ ❛✉ s❡♥s ❞❡ •✬❡s♣❛❝❡✿ D p ′ ∨2,-(s ′ +2) ✿ T (t) • X t -T (ǫ) • X ǫ = M (ǫ) t + t ǫ b(u) • (∂ x T u ) • X u du + 1 2 t ǫ σ(u) 2 • (∂ xx T u ) • X u du + t ǫ D u • X u du + t ǫ σ(u) • ∂ x V u • X u du ✭✼✶✮ ♦ù M (ǫ) t ǫ≤t≤1 ❡st ✉♥❡ ♠❛rt✐♥❣❛•❡ ❢❛✐❜•❡ s✉r •✬❡s♣❛❝❡ ❞❡ ❲✐❡♥❡r✱ ❡t ❛✉ s❡♥s ❞❡s ❞✐✈❡r❣❡♥❝❡s ♦♥ ❛✿ M (ǫ) t = t ǫ [b(u) • (∂ x T u ) • X u + V u • X u ] dW j u ✭✼✷✮ ◆♦✉s ♣r♦✉✈♦♥s é❣❛•❡♠❡♥t ✉♥❡ ✈❡rs✐♦♥ ❛♥t✐❝✐♣❛t✐✈❡ ❞❡ ♥♦tr❡ ❢♦r♠✉•❡ ❞✬■tô ❢❛✐❜•❡ ✭♦♥ ✉t✐•✐s❡ •❡s ♥♦t❛t✐♦♥s ❝•❛ss✐q✉❡s ❞✉ ❝❛•❝✉• st♦❝❤❛st✐q✉❡ ❛♥t✐❝✐♣❛t✐❢❀ ❝❡s ♥♦t❛t✐♦♥s s♦♥t ♣ré❝✐sé❡s ❞❛♥s •❡ ❝❤❛♣✐tr❡ ✹✮✳ ❖♥ ❝♦♥s✐❞èr❡ ✉♥ ♣r♦❝❡ss✉s ❛♥t✐❝✐♣❛t✐❢✿ X t = X 0 + t 0 b s ds + t 0 σ s dW s ✭✼✸✮ ■❝✐ X 0 ❡st ✉♥❡ ✈❛r✐❛❜•❡ ❛•é❛t♦✐r❡ ❡t b ❡t σ ♥❡ s♦♥t ♣❛s ♥é❝❡ss❛✐r❡♠❡♥t ❛❞❛♣tés✳ ❖♥ ❛ ❛•♦rs •❡✿ ❚❤é♦rè♠❡ ✵✳✸✳✹✳✹✳ ❙♦✐t T ∈ S q,-δ ✳ ❖♥ s✉♣♣♦s❡ q✉❡ ♣♦✉r t♦✉t p✿ s≤t E (Id + L) δ+3 2 D s σ t p dsdt < ∞ ✭✼✹✮ s≤t E (Id + L) δ+2 2 D s b t p dsdt < ∞ ✭✼✺✮ 1 0 1 Σ t p dt < ∞ ✭✼✻✮ ①①✈✐✐✐ ■◆❚|❖❉❯❈❚■❖◆ ♦ù Σ ❡st •❛ ♠❛tr✐❝❡ ❞❡ ▼❛••✐❛✈✐♥ ❞❡ X t ✳ ❆•♦rs✱ ❛✉ s❡♥s ❞❡ •✬❡s♣❛❝❡ D q,-δ ✱ ♦♥ ❛✿ T • X t = T • X 0 + t 0 b s • T ′ • X s + 1 2 σ 2 s • T ′′ • X s ds + t 0 σ s • T ′ • X s dW s + t 0 (D -X) s σ s • T ′′ • X s ds ✭✼✼✮ ✵✳✸✳✺ ❆♣♣•✐❝❛t✐♦♥ ❛✉ t❡♠♣s •♦❝❛• ◆♦✉s ❝♦♠♣❛r♦♥s ♥♦tr❡ ❢♦r♠✉•❡ ❞✬■tô à ❞✬❛✉tr❡s ❡①t❡♥s✐♦♥s ❞é•à ❝♦♥♥✉❡s✱ ♥♦t❛♠✲ ♠❡♥t ❝❡••❡s q✉❡ ♥♦✉s ❛✈♦♥s é✈♦q✉é❡s ♣•✉s ❤❛✉t✳ ❈❡ ❢❛✐s❛♥t✱ ♥♦✉s ♠♦♥tr♦♥s ❡♥ ♣❛rt✐❝✉•✐❡r ❝♦♠♠❡♥t •❡ t❡♠♣s •♦❝❛• ❞❡ ❝❡rt❛✐♥s ♣r♦❝❡ss✉s ♣❡✉t s❡ ❞és✐♥té❣r❡r ❡♥ ✉♥❡ ✐♥té❣r❛•❡ ❞❡ P❡tt✐s s✉r D ′ ✳ ❊♥ ❡✛❡t✱ ❡♥ ❞✐♠❡♥s✐♦♥ 1✱ ❝♦♥s✐❞ér♦♥s •❛ s♦•✉t✐♦♥ ❞❡ •✬❊❉❙✿ dX t = b(t, X t )dt + σ(t, X t )dW t ✭✼✽✮ ❖♥ é❝r✐t ❛•♦rs✱ s♦✉s ❞❡s ❤②♣♦t❤ès❡s ✭s✉r X✮ q✉✐ s♦♥t ♣ré❝✐sé❡s ❛✉ ❝❤❛♣✐tr❡ ✹✿ |X t -K| -|X ǫ -K| = t ǫ sgn(X s -K)b(s, X s )ds + t ǫ sgn(X s -K)σ(s, X s )dW s + 1 2 t ǫ σ(s, X s ) 2 δ K • X s ds ✭✼✾✮ ❖♥ ♣❡✉t ❛•♦rs ❞é❝♦♠♣♦s❡r •❡ t❡♠♣s •♦❝❛• ❛✐♥s✐✿ L K t (X) = t 0 σ(s, X s ) 2 δ K • X s ds ✭✽✵✮ ❖♥ ♣❡✉t ❣é♥ér❛•✐s❡r ❝❡tt❡ ✐❞é❡ à ❞❡s ❞✐♠❡♥s✐♦♥s s✉♣ér✐❡✉r❡s ❡♥ r❡♠♣•❛ç❛♥t •❛ ❢♦♥❝✲ t✐♦♥ ✈❛•❡✉r ❛❜s♦•✉❡ ♣❛r •❛ s♦•✉t✐♦♥ ❢♦♥❞❛♠❡♥t❛•❡ ❞❡ •✬éq✉❛t✐♦♥ ❞❡ •❛ ❝❤❛•❡✉r ❡♥ ❞✐♠❡♥s✐♦♥ q✉❡•❝♦♥q✉❡✳ ❈❡•❛ ♣❡r♠❡t ❞❡ ❞é✜♥✐r ✭❛✉ s❡♥s ❢❛✐❜•❡✮ ✉♥❡ ♥♦t✐♦♥ ❞❡ t❡♠♣s •♦❝❛• ♠✉•t✐✲❞✐♠❡♥s✐♦♥♥❡•✳ ❈❡s ✐❞é❡s ♣❡r♠❡tt❡♥t é❣❛•❡♠❡♥t ❞✬ét✉❞✐❡r •❛ ré❣✉•❛r✐té ❞✉ t❡♠♣s •♦❝❛•✱ ❡♥ ❡s♣❛❝❡ ❡t ❛✉ s❡♥s ❞❡ ▼❛••✐❛✈✐♥✳ ✵✳✹ ❆♣♣•✐❝❛t✐♦♥ à ✉♥ ♣r♦❜•è♠❡ ✈❛r✐❛t✐♦♥♥❡• ▲❡ ❝✐♥q✉✐è♠❡ ❡t ❞❡r♥✐❡r ❝❤❛♣✐tr❡ ❡st ❝♦♥s❛❝ré à •✬❛♣♣•✐❝❛t✐♦♥ ♣r✐♥❝✐♣❛•❡ q✉❡ ♥♦✉s ❞♦♥♥♦♥s à •❛ t❤é♦r✐❡ ❞é✈❡•♦♣♣é❡ ❞❛♥s ❝❡tt❡ t❤ès❡✱ à s❛✈♦✐r •✬ét✉❞❡ ❞✬✉♥ ♣r♦❜•è♠❡ ✵✳✹✳ ❆PP▲■❈❆❚■❖◆ ➚ ❯◆ P|❖❇▲➮▼❊ ❱❆|■❆❚■❖◆◆❊▲ ①①✐① ✈❛r✐❛t✐♦♥♥❡•✳ ❖♥ ❝♦♥s✐❞èr❡ •✬éq✉❛t✐♦♥ ❞✐✛ér❡♥t✐❡••❡ st♦❝❤❛st✐q✉❡✿ dX t = b(t, X t )dt + σ(t, X t )dW t ✭✽✶✮ ❖♥ ♥♦t❡ A t s♦♥ ❣é♥ér❛t❡✉r ❡t ♦♥ s❡ ❞♦♥♥❡ ❞❡✉① ❢♦♥❝t✐♦♥s ❝♦♥t✐♥✉❡s ❡t ❜♦r♥é❡s f ❡t g ❡t ✉♥❡ ❢♦♥❝t✐♦♥ ❝♦♥t✐♥✉❡ ❡t ♣♦s✐t✐✈❡ r ❡t ♦♥ ❝♦♥s✐❞èr❡ •❡ ♣r♦❜•è♠❡✿ K t u -ru ≤ -g ✭✽✷✮ u ≥ f ✭✽✸✮ (K t u -ru)(f -u) = 0 ✭✽✹✮ u(T, •) = f ✭✽✺✮ ❈❡ ♣r♦❜•è♠❡ ❛ été ét✉❞✐é ♣❛r ❞❡ ♥♦♠❜r❡✉① ❛✉t❡✉rs✱ ✈♦✐r ❡♥ ♣❛rt✐❝✉•✐❡r ❬✽(✱ ❬✾(✱ ❬✸✽(✱ ❬✸✸(✳✳✳ ❖♥ ♠♦♥tr❡ ❞❛♥s ❝❡s ♦✉✈r❛❣❡s q✉❡ s♦✉s ❝❡rt❛✐♥❡s ❤②♣♦t❤ès❡s ✭ré❣✉•❛r✐té ❞❡s ❝♦❡✣❝✐❡♥ts ❡t ❡••✐♣t✐❝✐té ❞✉ ❣é♥ér❛t❡✉r✮✱ •✬✉♥✐q✉❡ s♦•✉t✐♦♥ à ❝❡ ♣r♦❜•è♠❡ ❡st ❞♦♥♥é❡ ♣❛r •❛ ❢♦♥❝t✐♦♥✿ u(t, x) = sup τ ∈Z t,T E f (X t τ (x)) exp - τ t r(u, X t u (x))du + τ t g(s, X t s (x)) exp - s t r(u, X t u (x))du ds ✭✽✻✮ ◆♦tr❡ ❝♦♥tr✐❜✉t✐♦♥ ❡st ❞❡ ♠♦♥tr❡r q✉❡ •✬♦♥ ♣❡✉t ❛✛❛✐❜•✐r ❝❡s ❤②♣♦t❤ès❡s✱ ❝✬❡st à ❞✐r❡ •❡s ré❞✉✐r❡ à ❝❡••❡s q✉✐ ♣❡r♠❡tt❡♥t ❞✬❛♣♣•✐q✉❡r •❛ ❢♦r♠✉•❡ ❞✬■tô ❞✉ t❤é♦rè♠❡ ✹✳✹✳✸✳✶✳ ❊♥ ♣r❛t✐q✉❡ ✐• s✉✣t ❞❡ s✉♣♣♦s❡r q✉❡ •❡s ❝♦❡✣❝✐❡♥ts ❞❡ •✬❊❉❙ ❛✐♥s✐ q✉❡ f ✱ g ❡t r s♦♥t tr♦✐s ❢♦✐s ❞ér✐✈❛❜•❡s ❡♥ ❡s♣❛❝❡✱ q✉❡ •❡s ❝♦❡✣❝✐❡♥ts ❞❡ •✬❊❉❙ s♦♥t ❤♦•❞❡r✐❡♥s ❡♥ t❡♠♣s ❡t q✉❡ X ✈ér✐✜❡ ✉♥❡ ❝♦♥❞✐t✐♦♥ ❞❡ t②♣❡ ❍♦r♠❛♥❞❡r✳ ❈❡tt❡ ❞❡r♥✐èr❡ ❤②♣♦t❤ès❡ ❡st ♣•✉s ❢❛✐❜•❡ q✉❡ •❡s ❤②♣♦t❤ès❡s ❞✬❡••✐♣t✐❝✐té q✉❡ •✬♦♥ ❢❛✐t ❤❛❜✐t✉❡••❡♠❡♥t✳ ▲✬✐❞é❡ ♣r✐♥❝✐♣❛•❡ ❞❡ •❛ ♣r❡✉✈❡ ❡st q✉❡ •✬♦♥ ♣❡✉t ❛♣♣•✐q✉❡r ♥♦tr❡ ❢♦r♠✉•❡ ❞✬■tô ❢❛✐❜•❡ à •❛ ❢♦♥❝t✐♦♥ u✱ q✉✐✱ ❡♥ ❣é♥ér❛•✱ ♥✬❡st ❞ér✐✈❛❜•❡ ♥✐ ❡♥ t❡♠♣s ♥✐ ❡♥ ❡s♣❛❝❡✳ ①①① ■◆❚|❖❉❯❈❚■❖◆ ❈❤❛♣t❡r ✶ ■♥tr♦❞✉❝t✐♦♥ t♦ ▼❛••✐❛✈✐♥ ❝❛•❝✉•✉s ▼❛••✐❛✈✐♥ ❝❛•❝✉•✉s ✇✐•• ❜❡ ♦✉r ♠❛✐♥ t♦♦• t❤r♦✉❣❤♦✉t t❤❡ ✇❤♦•❡ ♦❢ t❤✐s ❞♦❝✉♠❡♥t✳ ❋♦r t❤❡ s❛❦❡ ♦❢ ❝♦♠♣•❡t❡♥❡ss ❛♥❞ ✐♥ ♦r❞❡r t♦ ✐♥tr♦❞✉❝❡ s♦♠❡ ♥♦t❛t✐♦♥✱ ✇❡ ♣r♦✈✐❞❡ ❛ ❜r✐❡❢ ✐♥tr♦❞✉❝t✐♦♥ t♦ t❤❡ t♦♣✐❝✳ ▼♦st ♦❢ t❤❡ r❡s✉•ts ✐♥ t❤✐s ❝❤❛♣t❡r ❛r❡ ❝•❛ss✐❝✱ ❛♥❞ ✇❡ ❢♦••♦✇❡❞ ❬✹✺(✱ ❬✺✸(✱ ❬✷✺(✱ ❬✻✻(✱ ❬✺✺(✱ ❬✺✻(✱ ❬✼✺( ❛♥❞ ❬✹✹(✳ ✶✳✶ ▼❛••✐❛✈✐♥ ❛♥❛•②s✐s ✐♥ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥ ❲❡ st❛rt ✇✐t❤ r❡s✉•ts ❢r♦♠ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛• ❣❛✉ss✐❛♥ ❛♥❛•②s✐s✳ ❚❤❡s❡ ✇✐•• •✐❢t ❛♥❞ ❡①t❡♥❞ t♦ t❤❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥♥❛• ❝❛s❡ ♦❢ t❤❡ ❲✐❡♥❡r s♣❛❝❡✳ ❋♦r t❤✐s s❡❝t✐♦♥ ✇❡ ❢♦••♦✇❡❞ ❬✹✹( ❛♥❞ ❬✺✺(✳ ✶✳✶✳✶ •❛✉ss✐❛♥ r❛♥❞♦♠ ✈❛r✐❛❜•❡s ❚❤r♦✉❣❤♦✉t t❤✐s ❞♦❝✉♠❡♥t✱ ✇❡ ✇✐•• ♠❛❦❡ ❛♥ ✐♥t❡♥s✐✈❡ ✉s❡ ♦❢ t❤❡ ♥♦r♠❛• ❣❛✉ss✐❛♥ •❛✇✱ ✇❤✐❝❤ ✇❡ ♥♦t❡ µ✳ ■t ✐s ❝❤❛r❛❝t❡r✐③❡❞ ❜② ✐ts ❞❡♥s✐t② ✇✐t❤ r❡❣❛r❞ t♦ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ♦♥ R✿ dµ(x) = 1 √ 2π • e -x 2 2 • dx ✭✶✳✶✮ ♦r✱ ❡q✉✐✈❛•❡♥t•②✱ ❜② ✐ts ❋♦✉r✐❡r tr❛♥s❢♦r♠✿ R e iξx dµ(x) = e -ξ 2 2 ✭✶✳✷✮ ❚❤❡ ❢♦••♦✇✐♥❣ ✐s str❛✐❣❤t❢♦r✇❛r❞ ❜✉t ♥♦t❡✇♦rt❤②✿ Pr♦♣♦s✐t✐♦♥ ✶✳✶✳✶✳✶✳ ▲❡t p ∈ [1, ∞[ ❛♥❞ Q ∈ R[X]❀ t❤❡♥ Q ∈ L p (R, µ)✳ ❚❤❡r❡ ❛•s♦ ✐s✿ ❚❤❡♦r❡♠ ✶✳✶✳✶✳✶✳ ❋♦r ❛♥② p ∈]1, ∞[✱ ♣♦•②♥♦♠✐❛•s ❛r❡ ❞❡♥s❡ ✐♥ L p (µ)✳ ✶ ✷ ❈❍❆P❚❊| ✶✳ ■◆❚|❖❉❯❈❚■❖◆ ❚❖ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ Pr♦♦❢✳ ▲❡t p ∈]1, ∞[✳ ❲❡ ❤❛✈❡ ❛•r❡❛❞② ♦❜s❡r✈❡❞ t❤❛t ❛♥② ♣♦•②♥♦♠✐❛• ✐s ✐♥ L p (µ)✳ ◆♦✇ •❡t f ∈ L p * ✱ s♦ ❜② ❍ö•❞❡r✬s t❤❡♦r❡♠ ❛•• t❤❡ x → x n f (x) ❛r❡ µ✲✐♥t❡❣r❛❜•❡ ❛♥❞ s✉♣♣♦s❡ t❤❛t ❢♦r ❛•• n ∈ N✿ x n f (x)dµ(x) = 0 ✭✶✳✸✮ ❚❤❡♥✱ ❜② ❛ ✈❡rs✐♦♥ ♦❢ ❋✉❜✐♥✐✬s t❤❡♦r❡♠✿ f (f )e iξx dµ(x) = ∞ n=0 (iξ) n n! x n f (x)dµ(x) = 0 ✭✶✳✹✮ s♦ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ t❤❡ ♠❡❛s✉r❡ f • dµ ✐s 0✱ ❤❡♥❝❡ ❜② ✐♥•❡❝t✐✈✐t② ♦❢ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ f = 0 µ✲❛✳s✳ ❇② ❞✉❛•✐t② t❤✐s ♣r♦✈❡s t❤❛t ♣♦•②♥♦♠✐❛•s ❛r❡ ❞❡♥s❡ ✐♥ (L p * ) ′ = L p ✳ ✶✳✶✳✷ •r❛❞✐❡♥t ❛♥❞ ❞✐✈❡r❣❡♥❝❡ ♦♣❡r❛t♦rs ♦♥ L p (R, µ) ■♥ t❤❡ s❡q✉❡• p ✐s ❛♥② r❡❛• ♥✉♠❜❡r ✐♥ ]1, ∞[✳ ❆ ✇❡❛❦ ✭✐❡✿ ❞✐str✐❜✉t✐♦♥❛•✮ ❞❡r✐✈❛t✐✈❡ ♦♣❡r❛t♦r ✐s ❝•❛ss✐❝❛••② ❞❡✜♥❡❞ ♦♥ L p (R, µ) ❛s ✐t ✐s ♦♥ L p (R, λ)❀ s❡❡✱ ❢♦r ❡①❛♠♣•❡✱ ❬✶(✱ ❬✼✵( ♦r ❬✼✶(✳ ❲❡ ✇✐•• ✐♥❞✐✛❡r❡♥t•② ♥♦t❡ f ′ ✱ df ♦r ∇f ❢♦r t❤❡ ✭✇❡❛❦ ♦r str♦♥❣✮ ❞❡r✐✈❛t✐✈❡ ♦❢ f ✱ ✇❤❡♥❡✈❡r ✐t ❡①✐sts✳ ❲❡ ✐♥tr♦❞✉❝❡ D p,k (R) t❤❡ s♣❛❝❡ ♦❢ t❤♦s❡ ❢✉♥❝t✐♦♥s ✐♥ L p ✇❤✐❝❤ ❛❞♠✐t k ✇❡❛❦ ❞❡r✐✈❛t✐✈❡s ✐♥ L p ❀ ✐♥ ♦♣❡r❛t♦r •❛♥❣✉❛❣❡ t❤❡♦r②✿ D p,k (R) = dom L p ∇ k ✭✶✳✺✮ ❲❡ ❡q✉✐♣ D p,k (R) ✇✐t❤ t❤❡ ❣r❛♣❤ ♥♦r♠✿ f D p,k (R) = f L p + ∇ k f L p ✭✶✳✻✮ ❚❤❡ ✭❣❛✉ss✐❛♥✮ ❞✐✈❡r❣❡♥❝❡ ♦♣❡r❛t♦r ✐s ❞❡✜♥❡❞ ❛s t❤❡ tr❛♥s♣♦s❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐♥ t❤❡ s❡♥s❡ ♦❢ t❤❡ ❞✉❛•✐t② ✐♥❞✉❝❡❞ ❜② t❤❡ ♠❡❛s✉r❡ µ✳ ❚♦ ❣❡t ❛ ✢❛✈♦✉r ♦❢ t❤✐s✱ ❝♦♥s✐❞❡r f, g ∈ S✳ ❚❤❡♥✱ ✐♥t❡❣r❛t✐♥❣ ❜② ♣❛rts✱ ♦♥❡ ♥♦t✐❝❡s t❤❛t✿ R f ′ (x)g(x)dµ(x) = R f ′ (x)g(x) e -x 2 2 √ 2π dx = - R f (x) d dx g(x) e -x 2 2 √ 2π dx = - R f (x)e x 2 2 d dx g(x)e -x 2 2 dµ(x) = R f (x) (-g ′ (x) + xg(x)) dµ(x) ✭✶✳✼✮ ✶✳✶✳ ▼❆▲▲■❆❱■◆ ❆◆❆▲(❙■❙ ■◆ ❋■◆■❚❊ ❉■▼❊◆❙■❖◆ ✸ ❚❤✐s •❡❛❞s ✉s t♦ t❤❡ ❢♦••♦✇✐♥❣ ❞❡✜♥✐t✐♦♥✿ ❉❡✜♥✐t✐♦♥ ✶✳✶✳✷✳✶ ✭•❛✉ss✐❛♥ ❞✐✈❡r❣❡♥❝❡ ♦♣❡r❛t♦r✮✳ ■t ✐s ❢♦r♠❛••② ❞❡✜♥❡❞ ♦♥ ✐ts ❞♦♠❛✐♥ dom L p (δ) t❤r♦✉❣❤ t❤❡ ❢♦r♠✉•❛✿ δf (x) = -f ′ (x) + xf (x) = -e x 2 2 d dx f (x)e -x 2 2 ✭✶✳✽✮ |❡♠❛r❦ ✶✳✶✳✷✳✶✳ ❚❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❞✐✈❡r❣❡♥❝❡ ✐s ❞❡♥s❡ ✐♥ L p ✳ ❉✐✈❡r❣❡♥❝❡s ❝♦✲ ✐♥❝✐❞❡ ✇❤❡♥ ❞❡✜♥❡❞ ✐♥ t❤❡ s❡♥s❡ ♦❢ t✇♦ ♦r ♠♦r❡ ♣ s♦ ✇❡ ♦♠✐t t❤❡ ✐♥❞❡①✳ ✶✳✶✳✸ ❍❡r♠✐t❡ ♣♦•②♥♦♠✐❛•s ❚❤❡ ♣✉r♣♦s❡ ♦❢ t❤✐s ♣❛r❛❣r❛♣❤ ✐s t♦ ✐♥tr♦❞✉❝❡ t❤❡✿ ❉❡✜♥✐t✐♦♥ ✶✳✶✳✸✳✶ ✭❍❡r♠✐t❡ ♣♦•②♥♦♠✐❛•s✮✳ H n (x) = (δ n 1)(x) = (-1) n e x 2 2 d n dx n e -x 2 2 ✭✶✳✾✮ ❇② ❛ str❛✐❣❤t❢♦r✇❛r❞ r❡❝✉rr❡♥❝❡ ♦♥❡ ❝❤❡❝❦s t❤❛t t❤❡s❡ ✐♥❞❡❡❞ ❛r❡ ♣♦•②♥♦♠✐❛• ❢✉♥❝t✐♦♥s ❛♥❞ t❤❛t H n ❤❛s ❞❡❣r❡❡ ♥✳ ■t ✐s ❡❛s② t♦ ❝♦♠♣✉t❡ t❤❡ ✜rst ❢❡✇ ❍❡r♠✐t❡ ♣♦•②♥♦♠✐❛•s✿ H 0 = 1 H 1 = X H 2 = X 2 -1 H 3 = X 3 -3X H 4 = X 4 -6X 2 + 3 . . . ❲❡ ✇✐•• ❛•s♦ ♥❡❡❞ t❤❡ ❢♦••♦✇✐♥❣ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❍❡r♠✐t❡ ♣♦•②♥♦♠✐❛•s✿ h n,a (x) := √ aH n x √ a ✭✶✳✶✵✮ ❚❤❡ ❍❡r♠✐t❡ ♣♦•②♥♦♠✐❛•s ❛r✐s❡ ❢r♦♠ ❛ ❣❡♥❡r❛t✐♥❣ s❡r✐❡s✿ (-1) n e x 2 2 d n dx n e -x 2 2 = e x 2 2 d n dt n |t=0 e -(x-t) 2 2 ✭✶✳✶✶✮ ❛♥❞ t❤❡r❡❢♦r❡ ✇❡ ✐♥❞❡❡❞ ❤❛✈❡✿ exp xt - t 2 2 = exp x 2 2 - 1 2 (x -t) 2 = ∞ n=0 t n n! H n (x) ✭✶✳✶✷✮ ✹ ❈❍❆P❚❊| ✶✳ ■◆❚|❖❉❯❈❚■❖◆ ❚❖ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ❛s ✇❡•• ❛s✿ exp xt -a t 2 2 = ∞ n=0 t n n! h n,a (x) ✭✶✳✶✸✮ ❚❤❡ ❢♦••♦✇✐♥❣ r❡s✉•t ✇✐•• ❜❡ ❝r✉❝✐❛• t♦ ♦✉r ♣✉r♣♦s❡✿ ❚❤❡♦r❡♠ ✶✳✶✳✸✳✶✳ Hn √ n! ✐s ❛♥ ♦rt❤♦♥♦r♠❛• ❜❛s✐s ♦❢ L 2 (µ)✳ Pr♦♦❢✳ ❲❡ ❤❛✈❡ s❡❡♥ t❤❛t ♣♦•②♥♦♠✐❛•s ❛r❡ ❞❡♥s❡✱ ✐♥ ♣❛rt✐❝✉•❛r✱ ✐♥ L 2 (µ)✳ ❆•s♦✱ ✇❡ ❦♥♦✇ t❤❛t H n ❤❛s ❞❡❣r❡❡ n✱ t❤❡r❡❢♦r❡ t❤❡ ❍❡r♠✐t❡ ♣♦•②♥♦♠✐❛•s ❛r❡ ❛ ❜❛s❡ ♦❢ R[X] ✭r❡s♣✳ C[X]✮✳ ❚❤❡r❡❢♦r❡ ✇❡ ♦♥•② ♥❡❡❞ t♦ ♣r♦✈❡ t❤❛t t❤❡ ❢❛♠✐•② ✇❡ ❝♦♥s✐❞❡r ✐s ♦rt❤♦♥♦r♠❛•✳ ❋✐rst •❡t n > m ❜❡ t✇♦ ✐♥t❡❣❡rs✳ ❚❤❡♥✿ E µ [H n H m ] = E µ [(δ n 1)H m ] = E µ [H (n) m ] = 0 ✭✶✳✶✹✮ s♦ ✇❡ ❤❛✈❡ ♣r♦✈❡♥ t❤❡ ♦rt❤♦❣♦♥❛•✐t②✳ ❋♦r t❤❡ ♥♦r♠❛•✐t②✱ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❣❡♥❡r❛t✐♥❣ s❡r✐❡s ❢♦r t❤❡ ❍❡r♠✐t❡ ♣♦•②♥♦♠✐❛•s ❛♥❞ t❛❦✐♥❣ t❤❡ ♦rt❤♦❣♦♥❛•✐t② ✐♥t♦ ❛❝❝♦✉♥t✱ ♦♥❡ s❡❡s t❤❛t✿ R exp xt - t 2 2 2 dµ(x) = ∞ n=0 t 2n n! 2 R H n (x) 2 dµ(x) ✭✶✳✶✺✮ ■t ✐s t❤❡♥ ❝•❛ss✐❝ t♦ ❝♦♠♣✉t❡ t❤❡ ✐♥t❡❣r❛• ♦❢ t❤❡ ▲❍❙ ✇❤✐❝❤ ✐s e t 2 ✱ t❤❡r❡❢♦r❡ ✐❞❡♥t✐❢②✲ ✐♥❣ ❝♦❡✣❝✐❡♥ts ✐♥ t❤❡ ❚❛②•♦r s❡r✐❡s ❡①♣❛♥s✐♦♥ •❡❛❞s t♦ E µ [H n ] = n! ❛s ❡①♣❡❝t❡❞✳ ❲❡ ♣r♦✈✐❞❡ t✇♦ ♦t❤❡r ❝❤❛r❛❝t❡r✐③❛t✐♦♥s ♦❢ t❤❡ ❍❡r♠✐t❡ ♣♦•②♥♦♠✐❛•s✿ Pr♦♣♦s✐t✐♦♥ ✶✳✶✳✸✳✶ ✭|❡❝✉rs✐✈❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❍❡r♠✐t❡ ♣♦•②♥♦♠✐❛•s✮✳ ❚❤❡ ❍❡r♠✐t❡ ♣♦•②♥♦♠✐❛•s ✈❡r✐❢② H 0 = 0✱ H 1 = X ❛♥❞✿ H n+1 = XH n -nH n-1 ✭✶✳✶✻✮ Pr♦♣♦s✐t✐♦♥ ✶✳✶✳✸✳✷ ✭❖❉❊ ✈❡r✐✜❡❞ ❜② t❤❡ ❍❡r♠✐t❡ ♣♦•②♥♦♠✐❛•s✮✳ ❚❤❡ ❍❡r♠✐t❡ ♣♦•②♥♦♠✐❛•s ✈❡r✐❢②✿ H ′′ n -XH ′ n + nH n = 0 ✭✶✳✶✼✮ ✶✳✶✳✹ ❚❤❡ ❖r♥st❡✐♥ ❯❤•❡♥❜❡❝❦ ♦♣❡r❛t♦r ❛♥❞ t❤❡ ❛ss♦❝✐❛t❡❞ s❡♠✐❣r♦✉♣ ■t ✐s str❛✐❣❤t❢♦r✇❛r❞ ❝❛•❝✉•✉s t♦ ❞❡r✐✈❡ t❤❡ ❢♦••♦✇✐♥❣ ❡q✉❛t✐♦♥✱ ✇❤✐❝❤ ✐s ❢❛♠✐•✐❛r t♦ q✉❛♥t✉♠ ♣❤②s✐❝✐sts✿ dδf -δdf = f ✭✶✳✶✽✮ ✶✳✶✳ ▼❆▲▲■❆❱■◆ ❆◆❆▲(❙■❙ ■◆ ❋■◆■❚❊ ❉■▼❊◆❙■❖◆ ✺ ❲❡ ✇❛♥t t♦ ❛♣♣•② t❤✐s ❢♦r♠✉•❛ t♦ ❛♥② ♦♥❡ ♦❢ t❤❡ H n ✳ ❋✐rst✱ ❝♦♥s✐❞❡r H ′ n+1 ✳ ❚❤✐s ❤❛s t♦ ❜❡ ❛ ♣♦•②♥♦♠✐❛• ♦❢ ❞❡❣r❡❡ n✳ ▲❡t ✉s ❝♦♠♣✉t❡ ✐t ❡①♣•✐❝✐t❡•② ❜② ♣r♦•❡❝t✐♥❣ ✐t ♦♥ t❤❡ H m ✱ m ≤ n✳ E µ [H ′ n+1 H n ] = E µ [H n+1 δH n ] = E µ [H 2 n+1 ] = (n + 1)! ✭✶✳✶✾✮ ❛♥❞ s✐♠✐•❛r•②✱ ❢♦r m < n✿ E µ [H ′ n+1 H m ] = 0 ✭✶✳✷✵✮ ❚❤❡r❡❢♦r❡ ❜② ❛♣♣•②✐♥❣ ❢♦r♠✉•❛ ✭✶✳✶✽✮ ♦♥❡ ♦❜t❛✐♥s✿ δdH n = nH n ✭✶✳✷✶✮ ❚❤✐s •❡❛❞s ✉s t♦ ✐♥tr♦❞✉❝✐♥❣ t❤❡✿ ❉❡✜♥✐t✐♦♥ ✶✳✶✳✹✳✶ ✭❖r♥st❡✐♥ ❯❤•❡♥❜❡❝❦ ♦♣❡r❛t♦r✮✳ Lf = δdf = d * df ✭✶✳✷✷✮ ❖❢ ❝♦✉rs❡ ✐t ✐s str❛✐❣❤t❢♦r✇❛r❞ t♦ ❝♦♠♣✉t❡✿ Lf (x) = -f ′′ (x) + xf ′ (x) ✭✶✳✷✸✮ ❚❤✐s ♦♣❡r❛t♦r ✐s s②♠♠❡tr✐❝❛• ❛♥❞ ♣♦s✐t✐✈❡ ❜② ❝♦♥str✉❝t✐♦♥❀ ✐t ✐s ❞✐❛❣♦♥❛•✐③❡❞ ❜② t❤❡ ♦rt❤♦♥♦r♠❛• ❜❛s✐s ♦❢ t❤❡ s❝❛•❡❞ ❍❡r♠✐t❡ ♣♦•②♥♦♠✐❛•s ✇❤✐❝❤ ✐s r❡•❡✈❛♥t t♦ ♦✉r st✉❞②✳ -L ✐s t❤❡ ❣❡♥❡r❛t♦r ♦❢ ❛ ❝♦♥tr❛❝t✐♦♥ s❡♠✐❣r♦✉♣ ✭s❡❡ ❛♣♣❡♥❞✐① ❈✮ ♦♥ L p (µ)✿ t❤❡ ❖r♥st❡✐♥✲❯❤•❡♥❜❡❝❦ s❡♠✐❣r♦✉♣✱ ✇❤✐❝❤ ✇❡ ♥♦t❡ P t ✳ ❇② ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ❝♦♥tr❛❝t✐♦♥ s❡♠✐❣r♦✉♣ P t ❡♥•♦②s t❤❡ t✇♦ ❢♦••♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿ P t P s f = P t+s f ✭✶✳✷✹✮ ❛♥❞ P t f L p ≤ f L p ✭✶✳✷✺✮ ❆•s♦✱ s❡♠✐❣r♦✉♣ t❤❡♦r② t❡••s ✉s t❤❛t P t f ✐s t❤❡ ✉♥✐q✉❡ s♦•✉t✐♦♥ ♦❢ t❤❡ ❈❛✉❝❤② ♣r♦❜•❡♠✿ ∂ ∂t g(t, x) = Lg(x) ✭✶✳✷✻✮ g(0, •) = f ✭✶✳✷✼✮ ❋r♦♠ t❤❡ ✉♥✐❝✐t② ♦❢ t❤❡ s♦•✉t✐♦♥ t♦ s✉❝❤ ❛ ❈❛✉❝❤② ♣r♦❜•❡♠ ♦♥❡ s❡❡s t❤❛t ❢♦r f ∈ L p 1 ∩ L p 2 ✱ t❤❡ t✇♦ ♣♦t❡♥t✐❛• ❞❡✜♥✐t✐♦♥s ♦❢ P t f ❝♦✐♥❝✐❞❡ ❛♥❞ ✇❡ ✇✐•• ♥♦ •♦♥❣❡r ♠❛❦❡ ❛ ❞✐st✐♥❝t✐♦♥✳ ❆❣❛✐♥ ❜❡❝❛✉s❡ ♦❢ t❤❡ ✉♥✐❝✐t② ❛♥❞ ♦❢ t❤❡ ❢♦r♠✉•❛ LH n = H n ✱ ♦♥❡ s❡❡s t❤❛t✿ P t H n = e -nt H n ✭✶✳✷✽✮ ❲❡ s❤❛•• ♥♦✇ ♣r♦✈❡ t❤❡ ❢♦••♦✇✐♥❣✿ ✻ ❈❍❆P❚❊| ✶✳ ■◆❚|❖❉❯❈❚■❖◆ ❚❖ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ❚❤❡♦r❡♠ ✶✳✶✳✹✳✶ ✭▼❡❤•❡r✬s ❢♦r♠✉•❛✮✳ ❚❤❡ ❖r♥st❡✐♥ ❯❤•❡♥❜❡❝❦ s❡♠✐❣r♦✉♣ ✐s ❞❡✲ ✜♥❡❞ ❜② ❛ ❞❡♥s✐t②✿ P t f (x) = f (y)p t (x, y)dµ(y) ✭✶✳✷✾✮ ✇✐t❤✱ ♠♦r❡ ♣r❡❝✐s❡•②✿ p t (x, y) = ∞ n=0 e -nt n! H n (x)H n (y) = 1 √ 1 -e -2t exp - e -2t x 2 -2xye -t + e -2t y 2 1 -e -2t ✭✶✳✸✵✮ ❚❤✐s ✐s ❡q✉✐✈❛•❡♥t t♦✿ P t f (x) = f e -t x + √ 1 -e -2t y dµ(y) = E f (e -t x + √ 1 -e -2t N ) ✭✶✳✸✶✮ ✇❤❡r❡ N ✐s ❛ ♥♦r♠❛• ❣❛✉ss✐❛♥ r❛♥❞♦♠ ✈❛r✐❛❜•❡✳ Pr♦♦❢✳ ❲❡ ♣r♦✈❡ t❤❡ r❡s✉•t ✐♥ t❤❡ ❝❛s❡ ✇❤❡r❡ p = 2 ❛♥❞ ❡①t❡♥❞ ✐t ❜② ❞❡♥s✐t② t♦ t❤❡ ♦t❤❡r ❝❛s❡s✳ ■♥❞❡❡❞ ✐❢ f ∈ L 2 ✱ ♦♥❡ ❤❛s ❛♥ ❡①♣❛♥s✐♦♥ ♦❢ t❤❡ t②♣❡✿ f = c n (f ) H n √ n! ✭✶✳✸✷✮ ✇❤❡r❡ c n (f ) = R f (x) H n (x) √ n! dx ✭✶✳✸✸✮ ❛♥❞ t❤❡r❡❢♦r❡✿ P t f = e -nt c n (f ) H n √ n! ✭✶✳✸✹✮ ❙♦ ✐❢ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ❢✉♥❝t✐♦♥✿ p t (x, y) = ∞ n=0 e -nt n! H n (x)H n (y) ✭✶✳✸✺✮ ✇❤✐❝❤ ✐s ❡❛s✐•② ✈❡r✐✜❡❞ t♦ ❜❡ ✇❡•• ❞❡✜♥❡❞ ✐♥ L 2 (µ ⊗2 )✱ ✇❡ s❡❡ t❤❛t✿ R f (y)p t (x, y)dy = e -nt H n (x) √ n! R f (y) H n (y) √ n! dy = e -nt c n (f ) H n √ n! = P t f (x) ✭✶✳✸✻✮ ❚❤❡ ❡①♣•✐❝✐t ❡①♣r❡ss✐♦♥ ❢♦r p t ✐s t❤❡♥ ❝♦♠♣✉t❡❞ t❤r♦✉❣❤ ✐♥✈❡rs❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠✳ ■♥❞❡❡❞✱ r❡❝❛•• t❤❛t✿ H n (x) = (-1) n e x 2 2 d dx e -x 2 2 ✭✶✳✸✼✮ ✶✳✶✳ ▼❆▲▲■❆❱■◆ ❆◆❆▲(❙■❙ ■◆ ❋■◆■❚❊ ❉■▼❊◆❙■❖◆ ✼ s♦ s✐♥❝❡✿ e -x 2 2 = e isx-x 2 2 ds √ 2π ✭✶✳✸✽✮ t❤❡r❡ ❝♦♠❡s✿ H n (x) = (-i) n e x 2 2 s n e isx-x 2 2 ds √ 2π ✭✶✳✸✾✮ ❚❤❡r❡❢♦r❡✿ p t (x, y) = ∞ n=0 e -nt n! H n (x)H n (y) = e x 2 +y 2 2 ∞ n=0 (-στ t) n n! e iσx-σ 2 2 e iτ y-τ 2 2 dσ √ 2π dτ √ 2π = e x 2 +y 2 2 exp -στ e -t + iσx - σ 2 2 + iτ y - τ 2 2 dσ √ 2π dτ √ 2π ✭✶✳✹✵✮ ❚❤❡♥ ✇❡ ❞♦ t❤❡ ❢♦••♦✇✐♥❣ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜•❡✱ ✇❤✐❝❤ s❡♣❛r❛t❡s t❤❡ ❞♦✉❜•❡ ✐♥t❡❣r❛•✿ (σ, τ ) → σ + τ √ 2 , σ -τ √ 2 ✭✶✳✹✶✮ ❛♥❞ ✇❡ ♦❜t❛✐♥✿ p t (x, y) = e x 2 +y 2 2 exp - σ 2 -τ 2 2 e -t + i σ + τ √ 2 - (σ + τ ) 2 4 + i σ -τ √ 2 - (σ -τ ) 2 4 dσ √ 2π dτ √ 2π = e x 2 +y 2 2 exp - 1 2 (1 + e -t )σ 2 + i x + y √ 2 σ dσ √ 2π • exp - 1 2 (1 -e -t )τ 2 + i x -y √ 2 τ dτ √ 2π = e
x 2 +y 2 2

• e

- (x+y) 2 2(1+e -t √ 1 + e t • e - (x-y) 2 2(1-e -t √ 1 -e -t = 1 √ 1 -e -2t • exp - 1 -e -t 1 + e -t • (x + y) 2 4 -- 1 + e -t 1 -e -t • (x -y) 2 4 ✭✶✳✹✷✮ s♦ ✇❡ ♦❜t❛✐♥ ♦✉r ❡①♣r❡ss✐♦♥ ♦❢ p t ✳ ❋✐♥❛••②✱ t❤❡ •❛st ✐❞❡♥t✐t② ❛r✐s❡s ❢r♦♠ ❛ s✐♠♣•❡ ❝❤❛♥❣•❡ ♦❢ ✈❛r✐❛❜•❡s✳ |❡♠❛r❦ ✶✳✶✳✹✳✶✳ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡s❡ ❞❡♥s✐t✐❡s ♠❛② ❛•s♦ ❜❡ ❞♦♥❡ t❤r♦✉❣❤ t❤❡ ❙❉❊ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❖r♥st❡✐♥✲❯❤•❡♥❜❡❝❦ ♣r♦❝❡ss✿ dX t = -X t dt + √ 2dW t ✭✶✳✹✸✮ ✇❤✐❝❤ s♦•✈❡s ❛s✿ X t = X 0 + √ 2 t 0 e -(t-s) dW s ✭✶✳✹✹✮ ❛♥❞ ♦♥❡ ♠❛② ❝❤❡❝❦ t❤❛t X t ❤❛s ❞❡♥s✐t② p t ✐❢ X 0 = 0✳ ✽ ❈❍❆P❚❊| ✶✳ ■◆❚|❖❉❯❈❚■❖◆ ❚❖ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ❋✐♥❛••② t❤❡ ❢♦••♦✇✐♥❣ r❡s✉•t ✐s s✐♠♣•❡ ❜✉t ✐♠♣♦rt❛♥t✿ ❚❤❡♦r❡♠ ✶✳✶✳✹✳✷✳ ❚❤❡ P t ❛♥❞ L ❛r❡ s②♠♠❡tr✐❝❛• ♦♣❡r❛t♦rs✳ ✶✳✶✳✺ ❋r❛❝t✐♦♥❛• ❙♦❜♦•❡✈ s♣❛❝❡s ❋r❛❝t✐♦♥❛• ♣♦✇❡rs ♦❢ Id -L ❛r❡ ❞❡✜♥❡❞ ❛s ✐s ❡①♣•❛✐♥❡❞ ✐♥ ❛♣♣❡♥❞✐① ❈❀ t❤❡♥ ♦♥❡ ❣❡ts t❤❡✿ ❚❤❡♦r❡♠ ✶✳✶✳✺✳✶✳ ▲❡t p ∈]1, ∞[ ❛♥❞ k ∈ N❀ t❤❡ t✇♦ ❢♦••♦✇✐♥❣ ♥♦r♠s ❛r❡ ❡q✉✐✈❛✲ •❡♥t ♦♥ D p,k ✿ f D p,k (R) = f L p + ∇ k f L p ✭✶✳✹✺✮ ❛♥❞✿ f = (Id -L) k/2 f L p ✭✶✳✹✻✮ ❚❤✐s ❛••♦✇s t♦ ❞❡✜♥❡ ❢r❛❝t✐♦♥❛• ❙♦❜♦•❡✈ s♣❛❝❡s ❜✉✐•t ♦♥ t❤❡ ❣❛✉ss✐❛♥ ♠❡❛s✉r❡ ✐♥ ❛ ♥❛t✉r❛• ✇❛②❀ ✇❡ ♦♠✐t t❤❡ ❞❡t❛✐•s ❛s ✇❡ ♠❛② r❡❝♦✈❡r t❤❡♠ ❛s ❛ s♣❡❝✐❛• ❝❛s❡ ♦❢ t❤❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥♥❛• r❡s✉•ts ✇❤✐❝❤ ❛r❡ ♠♦r❡ ❞✐r❡❝t•② r❡•❡✈❛♥t t♦ ♦✉r st✉❞② ✲ s❡❡ t❤❡ ♥❡①t s❡❝t✐♦♥✳ ✶✳✶✳✻ ❊①t❡♥s✐♦♥ t♦ d✲❞✐♠❡♥s✐♦♥❛• s♣❛❝❡s ■❢ n = (n 1 , • • • , n d ) ∈ N N ✱ ✇❡ ♥♦t❡✿ n! = N i=1 n i ! ✭✶✳✹✼✮ ❲❡ ❛•s♦ ❞❡✜♥❡ N ✲❞✐♠❡♥s✐♦♥♥❛• ❍❡r♠✐t❡ ♣♦•②♥♦♠✐❛•s ❛s✿ H n (x) = N i=1 H n i (x i ) ✭✶✳✹✽✮ ✇❤❡r❡ ✇❡ ❛❜✉s✐✈❡•② ✉s❡❞ t❤❡ s❛♠❡ ♥♦t❛t✐♦♥ ❢♦r 1✲❞✐♠❡♥s✐♦♥♥❛• ❛♥❞ N ✲❞✐♠❡♥s✐♦♥♥❛• ❍❡r♠✐t❡ ♣♦•②♥♦♠✐❛•s✳ ❲❡ ♥♦t❡ µ ⊗N ❢♦r t❤❡ N ✲❞✐♠❡♥s✐♦♥♥❛• ♥♦r♠❛• ❣❛✉ss✐❛♥ ♠❡❛✲ s✉r❡ ♦♥ R N ✳ ❚❤❡♥ ♦♥❡ ♣r♦✈❡s✿ ❚❤❡♦r❡♠ ✶✳✶✳✻✳✶✳ Hn √ n! ✐s ❛♥ ♦rt❤♦♥♦r♠❛• ❜❛s✐s ♦❢ L 2 (µ ⊗N )✳ Pr♦♦❢✳ ❙✐♥❝❡ ❡✈❡r② ♠♦♥♦♠✐❛• X k i ♠❛② ❜❡ ✇r✐tt❡♥ ❛s ❛ •✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ H n (X i )✱ n ∈ N✱ ✐t ✐s ❝•❡❛r t❤❛t ❡✈❡r② ♠♦♥♦♠✐❛• X k ♦♥ R N ♠❛② ❜❡ ✇r✐tt❡♥ ❛s ❛ •✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❡•❡♠❡♥ts ✐♥ ♦✉r ❢❛♠✐•②✳ ❚❤❡r❡❢♦r❡ ✇❡ ✇✐•• ❜❡ ❞♦♥❡ ❛s s♦♦♥ ✶✳✷✳ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ✾ t❤❛t ✇❡ ♣r♦✈❡ t❤❛t ♦✉r ❢❛♠✐•② ✐s ♦rt❤♦❣♦♥❛• ❛♥❞ t❤✐s •❛st ❢❛❝t ✐s str❛✐❣❤t❢♦r✇❛r❞ ❛s s♦♦♥ ❛s ♦♥❡ ♥♦t✐❝❡s t❤❛t✿ R N H n (x)H m (x)dµ ⊗N (x) = N i=1 R H m i (x i )H n i (x i )dµ(x i ) ✭✶✳✹✾✮ ❚❤✐s ❢❛❝t ❛•• t❤❡ r❡s✉•ts ✇❡ ❤❛✈❡ •✉st s❡❡♥ t♦ t❤❡ N ✲❞✐♠❡♥s✐♦♥♥❛• ❝❛s❡❀ ❛s t❤❡ ♣r♦♦❢s ❛r❡ str❛✐❣❤t❢♦r❛r❞ ✇❡ ♦♠✐t t❤❡♠✳ ✶✳✷ ▼❛••✐❛✈✐♥ ❝❛•❝✉•✉s ■♥ t❤✐ s❡❝t✐♦♥ ✇❡ ✇✐•• ❝♦♥str✉❝t ▼❛••✐❛✈✐♥ ❝❛•❝✉•✉s ❛s ❛♥ ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥♥❛• ❡①t❡♥s✐♦♥ ♦❢ t❤❡ r❡s✉•ts ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✳ ✶✳✷✳✶ ❚❤❡ ❲✐❡♥❡r ❛♥❞ ❈❛♠❡r♦♥✲▼❛rt✐♥ s♣❛❝❡s ❉❡✜♥✐t✐♦♥ ✶✳✷✳✶✳✶ ✭❲✐❡♥❡r s♣❛❝❡✮✳ ❚❤❡ ❲✐❡♥❡r s♣❛❝❡ ✐s t❤❡ ❢♦••♦✇✐♥❣ ✈❡❝t♦r s♣❛❝❡ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s✿ W = {f ∈ C ([0, 1], R) s.t.f (0) = 0} ✭✶✳✺✵✮ ❡q✉✐♣♣❡❞ ✇✐t❤ ✐ts ♥❛t✉r❛• ❇❛♥❛❝❤ s♣❛❝❡ str✉❝t✉r❡ ❛r✐s✐♥❣ ❢r♦♠ t❤❡ ❢♦••♦✇✐♥❣ ♥♦r♠✿ f ∞ = sup t∈[0,1] |f (t)| ✭✶✳✺✶✮ ❲❡ r❡❝❛•• t❤❡✿ ❚❤❡♦r❡♠ ✶✳✷✳✶✳✶ ✭|❛❞♦♥✮✳ ❚❤❡ str♦♥❣ ❞✉❛• s♣❛❝❡ W ′ ♦❢ W ✐s M/(R • δ 0 )✱ ✇❤❡r❡ M ✐s t❤❡ s❡t ♦❢ ❛•• |❛❞♦♥ ♠❡❛s✉r❡s ♦♥ [0, 1] ❛♥❞ δ 0 ✐s t❤❡ ❉✐r❛❝ ♠❛ss ❛t 0✳ ■t ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡ ✇❤❡♥ ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ❞✉❛• ♥♦r♠ ♦❢ • ∞ ✱ ✇❤✐❝❤ ✐s✿ ν W * = sup f ∞=1 f dν ✭✶✳✺✷✮ ❲❡ ✇✐•• ❝♦♥s✐❞❡r t❤❡ ❢♦••♦✇✐♥❣ ❡✈❛•✉❛t✐♦♥ ❢✉♥❝t✐♦♥❛•s✿ ❉❡✜♥✐t✐♦♥ ✶✳✷✳✶✳✷ ✭❈❛♥♦♥✐❝❛• ❢✉♥❝t✐♦♥❛•s ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡✮✳ W t = w ∈ W → w(t) ∈ R ✭✶✳✺✸✮ ✶✵ ❈❍❆P❚❊| ✶✳ ■◆❚|❖❉❯❈❚■❖◆ ❚❖ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ❈•❡❛r•②✱ t❤❡s❡ ❛r❡ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥❛•s ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡✿ t❤❡② ❛❝t✉❛••② ❝♦✐♥❝✐❞❡ ✇✐t❤ t❤❡ ❉✐r❛❝ ♠❛ss❡s✳ ❚❤❡ ❝♦r♥❡rst♦♥❡ ♦❢ ▼❛••✐❛✈✐♥ ❝❛•❝✉•✉s ✐s t❤❡ ❢♦•✲ •♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✶✳✷✳✶✳✷✳ ❚❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ♠❡❛s✉r❡ µ ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡ s✉❝❤ t❤❛t t❤❡ st♦❝❤❛st✐❝ ♣r♦❝❡ss (W t ) 0≤t≤1 ❞❡✜♥❡❞ ♦♥ t❤❡ ♣r♦❜❛❜✐•✐t② s♣❛❝❡ (W, µ) ✐s ❛ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳ ❲❡ ❝❛•• t❤❛t ♠❡❛s✉r❡ t❤❡ ❲✐❡♥❡r ♠❡❛s✉r❡✳ ❚❤❡ ❝♦♦r❞✐♥❛t❡ ♣r♦❝❡ss X ❝♦♥s✐❞❡r❡❞ ✉♥❞❡r t❤❡ ❲✐❡♥❡r ♠❡❛s✉r❡ ✐s t❤❡ ❝❛♥♦♥✐❝❛• ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳ Pr♦♦❢✳ ❲❡ ❞♦ ♥♦t r❡❝❛•• ❤♦✇ t♦ ❝♦♥str✉❝t ❛ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥❀ ✐♥st❡❛❞✱ ✇❡ s✐♠✲ ♣•② s❤♦✇ ❤♦✇✱ ❜❡✐♥❣ ❣✐✈❡♥ ❛ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ Y ♦♥ ❛ ❝❡rt❛✐♥ ♣r♦❜❛❜✐•✐t② s♣❛❝❡ (Ω, F, P)✱ ✇❡ ♠❛② ♠❛♣ ✐t t♦ t❤❡ ❡✈❛•✉❛t✐♦♥ ❢✉♥❝t✐♦♥❛•s ❛♥❞ ❝♦♥str✉❝t t❤❡ ❲✐❡♥❡r ♠❡❛s✉r❡✳ ■♥❞❡❡❞ •❡t ✉s ❞❡✜♥❡ ❛ ♠❡❛s✉r❛❜•❡ ❛♣♣•✐❝❛t✐♦♥✿ Φ = ω ∈ Ω → Φ(ω) ∈ W ✭✶✳✺✹✮ ❜② ♣✉tt✐♥❣✿ Φ(ω) = t ∈ [0, 1] → Y t (ω) = W t (Y (ω)) ∈ R ✭✶✳✺✺✮ ❈•❡❛r•② t❤❡ Φ(ω) ❛r❡ ❛✳s✳ ✐♥ W ❛s t❤❡ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ❛✳s✳ ❤❛s ❝♦♥t✐♥✉♦✉s tr❛•❡❝t♦r✐❡s✳ ❚❤❡♥ Φ ✐s ♠❡❛s✉r❛❜•❡ ❜❡❝❛✉s❡ ♦♥ ♦♥❡ ❤❛♥❞ Y ✐s ❛♥❞ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞ t❤❡ ❡✈❛•✉❛t✐♦♥ ❢✉♥❝t✐♦♥❛•s ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡ ❛r❡ ❝♦♥t✐♥✉♦✉s✱ ❤❡♥❝❡ ❜♦r❡•✐❛♥ ❢♦r t❤❡ ♥❛t✉r❛• t♦♣♦•♦❣②✳ ❲❡ ♠❛② t❤❡r❡❢♦r❡ ❞❡✜♥❡ µ ❛s t❤❡ ✐♠❛❣❡ ♠❡❛s✉r❡ ♦❢ P ❜② Φ ❛♥❞ ✇❡ t❤❡♥ ❤❛✈❡✱ ❢♦r ❛♥② ♠❡❛s✉r❛❜•❡ ❛♣♣•✐❝❛t✐♦♥ f ♦♥ W✿ E µ [f (W t , 0 ≤ t ≤ 1)] = E P [f ((W • Y ) t , 0 ≤ t ≤ 1)] = E P [f (Y t , 0 ≤ t ≤ 1)] ✭✶✳✺✻✮ ❙✐♥❝❡ Y ✐s ❛ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ✉♥❞❡r P t❤✐s ♣r♦✈❡s t❤❛t W ✐s ❛ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ✉♥❞❡r µ✳ ■t ✐s ✐♥t❡r❡st✐♥❣ t♦ ♥♦t❡ t❤❛t ❢♦r µ -a.s. w 1 , w 2 ∈ W ♦♥❡ ❤❛s✿ W t (w 1 + w 2 ) = W t (w 1 ) + W t (w 2 ) ✭✶✳✺✼✮ ❚❤❡ ❢♦••♦✇✐♥❣ s♣❛❝❡ ♦❢ ❢✉♥❝t✐♦♥s ✐s ❝•♦s❡•② r❡•❛t❡❞ t♦ t❤❡ ❲✐❡♥❡r s♣❛❝❡✿ ❉❡✜♥✐t✐♦♥ ✶✳✷✳✶✳✸ ✭❈❛♠❡r♦♥✲▼❛rt✐♥ s♣❛❝❡✮✳ H = t ∈ [0, 1] → t 0 f (s)ds ✭✶✳✺✽✮ ❚❤❛t ✐s✿ H ✐s t❤❡ s♣❛❝❡ ♦❢ ♣r✐♠✐t✐✈❡s ♦❢ L 2 ([0, 1] λ) ❢✉♥❝t✐♦♥s t❛❦✐♥❣ t❤❡ ✈❛•✉❡ 0 ❛t 0✳ ❊q✉✐✈❛•❡♥t•②✱ H ✐s t❤❡ s♣❛❝❡ ♦❢ t❤♦s❡ L 2 ❢✉♥❝t✐♦♥s ✇❤✐❝❤ ❝❛♥❝❡• ❛t 0 ❛♥❞ ❤❛✈❡ L 2 ✇❡❛❦ ❞❡r✐✈❛t✐✈❡s✳ ❲❡ ✇✐•• ✉s✉❛••② ♥♦t❡ ḣ ❢♦r t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ h ∈ H✳ ✶✳✷✳ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ✶✶ ❲❡ ✇✐•• ♠❛❦❡ ✉s❡ ♦❢ t❤❡ ❢♦••♦✇✐♥❣ ❝•❛ss✐❝ r❡s✉•ts✿ ❚❤❡♦r❡♠ ✶✳✷✳✶✳✸ ✭❚♦♣♦•♦❣✐❝❛• str✉❝t✉r❡ ♦❢ t❤❡ ❈❛♠❡r♦♥✲▼❛rt✐♥ s♣❛❝❡✮✳ ❚❤❡ ❢♦•✲ •♦✇✐♥❣ q✉❛♥t✐t② ✐s ❛ ♥♦r♠ ♦♥ t❤❡ ✈❡❝t♦r s♣❛❝❡ H✿ |h| H = 1 0 ḣ2 (s)ds ✭✶✳✺✾✮ ■t ❞❡✜♥❡s ❛ ❍✐•❜❡rt s♣❛❝❡ str✉❝t✉r❡ ♦♥ H✳ ❆•s♦✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦••♦✇✐♥❣ ✐♥•❡❝t✐♦♥s✿ W * ֒→ H ֒→ i W ✭✶✳✻✵✮ ✇✐t❤ i ❞❡♥s❡ ✐♥t♦ • ∞ ❛♥❞ ❝♦♠♣❛❝t ✐♥t♦ | • | H ✳ Pr♦♦❢✳ ❚❤❡ ❍✐•❜❡rt str✉❝t✉r❡ ❛♥❞ t❤❡ |❍❙ ✐♥•❡❝t✐♦♥ ❛r❡ ♦❜✈✐♦✉s✳ ❚❤❡ ❝♦♠♣❛❝t♥❡ss ♦❢ i ✐s ❛ s♣❡❝✐❛• ❝❛s❡ ♦❢ t❤❡ |❡••✐❝❤ t❤❡♦r❡♠✳ ❚❤❡ ❞❡♥s✐t② ♠❛② ❜❡ ♦❜t❛✐♥❡❞✱ ❢♦r ❡①❛♠♣•❡✱ ✈✐❛ t❤❡ ❙t♦♥❡✲❲❡✐❡rstr❛ss t❤❡♦r❡♠✳ ❋♦r t❤❡ ▲❍❙ ✐♥•❡❝t✐♦♥✱ s✐♠♣•② ❝♦♥s✐❞❡r t❤❡ ❛♣♣•✐❝❛t✐♦♥✿ Φ = ν → t → t 0 ν(]s, 1])ds ✭✶✳✻✶✮ ■t ✇❛s ♥❛t✉r❛• t♦ ✐♥tr♦❞✉❝❡ Φ ❜❡❝❛✉s❡ t❤❡ ❢♦••♦✇✐♥❣ ✐s ✈❛•✐❞ µ✲❛✳s✳ t❤r♦✉❣❤ ❛ s✐♠♣•❡ ✐♥t❡❣r❛t✐♦♥ ❜② ♣❛rts ✇❤✐❝❤ ❝❛♥ ❜❡ •✉st✐✜❡❞ ♣r♦♣❡r•② ❜② ✇♦r❦✐♥❣ ♦♥ |✐❡♠❛♥♥ s✉♠s ❛♥❞ t❛❦✐♥❣ •✐♠✐ts ✐♥ ♣r♦❜❛❜✐•✐t②✿ W * ν, w W = 1 0 w(s)dν(s) = 1 0 W s (w)dν(s) = W 1 (w)ν([0, 1]) - 1 0 ν([0, s])dW s (w) = 1 0 ν(]s, 1])dW s (w) ✭✶✳✻✷✮ ❆•s♦ ✇❡ ♥♦✇ s❡❡ t❤❛t✱ ✐♥ ❛❞❞✐t✐♦♥ t♦ • W * ✱ t❤❡r❡ ✐s ❛♥♦t❤❡r ♥❛t✉r❛• ♥♦r♠ ♦♥ W * ✿ ν W * ,2 = 1 0 ν(]s, 1]) 2 ds = E 1 0 ν(]s, 1])dW s 2 ✭✶✳✻✸✮ ✶✷ ❈❍❆P❚❊| ✶✳ ■◆❚|❖❉❯❈❚■❖◆ ❚❖ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ■t ❝•❡❛r•② ❛r✐s❡s ❢r♦♠ ❛ s❝❛•❛r ♣r♦❞✉❝t✱ ❛♥❞ ♣r♦✈✐❞❡s ❛♥ ✐s♦♠❡tr② ❜❡t✇❡❡♥ t❤❡ |❛❞♦♥ ♠❡❛s✉r❡s ❛♥❞ ❛ s✉❜s❡t ♦❢ t❤❡ st♦❝❤❛st✐❝ ✐♥t❡❣r❛•s ♦♥ ❞❡t❡r♠✐♥✐st✐❝ L 2 (λ) ❢✉♥❝t✐♦♥s✳ ■t ✐s ♥♦t ❛ ❝♦♠♣•❡t❡ s♣❛❝❡✿ ✐♥❞❡❡❞✱ t❤❡ s❡t ♦❢ t❤❡ ♣r✐♠✐t✐✈❡s ♦❢ |❛❞♦♥ ♠❡❛s✉r❡s✱ ✐❡ t❤❡ ✐♠❛❣❡ ♦❢ Φ✱ ✐s ❡❛s✐•② s❡❡♥ t♦ ❜❡ ❞❡♥s❡ ✐♥ H✱ t❤❡r❡❢♦r❡ t❤❡ s❡t ♦❢ t❤❡ st♦❝❤❛st✐❝ ✐♥t❡❣r❛•s ḣdW ✱ ✇❤❡r❡ h ❞❡s❝r✐❜❡s H✱ ✐s ✐s♦♠❡tr✐❝ t♦ t❤❡ ❝♦♠♣•❡t✐♦♥ ♦❢ t❤❡ s❡t ♦❢ |❛❞♦♥ ♠❡❛s✉r❡s W * ❢♦r t❤❡ ♥♦r♠ • W * ,2 ✳ ❲❡ ✜♥✐s❤ t❤✐s ♣❛r❛❣r❛♣❤ ✇✐t❤ t✇♦ s✐♠♣•❡ ❝♦♠♣✉t❛t✐♦♥s r❡•❛t❡❞ t♦ ❡q✉❛t✐♦♥ ✶✳✺✼✳ Pr♦♣♦s✐t✐♦♥ ✶✳✷✳✶✳✶✳ ▲❡t h ∈ L 2 ✳ ❋♦r ❛✳s✳ w 1 , w 2 ∈ W✿ h(s)dW s (w 1 + w 2 ) = h(s)dW s (w 1 ) + h(s)dW s (w 2 ) ✭✶✳✻✹✮ Pr♦♦❢✳ ❲r✐t❡ t❤❡ st♦❝❛st✐❝ ✐♥t❡❣r❛• ♦♥ t❤❡ ▲❍❙ ❛s t❤❡ •✐♠✐t ♦❢ ❛ |✐❡♠❛♥♥ s✉♠ ❛♥❞ ❛♣♣•② ❡q✉❛t✐♦♥ ✶✳✺✼✳ ❲✐t❤ s✐♠✐•❛r t❡❝❤♥✐q✉❡s✱ ♦♥❡ ♣r♦✈❡s✿ Pr♦♣♦s✐t✐♦♥ ✶✳✷✳✶✳✷✳ ▲❡t g, h ∈ H✳ ❚❤❡♥ ḣ(s)dW s (g) = (g, h) H ✭✶✳✻✺✮ ✶✳✷✳✷ ❈❤❛♦s ❞❡❝♦♠♣♦s✐t✐♦♥ ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡ ❲❡ r❡❝❛•• t❤❛t W ❞❡♥♦t❡s t❤❡ ❝❛♥♦♥✐❝❛• ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ♦♥ (W, µ) ❛♥❞ ✇❡ st❛rt ✇✐t❤ t❤❡ ❢♦••♦✇✐♥❣✿ ▲❡♠♠❛ ✶✳✷✳✷✳✶✳ ▲❡t (h i ) ❜❡ ❛♥② ❖◆❇ ♦❢ H✳ ❚❤❡♥✿ σ(W t , 0 ≤ t ≤ 1) = σ 1 0 ḣi (s)dW s , i ∈ N ✭✶✳✻✻✮ Pr♦♦❢✳ ❇② ❡•❡♠❡♥t❛r② st♦❝❤❛st✐❝ ❝❛•❝✉•✉s✱ ❛♥② st♦❝❤❛st✐❝ ✐♥t❡❣r❛• 1 0 ḣi (s)dW s ✐s ♠❡❛s✉r❛❜•❡ ✇✐t❤ r❡s♣❝t t♦ t❤❡ σ✲✜❡•❞ σ(W t , 0 ≤ t ≤ 1)✱ t❤❡r❡❢♦r❡ ✇❡ ❤❛✈❡ t❤❡ ✐♥❝•✉s✐♦♥✿ σ(W t , 0 ≤ t ≤ 1) ⊃ σ 1 0 ḣi (s)dW s ✭✶✳✻✼✮ ❈♦♥✈❡rs❡•② s✐♥❝❡ (h i ) ✐s ❛♥ ❖◆❇ ♦❢ H✱ ( ḣi ) ✐s ❛♥ ❖◆❇ ♦❢ L 2 (λ)✱ ❤❡♥❝❡ t❤❡ ✐♥❞✐❝❛t♦r ♦❢ ❛♥ ✐♥t❡r✈❛• ❤❛s ❛ ❞❡❝♦♠♣♦s✐t✐♦♥✿ 1 [0,t] = ∞ i=1 α i ḣi ✭✶✳✻✽✮ ✶✳✷✳ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ✶✸ ❚❤❡r❡❢♦r❡✱ ♦♥❡ s❡❡s t❤❛t✱ ✐♥ t❤❡ L 2 s❡♥s❡✿ W t = lim n→∞ n i=1 α i • 1 0 ḣi (s)dW s ✭✶✳✻✾✮ ❢r♦♠ ✇❤✐❝❤ t❤❡ ❝♦♥✈❡rs❡ ✐♥❝•✉s✐♦♥ ❛r✐s❡s✳ ❚❤❡ ♠❛✐♥ ✐♥t❡r❡st ♦❢ t❤✐s r❡s✉•t ✐s t❤❛t ✐t s❤♦✇s t❤❛t t❤❡ ✐♥❢♦r♠❛t✐♦♥ ✐♥ t❤❡ ❝❛♥♦♥✐❝❛• ❇r♦✇♥✐❛♥ ✜•tr❛t✐♦♥ ✐s ❡♥❝♦♠♣❛ss❡❞ ✐♥ ❛ ❝♦✉♥t❛❜•❡ ✜•tr❛t✐♦♥✳ ❚❤✐s ✇✐•• •❡t ✉s ❞❡✜♥❡ ♠♦r❡ ❤✐•❜❡rt✐❛♥ str✉❝t✉r❡s✳ ■♥ ♣❛rt✐❝✉•❛r✱ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡✿ ❉❡✜♥✐t✐♦♥ ✶✳✷✳✷✳✶ ✭P♦•②♥♦♠✐❛•s ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡✮✳ ❙✉♣♣♦s❡ t❤❛t ❛♥ ❖◆❇ (h i ) ♦❢ H ❤❛s ❜❡❡♥ ✜①❡❞✳ ❲❡ ❝❛•• ♣♦•②♥♦♠✐❛• ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡ ❛♥② r❛♥❞♦♠ ✈❛r✐❛❜•❡ ♦❢ t❤❡ t②♣❡✿ P 1 0 ḣ1 (s)dW s , . . . , 1 0 ḣn (s)dW s ✭✶✳✼✵✮ ✇❤❡r❡ n ✐s ❛♥ ✐♥t❡❣❡r ❛♥❞ P ✐s ❛ ♣♦•②♥♦♠✐❛• ❢✉♥❝t✐♦♥ ♦♥ R n ✭r❡s♣ C n ✮✳ ❲❡ ✇✐•• ♥♦t❡ P t❤❡ s♣❛❝❡ ♦❢ ❛•• ♣♦•②♥♦♠✐❛• r❛♥❞♦♠ ✈❛r✐❛❜•❡s ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡✳ ■t ✐s ♥♦t❡✇♦rt❤② t❤❛t s✐♥❝❡ t❤❡ ḣi ❛r❡ ♦rt❤♦❣♦♥❛• t♦ ❡❛❝❤ ♦t❤❡r ✐♥ L 2 (λ) ❛♥❞ ❤❛✈❡ ♥♦r♠ 1✱ t❤❡r❡ ✐s t❤❡ ✐❞❡♥t✐t② ✐♥ •❛✇✿ 1 0 ḣ1 (s)dW s , . . . , 1 0 ḣn (s)dW s = N (0, Id n ) ✭✶✳✼✶✮ ❛♥❞ ✐♥ ♣❛rt✐❝✉•❛r ♣♦•②♥♦♠✐❛• r❛♥❞♦♠ ✈❛r✐❛❜•❡s ❤❛✈❡ ♠♦♠❡♥ts ♦❢ ❛•• ♦r❞❡rs✳ P♦•②✲ ♥♦♠✐❛• ❢✉♥❝t✐♦♥s ❛r❡ t❤❡r❡❢♦r❡ ❡❛s② t♦ ✇♦r❦ ✇✐t❤✱ ❛♥❞ ♦✉r str❛t❡❣② ✐♥ ♠❛♥② ❝❛s❡s ✇✐•• ❜❡ t♦ ❜✉✐•❞ ❛♥ ♦❜•❡❝t ♦r ♣r♦✈❡ ❛ r❡s✉•t ✜rst ❢♦r t❤❡ ♣♦•②♥♦♠✐❛• ❢✉♥❝t✐♦♥s✱ ❛♥❞ t❤❡♥ t♦ ❡①t❡♥❞ ✐t t♦ ♠♦r❡ ❣❡♥❡r❛• r❛♥❞♦♠ ✈❛r✐❛❜•❡s ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡✳ ❚❤✐s ✇✐•• ❜❡ ♣♦ss✐❜•❡ ❜❡❝❛✉s❡ ♦❢ t❤❡✿ ❚❤❡♦r❡♠ ✶✳✷✳✷✳✶ ✭❞❡♥s✐t② ♦❢ t❤❡ ♣♦•②♥♦♠✐❛•s ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡✮✳ ▲❡t X ✐♥ L p (W, µ) ❛♥❞ ǫ > 0✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛♥ ✐♥t❡❣❡r n ❛♥❞ ❛ ♣♦•②♥♦♠✐❛• ❢✉♥❝t✐♦♥ ✐♥ n ✈❛r✐❛❜•❡s s✉❝❤ t❤❛t✿ X -P 1 0 ḣ1 (s)dW s , . . . , 1 0 ḣn (s)dW s L p (W,µ) < ǫ ✭✶✳✼✷✮ Pr♦♦❢✳ ▲❡t ✉s ❞❡✜♥❡ ❛ ❞✐s❝r❡t❡ ♠❛rt✐♥❣❛•❡ ❛s✿ M n = E X| 1 0 ḣi (s)dW s , i ≤ n ✭✶✳✼✸✮ ✶✹ ❈❍❆P❚❊| ✶✳ ■◆❚|❖❉❯❈❚■❖◆ ❚❖ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ❚❤❡♥✱ ❜② •❡♠♠❛ ✶✳✷✳✷✳✶ M n ❝♦♥✈❡r❣❡s ✐♥ L p t♦ X✱ ✐❡ ❢♦r ❜✐❣ ❡♥♦✉❣❤ n✿ M n -X L p < ǫ ✭✶✳✼✹✮ ❆•s♦✱ t❤❡r❡ ❡①✐sts ❛ ❢✉♥❝t✐♦♥ f n s✉❝❤ t❤❛t✿ M n = f n 1 0 ḣ1 (s)dW s , . . . , 1 0 ḣn (s)dW s ✭✶✳✼✺✮ ◆♦✇ s✐♥❝❡ M n ∈ L p (W, µ) ❛♥❞ 1 0 ḣ1 (s)dW s , . . . , 1 0 ḣn (s)dW s ✐s N (0, Id n ) ✉♥❞❡r µ✱ ♥❡❝❡ss❛r✐•② f n ∈ L p (µ n ) ❛♥❞✿ M n L p (W,µ) = f n L p (µn) ✭✶✳✼✻✮ ❋✐♥❛••② ❜② t❤❡♦r❡♠ ✶✳✶✳✶✳✶ t❤❡r❡ ❡①✐sts ❛ ♣♦•②♥♦♠✐❛• P n s✉❝❤ t❤❛t✿ f n -P n L p (µn) < ǫ ✭✶✳✼✼✮ ❛♥❞ ♦❢ ❝♦✉rs❡✿ P n 1 0 ḣ1 (s)dW s , . . . , 1 0 ḣn (s)dW s L p (W,µ) = P n L p (µn) ✭✶✳✼✽✮ s♦ ✇❡ ❛r❡ ❞♦♥❡✳ ❆s ✇❡ ❝♦♥s✐❞❡r ♣♦•②♥♦♠✐❛• r❛♥❞♦♠ ✈❛r✐❛❜•❡s ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡✱ s✐♥❝❡ t❤❡s❡ ❤❛✈❡ t❤❡ s❛♠❡ •❛✇ ❛s t❤❡ ✉♥❞❡r•②✐♥❣ ♣♦•②♥♦♠✐❛• ❡✈❛•✉❛t❡❞ ❛t ❛ ♥♦r♠❛• ❣❛✉ss✐❛♥ r❛♥❞♦♠ ✈❛r✐❛❜•❡✱ ✐t ✐s ♥❛t✉r❛• t♦ ❞❡❝♦♠♣♦s❡ t❤❡ ✉♥❞❡r•②✐♥❣ ♣♦•②♥♦♠✐❛• ✐♥ t❤❡ ❍❡r♠✐t❡ ❖◆❇ ❛♥❞ t♦ st✉❞② t❤❡ ❤✐•❜❡rt✐❛♥ ♣r♦♣❡rt✐❡s ✇❤✐❝❤ t❤✉s ❛r✐s❡✳ ■♥❞❡❡❞✱ str❛✐❣❤t❢♦r✇❛r❞ ❝♦♠♣✉t❛t✐♦♥s s❤♦✇ t❤❛t✿
Pr♦♣♦s✐t✐♦♥ ✶✳✷✳✷✳✶✳ ■❢ p, q = (p 1 , . . . , p n 1 ), (q 1 , . . . , q n 1 ) ❛r❡ t✇♦ ❞✐✛❡r❡♥t ♠✉•t✐✲

✐♥❞✐❝❡s✿ E H p 1 0 ḣ p1 (s)dW s , . . . , 1 0 ḣ np (s)dW s H q 1 0 ḣ q1 (s)dW s , . . . , 1 0 ḣ nq (s)dW s = 0 ✭✶✳✼✾✮ ❛♥❞ E H p 1 0 ḣ p 1 (s)dW s , . . . , 1 0 ḣ np (s)dW s 2 = p! ✭✶✳✽✵✮ ❇② ❝♦♠❜✐♥✐♥❣ t❤❡♦r❡♠ ✶✳✷✳✷✳✶ ❛♥❞ ♣r♦♣♦s✐t✐♦♥ ✶✳✷✳✷✳✶ ✱ ✇❡ ✐♠♠❡❞✐❛t❡•② ♦❜t❛✐♥ t❤❡✿ ✶✳✷✳ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ✶✺ ❚❤❡♦r❡♠ ✶✳✷✳✷✳✷ ✭❲✐❡♥❡r ❝❤❛♦s ❞❡❝♦♠♣♦s✐t✐♦♥ ✲ ✜rst st❛t❡♠❡♥t✮✳ ❚❤❡ ❝♦✉♥t❛❜•❡ ❢❛♠✐•② ♦❢ t❤❡ 1 p! • H p 1 0 ḣ p 1 (s)dW s , . . . , 1 0 ḣ np (s)dW s ✱ ✇❤❡r❡ p ❞❡s❝r✐❜❡s ❛•• ♠✉•t✐✲ ✐♥❞✐❝❡s✱ t❤❡ H p ❛r❡ t❤❡ ❍❡r♠✐t❡ ♣♦•②♥♦♠✐❛•s ❛♥❞ (h i ) ✐s ❛♥② ❖◆❇ ♦❢ H ❝♦♥st✐t✉t❡s ❛♥ ❖◆❇ ♦❢ L 2 (W, µ)✳ ❚❤❡ t❡r♠✐♥♦•♦❣② ✧❲✐❡♥❡r ❝❤❛♦s ❞❡❝♦♠♣♦s✐t✐♦♥✧ ✐s •✉st✐✜❡❞ ❜② t❤❡ ❢♦••♦✇✐♥❣✿ ❉❡✜♥✐t✐♦♥ ✶✳✷✳✷✳✷ ✭❲✐❡♥❡r ❝❤❛♦s✮✳ ❚❤❡ ♥t❤ ♦r❞❡r ❲✐❡♥❡r ❝❤❛♦s✱ ✇❤✐❝❤ ✇❡ ♥♦t❡ C n ✱ ✐s t❤❡ •✐♥❡❛r s♣❛♥ ♦❢ t❤❡ H p 1 0 ḣ p 1 (s)dW s , . . . , 1 0 ḣ np (s)dW s ✱ ✇❤❡r❡ t❤❡ H p ❛r❡ t❤❡ ❍❡r♠✐t❡ ♣♦•②♥♦♠✐❛•s ✇✐t❤ t♦t❛• ❞❡❣r❡❡ n✳ ❲❡ ✇✐•• ❝❤❛r❛❝t❡r✐③❡ t❤❡ ❲✐❡♥❡r ❝❤❛♦s❡s t❤❛♥❦s t♦ t❤❡ ❢♦••♦✇✐♥❣ r❡s✉•t✿ Pr♦♣♦s✐t✐♦♥ ✶✳✷✳✷✳✷✳ ❋♦r ❛♥② n ∈ N ❛♥❞ h ∈ H s✉❝❤ t❤❛t |h| H = 1✿ H n 1 0 ḣ(s)dW s = • • • 0<s 1 <•••<sn<1 ḣ(s 1 ) • • • ḣ(s n )dW s 1 • • • dW sn ✭✶✳✽✶✮ Pr♦♦❢✳ ❚❤❡ st❛t❡♠❡♥t ✐s tr✐✈✐❛• ❢♦r n = 0, 1✳ ■t ✐s t❤❡♥ ♣r♦✈❡❞ ❜② ❛ s✐♠♣•❡ r❡❝✉rs✐♦♥ ❛s t❤❡ ❍❡r♠✐t❡ ♣♦•②♥♦♠✐❛•s ❛♥❞ t❤❡ ♠✉•t✐♣•❡ ✐♥t❡❣r❛•s ✈❡r✐❢② t❤❡ s❛♠❡ r❡❝✉rs✐♦♥ r❡•❛t✐♦♥s❤✐♣✳ ❋r♦♠ t❤✐s ✇❡ ❞❡❞✉❝❡✿ ❚❤❡♦r❡♠ ✶✳✷✳✷✳✸ ✭❈❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ ♥t❤ ❲✐❡♥❡r ❝❤❛♦s✮✳ C n ✐s t❤❡ •✐♥❡❛r s♣❛♥ ♦❢ t❤❡ ♦r❞❡r n ♠✉•t✐♣•❡ ❲✐❡♥❡r ✐♥t❡❣r❛•s ❞❡s❝r✐❜❡❞ ✐♥ ♣r♦♣♦s✐t✐♦♥ ✶✳✷✳✷✳✷✱ ✇❤❡r❡ h ❞❡s❝r✐❜❡s ❛♥② ❣✐✈❡♥ ❖◆❇ ♦❢ H✳ C n ❛•s♦ ✐s t❤❡ ❡✐❣❡♥s♣❛❝❡ ♦❢ t❤❡ ♦♣❡r❛t♦rs P t ❛♥❞ L✱ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❡✐❣❡♥✈❛•✉❡s e -nt ❛♥❞ n r❡s♣❡❝t✐✈❡•②✳ ■♥ ♣❛rt✐❝✉•❛r✱ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❲✐❡♥❡r ❝❤❛♦s❡s ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ❝❤♦✐❝❡ ♦❢ ♦♥❡ s♣❡❝✐✜❝ ❖◆❇ ♦❢ H✳ ◆♦✇ t❤❡ ❲✐❡♥❡r ❝❤❛♦s ❞❡❝♦♠♣♦s✐t✐♦♥ t❤❡♦r❡♠ ♠❛② ❜❡ r❡♣❤r❛s❡❞✿ ❚❤❡♦r❡♠ ✶✳✷✳✷✳✹ ✭❲✐❡♥❡r ❝❤❛♦s ❞❡❝♦♠♣♦s✐t✐♦♥ ✲ s❡❝♦♥❞ st❛t❡♠❡♥t✮✳ L 2 (W, µ) = ⊥ C n ✭✶✳✽✷✮ ✶✳✷✳✸ •r❛❞✐❡♥t ❛♥❞ ❞✐✈❡r❣❡♥❝❡ ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡ ❲❡ ✇❛♥t t♦ ❞❡✜♥❡ ❛ ❢✉♥❝t✐♦♥❛• ❞❡r✐✈❛t✐✈❡ ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡✳ ❚❤❡ ♦♥•② ✇❛② t♦ ❞♦ t❤✐s ✐s t❤r♦✉❣❤ ♣❛rt✐❛•✱ ✐❡ ❞✐r❡❝t✐♦♥❛• ❞❡r✐✈❛t✐✈❡s✱ ❛♥❞ ✇❡ ✇✐•• ✇❛♥t t♦ ❝♦♥s✐❞❡r✱ ✇❤❡♥ t❤❡② ❡①✐st✱ L p ✲•✐♠✐ts ♦❢ t❤❡ t②♣❡✿ ∇ wX (w) = lim α→0 X(w + α w) -X(w) α ✭✶✳✽✸✮ ✶✻ ❈❍❆P❚❊| ✶✳ ■◆❚|❖❉❯❈❚■❖◆ ❚❖ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ❖❢ ❝♦✉rs❡✱ ❢♦r t❤❡ t❤❡♦r② t♦ ♠❛❦❡ s❡♥s❡✱ ✐t ✐s ♥❡❝❡ss❛r② t❤❛t✱ ✐❢ X = Y ❛•♠♦st s✉r❡•② ❛♥❞ X ✐s ❞✐✛❡r❡♥t✐❛❜•❡✱ t❤❡♥ Y ✐s ❛•s♦ ❞✐✛❡r❡♥t✐❛❜•❡ ❛♥❞ ∇X = ∇Y ❛✳s✳ ✐♥ ❡✈❡r② ❞✐r❡❝t✐♦♥✳ ❚❤❡ ❝❛s❡s ✇❤❡r❡ t❤✐s ✐s ♣♦ss✐❜•❡ ❛r❡ ❡①♣•❛✐♥❡❞ ✐♥ t❤❡ ❢♦••♦✇✐♥❣ ♣❛r❛❣r❛♣❤✿ ❚❤❡ •✐rs❛♥♦✈ ❛♥❞ ❈❛♠❡r♦♥✲▼❛rt✐♥ t❤❡♦r❡♠s ❋✐rst ✇❡ r❡❝❛•• t❤❡ ❝•❛ss✐❝✿ ❚❤❡♦r❡♠ ✶✳✷✳✸✳✶ ✭•✐rs❛♥♦✈✮✳ ❆ss✉♠❡ t❤❛t u ✐s ❛ ♠❡❛s✉r❛❜•❡ ♣r♦❝❡ss ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡ ❛❞❛♣t❡❞ t♦ t❤❡ ❝❛♥♦♥✐❝❛• ❇r♦✇♥✐❛♥ ✜•tr❛t✐♦♥✱ s✉❝❤ t❤❛t µ✲❛✳s✳ 1 0 u 2 s ds < ∞ ✭✶✳✽✹✮ ❛♥❞ •❡t Λ t = exp - t 0 u s dW s - 1 2 t 0 u 2 s ds ✭✶✳✽✺✮ ❆ss✉♠❡ t❤❛t E[Λ 1 ] = 1✳ ❚❤❡♥✱ t❤❡ ♣r♦❝❡ss (W t + t 0 u s ds, 0 ≤ t ≤ 1) ✐s ❛ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ✉♥❞❡r t❤❡ ♣r♦❜❛❜✐•✐t② Λ 1 • µ✳ Pr♦♦❢✳ ❖♠✐tt❡❞✱ s❡❡ ❬✻✵(✳ |❡♠❛r❦ ✶✳✷✳✸✳✶✳ ❚✇♦ ❛r❝❤❡t②♣❛• ❝❛s❡s ✇❤❡r❡ E[Λ 1 ] = 1 ❛r❡✱ ✜rst✱ ✇❤❡♥ t❤❡ ◆♦✈✐❦♦✈ ❝♦♥❞✐t✐♦♥ ✐s ✈❡r✐✜❡❞✿ E exp 1 2 1 0 u 2 s ds < ∞ ✭✶✳✽✻✮ ❛♥❞✱ s❡❝♦♥❞✱ ✇❤❡♥ t❤❡ ❑❛③❛♠❛❦✐ ❝♦♥❞✐t✐♦♥ ✐s ✈❡r✐✜❡❞✿ E exp 1 2 1 0 u s dW s < ∞ ✭✶✳✽✼✮ ❯s✐♥❣ ❡✐t❤❡r ♦♥❡ ♦❢ t❤❡ t✇♦ ❝r✐t❡r✐❛ ❛❜♦✈❡✱ ✐t ✐s ✐♠♠❡❞✐❛t❡ t♦ ♦❜t❛✐♥ t❤❡ ❢♦••♦✇✐♥❣ ❝♦r♦••❛r② ♦❢ t❤❡ •✐rs❛♥♦✈ t❤❡♦r❡♠✿ ❚❤❡♦r❡♠ ✶✳✷✳✸✳✷ ✭❈❛♠❡r♦♥✱ ▼❛rt✐♥✮✳ ▲❡t h ∈ H✳ ❚❤❡♥ W + h ✐s ❛ ❇r♦✇♥✲ ✐❛♥ ♠♦t✐♦♥ ✉♥❞❡r t❤❡ ♣r♦❜❛❜✐•✐t② ♠❡❛s✉r❡ ✇✐t❤ ❞❡♥s✐t② ✇✐t❤ r❡❣❛r❞ t♦ t❤❡ ❲✐❡♥❡r ♠❡❛s✉r❡ ❣✐✈❡♥ ❜②✿ exp - 1 0 ḣs dW s - 1 2 1 0 h 2 s ds ✭✶✳✽✽✮ ❊q✉✐✈❛•❡♥t•②✱ t❤✐s ♠❡❛♥s t❤❛t ❢♦r ❛♥② F ∈ L p (W, µ)✱ p > 1✱ ❛♥❞ ❛♥② h ∈ H✱ ♦♥❡ ❤❛s✿ E µ F (w + h) exp -ḣs dW s - 1 2 1 0 ḣ2 s ds = E µ [F ] ✭✶✳✽✾✮ ✶✳✷✳ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ✶✼ ❚❤❡ ❈❛♠❡r♦♥✲▼❛rt✐♥ t❤❡♦r❡♠ ✇❛s ❛❝t✉❛••② ♣r♦✈❡❞ ♣r✐♦r t♦ t❤❡ •✐rs❛♥♦✈ t❤❡✲ ♦r❡♠✳ ■t ❛❞♠✐ts ✈❛r✐♦✉s ✐♥t✉✐t✐✈❡ ♣r♦♦❢s✱ ♦♥❡ ♦❢ ✇❤✐❝❤ ✐s ❜❛s❡❞ ♦♥ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥s ❛♥❞ s✉✐ts ♦✉r s❡tt✐♥❣ ✇❡••✿ Pr♦♦❢✳ ❋✐rst r❡❝❛•• ❡q✉❛t✐♦♥ ✭✶✳✻✷✮✿ ❢♦r ❛♥② ν ∈ W * ✿ W * ν, w W = 1 0 ν(]t, 1])dW t (w) ✭✶✳✾✵✮ ❚❤❡r❡❢♦r❡ t❤❡ ❝❤❛r❛❝t❡r✐s✐t❝ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❲✐❡♥❡r ♠❡❛s✉r❡ ✐s✿ E[e i ν,w ] = E[e i 1 0 ν(]t,1])dWt ] = e -1 2 1 0 ν(]t,1]) 2 dt ✭✶✳✾✶✮ ❛♥❞ s✐♠✐•❛r•②✿ E[exp (i ν, w (w + h)) • Λ 1 ] = E exp i 1 0 ν(]t, 1])dW t + i 1 0 ν(]t, 1]) ḣ(t)dt - 1 0 ḣ(t)dW t - 1 2 1 0 ḣ(t) 2 dt = E exp 1 0 iν(]t, 1]) -ḣ(t) dW t exp i 1 0 ν(]t, 1]) ḣ(t)dt - 1 2 • 1 0 ḣ(t) 2 dt = exp 1 0 iν(]t, 1]) -ḣ(t) 2 dt exp i 1 0 ν(]t, 1]) ḣ(t)dt - 1 2 • 1 0 ḣ(t) 2 dt = exp - 1 2 • 1 0 ν(]t, 1]) 2 dt = E[exp (i ν, w )] ✭✶✳✾✷✮ s♦ ✇❡ ❛r❡ ❞♦♥❡✳ ❇② t❤❡ ❈❛♠❡r♦♥✲▼❛rt✐♥ t❤❡♦r❡♠✱ ✐t ✐s ❝•❡❛r t❤❛t ✐❢ X = Y ❛✳s✳ t❤❡♥ ❢♦r ❛♥② h ∈ H ♦♥❡ ❛•s♦ ❤❛s X(• + h) = Y (• + h) ❛✳s✳ ❚❤❡r❡❢♦r❡ ✐t ✐s r❡❛s♦♥❛❜•❡ t♦ tr② ❛♥❞ ❝♦♥s✐❞❡r ❞❡r✐✈❛t✐✈❡s ✐♥ t❤✐s ❞✐r❡❝t✐♦♥✳ ❆•s♦✱ t❤❡ ❢♦••♦✇✐♥❣ r❡s✉•t ✇❤✐❝❤ ✇❡ ❝♦♣② ❢r♦♠ ❬✹✶( ✐s ❛ ❝♦♥✈❡rs❡ t♦ t❤❡ ❈❛♠❡r♦♥✲▼❛rt✐♥ t❤❡♦r❡♠ ✇❤✐❝❤ •❡❛❞s ✉s ♥♦t t♦ ❝♦♥s✐❞❡r ♦t❤❡r ❞✐r❡❝t✐♦♥s t❤❛♥ t❤❡ s♣❛❝❡ H ❢♦r ♦✉r ❞❡r✐✈❛t✐✈❡s✿ ❚❤❡♦r❡♠ ✶✳✷✳✸✳✸✳ ▲❡t w ∈ W -H✳ ❚❤❡♥✿ µ(• + w) ✐s ♥♦t ❛❜s♦•✉t❡•② ❝♦♥t✐♥✉♦✉s ✇✐t❤ r❡❣❛r❞s t♦ µ✳ Pr♦♦❢✳ ❖♠✐tt❡❞ ❛s ✐t ✐s ❜❛s❡❞ ♦♥ ❡•❛❜♦r❛t❡ ❋♦✉r✐❡r ❛♥❛•②s✐s ❝♦♥s✐❞❡r❛t✐♦♥s ❛♥❞ ✇❡ ✇✐•• ♥♦t ♠❛❦❡ ❞✐r❡❝t ✉s❡ ♦❢ t❤❡ r❡s✉•ts❀ s❡❡ ❬✹✶( ❢♦r t❤❡ ❞❡t❛✐•s✳ ✶✽ ❈❍❆P❚❊| ✶✳ ■◆❚|❖❉❯❈❚■❖◆ ❚❖ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ❈♦♥str✉❝t✐♦♥ ♦❢ t❤❡ •r♦ss✲❙♦❜♦•❡✈ ❞❡r✐✈❛t✐✈❡ ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡ ▲❡t 1 < p < ∞✳ ❉❡✜♥✐t✐♦♥ ✶✳✷✳✸✳✶ ✭❞✐✛❡r❡♥t✐❛❜✐•✐t② ✐♥ ❛ ❞✐r❡❝t✐♦♥✮✳ ❲❡ ✇✐•• s❛② t❤❛t X ∈ L p ✐s ❞✐✛❡r❡♥t✐❛❜•❡ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ h ∈ H ✇❤❡♥ t❤❡ ❢♦••♦✇✐♥❣ •✐♠✐t ❡①✐sts ✐♥ t❤❡ L p s❡♥s❡✿ ∇ h X(w) = lim α→0 X(w + αh) -X(w) α ✭✶✳✾✸✮ ❙✉♣♣♦s❡ t❤❛t X ∈ L p ✐s ❞✐✛❡r❡♥t✐❛❜•❡ ✐♥ ❡✈❡r② ❞✐r❡❝t✐♦♥ h ∈ H✳ ❚❤❡♥ ✐t ✐s ❝•❡❛r t❤❛t ❛✳s✳ h ∈ H → ∇ h X ∈ R ✐s •✐♥❡❛r ❛♥❞ ✇❡ ✇♦✉•❞ •✐❦❡ t♦ ❝♦♥s✐❞❡r t❤❡ ✧✇❤♦•❡✧ ❣r❛❞✐❡♥t ∇X ♦❢ X ❛s ❛ r❛♥❞♦♠ ✈❛r✐❛❜•❡ ✇✐t❤ ✈❛•✉❡s ✐♥ L(H, R) ≈ H✳ ❋♦r t❤✐s t♦ ❜❡ ❛ tr❛❝t❛❜•❡ ❢r❛♠❡✇♦r❦✱ ✇❡ ♥❡❡❞ ❛♥ L p str✉❝t✉r❡ ❢♦r H✲✈❛•✉❡❞ r❛♥❞♦♠ ✈❛r✐❛❜•❡s✳ ▼♦r❡ ❣❡♥❡r❛••②✱ ✐❢ Ξ ✐s s♦♠❡ s❡♣❛r❛❜•❡ ❍✐•❜❡rt s♣❛❝❡✱ ♦♥❡ ♠❛② ❞❡✜♥❡✿ X L p (W,µ,Ξ) = |X| Ξ L p (µ) = sup |x| Ξ =1 (x, X) Ξ L p (µ) ✭✶✳✾✹✮ ❚❤✐s •❡❛❞s ✉s t♦ t❤❡ ❢♦••♦✇✐♥❣ ❞❡✜♥✐t✐♦♥ ♦❢ ❞✐✛❡r❡♥t✐❛❜✐•✐t②✱ ✇❤✐❝❤✱ ❜❡❝❛✉s❡ ✇❡ ✇♦r❦ ✐♥ ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥s✱ ❞❡♠❛♥❞s s•✐❣❤t•② ♠♦r❡ t❤❛♥ •✉st ❤❛✈✐♥❣ ❛ ♣❛rt✐❛• ❞❡r✐✈❛t✐✈❡ ✐♥ ❡❛❝❤ ❞✐r❡❝t✐♦♥✿ ❉❡✜♥✐t✐♦♥ ✶✳✷✳✸✳✷ ✭❞✐✛❡r❡♥t✐❛❜•❡ r❛♥❞♦♠ ✈❛r✐❛❜•❡s✮✳ ❲❡ s❛② t❤❛t X ∈ L p ✐s ❞✐✛❡r❡♥t✐❛❜•❡ ❛♥❞ ✇❡ ✇✐•• ♥♦t❡ X ∈ D p,1 ✱ ✐❢ X ✐s L p ✲❞✐✛❡r❡♥t✐❛❜•❡ ✐♥ ❡✈❡r② ❞✐r❡❝t✐♦♥ h ∈ H ❛♥❞ ✐❢ ∇X ∈ L p (W, µ, H)✱ ✐❡✿ sup |g| H =1 ∇ g X L p < ∞ ✭✶✳✾✺✮ |❡♠❛r❦ ✶✳✷✳✸✳✷✳ ■♥ ♦♣❡r❛t♦r t❤❡♦r② •❛♥❣✉❛❣❡ D p,1 ✐s t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ♦♣❡r❛t♦r ∇✳ ▲❡t ✉s ♥♦✇ ❣✐✈❡ ❛ ❢❡✇ ❛r❝❤❡t②♣❛• ❡①❛♠♣•❡s ♦❢ ❞✐✛❡r❡♥t✐❛❜•❡ r❛♥❞♦♠ ✈❛r✐❛❜•❡s t♦❣❡t❤❡r ✇✐t❤ t❤❡✐r ❞❡r✐✈❛t✐✈❡s✿ Pr♦♣♦s✐t✐♦♥ ✶✳✷✳✸✳✶✳ ▲❡t g, h ∈ H✳ ❚❤❡♥ ḣdW ∈ D p,1 ❛♥❞ ∇ g ḣdW = (g, h) H ✭✶✳✾✻✮ Pr♦♦❢✳ ḣdW ∈ L p ❜❡❝❛✉s❡ ✐t ✐s ❣❛✉ss✐❛♥✳ ❚❤❛t ✐t ❤❛s ❛ ❞❡r✐✈❛t✐✈❡ ✐♥ ❡✈❡r② ❞✐r❡❝t✐♦♥✱ ❛♥❞ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡s❡✱ ❛r❡ str❛✐❣❤t❢♦r✇❛r❞ ❝♦♥s❡q✉❡♥❝❡s ♦❢ t❤❡ •✐♥❡❛r✐t②✱ s❡❡ ♣r♦♣♦s✐t✐♦♥ ✶✳✷✳✶✳✷✳ ❚❤❡♥ ❜② t❤❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ ✐♥❡q✉❛•✐t② ✐t ✐s ❝•❡❛r t❤❛t✱ µ✲❛✳s✿ sup |g| H =1 ∇ g ḣdW = sup |g| H =1 (g, h) H ≤ |h| H ✭✶✳✾✼✮ s♦ ∇ ḣdW ✐s ✇❡•• ❞❡✜♥❡❞ ✐♥ L p (W, µ, H) ❛♥❞ ✇❡ ❛r❡ ❞♦♥❡✳ ✶✳✷✳ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ✶✾ ▲❡t ✉s ❛•s♦ ♥♦t❡ t❤❛t✱ ❛s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ♣r❡✈✐♦✉s ♣r♦♣♦s✐t✐♦♥✱ W t ∈ D p,1 ❛♥❞✿ ∇ g W t = ∇ g 1 [0,t] dW = 1 [0,t] ġ = g(t) ✭✶✳✾✽✮ ❆♥♦t❤❡r ❝r✉❝✐❛• ❡①❛♠♣•❡ ✐s t❤❡ ❢♦••♦✇✐♥❣✿ Pr♦♣♦s✐t✐♦♥ ✶✳✷✳✸✳✷✳ ▲❡t g, h ∈ H ❛♥❞ •❡t P ∈ C[X]✳ ❚❤❡♥ P ( ḣdW ) ∈ D p,1 ✱ ❛♥❞✿ ∇ g P ḣdW = P ′ ḣdW • (g, h) H ✭✶✳✾✾✮ ♦r ❡q✉✐✈❛•❡♥t•②✱ ❜② t❤❡ |✐❡s③ t❤❡♦r❡♠✿ ∇ ḣdW = h ✭✶✳✶✵✵✮ Pr♦♦❢✳ ❙✐♥❝❡ ḣdW ✐s ❣❛✉ss✐❛♥ ✐t ❤❛s ♠♦♠❡♥ts ♦❢ ❛•• ♦r❞❡rs ❛♥❞ P ( ḣdW ) ∈ L p ✳ ◆♦✇✱ ✉s✐♥❣ ♣r♦♣♦s✐t✐♦♥ ✶✳✷✳✶✳✷ ❛♥❞ ❛ ❚❛②•♦r ❡①♣❛♥s✐♦♥✱ ✇❡ ♦❜t❛✐♥✱ µ✲❛✳s✿ P ḣdW (w + αg) = P ḣdW (w) + α(g, h) H = degP n=0 P (n) (( ḣdW )(w)) n! α n (h, g) n H ✭✶✳✶✵✶✮ ❙✐♥❝❡ t❤❡ P (n) (( ḣdW )(w)) ❛r❡ ✐♥ L p ❛s ✇❡•• ✐t ✐s ❝•❡❛r t❤❛t t❤❡ ◆❡✇t♦♥ r❛t✐♦ ❝♦♥✈❡r❣❡s ✐♥ L p ❛s α t❡♥❞s t♦ 0 ❛♥❞ ✇❡ ♦❜t❛✐♥ t❤❡ ❡①✐st❡♥❝❡ ✐❢ ❞❡r✐✈❛t✐✈❡s ✐♥ ❡✈❡r② ❞✐r❡❝t✐♦♥✳ ◆♦✇✱ s✐♠✐•❛r•② t♦ ✇❤❛t ✇❡ ❤❛✈❡ ❞♦♥❡ ✐♥ t❤❡ ♣r❡✈✐♦✉s ♣r♦♦❢✱ ❛✳s✿ sup |g| H =1 ∇ g P ḣdW ≤ P ′ ḣdW • |h| H ✭✶✳✶✵✷✮ s♦ ✇❡ ♦❜t❛✐♥ ∇P ḣdW ∈ L p (W, µ, H) ❛♥❞ ∇P ḣdW L p ≤ P ′ ḣdW • |h| H ✭✶✳✶✵✸✮ ■t ✐s str❛✐❣❤t❢♦r✇❛r❞ t♦ ❣❡♥❡r❛•✐③❡ t❤✐s t♦ ♣♦•②♥♦♠✐❛•s ✐♥ s❡✈❡r❛• ✈❛r✐❛❜•❡s✱ ❛♥❞ t❤❡r❡❢♦r❡ ✇❡ ♦❜t❛✐♥ t❤❡ ❢♦••♦✇✐♥❣✿ ✷✵ ❈❍❆P❚❊| ✶✳ ■◆❚|❖❉❯❈❚■❖◆ ❚❖ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ Pr♦♣♦s✐t✐♦♥ ✶✳✷✳✸✳✸ ✭❞✐✛❡r❡♥t✐❛❜✐•✐t② ♦❢ t❤❡ ♣♦•②♥♦♠✐❛•s ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡✮✳ ❙✉♣♣♦s❡ t❤❛t ❛♥ ❖◆❇ (h i ) ♦❢ H ❤❛s ❜❡❡♥ ✜①❡❞✳ ❚❤❡♥✱ ❛♥② ♣♦•②♥♦♠✐❛• ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡ ✐s ✐♥ D p,1 ✱ ❛♥❞ ❢♦r ❛♥② g ∈ H✿ ∇ g P ḣ1 dW, • • • , ḣn dW = n i=1 ∂ i P ḣ1 dW, • • • , ḣn dW (g, h i ) H ✭✶✳✶✵✹✮ ♦r✱ ❡q✉✐✈❛•❡♥t•②✿ ∇P ḣ1 dW, • • • , ḣn dW = n i=1 ∂ i P ḣ1 dW, • • • , ḣn dW h i ✭✶✳✶✵✺✮ ❲❡ ♥♦✇ ♣r♦✈❡ t❤❡✿ ❚❤❡♦r❡♠ ✶✳✷✳✸✳✹✳ ❚❤❡ •r♦ss✲❙♦❜♦•❡✈ ❞❡r✐✈❛t✐✈❡ ✐s ❛ ❝•♦s❡❞ ♦♣❡r❛t♦r✱ ✐❡ ✐❢ (X n ) ✐s ❛ s❡q✉❡♥❝❡ ✐♥ D p,1 ✇❤✐❝❤ ❝♦♥✈❡r❣❡s t♦ 0 ✐♥ L p t❤❡♥ ✐❢ (∇X n ) ❝♦♥✈❡r❣❡s ✐♥ L p (W, µ, H) ✐ts •✐♠✐t ❤❛s t♦ ❜❡ 0✳ Pr♦♦❢✳ ❙✐♥❝❡ ✇❡ ❦♥♦✇ t❤❛t lim ∇X n ❡①✐sts ✐t ✐s ❡♥♦✉❣❤ t♦ ♣r♦✈❡ t❤❛t ✐t ✐s 0 ✐♥ ❡✈❡r② ❞✐r❡❝t✐♦♥✱ ✐❡ ❢♦r ❛♥② h ∈ H✱ ∇ h X n → 0✳ ❙✐♥❝❡ t❤❡ ♣♦•②♥♦♠✐❛• r❛♥❞♦♠ ✈❛r✐❛❜•❡s ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡ ❛r❡ ❞❡♥s❡ ✐♥ L p * ✱ ❜② ❞✉❛•✐t② ✐t ✐s ❡♥♦✉❣❤ t♦ ♣r♦✈❡ t❤❛t ❢♦r ❛♥② ♣♦•②♥♦♠✐❛• φ✿ E[φ∇ h X n ] → 0✳ ❚♦ ❞♦ t❤✐s✱ ✇❡ ♥♦t❡ t❤❛t✿ E[∇ h X n φ] = d dα |α=0 E[X n (• + αh) • φ] = d dα |α=0 E X n φ(• -αh) exp α 1 0 ḣ(s)dW s - α 2 2 1 0 ḣ2 (s)ds = E X n -∇ h φ + φ 1 0 ḣ(s)dW s ✭✶✳✶✵✻✮ s♦ ✇❡ ❛r❡ ❞♦♥❡ ❜② ❍ö•❞❡r✬s ✐♥❡q✉❛•✐t②✳ ❚❤❡♦r❡♠ ✶✳✷✳✸✳✺✳ D p,1 ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡ ✐❢ ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ♥♦r♠✿ X D p,1 = X L p + ∇X L p (W,µ,H) ✭✶✳✶✵✼✮ ▼♦r❡ ♣r❡❝✐s❡•②✱ D p,1 ✐s t❤❡ ❝♦♠♣•❡t✐♦♥ ♦❢ t❤❡ s❡t ♦❢ ♣♦•②♥♦♠✐❛•s ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡ ❢♦r t❤❛t ♥♦r♠✳ Pr♦♦❢✳ ❋♦r t❤❡ ✜rst ♣❛rt✱ ❛♥ ♦♣❡r❛t♦r ✐s ❝•♦s❡❞ ✐❢ ❛♥❞ ♦♥•② ✐❢ ✐ts ❞♦♠❛✐♥ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❢♦r ✐ts ❣r❛♣❤ ♥♦r♠ s♦ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠ ❣✐✈❡s t❤❡ ❝♦♥❝•✉s✐♦♥✳ ❆•t❡r♥❛t✐✈❡•② ✇❡ ♠❛② ♦❜t❛✐♥ t❤❡ r❡s✉•t ✐♥ ❛ ♠♦r❡ ✧❝♦♠♣✉t❛t✐♦♥❛•✧ ✇❛② ❜② ❝♦♥s✐❞❡r✐♥❣ ❛ ❈❛✉❝❤② s❡q✉❡♥❝❡ (X n )✱ ❛♥❞ ❜② ♥♦t✐♥❣ t❤❛t s✐♥❝❡ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡s ✶✳✷✳ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ✷✶ ❛r❡ ❇❛♥❛❝❤ s♣❛❝❡s✱ (X n ) ❤❛s t♦ ❝♦♥✈❡r❣❡ ✐♥ L p ✱ t♦✱ s❛②✱ X✱ ❛♥❞ (∇X n ) ❤❛s t♦ ❝♦♥✈❡r❣❡ ✐♥ L p (H)✱ t♦✱ s❛②✱ ξ✳ ◆♦✇ ✇❡ •✉st ♥❡❡❞ t♦ ♣r♦✈❡ t❤❛t X ∈ D p,1 ❛♥❞ t❤❛t ∇X = ξ✳ ❚♦ ❞♦ t❤✐s✱ •❡t h ∈ H ❛♥❞ (α k ) ❛ s❡q✉❡♥❝❡ ♦❢ r❡❛• ♥✉♠❜❡rs ❞❡❝r❡❛s✐♥❣ t♦ 0✳ ❚❤❡♥✱ s✐♥❝❡ L p ❝♦♥✈❡r❣❡♥❝❡ ✐♠♣•✐❡s t❤❡ ❛•♠♦st s✉r❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❛ s✉❜s❡q✉❡♥❝❡✱ ❜② ❛ ❞✐❛❣♦♥❛• ❡①tr❛❝t✐♦♥ ♣r♦❝❡❞✉r❡ t❤❡r❡ ❡①✐sts ❛ s✉❜s❡q✉❡♥❝❡ (X ψ(n) ) s✉❝❤ t❤❛t ❛✳s✳ ❢♦r ❡✈❡r② k✿ X ψ(n) (• + α k h) → X(• + α k h) ✭✶✳✶✵✽✮ ◆♦✇ ✇❡ s✐♠♣•② ✇r✐t❡✿ X ψ(m+n) (• + α k h) -X ψ(m+n) α k -(ξ, h) H ≤ X ψ(m+n) (• + α k h) -X ψ(m+n) α k - X ψ(n) (• + α k h) -X ψ(n) α k ≤ X ψ(n) (• + α k h) -X ψ(n) α k -∇ h X ψ(n) ≤ ∇ h X ψ(n) -(ξ, h) H ✭✶✳✶✵✾✮ ▲❡t ǫ > 0✳ ❚❤❡r❡ ❡①✐sts N s✉❝❤ t❤❛t ❢♦r ❛♥② n ≥ N t❤❡ t❤✐r❞ t❡r♠ ✐s ❝♦♥tr♦••❡❞ ❜② ǫ✳ ❚❛❦❡ n = N ❢r♦♠ ♥♦✇ ♦♥✳ ❚❤❡r❡ ❡①✐sts K s✉❝❤ t❤❛t ❢♦r ❛♥② k ≥ K✱ ❛t n = N ✱ t❤❡ s❡❝♦♥❞ t❡r♠ ✐s ❝♦♥tr♦••❡❞ ❜② ǫ✳ ❋✐♥❛••②✱ ❛t n = N ❛♥❞ k = K✱ ❜② t❤❡ ❈❛✉❝❤② ♣r♦♣❡rt② ❢♦r ❜✐❣ ❡♥♦✉❣❤ m t❤❡ ✜rst t❡r♠ ✐s ❝♦♥tr♦••❡❞ ❜② ǫ✳ ❚❤❡r❡❢♦r❡ ✐❢ ✇❡ •❡t m t❡♥❞ t♦ ✐♥✜♥✐t② ✇❡ ♦❜t❛✐♥✿ X(• + α K h) -X α K -(ξ, h) H < 3ǫ ✭✶✳✶✶✵✮ ❚❤✐s ♣r♦✈❡s t❤❛t X ❤❛s ❛ ❞✐✛❡r❡♥t✐❛• ✐♥ ❞✐r❡❝t✐♦♥ h ❛♥❞ t❤❛t t❤✐s ❞✐✛❡r❡♥t✐❛• ✐s (ξ, h) H ✳ ❚❤❡♥ s✐♥❝❡ ξ ∈ L p (H) ✇❡ ❛✉t♦♠❛t✐❝❛••② ♦❜t❛✐♥ X ∈ D p,1 ❛♥❞ ∇X = ξ s♦ ✇❡ ❛r❡ ❞♦♥❡✳ ❋♦r t❤❡ s❡❝♦♥❞ ♣❛rt✱ ✇❡ ♥♦t❡ t❤❛t t❤❡ ♣♦•②♥♦♠✐❛•s ❛r❡ ❞❡♥s❡ ✐♥ ❛♥② L p ✱ ❛♥❞ ❛❝t✉❛••② ❜② ✉s✐♥❣ t❤❡ s❛♠❡ t❡❝❤♥✐q✉❡ ❛s ❜❡❢♦r❡ ♦♥❡ ♠❛② ♣r♦✈❡ t❤❛t t❤❡② ❛r❡ ❞❡♥s❡ ✐♥ D p,1 ❛s ✇❡••✳ |❡♠❛r❦ ✶✳✷✳✸✳✸✳ ❚❤❡ ❝♦♥str✉❝t✐♦♥ ✐s s♦♠❡t✐♠❡s ❞♦♥❡ ❜② ❞❡✜♥✐♥❣ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦♥ t❤❡ ♣♦•②♥♦♠✐❛•s ♦♥•②✱ ❜② ♣r♦✈✐♥❣ ❛ ✈❡rs✐♦♥ ♦❢ t❤❡♦r❡♠ ✶✳✷✳✸✳✹ ❢♦r ♣♦•②♥♦♠✐❛•s ♦♥•②✱ ❛♥❞ ❜② ❞❡✜♥✐♥❣ D p,1 ❛s t❤❡ s❡t ♦❢ t❤♦s❡ r❛♥❞♦♠ ✈❛r✐❛❜•❡s X ❢♦r ✇❤✐❝❤ t❤❡r❡ ❡①✐sts ❛♥ ❛♣♣r♦①✐♠❛t✐♥❣ s❡q✉❡♥❝❡ ♦❢ ♣♦•②♥♦♠✐❛•s (P n ) s✉❝❤ t❤❛t t❤❡ ❣r❛❞✐❡♥ts (∇P n ) ❝♦♥✈❡r❣❡✳ ❚❤❡♥ ❜② t❤❡♦r❡♠ ✶✳✷✳✸✳✹ ✐t ♠❛❦❡s s❡♥s❡ t♦ ❞❡✜♥❡ ∇X = lim ∇P n ✳ ■♥ t❤✐s ❝♦♥str✉❝t✐♦♥ D p,1 ✐s ❛✉t♦♠❛t✐❝❛••② t❤❡ ❝♦♠♣•❡t✐♦♥ ♦❢ t❤❡ ♣♦•②♥♦♠✐❛•s ❢♦r t❤❡ ♦♣✲ ❡r❛t♦r ♥♦r♠✳ ❖❢ ❝♦✉rs❡ ✐t ✐s ♣♦ss✐❜•❡ ✭❜✉t t❡❞✐♦✉s✮ t♦ ♣r♦✈❡ t❤❛t ❜♦t❤ ❝♦♥str✉❝t✐♦♥s ❛r❡ ❡q✉✐✈❛•❡♥t✳ ✷✷ ❈❍❆P❚❊| ✶✳ ■◆❚|❖❉❯❈❚■❖◆ ❚❖ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ❊①t❡♥s✐♦♥ t♦ ❍✐•❜❡rt s♣❛❝❡ ✈❛•✉❡❞ r❛♥❞♦♠ ✈❛r✐❛❜•❡s ❛♥❞ ❤✐❣❤❡r ♦r❞❡r ❞❡r✐✈❛t✐✈❡s ■❢ Ξ ✐s ❛ s❡♣❛r❛❜•❡ ❍✐•❜❡rt s♣❛❝❡✱ t❤❡♥ t❤❡r❡ ✐s ♥♦ ❞✐✣❝✉•t② ✐♥ ❡①t❡♥❞✐♥❣ ♦✉r r❡s✉•ts t♦ L p (W, µ, Ξ)✱ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜•❡s t❛❦✐♥❣ ✈❛•✉❡s ✐♥ Ξ✳ ■❢ (x i ) ✐s ❛♥ ❖◆❇ ♦❢ Ξ ❛♥❞ ✐❢ (h i ) ✐s ❛♥ ❖◆❇ ♦❢ H t❤❡♥ ✇❡ ❞❡✜♥❡ t❤❡ ♣♦•②♥♦♠✐❛•s ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡ ❛s r❛♥❞♦♠ ✈❛r✐❛❜•❡s ♦❢ t❤❡ t②♣❡✿ n j=1 P j h 1 dW, • • • , h n j dW • x j ✭✶✳✶✶✶✮ ❛♥❞ ❛•• t❤❡ ✐❞❡❛s ❛♥❞ ♣r♦♦❢s ♠❛② ❜❡ ❛❞❛♣t❡❞ ✐♥ ❛ str❛✐❣❤t❢♦r✇❛r❞ ✇❛②✳ ❲❡ ❛r❡ ❡s♣❡❝✐❛••② ✐♥t❡r❡st❡❞ ✐♥ t❤❡ ❝❛s❡ ✇❤❡r❡ Ξ = H ⊗k ✱ t❤❡ ❦t❤ ♦r❞❡r s②♠✲ ♠❡tr✐③❡❞ t❡♥s♦r ♣♦✇❡r ♦❢ H ✭s❡❡ ❛♣♣❡♥❞✐① ❆✮✳ ■♥❞❡❡❞✱ t❤✐s ❝❛s❡ ❛••♦✇s ✉s t♦ ❝♦♥str✉❝t ❤✐❣❤❡r ♦r❞❡r •r♦ss✲❙♦❜♦•❡✈ s♣❛❝❡s ✈✐❛ ❛♥ ❡❛s② r❡❝✉rr❡♥❝❡✿ D p,k = X ∈ D p,k-1 ∇ k-1 X ∈ D p,1 H ⊗(k-1) ✭✶✳✶✶✷✮ ❛♥❞ ∇ k X = ∇ ∇ k-1 X ✭✶✳✶✶✸✮ ✐♥ t❤❡ s❡♥s❡ ♦❢ D p,1 H ⊗(k-1) ✳ ✶✳✷✳✹ ❚❤❡ ❞✐✈❡r❣❡♥❝❡ ♦♣❡r❛t♦r ❆s ✐♥ t❤❡ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥♥❛• ❝❛s❡✱ t❤❡ ❞✐✈❡r❣❡♥❝❡ ♦♣❡r❛t♦r ✇✐•• ❜❡ ❞❡✜♥❡❞ ❛s t❤❡ ❛❞•♦✐♥t ♦❢ t❤❡ ❣r❛❞✐❡♥t✱ ❛•t❤♦✉❣❤ ♦♥ t❡ ❲✐❡♥❡r s♣❛❝❡ t❤❡ ❢♦r♠❛• ❞❡✜♥✐t✐♦♥ ✐s s•✐❣❤t•② ♠♦r❡ ✐♥✈♦•✈❡❞✿ ❉❡✜♥✐t✐♦♥ ✶✳✷✳✹✳✶ ✭❉✐✈❡r❣❡♥❝❡ ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡✮✳ ▲❡t ξ ∈ L p (W, µ, H)✳ ❲❡ s❛② t❤❛t ξ ∈ dom p (δ) ✐❢ ❢♦r ❛♥② φ ∈ D p * ,1 ✇❡ ❤❛✈❡✿ E[(∇φ, ξ) H ] ≤ c φ D p * ,1 ✭✶✳✶✶✹✮ ❛♥❞ ✐♥ t❤✐s ❝❛s❡ ✇❡ ❞❡✜♥❡ δξ ✈✐❛ t❤❡ |✐❡s③ t❤❡♦r❡♠ ❜②✿ E[φδξ] = E[(ξ, ∇φ) H ] ✭✶✳✶✶✺✮ ❚❤❡ ❢♦••♦✇✐♥❣ r❡s✉•ts ✇✐•• ❜❡ ✉s❡❢✉•✿ Pr♦♣♦s✐t✐♦♥ ✶✳✷✳✹✳✶ ✭❞✐✈❡r❣❡♥❝❡ ♦❢ ❛ ♣r♦❞✉❝t✮✳ ▲❡t ξ ∈ dom p (δ) ❛♥❞ X ∈ D q,1 ❢♦r s♦♠❡ q > p * ✳ ❉❡✜♥❡ r > 1 s✉❝❤ t❤❛t✿ 1 p + 1 q = 1 r ✭✶✳✶✶✻✮ ✶✳✷✳ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ✷✸ ❚❤❡♥ Xξ ∈ dom r (δ) ❛♥❞✿ δ(Xξ) = Xδξ -(∇X, ξ) H ✭✶✳✶✶✼✮ Pr♦♦❢✳ ❚❤❡ ❝❤♦✐❝❡ ♦❢ r •✉st✐✜❡s t❤❛t ❜♦t❤ ❡①♣r❡ss✐♦♥s ♦♥ t❤❡ |❍❙ ❛r❡ ✇❡•• ❞❡✜♥❡❞ ✐♥ L r ❜② ❍ö•❞❡r✬s t❤❡♦r❡♠✳ ❆•s♦✱ •❡t φ ❜❡ ❛♥② ♣♦•②♥♦♠✐❛• ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡✳ ❚❤❡♥ φ ❤❛s ♠♦♠❡♥ts ♦❢ ❛•• ♦r❞❡rs✱ ❛♥❞ ❡s♣❡❝✐❛••② ♦❢ ♦r❞❡r r * ✳ ❚❤❡r❡❢♦r❡ ✇❡ ♠❛② ✇r✐t❡✿ E[δ(Xξ)φ] = E[(Xξ, ∇φ) H ] = E[(ξ, X∇φ) H ] = E[(ξ, ∇(Xξ) -φ∇X) H ] = E[(φ(Xδξ -(∇X, ξ) H )] ✭✶✳✶✶✽✮ s♦ ❜② ❞✉❛•✐t② t❤❡ ♣r♦♦❢ ✐s ✜♥✐s❤❡❞✳ Pr♦♣♦s✐t✐♦♥ ✶✳✷✳✹✳✷✳ ❚❤❡ ❞✐✈❡r❣❡♥❝❡ ♦❢ ❛ ❞❡t❡r♠✐♥✐st✐❝ h ∈ H ✐s t❤❡ st♦❝❤❛st✐❝ ✐♥t❡❣r❛• ♦❢ ✐ts ❞❡r✐✈❛t✐✈❡✿ δh = 1 0 ḣ(s)dW s ✭✶✳✶✶✾✮ Pr♦♦❢✳ ❇② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❞✐✈❡r❣❡♥❝❡✱ ❢♦r ❛♥② ♣♦•②♥♦♠✐❛• φ✿ E[φδh] = E[∇ h φ] ✭✶✳✶✷✵✮ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ s✐♠♣•✐❢②✐♥❣ ❢♦r♠✉•❛ ✭✶✳✶✵✻✮ ✇❡ ♦❜t❛✐♥✿ E[∇ h φ] = E φ • ḣdW ✭✶✳✶✷✶✮ s♦ ❜② t❤❡ ✉s✉❛• ❞✉❛•✐t② ❛r❣✉♠❡♥t ✇❡ ❛r❡ ❞♦♥❡✳ ✶✳✷✳✺ ❖r♥st❡✐♥✲❯❤•❡♥❜❡❝❦ ♦♣❡r❛t♦r ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡ ❉❡✜♥✐t✐♦♥ ❛♥❞ ✜rst ♣r♦♣❡rt✐❡s ▲❡t ✉s ❞❡✜♥❡ ❛ ❢❛♠✐•② ♦❢ ♦♣❡r❛t♦rs ♦♥ t❤❡ ♣♦•②♥♦♠✐❛•s ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡ ❜② t❤❡ ❢♦r♠✉•❛✿ P t f h 1 dW, . . . , h n dW = P (n) t f h 1 dW, . . . , h n dW ✭✶✳✶✷✷✮ ✇❤❡r❡ P (n) t ✐s t❤❡ ❖r♥st❡✐♥ ❯❤•❡♥❜❡❝❦ ♦♣❡r❛t♦r ♦♥ R n ✳ ❚❤❡ r❡s✉•ts ❢♦r t❤❡ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥♥❛• ❝❛s❡ ❛r❡ ❛✉t♦♠❛t✐❝❛••② •✐❢t❡❞ s✐♥❝❡ t❤❡ ( h 1 dW, . . . , H n dW ) ❛r❡ ✷✹ ❈❍❆P❚❊| ✶✳ ■◆❚|❖❉❯❈❚■❖◆ ❚❖ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ♥♦r♠❛• ❝❡♥t❡r❡❞ ❣❛✉ss✐❛♥s❀ ✐♥ ♣❛rt✐❝✉•❛r (P t ) ❞❡✜♥❡s ❛ ❝♦♥tr❛❝t✐✈❡ L p s❡♠✐❣r♦✉♣ ♦♥ t❤❡ ♣♦•②♥♦♠✐❛•s ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡✱ ❛♥❞ ❢♦r ❛♥② ♣♦•②♥♦♠✐❛• X✱ t❤❡ ▼❡❤•❡r ❢♦r♠✉•❛ ❤♦•❞s✿ P t X(w) = Ẽ X(e -t w + √ 1 -e -2t w) ✭✶✳✶✷✸✮ ■t ✐s str❛✐❣❤t❢♦r✇❛r❞ t♦ ❡①t❡♥❞ (P t ) t♦ ❛ ❝♦♥tr❛❝t✐♦♥ s❡♠✐❣r♦✉♣ ♦♥ ❛•• L p ❜② ❞❡♥✲ s✐t②❀ ▼❡❤•❡r✬s ❢♦r♠✉•❛ ✐s t❤❡♥ ✈❡r✐✜❡❞ ♦♥ ❛•• L p ❛♥❞✱ ❛s ✐♥ t❤❡ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥♥❛• ❝❛s❡✱ t❤❡r❡ ✐s ♥♦ ❛♠❜✐❣✉✐t② ❛❜♦✉t t❤❡ ❝❤♦✐❝❡ ♦❢ p✳ ❆s ❛ s❡♠✐❣r♦✉♣ (P t ) ❛❞♠✐ts ❛ ❣❡♥❡r❛t♦r ✇❤✐❝❤ ✇❡ ❞❡♥♦t❡ L✳ ❆s ✐♥ t❤❡ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥♥❛• ❝❛s❡✱ t❤❡ P t ❛♥❞ L ❛r❡ s②♠♠❡tr✐❝❛•✳ ❙✐♠✐•❛r•② t♦ ❡q✉❛t✐♦♥ ✶✳✶✷✷ ✇❡ ❤❛✈❡✿ L f h 1 dW, . . . , h n dW = L (n) f h 1 dW, . . . , h n dW ✭✶✳✶✷✹✮ ❲❡ ✇✐•• ❛•s♦ ♣r♦✈❡ t❤❛t✿ ❚❤❡♦r❡♠ ✶✳✷✳✺✳✶✳ ❲❤❡♥❡✈❡r ❜♦t❤ ❡①♣r❡ss✐♦♥s ❛r❡ ✇❡•• ❞❡✜♥❡❞✿ L = δ • ∇ ✭✶✳✶✷✺✮ Pr♦♦❢✳ ■t ✐s ❡♥♦✉❣❤ t♦ ♣r♦✈❡ t❤❡ ❡q✉❛t✐♦♥ ❢♦r t❤❡ ♣♦•②♥♦♠✐❛•s✱ ❛♥❞ ✐♥❞❡❡❞✿ ∇ P h 1 dW, . . . , h n dW = n i=1 ∂ i P h 1 dW, . . . , h n dW h i ✭✶✳✶✷✻✮ ❛♥❞ t❤❡r❡❢♦r❡ δ∇ P h 1 dW, . . . , h n dW = n i=1 ∂ i P h 1 dW, . . . , h n dW δh i - n i=1 ∂ 2 ii P h 1 dW, . . . , h n dW |h i | 2 = L (n) P h 1 dW, . . . , h n dW = L P h 1 dW, . . . , h n dW ✭✶✳✶✷✼✮ ❋✐♥❛••② t❤❡ ❢♦••♦✇✐♥❣ r❡s✉•t ✐s ❛♥ ✐♠♠❡❞✐❛t❡ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ❢♦r♠✉•❛ ✭✶✳✶✷✷✮✿ ❚❤❡♦r❡♠ ✶✳✷✳✺✳✷✳ ❋♦r ❛♥② X ∈ C n ✿ P t X = e -nt X ✭✶✳✶✷✽✮ ✶✳✷✳ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ✷✺ ❛♥❞ LX = -nX ✭✶✳✶✷✾✮ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❲✐❡♥❡r ❝❤❛♦s ❞❡❝♦♠♣♦s✐t✐♦♥ ❞✐❛❣♦♥❛•✐③❡s t❤❡ ❖r♥st❡✐♥ ❯❤•❡♥✲ ❜❡❝❦ s❡♠✐❣r♦✉♣✳ ❙✉❜♦r❞✐♥❛t✐♦♥ ♦❢ t❤❡ ❖r♥st❡✐♥✲❯❤•❡♥❜❡❝❦ s❡♠✐❣r♦✉♣ ■♥ ♦r❞❡r t♦ ♣r♦✈❡ s♦♠❡ ✐♥t❡r♣♦•❛t✐♦♥ r❡s✉•ts ❢♦r t❤❡ •r♦ss✲❙♦❜♦•❡✈ s♣❛❝❡s✱ ✇❡ ✇✐•• ♥❡❡❞ t♦ ♠❛❦❡ ❛♥ ✐♥t❡♥s✐✈❡ ✉s❡ ♦❢ ❛ s❡♠✐❣r♦✉♣ ✇❤✐❝❤ ✐s ❝•♦s❡•② r❡•❛t❡❞ t♦ t❤❡ ❖r♥st❡✐♥✲❯❤•❡♥❜❡❝❦ s❡♠✐❣r♦✉♣✳ ▼♦r❡ ♣r❡❝✐s❡•②✱ ♦♥❡ ♠❛② ✐♥tr♦❞✉❝❡ T t ❜❡ t❤❡ s❡♠✐❣r♦✉♣ ❣❡♥❡r❛t❡❞ ❜② -(Id -L) 1/2 ❀ t❤❡♥ ✇❡ ❦♥♦✇ t❤❛t Id -L ✐s t❤❡ ❣❡♥❡r❛t♦r ♦❢ t❤❡ s✉❜♠❛r❦♦✈✐❛♥ s❡♠✐❣r♦✉♣ e -t P t ❛♥❞ t❤❛t T t ✐s ✐ts 1/2✲s✉❜♦r❞✐♥❛t✐♦♥ ✭s❡❡ ❛♣♣❡♥❞✐① ❈ ❢♦r ❞❡t❛✐•s✮✱ t❤❡r❡❢♦r❡ ♦♥❡ ❤❛s t❤❡ ❢♦••♦✇✐♥❣ ❡①♣•✐❝✐t ❡①♣r❡ss✐♦♥ ❢♦r T t ✿ T t = ∞ 0 e -s P s λ 1/2 t (ds) ✭✶✳✶✸✵✮ ✇❤❡r❡ t❤❡ ♣r♦❜❛❜✐•✐t② ♠❡❛s✉r❡s λ ❛r❡ ❝❤❛r❛❝t❡r✐③❡❞ t❤r♦✉❣❤ t❤❡✐r ▲❛♣•❛❝❡ tr❛♥s✲ ❢♦r♠s✿ ❢♦r 0 < β < 1 ∞ 0 e -γs λ β t (ds) = e -γ β t ✭✶✳✶✸✶✮ ❛♥❞ ♦♥❡ ♠❛② ❝♦♠♣✉t❡ ❡①♣•✐❝✐t❡•② ❜② ✐♥✈❡rt✐♥❣ t❤❡ ▲❛♣•❛❝❡ tr❛♥s❢♦r♠✿ λ 1/2 t (ds) = 1 2 √ π ts -3/2 e -t 2 4s • ds ✭✶✳✶✸✷✮ ❛♥❞ t❤❡r❡❢♦r❡✿ T t = 1 2 √ π t ∞ 0 s -3/2 e -s e -t 2 4s P s ds ✭✶✳✶✸✸✮ ❲❡ ♥♦t❡ t❤❛t T t • 1 = e -t • 1 ❛♥❞ t❤❛t ♠♦r❡ ❣❡♥❡r❛••②✿ Pr♦♣♦s✐t✐♦♥ ✶✳✷✳✺✳✶✳ ■❢ F ∈ L p t❤❡♥ T t F L p ≤ e -t F L p ✭✶✳✶✸✹✮ ❆•s♦✿ Pr♦♣♦s✐t✐♦♥ ✶✳✷✳✺✳✷✳ ■❢ F ∈ L p t❤❡♥ T t F ∈ s D p,s ✭✶✳✶✸✺✮ ✷✻ ❈❍❆P❚❊| ✶✳ ■◆❚|❖❉❯❈❚■❖◆ ❚❖ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ✶✳✷✳✻ ❋r❛❝t✐♦♥❛• s♣❛❝❡s ❛♥❞ t❤❡ ▼❡②❡r ✐♥❡q✉❛•✐t✐❡s ❲❡ ✇♦✉•❞ •✐❦❡ t♦ ❞❡✜♥❡ ❢r❛❝t✐♦♥♥❛• ♣♦✇❡rs ♦❢ t❤❡ ♦♣❡r❛t♦r Id + L ❛s ♦♣❡❛t♦rs ♦♥ ❛ s✉❜s♣❛❝❡ ♦❢ L p ✳ ■♥ t❤❡ ❝❛s❡ ✇❤❡r❡ p = 2✱ ✐t ✐s ♥❛t✉r❛• t♦ ✉s❡ t❤❡ ❲✐❡♥❡r ❝❤❛♦s r❡♣r❡s❡♥t❛t✐♦♥ ❛♥❞ t♦ s❡t✿ (Id + L) α/2 I n (f n ) = (1 + n) α/2 • I n (f n ) ✭✶✳✶✸✻✮ ❚❤❡♥✱ t❤❡ ❞❡✜♥✐t✐♦♥ ✐s ❝♦♠♣•❡t❡❞ ❜② •✐♥❡❛r✐t② ❛♥❞ ❞❡♥s✐t②✳ ❖❢ ❝♦✉rs❡✱ ❢♦r ❛ ❣❡♥❡r❛• p t❤✐s str❛t❡❣② ✐s ♥♦ •♦♥❣❡r ✈❛•✐❞✱ ❜✉t ✇❡ ✐♥s♣✐r❡ ❢r♦♠ ✐t ❛♥❞ t❤❡ ❡q✉❛t✐♦♥✿ (1 + x) -α = 1 Γ(α) • ∞ 0 e -t(1+x) t -α dt ✭✶✳✶✸✼✮ ✇❤✐❝❤ ✐s ✈❛•✐❞ ❢♦r x, α > 0 ✐❢ Γ ❞❡♥♦t❡s t❤❡ ✉s✉❛• •❛♠♠❛ ❢✉♥❝t✐♦♥ ❛♥❞ ✇❡ ❞❡✜♥❡ ❛♥ ♦♣❡r❛t♦r ♦♥ L p ❛s✿ A α X = 1 Γ(α) • ∞ 0 e -t t -α P t Xdt ✭✶✳✶✸✽✮ ■t ✐s ❡❛s✐•② ✈❡r✐✜❡❞ t❤❛t t❤❡ ✐♥t❡❣r❛• ❛❜♦✈❡ ✐s ✇❡•• ❞❡✜♥❡❞ ♦♥ L p ✐♥ t❤❡ s❡♥s❡ ♦❢ ❇♦❝❤♥❡r ❛♥❞ t❤❛t A α ✐s ❛ ❝♦♥tr❛❝t✐♦♥ ♦♥ L p ✳ ❆•s♦✱ ❜② ❛ s✐♠♣•❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜•❡s✿ A α I n (f n ) = (1 + n) -α I n (f n ) ✭✶✳✶✸✾✮ ❚❤❡r❡❢♦r❡✱ ♦♥ ♦♥❡ ❤❛♥❞ ♦✉r ❞❡✜♥✐t✐♦♥ ♠❛❦❡s ✐♥t✉✐t✐✈❡ s❡♥s❡ ❛♥❞ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞ ✇❡ ♥♦t❡ t❤❛t✿ Pr♦♣♦s✐t✐♦♥ ✶✳✷✳✻✳✶✳ A α ♠❛♣s t❤❡ ♣♦•②♥♦♠✐❛•s ♦♥ t❤❡ ♣♦•②♥♦♠✐❛•s✳ ❛♥❞ ❈♦r♦••❛r② ✶✳✷✳✻✳✶✳ A α (L p ) ⊃ L p ✭✶✳✶✹✵✮ ❲❡ ✇✐•• ♥♦✇ ♣r♦✈❡ t❤❛t✿ Pr♦♣♦s✐t✐♦♥ ✶✳✷✳✻✳✷✳ A α ✐s ✐♥•❡❝t✐✈❡✳ Pr♦♦❢✳ ❋✐rst s✉♣♣♦s❡ t❤❛t p = 2✱ ❛♥❞ •❡t X ∈ L 2 ❜❡ s✉❝❤ t❤❛t A α X = 0✳ ❚❤❡♥ ✐❢ ✇❡ ✇r✐t❡ ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥ ❲✐❡♥❡r ❝❤❛♦s❡s✿ X = I n (f n ) ✭✶✳✶✹✶✮ ✇❡ ❤❛✈❡ A α X = I n (f n ) (1 + n) α ✭✶✳✶✹✷✮ s♦ t❤❡ ♦rt❤♦❣♦♥❛•✐t② ♦❢ t❤❡ ❲✐❡♥❡r ❝❤❛♦s❡s ②✐❡•❞ t❤❡ r❡s✉•t✳ ◆♦✇ ✐❢ p > 2✱ L p ⊂ L 2 ❛♥❞ ✇❡ ❛•s♦ ❣❡t t❤❡ r❡s✉•t✳ ❋✐♥❛••② •❡t ✉s s✉♣♣♦s❡ t❤❛t p < 2✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦••♦✇✐♥❣✿ ✶✳✷✳ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ✷✼ ▲❡♠♠❛ ✶✳✷✳✻✳✶✳ ▲❡t X ∈ L p ❛♥❞ Y ∈ L p * ✳ ❚❤❡♥✿ E [A α X • Y ] = E [X • A α Y ] ✭✶✳✶✹✸✮ ◆♦✇ •❡t E ❜❡ t❤❡ s❡t ♦❢ t❤♦s❡ ♣♦•②♥♦♠✐❛•s ✇✐t❤ L p * ✲♥♦r♠ 1✳ ❚❤❡♥✿ A α X L p = sup Y ∈E E [A α X • Y ] = sup Y ∈E E X • (1 + n) -α Y ✭✶✳✶✹✹✮ t❤❡r❡❢♦r❡ A α X L p = 0 ✐❢ ❛♥❞ ♦♥•② ✐❢ X ✐s ♦rt❤♦❣♦♥❛• t♦ ❡✈❡r② ♣♦•②♥♦♠✐❛•✱ ✐❡ ✐✛ X = 0 s♦ ✇❡ ❛r❡ ❞♦♥❡✳ ❚❤✐s •✉st✐✜❡s t❤❛t ✇❡ ✐♥tr♦❞✉❝❡✿ (Id + L) α = A α if α < 0 A -1 α if α > 0 ✭✶✳✶✹✺✮ ❚❤✐s ❞❡✜♥✐t✐♦♥ ♦❢ ♦✉r ❢r❛❝t✐♦♥❛• ♣♦✇❡r ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ✐♥t✉✐t✐✈❡ ♦♥❡ ✐♥ t❤❡ L 2 ❝❛s❡ ✭❜❡❝❛✉s❡ t❤❡ t✇♦ ♦♣❡r❛t♦rs ❝♦✐♥❝✐❞❡ ♦♥ ♣♦•②♥♦♠✐❛•s✮✳ ❆•s♦✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦••♦✇✐♥❣ ❝r✉❝✐❛•✿ ❚❤❡♦r❡♠ ✶✳✷✳✻✳✶ ✭▼❡②❡r ✐♥❡q✉❛•✐t✐❡s✮✳ ▲❡t 1 < p < ∞ ❛♥❞ n ∈ N✳ ❚❤❡♥✱ t❤❡ t✇♦ ❢♦••♦✇✐♥❣ ♥♦r♠s ❛r❡ ❡q✉✐✈❛•❡♥t✿ X = X L p + ∇ n X L p (H ⊗n ) ❛♥❞ X = (Id + L) n/2 X L p ■♥ t❤❡ p = 2 ❝❛s❡ ✐t ✐s ✈❡r② ❡❛s② t♦ ♣r♦✈❡ t❤❡ r❡s✉•t ❜② ❝♦♠♣❛r✐♥❣ t❤❡ r❡s♣❡❝t✐✈❡ ❛❝t✐♦♥s ♦❢ ∇ ❛♥❞ (Id + L) 1/2 ♦♥ ❛ ♣♦•②♥♦♠✐❛•✳ ❯♥❢♦rt✉♥❛t❡•②✱ t❤✐s ✐❞❡❛ ❞♦❡s ♥♦t ❡①t❡♥❞ t♦ ❛ ❣❡♥❡r❛• p ❢♦r t❤❡ •❛❝❦ ♦❢ ❛♥ ♦rt❤♦❣♦♥❛•✐t② ♣r♦♣❡rt②✳ ❲❡ ♦♠✐t t❤❡ ♣r♦♣❡r ♣r♦♦❢ ✇❤✐❝❤✱ ❛•t❤♦✉❣❤ ❝•❛ss✐❝✱ ✐s q✉✐t❡ ✐♥✈♦•✈❡❞ ❛♥❞ ♠❛② ❜❡ ❢♦✉♥❞ ✐♥ ❬✺✸( ♦r ❬✻✻( ❢♦r ❡①❛♠♣•❡✳ ❚❤❡ s❛♠❡ r❡❢❡r❡♥❝❡s ❛•s♦ ♣r♦✈✐❞❡ ❛ s•✐❣❤t•② ♠♦r❡ ♣r❡❝✐s❡ r❡s✉•t ✇❤✐❝❤ ✐s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ▼❡②❡r✬s ♠✉•t✐♣•✐❡r t❤❡♦r❡♠✿ ❚❤❡♦r❡♠ ✶✳✷✳✻✳✷✳ ▲❡t 1 < p < ∞✱ s ∈ R ❛♥❞ α > 0✳ ❚❤❡♥ t❤❡ t✇♦ ❢♦••♦✇✐♥❣ ♥♦r♠s ❛r❡ ❡q✉✐✈❛•❡♥t✿ X = (Id + L) s/2 X L p ❛♥❞ X = (α • Id + L) s/2 X L p ❚❤❡ ▼❡②❡r ✐♥❡q✉❛•✐t✐❡s ❛••♦✇ ✉s t♦ ❡①t❡♥❞ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ D p,k s♣❛❝❡s ❢♦r ❛♥② 1 < p < ∞ ❛♥❞ ❛♥② r❡❛• ♥✉♠❜❡r k✳ ❇❡❝❛✉s❡ t❤❡ A α ❛r❡ ❝♦♥tr❛❝t✐♦♥s✱ ♦♥❡ ♠❛② s❡❡ t❤❛t✿ ✷✽ ❈❍❆P❚❊| ✶✳ ■◆❚|❖❉❯❈❚■❖◆ ❚❖ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ Pr♦♣♦s✐t✐♦♥ ✶✳✷✳✻✳✸✳ ▲❡t 1 < p 1 < p 2 < ∞ ❛♥❞ s 1 < s 2 ✳ ❚❤❡♥✿ D p 1 ,s 1 ⊂ D p 2 ,s 2 ✭✶✳✶✹✻✮ ■♥ ♣❛rt✐❝✉•❛r✱ ❡•❡♠❡♥ts ♦❢ D p,-s ❢♦r ❛ ♣♦s✐t✐✈❡ s ❛r❡ ♥♦t ♥❡❝❡ss❛r✐•② r❛♥❞♦♠ ✈❛r✐❛❜•❡s ❛♥❞ ✇❡ r❡❢❡r t♦ t❤❡♠ ❛s ❞✐str✐❜✉t✐♦♥s ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡✳ ❆•s♦✱ ❛♥ ✐♠♠❡❞✐❛t❡ ❝♦r♦••❛r② ♦❢ •❡♠♠❛ ✶✳✷✳✻✳✶ ✐s✿ ❚❤❡♦r❡♠ ✶✳✷✳✻✳✸✳ ▲❡t 1 < p < ∞ ❛♥❞ s > 0✳ ❚❤❡♥✿ D * p,s = D p * ,-s ✭✶✳✶✹✼✮ ❆•s♦✱ ❢♦r p > 1✱ D p,s ✐s ✉♥✐❢♦r♠•② ❝♦♥✈❡① ❛s ✐t ✐s ♥❛t✉r❛••② ✐s♦♠❡tr✐❝ t♦ L p ✇❤✐❝❤ ✐♥ t✉r♥ ✐s ✉♥✐❢♦r♠•② ❝♦♥✈❡①❀ ✐♥ ♣❛rt✐❝✉•❛r t❤✐s ♣r♦✈❡s t❤❛t D p,s ✐s ✐s♦♠♦r♣❤✐❝ t♦ ✐ts ❜✐❞✉❛• ❛♥❞✿ ❈♦r♦••❛r② ✶✳✷✳✻✳✷✳ D * p * ,-s = D p,s ✭✶✳✶✹✽✮ ❆♥♦t❤❡r ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ r❡✢❡①✐✈✐t② ♦❢ D p,s ✐s t❤❡ ✈❡r② ❝♦♥✈❡♥✐❡♥t✿ ▲❡♠♠❛ ✶✳✷✳✻✳✷✳ ▲❡t p > 1 ❛♥❞ s > 0✳ ▲❡t (X n ) ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜•❡s ✇❤✐❝❤ ❝♦♥✈❡r❣❡s t♦ X ✐♥ L p ❛♥❞ ✇❤✐❝❤ ✐s ❜♦✉♥❞❡❞ ✐♥ D p,s ✳ ❚❤❡♥ (X n ) ❝♦♥✈❡r❣❡s t♦ X ✐♥ D p,s ✳ Pr♦♦❢✳ ❙✐♥❝❡ D p,s ✐s ❛ r❡✢❡①✐✈❡ ❇❛♥❛❝❤ s♣❛❝❡✱ ❛s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ❇❛♥❛❝❤✲ ❆•❛♦❣•✉ t❤❡♦r❡♠ t❤❡r❡ ❡①✐sts ❛ s✉❜s❡q✉❡♥❝❡ X φ(n) ✇❤✐❝❤ ❝♦♥✈❡r❣❡s ✇❡❛❦•② ✐♥ D p,s ✳ ■♥ ♣❛rt✐❝✉•❛r✱ ❢♦r ❛♥② ♣♦•②♥♦♠✐❛• r❛♥❞♦♠ ✈❛r✐❛❜•❡ P ✿ lim n→∞ E (Id + L) s/2 X φ(n) • P = E (Id + L) s/2 X n • P ✭✶✳✶✹✾✮ s♦ ✇❡ ❛r❡ ❞♦♥❡✳ ❲❡ ✜♥✐s❤ t❤✐s ♣❛r❛❣r❛♣❤ ✇✐t❤ ❛ r❡s✉•t ✇❤✐❝❤ ✇❡ ✇✐•• ♦❢t❡♥ ✉s❡✿ Pr♦♣♦s✐t✐♦♥ ✶✳✷✳✻✳✹✳ ▲❡t p ∈]1, ∞[✳ • ▲❡t h ∈ L p (H)✳ ❚❤❡♥✿ P t 1 0 ḣu dW u = e -t • 1 0 P t ḣu dW u ✭✶✳✶✺✵✮ • ▲❡t α > 0✱ s ∈ R ❛♥❞ h ∈ D p,s (H)✳ ❚❤❡♥✱ 1 0 ḣs dW s ∈ D p,s ❛♥❞✿ (α • Id + L) s/2 1 0 ḣu dW u = 1 0 ((1 + α) • Id + L) s/2 ḣu dW u ✭✶✳✶✺✶✮ ✶✳✷✳ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ✷✾ ✶✳✷✳✼ ❉✐str✐❜✉t✐♦♥s ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡ ❋✐rst ✇❡ ✐♥tr♦❞✉❝❡✿ ❉❡✜♥✐t✐♦♥ ✶✳✷✳✼✳✶ ✭❙♠♦♦t❤ r❛♥❞♦♠ ✈❛r✐❛❜•❡s✮✳ ❚❤❡ s♣❛❝❡ D ♦❢ t❤❡ s♠♦♦t❤ r❛♥✲ ❞♦♠ ✈❛r✐❛❜•❡s ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡ ✐s ❞❡✜♥❡❞ ❛s✿ D = p,s D p,s ✭✶✳✶✺✷✮ ❲❡ ❡q✉✐♣ ✐t ✇✐t❤ ✐ts ♥❛t✉r❛• ❋ré❝❤❡t s♣❛❝❡ t♦♣♦•♦❣② ✭s❡❡ ❛♣♣❡♥❞✐① ❆✮✱ ✐❡ t❤❡ ♦♥❡ ❣✐✈❡♥ ❜② t❤❡ ❞✐st❛♥❝❡✿ d(x, y) = ∞ p=1 ∞ k=0 1 2 p+k • X -Y D p,k 1 + X -Y D p,k ✭✶✳✶✺✸✮ ❚❤❡♥ ✐t ✐s ♥❛t✉r❛• t♦ ❝♦♥s✐❞❡r t❤❡✿ ❉❡✜♥✐t✐♦♥ ✶✳✷✳✼✳✷ ✭❉✐str✐❜✉t✐♦♥s ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡✮✳ ❚❤❡ ✈❡❝t♦r s♣❛❝❡ D ′ ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥s ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡ ✐s ❞❡✜♥❡❞ ❛s t❤❡ str♦♥❣ ❞✉❛• ♦❢ t❤❡ s♣❛❝❡ D ♦❢ s♠♦♦t❤ r❛♥❞♦♠ ✈❛r✐❛❜•❡s✿ D ′ = p>1,s∈R D p,s ✭✶✳✶✺✹✮ ❚❤✐s s♣❛❝❡ ✐s ♥❛t✉r❛••② ❡q✉✐♣♣❡❞ ✇✐t❤ ✐ts str♦♥❣ ❞✉❛• t♦♣♦•♦❣②✳ ❚❤❡ ✐♥t❡r❡st ♦❢ t❤❡ s♣❛❝❡ D ′ ✐s t❤❛t t❤❡ ♦❜•❡❝ts ♦❢ t❤❡ ▼❛••✐❛✈✐♥ ❝❛•❝✉•✉s✱ ✇❤✐❝❤ ✇❡r❡ ❞❡✜♥❡❞ ♦♥ D✱ ❡①t❡♥❞ t♦ D ′ ❜② ❞✉❛•✐t②✱ •✉st ❛s ♦♣❡r❛t♦rs ♦♥ s♣❛❝❡s ♦❢ t❡st ❢✉♥❝t✐♦♥s ❡①t❡♥❞ t♦ ❞✐str✐❜✉t✐♦♥s✳ ❲❡ ❡①❡♠♣•✐❢② t❤✐s ✐❞❡❛ ✇✐t❤ t❤❡ ❣r❛❞✐❡♥t ❛♥❞ t❤❡ ❞✐✈❡r❣❡♥❝❡✱ st❛rt✐♥❣ ✇✐t❤ t❤❡✿ ❚❤❡♦r❡♠ ✶✳✷✳✼✳✶ ✭❈♦♥t✐♥✉✐t② ♦❢ t❤❡ ❣r❛❞✐❡♥t✮✳ ❋♦r ❡✈❡r② p ∈]1, ∞[ ❛♥❞ s ∈ R✱ ∇ ❛❞♠✐ts ❛ ❝♦♥t✐♥✉♦✉s ❡①t❡♥s✐♦♥✿ ∇ : D p,s → D p,s-1 (H) ✭✶✳✶✺✺✮ Pr♦♦❢✳ ▲❡t X ∈ D✳ ❚❤❡♥ t❤❡r❡ ✐s✿ ∇X D p,s-1 (H) ≤ (Id + L) s-1 2 ∇X L p (H) = ∇ (2 • Id + L) s-1 2 X L p (H) ≤ C • (•Id + L) • (2 • Id + L) s-1 2 X L p ) ≤ C • (Id + L) s 2 X L p = C • X Dp,s ✭✶✳✶✺✻✮ ✸✵ ❈❍❆P❚❊| ✶✳ ■◆❚|❖❉❯❈❚■❖◆ ❚❖ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ❍❡r❡ ✇❡ ❤❛✈❡ ✉s❡❞ t❤❡ ▼❡②❡r ✐♥❡q✉❛•✐t✐❡s ✭❛♥❞ t❤❡✐r ❡①t❡♥s✐♦♥✮✳ ❚❤❡ ♣r♦✈❡ ✐s t❤❡♥ ✜♥✐s❤❡❞ ❛s D ✐s ❞❡♥s❡ ✐♥ ❛♥② ♦❢ t❤❡ D p,s ✳ ❆s ❛♥ ✐♠♠❡❞✐❛t❡ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤✐s ✇❡ ❣❡t✿ ❈♦r♦••❛r② ✶✳✷✳✼✳✶ ✭❲❡❛❦ ❣r❛❞✐❡♥t✮✳ ∇ ❛❞♠✐ts ❛ ❝♦♥t✐♥✉♦✉s ❡①t❡♥s✐♦♥✿ ∇ : D ′ → D ′ (H) ✭✶✳✶✺✼✮ ❆•s♦ ❜② ❞✉❛•✐t② ♦♥❡ ❞❡❞✉❝❡s✿ ❈♦r♦••❛r② ✶✳✷✳✼✳✷ ✭❲❡❛❦ ❞✐✈❡r❣❡♥❝❡✮✳ ❚❤❡ ❞✐✈❡r❣❡♥❝❡ ❛❞♠✐ts ❝♦♥t✐♥✉♦✉s ❡①t❡♥✲ s✐♦♥s✿ δ : D p,s (H) → D p,s-1 ✭✶✳✶✺✽✮ ❛♥❞✿ δ : D ′ → D ′ ✭✶✳✶✺✾✮ ✶✳✷✳✽ ❲❡❛❦ ♠❛rt✐♥❣❛•❡s ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ❞❡✜♥❡ ❛ ♥♦t✐♦♥ ♦❢ ♠❛rt✐♥❣❛•✐t② ❢♦r ✇❡❛❦ ♣r♦❝❡ss❡s✳ ❚❤❡ r❡s✉•ts ✐♥ t❤✐s ♣❛r❛❣r❛♣❤ ♣❡r❤❛♣s ❛r❡ •❡ss ❝•❛ss✐❝ t❤❛♥ t❤♦s❡ ✐♥ t❤❡ r❡st ♦❢ t❤✐s ❝❤❛♣t❡r✱ ❛♥❞ r❛t❤❡r t❤❛♥ ❣✐✈✐♥❣ t❤❡ ♠♦st ❣❡♥❡r❛• ♣♦ss✐❜•❡ st❛t❡♠❡♥ts ✇❡ ❝♦♥t❡♥t ♦✉rs❡•✈❡s ✇✐t❤ ♥♦t✐♦♥s ✇❤✐❝❤ ✇✐•• ❢✉•✜•• ♦✉r ♣✉r♣♦s❡ •❛t❡r ♦♥✳ ▼♦r❡ ❣❡♥❡r❛• r❡s✉•ts ♠❛② ❜❡ ❢♦✉♥❞ ✐♥ ❬✼✼( ♦r ❬✹✾(✳ ❋✐rst ✇❡ ♥❡❡❞ t♦ ❞❡✜♥❡ ❛ ♥♦t✐♦♥ ♦❢ ✇❡❛❦ ♠❡❛s✉r❛❜✐•✐t② ❛♥❞ ❛ ✇❡❛❦ ❝♦♥❞✐t✐♦♥♥❛• ❡①♣❡❝t❛t✐♦♥✳ ❚♦ ❞♦ s♦ ✇❡ s❤♦✇ t❤❛t ✐♥ r❡•❡✈❛♥t ❝❛s❡s t❤❡ str♦♥❣ ❝♦♥❞✐t✐♦♥♥❛• ❡①♣❡❝t❛t✐♦♥s ❝♦♠♠✉t❡s ✇✐t❤ ♦♣❡r❛t♦rs r❡•❛t❡❞ t♦ t❤❡ ❖r♥st❡✐♥✲❯❤•❡♥❜❡❝❦ ♦♣❡r❛t♦r✿ ▲❡♠♠❛ ✶✳✷✳✽✳✶✳ ■❢ X ∈ L p (µ)✱ p ≥ 1 ✐s F t ✲♠❡❛s✉r❛❜•❡ ❛♥❞ s ≥ 0 t❤❡♥ P s X ✐s F t ✲♠❡❛s✉r❛❜•❡ ❛s ✇❡••✳ Pr♦♦❢✳ ❋✐rst ✇❡ ♦❜t❛✐♥ t❤❡ r❡s✉•t ✐❢ X ✐s ✐♥ t❤❡ ♥t❤ ❲✐❡♥❡r ❝❤❛♦s s✐♥❝❡ t❤❡♥ P s X = e -ns X✳ ❚❤❡ r❡s✉•t t❤❡♥ ❤♦•❞s ❢♦r ❛ ♣♦•②♥♦♠✐❛• ❢✉♥❝t✐♦♥ ♦♥ W ✱ ❛♥❞ ✜✲ ♥❛••② ❢♦r ❛♥② X ✐♥ L p ❜② ❞❡♥s✐t② ♦❢ t❤❡ ♣♦•②♥♦♠✐❛•s✳ ▲❡♠♠❛ ✶✳✷✳✽✳✷✳ ■❢ X ∈ L p (µ)✱ p ≥ 1 ❛♥❞ s ≥ 0 t❤❡♥ P s E [X|F t ] = E [P s X|F t ] ✭✶✳✶✻✵✮ Pr♦♦❢✳ ■♥❞❡❡❞ ✐❢ Y ∈ L p ′ ❛♥❞ Y ∈ F t ✇❡ ❤❛✈❡✿ E [P s E [X|F t ] Y ] = E [E [X|F t ] P s Y ] = E [XP s Y ] = E [P s XY ] ✭✶✳✶✻✶✮ ❛♥❞ P s E [X|F t ] ∈ F t ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s r❡s✉•t✱ ✇❤✐❝❤ ❝♦♠♣•❡t❡s t❤❡ ♣r♦♦❢✳ ✶✳✷✳ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ✸✶ ❙✐♠✐•❛r r❡s✉•ts ❢♦r L ♦r ❢♦r (Id + L) k/2 ✱ k ∈ Z ❛r❡ ♦❜t❛✐♥❡❞ ❜② t❤❡ s❛♠❡ ♠❡t❤♦❞ ✐❢ X ✐s r❡❣✉•❛r ❡♥♦✉❣❤✳ ❚❤✐s •❡❛❞s t♦✿ ▲❡♠♠❛ ✶✳✷✳✽✳✸✳ ■❢ X ∈ D p,k ✱ p > 1✱ k ≥ 0 t❤❡♥ E [X|F t ] ∈ D p,k ✳ Pr♦♦❢✳ ■♥❞❡❡❞✱ Y := (Id + L) k/2 X ∈ L p ✱ s♦✿ E [X|F t ] = E (Id + L) -k/2 Y |F t = (Id + L) -k/2 E [Y |F t ] ✭✶✳✶✻✷✮ ❲❡ ♠❛② ♥♦✇ ❞❡✜♥❡ ❛ ♥♦t✐♦♥ ♦❢ ✇❡❛❦ ♠❡❛s✉r❛❜✐•✐t②✿ ❉❡✜♥✐t✐♦♥ ✶✳✷✳✽✳✶ ✭❲❡❛❦ ♠❡❛s✉r❛❜✐•✐t②✮✳ ❲❡ ✇✐•• s❛② t❤❛t T ∈ D ′ ✐s ✇❡❛❦•② F t ✲♠❡❛s✉r❛❜•❡ ✐❢ ∀X ∈ D✱ T, X = T, E [X|F t ] ✭✶✳✶✻✸✮ ■❢ X ✐s ❛ ♣r♦♣❡r r❛♥❞♦♠ ✈❛r✐❛❜•❡ t❤✐s ✐s ❡q✉✐✈❛•❡♥t t♦ ❜❡✐♥❣ ♠❡❛s✉r❛❜•❡✳ ❆•s♦ ✇❡ ❤❛✈❡✿ ▲❡♠♠❛ ✶✳✷✳✽✳✹✳ T ∈ D p,-k ✐s ✇❡❛❦•② F t ✲♠❡❛s✉r❛❜•❡ ✐❢ ❛♥❞ ♦♥•② ✐❢ (Id + L) -k/2 X ✐s ♠❡❛s✉r❛❜•❡✳ Pr♦♦❢✳ ■♥❞❡❡❞ ✐❢ T ✐s ✇❡❛❦•② ♠❡❛s✉r❛❜•❡ t❤❡ ♠❡❛s✉r❛❜✐•✐t② r❡✇r✐t❡s✱ ∀X ∈ D✿ E (Id + L) -k/2 T • (Id + L) k/2 X = E (Id + L) -k/2 T • (Id + L) k/2 E [X|F t ] = E (Id + L) -k/2 T • E (Id + L) k/2 X|F t = E E (Id + L) -k/2 T |F t • (Id + L) k/2 X ✭✶✳✶✻✹✮ ❛♥❞ D ✐s st❛❜•❡ ❜② (Id + L) k/2 ❛♥❞ ❞❡♥s❡ ✐♥ L p ′ ✱ s♦ ✇❡ ❣❡t✿ (Id + L) -k/2 T = E (Id + L) -k/2 T |F t ✭✶✳✶✻✺✮ ❛♥❞ t❤❡r❡❢♦r❡ (Id + L) -k/2 T ✐s F t ✲♠❡❛s✉r❛❜•❡✳ ❈♦♥✈❡rs❡•② ✐❢ (Id + L) -k/2 T ✐s F t ✲♠❡❛s✉r❛❜•❡ (Id + L) -k/2 T = E (Id + L) -k/2 T |F t ❤♦•❞s ❛♥❞ ✇❡ ❝❛♥ ❞♦ t❤❡ ❛❜♦✈❡ ❝♦♠♣✉t❛t✐♦♥s ❜❛❝❦✇❛r❞s t♦ ✜♥❞ t❤❛t T ✐s ✇❡❛❦•② F t ✲♠❡❛s✉r❛❜•❡✳ ✸✷ ❈❍❆P❚❊| ✶✳ ■◆❚|❖❉❯❈❚■❖◆ ❚❖ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ❲❡ ❛•s♦ ❞❡✜♥❡ t❤❡ ❝♦♥❞✐t✐♦♥♥❛• ❡①♣❡❝t❛t✐♦♥ ♦❢ T ∈ D ′ ❜②✿ E [T |F t ] , X = T, E [X|F t ] ✭✶✳✶✻✻✮ ■t ✐s ❡❛s✐•② ❝❤❡❝❦❡❞ t❤❛t t❤✐s ❞❡✜♥✐t✐♦♥ ❡①t❡♥❞s t❤❡ ✉s✉❛• ❝♦♥❞✐t✐♦♥♥❛• ❡①♣❡❝t❛t✐♦♥ ♦♥ L p ❛♥❞ t❤❛t ✐t ✐s ❡q✉✐✈❛•❡♥t t♦ ❞❡✜♥✐♥❣✿ E [T |F t ] = (Id + L) k/2 E (Id + L) -k/2 T |F t ✭✶✳✶✻✼✮ ❛♥❞ ❤❡♥❝❡ t❤❛t t❤❡ ♥❡❣❛t✐✈❡ •r♦ss✲❙♦❜♦•❡✈ s♣❛❝❡s ❛•s♦ ❛r❡ •❡❢t st❛❜•❡ ❜② t❤❡ ✇❡❛❦ ❝♦♥❞✐t✐♦♥♥❛• ❡①♣❡❝t❛t✐♦♥✳ ❲❡ ♥♦✇ ♠❛② ❞❡✜♥❡ ❛ ✇❡❛❦ ♠❛rt✐♥❣❛•❡ ❛s ❛ ✇❡❛❦ ❛❞❛♣t❡❞ ♣r♦❝❡ss M ✈❡r✐❢②✐♥❣ t❤❡ ♠❛rt✐♥❣❛•❡ ❡q✉❛•✐t②✿ E [M t |F s ] = M s ✭✶✳✶✻✽✮ ❢♦r s ≤ t✳ ■❢ M ✐s ❛ ✇❡❛❦ ♠❛rt✐♥❣❛•❡ ❛♥❞ M t ∈ D p,-k ✱ ✇❡ ♥♦t❡ t❤❛t✿ E (Id + L) -k/2 M t |F s = (Id + L) -k/2 M s ✭✶✳✶✻✾✮ ❢r♦♠ ✇❤✐❝❤ ✇❡ ❞❡❞✉❝❡ t❤❡ t✇♦ ❢♦••♦✇✐♥❣ r❡s✉•ts✿ ▲❡♠♠❛ ✶✳✷✳✽✳✺✳ ■❢ M ✐s ❛ ✇❡❛❦ ♠❛rt✐♥❣❛•❡ ❛♥❞ M t ∈ D p,-k t❤❡♥ ∀s ≤ t✱ M s ∈ D p,-k ❛♥❞ M s D p,-k ≤ M t D p,-k ✳ ▲❡♠♠❛ ✶✳✷✳✽✳✻✳ ❆ ✇❡❛❦ ♣r♦❝❡ss (M t ) 0≤t≤1 s✉❝❤ t❤❛t M 1 ∈ D p,-k ✐s ❛ ✇❡❛❦ ♠❛r✲ t✐♥❣❛•❡ ✐❢ ❛♥❞ ♦♥•② ✐❢ t❤❡ ♣r♦❝❡ss (Id + L) -k/2 M t 0≤t≤1 ✐s ❛♥ L p ♠❛rt✐♥❣❛•❡✳ ❲❡ ♥♦✇ s❤♦✇ ❤♦✇ ✇❡ ♠❛② ❞❡✜♥❡ ❇r♦✇♥✐❛♥ ✐♥t❡❣r❛•s ♦❢ ✇❡❛❦ ❛❞❛♣t❡❞ ♣r♦❝❡ss❡s ❛♥❞ ❤♦✇ t❤❡s❡ ♣r♦✈✐❞❡ ❛♥ ❡①❛♠♣•❡ ♦❢ ✇❡❛❦ ♠❛rt✐♥❣❛•❡s✳ ❋✐rst ✇❡ r❡❝❛•• t❤❛t t❤❡ ❞✐✈❡r❣❡♥❝❡ ❛s ❞❡✜♥❡❞ ♦♥ D p,1 (H) ❡①t❡♥❞s t❤❡ st♦❝❤❛st✐❝ ✐♥t❡❣r❛•s t♦ ♥♦♥ ❛❞❛♣t❡❞ ♣r♦❝❡ss❡s ❛♥❞ t❤❛t t❤❡ ❞✐✈❡r❣❡♥❝❡ ❡①t❡♥❞s ❜② ❞✉❛•✐t② t♦ D ′ (H)✳ ❲❡ ✇✐•• ✉s❡ t❤❡ ❢♦••♦✇✐♥❣ ♥♦t❛t✐♦♥✿ ✐❢ f ∈ D ′ (H)✱ ✇❡ ♥♦t❡ f |[0,t] t❤❡ ❡•❡♠❡♥t ♦❢ D ′ (H) ✇❤♦s❡ t✐♠❡ ❞❡r✐✈❛t✐✈❡ ✐s ḟ ❜❡❢♦r❡ t ❛♥❞ 0 ❛❢t❡r t ❛♥❞ ✇❡ ✐♥tr♦❞✉❝❡✿ t 0 ḟ (u)dW u := δ f |[0,t] ✳ ◆♦✇ •❡t ✉s s✉♣♣♦s❡ t❤❛t f ∈ D p,-k (H)✳ ❚❤❡♥ f |[0,t] ∈ D p,-k (H) ❛♥❞ ✇❡ ❦♥♦✇ t❤❛t✿ (Id + L) -k/2 δ f |[0,t] = δ (2 • Id + L) -k/2 f |[0,t] ✭✶✳✶✼✵✮ ▲❡t ✉s ♥♦t❡ M t ❢♦r t❤✐s q✉❛♥t✐t②✳ ❈•❡❛r•②✱ ✐❢ f ✐s ✇❡❛❦•② ❛❞❛♣t❡❞✱ M t = t 0 (2 • Id + L) -k/2 ḟ (u) dW u ✭✶✳✶✼✶✮ ✐s ❛ ♠❛rt✐♥❣❛•❡✱ s♦ ✇❡ ❤❛✈❡ ♣r♦✈❡❞✿ Pr♦♣♦s✐t✐♦♥ ✶✳✷✳✽✳✶✳ ■❢ f ∈ D ′ (H) ✐s ❛ ✇❡❛❦•② ❛❞❛♣t❡❞ ♣r♦❝❡ss✱ t❤❡♥ t❤❡ ✇❡❛❦ ♣r♦❝❡ss t 0 ḟ (u)dW u t ✐s ❛ ✇❡❛❦ ♠❛rt✐♥❣❛•❡✳ ✶✳✷✳ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ✸✸ ✶✳✷✳✾ ▼❛••✐❛✈✐♥ ❝❛•❝✉•✉s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ■t ✐s ♣♦ss✐❜•❡ t♦ ❡①t❡♥❞ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ♠♦st ♦❢ t❤❡ ♠❛t❤❡♠❛t✐❝❛• ♦❜•❡❝ts ✇❡ ❤❛✈❡ r❡✈✐❡✇❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r t♦ t❤❡ ❝❛s❡ ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜•❡s t❛❦✐♥❣ ✈❛•✉❡s ✐♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡ X ✐❢ t❤❛t s♣❛❝❡ ❡♥•♦②s t❤❡ s♦✲❝❛••❡❞ ✉♥✐❢♦r♠ ♠❛rt✐♥❣❛•❡✲❞✐✛❡r❡♥❝❡ ♣r♦♣❡rt②✱ ❯▼❉ ❢♦r s❤♦rt✿ ❉❡✜♥✐t✐♦♥ ✶✳✷✳✾✳✶ ✭❯▼❉ ♣r♦♣❡rt②✮✳ ❆ ❇❛♥❛❝❤ s♣❛❝❡ X ✐s s❛✐❞ t♦ ❤❛✈❡ t❤❡ ❯▼❉ ♣r♦♣❡rt② ✐❢ ❢♦r s♦♠❡ p ∈]1, ∞[✱ t❤❡r❡ ❡①✐sts ❛ ✉♥✐✈❡rs❛• ❝♦♥st❛♥t C(p, X) s✉❝❤ t❤❛t ❢♦r ❡✈❡r② ✭❞✐s❝r❡t❡✮ X✲✈❛•✉❡❞✱ L p ♠❛rt✐♥❣❛•❡✱ ❢♦r ❡✈❡r② ǫ ∈ {-1, 1} N ❛♥❞ ❡✈❡r② N ∈ N t❤❡r❡ ✐s✿ E N n=1 ǫ n (M n -M n-1 ) p X ≤ C(p, X) • E [ M N -M 0 p X ] ✭✶✳✶✼✷✮ ❲❡ ✇✐•• ♥♦t ♣r♦✈✐❞❡ ❛ ❢✉•• r❡✈✐❡✇ ♦❢ ❯▼❉ s♣❛❝❡s ❛♥❞ t❤❡✐r ❛♣♣•✐❝❛t✐♦♥s❀ ✐♥st❡❛❞ ✇❡ s✉❣❣❡st t❤❡ s✉r✈❡② ❬✺✽( ❛♥❞ t❤❡ r❡❢❡r❡♥❝❡s t❤❡r❡✐♥✳ ❲❡ ✇✐•• ❝♦♥t❡♥t ♦✉r❡•✈❡s ✇✐t❤ ✉s✐♥❣ t❤❡ ❢❛❝t t❤❛t t❤❡ ♠♦st ✐♠♣♦rt❛♥t r❡s✉•ts ✐♥ ▼❛••✐❛✈✐♥ ❝❛•❝✉•✉s✱ ❡s♣❡✲ ❝✐❛••② t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❣r❛❞✐❡♥t ❛♥❞ t❤❡ ▼❡②❡r ✐♥❡q✉❛•✐t✐❡s✱ ❤♦•❞ ❢♦r ❯▼❉ ✈❛•✉❡❞ r❛♥❞♦♠ ✈❛r✐❛❜•❡s✳ ❚❤❡ ✐♥t✉✐t✐♦♥ ❜❡❤✐♥❞ t❤✐s ✐s t❤❛t t❤❡ ❯▼❉ ♣r♦♣❡rt② ✐s ❛ ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❢♦r t❤❡ ❍✐•❜❡rt tr❛♥s❢♦r♠ t♦ ❜❡ ❜♦✉♥❞❡❞ ♦♥ t❤❛t s♣❛❝❡✳ ❲❡ ❛•s♦ ♠❡♥t✐♦♥ ❡①❛♠♣•❡s ♦❢ ❇❛♥❛❝❤ s♣❛❝❡s ❡♥•♦②✐♥❣ t❤❡ ❯▼❉ ♣r♦♣❡rt②✿ ❍✐•❜❡rt s♣❛❝❡s ♦♥ ♦♥❡ ❤❛♥❞ ✭✇❡ ❤❛✈❡ ❛•r❡❛❞② ❝♦♥tr✉❝t❡❞ t❤❡ ▼❛•✐❛✈✐♥ ❝❛•❝✉•✉s ♦♥ ❍✐•❜❡rt s♣❛❝❡s✮ ❛♥❞ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞ t❤❡ ❙♦❜♦•❡✈ s♣❛❝❡s W p,s ❢♦r 1 < p < ∞✱ ✇❤✐❝❤ ✇✐•• ❜❡ ♦✉r ❝❛s❡ ♦❢ ❛♣♣•✐❝❛t✐♦♥✳ ❚❤❡ C k b ✱ ❤♦✇❡✈❡r✱ ❛r❡ ♥♦t ❯▼❉✳ ✸✹ ❈❍❆P❚❊| ✶✳ ■◆❚|❖❉❯❈❚■❖◆ ❚❖ ▼❆▲▲■❆❱■◆ ❈❆▲❈❯▲❯❙ ❈❤❛♣t❡r ✷ ❚♦♣♦•♦❣✐❝❛• str✉❝t✉r❡ ♦❢ S ❛♥❞ S ′ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ r❡❝❛•• t❤❡ r❡s✉•ts ❛❜♦✉t t❤❡ t♦♣♦•♦❣② ♦❢ S ❛♥❞ S ′ ✇❤✐❝❤ ✇❡ ✇✐•• ♥❡❡❞ ✐♥ t❤❡ s❡q✉❡•✳ ❋✐rst✱ s❡✈❡r❛• ❢❛♠✐•✐❡s ♦❢ s❡♠✐✲♥♦r♠s ❛r❡ ✐♥tr♦❞✉❝❡❞ ❛♥❞ ✐t ✐s ♣r♦✈❡❞ t❤❛t t❤❡② ❛•• ❡♥❞♦✇ t❤❡ ❙❝❤✇❛rt③ ❝•❛ss ✇✐t❤ t❤❡ s❛♠❡ ✭✉s✉❛•✮ ❋ré❝❤❡t s♣❛❝❡ t♦♣♦•♦❣②✳ ❲❡ t❤❡♥ r❡❝❛•• s♦♠❡ ✐♠♣♦rt❛♥t ♣r♦♣❡rt✐❡s ♦❢ t❤✐s t♦♣♦•♦❣②✳ ✷✳✶ ❚✇♦ st❛♥❞❛r❞ ❢❛♠✐•✐❡s ♦❢ s❡♠✐ ♥♦r♠s ♦♥ S ❋♦r ❛ ♠✉•t✐✲✐♥❞❡① α ∈ N N ❛♥❞ x ∈ R N ✱ ✇❡ ✇✐•• ✉s❡ t❤❡ ❢♦••♦✇✐♥❣ ♥♦t❛t✐♦♥✿ α! = α 1 • • • α N x α = x α 1 1 • • • x α N N D α = ∂ α 1 1 • • • ∂ α N N ❚❤❡♥ t❤❡r❡ ✐s ❛ ❝•❛ss✐❝ ❢❛♠✐•② ♦❢ s❡♠✐✲♥♦r♠s ❞❡✜♥❡❞ ❜② t❤❡ ❢♦••♦✇✐♥❣ ❡q✉❛t✐♦♥ ❢♦r α, β ∈ N N ✿ f (∞) α,β = sup x∈R N |x α D β f (x)| ✭✷✳✶✮ ❲❡ r❡❝❛•• t❤❛t t❤❡ ❙❝❤✇❛rt③ ❝•❛ss✱ ❢♦r ✇❤✐❝❤ ✇❡ ♥♦t❡ S(R N )✱ ✐s ❞❡✜♥❡❞ ❛s t❤❡ s♣❛❝❡ ♦❢ t❤♦s❡ ❢✉♥❝t✐♦♥s ♦♥ R N s✉❝❤ t❤❛t ❢♦r ❡✈❡r② α, β ∈ N N ✿ f (∞) α,β < ∞✳ ❚❤❡♥ t❤✐s ❢❛♠✐•② ♦❢ s❡♠✐✲♥♦r♠s ♥❛t✉r❛••② ❞❡✜♥❡ ❛ ❋ré❝❤❡t s♣❛❝❡ t♦♣♦•♦❣② ✭s❡❡ ❛♣♣❡♥❞✐① ❆✮ t❤r♦✉❣❤ t❤❡ ❞✐st❛♥❝❡✿ d(f, g) = ∞ α=0 ∞ β=0 1 2 α+β • f -g (∞) α,β 1 + f -g (∞) α,β ✭✷✳✷✮ ❲❡ ❛•s♦ ✐♥tr♦❞✉❝❡ t❤❡ ❢♦••♦✇✐♥❣✱ ❢♦r α, β ∈ N N ✿ f (p) α,β = x → x α D β f (x) L p ✭✷✳✸✮ ❛♥❞ ✇❡ ❤❛✈❡ t❤❡✿ ✸✺ ✸✻ ❈❍❆P❚❊| ✷✳ ❚❖P❖▲❖•■❈❆▲ ❙❚|❯❈❚❯|❊ ❖❋ S ❆◆❉ S ′ Pr♦♣♦s✐t✐♦♥ ✷✳✶✳✵✳✶✳ ❋♦r p, q ∈ [1, ∞]✱ t❤❡ ❢❛♠✐•✐❡s ( • (p) α,β , α, β ∈ N N ) ❛♥❞ ( • (q) α,β , α, β ∈ N N ) ❞❡✜♥❡ t❤❡ s❛♠❡ t♦♣♦•♦❣② ♦♥ S R N ✳ Pr♦♦❢✳ ■♥❞❡❡❞✱ ♦♥ ♦♥❡ ❤❛♥❞✱ ♠✉•t✐♣•②✐♥❣ ❛♥❞ ❞✐✈✐❞✐♥❣ ❜② (1 + |x| 2 ) s ②✐❡•❞s✿ f L p ≤ dx (1 + |x| 2 ) s 1/p • sup x∈R N |(1 + |x| 2 ) s/p f (x)| ✭✷✳✹✮ ❛♥❞ t❤❡ ✐♥t❡❣r❛• ❛❜♦✈❡ ✐s ✜♥✐t❡ ❢♦r ❜✐❣ ❡♥♦✉❣❤ s(N )✳ p ❛♥❞ s✉❝❤ ❛♥ s ❜❡✐♥❣ ✜①❡❞✱ t❤❡ ❡①✐sts ❛ ♣♦•②♥♦♠✐❛• ❞♦♠✐♥❛t✐♥❣ (1 + |x| 2 ) s/p ✳ ❍❡♥❝❡ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t C(N, p) ❛♥❞ ❛♥ ✐♥t❡❣❡r k(N, p) s✉❝❤ t❤❛t✿ f (p) α,β ≤ C |γ|≤k f (∞) α+γ,β ✭✷✳✺✮ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ♦♥❡ ❝❛♥ ✉s❡ t❤❡ ❢♦••♦✇✐♥❣ ✭♥♦♥✲♦♣t✐♠❛•✱ ❜✉t s✉✣❝✐❡♥t•② ♣r❡❝✐s❡ ❢♦r ♦✉r ✉s❡✮ ❙♦❜♦•❡✈ ❡♠❜❡❞❞✐♥❣ r❡s✉•t❀ s❡❡✱ ❢♦r ❡①❛♠♣•❡✱ ❬✶(✿ Pr♦♣♦s✐t✐♦♥ ✷✳✶✳✵✳✷ ✭❙♦❜♦•❡✈ ❡♠❜❡❞❞✐♥❣✮✳ ■❢ k > N/p✱ W k,p ֒→ C 0 ✳ ✇❤✐❝❤✱ ❝❤♦♦s✐♥❣ ❛ ❜✐❣ ❡♥♦✉❣❤ ✐♥t❡❣❡r k ❛t ✜①❡❞ p ❛♥❞ N ❛♥❞ ✉s✐♥❣ t❤❡ ▲❡✐❜♥✐③ ❢♦r♠✉•❛✱ ♣r♦✈❡s t❤❛t t❤❡r❡ ❡①✐sts ❝♦♥st❛♥ts C i (p, N ) s✉❝❤ t❤❛t✿ f (∞) α,β ≤ C 1 • x α D β f W k,p ≤ C 2 • |γ|≤k,δ≤α f (p) δ,β+γ ✭✷✳✻✮ ❚❤✐s ❝♦♥❝•✉❞❡s ♦✉r ♣r♦♦❢✳ ✷✳✷ ❚❤❡ ♦♣❡r❛t♦r K ❛♥❞ ❛ss♦❝✐❛t❡❞ s❡♠✐ ♥♦r♠s ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✐♥tr♦❞✉❝❡ ❢❛♠✐•✐❡s ♦❢ s❡♠✐✲♥♦r♠s ✇❤✐❝❤ ❞❡✜♥❡ t❤❡ s❛♠❡ t♦♣♦•♦❣② ♦♥ S t❤❛♥ t❤❡ ♦♥❡s ✇❡ ❤❛✈❡ st✉❞✐❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s ♣❛r❛❣r❛♣❤✳ ❚❤❡ ♣✉r♣♦s❡ ♦❢ t❤❡s❡ ✐s t❤❛t t❤❡s❡ •❡ss ❝•❛ss✐❝❛• ❢❛♠✐•✐❡s ✇✐•• ❜❡ ♠♦r❡ ❝♦♥✈❡♥✐❡♥t ❢♦r t❤❡ st✉❞② ♦❢ S ′ ✳ ❚❤✐s ✇✐•• ❜❡ ✉s❡❢✉• ❛s ✇❡ t✉r♥ t♦ t❤❡ •✐❢t✐♥❣ ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜•❡s ✐♥ t❤❡ ♥❡①t ❝❤❛♣t❡r✳ ❙✉❝❤ r❡s✉•ts ❛r❡ ✇❡••✲❦♥♦✇♥ ✭❝❢✱ ❢♦r ❡①❛♠♣•❡✱ ❬✽✶( ♦r ❬✷✺(✮ ❜✉t ♥♦t ✈❡r② ❞♦❝✉♠❡♥t❡❞✱ s♦ ✇❡ ♣r♦✈✐❞❡ ❞❡t❛✐•s ❢♦r t❤❡ s❛❦❡ ♦❢ ❝♦♠♣•❡t❡♥❡ss✳ ❆•• ♦✉r st✉❞② ✇✐•• ❜❡ ❜❛s❡❞ ♦♥ t❤❡ ❢♦••♦✇✐♥❣ ❞✐✛❡r❡♥t✐❛• ♦♣❡r❛t♦r✿ K = Id -∆ + |x| 2 ✭✷✳✼✮ ✐❡✿ Kf (x) = f (x) -∆f (x) + |x| 2 f (x) ✭✷✳✽✮ ❲❡ ❛•s♦ ✐♥tr♦❞✉❝❡✿ K = K -Id ✭✷✳✾✮ ▲❡t ✉s ♥♦t❡ t❤❛t K ❛♥❞ K ❛r❡ ❝♦♥t✐♥✉♦✉s ♦♣❡r❛t♦rs ♦♥ S✳ ❖✉r ❛❣❡♥❞❛ ✐s t♦ ❢♦r♠❛••② ❞❡✜♥❡ ❢r❛❝t✐♦♥❛• ♣♦✇❡rs ♦❢ K ❛♥❞ t♦ st✉❞② t❤❡✐r r❡❣✉•❛r✐t②✳ ✷✳✷✳ ❚❍❊ ❖P❊|❆❚❖| K ❆◆❉ ❆❙❙❖❈■❆❚❊❉ ❙❊▼■ ◆❖|▼❙ ✸✼ ✷✳✷✳✶ ❙❡♠✐ ♥♦r♠s ❜✉✐•t ♦♥ L 2 ❲❡ st❛rt ✇✐t❤ t❤❡ ❢♦••♦✇✐♥❣ ♦❜s❡r✈❛t✐♦♥✿ K ✐s s②♠♠❡tr✐❝ ❛♥❞ ♣♦s✐t✐✈❡ ❢♦r t❤❡ ❍✐•❜❡rt s♣❛❝❡ str✉❝t✉r❡ ✐♥❞✉❝❡❞ ❜② L 2 ♦♥ S✱ ✐❡ ❢♦r ❡✈❡r② f, g ∈ S✿ (Kf ) • g = f • (Kg) ✭✷✳✶✵✮ ❛♥❞✿ (Kf ) • f = (1 + |x| 2 ) • |f | 2 + |Df | 2 ≥ |f | 2 ✭✷✳✶✶✮ ■t ✐s t❤❡r❡❢♦r❡ ♥❛t✉r❛• t♦ tr② ❛♥❞ ❞✐❛❣♦♥❛•✐③❡ K ❛♥❞ ✐♥❞❡❡❞ ✇❡ ♣r♦✈❡ t❤❛t t❤❡r❡ ❡①✐sts ❛♥ ♦rt❤♦❣♦♥❛• ❜❛s✐s ♦❢ L 2 ♠❛❞❡ ♦❢ ❡•❡♠❡♥ts ♦❢ S(R N ) ✇❤✐❝❤ ❛r❡ ❡✐❣❡♥✈❡❝t♦rs ♦❢ K✳ ❚❤❡s❡ ❡✐❣❡♥✈❡❝t♦rs ❛r❡ ❝•♦s❡•② r❡•❛t❡❞ t♦ t❤❡ s♦✲❝❛••❡❞ ❍❡r♠✐t❡ ❢✉♥❝t✐♦♥s✳ ❲❡ ❤❛✈❡ r❡❝❛••❡❞ t❤❡ ❢❛❝ts ✇❡ ♥❡❡❞ ♦♥ ❍❡r♠✐t❡ ♣♦•②♥♦♠✐❛•s ✐♥ ❝❤❛♣t❡r ✶ ❛♥❞ ♥♦✇✱ ❢♦r n ∈ N N ✱ ✇❡ ✐♥tr♦❞✉❝❡✿ fn (x) = H n ( √ 2x) exp(-|x| 2 /2) ✭✷✳✶✷✮ ■t ✐s ♥♦t❡✇♦rt❤② t❤❛t✱ ❛❜✉s✐✈❡•② ✉s✐♥❣ t❤❡ s❛♠❡ ♥♦t❛t✐♦♥ ❢♦r t❤❡ ❍❡r♠✐t❡ ❢✉♥❝✲ t✐♦♥s ✐♥ ❞✐♠❡♥s✐♦♥s 1 ❛♥❞ N ✿ fn (x) = fn 1 (x 1 ) . . . fn N (x N ) ✭✷✳✶✸✮ ❚❤❡s❡ ❜❡•♦♥❣ t♦ t❤❡ ❙❝❤✇❛rt③ ❝•❛ss ❛♥❞ ❛s ❝❛♥ ❜❡ ❝❤❡❝❦❡❞ ✉s✐♥❣ t❤❡ P❉❊ ✈❡r✐✜❡❞ ❜② t❤❡ ❍❡r♠✐t❡ ♣♦•②♥♦♠✐❛•s✱ t❤❡② ❛r❡ ❡✐❣❡♥❢✉♥❝t✐♦♥s ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛• ♦♣❡r❛t♦r K ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❡✐❣❡♥✈❛•✉❡✿ λ n = 2|n| + N + 1 ≥ 1 > 0 ✭✷✳✶✹✮ r❡s♣❡❝t✐✈❡•②✳ ■♥ ❞✐♠❡♥s✐♦♥ ✶✱ t❤❡ fn ❢♦r♠ ❛♥ ♦rt❤♦❣♦♥❛• ❢❛♠✐•② s✐♥❝❡ ❡❛❝❤ fn ✐s ❛♥ ❡✐❣❡♥✈❡❝t♦r ❛ss♦❝✐❛t❡❞ t♦ ❛ ❞✐✛❡r❡♥t ❡✐❣❡♥✈❛•✉❡ ♦❢ t❤❡ s②♠♠❡tr✐❝ ♦♣❡r❛t♦r K✳ ❚❤❡ ♦rt❤♦❣♦♥❛•✐t② ❡①t❡♥❞s t♦ ♠✉•t✐♣•❡ ❞✐♠❡♥s✐♦♥s ❛s t❤❡ r❡•❡✈❛♥t ♠✉•t✐♣•❡ ✐♥t❡❣r❛• ❝❛♥ ❜❡ s❡♣❛r❛t❡❞ ✐♥t♦ ❛ ♣r♦❞✉❝t ♦❢ ♠✉•t✐♣•❡ ✐♥t❡❣r❛•s ✉s✐♥❣ ✭✷✳✶✸✮✳ ❯s✐♥❣ t❤✐s ❢❛❝t ✐t ✐s ❡❛s② t♦ ❝♦♠♣✉t❡ t❤❡ L 2 ♥♦r♠ ♦❢ fn ✱ ❤❡♥❝❡ t❤❡ ❢♦••♦✇✐♥❣ ❢✉♥❝t✐♦♥s ❢♦r♠ ❛♥ ♦rt❤♦♥♦r♠❛• ❢❛♠✐•②✿ f n (x) = fn (x) π N/4 √ n! = H n ( √ 2x) exp(-|x| 2 /2) π N/4 √ n! ✭✷✳✶✺✮ ❋✐♥❛••②✱ (f n ) ✐s ❛♥ ♦rt❤♦❣♦♥❛• ❜❛s✐s ♦❢ L 2 ✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ❝❤❡❝❦❡❞ ❜② ✉s✐♥❣ t❤❡ ❢❛❝t t❤❡ t❤❡ ♥♦r♠❛•✐③❡❞ ❍❡r♠✐t❡ ♣♦•②♥♦♠✐❛•s ❛r❡ ❛♥ ♦rt❤♦❣♦♥❛• ❜❛s✐s ♦❢ L 2 (µ N ) ❛♥❞ t❤❛t t❤❡ ❢♦••♦✇✐♥❣ ✐s ❛♥ ✐s♦♠❡tr✐❝ ✐s♦♠♦r♣❤✐s♠ ♠❛♣♣✐♥❣ f n ♦♥ H n / √ n!✿ L 2 -→ L 2 (µ) f -→ x → π N/4 e |x| 2 /4 f (x/ √ 2) ✭✷✳✶✻✮ ❛♥❞ ✇❡ ❤❛✈❡ ♣r♦✈❡❞ t❤❡✿ ✸✽ ❈❍❆P❚❊| ✷✳ ❚❖P❖▲❖•■❈❆▲ ❙❚|❯❈❚❯|❊ ❖❋ S ❆◆❉ S ′ ❚❤❡♦r❡♠ ✷✳✷✳✶✳✶ ✭|❡❞✉❝t✐♦♥ ♦❢ K✮✳ (f n ) ✐s ❛♥ ♦rt❤♦♥♦r♠❛• ❜❛s✐s ♦❢ L 2 ♠❛❞❡ ♦❢ ❡•❡♠❡♥ts ♦❢ S(R N )✳ ▼♦r❡♦✈❡r f n ✐s ❛♥ ❡✐❣❡♥✈❡❝t♦r ♦❢ K ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❡✐❣❡♥✈❛•✉❡ λ n = 2|n| + N + 1✳ ◆♦✇✱ ✇❡ ♠❛② ❞❡❝♦♠♣♦s❡ ❛♥② f ∈ S ✭♦r ♠♦r❡ ❣❡♥❡r❛••②✿ ❛♥② f ∈ L 2 ✮ ❛s✿ f = n∈N N c n (f ) • f n ✭✷✳✶✼✮ ❛♥❞ t❤❡♥✿ f L 2 = n∈N N c n (f ) 2 ✭✷✳✶✽✮ ❚❤✐s ❛••♦✇s ✉s t♦ ❞❡✜♥❡ t❤❡ ❢r❛❝t✐♦♥❛• ♣♦✇❡r K s ❢♦r s ∈ R ✐♥ t❤❡ ❢♦••♦✇✐♥❣ ✇❛②✿ K s f = n∈N N c n (f ) • λ s n • f n ✭✷✳✶✾✮ ♦♥ t❤❡ ✈❡❝t♦r s♣❛❝❡ dom L 2 (K ∫ ) ♦❢ t❤♦s❡ f ❢♦r ✇❤✐❝❤ t❤✐s q✉❛♥t✐t② ✐s ✇❡•• ❞❡✜♥❡❞ ✭s❡❡ ❛♣♣❡♥❞✐① ❈ ❢♦r t❤❡ ❞❡t❛✐•s✮✳ ❲❡ ❛•s♦ ✐♥tr♦❞✉❝❡ t❤❡ ♥♦r♠✿ f S 2,s = K s/2 f L 2 ✭✷✳✷✵✮ ❛♥❞ t❤❡ s♣❛❝❡ S 2,s = dom L 2 (K s/2 ) ✐s t❤❡ ❝♦♠♣•❡t✐♦♥ ♦❢ S ❢♦r t❤✐s ♥♦r♠✳ ❲❤❡♥ t❤✐s ✐s ❞❡✜♥❡❞✱ t❤❡r❡ ✐s✿ f S 2,s = n∈N N c n (f ) 2 • λ s n ✭✷✳✷✶✮ ❲❡ ♥♦✇ st✉❞② t❤❡ ❡✛❡❝t ♦❢ ❞❡r✐✈✐♥❣ ♦r ♠✉•t✐♣•②✐♥❣ ❜② ❛ ♠♦♥♦♠✐❛• ❛♥ ❡•❡♠❡♥t ♦❢ ♦✉r ❖◆❇❀ •❡t ✉s ♥♦t❡ ǫ i ❢♦r t❤❡ N ✲t✉♣•❡ ✇✐t❤ ❛ 1 ✐♥ t❤❡ ✐t❤ ❝♦♦r❞✐♥❛t❡ ❛♥❞ 0 ❡•s❡✇❤❡r❡✱ t❤❡♥✿ x i f n (x) = - n i 2 f n-ǫ i (x) + n i + 1 2 f n+ǫ i (x) ✭✷✳✷✷✮ ❛♥❞ ∂ i f n (x) = 3 √ 2n i 2 f n-ǫ i (x) - n i + 1 2 f n+ǫ i (x) ✭✷✳✷✸✮ ❖♥❝❡ ❛❣❛✐♥ t❤❡ ❞❡t❛✐•s ♦❢ t❤❡ ❝♦♠♣✉t❛t✐♦♥s r❡•② ♦♥ t❤❡ r❡❝✉rs✐♦♥ ✈❡r✐✜❡❞ ❜② t❤❡ ❍❡r♠✐t❡ ♣♦•②♥♦♠✐❛•s✳ ❯s✐♥❣ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✭✷✳✶✼✮ ✐♥ ❝♦♠❜✐♥❛t✐♦♥ ✇✐t❤ t❤❡ t✇♦ ❡q✉❛t✐♦♥s ❛❜♦✈❡ ♦♥❡ ♠❛② ❡①♣•✐❝✐t❡•② ❝♦♠♣✉t❡ t❤❡ ❝♦❡✣❝✐❡♥ts m n ❛♥❞ d n s✉❝❤ t❤❛t✿ x • f (x) = n∈N N m n f n (x) ✭✷✳✷✹✮ ✷✳✷✳ ❚❍❊ ❖P❊|❆❚❖| K ❆◆❉ ❆❙❙❖❈■❆❚❊❉ ❙❊▼■ ◆❖|▼❙ ✸✾ ❛♥❞ ∂ i f (x) = n∈N N d n f n (x) ✭✷✳✷✺✮ ❚❤❡♥ ♦♥❡ ♦❜t❛✐♥s t❤❡ ❢♦••♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✷✳✷✳✶✳✷✳ ❚❤❡r❡ ❡①✐sts ❛ ✉♥✐✈❡rs❛• ❝♦♥st❛♥t C(N ) s✉❝❤ t❤❛t t❤❡ ❢♦••♦✇✐♥❣ ❝♦♥tr♦•s ❤♦•❞✿ x i • f L 2 ≤ C • f S 2,1 ✭✷✳✷✻✮ ∂ i • f L 2 ≤ C • f S 2,1 ✭✷✳✷✼✮ ❋r♦♠ ✇❤✐❝❤ ♦♥ ❞❡❞✉❝❡s t❤❡✿ ❚❤❡♦r❡♠ ✷✳✷✳✶✳✸✳ ❚❤❡ ❢❛♠✐•✐❡s ♦❢ s❡♠✐ ♥♦r♠s ( • (2) α,β , α, β ∈ N N ) ❛♥❞ ( • S 2,s , s ∈ N) ❞❡✜♥❡ t❤❡ s❛♠❡ t♦♣♦•♦❣② ♦♥ S R N ✳ ❲❡ ❞♦ ♥♦t ✇♦r❦ ♦✉t t❤❡ ❞❡t❛✐•s ♦❢ t❤❡ ♣r♦♦❢✱ ✜rst t❤❡② ♠❛② ❜❡ ❢♦✉♥❞ ✐♥ ❬✺✾(✱ s❡❝♦♥❞ ❜❡❝❛✉s❡ ✐♥ t❤❡ ♥❡①t ♣❛r❛❣r❛♣❤ ✇❡ ✇✐•• ♣r♦✈❡ r❡s✉•ts ✐♥ t❤❡ L p ❢r❛♠❡✇♦r❦ ✇❤✐❝❤ ❛r❡ ❛♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ L 2 ❝❛s❡ ✇❡ st✉❞✐❡❞ ✐♥ t❤✐s ♣❛r❛❣r♣❤✳ ✷✳✷✳✷ ❙❡♠✐ ♥♦r♠s ❜✉✐•t ♦♥ L p ✱ 1 < p < ∞ ■♥ ♦r❞❡r t♦ ❞❡✜♥❡ t❤❡ ❢r❛❝t✐♦♥❛• ♣♦✇❡rs ♦❢ K✱ ✇❡ ✇✐•• ♣r♦✈❡ t❤❛t t❤✐s ♦♣❡r❛t♦r ♠❛② ❜❡ ✉♥❞❡rst♦♦❞ ❛s t❤❡ ❣❡♥❡r❛t♦r ♦❢ ❛ s❡♠✐❣r♦✉♣ ♦♥ L p ❀ t❤❡♥ ✇❡ ✇✐•• ❜❡ ❛❜•❡ t♦ ✉s❡ t❤❡ r❡s✉•ts ✐♥ ❛♣♣❡♥❞✐① ❈✳ ❚❤❡ ❢♦••♦✇✐♥❣ r❡s✉•ts ❛r❡ ❞❡t❛✐•❡❞ ✐♥ ❬✷✺(✱ ❢♦r ❡①❛♠♣•❡✳ ❋✐rst ✇❡ ♥♦t❡ t❤❛t K ✐s t❤❡ ❣❡♥❡r❛t♦r ♦❢ ❛ s❡♠✐❣r♦✉♣ K t ♦♥ L p ❀ t❤❡ tr❛♥s✐t✐♦♥ ❞❡♥s✐t② k t ♦❢ K t ✐s ❣✐✈❡♥ ❜②✿ k t (x, y) = n∈N N exp(-λ n • t) • f n (x) • f n (y) ✭✷✳✷✽✮ ■♥ t❤❡ ❝❛s❡ ♦❢ ❞✐♠❡♥s✐♦♥ 1✱ ❜② ▼❡❤•❡r✬s ❢♦r♠✉•❛ t❤✐s ✐s✿ k (1) t (x, y) = exp(-2t) • ∞ n=0 (exp(-2t)) n • f n (x) • f n (y) = e -4t π(1 -e -4t ) • exp 4xye -2t -(x 2 + y 2 )(1 + e -4t ) 2(1 -e -4t ) = e -t 2 sinh(2t) • exp - 1 2 • coth(2t) • x 2 - 2xy cosh(2t) + y 2 ✭✷✳✷✾✮ ✹✵ ❈❍❆P❚❊| ✷✳ ❚❖P❖▲❖•■❈❆▲ ❙❚|❯❈❚❯|❊ ❖❋ S ❆◆❉ S ′ ❛♥❞ ✐♥ ❞✐♠❡♥s✐♦♥ N t❤❡r❡ ✐s✿ k t (x, y) = N i=1 p (1) t (x i , y i ) ✭✷✳✸✵✮ ❚❤❡ ❢r❛❝t✐♦♥❛• ♣♦✇❡rs ♦❢ K ♠❛② t❤❡r❡❢♦r❡ ❜❡ ❞❡✜♥❡❞ ❛s ❞♦♠❛✐♥ ♦♣❡r❛t♦rs ♦♥ L p ❛s ✐s ❡①♣•❛✐♥❡❞ ✐♥ ❛♣❡♥❞✐① ❈✳ ❲❡ ♠❛② t❤❡r❡❢♦r❡ ❞❡✜♥❡ t❤❡ s♣❛❝❡s S p,s ❛s dom L p (K s/2 )✱ ✐❡ t❤❡ ❝♦♠♣•❡t✐♦♥ ♦❢ S ❢♦r t❤❡ ♥♦r♠✿ f Sp,s = K s/2 f L p ✭✷✳✸✶✮ ❙✐♠✐•❛r•②✱ K ✐s t❤❡ ❣❡♥❡r❛t♦r ♦❢ ❛ s❡♠✐❣r♦✉♣ Kt ✇✐t❤ tr❛♥s✐t✐♦♥ ❞❡♥s✐t②✿ k(1) t (x, y) = exp(t) • k (1) t (x, y) ✭✷✳✸✷✮ ❛♥❞ ✐t ✐s s❤♦✇♥ ✐♥ ❬✷✺( t❤❛t t❤❡r❡ ❡①✐sts ❛ ✉♥✐✈❡rs❛• ❝♦♥st❛♥t C(N ) s✉❝❤ t❤❛t✿ kt (x, y) ≤ C • q t (x, y) ✭✷✳✸✸✮ ✇❤❡r❡ q t ✐s t❤❡ tr❛♥s✐t✐♦♥ ❞❡♥s✐t② ♦❢ t❤❡ ❤❡❛t ❦❡r♥❡•✳ ❚❤❡r❡❢♦r❡ t❤❡r❡ ✐s t❤❡✿ Pr♦♣♦s✐t✐♦♥ ✷✳✷✳✷✳✶✳ ❋♦r s > 0✱ ❛♥❞ f ∈ L p ✱ t❤❡r❡ ✐s✿ f S p,-s ≤ C • (Id -∆) -s/2 |f | L p ≤ C • f L p ✭✷✳✸✹✮ ❛♥❞ ❝♦♥✈❡rs❡•②✿ f L p ≤ C -1 • f Sp,s ✭✷✳✸✺✮ Pr♦♦❢✳ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ r❡s✉•ts ✐♥ ❛♣♣❡♥❞✐① ❈ •❡t ✉s ✇r✐t❡✿ K -s/2 f (x) = 1 Γ(s/2) ∞ 0 t s/2-1 e -t Kt f (x)dt = 1 Γ(s/2) ∞ 0 t s/2-1 e -t R N kt (x, y)f (y)dydt ✭✷✳✸✻✮ ❚❤❡♥ ❜② ❍ö•❞❡r✬s ✐♥❡q✉❛•✐t② ❛♥❞ ❜❡❝❛✉s❡ k ✐s ❝♦♥tr♦••❡❞ ❜② t❤❡ ❤❡❛t ❦❡r♥❡•✿ K -s/2 f (x) ≤ 1 Γ(s/2) ∞ 0 t s/2-1 e -t R N kt (x, y)|f |(y)dydt ≤ C Γ(s/2) ∞ 0 t s/2-1 e -t R N q t (x, y)|f |(y)dydt = C • (Id -∆) -s/2 |f | L p ≤ C • |f | L p = C • f L p ✭✷✳✸✼✮ ❋✐♥❛••②✱ t❤❡ s❡❝♦♥❞ ✐♥❡q✉❛•✐t② ✐s ♦❜t❛✐♥❡❞ ❜② ❛♣♣•②✐♥❣ t❤❡ ✜rst ♦♥❡ t♦ K s/2 f ✳ ✷✳✷✳ ❚❍❊ ❖P❊|❆❚❖| K ❆◆❉ ❆❙❙❖❈■❆❚❊❉ ❙❊▼■ ◆❖|▼❙ ✹✶ |❡♠❛r❦ ✷✳✷✳✷✳✶✳ ❲❡ ❝♦✉•❞ ♥♦t ❞✐r❡❝t•② ♦❜t❛✐♥ ❛ r❡s✉•t s✉❝❤ ❛s✿ f S p,-s ≤ C • (Id -∆) -s/2 f L p ✭✷✳✸✽✮ ✐❡ ❣❡tt✐♥❣ r✐❞ ♦❢ t❤❡ ❛❜s♦•✉t❡ ✈❛•✉❡ ✐♥s✐❞❡ t❤❡ s❡❝♦♥❞ t❡r♠ ✐s ♥♦t tr✐✈✐❛•❀ ✐t ✐s t❤❡ ♣♦✐♥t ♦❢ t❤❡ ♥❡①t r❡s✉•t✳ ❲❡ ❞❡✜♥❡ ❝♦♥t✐♥✉♦✉s ♦♣❡r❛t♦rs X i ❛♥❞ X 2 ♦♥ S ❛s✿ (X i f )(x) = x i f (x) ✭✷✳✸✾✮ ❛♥❞✿ (X 2 f )(x) = |x| 2 f (x) ✭✷✳✹✵✮ ❚❤❡♥ ✇❡ ♣r♦✈❡ t❤❡✿ ❚❤❡♦r❡♠ ✷✳✷✳✷✳✶✳ ❚❤❡r❡ ❡①✐sts ❛ ✉♥✐✈❡rs❛• ❝♦♥st❛♥t C✱ ❞❡♣❡♥❞✐♥❣ ♦♥•② ♦♥ 1 < p < ∞ ❛♥❞ N ✱ s✉❝❤ t❤❛t t❤❡ ❢♦••♦✇✐♥❣ ❡st✐♠❛t❡s ❤♦•❞✿ ∂ i u L p ≤ C K 1/2 u L p ✭✷✳✹✶✮ X i u L p ≤ C K 1/2 u L p ✭✷✳✹✷✮ Pr♦♦❢✳ ❆•• t❤❡ ♥♦t❛t✐♦♥ ✐♥ t❤✐s ♣r♦♦❢ ✐s t❤❛t ♦❢ ❛♣♣❡♥❞✐① ❊✳ ❲❡ ✇✐•• ❛♣♣•② t❤❡♦r❡♠ ❊✳✵✳✵✳✷✻ t♦ t❤❡ ♦♣❡r❛t♦r K❀ t♦ ❞♦ s♦✱ ✜rst ♥♦t✐❝❡ t❤❛t ♦♥❡ ❣❡ts ❛ ❣♦♦❞ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ❜② s❡tt✐♥❣✿ ρ(x) = 1 + |x| 2 1 3 ✭✷✳✹✸✮ ❚❤❡♥ t❛❦❡ m = 1✱ b 0 = 1✱ b 1 = -1✱ a β = 0✱ µ = 0 ❛♥❞ ν = 3✱ s♦ t❤❛t χ 0 = ν = 3 ❛♥❞ χ 2 = µ = 0✳ ❍❡♥❝❡✱ A = ρ ν -ρ µ ∆ = K ✭✷✳✹✹✮ ❛♥❞ ✇❡ ❤❛✈❡ ♣r♦✈❡❞ t❤❡✿ ▲❡♠♠❛ ✷✳✷✳✷✳✶✳ K ∈ A 1 0,3 R N , ρ ✳ ❚❤❡r❡❢♦r❡ ✇❡ ♠❛② ❛♣♣•② t❤❡♦r❡♠ ❊✳✵✳✵✳✷✻ ✭✐♥ t❤✐s ❝❛s❡ ✇✐t❤ χ = 0✮❀ ❢♦r ❛ ❣❡♥❡r✐❝ ❝♦♥st❛♥t C i ❛♥❞ s♦♠❡ λ ✇✐t❤ s♠❛•• ❡♥♦✉❣❤ r❡❛• ♣❛rt ✇❡ ♦❜t❛✐♥✿ ∆u L p ≤ u W 2 p ≤ C 1 u W 2 p (1,ρ 3p ) ≤ C 2 Ku -λu L p ≤ C 2 • ( Ku L p + |λ| • u L p ) ≤ C 3 • Ku L p ✭✷✳✹✺✮ ❋♦r t❤❡ •❛st ✐♥❡q✉❛•✐t② ✇❡ ✉s❡❞ ♣r♦♣♦s✐t✐♦♥ ✷✳✷✳✷✳✶✳ ✹✷ ❈❍❆P❚❊| ✷✳ ❚❖P❖▲❖•■❈❆▲ ❙❚|❯❈❚❯|❊ ❖❋ S ❆◆❉ S ′ ❚❤❡♥ ❜② ❝♦♠♣•❡① ✐♥t❡r♣♦•❛t✐♦♥ ♦♥❡ ❣❡ts✿ √ -∆u L p ≤ C √ Ku L p = C u S p,1 ✭✷✳✹✻✮ s♦ ✇❡ ♦❜t❛✐♥ t❤❡ ✜rst ♣❛rt ♦❢ t❤❡ r❡s✉•t t❤r♦✉❣❤ t❤❡ ✇❡••✲❦♥♦✇♥ ✭❝❢✱ ❢♦r ❡①❛♠♣•❡✱ ❬✻✼(✮ ✐♥❡q✉❛•✐t②✿ ∂ i u L p ≤ C √ -∆u L p ✭✷✳✹✼✮ ❆•s♦✱ ♦♥❡ s❡❡s t❤❛t✿ X 2 u L p ≤ Ku L p + u L p + ∆u L p ≤ C Ku L p ✭✷✳✹✽✮ ❛♥❞ s✐♠✐•❛r•②✱ ♦♥❡ ❣❡ts t❤❡ s❡❝♦♥❞ ♣❛rt ♦❢ t❤❡ r❡s✉•t ❜② ✐♥t❡r♣♦•❛t✐♦♥✳ ❲❡ ✜♥✐s❤ t❤✐s ♣❛r❛❣r❛♣❤ ✇✐t❤ ❛ ❝♦✉♣•❡ ♦❢ r❡s✉•ts r❡•❛t❡❞ t♦ t❤❡ ✐♥t❡r♣♦•❛t✐♦♥ ♦❢ t❤❡ S p,s s♣❛❝❡s✳ ❋✐rst ❛s ❛♥ ❛♣♣•✐❝❛t✐♦♥ ♦❢ t❤❡ r❡s✉•ts ❢♦r s❡♠✐❣r♦✉♣ ✐♥t❡r♣♦•❛t✐♦♥ ✇❤✐❝❤ ♠❛② ❜❡ ❢♦✉♥❞ ✐♥ ❛♣♣❡♥❞✐① ❉ t❤❡r❡ ✐s t❤❡✿ ❚❤❡♦r❡♠ ✷✳✷✳✷✳✷✳ ❲❡ ❝♦♥s✐❞❡r p 1 , p 2 > 1✱ s 1 , s 2 ∈ R ❛♥❞ 0 < θ < 1✳ ▲❡t p ❛♥❞ s ❜❡ s✉❝❤ t❤❛t✿ 1 p = θ p 1 + 1-θ p 2 ❛♥❞ s = θs 1 + (1 -θ)s 2 ✳ ❚❤❡♥ [S p 1 ,s 1 , S p 2 ,s 2 ] θ = S p,s ✳ ❆•s♦ ✇❡ ♦❜t❛✐♥ t❤❡ ❢♦••♦✇✐♥❣ t❤r♦✉❣❤ ❝•❛ss✐❝ s❡♠✐❣r♦✉♣ ✐♥t❡r♣♦•❛t✐♦♥ ❛r❣✉✲ ♠❡♥ts✱ ❝❢ ❛♣♣❡♥❞✐① ❈✿ ❚❤❡♦r❡♠ ✷✳✷✳✷✳✸ ✭❉✉❛• s♣❛❝❡ ♦❢ S p,s ✮✳ ❋♦r 1 < p < ∞ ❛♥❞ s ∈ R t❤❡r❡ ✐s✿ S ′ p,s = S p * ,-s ✭✷✳✹✾✮ ✇❤❡r❡ 1/p * + 1/p = 1✳ ❲❡ ♠❛② ❛•s♦ ✐♥tr♦❞✉❝❡ s♦♠❡ r❡❛• ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡s✿ ❉❡✜♥✐t✐♦♥ ✷✳✷✳✷✳✶ ✭|❡❛• ✐♥t❡r♣♦•❛t✐♦♥ ♦❢ t❤❡ S p,s ✮✳ ▲❡t p > 1 ❛♥❞ s ∈ R✳ ❲❡ ❞❡✜♥❡ ❛ ❢❛♠✐•② ♦❢ ❇❛♥❛❝❤ s♣❛❝❡s T p,s ❛s ❢♦••♦✇s✿ • ✐❢ s ∈ Z t❤❡♥ T p,s = S p,s • ❖t❤❡r✇✐s❡ •❡t k = [s]✱ σ = {s}✳ ❚❤❡♥ T p,s = (S p,k+1 , S p,k ) 1-σ,p ✳ ❛♥❞ t❤❡r❡ ✐s t❤❡✿ ❚❤❡♦r❡♠ ✷✳✷✳✷✳✹✳ ❋♦r ❛♥② ǫ > 0✱ t❤❡r❡ ✐s✿ T p,s-ǫ ֒→ S p,s ֒→ T p,s+ǫ ✭✷✳✺✵✮ Pr♦♦❢✳ ❖♠✐tt❡❞ ❛s ✐t ✐s ✈❡r② s✐♠✐•❛r t♦ t❤❛t ♦❢ t❤❡♦r❡♠ ✸✳✶✳✵✳✾✱ ✇❤✐❝❤ ✇❡ ♣r♦✈❡ ✐♥ ❞❡t❛✐•✳ ❲❡ ❛•s♦ ❤❛✈❡ ❛ ❞✉❛•✐t② r❡s✉•t ❢♦r t❤❡ r❡❛• ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡s✿ ❚❤❡♦r❡♠ ✷✳✷✳✷✳✺ ✭❉✉❛• s♣❛❝❡ ♦❢ T p,s ✮✳ ❋♦r 1 < p < ∞ ❛♥❞ s ∈ R t❤❡r❡ ✐s✿ T ′ p,s = T p * ,-s ✭✷✳✺✶✮ ✇❤❡r❡ 1/p * + 1/p = 1✳ ✷✳✷✳ ❚❍❊ ❖P❊|❆❚❖| K ❆◆❉ ❆❙❙❖❈■❆❚❊❉ ❙❊▼■ ◆❖|▼❙ ✹✸ ✷✳✷✳✸ ❙❡♠✐ ♥♦r♠s ❜✉✐•t ♦♥ L ∞ ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ❡①t❡♥❞ t❤❡ L p r❡s✉•ts ♦❢ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥ t♦ t❤❡ L ∞ ❝❛s❡✳ ❲❡ ✇❡r❡ ♥♦t ❛❜•❡ t♦ ✜♥❞ t❤❡ ❡①❛❝t r❡s✉•ts ✇❡ ♥❡❡❞❡❞ ✐♥ t❤❡ •✐t❡r❛t✉r❡✱ s♦ ✇❡ ❜✉✐•t ♦✉r ♦✇♥ ♣r♦♦❢s✳ ▼♦r❡ ♣r❡❝✐s❡•②✱ ✇❡ ✇✐•• ♣r♦✈❡ t❤❡✿ ❚❤❡♦r❡♠ ✷✳✷✳✸✳✶✳ ❚❤❡r❡ ❡①✐sts ❛ ✉♥✐✈❡rs❛• ❝♦♥st❛♥t C s✉❝❤ t❤❛t t❤❡ ❢♦••♦✇✐♥❣ ❡st✐♠❛t❡s ❤♦•❞✿ √ -∆u ∞ ≤ C K 1/2 u ∞ ✭✷✳✺✷✮ X i u ∞ ≤ C K 1/2 u ∞ ✭✷✳✺✸✮ Pr♦♦❢✳ ❲❡ ♣r♦✈❡ t❤❡ t❤❡♦r❡♠ ♦♥•② ✐♥ ❞✐♠❡♥s✐♦♥ ✶❀ r❡s✉•ts ❣❡♥❡r❛•✐③❡ t♦ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥ ✇✐t❤ ♥♦ ♦t❤❡r ❞✐✣❝✉•t② t❤❛♥ t❡❞✐♦✉s ♥♦t❛t✐♦♥✳ ❲❡ st❛rt ✇✐t❤ t❤❡ s❡❝♦♥❞ ♣❛rt ♦❢ t❤❡ t❤❡♦r❡♠✳ ❲❡ r❡❝❛•• t❤❛t K ✐s ✐♥✈❡rt✐❜•❡ ♦♥ ❛♥② L p ❛♥❞ ✐ts ✐♥✈❡rs❡ ❤❛s ❛ ❦❡r♥❡• Π ✇❤✐❝❤ ✐s ❣✐✈❡♥ ❜②✿ K -1 f (x) = ∞ -∞ Π(x, y)f (y)dy ✭✷✳✺✹✮ Π(x, y) = ∞ n=0 f n (x)f n (y) 2n + 2 ✭✷✳✺✺✮ ✇❤❡r❡ t❤❡ f n ❛r❡ t❤❡ ❍❡r♠✐t❡ ❢✉♥❝t✐♦♥s✳ ❚❤❡♥ ❜② ❋✉❜✐♥✐✬s t❤❡♦r❡♠✿ Π(x, y) = ∞ n=0 f n (x)f n (y) 1 0 r 2n+1 dr = 1 0 r • ∞ n=0 r 2n f n (x)f n (y)dr ✭✷✳✺✻✮ s♦ ❜② ▼❡❤•❡r✬s ❢♦r♠✉•❛ ♦♥❡ ♦❜t❛✐♥s✿ Π(x, y) = 1 0 r 2 π(1 -r 4 ) • exp 4xyr 2 -(x 2 + y 2 )(1 + r 4 ) 2(1 -r 4 ) dr ✭✷✳✺✼✮ ✹✹ ❈❍❆P❚❊| ✷✳ ❚❖P❖▲❖•■❈❆▲ ❙❚|❯❈❚❯|❊ ❖❋ S ❆◆❉ S ′ ❛♥❞ ❤❡♥❝❡✿ K -1 f (x) = 1 0 r 2 π(1 -r 4 ) ∞ -∞ exp 4xyr 2 -(x 2 + y 2 )(1 + r 4 ) 2(1 -r 4 ) f (y)dr = 1 0 r 2 π(1 -r 4 ) ∞ -∞ exp - 1 2 1 + r 4 1 -r 4 x 2 + y 2 - 4r 2 1 + r 4 xy f (y)dr = 1 0 r 2 π(1 -r 4 ) exp - 1 2 1 -r 4 1 + r 4 x 2 × ∞ -∞ exp - 1 2 1 + r 4 1 -r 4 y - 2r 2 1 + r 4 x 2 f (y)dydr = 1 0 r 2 π(1 -r 4 ) exp - 1 2 1 -r 4 1 + r 4 x 2 × ∞ -∞ exp - 1 2 1 + r 4 1 -r 4 y 2 f y + 2r 2 1 + r 4 x dydr ✭✷✳✺✽✮ ❙♦ ✇❡ ♦❜t❛✐♥✿ |K -1 f (x)| ≤ f ∞ √ π • 1 0 r 2 π(1 -r 4 ) exp - 1 2 1 -r 4 1 + r 4 x 2 ∞ -∞ exp - 1 2 1 + r 4 1 -r 4 y 2 dydr = √ 2 • f ∞ • 1 0 r 2 π(1 + r 4 ) exp - 1 2 1 -r 4 1 + r 4 x 2 dr = √ 2 • f ∞ • e x 2 2 • 1 0 r 2 π(1 + r 4 ) exp - 1 1 + r 4 x 2 dr ≤ √ 2 • f ∞ • e x 2 2 • 1 0 1 π(1 + r 4 ) exp - 1 1 + r 4 x 2 dr ✭✷✳✺✾✮ ❋✐rst✱ ✇❡ ♥♦t✐❝❡ t❤❡ ❢♦••♦✇✐♥❣ ♠❛•♦r✐③❛t✐♦♥✿ |K -1 f (x)| ≤ 2 π • f ∞ • e x 2 2 ✭✷✳✻✵✮ ❋r♦♠ t❤✐s✱ ✇❡ ❞❡❞✉❝❡ t❤❛t K -1 f ✐s ❝♦♥tr♦••❡❞ ❜② f ∞ ♦♥ ❛♥② ❝♦♠♣❛❝t✳ ❚❤❡♥ ✇❡ ❞♦ t❤❡ ❢♦••♦✇✐♥❣ ❝❤❛♥❣❡s ♦❢ ✈❛r✐❛❜•❡s ✐♥ t❤❡ •❛st ✐♥t❡❣r❛•✿ s = 1 1 + r 4 ✭✷✳✻✶✮ t = sx 2 ✭✷✳✻✷✮ ✷✳✷✳ ❚❍❊ ❖P❊|❆❚❖| K ❆◆❉ ❆❙❙❖❈■❆❚❊❉ ❙❊▼■ ◆❖|▼❙ ✹✺ ❛♥❞ t❤❡s❡ ②✐❡•❞✿ |K -1 f (x)| ≤ √ 2 4 • f ∞ • e x 2 2 • 1 1/2 [s(1 -s)] -3/4 e -sx 2 ds = √ 2 4 • f ∞ • xe x 2 2 • x 2 x 2 2 t(x 2 -t) -3/4 e -t dt ✭✷✳✻✸✮ ❋r♦♠ t❤✐s ♦♥❡ ♠❛② ❛•r❡❛❞② s❡❡ t❤❛t Kf ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ❜♦✉♥❞❡❞ ♦♥ ❛♥② ❝♦♠♣❛❝t✳ ◆♦✇ ✇❡ ❝✉t t❤✐s •❛st ✐♥t❡❣r❛• ✐♥t♦ t✇♦ ♣❛rts✿ x 2 x 2 2 • • • = 3x 2 4 x 2 2 • • • + x 2 3x 2 4 . . . ✭✷✳✻✹✮ ❋♦r t❤❡ ✜rst ✐♥t❡❣r❛•✱ ✇❡ ♥♦t❡ t❤❛t ♦♥ t❤❡ ❞♦♠❛✐♥ ♦❢ ✐♥t❡❣r❛t✐♦♥✱ t❤❡r❡ ✐s✿ t(x 2 -t) ≥ 3 16 x 4 ✭✷✳✻✺✮ s♦ 3x 2 4 x 2 2 t(x 2 -t) -3/4 e -t dt ≤ 3 16 -3 4 x -3 • 3x 2 4
x 2 2 e -t dt ≤ 3 16

-3 4 x -3 e -x 2 2 ✭✷✳✻✻✮ ❛♥❞ ❢♦r t❤❡ s❡❝♦♥❞ ♦♥❡✿ x 2 3x 2 4 t(x 2 -t) -3/4 e -t dt ≤ 3 4 -3 4 x -3/2 e -3 4 x 2 • x 2 3x 2 4 (x 2 -t) -3/4 dt = 3 4 -3 4 x -1 e -3 4 x 2 1 3/4 (1 -τ ) -3 4 dτ ✭✷✳✻✼✮ ❛♥❞ t❤✐s •❛st ✐♥t❡❣r❛• ✐s ✜♥✐t❡ ✭✐ts ✈❛•✉❡ ✐s 4/ √ 2✮✳ ❖✈❡r❛••✱ ✇❡ ♦❜t❛✐♥ ❛ ♠❛•♦r✐③❛t✐♦♥ ♦❢ t❤❡ ❢♦r♠✿ |K -1 f (x)| ≤ C 1 x -3 e -x 2 2 + C 2 x -1 e -3 4 x 2 xe x 2 2 f ∞ ✭✷✳✻✽✮ ◆♦✇ ✇❡ ❤❛✈❡ ♦♥ ♦♥❡ ❤❛♥❞✱ ❜❡❝❛✉s❡ ♦❢ ✭✷✳✻✵✮✿ sup |x|≤1 |K -1 f (x)| ≤ 2e π • f ∞ ✭✷✳✻✾✮ ❛♥❞ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❜❡❝❛✉s❡ ♦❢ ✭✷✳✻✽✮✿ sup |x|≥1 |K -1 f (x)| ≤ sup |x|≥1 (C 1 x -2 + C 2 xe -1 4 x ) • f ∞ ✭✷✳✼✵✮ ✹✻ ❈❍❆P❚❊| ✷✳ ❚❖P❖▲❖•■❈❆▲ ❙❚|❯❈❚❯|❊ ❖❋ S ❆◆❉ S ′ ❆•s♦ ❜❡❝❛✉s❡ ♦❢ ✭✷✳✻✽✮✿ sup |x|≥1 |x 2 | • |K -1 f (x)| ≤ sup |x|≥1 (C 1 + C 2 x 3 e -1 4 x 2 ) • f ∞ ✭✷✳✼✶✮ ✐❡✿ X 2 K -1 f ∞ ≤ C • f ∞ ✭✷✳✼✷✮ ❚❤❡ ♣r♦♦❢ ✐s t❤❡♥ ✜♥✐s❤❡❞ ❛s ✐♥ t❤❡ L p ❝❛s❡✳ ❲❡ ♥♦t❡ t❤❛t ✇❡ ❝♦✉•❞ ♥♦t ❣❡t ❛ ❝♦♥tr♦• ♦❢ t❤❡ t②♣❡✿ ∂ i u ∞ ≤ C • K 1/2 u ∞ ✳ ❚❤✐s ✐s ❜❡❝❛✉s❡ ∂ i u ∞ ≤ C • √ -∆u ∞ ❞♦❡s ♥♦t ❤♦•❞ ✐♥ ❣❡♥❡r❛•✳ ❍♦✇❡✈❡r✱ s✉❝❤ ❛♥ ✐♥❡q✉❛•✐t② ❞♦❡s ❤♦•❞ ❢♦r t❤❡ ❍ö•❞❡r s♣❛❝❡s Λ γ ✱ γ ∈ R + -N✱ ❝❢ ❬✻✼(✳ ❲❡ s❤❛•• t❤❡r❡❢♦r❡ ♣r♦✈❡ t❤❡✿ ❚❤❡♦r❡♠ ✷✳✷✳✸✳✷✳ ❋♦r γ ∈ R + -N✱ t❤❡r❡ ❡①✐sts ❛ ✉♥✐✈❡rs❛• ❝♦♥st❛♥t C s✉❝❤ t❤❛t t❤❡ ❢♦••♦✇✐♥❣ ❡st✐♠❛t❡s ❤♦•❞✿ ∂ i u Λ γ ≤ √ -∆u Λ γ ≤ C K 1/2 u Λ γ ✭✷✳✼✸✮ X i u Λ γ ≤ C K 1/2 u Λ γ ✭✷✳✼✹✮ Pr♦♦❢✳ ❲❡ ❞♦ t❤❡ ♣r♦♦❢ ❢♦r 0 < γ < 1✳ ❖♥❝❡ ❛❣❛✐♥✱ ✇❡ ✇♦r❦ ✇✐t❤ t❤❡ ♦♣❡r❛t♦r K✳ ▲❡t 0 < ǫ < 1❀ ✇❡ ✇❛♥t t♦ ❝♦♥tr♦• t❤❡ ❞✐✛❡r❡♥❝❡✿ (x + ǫ) 2 K -1 f (x + ǫ) -x 2 K -1 f (x) = (x + ǫ) 2 -x 2 K -1 f (x + ǫ) + x 2 K -1 f (x + ǫ) -K -1 f (x) ✭✷✳✼✺✮ ❚❤❡ ✜rst t❡r♠ ♦❢ t❤❡ s✉♠ ♦♥ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ s✐♠♣•② ✐s✿ (2xǫ + ǫ 2 ) K -1 f (x + ǫ)❀ ✉s✐♥❣ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠ ♦♥❡ ❡❛s✐•② ❝♦♥tr♦•s t❤✐s ❜② Cǫ f ∞ ✳ ❋♦r t❤❡ s❡❝♦♥❞ t❡r♠✱ ✇r✐t❡✿ K -1 f (x + ǫ) -K -1 f (x) = 1 0 1 π(1 -r 4 ) exp - 1 2 1 -r 4 1 + r 4 (x + ǫ) 2 -exp - 1 2 1 -r 4 1 + r 4 x 2 × ∞ -∞ exp - 1 2 1 + r 4 1 -r 4 y 2 f y + 2r 2 1 + r 4 x dydr - 1 0 1 π(1 -r 4 ) exp - 1 2 1 -r 4 1 + r 4 x 2 × ∞ -∞ exp - 1 2 1 + r 4 1 -r 4 y 2 f y + 2r 2 1 + r 4 (x + ǫ) -f y + 2r 2 1 + r 4 x dydr ✭✷✳✼✻✮ ❚♦ ❝♦♥tr♦• t❤❡ ✜rst ✐♥t❡❣r❛•✱ ♦♥❡ ♠❛② ✉s❡ t❤❡ ❢♦••♦✇✐♥❣ s✐♠♣•❡ ❝❛•❝✉•✉s •❡♠♠❛✿ ✷✳✷✳ ❚❍❊ ❖P❊|❆❚❖| K ❆◆❉ ❆❙❙❖❈■❆❚❊❉ ❙❊▼■ ◆❖|▼❙ ✹✼ ▲❡♠♠❛ ✷✳✷✳✸✳✶✳ ▲❡t 0 < η < 1✳ ❚❤❡♥ ♦♥❡ ❤❛s ❛ ✉♥✐❢♦r♠ ❝♦♥tr♦• ♦❢ t❤❡ t②♣❡✿ exp - 1 2 1 -r 4 1 + r 4 (x + ǫ) 2 -exp - 1 2 1 -r 4 1 + r 4 x 2 ≤ C(η) 1 -r 4 1 + r 4 ǫ exp - 1 -η 2 1 -r 4 1 + r 4 x 2 ✭✷✳✼✼✮ ❛♥❞ t❤❡r❡❢♦r❡ t❤❡ ✜rst ✐♥t❡❣r❛• ✐s ❝♦♥tr♦•❡❞ ❜②✿ Cǫ f ∞ 1 0 1 1 -r 4 1 -r 4 1 + r 4 exp - 1 -η 2 1 -r 4 1 + r 4 x 2 ∞ -∞ exp - 1 2 1 + r 4 1 -r 4 y 2 dydr = Cǫ f ∞ 1 0 (1 -r 4 ) 3/2 (1 + r 4 ) 2 exp - 1 -η 2 1 -r 4 1 + r 4 x 2 dr ≤ ǫ f ∞ C 1 (η)x -2 + C 2 (η)e -1-η 4 x 2 x 2 ✭✷✳✼✽✮ ❜② ♠❡t❤♦❞s s✐♠✐•❛r t♦ t❤♦s❡ ♦❢ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠✱ ♥♦t✐♥❣ t❤❛t✿ (1-r 4 ) 3/2 (1+r 4 ) 2 ≤ 1 1+r 4 ✳ ■t ✐s ♣♦ss✐❜•❡✱ ❛•t❤♦✉❣❤ ♥♦♥✲♥❡❝❡ss❛r②✱ t♦ ♦♣t✐♠✐③❡ t❤❡ ❛❜♦✈❡ ✐♥ η✳ ❚❤❡♥✱ t❤❡ s❡❝♦♥❞ ✐♥t❡❣r❛• ✐s ❝♦♥tr♦•❡❞ ❜②✿ Cǫ γ f Λ γ 1 0 1 π(1-r 4 ) 2r 2 1+r 4 γ exp -1 2 1-r 4 1+r 4 x 2 ∞ -∞ exp -1 2 1+r 4 1-r 4 y 2 dydr ≤ Cǫ γ f Λ γ 1 0 1 (1-r 4 ) exp -1 2 1-r 4 1+r 4 x 2 dr ✭✷✳✼✾✮ ❜❡❝❛✉s❡ 2r 2 1+r 4 ≤ 1✳ ◆♦✇ t❤❡ •❛st ✐♥t❡❣r❛• ♦♥ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ ❤❛s ❛•r❡❛❞② ❜❡❡♥ ❡♥❝♦✉♥t❡r❡❞ ❛♥❞ ♠❛•♦r✐③❡❞ ✐♥ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠✱ s♦ ✇❡ ❛r❡ ❞♦♥❡ ❛s✱ ♣✉tt✐♥❣ t❤❡ ♣❛rts t♦❣❡t❤❡r✱ ✇❡ ❤❛✈❡ ♣r♦✈❡❞✿ (x + ǫ) 2 K -1 f (x + ǫ) -x 2 K -1 f (x) ≤ Cǫ γ f Λ γ ✭✷✳✽✵✮ ✇❤✐❝❤ ♠❡❛♥s t❤❛t✿ X 2 K -1 f Λ γ ≤ f Λ γ ✭✷✳✽✶✮ ❛♥❞ ✇❡ ✜♥✐s❤ t❤❡ ♣r♦♦❢ ❜② t❤❡ ✉s✉❛• ♠❡t❤♦❞✳ ❲❡ ❛•s♦ r❡•❛t❡ t❤❡ S ∞,s s♣❛❝❡s t♦ t❤❡ S p,s s♣❛❝❡s ✐♥ t❤❡ ❢♦••♦✇✐♥❣ ✇❛②✿ Pr♦♣♦s✐t✐♦♥ ✷✳✷✳✸✳✶ ✭❙♦♠❡ ❙♦❜♦•❡✈ t②♣❡ ✐♥•❡❝t✐♦♥s✮✳ ❚❤❡ ❢♦••♦✇✐♥❣ ✐♥•❡❝t✐♦♥s ❤♦•❞✿ • S p,s ֒→ C 0 b ❢♦r s > N p • S ∞,s ֒→ L p ❢♦r s > N 2p ✹✽ ❈❍❆P❚❊| ✷✳ ❚❖P❖▲❖•■❈❆▲ ❙❚|❯❈❚❯|❊ ❖❋ S ❆◆❉ S ′ Pr♦♦❢✳ ❋♦r t❤❡ ✜rst ✐♥•❡❝t✐♦♥✱ s✐♠♣•② ♦❜s❡r✈❡ t❤❛t S p,s ֒→ W p,s ✳ ❋♦r t❤❡ s❡❝♦♥❞ ♦♥❡✱ ✇r✐t❡✿ f p L p ≤ dx (1 + |x|) s • sup(1 + |x|) s p |f (x)| p ✭✷✳✽✷✮ ❛♥❞ ♥♦t✐❝❡ t❤❛t t❤❡ ✐♥t❡❣r❛• ✐❢ ✜♥✐t❡ ✐✛ s > N ❛♥❞ t❤❛t ❛❝❝♦r❞✐♥❣ t♦ t❤❡♦r❡♠ ✷✳✷✳✸✳✷✿ sup(1 + |x|) s p |f (x)| ≤ C • K s 2p f ∞ = C f S ∞, s 2p ✭✷✳✽✸✮ ◆♦✇ s✐♥❝❡ C 0 b = S ∞,0 ❛♥❞ L p = S p,0 ✇❡ ✐♠♠❡❞✐❛t❡•② ♦❜t❛✐♥ t❤❡✿ ❈♦r♦••❛r② ✷✳✷✳✸✳✶✳ ❋♦r ❛♥② ǫ > 0✱ t❤❡ ❢♦••♦✇✐♥❣ ✐♥•❡❝t✐♦♥s ❤♦•❞✿ • S p,s+ N p +ǫ ֒→ S ∞,s • S ∞,s+ N 2p +ǫ ֒→ S p,s ❲❡ ♠❛② ♥♦✇ ❞❡❞✉❝❡ t❤❡✿ Pr♦♣♦s✐t✐♦♥ ✷✳✷✳✸✳✷ ✭❉✉❛• s♣❛❝❡ ❢♦r S ∞,s ✮✳ ❋♦r ❛♥② ǫ > 0✱ t❤❡ ❢♦••♦✇✐♥❣ ✐♥•❡❝t✐♦♥ ❤♦•❞s✿ S ′ ∞,s ֒→ S ∞,-s-N -ǫ ✭✷✳✽✹✮ Pr♦♦❢✳ ❲❡ ❛♣♣•② t❤❡ ✜rst ✐♥❝•✉s✐♦♥ ♦❢ t❤❡ ❛❜♦✈❡ ❝♦r♦••❛r② t✇✐❝❡ ✐♥ ❛ r♦✇ ❛♥❞ ♦❜t❛✐♥✿ S ′ ∞,s ֒→ S ′ p,s+ N p +ǫ = S p * ,-s-N p -ǫ ֒→ S ∞,-s-N ( 1 p + 1 p * )-ǫ ′ ✭✷✳✽✺✮ ✷✳✸ ❚❤❡ t♦♣♦•♦❣✐❡s ♦♥ S ❛♥❞ S ′ ❚❤❡ r❡s✉•ts ✐♥ t❤❡ ♣r❡✈✐♦✉s ♣❛r❛❣r❛♣❤s ❛••♦✇ ✉s t♦ ✉♥❞❡rst❛♥❞ ❤♦✇ t❤❡ t♦♣♦•♦❣② ♦♥ S ✐s ❞❡✜♥❡❞✳ ▼♦r❡ ♣r❡❝✐s❡•②✱ t❤❡r❡ ✐s t❤❡✿ ❚❤❡♦r❡♠ ✷✳✸✳✵✳✸ ✭❊q✉✐✈❛•❡♥❝❡ ♦❢ s♦♠❡ t♦♣♦•♦❣✐❡s ♦♥ S✮✳ ❆♥② ♦♥❡ ♦❢ t❤❡ ❢♦••♦✇✐♥❣ ❢❛♠✐•✐❡s ♦❢ s❡♠✐ ♥♦r♠s ❡q✉✐♣s S ✇✐t❤ t❤❡ s❛♠❡ ❋ré❝❤❡t s♣❛❝❡ t♦♣♦•♦❣②✿ • ❢♦r s♦♠❡ ✜①❡❞ 1 ≤ p ≤ ∞✿ • (p) α,β ✱ ✇❤❡r❡ α, β ❞❡s❝r✐❜❡ N N ❀ • ❢♦r s♦♠❡ ✜①❡❞ 1 < p ≤ ∞✿ • Sp,s ✱ ✇❤❡r❡ s ❞❡s❝r✐❜❡s N ✭♦r Z✮❀ • ❢♦r s♦♠❡ ✜①❡❞ 1 < p ≤ ∞✿ • Tp,s ✱ ✇❤❡r❡ s ❞❡s❝r✐❜❡s N ✭♦r Z✮✳ ✷✳✸✳ ❚❍❊ ❚❖P❖▲❖•■❊❙ ❖◆ S ❆◆❉ S ′ ✹✾ ■t ✇✐•• ❜❡ ✉s❡❢✉• t♦ ♥♦t❡ t❤❛t ❢♦r ❛♥② 1 < p ≤ ∞✿ S = s∈R S p,s ✭✷✳✽✻✮ ❛♥❞ ❜❡❝❛✉s❡ ♦❢ t❤❡♦r❡♠s ✷✳✷✳✷✳✶ ❛♥❞ ✷✳✷✳✸✳✷✿ S ′ = s∈R S p,s ✭✷✳✽✼✮ ❚❤❡ t♦♣♦•♦❣② ♦♥ S ′ ✐s t❤❡♥ ♦❜t❛✐♥❡❞ ❛s ❛ str♦♥❣ ❞✉❛•✲❋ré❝❤❡t s♣❛❝❡ t♦♣♦•♦❣②✳ ■♥ t❤❡ ✜♥❛• ♣❛r❛❣r❛♣❤s ♦❢ t❤✐s ❝❤❛♣t❡r✱ ✇❡ •✐st ❛ ♥✉♠❜❡r ♦❢ ✉s❡❢✉• ♣r♦♣❡rt✐❡s ♦❢ t❤❡ t♦♣♦•♦❣② ❞❡s❝r✐❜❡❞ ✐♥ t❤❡♦r❡♠ ✷✳✸✳✵✳✸✳ ■♥ t❤❡ r❡st ♦❢ t❤✐s ❞♦❝✉♠❡♥t ✇❡ ✇✐•• ❛•✇❛②s ❝♦♥s✐❞❡r t❤❛t S ✐s ❡q✉✐♣♣❡❞ ✇✐t❤ t❤✐s t♦♣♦•♦❣② ❛♥❞ t❤❛t S ′ ✐s ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ❛ss♦❝✐❛t❡❞ str♦♥❣ ❞✉❛• s♣❛❝❡ t♦♣♦•♦❣②✳ ✷✳✸✳✶ Pr♦♣❡rt✐❡s ♦❢ t❤❡ t♦♣♦•♦❣✐❡s ♦♥ S ❛♥❞ S ′ ❚❤❡ r❡s✉•ts ✐♥ t❤✐s s❡❝t✐♦♥s ❛r❡ ❝♦♥s❡q✉❡♥❝❡s ♦❢ ♠♦r❡ ❣❡♥❡r❛• t❤❡♦r❡♠s ❢r♦♠ t❤❡ t❤❡♦r② ♦❢ t♦♣♦•♦❣✐❝❛• ✈❡❝t♦r s♣❛❝❡s❀ ✇❡ r❡❢❡r t♦ t❤✐s ❛♣♣❡♥❞✐① ❆ ❢♦r t❤❡ r❡•❡✈❛♥t ❞❡✜♥✐t✐♦♥s ❛♥❞ r❡s✉•ts✳ ❋✐rst✱ ❢♦r s♦♠❡ f ∈ L 2 ✱ r❡❝❛•• t❤❛t t❤❡r❡ ✐s ❛ ❞❡❝♦♠♣♦s✐t✐♦♥✿ K -s f (x) = c n (f ) λ s n f n (x) ✭✷✳✽✽✮ ❛♥❞ t❤❡r❡❢♦r❡✿ Pr♦♣♦s✐t✐♦♥ ✷✳✸✳✶✳✶✳ ❋♦r s > N N 2 ✱ K -s ✐s ❛ ❍✐•❜❡rt✲❙❝❤♠✐❞t ♦♣❡r❛t♦r✱ ❛♥❞✿ K -s HS = n∈N N 1 λ 2s n 1 2 ✭✷✳✽✾✮ ✐❡ t❤❡ ✐♥❝•✉s✐♦♥ S 2,k ֒→ S 2,k+s ✐s ❍✐•❜❡rt✲❙❝❤♠✐❞t ✭♥✉❝•❡❛r✮ ❢♦r ❛♥② k✳ ❲❡ ❞❡❞✉❝❡ ❢r♦♠ t❤✐s ❛♥❞ t❤❡♦r❡♠ ❆✳✹✳✹✳✷ t❤❛t✿ ❚❤❡♦r❡♠ ✷✳✸✳✶✳✶✳ S ❡q✉✐♣♣❡❞ ✇✐t❤ ✐ts st❛♥❞❛r❞ ❋ré❝❤❡t s♣❛❝❡ t♦♣♦•♦❣② ✐s ❛ ♥✉❝•❡❛r t♦♣♦•♦❣✐❝❛• ✈❡❝t♦r s♣❛❝❡✳ ❆•s♦✱ ❜❡❝❛✉s❡ ♦❢ t❤❡♦r❡♠ ❆✳✹✳✹✳✸ t❤❡r❡ ✐s t❤❡✿ ❚❤❡♦r❡♠ ✷✳✸✳✶✳✷✳ S ′ ❡q✉✐♣♣❡❞ ✇✐t❤ ✐ts st♦♥❣ ❞✉❛• t♦♣♦•♦❣② ✐s ❛ ♥✉❝•❡❛r s♣❛❝❡✳ ❚❤❡r❡ ❛•s♦ ✐s t❤❡✿ ✺✵ ❈❍❆P❚❊| ✷✳ ❚❖P❖▲❖•■❈❆▲ ❙❚|❯❈❚❯|❊ ❖❋ S ❆◆❉ S ′ ❚❤❡♦r❡♠ ✷✳✸✳✶✳✸✳ S ✐s ❛ ▼♦♥t❡• s♣❛❝❡✳ Pr♦♦❢✳ S ✐s ❛ ♥✉❝•❡❛r ❋ré❝❤❡t s♣❛❝❡✳ ❛♥❞ t❤❡r❡❢♦r❡✿ ❚❤❡♦r❡♠ ✷✳✸✳✶✳✹✳ S ′ ✐s ❛ ▼♦♥t❡• s♣❛❝❡✳ Pr♦♦❢✳ S ′ ✐s t❤❡ str♦♥❣ ❞✉❛• ♦❢ ❛ ▼♦♥t❡• s♣❛❝❡✳ ▲❡t ✉s ❛•s♦ ♥♦t❡ t❤❛t S ✐s ❜❛r❡••❡❞ ❛s ❛ ❋ré❝❤❡t s♣❛❝❡ ❛♥❞ t❤❛t S ′ ✐s ❜❛r❡••❡❞ ❛s ❛ ▼♦♥t❡• s♣❛❝❡✳ ✷✳✸✳✷ S ′ ✲✈❛•✉❡❞ ♣❛t❤s ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ♠❡♥t✐♦♥ t❤♦s❡ r❡s✉•ts ♦❢ t♦♣♦•♦❣✐❝❛• ♥❛t✉r❡ ✇❤✐❝❤ ✇✐•• ♣r♦✈✐❞❡ ✉s ✇✐t❤ ❛♥ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ str✉❝t✉r❡ ♦❢ r❡❣✉•❛r S ′ ✲✈❛•✉❡❞ ♣❛t❤s✳ ❲❡ r❡❢❡r t♦ ❛♣♣❡♥❞✐① ❆ ❛♥❞ t♦ t❤❡ r❡❢❡r❡♥❝❡s t❤❡r❡✐♥ ❢♦r ❞❡t❛✐•s✳ ▼♦r❡ ♣r❡❝✐s•② ✇❡ ✇♦✉•❞ •✐❦❡ t♦ ✉♥❞❡rst❛♥❞ t❤❡ str✉❝t✉r❡ ♦❢ ❝♦♥t✐♥✉♦✉s ♣❛t❤s ❛♥❞ ♦❢ ♣❛t❤s ✇✐t❤ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥ ✭❇❱ ❢♦r s❤♦rt✮ t❛❦✐♥❣ ✈❛•✉❡s ✐♥ S ′ ✳ ❋✐rst✱ ✇❡ ♥♦t❡ t❤❛t s✐♥❝❡ S(R N ) ✐s ❛ ▼♦♥t❡• s♣❛❝❡✱ ❡✈❡r② ✇❡❛❦•② ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ✐♥ S ′ (R N ) ❛•s♦ ✐s str♦♥❣•② ❝♦♥✈❡r❣❡♥t✳ ▼♦r❡♦✈❡r✱ ✇❡ r❡❝❛•• t❤❛t ❛ ❢✉♥❝t✐♦♥ ❢r♦♠ ❛ ♠❡tr✐❝ s♣❛❝❡ t♦ ❛ t♦♣♦•♦❣✐❝❛• s♣❛❝❡ ✐s ❝♦♥t✐♥✉♦✉s ✐❢ ❛♥❞ ♦♥•② ✐❢ ✐t ✐s s❡q✉❡♥t✐❛••② ❝♦♥t✐♥✉♦✉s✳ ❚❤❡r❡❢♦r❡ t❤❡ ♥♦t✐♦♥ ♦❢ ❛ ❝♦♥t✐♥✉♦✉s ♣❛t❤ ♦♥ S ′ (R N ) ✇✐•• ♥♦t ❞❡♣❡♥❞ ♦♥ ✇❤✐❝❤ t♦♣♦•♦❣② ✇❡ ♣✐❝❦✱ ❛♥❞ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ❢♦••♦✇✐♥❣✿ ❉❡✜♥✐t✐♦♥ ✷✳✸✳✷✳✶ ✭❈♦♥t✐♥✉♦✉s S ′ ✲✈❛•✉❡❞ ♣❛t❤s✮✳ ❚❤❡ ♣❛t❤✿ T : [0, 1] -→ S ′ (R N ) ✭✷✳✾✵✮ ✐s s❛✐❞ t♦ ❜❡ ❝♦♥t✐♥✉♦✉s ✐❢ ❛♥❞ ♦♥•② ✐❢ ❢♦r ❛•• φ ∈ S(R N )✱ t → T (t), φ ✐s ❛ ❢✉♥❝t✐♦♥ ✐♥ C [0, 1], R N ✳ ❆ s✐♠✐•❛r ❞❡✜♥✐t✐♦♥ ❤♦•❞s ❢♦r ❛ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥ ♣❛t❤✳ ❲❡ ♥♦✇ t✉r♥ ♦♥ ❛ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥ ♣❛t❤s ♦♥ S ′ (R N ) ✇❤✐❝❤ ✇✐•• ❜❡ ❝♦♥✈❡♥✐❡♥t t♦ ♦✉r ♣✉r♣♦s❡✳ ▲❡t T ❜❡ ❛ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥ ♣❛t❤✳ ❚❤❡♦r❡♠ ✷✳✸✳✷✳✶ ✭•r♦t❤❡♥❞✐❡❝❦ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❇❱ ♣❛t❤s✮✳ ❈♦♥s✐❞❡r T ∈ BV ([0, 1], S ′ )✳ ❚❤❡♥✱ t❤❡r❡ ❡①✐st s❡q✉❡♥❝❡s (λ n ) ∈ l 1 ✱ (V n ) ❜♦✉♥❞❡❞ ✐♥ ❇❱✱ ❛♥❞ (F n ) ❛♥ ❡q✉✐❝♦♥t✐♥✉♦✉s ❢❛♠✐•② ♦❢ S ′ (R N ) ❞❡♣❡♥❞✐♥❣ ♦♥•② ♦♥ T s✉❝❤ t❤❛t ∀φ ∈ S(R N )✿ T (t), φ = ∞ n=1 λ n • V n (t) • F n , φ ✭✷✳✾✶✮ ❆♥ ❛♥❛•♦❣✉❡ r❡s✉•t ❤♦•❞s ❢♦r ❝♦♥t✐♥✉♦✉s ♣❛t❤s✳ ✷✳✸✳ ❚❍❊ ❚❖P❖▲❖•■❊❙ ❖◆ S ❆◆❉ S ′ ✺✶ Pr♦♦❢✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ ❢♦••♦✇✐♥❣ •✐♥❡❛r ♠❛♣♣✐♥❣✿ l = S ′ (R N ) → BV φ → (t → T (t), φ ) ✭✷✳✾✷✮ ◆♦✇ •❡t (φ n ) ∈ S(R N ) N ❛♥❞ φ ∈ S(R N ) s✉❝❤ t❤❛t φ n S → φ ❛♥❞ t❤❡r❡ ❡①✐sts g ∈ BV s✉❝❤ t❤❛t l(φ) BV → g✳ ❚❤❡♥ s✐♥❝❡ ❇❱ ❝♦♥✈❡r❣❡♥❝❡ ✐♠♣•✐❡s ♣♦✐♥t✇✐s❡ ❝♦♥✈❡r❣❡♥❝❡ ✐t ✐s ✐♠♠❡❞✐❛t❡•② ❝❤❡❝❦❡❞ t❤❛t g = l(φ)✱ ❤❡♥❝❡ t❤❡ ❣r❛♣❤ ♦❢ l ✐s ❝•♦s❡❞✳ ◆♦✇ s✐♥❝❡ S(R N ) ✐s ❛ ❋ré❝❤❡t s♣❛❝❡ ❛♥❞ ❇❱ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡ ✇❡ ❝❛♥ ✉s❡ ❛ ✈❡rs✐♦♥ ♦❢ t❤❡ ❝•♦s❡❞ ❣r❛♣❤ t❤❡♦r❡♠ t♦ ❝♦♥❝•✉❞❡ t❤❛t l ✐s ❝♦♥t✐♥✉♦✉s✳ ❋✐♥❛••②✱ s✐♥❝❡ S(R N ) ✐s ❛ ♥✉❝•❡❛r s♣❛❝❡ l ✐s ❛ ♥✉❝•❡❛r ♠❛♣♣✐♥❣✳ ❚❤❡r❡❢♦r❡ ✇❡ ✜♥✐s❤ t❤❡ ♣r♦♦❢ ❜② ✉s✐♥❣ t❤❡ •r♦t❤❡♥❞✐❡❝❦ t❤❡♦r❡♠ ❆✳✹✳✸✳✶✳ ❚❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ✐♥ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠ ✇✐•• ❜❡ ✉s❡❢✉• ❜❡❝❛✉s❡ ♦❢ t❤❡ ❢♦•✲ •♦✇✐♥❣ ❢❛❝t✿ Pr♦♣♦s✐t✐♦♥ ✷✳✸✳✷✳✶✳ ❈♦♥s✐❞❡r s♦♠❡ p > 1✳ ❆♥② ❡q✉✐❝♦♥t✐♥✉♦✉s s❡t E ⊂ S(R N ) ✐s ✐♥❝•✉❞❡❞ ✐♥ ♦♥❡ ♦❢ t❤❡ S p,-ν ❢♦r s♦♠❡ ν > 0✳ Pr♦♦❢✳ ❚❤❡ ❡q✉✐❝♦♥t✐♥✉✐t② ♦❢ E ✐♥ S ♠❡❛♥s t❤❛t t❤❡ ❢♦••♦✇✐♥❣ s❡t✿ φ ∈ S(R N )|∀T ∈ E, | T, φ | < ǫ ✭✷✳✾✸✮ ✐s ❛ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ 0✳ ❚❤❡r❡❢♦r❡ ✐t ❝♦♥t❛✐♥s ❛ ❜❛•• ❢♦r ♦♥❡ ♦❢ t❤❡ s❡♠✐♥♦r♠s • S k,2 ✱ ✐❡ t❤❡r❡ ❡①✐sts k ∈ Z ❛♥❞ η > 0 s✉❝❤ t❤❛t✿ φ S k,2 < η ⇒ ∀T ∈ E | T, φ | < ǫ ✭✷✳✾✹✮ ❲❡ ♥♦✇ ✐♥tr♦❞✉❝❡✿ g n := η 2λ k n • f n ✭✷✳✾✺✮ ❈•❡❛r•②✿ g n S k,2 = η 2 < η ✭✷✳✾✻✮ ❛♥❞ ✐❢ T ∈ E✿ T, fn = η 2λ k n • c n (T ) ✭✷✳✾✼✮ s♦ ✇❡ ❡✈❡♥t✉❛••② ❣❡t✿ |c n (T )| < 2ǫ η • λ k n ✭✷✳✾✽✮ ❛♥❞ t❤❡r❡❢♦r❡✿ T 2 S k,2 < 2ǫ η • n λ k-ν n ✭✷✳✾✾✮ ❛♥❞ t❤✐s ✐s ❛ ✜♥✐t❡ s✉♠ ❢♦r ❜✐❣ ❡♥♦✉❣❤ ν✱ s♦ E ⊂ S 2,-ν ✳ ❚❤❡ ♣r♦♦❢ ✐s ✜♥✐s❤❡❞ ❜② ✉s✐♥❣ t❤❡ ❡q✉✐✈❛•❡♥❝❡ ♦❢ t❤❡ ❋ré❝❤❡t s♣❛❝❡ t♦♣♦•♦❣✐❡s ♦♥ S ❛♥❞ ♦♥ ✐ts ❞✉❛• ❛s ✇❛s st❛t❡❞ ✐♥ t❤❡♦r❡♠ ✷✳✸✳✵✳✸✳ ✺✷ ❈❍❆P❚❊| ✷✳ ❚❖P❖▲❖•■❈❆▲ ❙❚|❯❈❚❯|❊ ❖❋ S ❆◆❉ S ′ ❆s ❛♥ ✐♠♠❡❞✐❛t❡ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ t✇♦ ♣r❡✈✐♦✉s r❡s✉•ts✱ t❤❡r❡ ✐s t❤❡✿ ❚❤❡♦r❡♠ ✷✳✸✳✷✳✷✳ ▲❡t T ∈ BV ([0, 1], S ′ )✳ ❋♦r ❡✈❡r② p ∈ [1, ∞] t❤❡r❡ ❡①✐sts s ∈ R s✉❝❤ t❤❛t T ∈ BV ([0, 1], S p,s )✳ ❛♥❞ ♦♥❝❡ ❛❣❛✐♥ t❤❡r❡ ✐s ❛ s✐♠✐•❛r r❡s✉•t ❢♦r ❝♦♥t✐♥✉♦✉s ♣❛t❤s✳ ❈❤❛♣t❡r ✸ ▲✐❢t✐♥❣ ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜•❡s ❜② ❞✐str✐❜✉t✐♦♥s ❚❤❡ •✐❢t✐♥❣ ♦❢ s♠♦♦t❤ r❛♥❞♦♠ ✈❛r✐❛❜•❡s ✐s ❛ ✇❡•• ❞♦❝✉♠❡♥t❡❞ t♦♣✐❝✱ s❡❡ ❢♦r ❡①❛♠♣•❡✿ ❬✽✶(✱ ❬✷✺(✱ ❬✼✺( ♦r ❬✽✸(✳ ❍❡r❡ ✇❡ tr② t♦ ✐♠♣r♦✈❡ t❤❡ ✉s✉❛• r❡s✉•ts ❜② ✉s✐♥❣ t❤❡ ✐♥t❡r♣♦•❛t✐♦♥ t❤❡♦r② t♦ ❡①t❡♥❞ t❤❡♠ t♦ ❢r❛❝t✐♦♥❛• ✐♥❞✐❝❡s ❢♦r t❤❡ ❞✐str✐❜✉t✐♦♥ s♣❛❝❡ ❛♥❞ t❤❡ •r♦ss✲❙♦❜♦•❡✈ s♣❛❝❡✳ ❚♦ ❞♦ s♦ ✇❡ ❢♦••♦✇ t❤❡ ✐❞❡❛s ✐♥ ❬✽✸( ❜✉t ✇❡ ❡①t❡♥❞ t❤❡♠ ✐♥ s❡✈❡r❛• ✇❛②s✳ ❋✐rst ✇❡ ✇✐•• ❝♦♥s✐❞❡r ❛♥② t❡♠♣❡r❡❞ ❞✐str✐❜✉t✐♦♥ ✐♥st❡❛❞ ♦❢ •✉st ❡•❡♠❡♥ts ✐♥ ❛ ♥❡❣❛t✐✈❡ ❇❡ss❡• ♣♦t❡♥t✐❛• s♣❛❝❡✳ ❲❡ ❛❝❤✐❡✈❡ t❤✐s ❜② ✇♦r❦✐♥❣ ✐♥ t❤❡ s♣❛❝❡s S p,δ ❀ ✐♥❞❡❡❞✿ S = S p,δ ✭✸✳✶✮ s♦✿ S ′ = S p,δ ✭✸✳✷✮ ✇❤✐•❡ t❤❡ ✉♥✐♦♥ ♦❢ ❛•• ❇❡ss❡• ♣♦t❡♥t✐❛• s♣❛❝❡s ✐s str✐❝t•② ✐♥❝•✉❞❡❞ ✐♥ S ′ ✳ ❙❡❝♦♥❞ ✇❡ ♣❛② ❡①tr❛ ❛tt❡♥t✐♦♥ t♦ s♦♠❡ ❝♦♥st❛♥ts ✐♥ ♦✉r ♠❛•♦r✐③❛t✐♦♥s ❛s ✇❡ ✇✐•• ♥❡❡❞ t❤❛t ♠✉❝❤ ♣r❡❝✐s✐♦♥ ❢✉rt❤❡r ♦♥✳ ❚❤✐r❞✱ ✇❡ ❣✐✈❡ r❡s✉•ts ❢♦r t❤❡ •✐❢t✐♥❣ ♦❢ t❤❡ S ∞ -,-δ s♣❛❝❡s✳ ❚❤❡ r❡s✉•ts ❛r❡ ♠♦r❡ ❝♦♠♣•✐❝❛t❡❞ t♦ st❛t❡ ❛♥❞ ♣r♦✈❡ ❜✉t ❞♦ ♥♦t r❡q✉✐r❡ ❛ ♣r✐♦r✐ ❛ss✉♠♣t✐♦♥s ♦♥ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ❞❡♥s✐t② ❢♦r t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜•❡ ✇❡ ❝♦♥s✐❞❡r✳ ❚❤✐s ❛••♦✇s ✉s t♦ ❣✐✈❡ r❡s✉•ts ❢♦r t❤❡ •✐❢t✐♥❣ ♦❢ r❡❣✉•❛r ❡♥♦✉❣❤ r❛♥❞♦♠ ✈❛r✐❛❜•❡s ❜② r❡❣✉•❛r ❡♥♦✉❣❤ r❛♥❞♦♠ ❞✐str✐❜✉t✐♦♥s✳ ❋♦r ❛ ♣r✐♠❡r ✐♥ ✐♥t❡r♣♦•❛t✐♦♥ t❤❡♦r②✱ ✇❡ r❡❢❡r t♦ ❛♣♣❡♥❞✐① ❉ ❛♥❞ t♦ t❤❡ r❡❢❡r✲ ❡♥❝❡s t❤❡r❡✐♥✳ ✺✸ ✺✹ ❈❍❆P❚❊| ✸✳ ▲■❋❚■◆• ❖❋ |❆◆❉❖▼ ❱❆|■❆❇▲❊❙ ❇( ❉■❙❚|■❇❯❚■❖◆❙ ✸✳✶ ■♥t❡r♣♦•❛t✐♦♥ ♦❢ •r♦ss✲❙♦❜♦•❡✈ s♣❛❝❡s ❲❡ st❛rt ❜② r❡❝❛••✐♥❣ s♦♠❡ r❡s✉•ts ❢♦r t❤❡ ✐♥t❡r♣♦•❛t✐♦♥ ♦❢ •r♦ss✲❙♦❜♦•❡✈ s♣❛❝❡s✳ ❲❡ ❝♦♣② t❤❡ ✜rst ♦♥❡ ❢r♦♠ ❬✽✷(✿ ❚❤❡♦r❡♠ ✸✳✶✳✵✳✸ ✭❈♦♠♣•❡① ✐♥t❡r♣♦•❛t✐♦♥ ♦❢ •r♦ss✲❙♦❜♦•❡✈ s♣❛❝❡s✮✳ ❲❡ ❝♦♥s✐❞❡r p 1 , p 2 > 1✱ s 1 , s 2 ∈ R ❛♥❞ 0 < θ < 1✳ ▲❡t p ❛♥❞ s ❜❡ s✉❝❤ t❤❛t✿ 1 p = θ p 1 + 1-θ p 2 ❛♥❞ s = θs 1 + (1 -θ)s 2 ✳ ❚❤❡♥ [D p 1 ,s 1 , D p 2 ,s 2 ] θ = D p,s ✳ ■t ✐s ✐♥t❡r❡st✐♥❣ t♦ ♥♦t❡ t❤❛t t❤✐s ✐s r❡•❛t❡❞ t♦ t❤❡ ❢♦••♦✇✐♥❣ r❡s✉•t ✇❤✐❝❤ ✇❡ ❝♦♣② ❢r♦♠ ❬✻✻(✿ ❚❤❡♦r❡♠ ✸✳✶✳✵✳✹✳ ▲❡t 1 < p < ∞ ❛♥❞ s 1 < s 2 ∈ R❀ •❡t λ ∈ [0, 1] ❛♥❞ s❡t✿ s = λ • s 1 + (1 -λ) • s 2 ✭✸✳✸✮ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ✉♥✐✈❡rs❛• ❝♦♥st❛♥t C(p, s 1 , s 2 , λ) s✉❝❤ t❤❛t ❢♦r ❡✈❡r② φ ∈ D t❤❡ ❢♦••♦✇✐♥❣ ❤♦•❞s✿ X Dp,s ≤ C • X λ Dp,s 1 • X 1-λ Dp,s 2 ✭✸✳✹✮ Pr♦♦❢✳ ❚❤✐s ✐s ❛♥ ✐♠♠❡❞✐❛t❡ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠ ❛♥❞ t❤❡ ❞❡✜♥✐✲ t✐♦♥ ♦❢ ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ ❢✉♥❝t♦r✳ ❲❡ ❛•s♦ r❡❢❡r t♦ ❬✻✻( ❢♦r ❛♥ ❡①♣•❝✐t ♣r♦♦❢ ✇❤✐❝❤ ❞♦❡s ♥♦t ❞✐r❡❝t•② ✉s❡ ✐♥t❡r♣♦•❛t✐♦♥ t❤❡♦r②✱ ❛•t❤♦✉❣❤ t❤❡ ✐❞❡❛s ❛r❡ t❤❡ s❛♠❡✳ ❚❤❡♦r❡♠ ✸✳✶✳✵✳✹ ❤❛s ❛♥ ♦❜✈✐♦✉s ❜✉t ✈❡r② ✉s❡❢✉• ❝♦r♦••❛r②✿ ❈♦r♦••❛r② ✸✳✶✳✵✳✶✳ ▲❡t p, s 1 , s 2 ❛♥❞ λ ❜❡ ❛s ✐♥ t❤❡♦r❡♠ ✸✳✶✳✵✳✹ ❛♥❞ •❡t (X n ) ❛ s❡q✉❡♥❝❡ ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜•❡s s✉❝❤ t❤❛t (X n ) ❝♦♥✈❡r❣❡s ✐♥ D p,s 1 ❛♥❞ ✐s ❜♦✉♥❞❡❞ ✐♥ D p,s 2 ✳ ❚❤❡♥ ❢♦r ❡✈❡r② s < s 2 (X n ) ❝♦♥✈❡r❣❡s ✐♥ D p,s ✳ |❡♠❛r❦ ✸✳✶✳✵✳✶✳ ■t ✐s ✐♥t❡r❡st✐♥❣ t♦ ❝♦♠♣❛r❡ t❤✐s •❛st r❡s✉•t t♦ •❡♠♠❛ ✶✳✷✳✻✳✷✳ ❆s ❛♥♦t❤❡r ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ❝♦♠♣•❡① ✐♥t❡r♣♦•❛t✐♦♥ ♦❢ •r♦ss✲❙♦❜♦•❡✈ s♣❛❝❡s ✇❡ ❣✐✈❡ t✇♦ ❡①t❡♥s✐♦♥s ♦❢ ❍ö•❞❡r✬s t❤❡♦r❡♠ ✇❤✐❝❤ ✇❡ ❛•s♦ t❛❦❡ ❢r♦♠ ❬✽✷(✿ ❚❤❡♦r❡♠ ✸✳✶✳✵✳✺ ✭❍ö•❞❡r t❤❡♦r❡♠ ❢♦r •r♦ss✲❙♦❜♦•❡✈ s♣❛❝❡s✮✳ ❈♦♥s✐❞❡r s > 0 ❛♥❞ 1 < p, q, r < ∞ s✉❝❤ t❤❛t 1/p = 1/q + 1/r✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ✉♥✐✈❡rs❛• ❝♦♥st❛♥t C(s, q, r) s✉❝❤ t❤❛t ❢♦r ❡✈❡r② X, Y ∈ D✿ X • Y Dp,s ≤ C • X Dq,s • Y Dr,s ✭✸✳✺✮ Pr♦♦❢✳ ❋✐rst ✇❡ ♣r♦✈❡ t❤❡ r❡s✉•t ❢♦r s ∈ N✳ ■♥❞❡❡❞ ∇(X • Y ) ✐s ❝♦♠♣✉t❡❞ ❜② t❤❡ ▲❡✐❜♥✐③ ❢♦r♠✉•❛ ❛♥❞ t❤❡♥ t❤❡ r❡s✉•t ✐s •✉st ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ✉s✉❛• ❍ö•❞❡r t❤❡♦r❡♠✳ ❚❤❡ ❣❡♥❡r❛• r❡s✉•t ✐s t❤❡♥ ♦❜t❛✐♥❡❞ ❜② ❝♦♠♣•❡① ✐♥t❡r♣♦•❛t✐♦♥✳ ✸✳✶✳ ■◆❚❊|P❖▲❆❚■❖◆ ❖❋ •|❖❙❙✲❙❖❇❖▲❊❱ ❙P❆❈❊❙ ✺✺ ❚❤❡♦r❡♠ ✸✳✶✳✵✳✻ ✭❍ö•❞❡r t❤❡♦r❡♠ ❢♦r •r♦ss✲❙♦❜♦•❡✈ ❞✐str✐❜✉t✐♦♥ s♣❛❝❡s✮✳ ▲❡t 1 < p, q, r < ∞ s✉❝❤ t❤❛t 1/p = 1/q + 1/r ❛♥❞ s > 0✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ✉♥✐✈❡rs❛• ❝♦♥st❛♥t C(s, q, r) s✉❝❤ t❤❛t ❢♦r ❡✈❡r② X, Y ∈ D✿ X • Y D p,-s ≤ C • X Dq,s • Y D r,-s ✭✸✳✻✮ Pr♦♦❢✳ ❇② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❞✉❛•✐t② ♦♥ t❤❡ •r♦ss✲❙♦❜♦•❡✈ s♣❛❝❡s✱ t❤❡r❡ ✐s ♦♥ ♦♥❡ ❤❛♥❞✿ X • Y D p,-s = sup φ D p * ,s =1 X • Y, φ ✭✸✳✼✮ ❛♥❞ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞✿ X • Y, φ = Y, X • φ ≤ Y D r,-s • X • φ D r * ,s ✭✸✳✽✮ ◆♦✇ s✐♥❝❡ 1/r * = 1/p * + 1/q✱ ❛♣♣•②✐♥❣ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠ t❤❡r❡ ✐s✿ X • φ D r * ,s ≤ X Dq,s • φ D p * ,-s ✭✸✳✾✮ ❛♥❞ t❤❡r❡❢♦r❡✿ X • Y, φ ≤ X Dq,s • Y D r,-s • φ D p * ,-s ✭✸✳✶✵✮ s♦ ✇❡ ❛r❡ ❞♦♥❡✳ ❖❢t❡♥ ✇❡ ✇✐•• s✐♠♣•② r❡❢❡r t♦ t❤❡s❡ •❛st t✇♦ t❤❡♦r❡♠s ❛s ❍ö•❞❡r✬s ✐♥❡q✉q•✐t②❀ ❤♦♣❡❢✉••② t❤❡ ❝♦♥t❡①t ✇✐•• ❛•✇❛②s ❝•❛r✐❢② ✇❤✐❝❤ ♦♥❡ ✇❡ ❛r❡ ❛♣♣•②✐♥❣✳ •♦✐♥❣ ❢✉rt❤❡r ✇✐t❤ t❤❡ ♥♦t✐♦♥ ♦❢ ✐♥t❡r♣♦•❛t✐♦♥✱ ✐t ✐s ✐♥t❡r❡st✐♥❣ t♦ ❝♦♥s✐❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ r❡❛• ✐♥t❡r♣♦•❛t✐♦♥ ❢✉♥❝t♦rs ♦♥ •r♦ss✲❙♦❜♦•❡✈ s♣❛❝❡s✳ ❚❤❡r❡❢♦r❡ ❢♦••♦✇✐♥❣ ❬✽✸( ✇❡ ✐♥tr♦❞✉❝❡✿ ❉❡✜♥✐t✐♦♥ ✸✳✶✳✵✳✷ ✭|❡❛• ✐♥t❡r♣♦•❛t✐♦♥ ♦❢ •r♦ss✲❙♦❜♦•❡✈ s♣❛❝❡s✮✳ ▲❡t p > 1 ❛♥❞ s ∈ R✳ ❲❡ ❞❡✜♥❡ ❛ ❢❛♠✐•② ♦❢ ❇❛♥❛❝❤ s♣❛❝❡s E p,s ❛s ❢♦••♦✇s✿ • ✐❢ s ∈ Z t❤❡♥ E p,s = D p,s • ❖t❤❡r✇✐s❡ •❡t k = [s]✱ σ = {s}✳ ❚❤❡♥ E p,s = (D p,k+1 , D p,k ) 1-σ,p ✳ ❲❡ s♣❡❝✐❛•✐③❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ r❡❛• ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡s t♦ ♦✉r ❝❛s❡ ❛♥❞ ♦❜t❛✐♥ ❛ ❝❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ s♣❛❝❡ E p,s ❛♥❞ ❛♥ ❡①♣r❡ss✐♦♥ ❢♦r ✐ts ♥♦r♠❀ ✇❡ r❡❢❡r t♦ ❛♣♣❡♥❞✐① ❉ ❛♥❞ t❤❡ r❡❢❡r❡♥❝❡s t❤❡r❡✐♥ ♦r t♦ ❬✽✸( ❢♦r t❤❡ ❞❡t❛✐•s✳ ■♥ ♣r❛❝t✐❝❡✱ ✇❡ ✇✐•• ♠♦st•② r❡•② ♦♥ t❤❡ ❢♦••♦✇✐♥❣ r❡s✉•t✿ ❚❤❡♦r❡♠ ✸✳✶✳✵✳✼ ✭s❡♠✐❣r♦✉♣ ❝❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ s♣❛❝❡ E p,s ✮✳ F p Ep,s = F p D p,k + 1 0 t -1-pσ F -T t F p D p,k dt ✺✻ ❈❍❆P❚❊| ✸✳ ▲■❋❚■◆• ❖❋ |❆◆❉❖▼ ❱❆|■❆❇▲❊❙ ❇( ❉■❙❚|■❇❯❚■❖◆❙ Pr♦♦❢✳ ❚❤✐s ✐s ❛ s♣❡❝✐❛•✐③❛t✐♦♥ ♦❢ ❛ ❣❡♥❡r❛• r❡s✉•t ♦♥ s❡♠✐❣r♦✉♣ ✐♥t❡r♣♦•❛t✐♦♥ t♦ t❤❡ ♦♣❡r❛t♦r (Id + L) 1/2 ✳ ❙❡❡ ❛♣♣❡♥❞✐① ❉ ❢♦r t❤❡ ❞❡t❛✐•s✳ ❆•s♦ ❛s ❛♥ ❛♣♣•✐❝❛t✐♦♥ ♦❢ t❤❡ ❞✉❛•✐t② t❤❡♦r❡♠ ✐♥ ❉ ✇❡ ❣❡t t❤❡✿ ❚❤❡♦r❡♠ ✸✳✶✳✵✳✽ ✭❉✉❛•✐t② ❢♦r t❤❡ E p,s s♣❛❝❡s✮✳ ■❢ 1 < p < ∞ ❛♥❞ s ∈ R t❤❡♥✿ E * p,s = E p * ,-s ✭✸✳✶✶✮ ❚❤❡ ❢♦••♦✇✐♥❣ r❡s✉•t ❤♦✇ ❝•♦s❡•② r❡•❛t❡❞ t❤❡ ❝♦♠♣•❡① ❛♥❞ r❡❛• ✐♥t❡r♣♦•❛t❡❞ •r♦ss✲❙♦❜♦•❡✈ s♣❛❝❡s ❛r❡✳ ■t ✇✐•• ❛•s♦ t✉r♥ ♦✉t t♦ ❜❡ ✈❡r② ❝♦♥✈❡♥✐❡♥t ❛s ✐t ✇✐•• ❛••♦✇ ✉s t♦ ♣r♦✈❡ ❛ ♥✉♠❜❡r ♦❢ r❡s✉•ts ❢♦r ✇❤✐❝❤❡✈❡r ❢❛♠✐•② ♦❢ s♣❛❝❡s ✐s ♠♦r❡ ❝♦♥✈❡♥✐❡♥t t♦ ✉s❡ ❛♥❞ t♦ ❛✉t♦♠❛t✐❝❛••② ♦❜t❛✐♥ t❤❡♠ ❢♦r t❤❡ ♦t❤❡r ❢❛♠✐•② ✧✉♣ t♦ ǫ✧✳ ❚❤❡♦r❡♠ ✸✳✶✳✵✳✾ ✭✐♥❝•✉s✐♦♥s ❜❡t✇❡❡♥ t❤❡ r❡❛• ❛♥❞ ❝♦♠♣•❡① s♣❛❝❡s✮✳ ❋♦r ❡✈❡r② p > 1✱ s ∈ R ❛♥❞ ǫ > 0 ♦♥❡ ❤❛s t❤❡ ❢♦••♦✇✐♥❣ ❝♦♥t✐♥✉♦✉s ✐♥•❡❝t✐♦♥s✿ E p,s+ǫ ֒→ D p,s ֒→ E p,s-ǫ ✭✸✳✶✷✮ Pr♦♦❢✳ ❲❡ ❝♦✉•❞ ✉s❡ ❛ ❣❡♥❡r❛• r❡s✉•t ❢r♦♠ s❡♠✐❣r♦✉♣ ✐♥t❡r♣♦•❛t✐♦♥ t❤❡♦r② ❜✉t ❞✉❡ t♦ t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤✐s t❤❡♦r❡♠ ❢♦r t❤❡ s❛❦❡ ♦❢ ❝♦♠♣•❡t❡♥❡ss ✇❡ ♣r♦✈✐❞❡ ❛ ❢✉•• ♣r♦♦❢ ❢♦••♦✇✐♥❣ ❬✽✸(✳ ❲❡ ♥♦t❡ t❤❛t ✇❡ ♦♥•② ♥❡❡❞ t♦ ♣r♦✈❡ t❤❡ ✜rst ✐♥❝•✉s✐♦♥❀ ✐♥❞❡❡❞ t❤❡ s❡❝♦♥❞ ♦♥❡ t❤❡♥ ❛✉t♦♠❛t✐❝❛••② ❢♦••♦✇s ❜② t❤❡ ❞✉❛•✐t② t❤❡♦r❡♠✳ ■❢ s ∈ Z t❤❡r❡ ✐s ♥♦t❤✐♥❣ t♦ ♣r♦✈❡✳ ❖t❤❡r✇✐s❡ •❡t ✉s ✇r✐t❡✿ [s] = k ∈ Z ❛♥❞ {s} = σ ∈]0, 1[ ❛♥❞ •❡t X ∈ D✳ ❚❤❡♥✱ ❛s ♣❡r t❤❡ ❢r❛❝t✐♦♥♥❛• ♣♦✇❡r r❡s✉•ts ♣r♦✈✐❞❡❞ ✐♥ ❛♣♣❡♥❞✐① ❈ t❤❡r❡ ✐s t❤❡ ❢♦••♦✇✐♥❣ r❡•❛t✐♦♥✱ ✇❤❡r❡ t❤❡ |❍❙ ✐s ❛ ❇♦❝❤♥❡r ✐♥t❡❣r❛•✿ (I -L) σ/2 X = σ Γ(1 -σ) • ∞ 0 t -1+σ • [X -T t X] dt ✭✸✳✶✸✮ ■♥ t❤❡ r❡♠❛✐♥❞❡r ♦❢ t❤✐s ♣r♦♦❢✱ C ✇✐•• ❞❡♥♦t❡ s♦♠❡ ✉♥✐✈❡rs❛• ❝♦♥st❛♥t ♣♦ss✐❜•② ❞❡♣❡♥❞✐♥❣ ♦♥ p, k, σ, ǫ✱ ❡t❝✳ ❜✉t ♥♦t ♦♥ X✳ ❚❤❡r❡ ✐s✿ X p Dp,s = (I -L) σ/2 X p D p,k ≤ C • ∞ 0 t -1+σ • X -T t X p D p,k dt p = C • 1 0 t -1+σ • X -T t X p D p,k dt + ∞ 1 t -1+σ • X -T t X p D p,k dt p ✭✸✳✶✹✮ ✸✳✶✳ ■◆❚❊|P❖▲❆❚■❖◆ ❖❋ •|❖❙❙✲❙❖❇❖▲❊❱ ❙P❆❈❊❙ ✺✼ ■♥ t❤❡ ❛❜♦✈❡✱ t❤❡ ∞ 1 ✐♥t❡❣r❛• ✐s ❞♦♠✐♥❛t❡❞ ❜② C • X D p,k ❛♥❞ ❢♦r t❤❡ 1 0 ✐♥t❡❣r❛• ✇❡ ✇r✐t❡✿ 1 0 t -1+σ • X -T t X p D p,k dt = 1 0 t -1 p * -ǫ • t -1 p +ǫ+σ • X -T t X p D p,k dt ≤ C • t -(1+(σ+ǫ)p) • X -T t X p D p,k dt 1 p ✭✸✳✶✺✮ ❜② ❍ö•❞❡r✬s t❤❡♦r❡♠✳ ❲❡ ✜♥✐s❤ t❤❡ ♣r♦♦❢ ❜② ✐♥•❡❝t✐♥❣ t❤❡s❡ t✇♦ ♠❛•♦r✐③❛t✐♦♥s ✐♥ ✭✸✳✶✹✮ ❛♥❞ ✉s✐♥❣ t❤❡♦r❡♠ ✸✳✶✳✵✳✼✳ ◆♦✇ •❡t ✉s ✐♥tr♦❞✉❝❡ t❤❡ ❢♦••♦✇✐♥❣ q✉❛♥t✐t✐❡s✿ ❢♦r F ∈ L p ✇❡ s❡t✿ • f (0) = F • f (t) = 1 t • t 0 T τ F dτ ✐❢ t > 0 ❙✐♥❝❡ (I -L) 1/2 ❝♦♠♠✉t❡s t♦ ❛♥② T τ ✐t ✐s ❡❛s✐•② ✈❡r✐✜❡❞ t❤❛t t❤❡ f (t) ❛r❡ s♠♦♦t❤ ❢♦r t > 0 ❛♥❞✿ Pr♦♣♦s✐t✐♦♥ ✸✳✶✳✵✳✷✳ ■❢ F ∈ D p,s t❤❡♥✿ • f (t) Dp,s ≤ F Dp,s • f (t) -F Dp,s → 0 ❲❡ ✇✐•• ♥❡❡❞ t❤❡ ❢♦••♦✇✐♥❣ ❡st✐♠❛t❡s ♦♥ f ✿ ▲❡♠♠❛ ✸✳✶✳✵✳✶✳ ▲❡t F ∈ E p,s ❢♦r s♦♠❡ p > 1 ❛♥❞ s > 0✳ ❲❡ ♥♦t❡ k = [s] ❛♥❞ σ = {s}✳ ❚❤❡♥ t❤❡r❡ ❡①✐st ❝♦♥st❛♥ts C i ❞❡♣❡♥❞✐♥❣ ♦♥•② ♦♥ p ❛♥❞ s s✉❝❤ t❤❛t✿ 1 0 t -1+p(1-σ) f (t) p D p,k dt ≤ C 1 F p Ep,s ✭✸✳✶✻✮ 1 0 t -1+p(1-σ) f ′ (t) p D p,k dt ≤ C 2 F p Ep,s ✭✸✳✶✼✮ 1 0 t -1+p(1-σ) f (t) p D p,k+1 dt ≤ C 3 F p Ep,s ✭✸✳✶✽✮ Pr♦♦❢✳ ❚❤❡ ✜rst ♣♦✐♥t ✐s ♦❜✈✐♦✉s ❜❡❝❛✉s❡ f ✐s ❜♦✉♥❞❡❞ ✐♥ D p,k ❛s ✇❛s st❛t❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s •❡♠♠❛ ❛♥❞ ❜❡❝❛✉s❡ ♦❢ t❤❡♦r❡♠ ✸✳✶✳✵✳✾✳ ❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ s❡❝♦♥❞ ♣♦✐♥t ✇✐•• r❡q✉✐r❡ t❤❡ ❢♦••♦✇✐♥❣ s✐♠♣•❡ r❡s✉•t ✇❤✐❝❤ ✇❡ ❜♦rr♦✇ ❢r♦♠ ❬✶(✿ ✺✽ ❈❍❆P❚❊| ✸✳ ▲■❋❚■◆• ❖❋ |❆◆❉❖▼ ❱❆|■❆❇▲❊❙ ❇( ❉■❙❚|■❇❯❚■❖◆❙ ▲❡♠♠❛ ✸✳✶✳✵✳✷✳ ▲❡t φ ❜❡ ❛ s❝❛•❛r✲✈❛•✉❡❞ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ R + ❛♥❞ •❡t ψ(t) = 1 t t 0 φ(s)ds ✭✸✳✶✾✮ ❚❤❡♥ ❢♦r 1 ≤ p < ∞ ❛♥❞ ❢♦r ν✱ θ s✉❝❤ t❤❛t ν + 1/p = θ < 1 ♦♥❡ ❤❛s✿ ∞ 0 t νp |ψ(t)| p dt ≤ 1 1 -θ p • ∞ 0 t νp |φ(t)| p dt ✭✸✳✷✵✮ ❚❤❡♥ ✇❡ ♥♦t✐❝❡ t❤❛t ❢♦r t > 0✿ f ′ (t) = - 1 t 2 • t 0 T τ F dτ + 1 t T t F = 1 t (T t F -F ) - 1 t 2 • t 0 (T τ F -F )dτ ✭✸✳✷✶✮ t❤❡r❡❢♦r❡✿ 1 0 t -1+p(1-σ) f ′ (t) p D p,k dt ≤ C • 1 0 t -1-pσ T t F -F p D p,k dt + 1 0 t -1-p-pσ t 0 (T τ F -F )dτ p D p,k dt ✭✸✳✷✷✮ ❚❤❡ ✜rst t❡r♠ ✐♥ t❤✐s s✉♠ ✐s ❝♦♥tr♦••❡❞ ❜② ✉s✐♥❣ t❤❡ s❡♠✐❣r♦✉♣ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ✐♥ t❤❡♦r❡♠ ✸✳✶✳✵✳✼✳ ❋♦r t❤❡ s❡❝♦♥❞ t❡r♠ ✇❡ ❛♣♣•② •❡♠♠❛ ✸✳✶✳✵✳✷ ✇✐t❤ ν = -1/p -σ✱ ❤❡♥❝❡ 1 -θ = 1 + σ ❛♥❞ ✇❡ ♦❜t❛✐♥✿ 1 0 t -1-pσ t 0 T τ F -F t dτ p D p,k dt ≤ 1 (1 + σ) p 1 0 t -1-pσ T t F -F p D p,k dt ✭✸✳✷✸✮ s♦ t❤❡♥ ❛❣❛✐♥ t❤❡ s❡♠✐❣r♦✉♣ ❝❛r❛❝t❡r✐③❛t✐♦♥ ❝♦♥tr♦•s t❤❡ s❡❝♦♥❞ t❡r♠ ❛s ✇❛s ❞❡✲ s✐r❡❞✳ ❋♦r t❤❡ t❤✐r❞ ♣♦✐♥t ♦♥❡ ♥♦t✐❝❡s t❤❛t✿ (I + L) 1/2 f (t) = 1 t t 0 (I + L) 1/2 τ F dτ = - 1 t 1 0 T ′ τ F dτ = - 1 t (T t F -F ) ✭✸✳✷✹✮ t❤❡r❡❢♦r❡ f (t) D p,k+1 = 1 t (T t F -F ) D p,k+1 ✭✸✳✷✺✮ ❛♥❞ ❛s ♣r❡✈✐♦✉s•② t❤❡ s❡♠✐❣r♦✉♣ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ❝♦♥❝•✉❞❡s✳ ❲❡ ❛•s♦ ♥❡❡❞ ❡st✐♠❛t❡s ♦♥ 1/f (t)✿ ✸✳✶✳ ■◆❚❊|P❖▲❆❚■❖◆ ❖❋ •|❖❙❙✲❙❖❇❖▲❊❱ ❙P❆❈❊❙ ✺✾ ▲❡♠♠❛ ✸✳✶✳✵✳✸✳ ▲❡t F ∈ D ∞ -,k ❢♦r s♦♠❡ k ∈ N✳ ■❢ F > 0 ❛✳s✳ ❛♥❞ 1/F ∈ L ∞ - t❤❡♥ ❢♦r ❛♥② p > 1 ❛♥❞ ❛♥② p ′ > p t❤❡r❡ ❡①✐sts q > 1 ❞❡♣❡♥❞✐♥❣ ♦♥•② ♦♥ p✱ p ′ ❛♥❞ k s✉❝❤ t❤❛t t❤❡ ❢♦••♦✇✐♥❣ ❡st✐♠❛t❡ ❤♦•❞s✿ sup 0≤t≤1 1 f (t) D p,k ≤ 1 1 -e -1 k+1 • 1 F k+1 L p ′ + 1 1 -e -1 • 1 F L p • F k D q,k ✭✸✳✷✻✮ Pr♦♦❢✳ ❋✐rst✱ ❛♣♣•②✐♥❣ t❤❡ ▼❛••✐❛✈✐♥ ❝❛•❝✉•✉s ✈❡rs✐♦♥ ♦❢ ❋❛❛ ❞✐ ❇r✉♥♦✬s ❢♦r♠✉•❛✱ ✇❡ s❡❡ t❤❛t✿ ∇ k 1 f (t) = 1 f (t) k+1 • Π ✭✸✳✷✼✮ ✇❤❡r❡ Π ✐s ❛ •✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ∇ i 1 f (t)⊗• • •⊗∇ in f (t) ✇✐t❤ i 1 +• • •+i n = k✳ ❚❤❡♥✱ s✉❝❝❡ss✐✈❡ ❛♣♣•✐❝❛t✐♦♥s ♦❢ ❍ö•❞❡r✬s ✐♥❡q✉❛•✐t② ❛♥❞ ▼❡②❡r✬s ✐♥❡q✉❛•✐t✐❡s •❡❛❞ t♦✿ 1 f (t) D p,k ≤ 1 f (t) L p + 1 f (t) k+1 L p ′ • F k D q,k ✭✸✳✷✽✮ ❲❡ ♠❛② t❛❦❡ k ❛s t❤❡ ❡①♣♦♥❡♥t ❜❡❝❛✉s❡ ❡✈❡r② t❡r♠ ✐♥ Π ✐s ♦❢ ♦r❞❡r ❛t ♠♦st k ✐♥ t❡r♠s ♦❢ t❡♥s♦r ♣r♦❞✉❝ts ❛♥❞ ❜② ❛♣♣•②✐♥❣ ❍ö•❞❡r✬s ✐♥❡q✉❛•✐t②✳ ❆ ♣♦ss✐❜✐•❡ ✈❛•✉❡ ❢♦r q ✐s✿ 1/p = 1/p ′ + k/q✳ ◆♦✇ ✇❡ ♦♥•② ♥❡❡❞ t♦ ❝♦♥tr♦• 1/f (t) k+1 L p ′ ✳ ❚♦ ❞♦ t❤✐s ✇❡ ♥♦t✐❝❡ t❤❛t ✇❡ ❞❡✜♥❡ ❛ ❢❛♠✐•② ♦❢ ♠❛r❦♦✈✐❛♥ ❦❡r♥❡•s ✭❛♥❞ ♦❢ ❝♦♥tr❛❝t✐♦♥s✮ ♦♥ t❤❡ L p s♣❛❝❡s ❜② s❡tt✐♥❣✿ G(t)F = t 1 -e -t f (t) = 1 1 -e -t • t 0 T τ F dτ ✭✸✳✷✾✮ ❛♥❞ ❛♣♣•②✐♥❣ ❏❡♥s❡♥✬s ✐♥❡q✉❛•✐t② •❡❛❞s t♦✿ 1 G(t)F ≤ G(t) 1 F ✭✸✳✸✵✮ |❡t✉r♥✐♥❣ t♦ f ♦♥❡ ❣❡ts✿ 1 f (t) ≤ t (1 -e -t ) • G(t)F ≤ t 1 -e -t • G(t) 1 F ≤ 1 1 -e -1 • G(t) 1 F ✭✸✳✸✶✮ ❛♥❞ ✇❡ ❝♦♥❝•✉❞❡ s✐♥❝❡ t❤❡ G t ❛r❡ ❝♦♥tr❛❝t✐♦♥s✳ |❡♠❛r❦ ✸✳✶✳✵✳✷✳ ■♥ t❤❡ ❝❛s❡ ✇❤❡r❡ k = 0 t❤❡ ♣♦•②♥♦♠✐❛• Π ✐♥ t❤❡ ❛❜♦✈❡ ♣r♦♦❢ ✐s 1 ❛♥❞ t❤❡r❡❢♦r❡ t❤❡ r❡s✉•t ✐♠♣r♦✈❡s t♦✿ sup 0≤t≤1 1 f (t) L p ≤ 1 1 -e -1 • 1 F L p ✭✸✳✸✷✮ ✻✵ ❈❍❆P❚❊| ✸✳ ▲■❋❚■◆• ❖❋ |❆◆❉❖▼ ❱❆|■❆❇▲❊❙ ❇( ❉■❙❚|■❇❯❚■❖◆❙ ❲❡ ♠❛② ♥♦✇ st❛t❡ t❤❡ ❢♦••♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✸✳✶✳✵✳✶✵✳ ▲❡t s > 0 ❛♥❞ F ∈ E ∞ -,s s✉❝❤ t❤❛t F > 0 ❛✳s✳ ❛♥❞ 1/F ∈ L ∞ -✳ ❚❤❡♥ ❢♦r ❡✈❡r② s ′ < s 1/F ∈ E ∞ -,s ′ ❛♥❞ ❢♦r ❛♥② p ′ > p t❤❡r❡ ❡①✐sts q(p, p ′ , s) > 1 ❛♥❞ ❛ ❝♦♥st❛♥t C(p, p ′ , s) s✉❝❤ t❤❛t t❤❡ ❢♦••♦✇✐♥❣ ❡st✐♠❛t❡ ❤♦•❞s✿ 1 F E p,s ′ ≤ C • 1 F 2(k+1) p L p ′ + 1 F p L p ′ • F kp+ p p ′ Eq,s ✭✸✳✸✸✮ ✇❤❡r❡ k = [s]✳ ❖♥❝❡ t❤❡ t❤❡♦r❡♠ ✐s ♣r♦✈❡❞ t❤❡ ❢♦••♦✇✐♥❣ ✐s ❛♥ ✐♠♠❡❞✐❛t❡ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡✲ ♦r❡♠ ✸✳✶✳✵✳✾✿ ❈♦r♦••❛r② ✸✳✶✳✵✳✷✳ ❆ s✐♠✐•❛r st❛t❡♠❡♥t ❤♦•❞s ❢♦r t❤❡ s♣❛❝❡s D ∞ -,s Pr♦♦❢✳ ❲✐t❤ t❤❡ ♥♦t❛t✐♦♥ ♦❢ t❤❡ t❤❡♦r❡♠✱ ✇❡ s❡t k = [s] ❛♥❞ σ = {s}✳ ▲❡t ✉s ❝♦♥s✐❞❡r s ′ < s❀ ✐❢ s ✐s ❛♥ ✐♥t❡❣❡r t❤❡ r❡s✉•t ✐s ✐♥❝•✉❞❡❞ ✐♥ •❡♠♠❛ ✸✳✶✳✵✳✸ s♦ ✇❡ ❛② s✉♣♣♦s❡ [s ′ ] = k ❛s ✇❡•• ❛♥❞ ✇❡ s❡t σ ′ = {s ′ }✳ ❲❡ ♥♦✇ ✐♥tr♦❞✉❝❡ g(t) = 1/f (t) ❢♦r t ∈ [0, 1] ❛♥❞ ✇❡ ❡①t❡♥❞ g t♦ ❛ s♠♦♦t❤ ❢✉♥❝t✐♦♥ ✇✐t❤ ❝♦♠❛♣❝t s✉♣♣♦rt✳ ❖❢ ❝♦✉rs❡ g(0) = 1/F s♦ ❜② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ r❡❛• ✐♥t❡r♣♦•❛t✐♦♥ ❢✉♥❝t♦r✱ t♦ ♦❜t❛✐♥ ❛ ♠❛•♦r✐③❛t✐♦♥ ♦❢ 1/F E p,s ′ ✐t ✐s ❡♥♦✉❣❤ t♦ ❝♦♥tr♦• t❤❡ t✇♦ ❢♦••♦✇✐♥❣ t❡r♠s✿ 1 0 t νp g(t) p D p,k+1 dt ❛♥❞ 1 0 t νp g ′ (t) p D p,k dt ✇❤❡r❡ ν ✐s s✉❝❤ t❤❛t p -1 + ν = 1 -σ ′ ✳ ■♥ t❤❡ s❡q✉❡• C i ✇✐•• ❛•✇❛②s ❜❡ ❛ ✉♥✐✈❡rs❛• ❝♦♥st❛♥t✱ ♣♦ss✐❜•② ❞❡♣❡♥❞✐♥❣ ♦♥ s♦♠❡ s♣❡❝✐✜❡❞ ♣❛r❛♠❡t❡rs✳ ❲❡ st❛rt ✇✐t❤ t❤❡ s❡❝♦♥❞ t❡r♠✳ ❙✐♥❝❡ g ′ (t) = -f ′ (t)/f (t) 2 ❜② r❡♣•❛❝✐♥❣✱ ❛♣♣•②✲ ✐♥❣ ❍ö•❞❡r✬s ✐♥❡q✉❛•✐t② ❛♥❞ •❡♠♠❛ ✸✳✶✳✵✳✸ ✇❡ ♦❜t❛✐♥✱ ❢♦r ❛♥② δ > 0✿ 1 0 t νp g ′ (t) p D p,k dt = 1 0 t νp f ′ (t) f (t) 2 p D p,k dt ≤ C 1 (p, k, δ) • 1 0 t νp 1 f (t) 2 p D p(1+δ)/δ,k • f ′ (t) p D p(1+δ),k dt ≤ C 2 (p, k, δ) • 1 F 2(k+1) p L p 1+δ ′ δ ′ + 1 F p L p 1+δ δ • F N k D q,k • 1 0 t νp f ′ (t) p D p(1+δ),k dt ✭✸✳✸✹✮ ✸✳✷✳ ■◆❚❊•|❆❚■❖◆ ❇( P❆|❚❙ ✻✶ ❢♦r ❛♥② 0 < δ ′ < δ ❛♥❞ ❢♦r s♦♠❡ q ❛s ✐♥ •❡♠♠❛ ✸✳✶✳✵✳✸✳ ◆♦✇ ❜② ❏❡♥s❡♥✬s ✐♥❡q✉❛•✐t②✿ 1 0 t νp f ′ (t) p D p(1+δ),k dt ≤ 1 0 t νp(1+δ) f ′ (t) p(1+δ) D p(1+δ),k dt 1 1+δ ≤ F 1 1+δ E p(1+δ),s ✭✸✳✸✺✮ s✐♥❝❡ νp(1 + δ) = -1 + p(1 + δ)(1 -σ)✳ ❋✐♥❛••②✿ 1 0 t νp g(t) p D p,k+1 dt ≤ C 3 • 1 F 2(k+1) p L p 1+δ ′ δ ′ + 1 F p L p 1+δ δ • F kp D q,k • F 1 1+δ E p(1+δ),s ✭✸✳✸✻✮ ✇❤❡r❡ C 3 = C 3 (p, k, δ, δ ′ , ν)✳ ❚❤❡ ✜rst t❡r♠ ✐s ❞❡❛•t ✇✐t❤ ✐♥ ❛ s✐♠✐•❛r ✇❛②✳ ❋✐rst✱ ❜② ▼❡②❡r✬s ✐♥❡q✉❛•✐t✐❡s✿ g(t) D p,k+1 ≤ C 4 (p, k) g(t) D p,k + ∇g(t) D p,k = C 4 (p, k) g(t) D p,k + ∇f (t) f (t) 2 D p,k ✭✸✳✸✼✮ ❲❡ ❝♦♥s✐❞❡r t❤❡ t✇♦ t❡r♠s ✐♥ t❤❡ s✉♠ s❡♣❛r❛t❡•②✳ ❚❤❡ ✜rst t❡r♠ ✐s ❞❡❛•t ✇✐t❤ ❜② ✉s✐♥❣ •❡♠♠❛ ✸✳✶✳✵✳✸✳ ❋♦r t❤❡ s❡❝♦♥❞ t❡r♠ ❍ö•❞❡r✬s ✐♥❡q✉❛•✐t② •❡❛❞s t♦✿ ∇f (t) f (t) 2 D p,k ≤ C 5 (p, k) • 1 f (t) 2 D p 1+δ δ ,k • ∇f (t) D p(1+δ),k ≤ 1 f (t) 2 D p 1+δ δ ,k • f (t) D p(1+δ),k+1 ✭✸✳✸✽✮ ❛♥❞ ✇❡ ♣r♦❝❡❡❞ ❛s ♣r❡✈✐♦✉s•② t♦ ♦❜t❛✐♥✱ ❢♦r s♦♠❡ C 6 (p, k, δ, δ ′ , ν)✿ 1 0 t νp g(t) p D p,k+1 dt ≤ C 6 • 1 F 2(k+1) p L p 1+δ ′ δ ′ + 1 F p L p 1+δ δ • F kp D q,k • F 1 1+δ E p(1+δ),s ✭✸✳✸✾✮ ❲❡ t❤❡♥ ✜♥✐s❤ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ t❤❡♦r❡♠ ❜② ♣✉tt✐♥❣ t❤❡ t❡r♠s t♦❣❡t❤❡r ❛♥❞ s❡tt✐♥❣ p ′ = p(1 + δ)✳ ✸✳✷ ■♥t❡❣r❛t✐♦♥ ❜② ♣❛rts ❋✐rst ✇❡ ✐♥tr♦❞✉❝❡ ❛ ❝r✉❝✐❛• q✉❛♥t✐t②✿ ❉❡✜♥✐t✐♦♥ ✸✳✷✳✵✳✸ ✭▼❛••✐❛✈✐♥ ♠❛tr✐①✮✳ ▲❡t F ∈ D ∞ -,1 (R N )✳ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ▼❛••✐❛✈✐♥ ♠❛tr✐① ♦❢ F ❛s t❤❡ r❛♥❞♦♠ ♠❛tr✐① ✇✐t❤ ❝♦❡✣❝✐❡♥ts✿ Σ ij = (∇F i , ∇F j ) H ✭✸✳✹✵✮ ✻✷ ❈❍❆P❚❊| ✸✳ ▲■❋❚■◆• ❖❋ |❆◆❉❖▼ ❱❆|■❆❇▲❊❙ ❇( ❉■❙❚|■❇❯❚■❖◆❙ ❲❡ ♥♦t❡ t❤❛t t❤❡ ▼❛••✐❛✈✐♥ ♠❛tr✐① ✐s ❛✳s✳ s②♠♠❡tr✐❝ ❛♥❞ ♣♦s✐t✐✈❡ ❛s ❛ •r❛♠ ♠❛tr✐①✳ ❆•s♦✱ ✐❢ F ∈ D ∞ -,s (R N ) t❤❡♥ Σ ij ∈ D ∞ -,s-1 (R) ❛♥❞ ❜② ❍ö•❞❡r✬s t❤❡♦r❡♠✿ Σ ij D p,s-1 ≤ C(p, s) • F 2N D 2N p,s ✭✸✳✹✶✮ ❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ t❤❡ ✐♥✈❡rt✐❜✐•✐t② ♦❢ Σ ❛♥❞ ✐♥ t❤❡ r❡❣✉•❛r✐t② ♦❢ ✐ts ♣♦t❡♥t✐❛• ✐♥✈❡rs❡✳ ❲❡ ✇✐•• ♥❡❡❞ ❛♥♦t❤❡r st❛♥❞❛r❞ ❞❡✜♥✐t✐♦♥✿ ❉❡✜♥✐t✐♦♥ ✸✳✷✳✵✳✹ ✭♥♦♥✲❞❡❣❡♥❡r❛❝②✮✳ ❚❤❡ r❛♥❞♦♠ ✈❛r✐❛❜•❡ F ∈ D ∞ -,1 (R N ) ✐s ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐❢✿ 1 det Σ ∈ L ∞ - ✭✸✳✹✷✮ ■❢ ❛ r❛♥❞♦♠ ✈❛r✐❛❜•❡ ✐s ♥♦♥✲♥❡❣❡♥❡r❛t❡ t❤❡♥ ✐ts ▼❛••✐❛✈✐♥ ♠❛tr✐① Σ ✐s ✐♥✈❡rt✲ ✐❜•❡ ❛♥❞ ✇❡ ✇✐•• ♦❢t❡♥ ♥♦t❡ γ ❢♦r t❤✐s ✐♥✈❡rs❡✳ ❚❤❡ ❢♦••♦✇✐♥❣ ✐s ❛ str❛✐❣❤t❢♦r✇❛r❞ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡♦r❡♠ ✸✳✶✳✵✳✶✵ ❛♥❞ ❍ö•❞❡r✬s ✐♥❡q✉❛•✐t② ❜✉t ✇✐•• ❜❡ ❝r✉❝✐❛• t♦ ♦✉r ♣✉r♣♦s❡✿ Pr♦♣♦s✐t✐♦♥ ✸✳✷✳✵✳✸✳ ❲❡ ❝♦♥s✐❞❡r ❛ ♥♦♥✲❞❡❣❡♥❡r❛t❡ F ∈ D ∞-,s (R N ) ❢♦r s♦♠❡ s > 1 ❛♥❞ Σ ✐ts ▼❛••✐❛✈✐♥ ♠❛tr✐①✳ ❚❤❡♥ ❢♦r ❛♥② 1 ≤ s ′ < s✱ 1 det Σ ∈ D ∞ -,s ′ -1 (R) ✭✸✳✹✸✮ ❆•s♦✱ ❢♦r ❛•• p ′ > p t❤❡r❡ ❡①✐sts q > 1 ❛♥❞ α > 0 s✉❝❤ t❤❛t t❤❡ ❢♦••♦✇✐♥❣ ❡st✐♠❛t❡ ❤♦•❞s✿ 1 detΣ D p,s ′ -1 ≤ C • 1 (detΣ) 2(k+1) p L p ′ + 1 detΣ p L p ′ • F α Dq,s ✭✸✳✹✹✮ ✇❤❡r❡ k = [s] ❛♥❞ C ✐s ❛ ✉♥✐✈❡rs❛• ❝♦♥st❛♥t ❞❡♣❡♥❞✐♥❣ ♦♥•② ♦♥ p, p ′ , s, s ′ , q✳ Pr♦♦❢✳ ❆♣♣•②✐♥❣ t❤❡♦r❡♠ ✸✳✶✳✵✳✶✵ ②✐❡•❞s✿ 1 detΣ D p,s ′ -1 ≤ C • 1 (detΣ) 2(k+1) p L p ′ + 1 detΣ p L p ′ • detΣ kp+ p p ′ D q,s-1 ✭✸✳✹✺✮ ❚❤❡♥ s✐♥❝❡ detΣ ✐s ❛ ♣♦•②♥♦♠✐❛• ✐♥ t❤❡ ∇F i ✇❡ ✜♥✐s❤ t❤❡ ♣r♦♦❢ ✇✐t❤ ❍ö•❞❡r✬s ✐♥❡q✉❛•✐t②✳ |❡♠❛r❦ ✸✳✷✳✵✳✸✳ ❆ s✐♠✐•❛r r❡s✉•t ❤♦•❞s ❢♦r r❡❛• ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡s✳ ❲❡ ♥♦✇ r❡❝❛•• t❤❡ ❢♦••♦✇✐♥❣ ❝•❛ss✐❝❛• r❡s✉•t✿ ✸✳✷✳ ■◆❚❊•|❆❚■❖◆ ❇( P❆|❚❙ ✻✸ ❚❤❡♦r❡♠ ✸✳✷✳✵✳✶✶ ✭■♥t❡❣r❛t✐♦♥ ❜② ♣❛rts ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡✮✳ ❈♦♥s✐❞❡r t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜•❡s F ∈ D ∞ -,2 (R N ) ❛♥❞ G ∈ D ∞ -,1 (R N )✳ ❙✉♣♣♦s❡ t❤❛t F ✐s ♥♦♥✲ ❞❡❣❡♥❡r❛t❡✳ ❚❤❡♥ ❢♦r ❛♥② i = 1, . . . , N t❤❡ ❢♦••♦✇✐♥❣ q✉❛♥t✐t② ✐s ✇❡•• ❞❡✜♥❡❞✿ l i (G, F ) = δ G • N j=1 γ ij ∇F j ✭✸✳✹✻✮ ❛♥❞ ✐❢ φ ∈ C 1 (R N ) t❤❡♥✿ E[(∂ i φ) • F • G] = E[φ • F • l i (G, F )] ✭✸✳✹✼✮ Pr♦♦❢✳ ❋✐rst ✇❡ ♣r♦✈❡ t❤❛t l i (G, F ) ✐s ✇❡•• ❞❡✜♥❡❞ ✐♥ L p ✳ ■♥❞❡❡❞✱ ♥♦t✐♥❣ C ❢♦r ❛ ❣❡♥❡r✐❝ ❝♦♥st❛♥t✿ l i (G, F ) L p ≤ G • N i=1 γ ij ∇F j D p,1 ≤ C • G D q,1 • N i=1 γ ij ∇F j D r,1 ✭✸✳✹✽✮ ❚❤❡♥ ✇❡ ✉s❡ t❤❡ ❢❛❝t t❤❛t✿ γ = 1 detΣ t (comatΣ) ✭✸✳✹✾✮ t❤❡r❡❢♦r❡ ❜② ❍ö•❞❡r✬s ✐♥❡q✉❛•✐t②✿ N i=1 γ ij ∇F j D r,1 ≤ C • 1 detΣ Dr 1 ,1 • N i=1 t (comatΣ) ij ∇F j D r 2 ,1 ✭✸✳✺✵✮ ◆♦✇ t❤❡ t❡r♠ ✇✐t❤ t❤❡ ❞❡t❡r♠✐♥❛♥t ♠❛② ❜❡ ❝♦♥tr♦••❡❞ ✐♥ s♦♠❡ D r,2 ❜② ♣r♦♣♦✲ s✐t✐♦♥ ✸✳✷✳✵✳✸✱ ❛♥❞ t❤❡ ♦t❤❡r t❡r♠s ♠❛② ❜❡ ❝♦♥tr♦••❡❞ ✐♥ s♦♠❡ D r,2 ❜❡❝❛✉s❡ t❤❡ t (comatΣ) ij ∇F j ❛r❡ ♠✉•t✐✈❛r✐❛t❡ ♣♦•②♥♦♠✐❛•s ✐♥ ∇F ✳ ◆♦✇ ✇❡ ♣r♦✈❡ t❤❡ ❛❝t✉❛• ✐♥t❡❣r❛t✐♦♥ ❜② ♣❛rts ❢♦r♠✉•❛✳ ❇② t❤❡ ▼❛••✐❛✈✐♥ ❝❛•✲ ❝✉•✉s ❝❤❛✐♥ r✉•❡✱ φ(F ) ∈ D p,1 ❛♥❞✿ ∇(φ(F )) = N i=1 (∂ i φ)(F ) • ∇F i ✭✸✳✺✶✮ t❤❡r❢♦r❡✿ (∇(φ(F )), ∇F j ) H = N i=1 (∂ i φ)(F ) • σ ij ✭✸✳✺✷✮ ❛♥❞ s✐♥❝❡ γ ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ▼❛••✐❛✈✐♥ ♠❛tr✐①✱ ❡•❡♠❡♥t❛r② •✐♥❡❛r ❛•❣❡❜r❛ ②✐❡•❞s✿ (∂ i φ)(F ) = N j=1 γ ij (∇(φ(F )), ∇F j ) H ✭✸✳✺✸✮ ❚❤❡ r❡s✉•t ✐s ✜♥❛••② ♦❜t❛✐♥❡❞ ❜② ✐♥t❡❣r❛t✐♥❣ ❜② ♣❛rts✳ ✻✹ ❈❍❆P❚❊| ✸✳ ▲■❋❚■◆• ❖❋ |❆◆❉❖▼ ❱❆|■❆❇▲❊❙ ❇( ❉■❙❚|■❇❯❚■❖◆❙ ■t ✐s str❛✐❣❤t❢♦r✇❛r❞ ❜✉t ♥♦t❡✇♦rt❤② t❤❛t t❤❡ ♦♣❡r❛t♦rs l i ❛r❡ •✐♥❡❛r ✐♥ t❤❡ ✜rst ✈❛r✐❛❜•❡ ❜✉t ♥♦t ✐♥ t❤❡ s❡❝♦♥❞✳ ❚❤✐s ✐s t❤❡ ♠❛✐♥ ❞✐✣❝✉•t② ✐♥ t❤❡ st✉❞② ♦❢ t❤❡✐r r❡❣✉•❛r✐t②✳ ❙♦♠❡ r❡s✉•ts ❛r❡ ✇❡•• ❦♥♦✇♥ ❢♦r •r♦ss✲❙♦❜♦•❡✈ s♣❛❝❡s ✇✐t❤ ❛♥ ✐♥t❡❣❡r ♦r❞❡r ♦❢ ❞❡r✐✈❛❜✐•✐t②✳ ❍❡r❡✱ ❢♦••♦✇✐♥❣ t❤❡ •♦❣✐❝ ♦❢ ❲❛t❛♥❛❜❡ ✇❡ st❛t❡ t❤❡ ♠♦r❡ ♣r❡❝✐s❡✿ ❚❤❡♦r❡♠ ✸✳✷✳✵✳✶✷ ✭❋r❛❝t✐♦♥❛• r❡❣✉•❛r✐t② ♦❢ t❤❡ ♦♣❡r❛t♦r l i ✮✳ ❋♦r ❛♥② p > 1 ❛♥❞ s > 0✱ ❢♦r ❛♥② p ′ > p ❛♥❞ s ′ > s t❤❡r❡ ❡①✐sts q, r > 1✱ θ > 0 ❛♥❞ ❛ ✉♥✐✈❡rs❛• ❝♦♥st❛♥t ❞❡♣❡♥❞✐♥❣ ♦♥•② ♦♥ t❤❡ ❛❜♦✈❡ ♣❛r❛♠❡t❡rs s✉❝❤ t❤❛t t❤❡ ❢♦••♦✇✐♥❣ ❡st✐♠❛t❡ ❤♦•❞s ❢♦r ❛♥② G ∈ D ∞ -,s+1 ❛♥❞ ♥♦♥✲❞❡❣❡♥❡r❛t❡ F ∈ D ∞ -,s ′ +2 ✿ l i (G, F ) Dp,s ≤ C • G D p ′ ,s+1 • 1 (detΣ) 2(k+1) q L q + 1 detΣ q L q • F θ D r,s ′ +2 ✭✸✳✺✹✮ ✇❤❡r❡ k = [s]✳ |❡♠❛r❦ ✸✳✷✳✵✳✹✳ ■♥ t❤❡ ❝❛s❡ ✇❤❡r❡ s ✐s ❛♥ ✐♥t❡❣❡r✱ t❤❡ ✉s✉❛•✱ s✐♠♣•❡r r❡s✉•t •❡ts ♦♥❡ t❛❦❡ s ✐♥st❡❛❞ ♦❢ s ′ ✐♥ t❤❡ ♠❛•♦r✐③❛t✐♦♥ ♦❢ t❤❡ ❛❜♦✈❡ t②♣❡✳ Pr♦♦❢✳ ❯s❡ t❤❡ ❡①❛❝t s❛♠❡ t❡❝❤♥✐q✉❡ t❤❛♥ t♦ ♣r♦✈❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢ l i (G, F ) ✐♥ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠✱ ♦♥•② ✇✐t❤ ❤✐❣❤❡r ♦r❞❡r s♣❛❝❡s✳ ■♥ t❤❡ s❡q✉❡•✱ ✐t ✇✐•• ❜❡ ❝♦♥✈❡♥✐❡♥t t♦ ✉s❡ x i t❤❡ ♦♣❡r❛t♦r ♠❛♣♣✐♥❣ S t♦ ✐ts❡•❢✱ ❞❡✜♥❡❞ ❜② (x i f )(x) = x i f (x)✳ ❍❡r❡ ✐s ❛ ❝♦♥t✐♥✉✐t② r❡s✉•t ❛♥❛•♦❣✉❡ t♦ t❤❡ ♣r❡✈✐♦✉s ♦♥❡ ✇❤✐❝❤ ✐s •♦♥❣❡r t♦ st❛t❡ t❤❛♥ t♦ ♣r♦✈❡✿ ▲❡♠♠❛ ✸✳✷✳✵✳✹✳ ❋♦r ❛♥② r❛♥❞♦♠ ✈❛r✐❛❜•❡s F ❛♥❞ G ❛♥❞ ❢♦r ❛♥② ♠❡❛s✉r❛❜•❡ ❢✉♥❝t✐♦♥ f ♦♥❡ ❤❛s✿ E[(x i f )(F ) • G] = E[f (F ) • m i (G, F )] ✇✐t❤ m i (G, F ) = F i • G ❛♥❞ ❢♦r ❛♥② p ′ > p > 1 ✐❢ 1/p = 1/p ′ + 1/q✿ m i (G, F ) L p ′ ≤ G L q F L p ′ ◆♦✇ •❡t ✉s ❝♦♥s✐❞❡r ❛ ❢❛♠✐•② ♦❢ ❢✉♥❝t✐♦♥❛•s µ j ❢♦r j = 1, . . . , m s✉❝❤ t❤❛t ❡✈❡r② µ i ✐s ❡✐t❤❡r ♦♥❡ ♦❢ t❤❡ l i ♦r ♦♥❡ ♦❢ t❤❡ m i ✱ i = 1, . . . , N ✳ ❲❡ ❞❡✜♥❡ λ(G, F ) = µ m (G, µ m-1 (G, µ m-2 (G, . . . , F ) . . . )) ✭✸✳✺✺✮ ❢♦r r❡❣✉•❛r ❡♥♦✉❣❤ G ❛♥❞ ♥♦♥✲❞❡❣❡♥❡r❛t❡ F ✳ ❚❤❡♥✱ ❝♦♠❜✐♥✐♥❣ t❤❡♦r❡♠ ✸✳✷✳✵✳✶✷ ❛♥❞ •❡♠♠❛ ✸✳✷✳✵✳✹ ✇✐t❤ ❛ str❛✐❣❤t♦r✇❛r❞ r❡❝✉rr❡♥❝❡ ♦♥❡ ♦❜t❛✐♥s✿ ✸✳✷✳ ■◆❚❊•|❆❚■❖◆ ❇( P❆|❚❙ ✻✺ Pr♦♣♦s✐t✐♦♥ ✸✳✷✳✵✳✹✳ ▲❡t p ′ > p > 1 ❛♥❞ s ′ > s > 0✳ ❙✉♣♣♦s❡ t❤❛t λ ✐s ❛s ❛❜♦✈❡✱ ✇✐t❤ ❛t ♠♦st n ♦❢ t❤❡ µ j ❜❡✐♥❣ ❛♥ l i ✳❚❤❡♥ t❤❡r❡ ❡①✐sts q, r > 1✱ θ > 0 ❛♥❞ ❛ ✉♥✐✈❡rs❛• ❝♦♥st❛♥t C s✉❝❤ t❤❛t t❤❡ ❢♦••♦✇✐♥❣ ❡st✐♠❛t❡ ❤♦•❞s ❢♦r ❛♥② r❡❣✉•❛r ❡♥♦✉❣❤ F, G✱ F ❜❡✐♥❣ ♥♦♥❞❡❣❡♥❡r❛t❡✿ λ(G, F ) Dp,s ≤ C • G D p ′ ,s+m • 1 (detΣ) 2(k+1) q L q + 1 detΣ q L q m • F θ D r,s ′ +m+1 ✭✸✳✺✻✮ ✇❤❡r❡ k = [s] ❛♥❞ σ ✐s t❤❡ ▼❛••✐❛✈✐♥ ♠❛tr✐① ♦❢ F ✳ ❲❡ ♥♦✇ st❛t❡ ❛ ✜rst r❡s✉•t r❡•❛t❡❞ t♦ t❤❡ ✐♥t❡r♣♦•❛t✐♦♥ ♦❢ •r♦ss✲❙♦❜♦•❡✈ s♣❛❝❡s ❛♥❞ t❤❡ s♣❛❝❡s ✐♥tr♦❞✉❝❡❞ ✐♥ s❡❝t✐♦♥ ✷✿ ▲❡♠♠❛ ✸✳✷✳✵✳✺✳ ❙✉♣♣♦s❡ t❤❛t F ∈ D ∞ -,1 ❤❛s ❛ ❜♦✉♥❞❡❞ ❞❡♥s✐t② p F ✳ ❚❤❡♥ ❢♦r ❡✈❡r② 0 ≤ ρ < ρ ′ ≤ 1 ❛♥❞ 1 < p < p ′ < ∞ t❤❡r❡ ❡①✐sts ❛ ✉♥✐✈❡rs❛• ❝♦♥st❛♥t C(ρ, ρ ′ , p, p ′ ) s✉❝❤ t❤❛t ❢♦r ❛♥② φ ∈ S t❤❡ ❢♦••♦✇✐♥❣ ❡st✐♠❛t❡ ❤♦•❞s✿ φ • F Dp,ρ ≤ C • p F ∞ • (1 + ∇F L q ) • φ S p ′ ,ρ ′ ✭✸✳✺✼✮ Pr♦♦❢✳ ❋✐rst ✇❡ ♥♦t❡ t❤❛t✿ φ • F L p ≤ p F ∞ • φ L p ✭✸✳✺✽✮ ❛♥❞ ❢♦r q s✉❝❤ t❤❛t 1/p ′ +1/q = 1/p✱ ❜② ▼❡②❡r✬s ✐♥❡q✉❛•✐t✐❡s ❛♥❞ ❍ö•❞❡r✬s t❤❡♦r❡♠✿ φ • F D p,1 ≤ c(p) • φ • F L p + N i=1 ∂ i φ • F • ∇F i L p ≤ c(p) • φ • F L p + N i=1 ∂ i φ • F L p ′ • ∇F L q ≤ c(p) • [ p F ∞ • φ L p + p F ∞ • ∇φ L p ′ • ∇F L q ] ≤ c(p) p F ∞ (1 + ∇F L q ) • |φ| S p,1 ✭✸✳✺✾✮ ❋♦r t❤❡ •❛st ✐♥❡q✉❛•✐t② ✇❡ ✉s❡❞ t❤❡♦r❡♠ ✷✳✷✳✷✳✶✳ ❲❡ s❡❡ t❤❛t ✇❡ ❤❛✈❡ ♣r♦✈❡❞ ♦✉r r❡s✉•t ❢♦r t❤❡ ❝❛s❡s ρ = 0 ♦r ρ ′ = 1✳ ❲❡ ♥♦✇ s✉♣♣♦s❡ 0 < ρ < ρ ′ < 1✳ ❲❡ ♠❛② t❤❡♥ ❝❤♦♦s❡ p < p 1 < p ′ ❛♥❞ ρ < ρ 1 < ρ 2 < ρ ′ s✉❝❤ t❤❛t✿ 1 -ρ 2 - 1 p ′ = 1 -ρ 1 - 1 p 1 ❚❤❡♥ ❜② r❡❛• ✐♥t❡r♣♦•❛t✐♦♥ ♦♥❡ ♦❜t❛✐♥s✿ φ • F Ep 1 ,ρ 1 ≤ c(p) p F ∞ (1 + ∇F L q ) • φ Tp 1 ,ρ 1 ✭✸✳✻✵✮ ❚❤❡ ♣r♦♦❢ ✐s t❤❡♥ ✜♥✐s❤❡❞ ❜② ❝♦♠❜✐♥✐♥❣ t❤❡ ❛❜♦✈❡ ✇✐t❤ t❤❡♦r❡♠s ✸✳✶✳✵✳✾ ❛♥❞ ✷✳✷✳✷✳✹✳ ✻✻ ❈❍❆P❚❊| ✸✳ ▲■❋❚■◆• ❖❋ |❆◆❉❖▼ ❱❆|■❆❇▲❊❙ ❇( ❉■❙❚|■❇❯❚■❖◆❙ ❲❡ ♥♦✇ ♣r♦✈❡ ❛ r❡s✉•t ✇❤✐❝❤ ✇✐•• •❡t ✉s •✐❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ r❛♥❞♦♠ ✈❛r✐❛❜•❡s ❜② t❡♠♣❡r❡❞ ❞✐str✐❜✉t✐♦♥s ✇✐t❤ ♠✐♥✐♠❛• ❝♦♥❞✐t✐♦♥s✿ ❚❤❡♦r❡♠ ✸✳✷✳✵✳✶✸✳ ▲❡t F ∈ D ∞ -,1+δ ❢♦r s♦♠❡ δ > 0 ❜❡ ❛ ♥♦♥✲❞❡❣❡♥❡r❛t❡ R N ✲ ✈❛•✉❡❞ r❛♥❞♦♠ ✈❛r✐❛❜•❡ ❛❞♠✐tt✐♥❣ ❛ ❜♦✉♥❞❡❞ ❞❡♥s✐t②✳ ❚❤❡♥ ❢♦r ❡✈❡r② δ ′ < δ ❛♥❞ 1 < p < p ′ < ∞ t❤❡r❡ ❡①✐sts ❛ ✉♥✐✈❡rs❛• ❝♦♥st❛♥t C ❛♥❞ s♦♠❡ q, r > 1 ❛♥❞ θ > 0 s✉❝❤ t❤❛t t❤❡ ❢♦••♦✇✐♥❣ ❡st✐♠❛t❡ ❤♦•❞s ❢♦r ❡✈❡r② φ ∈ S(R N )✿ φ • F D p,-δ ≤ C • 1 (detΣ) 2(k+1) q L q + 1 detΣ q L q 2m • F θ D r,1+δ ′ • p F ∞ • φ S p ′ ,-δ ′ ✭✸✳✻✶✮ ❍❡♥❝❡ ❢♦r ❛♥② T ∈ S p ′ ,-δ ′ ✱ T • F ♠❛② ❜❡ ❞❡✜♥❡❞ ✐♥ D p,-δ ❛♥❞ t❤❡ ❛❜♦✈❡ ❡st✐♠❛t❡ st✐•• ❤♦•❞s✳ Pr♦♦❢✳ ❲❡ st✐•• ♥♦t❡ Σ ❢♦r t❤❡ ▼❛••✐❛✈✐♥ ♠❛tr✐① ♦❢ F ✳ ❆•s♦✱ ✇❡ ♥♦t❡ k ❛♥❞ σ ❢♦r [δ] ❛♥❞ {δ} r❡s♣❡❝t✐✈❡•②✳ ■❢ δ ✐s ❛♥ ✐♥t❡❣❡r✱ t❤❡ r❡s✉•t ✐s ❝•❛ss✐❝❛•✱ s♦ ✇❡ ♦♥•② ♣r♦✈❡ t❤❡ ❝❛s❡ ✇❤❡r❡ σ = 0✳ ■♥ t❤✐s ❝❛s❡ δ ′ ♠❛② ❜❡ ❝❤♦s❡♥ s✉❝❤ t❤❛t [δ ′ ] = k✳ ■❢ k ✐s ♦❞❞✱ ✇❡ ✇r✐t❡ k = 2l -1 ❛♥❞ φ = K l K -l φ✳ ❲❡ t❤❡♥ ❡①♣❛♥❞ K l ✿ K l = A i 1 • • • • • A i 2m ✇❤❡r❡✱ ❢♦r ❡❛❝❤ t❡r♠ ✐♥ t❤❡ s✉♠✱ m ≤ l ❛♥❞ ❡❛❝❤ A i j ✐s ❡✐t❤❡r ❛ ♣❛rt✐❛• ❞❡r✐✈❛t✐✈❡ ∂ k ♦r ❛ ♠✉•t✐♣•✐❝❛t✐♦♥ ❜② ❛ ❝♦♦r❞✐♥❛t❡ x k ✳ ❆♥ ❛♣♣•✐❝❛t✐♦♥ ♦❢ ♣r♦♣♦s✐t✐♦♥ ✸✳✷✳✵✳✹ t♦ ❛ r❡❣✉•❛r ❡♥♦✉❣❤ ψ •❡❛❞s t♦✿ E[(A i 1 • • • • • A i 2m ψ) • F • G] = E[ψ • F λ(G, F )] ✭✸✳✻✷✮ ❢♦r s♦♠❡ λ ❛s ✐♥ ♣r♦♣♦s✐t✐♦♥ ✸✳✷✳✵✳✹✱ ❛♥❞ ✐❢ 1/π + 1/p = 1 ❛♥❞ 1 < π ′ < π t❤❡♥ ❢♦r s♦♠❡ q, r, θ ♦♥❡ ❤❛s✿ λ(G, F ) D π ′ ,δ ′ -k-1 ≤ C • G D π,δ • 1 (detΣ) 2(k+1) q L q + 1 detΣ q L q 2m • F θ D r,1+δ ′ ✭✸✳✻✸✮ ◆♦✇ ✇❡ r❡❝❛•• t❤❛t✿ (A i 1 • • • • • A i 2m φ) • F D p,-δ = sup E[(A i 1 • • • • • A i 2m φ) • F • G], G D π,δ ≤ 1 = sup E[ψ • F • λ(G, F )], G D π,δ ≤ 1 ✭✸✳✻✹✮ s♦ ❜② t❛❦✐♥❣ •✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ❛♥❞ r❡♣•❛❝✐♥❣ ψ ❜② K -l φ ✇❡ ♦❜t❛✐♥✿ φ•F D p,-δ ≤ C 1 (detΣ) 2(k+1) q L q + 1 detΣ q L q 2m F θ D r,1+δ ′ (K -l φ)•F D p ′ ,k+1-δ ′ ✭✸✳✻✺✮ ✸✳✷✳ ■◆❚❊•|❆❚■❖◆ ❇( P❆|❚❙ ✻✼ ❛♥❞ s✐♥❝❡ 0 < k + 1 -δ ′ < 1 •❡♠♠❛ ✸✳✷✳✵✳✺ ❛♣♣•✐❡s t♦ t❤❡ •❛st ❢❛❝t♦r ♦❢ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ s♦ t❤❡ ♣r♦♦❢ ✐s ✜♥✐s❤❡❞ ✐♥ t❤✐s ❝❛s❡ s✐♥❝❡ t❤❡ ∇F L q t❡r♠ ♠❛② ❜❡ ✧✐♥❝•✉❞❡❞✧ ✐♥ t❤❡ F θ D r,1+δ ′ t❡r♠ ❜② ▼❡②❡r✬s ✐♥❡q✉❛•✐t✐❡s✳ ■❢ k ✐s ❡✈❡♥✱ •❡t ✉s ✇r✐t❡ k = 2l✳ ❆s ✐♥ t❤❡ ♣r❡✈✐♦✉s ❝❛s❡✱ ✇❡ ❡①♣❛♥❞ K l+1 s♦ ✇❡ ♠❛② ✇r✐t❡ Id = K l+1 K -(l+1) ❛s ❛ •✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❢❛❝t♦rs ♦❢ t❤❡ ❢♦••♦✇✐♥❣ t②♣❡✿ A i 1 • • • • • A i 2m+2 • K -(l+1) = A i 1 • • • • • A i 2m+1 • K -(l+1) • A 2m+2 + A i 1 • • • • • A i 2m+1 • [K -(l+1) , A 2m+2 ] ✭✸✳✻✻✮ ✇❤❡r❡ [, ] ✐s t❤❡ ❝♦♠♠✉t❛t♦r ♦❢ t✇♦ •✐♥❡❛r ♦♣❡r❛t♦rs✳ ❲❡ ♠❛② t❤❡♥ ♣r♦❝❡❡❞ ❡①❛❝t•② ❛s ✐♥ t❤❡ ♦❞❞ ❝❛s❡✱ ♥♦t✐♥❣ t❤❛t t❤❡ ♦♣❡r❛t♦rs K -(l+1) • A ❛♥❞ [K -(l+1) , A] ♠❛♣ L p t♦ ✐ts❡•❢ ✇❤❡♥ A ✐s ❛ ♣❛rt✐❛• ❞❡r✐✈❛t✐✈❡ ♦r ❛ ♠✉•t✐♣•✐❝❛t✐♦♥ ❜② ❛ ❝♦♦r❞✐♥❛t❡✱ s❡❡ ❬✼✵( ♦r ❬✼✶(✳ ■♥ ♦r❞❡r t♦ s✐♠♣•✐❢② t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠✱ ✇❡ r❡❝❛•• t❤❛t r❡❣✉•❛r ❡♥♦✉❣❤ r❛♥❞♦♠ ✈❛r✐❛❜•❡s ❛•✇❛②s ❛❞♠✐t ❛ ❜♦✉♥❞❡❞ ✭❛♥❞ ❡✈❡♥ ❝♦♥t✐♥✉♦✉s✮ ❞❡♥s✐t②✳ ❲❡ ❝♦♣② t❤❡ ❢♦••♦✇✐♥❣ r❡s✉•t ❢r♦♠ ❬✻✻(✿ ❚❤❡♦r❡♠ ✸✳✷✳✵✳✶✹✳ ❙✉♣♣♦s❡ t❤❛t p > N ❛♥❞ t❤❛t F ∈ D 8N p,2 ✐s ♥♦♥✲❞❡❣❡♥❡r❛t❡✳ ❚❤❡♥ F ❛❞♠✐ts ❛ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❜♦✉♥❞❡❞ ❞❡♥s✐t② p f ❛♥❞✿ p F ∞ ≤ C • ∇F D 8N p,1 • 1 Σ 2N L 4p ✭✸✳✻✼✮ ❈♦♠❜✐♥✐♥❣ t❤❡s❡ t✇♦ t❤❡♦r❡♠s ✇❡ ♦❜t❛✐♥✿ ❚❤❡♦r❡♠ ✸✳✷✳✵✳✶✺✳ ▲❡t F ∈ D ∞ -,1+δ ❢♦r s♦♠❡ δ > 1 ❜❡ ❛ ♥♦♥✲❞❡❣❡♥❡r❛t❡ R N ✲ ✈❛•✉❡❞ r❛♥❞♦♠ ✈❛r✐❛❜•❡✳ ❚❤❡♥ ❢♦r ❡✈❡r② δ ′ < δ ❛♥❞ 1 < p < p ′ < ∞ t❤❡r❡ ❡①✐sts ❛ ✉♥✐✈❡rs❛• ❝♦♥st❛♥t C ❛♥❞ s♦♠❡ q, r > 1 ❛♥❞ θ > 0 s✉❝❤ t❤❛t t❤❡ ❢♦••♦✇✐♥❣ ❡st✐♠❛t❡ ❤♦•❞s ❢♦r ❡✈❡r② φ ∈ S(R N )✿ φ • F D p,-δ ≤ C • 1 (detΣ) 2(k+1) q L q + 1 detΣ q L q 2m • F θ D r,1+δ ′ • φ S p ′ ,-δ ′ ✭✸✳✻✽✮ ❍❡♥❝❡ ❢♦r ❛♥② T ∈ S p ′ ,-δ ′ ✱ T • F ♠❛② ❜❡ ❞❡✜♥❡❞ ✐♥ D p,-δ ❛♥❞ t❤❡ ❛❜♦✈❡ ❡st✐♠❛t❡ st✐•• ❤♦•❞s✳ |❡♠❛r❦ ✸✳✷✳✵✳✺✳ ❖♥❝❡ ❛❣❛✐♥ t❤❡r❡ ❛r❡ s✐♠✐•❛r r❡s✉•ts ❢♦r t❤❡ s♣❛❝❡s E ❛♥❞ T ✳ ❚♦ ❝♦♥❝•✉❞❡ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❣✐✈❡ ❛♥♦t❤❡r r❡s✉•t ❢♦r t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❞❡♥s✐t✐❡s✳ ❚❤✐s ✐s ♠♦r❡ ♣r❡❝✐s❡ t❤❛t t❤❡♦r❡♠ ✸✳✷✳✵✳✶✹ ❜✉t ✐t ♦♥•② ✇♦r❦s ✐♥ ❞✐♠❡♥s✐♦♥ ✶ ❛s ♦❢ ♥♦✇✳ ❲❡ st❛rt ✇✐t❤ ❛ •❡♠♠❛ ✇❤✐❝❤ ✐s ❛ s•✐❣❤t•② ♠♦r❡ ❣❡♥❡r❛• ✈❡rs✐♦♥ ♦❢ ❛ r❡s✉•t ✐♥ ❬✹(✿ ✻✽ ❈❍❆P❚❊| ✸✳ ▲■❋❚■◆• ❖❋ |❆◆❉❖▼ ❱❆|■❆❇▲❊❙ ❇( ❉■❙❚|■❇❯❚■❖◆❙ ▲❡♠♠❛ ✸✳✷✳✵✳✻✳ ❙✉♣♣♦s❡ t❤❛t X ∈ D ∞ -,1 (R) ❤❛s ❛ ❝✉♠✉•❛t✐✈❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝✲ t✐♦♥ ✇❤✐❝❤ ✐s γ✲❍ö•❞❡r✳ ❚❤❡♥✱ ❢♦r ❛♥② x✱ ❢♦r ❛♥② p > 1 ❛♥❞ ❢♦r ❛♥② α < γ p ✿ 1 X>x ∈ D p,α ✭✸✳✻✾✮ ❛♥❞✱ ❢♦r s♦♠❡ r ❛♥❞ ❛ ❝♦♥st❛♥t C(p, α) > 0✱ t❤❡ ❢♦••♦✇✐♥❣ ✐♥❡q✉❛•✐t② t❤❡♥ ❤♦•❞s✿ 1 X>x Dp,α ≤ C(p, α) (1 + ∇X L r ) F X Λ γ ✭✸✳✼✵✮ Pr♦♦❢✳ ❇❡✐♥❣ ✐♥s♣✐r❡❞ ❜② ❬✹( ✇❡ ✉s❡ t❤❡ ❑ ✐♥t❡r♣♦•❛t✐♦♥ ♠❡t❤♦❞ ❛♥❞ ✇❡ ✇r✐t❡✿ H = H ǫ + H -H ǫ ✭✸✳✼✶✮ ✇❤❡r❡ H ✐s t❤❡ ❍❡❛✈✐s✐❞❡ ❢✉♥❝t✐♦♥ ❛♥❞ H ǫ ✐s t❤❡ ♣✐❡❝❡✇✐s❡ •✐♥❡❛r ❢✉♥❝t✐♦♥ ✇✐t❤ ✈❛•✉❡s 0 ♦♥ ] -∞, -ǫ] ❛♥❞ 1 ♦♥ [0, ∞[✳ ❲❡ ♥♦t❡ t❤❛t H ǫ ✐s 1/ǫ✲▲✐♣s❝❤✐t③✱ s♦ H ǫ (X -x) ∈ D p,1 ❛♥❞ ✇❡ ❛♣♣•② t❤❡ ❑✲♠❡t❤♦❞ t♦ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥✿ 1 X>x = H ǫ (X -x) + 1 X>x -H ǫ (X -x) ✭✸✳✼✷✮ ❋✐rst✱ ❜② t❤❡ ❝❤❛✐♥ r✉•❡ ❛♥❞ ❍ö•❞❡r✬s ✐♥❡q✉❛•✐t② ♦♥❡ ❤❛s✱ ❢♦r ❛♥② q > 1✿ ∇H ǫ (X -x) L p ≤ ǫ -1 ∇X L pq q-1 1 x-ǫ<X<x L pq ≤ ǫ -1 ∇X L pq q-1 [P (X < x) -P (X < x -ǫ)] 1 pq ≤ F X Λ γ ∇X L pq q-1 ǫ -1+ γ pq ✭✸✳✼✸✮ ❛♥❞ s✐♠✐•❛r•②✿ 1 X>x -H ǫ (X -x) L p = P (x -ǫ < X < x) 1 p ≤ F X Λ γ ǫ γ p ✭✸✳✼✹✮ s♦ ♦✈❡r❛••✿ K ǫ ≤ ǫ H ǫ (X -x) D p,1 + 1 X>x -H ǫ (X -x) L p ≤ F X Λ γ ǫ γ pq ✭✸✳✼✺✮ ✇❤❡r❡ K ǫ = K(ǫ, 1 X>x ❢♦••♦✇✐♥❣ t❤❡ ♥♦t❛t✐♦♥ ♦❢ ❛♣♣❡♥❞✐① ❉ ❢♦r t❤❡ ❑ ♠❡t❤♦❞ ♦❢ ✐♥t❡r♣♦•❛t✐♦♥ ❛♥❞ ✜♥❛••②✿ 1 X>x p Dp,α = 1 0 ǫ -α K ǫ p dǫ ǫ ≤ C 1 0 ǫ -αp-1+ γ q dǫ ✭✸✳✼✻✮ ❛♥❞ ❝•❡❛r•② t❤✐s •❛st ✐♥t❡❣r❛• ❝♦♥✈❡r❣❡s ✐❢ ❛♥❞ ♦♥•② ✐❢ α < γ pq ✳ ❙✐♥❝❡ ♦♥❡ ♠❛② ❝❤♦♦s❡ q ❛r❜✐tr❛r✐•② ❝•♦s❡ t♦ 1 t❤❡ ♣r♦♦❢ ✐s ❝♦♠♣•❡t❡✳ ❖✉r •❡♠♠❛ ♥♦✇ ❛••♦✇s ✉s t♦ ♣r♦✈❡ t❤❡ ❢♦••♦✇✐♥❣✿ ✸✳✸✳ ❙❖▼❊ |❊❙❯▲❚❙ ❋❖| ❚❍❊ S ∞,S ❙P❆❈❊❙ ✻✾ ❚❤❡♦r❡♠ ✸✳✷✳✵✳✶✻✳ ❙✉♣♣♦s❡ t❤❛t X ∈ D ∞ -,2-γ p +ǫ ✐s ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✇✐t❤ ❛ γ✲ ❍ö•❞❡r ❝✉♠✉•❛t✐✈❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥✳ ❚❤❡♥ X ❤❛s ❛ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❜♦✉♥❞❡❞ ❞❡♥s✐t② ❣✐✈❡♥ ❜② t❤❡ ❢♦r♠✉•❛✿ p X (x) = E ∇1 X>x , ∇X |∇X| 2 H = E 1 X>x δ ∇X |∇X| 2 ✭✸✳✼✼✮ Pr♦♦❢✳ ❚❤❡ ✐❞❡♥t✐t② ❜❡t✇❡❡♥ t❤❡ t✇♦ ❡①♣❡❝t❛t✐♦♥s ✐s •✉st ❛♥ ✐♥t❡❣r❛t✐♦♥ ❜② ♣❛rts✳ ❲❡ ♦♥•② ♥❡❡❞ t♦ ♣r♦✈❡ t❤❛t t❤❡s❡ q✉❛♥t✐t✐❡s ❛r❡ ✇❡•• ❞❡✜♥❡❞✳ ❋✐rst✱ ✇❡ ♠❛② ❛♣♣•② •❡♠♠❛ ✸✳✷✳✵✳✻ t♦ s❡❡ t❤❛t ∇1 X>x ∈ D p,-(1-α) ✭✸✳✼✽✮ ❢♦r ❛♥② α < γ p ✳ ❙❡❝♦♥❞✱ ❜② ❍ö•❞❡r✬s ✐♥❡q✉❛•✐t② ❛♥❞ t❤❡♦r❡♠ ✸✳✶✳✵✳✶✵ ✇❡ s❡❡ t❤❛t ∇X |∇X| 2 ∈ D p * ,1-γ p + ǫ 2 ✭✸✳✼✾✮ s♦ ♦♥❝❡ ❛❣❛✐♥ ❛ ✈❡rs✐♦♥ ♦❢ ❍ö•❞❡r✬s ✐♥❡q✉❛•✐t② ♣r♦✈❡s t❤❛t t❤❡ ✜rst ❡①♣❡❝t❛t✐♦♥ ✐s ✇❡•• ❞❡✜♥❡❞✳ ❚❤❡ s❡❝♦♥❞ ♦♥❡ ✐s ❞❡❛•t ✇✐t❤ ✐♥ ❛ s✐♠✐•❛r ✇❛② ❛♥❞ ❛••♦✇s ♦♥❡ t♦ s❡❡ t❤❛t t❤❡ ❞❡♥s✐t② ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ❜♦✉♥❞❡❞✳ |❡♠❛r❦ ✸✳✷✳✵✳✻✳ ■❢ γ > 1 t❤❡♥ t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜•❡ X ❤❛s ❛ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ✐♥ C 1 b ✱ ❤❡♥❝❡ ✇❡ ❦♥♦✇ t❤❛t ✐t ❤❛s ❛ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❜♦✉♥❞❡❞ ❞❡♥s✐t② ✇✐t❤♦✉t ❛♥② ❤②♣♦t❤❡s✐s ♦♥ t❤❡ ▼❛••✐❛✈✐♥ r❡❣✉•❛r✐t② ♦❢ X✳ ❈♦♥✈❡rs❡•②✱ ✐❢ γ = 0✱ ✐t ✐s ❜❡tt❡r t♦ ✉s❡ t❤❡♦r❡♠ ✸✳✷✳✵✳✶✹✳ ✸✳✸ ❙♦♠❡ r❡s✉•ts ❢♦r t❤❡ S ∞,s s♣❛❝❡s ❚❤❡ r❡s✉•ts ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥ ✇❡r❡ ♦♥•② ❢♦r 1 < p < ∞✳ ❍❡r❡ ✇❡ ♣r♦✈✐❞❡ ❛♥❛•♦❣✉❡ r❡s✉•ts ❢♦r p = ∞✳ ❲❡ st❛rt ✇✐t❤ ❛ r❡s✉•t s✐♠✐•❛r t♦ •❡♠♠❛ ✸✳✷✳✵✳✺✿ ▲❡♠♠❛ ✸✳✸✳✵✳✼✳ ❚❛❦❡ F ∈ D ∞ -,1 ✳ ❚❤❡♥ ❢♦r 0 < ρ < 1 ❛♥❞ ǫ > 0 t❤❡r❡ ❡①✐sts ❛ ✉♥✐✈❡rs❛• ❝♦♥st❛♥t C(ρ, p, ǫ) s✉❝❤ t❤❛t ❢♦r ❛♥② φ ∈ S ∞,ρ+ 1 p +ǫ ✱ t❤❡ ❢♦••♦✇✐♥❣ ❡st✐♠❛t❡ ❤♦•❞s✿ φ(F ) Dp,ρ ≤ C (1 + ∇F L p ) f S ∞,ρ+ 1 p +ǫ ✭✸✳✽✵✮ ❇❡❢♦r❡ ✇❡ ♣r♦✈❡ t❤✐s r❡s✉•ts •❡t ✉s ♠❛❦❡ ❛ ❢❡✇ ❝♦♠♠❡♥ts✳ ❋✐rst✱ t❤✐s ✐s ❡❛s✐❡r t♦ ❛♣♣•② t❤❛♥ •❡♠♠❛ ✸✳✷✳✵✳✺ ❜❡❝❛✉s❡ t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜•❡ F ✐s ♥♦t r❡q✉✐r❡❞ t♦ ❤❛✈❡ ❛ ❞❡♥s✐t② ❛ ♣r✐♦r✐✳ ❍♦✇❡✈❡r✱ t❤✐s ✐s ✐♥t❡r❡st✐♥❣ ♦♥•② ✐❢ ρ + 1/p < 1✿ ♦t❤❡r✇✐s❡ φ ✐s ▲✐♣s❝❤✐t③ ❛♥❞ ♦♥❡ ♠❛② •✉st ❛♣♣•② t❤❡ ✉s✉❛• ❝❤❛✐♥ r✉•❡✳ ❋✐♥❛••②✱ s✐♥❝❡ ♦♥❡ ♠❛② t❛❦❡ p ❛s ❜✐❣ ❛s ♦♥❡ ✇❛♥ts ♦♥ t❤❡ |❍❙ ❛♥❞ s✐♥❝❡ t❤❡ • D p,k ✐♥❝r❡❛s❡ ✐♥ p✱ ♦♥❡ ❣❡ts t❤❡ ♦❜✈✐♦✉s✿ ✼✵ ❈❍❆P❚❊| ✸✳ ▲■❋❚■◆• ❖❋ |❆◆❉❖▼ ❱❆|■❆❇▲❊❙ ❇( ❉■❙❚|■❇❯❚■❖◆❙ ❈♦r♦••❛r② ✸✳✸✳✵✳✸✳ ❚❛❦❡ F ∈ D ∞ -,1 ✳ ❚❤❡♥ ❢♦r 0 < ρ < 1 ❛♥❞ ǫ > 0 t❤❡r❡ ❡①✐sts s♦♠❡ q(ǫ) > p ❛♥❞ ❛ ✉♥✐✈❡rs❛• ❝♦♥st❛♥t C(ρ, ǫ) s✉❝❤ t❤❛t ❢♦r ❛♥② φ ∈ S ∞,ρ+ǫ ✱ t❤❡ ❢♦••♦✇✐♥❣ ❡st✐♠❛t❡ ❤♦•❞s✿ φ(F ) Dp,ρ ≤ C (1 + ∇F L q(ǫ) ) f S ∞,ρ+ǫ ✭✸✳✽✶✮ ◆♦✇ ✇❡ t✉r♥ t♦ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ •❡♠♠❛✿ Pr♦♦❢✳ ❋✐rst •❡t ψ ❜❡ ❛♥② s♠♦♦t❤ ❢✉♥❝t✐♦♥✳ ❚❤❡♥✿ ψ(F ) L p ≤ ψ ∞ ✭✸✳✽✷✮ ❛♥❞ ψ(F ) D p,1 ≤ C [ ψ ∞ + ∇F L p ψ C 1 ] ≤ C (1 + ∇F L p ) ψ S ∞,1+ǫ ✭✸✳✽✸✮ ❜❡❝❛✉s❡ ψ C 1 ≤ ψ Λ 1+ǫ ❛♥❞ ❜❡❝❛✉s❡ ♦❢ t❤❡♦r❡♠ ✷✳✷✳✸✳✷✳ ◆♦✇ •❡t ψ(t) ❜❡ ❛ s♠♦♦t❤ r❡❛• ✐♥t❡r♣♦•❛t✐♦♥ ♣❛t❤ ❢♦r φ ❛♥❞ •❡t f (t) = ψ(t)(F )✳ ❚❤❡♥✱ s❡tt✐♥❣ ν = -1 p + 1 -ρ✿ 1 0 t pν f (t) D p,1 dt ≤ C [1 + ∇F L p ] p 1 0 t -1+(1-ρ)p ψ(t) p S ∞,1+ǫ dt ✭✸✳✽✹✮ ❛♥❞ t❤❡r❡❢♦r❡✿ 1 0 t pν f (t) D p,1 dt 1 p ≤ C [1 + ∇F L p ] sup 0≤t≤1 t 1-(ρ+ 1 p ) ψ(t) S ∞,1+ǫ ✭✸✳✽✺✮ ❚❤❡♥ ✇❡ ♠❛② ❝♦♥tr♦• 1 0 t pν d dt f (t) L p dt ✐♥ ❛ s✐♠✐•❛r ✇❛②✱ ❛♥❞ ❜② t❛❦✐♥❣ ✐♥✜♠✉♠s r❡❛• ✐♥t❡r♣♦•❛t✐♦♥ ②✐❡•❞s✿ φ(F ) Ep,ρ ≤ C [1 + ∇F L p ] • φ T ∞,(1+ǫ)(ρ+ 1 p ) ✭✸✳✽✻✮ ❚❤❡♥ ✇❡ ❛r❡ ❞♦♥❡ ❜❡❝❛✉s❡ t❤✐s ✐s ✈❛•✐❞ ❢♦r ❛♥② ǫ > 0 ❛♥❞ ❜❡❝❛✉s❡ ♦❢ t❤❡♦r❡♠s ✸✳✶✳✵✳✾ ❛♥❞ ✷✳✷✳✷✳✹✳ ❋✐♥❛••②✱ ❜② s✐♠♣•② ❝♦♣②✐♥❣ t❤❡ ♣r♦♦❢ ♦❢ t❤❡♦r❡♠ ✸✳✷✳✵✳✶✸ ❛♥❞ ✉s✐♥❣ t❤❡♦r❡♠ ✷✳✷✳✸✳✷ ✐♥st❡❛❞ ♦❢ t❤❡♦r❡♠ ✷✳✷✳✷✳✶ ♦♥❡ ♦❜t❛✐♥s t❤❡ ❢♦••♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✸✳✸✳✵✳✶✼✳ ▲❡t F ∈ D ∞ -,1+δ ❢♦r s♦♠❡ δ > 0 ❜❡ ❛ ♥♦♥✲❞❡❣❡♥❡r❛t❡ R N ✲ ✈❛•✉❡❞ r❛♥❞♦♠ ✈❛r✐❛❜•❡✳ ❚❤❡♥ ❢♦r ❡✈❡r② δ ′ < δ ❛♥❞ 1 < p < ∞ t❤❡r❡ ❡①✐sts ❛ ✉♥✐✈❡rs❛• ❝♦♥st❛♥t C ❛♥❞ s♦♠❡ q, r > 1 ❛♥❞ θ > 0 s✉❝❤ t❤❛t t❤❡ ❢♦••♦✇✐♥❣ ❡st✐♠❛t❡ ❤♦•❞s ❢♦r ❡✈❡r② φ ∈ S(R N )✿ φ • F D p,-δ ≤ C • 1 (detΣ) 2(k+1) q L q + 1 detΣ q L q 2m • F θ D r,1+δ ′ • φ S ∞,-δ ′ ✭✸✳✽✼✮ ❍❡♥❝❡ ❢♦r ❛♥② T ∈ S p ′ ,-δ ′ ✱ T • F ♠❛② ❜❡ ❞❡✜♥❡❞ ✐♥ D p,-δ ❛♥❞ t❤❡ ❛❜♦✈❡ ❡st✐♠❛t❡ st✐•• ❤♦•❞s✳ ✸✳✹✳ ❊❳■❙❚❊◆❈❊ ❆◆❉ |❊•❯▲❆|■❚( ❖❋ ❉❊◆❙■❚■❊❙ ✼✶ ❚❤❡♥ ❛❣❛✐♥✱ ❤❡r❡ ✇❡ ❞♦ ♥♦t ♥❡❡❞ t♦ ♠❛❦❡ ❛♥ ❛ss✉♠♣t✐♦♥ ♦♥ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ❞❡♥s✐t② ❢♦r t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜•❡ X✳ ✸✳✹ ❊①✐st❡♥❝❡ ❛♥❞ r❡❣✉•❛r✐t② ♦❢ ❞❡♥s✐t✐❡s ❈♦♥s✐❞❡r s♦♠❡ δ > 0✳ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ s✉♣♣♦s❡ t❤❛t X ∈ D ∞ -,1+δ (R N ) ✐s ❛ ♥♦♥❞❡❣❡♥❡r❛t❡ r❛♥❞♦♠ ✈❛r✐❛❜•❡ ✇❤✐❝❤ ❛❞♠✐ts ❛ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❜♦✉♥❞❡❞ ❞❡♥s✐t② p X ✳ ❆s ❛♥ ✐♠♠❡❞✐❛t❡ ❝♦r♦••❛r② ♦❢ t❤❡♦r❡♠ ✸✳✷✳✵✳✶✸✱ ✇❡ s❡❡ t❤❛t t❤❡ ❢♦••♦✇✐♥❣ •✐♥❡❛r ❛♣♣•✐❝❛t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s✿ S p ′ ,-δ ′ → D p,-δ T → T • X ✭✸✳✽✽✮ ❢♦r ❛♥② p ′ > p✱ δ ′ < δ✳ ◆♦✇ •❡t φ ∈ D ′ p,-δ = D p * ,δ ✳ ❈•❡❛r•② t❤❡ ❢♦••♦✇✐♥❣ ✐s ❛ ❝♦♥t✐♥♦✉s •✐♥❡❛r ❢♦r♠✿ S p ′ ,-δ ′ → R T → T • X, φ ✭✸✳✽✾✮ ❚❤❡r❡❢♦r❡ ❜② ❞✉❛•✐t②✱ t❤❡r❡ ❡①✐sts s♦♠❡ ❢✉♥❝t✐♦♥ p X,φ ∈ S ′ p ′ ,-δ ′ = S (p ′ ) * ,δ s✉❝❤ t❤❛t ❢♦r ❛♥② T ∈ S p ′ ,-δ ′ t❤❡ ❢♦••♦✇✐♥❣ ❤♦•❞s✿ T • X, φ = T, p X,φ ✭✸✳✾✵✮ ❙✐♥❝❡ p ′ ❛♥❞ δ ′ ♠❛② ❛♣♣r♦❛❝❤ p ❛♥❞ δ ✐♥✜♥✐t❡s✐♠❛••②✱ ❜② ❞❡♥s✐t② ♦♥❡ ❝❤❡❝❦s t❤❛t p X,φ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡s❡ ♣❛r❛♠❡t❡rs ❛♥❞ t❤❛t p X,φ ∈ S (p * ) -,δ -✳ ❇② t❤❡ ❙♦❜♦•❡✈ ✐♥•❡❝t✐♦♥s✱ ✇❡ ❞❡❞✉❝❡ t❤❛t p X,φ ∈ C (δ-N p * ) -✐❢ t❤t ❡①♣♦♥❡♥t ✐s ♣♦s✐t✐✈❡✳ ❆•s♦✱ s✐♥❝❡ t❤❡ ❉✐r❛❝ ♠❛ss δ x ∈ S p,-N p * -ǫ ❢♦r ❛♥② x ❛♥❞ ❛♥② ǫ > 0✱ ♦♥❡ s❡❡s t❤❛t ✐t ♠❛❦❡s s❡♥s❡✱ ❛t •❡❛st ❢♦r p s♠❛•• ❡♥♦✉❣❤✱ t♦ t❛❦❡ T = δ x ✐♥ t❤❡ ❡①♣r❡ss✐♦♥ ❛❜♦✈❡✳ ❚❤❡♥ t❤❡r❡ ❝♦♠❡s ❛♥ ❡①♣•✐❝✐t ❡①♣r❡ss✐♦♥ ❢♦r p X,φ ✿ p X,φ (x) = δ x • X, φ ✭✸✳✾✶✮ ■♥ t❤❡ ❝❛s❡ ✇❤❡r❡ T ✐s ❛ ♣r♦♣❡r ❢✉♥❝t✐♦♥✱ ✇❡ ♥♦t❡ t❤❛t ✇❡ ❤❛✈❡ t❤❡ ✐❞❡♥t✐t②✿ E [T (X)φ] = T (x)p X,φ (x)dx ✭✸✳✾✷✮ ■❞❡♥t✐t②✐♥❣ ✇✐t❤ t❤❡ ✉s✉❛• ❞❡✜♥✐t✐♦♥s ❢♦r t❤❡ ❞❡♥s✐t② ❛♥❞ t❤❡ ❝♦♥❞✐t✐♦♥♥❛• ❞❡♥s✐t② ♦❢ ❛ r❛♥❞♦♠ ✈❛r✐❛❜•❡✱ ✇❡ t❤❡♥ ♦❜t❛✐♥✿ p X,1 = p X ✭✸✳✾✸✮ ❛♥❞ p X,φ (x) = p X (x) • E [φ|X = x] ✭✸✳✾✹✮ ❲❡ s✉♠ ✉♣ ♦✉r r❡s✉•ts ✐♥ t❤❡ ❢♦••♦✇✐♥❣✿ ✼✷ ❈❍❆P❚❊| ✸✳ ▲■❋❚■◆• ❖❋ |❆◆❉❖▼ ❱❆|■❆❇▲❊❙ ❇( ❉■❙❚|■❇❯❚■❖◆❙ ❚❤❡♦r❡♠ ✸✳✹✳✵✳✶✽✳ ▲❡t δ > 0 ❛♥❞ •❡t X ∈ D ∞ -,1+δ (R N ) ❜❡ ❛ ♥♦♥❞❡❣❡♥❡r❛t❡ r❛♥❞♦♠ ✈❛r✐❛❜•❡ ✇❤✐❝❤ ❛❞♠✐ts ❛ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❜♦✉♥❞❡❞ ❞❡♥s✐t② p X ✳ ▲❡t p > 1 ❛♥❞ φ ∈ D p,δ ✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ❢✉♥❝t✐♦♥ p X,φ ∈ S p -,δ -s✉❝❤ t❤❛t ❢♦r ❛♥② T ∈ S p * ,-δ ✿ T • X, φ = T, p X,φ ✭✸✳✾✺✮ ❆•s♦✱ ✐❢ δ > N p t❤❡♥ p X,φ ∈ C (δ-N p ) -✳ ❖♥❡ ❤❛s t❤❡ ❡①♣r❡ss✐♦♥✿ p X,φ (x) = p X (x) • E [φ|X = x] = δ x • X, φ ✭✸✳✾✻✮ ■♥ ♣❛rt✐❝✉•❛r✿ p X (x) = p X,1 (x) = δ x • X, 1 ✭✸✳✾✼✮ ❛♥❞ ❢♦r ❛•• p > 1 p X ∈ S p,δ ✱ t❤❡r❡❢♦r❡ p X ∈ C δ -✳ ❱❡r② s✐♠✐•❛r•②✱ ✇✐t❤♦✉t ❛ss✉♠✐♥❣ ❛ ♣r✐♦r✐ t❤❛t X ❤❛s ❛ ❝♦♥t✐♥✉♦✉s ❞❡♥s✐t② ❜② ✉s✐♥❣ t❤❡♦r❡♠ ✸✳✷✳✵✳✶✺ ✇❡ ❝♦✉•❞ ❤❛✈❡ ♣r♦✈❡❞✿ ❚❤❡♦r❡♠ ✸✳✹✳✵✳✶✾✳ ▲❡t δ > 1 ❛♥❞ •❡t X ∈ D ∞ -,1+δ (R N ) ❜❡ ❛ ♥♦♥❞❡❣❡♥❡r❛t❡ r❛♥❞♦♠ ✈❛r✐❛❜•❡✳ ❚❤❡♥ X ❛❞♠✐ts ❛ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❜♦✉♥❞❡❞ ❞❡♥s✐t② p X ❀ ❛❝t✉❛••② p X ∈ S p,δ ❢♦r ❛•• p ❛♥❞ p X ∈ C s -✱ t❤❡r❡❢♦r❡ ❛•• t❤❡ ❝♦♥❝•✉s✐♦♥s ♦❢ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠ ❤♦•❞✳ ❆•s♦✱ ✐♥ t❤❡ ❝❛s❡ ✇❤❡r❡ X, φ ❛r❡ ✐♥ D✱ ✐t ✐s ♥♦t❡✇♦rt❤② t❤❛t p X,φ ✱ ❛♥❞ ❡s♣❡❝✐❛••② p X ✱ ❛r❡ ✐♥ S✱ ✇❤✐❝❤ ✐s ♠✉❝❤ str♦♥❣❡r t❤❛♥ •✉st C ∞ ✳ ❋✐♥❛••②✱ t❤❡ ❢♦••♦✇✐♥❣ ✐s ❛♥ ✐♠♠❡❞✐❛t❡ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ❡q✉❛t✐♦♥ ✭✸✳✾✹✮ ✇❤✐❝❤ ✇✐•• ❜❡ ✉s❡❢✉• •❛t❡r ♦♥✿ Pr♦♣♦s✐t✐♦♥ ✸✳✹✳✵✳✺✳ ❯♥❞❡r t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ ❡✐t❤❡r ♦♥❡ ♦❢ t❤❡ t✇♦ t❤❡♦r❡♠s ❛❜♦✈❡✱ p X ❛♥❞ p X,φ ❤❛✈❡ t❤❡ s❛♠❡ s✉♣♣♦rts✳ ✸✳✺ ❚❤❡ ❝❛s❡ ♦❢ r❛♥❞♦♠ ✜❡•❞s ■♥ t❤✐s ♣❛r❛❣r❛♣❤ ✇❡ st✉❞② t❤❡ ▼❛••✐❛✈✐♥ r❡❣✉•❛r✐t② ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜•❡s ♦❢ t❤❡ t②♣❡✿ f (X) = f (w, X(w)) ✭✸✳✾✽✮ ✇❤❡r❡ X ✐s ❛ r❡❣✉•❛r ❡♥♦✉❣❤ r❛♥❞♦♠ ✈❛r✐❛❜•❡ ❛♥❞ f ✐s ❛ r❛♥❞♦♠ ✢♦✇ ♦♥ R N ✐♥ t❤❡ s❡♥s❡ ❞❡✜♥❡❞ ❜② ❬✹✵(✱ ❡♥•♦②✐♥❣ s♦♠❡ r❡❣✉•❛r✐t② ❜♦t❤ ♦♥ ✐ts ♣❛t❤s ❛♥❞ ✐♥ t❤❡ s❡♥s❡ ♦❢ ▼❛••✐❛✈✐♥✳ ■t ✐s ❝•❛ss✐❝ t♦ ✐♥tr♦❞✉❝❡ ♥♦r♠s s✉❝❤ ❛s✿ f L p (C 0 b ) = sup x∈R |f (x)| L p ✭✸✳✾✾✮ ♦r✿ f L p (C k b ) = sup x∈R |f (x)| + sup x∈R |f (k) (x)| L p ✭✸✳✶✵✵✮ ✸✳✺✳ ❚❍❊ ❈❆❙❊ ❖❋ |❆◆❉❖▼ ❋■❊▲❉❙ ✼✸ ❆❝❝♦r❞✐♥❣•②✱ ✇❡ ♥♦t❡ L p (C k b ) ❢♦r t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❇❛♥❛❝❤ s♣❛❝❡s✳ ❋♦r s❤♦rt✱ ✇❡ ✇✐•• r❡❢❡r t♦ s✉❝❤ ✢♦✇s ❛s C k b ✱ L p ✢♦✇s✱ ♦r ❛s L p (C k b ) ✢♦✇s✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ C k b ❛r❡ s❡♣❛r❛❜•❡ ❇❛♥❛❝❤ s♣❛❝❡s✱ s♦ •r♦ss✲❙♦❜♦•❡✈ s♣❛❝❡s ♠❛② ❜❡ ❜✉✐•t ♦♥ t❤❡♠✱ ❛♥❞ ❢♦••♦✇✐♥❣ ▼❡②❡r✬s ✐♥❡q✉❛•✐t✐❡s ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ❢♦••♦✇✐♥❣ ♥♦r♠s✿ f Dp,s(C 0 b ) = sup x∈R |(Id + L) s 2 f (x)| L p ✭✸✳✶✵✶✮ ❛♥❞✿ f Dp,s(C k b ) = sup x∈R |(Id + L) s 2 f (x)| + sup x∈R |(Id + L) s 2 f (k) (x)| L p ✭✸✳✶✵✷✮ ❛s ✇❡•• ❛s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s♣❛❝❡s ❛♥❞ ♦❜✈✐♦✉s ♥♦t❛t✐♦♥✳ ❖✉r ✜rst r❡s✉•t ✐s t❤❡ ❢♦••♦✇✐♥❣✿ Pr♦♣♦s✐t✐♦♥ ✸✳✺✳✵✳✻ ✭▼❛••✐❛✈✐♥ ❝❤❛✐♥ r✉•❡ ❢♦r r❛♥❞♦♠ ✢♦✇s✮✳ ▲❡t p, q, r > 1 s✉❝❤ t❤❛t 1/p + 1/q = 1/r ❛♥❞ s✉♣♣♦s❡ t❤❛t t❤❡ ❢♦••♦✇✐♥❣ ❤②♣♦t❤❡s❡s ❤♦•❞✿ • f ∈ L p (C 1 b ) • f ∈ D p,1 (C 0 b ) • X ∈ D q,1 (R N ) ❚❤❡♥✱ f (X) ∈ D r,1 ❛♥❞ t❤❡ ❢♦••♦✇✐♥❣ ❝❤❛✐♥ r✉•❡ ❤♦•❞s✿ ∇(f (X)) = N k=1 ∂ k f (X)∇X k + (∇f )(X) ✭✸✳✶✵✸✮ ❲❡ ❜♦rr♦✇ t❤❡ ✐❞❡❛ ❜❡❤✐♥❞ t❤✐s ♣r♦♦❢ ❢r♦♠ ❬✺✸(✿ Pr♦♦❢✳ ▲❡t ψ n ❜❡ ❛ s♠♦♦t❤ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❉✐r❛❝ ♠❛ss✳ ❚❤❡♥✱ ❛•♠♦st s✉r❡•②✱ ❢♦r ❛♥② x✿ f (w, X(w)) = lim n→∞ f (w, x)ψ n (X(w) -x) ✭✸✳✶✵✹✮ ❆•s♦✱ ❜② t❤❡ ✉s✉❛• ❝❤❛✐♥ r✉•❡✿ ∇ [f (w, x)ψ n (X(w) -x)] = ψ n (X(w) -x)(∇f )(w, x) -f (w, x) • N k=1 ∂ k ψ n (X(w) -x)∇X i ✭✸✳✶✵✺✮ ❛♥❞ ❝•❡❛r•② t❤✐s ❝♦♥✈❡r❣❡s✱ ❛t •❡❛st ✐♥ t❤❡ ❛•♠♦st s✉r❡ s❡♥s❡✱ t♦ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ ♦❢ ❡q✉❛t✐♦♥ ✸✳✶✵✸✳ ❆•s♦✱ ❜② t❤❡ ❤②♣♦t❤❡s❡s ✇❡ ♠❛❞❡ ❛♥❞ ❍ö•❞❡r✬s ✐♥❡q✉❛•✐t②✱ t❤✐s q✉❛♥t✐t② ✐s ❜♦✉♥❞❡❞ ✐♥ L r (C 0 b )✱ s♦ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ t❛❦❡s ♣•❛❝❡ ✐♥ L r ❛♥❞ ✇❡ ❛r❡ ❞♦♥❡✳ ✼✹ ❈❍❆P❚❊| ✸✳ ▲■❋❚■◆• ❖❋ |❆◆❉❖▼ ❱❆|■❆❇▲❊❙ ❇( ❉■❙❚|■❇❯❚■❖◆❙ ❆•t❡r♥❛t✐✈❡•② ❛♥ ✐♥t✉✐t✐✈❡ ✐❞❡❛ ✐s t♦ ✧s❡♣❛r❛t❡ t❤❡ ✈❛r✐❛❜•❡s✧ ❜❡t✇❡❡♥ t❤❡ r❛♥✲ ❞♦♠♥❡ss ✐♥ t❤❡ ✢♦✇ ❛♥❞ t❤❡ r❛♥❞♦♠♥❡ss ✐♥ t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜•❡✳ ❲❡ s❦❡t❝❤ ❛♥♦t❤❡r ♣r♦♦❢ ♦❢ ♣r♦♣♦s✐t✐♦♥ ✸✳✺✳✵✳✻ ✇❤✐❝❤ ✐s ✐♥ t❤❛t s♣✐r✐t ❛♥❞ ✐s ✐♥t❡r❡st✐♥❣ ✐♥ ✐ts ♦✇♥ r✐❣❤t✿ Pr♦♦❢✳ ▲❡t ✉s t❛❦❡ ✐♥✈❡rs❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠s ❛♥❞ ✇r✐t❡✿ f (w, X(w)) = e iξX(w) f (w, ξ)dξ ✭✸✳✶✵✻✮ ❚❤❡♥ ❜② ✉s✐♥❣ ❛ ❍✐•❜❡rt s♣❛❝❡ ✈❡rs✐♦♥ ♦❢ t❤❡ ❋✉❜✐♥✐ t❤❡♦r❡♠ ✭❝❢ ❬✹✵( ❢♦r ❡①❛♠♣•❡✮ ❛♥❞ s✐♠✐•❛r ❛r❣✉♠❡♥ts ❛s ❛❜♦✈❡✿ ∇(f (X)) = N k=1 iξ k e iξX(w) f (w, ξ)dξ ∇X k (w) + e iξX(w) (∇ f )(w, ξ)dξ ✭✸✳✶✵✼✮ ✇❤✐❝❤ ✐s ✐♥❞❡❡❞ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ ♦❢ ❡q✉❛t✐♦♥ ✸✳✶✵✸ ❛s ❝❛♥ ❜❡ s❡❡♥ ❜② t❛❦✐♥❣ ✐♥✈❡rs❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠s✳ ❲❡ ♥♦✇ t✉r♥ t♦ ❛♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ✐♥t❡❣r❛t✐♦♥ ❜② ♣❛rts ❢♦r♠✉•❛✳ ❲❡ st❛rt ✇✐t❤ s♦♠❡ ♣r❡•✐♠✐♥❛r② ❝♦♠♣✉t❛t✐♦♥s✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ ❤②♣♦t❤❡s❡s ♦❢ ♣r♦♣♦s✐t✐♦♥ ✸✳✺✳✵✳✻ ❤♦•❞ ❛♥❞ t❤❛t X ✐s ♥♦♥✲❞❡❣❡♥❡r❛t❡✳ ❆s ✉s✉❛• ✇❡ ♥♦t❡ σ ❢♦r ✐ts ▼❛••✐❛✈✐♥ ♠❛tr✐① ❛♥❞ γ ❢♦r t❤❡ ✐♥✈❡rs❡ ♦❢ σ✳ ❚❤❡♥ ❢♦r ❛♥② j ✇❡ ♠❛② ✇r✐t❡✿ (∇(f (X)), ∇X j ) = n k=1 σ kj ∂ k f (X) + ((∇f )(X), ∇X j ) ✭✸✳✶✵✽✮ t❤❡r❡❢♦r❡ ❜② str❛✐❣❤t❢♦r✇❛r❞ •✐♥❡❛r ❛•❣❡❜r❛✿ ∂ k f (X) = N j=1 γ kj (∇(f (X)) -(∇f )(X), ∇X j ) ✭✸✳✶✵✾✮ ◆♦✇ ✇❡ ❞❡✜♥❡ N ♥❡✇ r❛♥❞♦♠ ✢♦✇s ❜② t❤❡ ❡q✉❛t✐♦♥s✿ T k (f, X) = x → N j=1 γ kj ((∇f )(x), ∇X j ) ✭✸✳✶✶✵✮ ◆♦t❡ t❤❛t t❤✐s ❞❡♣❡♥❞s •✐♥❡❛r•② ♦♥ f ✱ ❛♥❞ ❞❡♣❡♥❞s ♦♥ X •✐♥❡❛r•② t❤r♦✉❣❤ ∇X j ❜✉t ❛•s♦ ♥♦♥✲•✐♥❡❛r•② t❤r♦✉❣❤ γ✳ ■t ✐s ♥♦t ❞✐✣❝✉•t t♦ s❡❡ t❤❛t T k (f, X) ✐s ❛♥ L p r❛♥❞♦♠ ✢♦✇ ✇✐t❤ t❤❡ s❛♠❡ s♠♦♦t❤♥❡ss ❛s (∇f )(x) ✭❛t •❡❛st C 0 b ❛s ♣❡r ♦✉r s❡t ♦❢ ❤②♣♦t❤❡s❡s✮✳ ❖❢ ❝♦✉rs❡ ✐❢ f ✐s ❞❡t❡r♠✐♥✐st✐❝ t❤❡♥ T k (f, X) ✐s ♥✉••✳ ◆♦✇ ✇❡ ❛r❡ r❡❛❞② t♦ st❛t❡ t❤❡ ❢♦••♦✇✐♥❣✿ ✸✳✺✳ ❚❍❊ ❈❆❙❊ ❖❋ |❆◆❉❖▼ ❋■❊▲❉❙ ✼✺ Pr♦♣♦s✐t✐♦♥ ✸✳✺✳✵✳✼ ✭❋•♦✇ ✐♥t❡❣r❛t✐♦♥ ❜② ♣❛rts ❢♦r♠✉•❛✮✳ ❯♥❞❡r t❤❡ ❤②♣♦t❤❡s❡s ♦❢ ♣r♦♣♦s✐t✐♦♥ ✸✳✺✳✵✳✻✱ ❢♦r ❛♥② φ ∈ D p * ,1 ✿ E [∂ k f (X) • φ] = E [f (X) • l k (X, φ)] -E [φ • T k (f, X)(X)] ✭✸✳✶✶✶✮ ❲❡ ♥♦t❡ t❤❛t ✐❢ f ✐s ❞❡t❡r♠✐♥✐st✐❝ t❤✐s ❜♦✐•s ❞♦✇♥ t♦ t❤❡ ✉s✉❛• ▼❛••✐❛✈✐♥ ✐♥t❡✲ ❣r❛t✐♦♥ ❜② ♣❛rts ❢♦r♠✉•❛✳ ❲❡ ✇✐•• ♥♦✇ ❜❡ ❢♦••♦✇✐♥❣ t❤❡ s❛♠❡ ❛❣❡♥❞❛ ❛s ✐♥ t❤❡ ♣r❡✈✐♦✉s ♣❛r❛❣r❛♣❤s✳ ❙✐♥❝❡ ✭❢r♦♠ ♣r❡✈✐♦✉s ♣❛r❛❣r❛♣❤s ✐♥ t❤✐s ❝❤❛♣t❡r✮ ✇❡ ❛•r❡❛❞② ❦♥♦✇ ❤♦✇ t♦ ❝♦♥tr♦• l k (X, φ) ✇❡ ✇✐•• ❢♦❝✉s ♦♥ T k (f, X) ✲ ❛♥❞ ♠♦r❡ s♣❡❝✐✜❝❛••② ♦♥ T k (f, X)(X)✳ ❲❡ st❛rt ✇✐t❤ t❤❡ ❢♦••♦✇✐♥❣✿ ▲❡♠♠❛ ✸✳✺✳✵✳✽✳ ▲❡t p > 1✱ ❛♥❞ f ❛♥❞ X ❜❡ s✉❝❤ t❤❛t t❤❡ ❢♦••♦✇✐♥❣ ❤②♣♦t❤❡s❡s ❤♦•❞✿ • f ✐s ❛ C n b ✢♦✇ • ❚❤❡r❡ ❡①✐sts p < r < ∞ s✉❝❤ t❤❛t ❢♦r ❡✈❡r② k ≥ 1 ❛♥❞ ❡✈❡r② ♠✉•t✐✲✐♥❞❡① j s✉❝❤ t❤❛t |j| + k = n + 1✱ ∇ k ∂ j 1 ,...,j N f ❡①✐sts ✐♥ L r ❛s ❛ C 0 b ✢♦✇✱ ✐❡✿ f ∈ |j|+k=n+1 D r,k C j b (R N ) ✭✸✳✶✶✷✮ • X ∈ D ∞ -,n ❚❤❡♥✱ (∇f )(X) ∈ D p,n (H)✱ ❛♥❞ ❢♦r s♦♠❡ ρ > p ❛♥❞ s♦♠❡ ✉♥✐✈❡rs❛• ❝♦♥st❛♥t C✿ ∇ n ((∇f ) (X)) L p (H) ≤ C •   |j|+k=n+1;k≥1 (∇ k ∂ j 1 ,...,j N f )(X) L r   • X n Dρ,n ≤ C •   |j|+k=n+1;k≥1 f D r,k (C j b (R N ))   • X n Dρ,n ✭✸✳✶✶✸✮ Pr♦♦❢✳ ❆ s✐♠♣•❡ r❡❝✉rr❡♥❝❡ ❜❛s❡❞ ♦♥ ♣r♦♣♦s✐t✐♦♥ ✸✳✺✳✵✳✻ ②✐❡•❞s t❤❛t (∇f )(X) ✐s n t✐♠❡s ❞✐✛❡r❡♥t✐❛❜•❡✳ ❚❤❡ s❡❝♦♥❞ ❜✉••❡t ♣♦✐♥t ✐♥ ♦✉r ❤②♣♦t❤❡s❡s ❡♥s✉r❡ ✉s t❤❛t ❢♦r ❛♥② j, k✿ sup x (∇ k ∂ j 1 ,...,j N f )(x) H ⊗k L r < ∞ ✭✸✳✶✶✹✮ s♦ ❛ ❢♦rt✐♦r✐ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ ✐♥ ♦✉r ♠❛•♦r✐③❛t✐♦♥ ✐s ✜♥✐t❡✳ ❚♦ ♦❜t❛✐♥ t❤❡ ❛❝t✉❛• ✐♥❡q✉❛•✐t②✱ ❝♦♠♣✉t❡ ∇ n ((∇f ) (X)) ✇✐t❤ ❋❛❛ ❞✐ ❇r✉♥♦✬s ❢♦r♠✉•❛ ❛♥❞ ❛♣♣•② ❍ö•❞❡r✬s ✐♥❡q✉❛•✐t②✳ ✼✻ ❈❍❆P❚❊| ✸✳ ▲■❋❚■◆• ❖❋ |❆◆❉❖▼ ❱❆|■❆❇▲❊❙ ❇( ❉■❙❚|■❇❯❚■❖◆❙ ❲❡ ✇♦✉•❞ •✐❦❡ t♦ ✐♠♣r♦✈❡ t❤✐s r❡s✉•t ✐♥t♦ ❛ ♠♦r❡ ♣r❡❝✐s❡ ♦♥❡ ❢♦r t❤❡ ❢r❛❝t✐♦♥❛• ❝❛s❡✳ ❲❡ st❛rt ✇✐t❤ ❛ r❡s✉•t ✇❤✐❝❤ ✐s ❛♥❛•♦❣✉❡ t♦ •❡♠♠❛ ✸✳✸✳✵✳✼ ❛♥❞ ❝♦r♦••❛r② ✸✳✸✳✵✳✸✿ ▲❡♠♠❛ ✸✳✺✳✵✳✾✳ ▲❡t X ∈ D ∞ -,1 ✳ ❚❤❡♥✱ ❢♦r ❡✈❡r② 0 < ρ < 1✱ ❢♦r ❡✈❡r② α ∈]1, ∞[✱ ❢♦r ❡✈❡r② ǫ > 0✱ ❢♦r ❡✈❡r② p > 1✱ t❤❡r❡ ❡①✐st q, r > 1 ❛♥❞ ❛ ✉♥✐✈❡rs❛• ❝♦♥st❛♥t C s✉❝❤ t❤❛t ❢♦r ❡✈❡r② f ∈ D ∞ -,1 (S) t❤❡ ❢♦••♦✇✐♥❣ ❡st✐♠❛t❡ ❤♦•❞s✿ f (X) Dp,ρ ≤ C • (1 + ∇X L r ) • f L q (S ∞,ρ+ǫ ) + f Dp,ρ(W α, N α +ǫ ) ✭✸✳✶✶✺✮ Pr♦♦❢✳ ❲❡ ✇✐•• ✉s❡ t❤❡ ❙♦❜♦•❡✈ ✐♥•❡❝t✐♦♥✿ W α, N α +ǫ ֒→ C 0 b ✭✸✳✶✶✻✮ ◆♦✇ •❡t ✉s ✇r✐t❡✿ f (X) L p ≤ f L p (C 0 b ) ≤ f L p (W α, N α +ǫ ) ✭✸✳✶✶✼✮ ❛♥❞✱ ✉s✐♥❣ t❤❡ ✢♦✇ ✈❡rs✐♦♥ ♦❢ t❤❡ ❝❤❛✐♥ r✉•❡ ❛♥❞ ❍ö•❞❡r✬s ✐♥❡q✉❛•✐t②✿ f (X) D p,1 ≤ C • f L p (C 0 b ) + f ′ L q (C 0 b ) • ∇X L r + ∇f L p (C 0 b ) ≤ C • f L p (W α, N α +ǫ ) + f L q (S ∞,1 ) • ∇X L r + ∇f L p (W α, N α +ǫ ) ✭✸✳✶✶✽✮ ◆♦✇ s✐♥❝❡ W α, N α +ǫ ✐s ❛ ❯▼❉ ❇❛♥❛❝❤ s♣❛❝❡ t❤❡ ▼❡②❡r ✐♥❡q✉❛•✐t✐❡s ❛r❡ ✈❛•✐❞ ❢♦r W α, N α +ǫ ✲✈❛•✉❡❞ r❛♥❞♦♠ ✈❛r✐❛❜•❡s✱ s♦ ✇❡ ♠❛② ✜♥✐s❤ t❤❡ ♣r♦♦❢ ❜② ✐♥t❡r♣♦•❛t✐♦♥ ❛s ❢♦r •❡♠♠❛ ✸✳✸✳✵✳✼✳ ◆♦✇ ✇❡ ❣❡t ❜❛❝❦ t♦ ❝♦♥tr♦••✐♥❣ (∇f )(X) Dp,s ✿ ▲❡♠♠❛ ✸✳✺✳✵✳✶✵✳ ▲❡t p > 1✱ s > 0✱ ǫ > 0 ❛♥❞ f ❛♥❞ X ❜❡ s✉❝❤ t❤❛t t❤❡ ❢♦••♦✇✐♥❣ ❤②♣♦t❤❡s❡s ❤♦•❞✿ • ❚❤❡r❡ ❡①✐sts p < q < ∞ s✉❝❤ t❤❛t✿ f ∈ k+l=[s]+1 D q,k S ∞,{s}+l+ǫ ✭✸✳✶✶✾✮ • ❚❤❡r❡ ❡①✐sts 1 < α < ∞ s✉❝❤ t❤❛t✿ f ∈ k+l=[s]+1 D p,k+{s} W α, N α +ǫ+l ✭✸✳✶✷✵✮ ✸✳✺✳ ❚❍❊ ❈❆❙❊ ❖❋ |❆◆❉❖▼ ❋■❊▲❉❙ ✼✼ • X ∈ D ∞ -,s ❚❤❡♥✱ (∇f )(X) ∈ D p,s (H) ❛♥❞ ❛♥❞ ❢♦r s♦♠❡ ρ > p ❛♥❞ s♦♠❡ ✉♥✐✈❡rs❛• ❝♦♥st❛♥t C t❤❡ ❢♦••♦✇✐♥❣ ❝♦♥tr♦• ❤♦•❞s✿ (∇f )(X) Dp,s ≤ C • (1 + ∇ [s]+1 X [s]+1 ) • k+l=[s]+1 f D q,k (S ∞,{s}+l+ǫ ) + f D p,k+{s} (W α, N α +ǫ+l ) ✭✸✳✶✷✶✮ Pr♦♦❢✳ ❇② ▼❡②❡r✬s ✐♥❡q✉❛•✐t✐❡s✿ (∇f )(X) Dp,s ≤ (Id + L) {s} 2 ∇ [s] (∇f )(X) L p ✭✸✳✶✷✷✮ ◆♦✇ ∇ [s] (∇f )(X) ♠❛② ❜❡ ❡①♣❛♥❞❡❞ ❜② ❋❛❛ ❞✐ ❇r✉♥♦✬s ❢♦r♠✉•❛❀ ❡❛❝❤ ♦❢ t❤❡ t❡r♠s ♦❢ t❤❡ ❡①♣❛♥s✐♦♥ ✐s ♦❢ t❤❡ ❢♦r♠✿ (∇ k ∂ l 1 • • • ∂ l N f )(X) ⊗ P ✭✸✳✶✷✸✮ ✇❤❡r❡ k + |l| = [s] + 1 ❛♥❞ P ✐s ❛ ♣♦•②♥♦♠✐❛• ✐♥ t❤❡ ∇ k X i ✇✐t❤ t♦t❛• ❞❡❣r❡❡ •♦✇❡r t❤❛♥ [s]✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ✜♥✐s❤ t❤❡ ♣r♦♦❢ ❜② s❡♣❛r❛t✐♥❣ t❤❡ t❡r♠s ✉s✐♥❣ t❤❡ tr✐❛♥❣•❡ ✐♥❡q✉❛•✐t② ❛♥❞ t❤❡♥ ✉s✐♥❣ ❍ö•❞❡r✬s ✐♥❡q✉❛•✐t② ❛♥❞ t❤❡ ♣r❡✈✐♦✉s •❡♠♠❛ ♦♥ ❡❛❝❤ t❡r♠ ✐♥ t❤❡ s✉♠✳ ❖✉r •❡♠♠❛ ♣r♦✈✐❞❡s ✉s ✇✐t❤ ❛ ❝♦♥tr♦• ❢♦r T k (f, X)(X)✿ Pr♦♣♦s✐t✐♦♥ ✸✳✺✳✵✳✽✳ ❙✉♣♣♦s❡ t❤❛t f ❛♥❞ X ✈❡r✐❢② ❛•• t❤❡ ❤②♣♦t❤❡s❡s ✐♥ t❤❡ ♣r❡✲ ✈✐♦✉s •❡♠♠❛✳ ❙✉♣♣♦s❡ ✐♥ ❛❞❞✐t✐♦♥ t❤❛t X ✐s ♥♦♥❞❡❣❡♥❡r❛t❡✳ ❚❤❡♥ ❢♦r s♦♠❡ s ′ > s✱ θ > 0 ❛♥❞ ❢♦r ❛♥② p 1 , p 2 , p 3 s✉❝❤ t❤❛t 1/p 1 + 1/p 2 + 1/p 3 < 1/p✿ T k (f, X)(X) Dp,s ≤ C •   k+l=[s]+1;k≥1 f D q,k (S ∞,{s}+l+ǫ ) + f D p,k+{s} (W α, N α +ǫ+l )   • 1 det Σ p 2 L p 2 + 1 det Σ 2([s]+1) p 2 L p 2 • X θ D p 3 ,1+s ′ ✭✸✳✶✷✹✮ Pr♦♦❢✳ ❲❡ st✐•• ♥♦t❡ γ ❢♦r t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ▼❛••✐❛✈✐♥ ♠❛tr✐① ♦❢ X✳ ❇② ❍ö•❞❡r✬s ✐♥❡q✉❛•✐t②✿ T k (f, X)(X) Dp,s ≤ C • (∇f )(X) Dq,s • δ j γ kj ∇X j Dr,s ✭✸✳✶✷✺✮ ❋✐rst✱ ✇❡ ❤❛✈❡ •✉st s❡❡♥ ❤♦✇ t♦ ❝♦♥tr♦• t❤❡ ✜rst t❡r♠ ♦❢ t❤❡ ♣r♦❞✉❝t ✐♥ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡ s❡❡♥ ❤♦✇ t♦ ❝♦♥tr♦• t❤❡ s❡❝♦♥❞ t❡r♠ ✐♥ t❤❡ ♣r♦♦❢ ♦❢ t❤❡♦r❡♠ ✸✳✷✳✵✳✶✶✳ ❚❤❡r❡❢♦r❡ ❜② r❡❝♦♠❜✐♥✐♥❣ ✇❡ ❛r❡ ❞♦♥❡✳ ✼✽ ❈❍❆P❚❊| ✸✳ ▲■❋❚■◆• ❖❋ |❆◆❉❖▼ ❱❆|■❆❇▲❊❙ ❇( ❉■❙❚|■❇❯❚■❖◆❙ ❲❡ ♠❛② ♥♦✇ st❛t❡ t❤❡ ♠❛✐♥ r❡s✉•t ✐♥ t❤✐s s❡❝t✐♦♥✿ ❚❤❡♦r❡♠ ✸✳✺✳✵✳✷✵ ✭▲✐❢t✐♥❣ ♦❢ ❛ r❛♥❞♦♠ ❞✐str✐❜✉t✐♦♥ ❜② ❛ ♥♦♥ ❞❡❣❡♥❡r❛t❡ |❱✮✳ ▲❡t X ∈ D ∞ -,1+δ ❢♦r s♦♠❡ δ > 0 ❜❡ ❛ ♥♦♥✲❞❡❣❡♥❡r❛t❡ R N ✲✈❛•✉❡❞ r❛♥❞♦♠ ✈❛r✐❛❜•❡✳ ❚❤❡♥ ❢♦r ❡✈❡r② δ ′ < δ ❛♥❞ 1 < p < ∞ t❤❡r❡ ❡①✐sts ❛ ✉♥✐✈❡rs❛• ❝♦♥st❛♥t C ❛♥❞ s♦♠❡ p 1 , p 2 , p 3 > p ❛♥❞ θ > 0 s✉❝❤ t❤❛t t❤❡ ❢♦••♦✇✐♥❣ ❡st✐♠❛t❡ ❤♦•❞s ❢♦r ❡✈❡r② φ ∈ S(D)✿ φ • X D p,-δ ≤ C • 1 (detΣ) 2(k+1) p 1 L p 1 + 1 detΣ p 1 L p 1 θ • X θ D p 2 ,1+δ ′ × [δ] k=0 φ D p 3 ,k (S ∞,-(δ ′ -k) ) + φ D p 3 ,k (W α,-(δ ′ -k-N α ) ) ✭✸✳✶✷✻✮ ❍❡♥❝❡ ❢♦r ❛♥② T ∈ D r,k (S p ′ ,-δ ′ )✱ T • X ♠❛② ❜❡ ❞❡✜♥❡❞ ✐♥ D p,-δ ❛♥❞ t❤❡ ❛❜♦✈❡ ❡st✐♠❛t❡ st✐•• ❤♦•❞s✳ Pr♦♦❢✳ ❲❡ ❢♦••♦✇ t❤❡ s❛♠❡ str❛t❡❣② ❛s ❢♦r t❤❡ ♣r♦♦❢ ♦❢ t❤❡♦r❡♠ ✸✳✸✳✵✳✶✼ s♦ ✇❡ ♦♠✐t t❤❡ ❞❡t❛✐•s✳ ❲❡ ❝♦♥❝•✉❞❡ t❤✐s s❡❝t✐♦♥ ✇✐t❤ t✇♦ s✐♠♣•❡ r❡♠❛r❦s✿ ✜rst t❤❛t ✐♥ ♣r❛❝t✐❝❡ ✐t ✇✐•• ❜❡ ❜❡st t♦ t❛❦❡ α ❛s ❜✐❣ ❛s ♣♦ss✐❜•❡❀ s❡❝♦♥❞ t❤❛t ✐❢ ✇❡ st✐❝❦ t♦ t❤❡ ❝❛s❡ ✇❤❡r❡ s ∈ N ✐t ✐s ♣♦ss✐❜•❡ t♦ ✇r✐t❡ ♠✉❝❤ s✐♠♣•❡r r❡s✉•ts ✇❤✐❝❤ ❞♦ ♥♦t ✐♥✈♦•✈❡ t❤❡ W α, N α +ǫ s♣❛❝❡s✳ ❈❤❛♣t❡r ✹ ❆ ✇❡❛❦ ■tô ❢♦r♠✉•❛ ■♥ t❤❡ ♣r❡✈✐♦✉s ❝❤❛♣t❡r✱ ✇❡ ❤❛✈❡ st❛t❡❞ ♣r❡❝✐s❡ ❝♦♥❞✐t✐♦♥s ✉♥❞❡r ✇❤✐❝❤ ❛ t❡♠♣❡r❡❞ ❞✐str✐❜✉t✐♦♥ ♠❛② ❜❡ •✐❢t❡❞ ❜② ❛ r❡❣✉•❛r✱ ♥♦♥✲❞❡❣❡♥❡r❛t❡ r❛♥❞♦♠ ✈❛r✐❛❜•❡✱ t❤❡ r❡s✉•t ❜❡✐♥❣ ❛ ❞✐str✐❜✉t✐♦♥ ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡✳ ❖❢ ❝♦✉rs❡✱ ✇❡ ❝♦✉•❞ ♠❛❦❡ ❜♦t❤ t❤❡ ❞✐str✐❜✉t✐♦♥ ❛♥❞ t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜•❡ ❞❡♣❡♥❞ ♦♥ t✐♠❡ ❛♥❞ t❤❡♥ t❤❡ r❡s✉•t ✇♦✉•❞ ❜❡ ❛ ✇❡❛❦ ♣r♦❝❡ss ❛s ♣❡r t❤❡ ♥♦t✐♦♥ ♦❢ ✇❡❛❦ ♠❡❛s✉r❛❜✐•✐t② ✇❤✐❝❤ ✇❡ ❞❡✜♥❡❞ ✐♥ s❡❝t✐♦♥ ✶✳✷✳✽✳ ■t ✐s t❤❡♥ ♥❛t✉r❛• t♦ t❤✐♥❦ ♦❢ ♣r♦✈✐♥❣ ❛♥ ■tô ❢♦r♠✉•❛ ❢♦r ❞✐str✐❜✉t✐♦♥s ❛♥❞ t❤✐s ❤❛❞ ✜rst ❜❡❡♥ ❞♦♥❡ ✐♥ ❬✼✽(✳ ■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ✇✐•• ✉s❡ s♦♠❡❤♦✇ ❞✐✛❡r❡♥t t❡❝❤♥✐q✉❡s ❜❛s❡❞ ♦♥ t❤❡ ♣r❡❝✐s❡ ❡st✐♠❛t❡s ✇❡ ♦❜t❛✐♥❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s ❝❤❛♣t❡r t♦ ♦❜t❛✐♥ ❛♥ ■tô ❢♦r♠✉•❛ ✇❤✐❝❤ ♠❛② ❜❡ ❛♣♣•✐❡❞ ✉♥❞❡r •❡ss ❞❡♠❛♥❞✐♥❣ ❝♦♥❞✐t✐♦♥s t❤❛♥ t❤❡ ♦♥❡ st❛t❡❞ ✐♥ ❬✼✽(✳ ❲❡ s❤❛•• ❛•s♦ ♣r♦✈❡ ❛ ❢♦r♠✉•❛ ♦❢ t❤❡ ■tô✲❲❡♥t③❡•• t②♣❡ ✇❤✐❝❤ ❤♦•❞s ✇❤❡♥ ❛ r❡❣✉•❛r ❡♥♦✉❣❤ r❛♥❞♦♠ ♣r♦❝❡ss ✐s •✐❢t❡❞ ❜② ❛ t❡♠♣❡r❡❞✲❞✐str✐❜✉t✐♦♥ ✈❛•✉❡❞ s❡♠✐✲♠❛rt✐♥❣❛•❡✳ ❚❤❡s❡ r❡s✉•ts ✇✐•• •❡❛❞ t♦ ❛♣♣•✐❝❛t✐♦♥s ✇❤✐❝❤ ❛r❡ ❞❡t❛✐•❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r ❛♥❞ t❤❡ ♥❡①t ♦♥❡✱ ❛♥❞ ✇♦✉•❞ ❤❛✈❡ r❡♠❛✐♥❡❞ ♦✉t ♦❢ r❡❛❝❤ ✇✐t❤ t❤❡ ♣r❡✈✐♦✉s•② ❡①✐st✐♥❣ ■tô ❢♦r♠✉•❛❡✳ ❆t ✈❛r✐♦✉s ♣♦✐♥ts ✐♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ✇✐•• ♥❡❡❞ s♦♠❡ r❡s✉•ts ♦♥ t❤❡ t♦♣♦•♦❣✐❡s ♦❢ S ❛♥❞ S ′ ✇❤✐❝❤ ❛r❡ q✉✐t❡ ✐♥✈♦•✈❡❞✳ ❋♦r t❤❡s❡ ✇❡ r❡❢❡r t♦ ❝❤❛♣t❡r ✷ ❛♥❞ ❛♣♣❡♥❞✐① ❆✳ ✹✳✶ ❚❤❡ ❢♦r♠✉•❛ ❢♦r ❛ ❣❡♥❡r❛• ■tô ♣r♦❝❡ss ✹✳✶✳✶ Pr♦❝❡ss❡s ✇✐t❤ ❛♥ ❛❜s♦•✉t❡•② ❝♦♥t✐♥✉♦✉s ❞r✐❢t ■♥ t❤✐s ♣❛r❛❣r❛♣❤ W ❞❡♥♦t❡s t❤❡ ❝❛♥♦♥✐❝❛• d✲❞✐♠❡♥s✐♦♥♥❛• ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡✳ ❲❡ ✇♦r❦ ✇✐t❤ t❤❡ ✜•tr❛t✐♦♥ ♦❢ W ✳ ▲❡t X ❜❡ t❤❡ N ✲❞✐♠❡♥s✐♦♥♥❛• ■tô ♣r♦❝❡ss s✉❝❤ t❤❛t✿ dX t = b t dt + σ t dW t ✭✹✳✶✮ X 0 = x ✭✹✳✷✮ ✼✾ ✽✵ ❈❍❆P❚❊| ✹✳ ❆ ❲❊❆❑ ■❚Ô ❋❖|▼❯▲❆ ✇❤❡r❡ b ✐s ❛♥ ❛❞❛♣t❡❞ N ✲❞✐♠❡♥s✐♦♥♥❛• ♣r♦❝❡ss ❛♥❞ σ ✐s ❛♥ ❛❞❛♣t❡❞ ♣r♦❝❡ss ♦❢ N ⊗ d✲❞✐♠❡♥s✐♦♥♥❛• r❛♥❞♦♠ ♠❛tr✐❝❡s✳ ❲❡ ✇✐•• ♥♦t❡ a = σ t σ✳ ❆•s♦ •❡t t ∈ [0, 1] → T (t) ∈ S ′ (R N ) ❜❡ ❛ ✭❞❡t❡r♠✐♥✐st✐❝✮ ♣❛t❤ ✇✐t❤ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥s✳ ❋✐rst ✇❡ ♥❡❡❞ t♦ ✜♥❞ tr❛❝t❛❜•❡ ❝♦♥❞✐t✐♦♥s ✉♥❞❡r ✇❤✐❝❤ ✐t ✐s •✐❝✐t t♦ ❞❡✜♥❡ t❤❡ ❝♦♠♣♦s❡s T t • X t ✱ ∂ i T t • X t ❛♥❞ ∂ 2 ij • X t ❢♦r t > 0✳ ▲❡t 1 < p < ∞✳ ❇② t❤❡ •r♦t❤❡♥❞✐❡❝❦ r❡♣r❡s❡♥t❛t✐♦♥✱ ❢♦r ❛♥② 1 < p < ∞ t❤❡r❡ ❡①✐sts s ′ > 0 s✉❝❤ t❤❛t T ✐s ❛ ♣❛t❤ ✇✐t❤ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥s ✐♥ S p,-s ✱ ❛s ✐s ❡①♣•❛✐♥❡❞ ✐♥ t❤❡♦r❡♠ ✷✳✸✳✷✳✷ ✲ t❤❡ ❝❛s❡ ✇❤❡r❡ t❤❡ T t ❛r❡ ❛❝t✉❛• ❢✉♥❝t✐♦♥s ❛♥❞ ♥♦t ❞✐str✐❜✉t✐♦♥s ✐♥ t❤❡ str✐❝t s❡♥s❡ ✐s ❡❛s✐❡r t♦ ❞❡❛• ✇✐t❤✳ ❚❤❡♥ t❤❡ ∂ i T t ❛r❡ ❛♥❞ t❤❡ ∂ 2 ij ❛r❡ ❇❱ ♣❛t❤s t❛❦✐♥❣ ✈❛•✉❡s ✐♥ S p,-(s+1) ❛♥❞ S p,-(s+2) r❡s♣❡❝t✐✈❡•②✳ ❋r♦♠ ♥♦✇ ♦♥✱ s✐♥❝❡ t❤✐s ✐s ♥♦t ❛ •♦ss ♦❢ ❣❡♥❡r❛•✐t②✱ ✇❡ ✇✐•• r❡str✐❝t ♦✉r st✉❞② t♦ t❤❛t ♦❢ ❛ ♣❛t❤✿ T t ∈ BV (S p,-s ) ✭✹✳✸✮ ■♥ ♣❛rt✐❝✉•❛r✱ t → T t S p,-s ✐s ❜♦✉♥❞❡❞✳ ❚❤❡♥ ✐❢ T t ✐s ❛s ✐♥ ✭✹✳✸✮ ❛❝❝♦r❞✐♥❣ t♦ t❤❡♦r❡♠ ✸✳✷✳✵✳✶✺✱ ❢♦r ❛•• t❤❡ ❞✐str✐❜✉t✐♦♥s ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ t♦ ❡①✐st ✇❡ ♥❡❡❞ t❤❛t X t ∈ D ∞ -,s ′ +3 ❢♦r s♦♠❡ s ′ > s ❛♥❞ t❤❛t X t ❜❡ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ❢♦r t > 0✳ ■♥ ❢❛❝t ❢♦r t❡❝❤♥✐❝❛• r❡❛s♦♥s ✇❡ ♥❡❡❞ t♦ ♠❛❦❡ ✉♥✐❢♦r♠ ❛ss✉♠♣t✐♦♥s ♦♥ X❀ ♠♦r❡ ♣r❡❝✐s❡•② •❡t ✉s s✉♣♣♦s❡ t❤❛t ❢♦r ❛•• r > 1✿ t 0 b u r D r,s ′ +3 du < ∞ ✭✹✳✹✮ t 0 σ u r D r,s ′ +3 du < ∞ ✭✹✳✺✮ sup ǫ≤u≤t 1 Σ(u) L r < ∞ ✭✹✳✻✮ ❢♦r ❛♥② 0 < ǫ < 1✱ ✇❤❡r❡ Σ(u) ✐s t❤❡ ▼❛••✐❛✈✐♥ ♠❛tr✐① ♦❢ X u ✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿ Pr♦♣♦s✐t✐♦♥ ✹✳✶✳✶✳✶✳ ❯♥❞❡r t❤❡ t❤r❡❡ ❛❜♦✈❡ ❤②♣♦t❤❡s❡s✱ ♦♥❡ ❤❛s✿ • ∀u ≥ 0✱ X u ∈ D ∞ -,s ′ +3 • ∀r > 0✱ ∀ǫ > 0✱ t ǫ X u r D r,s ′ +3 ds < ∞ • ∀u > 0✱ X u ✐s ♥♦♥✲❞❡❣❡♥❡r❛t❡✳ ■♥ ♣❛rt✐❝✉•❛r✱ t❤❡ T t • X t ❛r❡ ✇❡•• ❞❡✜♥❡❞ ✐♥ D p ′ ,-s ′ ✱ t❤❡ ∂ i T t • X t ❛r❡ ✇❡•• ❞❡✜♥❡❞ ✐♥ D p,-(s ′ +1) ❛♥❞ t❤❡ ∂ 2 ij T t • X t ❛r❡ ✇❡•• ❞❡✜♥❡❞ ✐♥ D p,-(s ′ +2) ✳ Pr♦♦❢✳ ❚❤❡ ✜rst t✇♦ ♣♦✐♥ts ❛r❡ ❡❛s② ❝♦♥s❡q✉❡♥❝❡s ♦❢ t❤❡ ❍ö•❞❡r ✐♥❡q✉❛•✐t✐❡s✳ ❋♦r t❤❡ t❤✐r❞ t❤✐r❞ ♣♦✐♥t✱ ✇❡ ♥♦t✐❝❡ t❤❛t ❝♦♥❞✐t✐♦♥ ✭✹✳✻✮ ✐♠♣•✐❡s t❤❛t ❢♦r ▲❡❜❡s❣✉❡✲❛✳s✳ u ∈ [ǫ, 1]✱ X u ✐s ♥♦♥✲❞❡❣❡♥❡r❛t❡❀ t❤❡♥ ✇❡ ✉s❡ t❤❛t t❤❡ ✐♥✈❡rs❡ ▼❛••✐❛✈✐♥ ♠❛tr✐① ✐s ❛♥ ✹✳✶✳ ❚❍❊ ❋❖|▼❯▲❆ ❋❖| ❆ •❊◆❊|❆▲ ■❚Ô P|❖❈❊❙❙ ✽✶ ■tô ♣r♦❝❡ss✱ ❤❡♥❝❡ ✐t ❤❛s ❝♦♥t✐♥✉♦✉s tr❛•❡❝t♦r✐❡s t♦ ❞❡❞✉❝❡ t❤❛t X u ✐s ♥♦♥❞❡❣❡♥❡r❛t❡ ❢♦r ❡✈❡r② u✳ ❚❤❡♥ t❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ ✇❡❛❦ ♣r♦❝❡ss❡s ✐s ❛ ❞✐r❡❝t ❛♣♣•✐❝❛t✐♦♥ ♦❢ t❤❡ r❡s✉•ts ✐♥ t❤❡ ♣r❡✈✐♦✉s ❝❤❛♣t❡r✳ |❡♠❛r❦ ✹✳✶✳✶✳✶✳ ■t ✐s ♥♦t❡✇♦rt❤② t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥s ✇❡ t❛❦❡ ❡♥s✉r❡ ✉s t❤❛t ❢♦r ❡✈❡r② u > ǫ✱ X u ❤❛s ❛ ❜♦✉♥❞❡❞ ❛♥❞ ❝♦♥t✐♥✉♦✉s ❞❡♥s✐t②✳ ❖✉r str❛t❡❣② ✐♥ ♦r❞❡r t♦ ♦❜t❛✐♥ ❛ ✇❡❛❦ ■tô ❢♦r♠✉•❛ ✇✐•• ♥♦✇ ❜❡ t❤❡ ❢♦••♦✇✐♥❣✿ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ❝•❛ss✐❝ s❡q✉❡♥❝❡ ♦❢ ♠♦••✐✜❡rs✿ ρ n (x) = 1 (2πn) N/2 • exp - |x| 2 2n ✭✹✳✼✮ ❛♥❞ ✇❡ r❡❣✉•❛r✐③❡ ♦✉r ❞✐str✐❜✉t✐♦♥ ✈❛•✉❡❞✲♣❛t❤❀ ♠♦r❡ ♣r❡❝✐s❡•② ✇❡ ✐♥tr♦❞✉❝❡ ❛ s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s T (n) ∈ S ❞❡✜♥❡❞ ❛s✿ T (n) = ρ n * T ✭✹✳✽✮ s♦ ✇❡ ❛r❡ ❛❜•❡ t♦ ❛♣♣•② t❤❡ ✭❝•❛ss✐❝✱ C 2 ✈❡rs✐♦♥ ♦❢ t❤❡✮ ■tô ❢♦r♠✉•❛✿ T (n) (t, X t ) -T (n) (ǫ, X ǫ ) = i t ǫ b i (u) • ∂ i T (n) (u, X u )du + 1 2 i,j t ǫ a ij (u) • ∂ ij T (n) (u, X u )du + t ǫ T (n) (du, X u ) + j t ǫ i b i (u) • ∂ i T (n) (u, X u ) dW j u ✭✹✳✾✮ ❛♥❞ s✐♥❝❡ ❜② ❞❡s✐❣♥✿ T (n) S p,-s → T ✭✹✳✶✵✮ ✇❡ ✇✐•• st✉❞② t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❡❛❝❤ t❡r♠ ❛s n t❡♥❞s t♦ ✐♥✜♥✐t②✳ ◆♦t❡ t❤❛t ✇❡ ❞♦ ♥♦t st❛rt ❢r♦♠ 0 ❛s t❤❡ ✐♥✐t✐❛• ❝♦♥❞✐t✐♦♥ x ✐s tr✐✈✐❛••② ❞❡❣❡♥r❛t❡✱ ❤❡♥❝❡ t❤❡r❡ ✐s ♥♦ ❤♦♣❡ ♦❢ ❞❡✜♥✐♥❣ T t • x✳ ❲❡ st❛rt ✇✐t❤ t❤❡ ❢♦••♦✇✐♥❣✿ ▲❡♠♠❛ ✹✳✶✳✶✳✶ ✭❈♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ▲❍❙✮✳ T (n) (t, X t ) -T (n) (ǫ, X ǫ ) D p ′ ,-s ′ → T (t) • X t -T (ǫ) • X ǫ ✭✹✳✶✶✮ Pr♦♦❢✳ ❚❤✐s ✐s ❛♥ ✐♠♠❡❞✐❛t❡ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡♦r❡♠ ✸✳✷✳✵✳✶✺❀ •❡t ✉s ♥♦t❡ t❤❛t ❤❡r❡ ✇❡ ♦♥•② ✉s❡❞ t❤❡ D ∞ -,s ′ r❡❣✉•❛r✐t② ♦❢ t❤❡ X u ✳ ✽✷ ❈❍❆P❚❊| ✹✳ ❆ ❲❊❆❑ ■❚Ô ❋❖|▼❯▲❆ ❚❤❡♥ ✇❡ ♣r♦✈❡✿ ▲❡♠♠❛ ✹✳✶✳✶✳✷ ✭❈♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✜rst t❡r♠✮✳ t ǫ b i (u) • ∂ i T (n) (u, X u )du D p ′ ,-(s ′ +1) -→ t ǫ b i (u) • ∂ i T u • X u du ✭✹✳✶✷✮ Pr♦♦❢✳ ❋✐rst •❡t ✉s ♥♦t❡ t❤❛t t❤❡♦r❡♠ ✸✳✷✳✵✳✶✺ ❛••♦✇s ✉s t♦ ❝❤❡❝❦ t❤❛t t❤❡ |❍❙ ❛❜♦✈❡ ✐s ✇❡•• ❞❡✜♥❡❞ ❛s ❛ ❇♦❝❤♥❡r ✐♥t❡❣r❛• ✐♥ D p ′ ,-(s ′ +1) ✳ ❙✐♥❝❡ t❤❡ t❡❝❤♥✐q✉❡s ✐♥✈♦•✈❡❞ ❛r❡ t❤❡ s❛♠❡ t❤❛♥ t❤♦s❡ ✇❡ ✇✐•• ✉s❡ t♦ ♣r♦✈❡ t❤❡ ❛❝t✉❛• ❝♦♥✈❡r❣❡♥❝❡ ✇❡ ❞♦ ♥♦t ✇r✐t❡ t❤❡ ❞❡t❛✐•s✳ ◆♦✇ •❡t ✉s t✉r♥ t♦ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❝♦♥✈❡r❣❡♥❝❡❀ ❜② ❍ö•❞❡r✬s t❤❡♦r❡♠✱ ❢♦r s♦♠❡ r > 1 ❛♥❞ p ′ > p ′′ > p > 1✿ b i (u) • ∂ i T (n) (u, X u ) -b i (u) • (∂ i T u ) • X u D p,-(s+1) ≤ C b i (u) D r,s+1 ∂ i T (n) (u, X u ) -(∂ i T u ) • X u D p ′′ ,-(s ′ +1) ✭✹✳✶✸✮ ❚❤❡ b i (u) D r,s+1 ❛r❡ ❜♦✉♥❞❡❞ ✐♥ u ❜② ❤②♣♦t❤❡s✐s✳ ❇② t❤❡♦r❡♠ ✸✳✷✳✵✳✶✺✱ ✇❡ ❤❛✈❡ ❛ ❜♦✉♥❞ ♦❢ t❤❡ t②♣❡✿ ∂ i T (n) (u, X u ) -(∂ i T u ) • X u D p ′′ ,-(s ′ +1) ≤ C(X u ) • ∂ i T (n) (u) -∂ i T (u) S p ′ ,-(s ′ +1) ✭✹✳✶✹✮ ✇❤❡r❡ u → C(X u ) ✐s ❛♥ ✐♥t❡❣r❛❜•❡ ❢✉♥❝t✐♦♥ ♦♥ [0, 1] ❜❡❝❛✉s❡ ♦❢ t❤❡ ❤②♣♦t❤❡s❡s ♦♥ X✱ ❛♥❞ t❤❡ ♦t❤❡r t❡r♠ ✐s ❜♦✉♥❞❡❞ ✐♥ n ❜② ❛♥ ✐♥t❡❣r❛❜•❡ ❢✉♥❝t✐♦♥ ♦❢ u ❜❡❝❛✉s❡ ♦❢ ❤♦✇ t❤❡ T (n) ✇❡r❡ ❝❤♦s❡♥✳ ❆•s♦✱ ❛t ✜①❡❞ u✱ t❤❡ t❡r♠ ✉♥❞❡r t❤❡ ✐♥t❡❣r❛• ❝♦♥✈❡r❣❡s t♦ 0 ✐♥ D p,-(s ′ +1) ❜❡❝❛✉s❡ ♦❢ t❤❡♦r❡♠ ✸✳✷✳✵✳✶✺✳ ❚❤❡r❡❢♦r❡ ✇❡ ♠❛② ❝♦♥❝•✉❞❡ t♦ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✜rst t❡r♠ ❜② t❤❡ ❞♦♠✐♥❛t❡❞ ❝♦♥✈❡r❣❡♥❝❡ t❤❡♦r❡♠ ❢♦r ❇♦❝❤♥❡r ✐♥t❡❣r❛•s✳ ▲❡t ✉s ♥♦t❡ t❤❛t C(X u ) ♦♥•② ✐♥✈♦•✈❡s t❤❡ D ∞ -,s ′ +2 r❡❣✉•❛r✐t② ♦❢ t❤❡ X u ❛♥❞ ✇❡ ❞✐❞ ♥♦t ✉s❡ ♠♦r❡ r❡❣✉•❛r✐t② ✐♥ ♦✉r ♣r♦♦❢✳ ❙✐♠✐•❛•r② ♦♥❡ ♠❛② ♣r♦✈❡✿ ▲❡♠♠❛ ✹✳✶✳✶✳✸ ✭❈♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ s❡❝♦♥❞ t❡r♠✮✳ t ǫ a ij (u) • ∂ ij T (n) (u, X u )du D p ′ ,-(s ′ +2) -→ t ǫ a ij (u) • ∂ ij T u • X u du ✭✹✳✶✺✮ Pr♦♦❢✳ ❚❤❡ ♣r♦♦❢ ✐s t❤❡ ❡①❛❝t s❛♠❡✱ ❤♦✇❡✈❡r ✇❡ ♥♦t❡ t❤❛t ❢♦r t❤✐s t❡r♠ t❤❡ D ∞ -,s ′ +3 r❡❣✉•❛r✐t② ♦❢ t❤❡ X u ✐s r❡q✉✐r❡❞ ✐♥❞❡❡❞✳ ❛♥❞✿ ✹✳✶✳ ❚❍❊ ❋❖|▼❯▲❆ ❋❖| ❆ •❊◆❊|❆▲ ■❚Ô P|❖❈❊❙❙ ✽✸ ▲❡♠♠❛ ✹✳✶✳✶✳✹ ✭❈♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❢♦✉rt❤ t❡r♠✮✳ j t ǫ i b i (u) • ∂ i T (n) (u, X u ) dW j u D p ′ ,-(s ′ +1) → M (ǫ) t ✭✹✳✶✻✮ ✇❤❡r❡ M (ǫ) t ǫ≤t≤1 ✐s ❛ ✇❡❛❦ ♠❛rt✐♥❣❛•❡ ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡✱ ❛♥❞ ✐♥ t❤❡ s❡♥s❡ ♦❢ ❞✐✈❡r❣❡♥❝❡s ✇❡ ❤❛✈❡✿ M (ǫ) t = j t ǫ i [b i (u) • (∂ i T u ) • X u ] dW j u ✭✹✳✶✼✮ Pr♦♦❢✳ ❚❤❡ ❣❡♥❡r❛• ✐❞❡❛ ✐s t❤❡ s❛♠❡✱ ❡①❝❡♣t t❤❛t ✇❡ ♥❡❡❞ t♦ ✉s❡ t❤❡ ❇✉r❦❤♦•❞❡r ❉❛✈✐s •✉♥❞② ✐♥❡q✉❛•✐t✐❡s✿ t ǫ b i (u) • ∂ i T (n) (u, X u )dW j u - t ǫ b i (u) • ∂ i T u • X u dW j u Dp,s = t ǫ (2 • Id + L) s/2 b i (u) • ∂ i T (n) -∂ i T u • X u (u, X u ) dW j u L p ≤ C(p) • t ǫ (2 • Id + L) s/2 b i (u) • ∂ i T (n) -∂ i T u • X u (u, X u ) 2 du 1/2 L p ≤ C(p, s) t ǫ b i (u) • ∂ i T (n) -∂ i T u • X u (u, X u ) p Dp,s du 1/p ✭✹✳✶✽✮ ❚❤❡ p ≥ 2 r❡str✐❝t✐♦♥ ✇❛s ♥❡❡❞❡❞ t♦ ♦❜t❛✐♥ t❤✐s •❛st ✐♥❡q✉❛•✐t②✳ ❲❡ ♠❛② t❤❡♥ ❝♦♠♣•❡t❡ t❤❡ ♣r♦♦❢ ✇✐t❤ t❤❡ ❡①❛❝t s❛♠❡ t❡❝❤♥✐q✉❡s t❤❛♥ ❢♦r t❤❡ ✜rst ❛♥❞ s❡❝♦♥❞ t❡r♠s✳ ❖♥❝❡ ❛❣❛✐♥✱ ✇❡ ♥♦t❡ t❤❛t ♦♥•② t❤❡ D ∞ -,s ′ +2 r❡❣✉•❛r✐t② ♦❢ t❤❡ X u ✇❛s r❡q✉✐r❡❞✳ ❋✐♥❛••② ✇❡ ❤❛✈❡ t♦ ❣✐✈❡ ❛ r❡s✉•t ❢♦r t❤❡ t❤✐r❞ t❡r♠✳ ❚❤✐s ♦♥❡ ✐s t❤❡ tr✐❝❦✐❡st ❜❡❝❛✉s❡ t❤❡ r✐❣❤t ❝❛♥❞✐❞❛t❡ ❢♦r t❤❡ •✐♠✐t ♦❢ t❤❡ ❙t✐❡•t•❡s ✐♥t❡❣r❛•s ✐s •❡ss ♦❜✈✐♦✉s✳ ❲❡ st❛rt ✇✐t❤ t❤❡ ❢♦••♦✇✐♥❣ t❡❝❤♥✐❝❛• r❡s✉•t✿ ▲❡♠♠❛ ✹✳✶✳✶✳✺✳ ❋♦r ❛♥② p > 1✱ s > 0✱ ❢♦r ❛♥② n ∈ N ❛♥❞ ❢♦r ❛♥② f ∈ S✿ ρ n * f S p,-s ≤ f S p,-s ✭✹✳✶✾✮ Pr♦♦❢✳ ❋✐rst✿ ρ n * f (x) = 1 (2πn) N/2 • exp - |y| 2 2n f (x -y)dy ✭✹✳✷✵✮ ✽✹ ❈❍❆P❚❊| ✹✳ ❆ ❲❊❆❑ ■❚Ô ❋❖|▼❯▲❆ ❛♥❞ t❤❡r❡❢♦r❡ K -s 2 (ρ n * f ) (x) = 1 (2πn) N/2 • exp - |y| 2 2n K -s 2 f (x -y)dy = ρ n * K -s 2 f (x) ✭✹✳✷✶✮ s♦ ✜♥❛••②✿ ρ n * f p S p,-s = ρ n * K -s 2 f p L p ≤ K -s 2 f p L p = f p S p,-s ✭✹✳✷✷✮ ✇❤❡r❡ t❤❡ ✐♥❡q✉❛•✐t② ✐s ❛ ❝•❛ss✐❝ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ❏❡♥s❡♥✬s ✐♥❡q✉❛•✐t② ♦♥ t❤❡ ♣r♦❜❛✲ ❜✐•✐t② s♣❛❝❡ L p (ρ n )✳ ❲❡ ❛r❡ ♥♦✇ r❡❛❞② t♦ st❛t❡✿ ▲❡♠♠❛ ✹✳✶✳✶✳✻ ✭❈♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ t❤✐r❞ t❡r♠✮✳ ❚❤❡ ❢♦••♦✇✐♥❣ •✐♠✐t ❡①✐sts ✐♥ D p ′ ,-s ′ ✿ t ǫ T (du, •X u ) := lim n→∞ t ǫ T (n) (du, X u ) ✭✹✳✷✸✮ ❛♥❞ ✐t ✐s ❞❡✜♥❡❞ ❜②✿ t ǫ T (du, •X u ), φ = t ǫ T (du, p Xu,φ ) ✭✹✳✷✹✮ Pr♦♦❢✳ ▲❡t ✉s ❞❡♥♦t❡ (u i ) ❢♦r t❤❡ ❡•❡♠❡♥ts ♦❢ ❛ ♣❛rt✐t✐♦♥ π ♦❢ [ǫ, t] ❛rr❛♥❣❡❞ ✐♥ ✐♥❝r❡❛s✐♥❣ ♦r❞❡r✳ ❚❤❡♥ ❛ ❙t✐❡•t•❡s ✐♥t❡❣r❛• ♦♥ [ǫ, t] ✐s ❛♣♣r♦❛❝❤❡❞ ❜② |✐❡♠❛♥♥ s✉♠s✿ t ǫ T (n) (du, X u ) = lim |π|→0 i T (n) (u i+1 , X u i ) -T (n) (u i , X u i ) ✭✹✳✷✺✮ ✇❤❡r❡ t❤❡ •✐♠✐t ✐s t❛❦❡♥ ✐♥ ♣r♦❜❛❜✐•✐t✐❡s✳ ◆♦✇✱ ❛♣♣•②✐♥❣ t❤❡♦r❡♠ ✸✳✷✳✵✳✶✺ ❛s ✐♥ t❤❡ ♣r❡✈✐♦✉s ♣r♦♦❢s✱ ♦♥❡ ♦❜t❛✐♥s ❛ ❝♦♥tr♦• ♦❢ t❤❡ t②♣❡✿ T (n) (u i+1 , X u i ) -T (n) (u i , X u i ) -T u i+1 • X u i -T u i • X u i D p ′ ,-s ′ ≤ B(u i ) • T (n) u i+1 -T (n) u i -T u i+1 -T u i S p ′ ,-s ′ ✭✹✳✷✻✮ ✇❤❡r❡ B ✐s ❛ ❢✉♥❝t✐♦♥ ✇❤✐❝❤ ✐s ❜♦✉♥❞❡❞ ♦♥ [ǫ, t] ✐♥❞❡♣❡♥❞❡♥t•② ♦❢ n✳ ❆•s♦ ❜② t❤❡ tr✐❛♥❣•❡ ✐♥❡q✉❛•✐t② ❛♥❞ •❡♠♠❛ ✹✳✶✳✶✳✺✿ T (n) u i+1 -T (n) u i -T u i+1 -T u i S p ′ ,-s ′ ≤ 2 • T u i+1 -T u i S p ′ ,-s ′ ✭✹✳✷✼✮ ✹✳✶✳ ❚❍❊ ❋❖|▼❯▲❆ ❋❖| ❆ •❊◆❊|❆▲ ■❚Ô P|❖❈❊❙❙ ✽✺ ✇❤✐❝❤ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ n ❡✐t❤❡r✳ ◆♦✇ s✐♥❝❡✿ i T u i+1 -T u i S p ′ ,-s ′ ≤ V ar(T ) < ∞ ✭✹✳✷✽✮ ❛♥❞ t❤✐s ❝♦♥tr♦• ✐s ✐♥❞❡♣❡♥❞❛♥t ♦❢ π ❜② t❤❡ ❞♦♠✐♥❛t❡❞ ❝♦♥✈❡r❣❡♥❝❡ t❤❡♦r❡♠ ♦♥❡ ♦❜t❛✐♥s t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❢♦••♦✇✐♥❣ |✐❡♠❛♥♥ s✉♠ ✐♥ D p ′ ,-s ′ ✿ i T u i+1 • X u i -T u i • X u i ✭✹✳✷✾✮ ❛♥❞ ✇❡ ❞❡♥♦t❡ ✐ts •✐♠✐t ❜②✿ t ǫ T (du, •X u ) := lim |π|→0 i T u i+1 • X u i -T u i • X u i ✭✹✳✸✵✮ ◆♦✇ •❡t φ ∈ D✳ ❇② t❤❡♦r❡♠ ✸✳✹✳✵✳✶✽ ❢♦r ❡✈❡r② u✱ p Xu,φ ❡①✐sts ❛♥❞ ❤❛s ❛t •❡❛st S (p ′ ) * ,s ′ r❡❣✉•❛r✐t②✳ ❚❤❡r❡❢♦r❡ ❜② ❞✐s✐♥t❡❣r❛t✐♥❣ ❡❛❝❤ t❡r♠ ♦❢ t❤❡ |✐❡♠❛♥♥ s✉♠ s❡♣❛r❛t❡•② ♦♥❡ ♦❜t❛✐♥s✿ E φ • T u i+1 • X u i -T u i • X u i = S ′ T u i +1 -T u i , p Xu,φ S ✭✹✳✸✶✮ ❙✐♥❝❡ V ar(T ) < ∞ ✐♥ S p ′ ,-s ′ ❛♥❞ u → p Xu,φ ✐s ❜♦✉♥❞❡❞ ✐♥ S (p ′ ) * ,s ′ ❜② s✐♠✐•❛r t❡❝❤♥✐q✉❡s ♦♥❡ s❤♦✇s t❤❛t t❤✐s ❝♦♥✈❡r❣❡s t♦✿ t ǫ T (du, p Xu,φ ) := t ǫ R N K s ′ 2 T (du, x) • K -s ′ 2 p Xu,φ (x)dx ✭✹✳✸✷✮ s♦ ✇❡ ❛r❡ ❞♦♥❡✳ ❖♥❝❡ ❛❣❛✐♥✱ ✇❡ ❞✐❞ ♥♦t ✉s❡ t❤❡ D ∞ -,s ′ +3 r❡❣✉•❛r✐t② ♦❢ t❤❡ X u ✿ ♦♥•② D ∞ -,s ′ +1 ✇❛s r❡q✉✐r❡❞ ❢♦r t❤✐s ♣❛rt ♦❢ t❤❡ ♣r♦♦❢✳ |❡♠❛r❦ ✹✳✶✳✶✳✷✳ ▲❡t ✉s r❡♠❡♠❜❡r t❤❛t ✇❡ r❡str✐❝t❡❞ ♦✉rs❡•✈❡s t♦ BV (S p ′ ,-s ′ ) ❜❡❝❛✉s❡ ♦❢ t❤❡ •r♦t❤❡♥❞✐❡❝❦ t❤❡♦r❡♠✳ ❚❤❡r❡❢♦r❡ ✐t ✐s ✇♦rt❤ ♠❡♥t✐♦♥♥✐♥❣ t❤❛t t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♦✉r ❙t✐❡•t•❡s ✐♥t❡❣r❛• ♠❛② ❛•s♦ ❜❡ ♦❜t❛✐♥❡❞ ❞✐r❡❝t•② ❢r♦♠ t❤❡ •r♦t❤❡♥❞✐❡❝❦ r❡♣r❡s❡♥t❛t✐♦♥✳ ■♥❞❡❡❞ s✐♥❝❡ (F j ) ✐s ❛ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ ✐♥ S p ′ ,-s ′ ❜② ❛ ♠❡t❤♦❞ s✐♠✐•❛r t♦ t❤❡ ♦♥❡ ✉s❡❞ ❢♦r t❤❡ ✜rst t❡r♠ ♦♥❡ ❤❛s ❛ ❝♦♥tr♦• ♦❢ t❤❡ t②♣❡✿ F (n) j (X u ) -F j • X u D p ′′ ,-s ≤ C(X u ) • F (n) j -F j S p ′ ,-s ′ ✭✹✳✸✸✮ ✇❤❡r❡ F (n) j = ρ n * F j ❛♥❞ u → C(X u ) ✐s ❛♥ ✐♥t❡❣r❛❜•❡ ❢✉♥❝t✐♦♥ ♦❢ u ∈ [0, 1] ✇❤✐❝❤ ✐s ✐♥❞❡♣❡♥❞❛♥t ♦❢ n✳ ❚❤❡r❡❢♦r❡ ♦♥❡ ♦❜t❛✐♥s t❤❛t ❢♦r ❡✈❡r② j✿ 1 ǫ F (n) j (X u ) -F j • X u D p ′′ ,-s d|V j |(u) → 0 ✭✹✳✸✹✮ ✽✻ ❈❍❆P❚❊| ✹✳ ❆ ❲❊❆❑ ■❚Ô ❋❖|▼❯▲❆ ❜② t❤❡ ❞♦♠✐♥❛t❡❞ ❝♦♥✈❡r❣❡♥❝❡ t❤❡♦r❡♠ ❢♦r ❇♦❝❤♥❡r ✐♥t❡❣r❛•s✳ ❙✐♥❝❡ t❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡s❡ ✐♥t❡❣r❛•s ❛•s♦ ✐s ❜♦✉♥❞❡❞ ✐♥ j ♦♥❡ ❝♦♥❝•✉❞❡s t♦ t❤❡ ❞♦♠✐♥❛t❡❞ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ t❤✐r❞ t❡r♠ ❜② t❤❡ ❞♦♠✐♥❛t❡❞ ❝♦♥✈❡r❣❡♥❝❡ t❤❡♦r❡♠ ❢♦r s❡r✐❡s✳ ❆•s♦✱ ♦♥❡ t❤❡♥ ❤❛s ❛ ♠♦r❡ ❡①♣•✐❝✐t ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ❙t✐❡•t•❡s ✐♥t❡❣r❛•✿ t ǫ T (du, •X u ) = ∞ n=0 λ n • t ǫ F n • X u dV n (u) ✭✹✳✸✺✮ ❲❡ ❛r❡ ♥♦✇ r❡❛❞② t♦ st❛t❡ t❤❡ ♠❛✐♥ r❡s✉•t ✐♥ t❤✐s ♣❛r❛❣r❛♣❤✿ ❚❤❡♦r❡♠ ✹✳✶✳✶✳✶ ✭❲❡❛❦ ■tô ❢♦r♠✉•❛✮✳ ❯♥❞❡r t❤❡ ❝♦♥❞✐t✐♦♥s ✭✹✳✸✮✱ ✭✹✳✺✮✱ ✭✹✳✻✮ ❛♥❞ ✭✹✳✻✮✱ ❢♦r ❛♥② ǫ > 0✱ ❢♦r ❛•• t ∈ [ǫ, 1]✱ t❤❡ ❢♦••♦✇✐♥❣ ❤♦•❞s ✐♥ t❤❡ s♣❛❝❡ D p ′ ∨2,-(s ′ +2) ✿ T (t) • X t -T (ǫ) • X ǫ = M (ǫ) t + i t ǫ b i (u) • (∂ i T u ) • X u du + 1 2 i,j t ǫ a ij (u) • (∂ ij T u ) • X u du + t ǫ T (du) • X u ✭✹✳✸✻✮ ✇❤❡r❡ M (ǫ) t ǫ≤t≤1 ✐s ❛ ✇❡❛❦ ♠❛rt✐♥❣❛•❡ ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡✱ ❛♥❞ ✐♥ t❤❡ s❡♥s❡ ♦❢ ❞✐✈❡r❣❡♥❝❡s ✇❡ ❤❛✈❡✿ M (ǫ) t = j t ǫ i [b i (u) • (∂ i T u ) • X u ] dW j u ✭✹✳✸✼✮ Pr♦♦❢✳ ❲❡ ♦♥•② ♥❡❡❞ t♦ s❤♦✇ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❡✈❡r② t❡r♠✱ ✇❤✐❝❤ ✐s ❞♦♥❡ ❜② ❛♣♣•②✐♥❣ t❤❡ ♣r❡✈✐♦✉s •❡♠♠❛❡✳ ▲❡t ✉s ❛•s♦ ♥♦t❡ t❤❛t t❤❡ •❡♠♠❛❡ ♣r♦✈✐❞❡ ♣r❡❝✐s❡ ❡st✐♠❛t❡s ♦❢ t❤❡ D p ′ ∨2,-(s ′ +2) ♥♦r♠ ♦❢ ❡✈❡r② t❡r♠ ✲ ✇❡ ❤❛✈❡ p ′ ∨2 r❛t❤❡r t❤❛♥ s✐♠♣•② p ′ ❜❡❝❛✉s❡ ♦❢ t❤❡ ♠❛rt✐♥❣❛•❡ t❡r♠❀ s❡❡ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ •❡♠♠❛✳ ◆♦✇ ✇❡ ♠❛❦❡ t❤❡ ❢♦••♦✇✐♥❣ ♦❜s❡r✈❛t✐♦♥✿ ✐❢ ✇❡ ♣r♦✈❡ t❤❛t ❛•• ❜✉t ♦♥❡ t❡r♠ ✐♥ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ■tô ❢♦r♠✉•❛ ❛❜♦✈❡ ❝♦♥✈❡r❣❡✱ t❤❡♥ ❜② ❡•❡♠❡♥t❛r② ❛•❣❡❜r❛ t❤❡ •❛st ♦♥❡ ❤❛s t♦ ❝♦♥✈❡r❣❡ t♦♦ ❛♥❞ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ t❛❦❡s ♣•❛❝❡ ✐♥ t❤❡ s❛♠❡ s♣❛❝❡✳ ❚❤❡r❡❢♦r❡ ♦♥❡ ♠❛② ♦❜t❛✐♥ t❤❡ t❤r❡❡ ❢♦••♦✇✐♥❣ r❡s✉•ts ✇❤✐❝❤ ❛r❡ s•✐❣❤t•② ♠♦r❡ ♣r❡❝✐s❡✳ ❚❤❡ ✜rst ♦♥❡ ✐s ✐♥t❡r❡st✐♥❣ ✇❤❡♥ T t ✐s ♠♦r❡ r❡❣✉•❛r ✐♥ t❤❡ ❝♦♥t✐♥✉♦✉s s❡♥s❡ t❤❛♥ ✐♥ t❤❡ ❇❱ s❡♥s❡✳ ❈♦r♦••❛r② ✹✳✶✳✶✳✶✳ |❡♣•❛❝❡ ❤②♣♦t❤❡s✐s ✭✹✳✸✮ ✇✐t❤✿ T t ∈ C 0 (S p,-σ ) ✭✹✳✸✽✮ ✹✳✶✳ ❚❍❊ ❋❖|▼❯▲❆ ❋❖| ❆ •❊◆❊|❆▲ ■❚Ô P|❖❈❊❙❙ ✽✼ ✇❤❡r❡ ♣♦ss✐❜•② σ > s ❛♥❞ r❡♣•❛❝❡ s ❜② σ ✐♥ ❤②♣♦t❤❡s❡s ✭✹✳✺✮✱ ✭✹✳✻✮ ❛♥❞ ✭✹✳✻✮✳ ❚❤❡♥✱ t❤❡ ❢♦r♠✉•❛ ✐♥ t❤❡♦r❡♠ ✹✳✶✳✶✳✶ st✐•• ❤♦•❞s ✐♥ t❤❡ s♣❛❝❡ D p ′ ∨2,-(σ ′ +2) ✳ Pr♦♦❢✳ ❚❤❡ ♣r♦♦❢ ✐s t❤❡ s❛♠❡ ❛s ❢♦r t❤❡♦r❡♠ ✹✳✶✳✶✳✶ ❡①❝❡♣t t❤❛t t❤❡ ♣❛rt ❛❜♦✉t t❤❡ ❙t✐❡•t•❡s ✐♥t❡❣r❛• ✐s ♥♦t r❡q✉✐r❡❞✳ ❚❤❡ s❡❝♦♥❞ r❡s✉•t ❛••♦✇s ♦♥❡ t♦ ✧s❛✈❡✧ ♦♥❡ ♦r❞❡r ♦❢ r❡❣✉•❛r✐t②✿ ❈♦r♦••❛r② ✹✳✶✳✶✳✷✳ ❙✉♣♣♦s❡ t❤❛t ✭✹✳✸✮ ❤♦•❞s ❛♥❞ r❡♣•❛❝❡ s+3 ❜② s+2 ✐♥ ❤②♣♦t❤❡s❡s ✭✹✳✺✮✱ ✭✹✳✻✮ ❛♥❞ ✭✹✳✻✮✳ ❚❤❡♥ t❤❡ ❝♦♥❝•✉s✐♦♥s ♦❢ t❤❡♦r❡♠ ✹✳✶✳✶✳✶ st✐•• ❤♦•❞ ✐♥ t❤❡ s♣❛❝❡ D p ′ ∨2,-(s ′ +1) ✳ Pr♦♦❢✳ ❚❤❡ ♣r♦♦❢ ✐s t❤❡ s❛♠❡ ❛s ❢♦r t❤❡♦r❡♠ ✹✳✶✳✶✳✶ ❡①❝❡♣t t❤❛t t❤❡ ♣❛rt ❛❜♦✉t t❤❡ ❝r♦❝❤❡t ✐♥t❡❣r❛• ✭✇❤✐❝❤ ❝♦♥t❛✐♥ t❤❡ ♦r❞❡r 2 ❞❡r✐✈❛t✐✈❡s✮ ✐s ♥♦t r❡q✉✐r❡❞✳ ❋✐♥❛••② t❤❡ t❤✐r❞ ❝♦r♦••❛r② ❛••♦✇ ✉s t♦ ♦❜t❛✐♥ ❝♦♥tr♦•s ✐♥ t❤❡ s♣❛❝❡ D p ′ ,-(s ′ +2) ❢♦r p ′ < 2✿ ❈♦r♦••❛r② ✹✳✶✳✶✳✸✳ ❙✉♣♣♦s❡ ✭✹✳✸✮✱ ✭✹✳✺✮✱ ✭✹✳✻✮ ❛♥❞ ✭✹✳✻✮ ❛s ✐♥ t❤❡♦r❡♠ ✹✳✶✳✶✳✶✳ ❚❤❡♥ t❤❡ ❝♦♥❝•✉s✐♦♥s ♦❢ t❤❡♦r❡♠ ✹✳✶✳✶✳✶ ❤♦•❞ ✐♥ t❤❡ s♣❛❝❡ D p ′ ,-(s ′ +2) ❢♦r p ′ < 2✳ Pr♦♦❢✳ ❚❤✐s t✐♠❡ ✐t ✐s t❤❡ ♣❛rt ❛❜♦✉t t❤❡ ♠❛rt✐♥❣❛•❡ t❡r♠ ✇❤✐❝❤ ✐s ♥♦t r❡q✉✐r❡❞✳ ❋✐♥❛••② t❤❡r❡ ✐s ❛ str❛✐❣❤t❢♦r✇❛r❞ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ■tô ❢♦r♠✉•❛ ✐♥ t❤❡♦r❡♠ ✹✳✶✳✶✳✶✱ ✇❤✐❝❤ ❛••♦✇s ♦♥❡ t♦ ♠✉•t✐♣•② T t • X t ❜② ❛ r❡❣✉•❛r ❡♥♦✉❣❤ ♣r♦❝❡ss Y t ✳ ▼♦r❡ ♣r❡❝✐s❡•② ❝♦♥s✐❞❡r ❛ ♦♥❡✲❞✐♠❡♥s✐♦♥♥❛• ■tô ♣r♦❝❡ss✿ dY t = b ′ t dt + σ ′ t dW t ✭✹✳✸✾✮ Y 0 = y ✭✹✳✹✵✮ ❛♥❞ ♠❛❦❡ t❤❡ ❢♦••♦✇✐♥❣ ❤②♣♦t❤❡s❡s ♦♥ Y ✿ ❢♦r ❛•• r > 1✱ 1 0 b ′ u r Dr,s du < ∞ ✭✹✳✹✶✮ 1 0 σ ′ u r Dr,s du < ∞ ✭✹✳✹✷✮ ✭✹✳✹✸✮ ❲❡ ❞♦ ♥♦t ♥❡❡❞ t♦ s✉♣♣♦s❡ t❤❛t Y ✐s ♥♦♥✲❞❡❣❡♥❡r❛t❡✳ ❚❤❡♥✿ ✽✽ ❈❍❆P❚❊| ✹✳ ❆ ❲❊❆❑ ■❚Ô ❋❖|▼❯▲❆ ❚❤❡♦r❡♠ ✹✳✶✳✶✳✷✳ ❙✉♣♣♦s❡ t❤❛t X ❛♥❞ T ❛r❡ ❛s ✐♥ t❤❡♦r❡♠ ✹✳✶✳✶✳✶ ❛♥❞ t❤❛t Y ✐s ❛s ❛❜♦✈❡✳ ❚❤❡♥ t❤❡ ❢♦••♦✇✐♥❣ st❛♥❞s ✐♥ D p,s ✿ Y t • T (t) • X t -Y ǫ • T (ǫ) • X ǫ = M (ǫ) t + i t ǫ b i (u)Y u • (∂ i T u ) • X u du + 1 2 i,j t ǫ Y u a 2 ij (u) • (∂ ij T u ) • X u du + t ǫ b ′ (u)Y u • T u • X u du + t ǫ Y u T (du) • X u ✭✹✳✹✹✮ ✇❤❡r❡ M (ǫ) t = j t ǫ Y u i [b i (u) • (∂ i T u ) • X u ] dW j u + j t ǫ σ ′ j (u)T u • X u dW j u ✭✹✳✹✺✮ ❚❤✐s r❡s✉•t ✐s ♣r♦✈❡❞ ❡①❛❝t•② ❛s t❤❡♦r❡♠ ✹✳✶✳✶✳✶✳ ❖❢ ❝♦✉rs❡ ❝♦r♦••❛r✐❡s ✹✳✶✳✶✳✶✱ ✹✳✶✳✶✳✷ ❛♥❞ ✹✳✶✳✶✳✸ ❛•s♦ ❛❞♠✐t s✐♠✐•❛r ❡①t❡♥s✐♦♥s✳ ✹✳✶✳✷ Pr♦❝❡ss❡s ✇✐t❤ ❛ ❣❡♥❡r❛• ❞r✐❢t ■♥ t❤✐s ♣❛r❛❣r❛♣❤ ✇❡ ❝♦♥s✐❞❡r t❤❡ ♣r♦❝❡ss X s✉❝❤ t❤❛t✿ dX t = dA t + σ t dW t ✭✹✳✹✻✮ X 0 = x ✭✹✳✹✼✮ ✇❤❡r❡ σ ✐s ❛s ✐♥ t❤❡ ♣r❡✈✐♦✉s ♣❛r❛❣r❛♣❤ ❛♥❞ A ✐s ❛ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥ ♣r♦❝❡ss✳ ❲❡ st✐•• ♠❛❦❡ t❤❡ ❤②♣♦t❤❡s❡s ✭✹✳✻✮✱ ✭✹✳✻✮ ❛♥❞ ✭✹✳✸✮ ❛♥❞ ✐♥st❡❛❞ ♦❢ ✭✹✳✺✮ ✇❡ s✉♣♣♦s❡ t❤❛t ❢♦r ❡✈❡r② r > 1✿ t 0 A u r D r,s ′ +3 du < ∞ ✭✹✳✹✽✮ ✇❤✐❝❤ ✐s ❡♥♦✉❣❤ t♦ ❡♥s✉r❡ t❤❛t t❤❡ q✉❛♥t✐t✐❡s ✇❡ ❝♦♥s✐❞❡r ❛r❡ ✇❡•• ❞❡✜♥❡❞✳ ❲❡ ♥♦✇ st❛t❡ t❤❡ ♠❛✐♥ r❡s✉•t ✐♥ t❤✐s ♣❛r❛❣r❛♣❤✿ ❚❤❡♦r❡♠ ✹✳✶✳✷✳✶✳ ❯♥❞❡r t❤❡ ❤②♣♦t❤❡s❡s ✭✹✳✺✮✱ ✭✹✳✻✮✱ ✭✹✳✻✮ ❛♥❞ ✭✹✳✸✮✱ ❢♦r ❛♥② ✹✳✶✳ ❚❍❊ ❋❖|▼❯▲❆ ❋❖| ❆ •❊◆❊|❆▲ ■❚Ô P|❖❈❊❙❙ ✽✾ ǫ > 0✱ ❢♦r ❛•• t ∈ [ǫ, 1]✱ t❤❡ ❢♦••♦✇✐♥❣ ❤♦•❞s ✐♥ t❤❡ s♣❛❝❡ D p ′ ∨2,-(s ′ +2) ✿ T (t) • X t -T (ǫ) • X ǫ = M (ǫ) t + t ǫ (∂ i T u ) • dA(u) + 1 2 i,j t ǫ a 2 ij (u) • (∂ ij T u ) • X u du + t ǫ T (du) • X u ✭✹✳✹✾✮ ✇❤❡r❡ M (ǫ) t ǫ≤t≤1 ✐s ❛s ✐♥ t❤❡♦r❡♠ ✹✳✶✳✶✳✶ ❛♥❞ t❤❡ ✐♥t❡❣r❛• ❛❣❛✐♥st A ✐s ❞❡✜♥❡❞ ❛s✿ t ǫ (∂ i T u ) • X u dA(u) = lim n→∞ t ǫ (∂ i (ρ n * T u )) (X u )dA(u) ✭✹✳✺✵✮ Pr♦♦❢✳ ❆s ✐♥ t❤❡ ♣r♦♦❢ ♦❢ t❤❡♦r❡♠ ✹✳✶✳✶✳✶ ✇❡ ✐♥tr♦❞✉❝❡ T (n) = ρ n * T ✐♥ ♦r❞❡r ❛♣♣•② t❤❡ ✉s✉❛• ■tô ❢♦r♠✉•❛ t♦ t❤❛t ❢✉♥❝t✐♦♥ ❛♥❞ ✇❡ st✉❞② t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❡❛❝❤ t❡r♠ ✐♥ t❤❡ ❢♦r♠✉•❛✳ ❊①❝❡♣t ❢♦r t❤❡ ✐♥t❡❣r❛• ❛❣❛✐♥st A✱ ❡✈❡r② t❡r♠ ✐s ❛♥❛•♦❣✉❡ t♦ t❤❛t ✐♥ t❤❡ ♣r♦♦❢ ♦❢ t❤❡♦r❡♠ ✹✳✶✳✶✳✶ ❛♥❞ s✐♥❝❡ ✇❡ ❤❛✈❡ t❤❡ s❛♠❡ r❡❣✉•❛r✐t② ❢♦r X ✇❡ ♦❜t❛✐♥ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡s❡ t❡r♠s s✐♠✐•❛r•②✳ ❋✐♥❛••② t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✐♥t❡❣r❛• ❛❣❛✐♥st A ✐s t❤❡♥ ❛✉t♦♠❛t✐❝✳ ❚❤❡ ♣r♦❜•❡♠ ✇✐t❤ t❤✐s ✐❞❡❛ ✐s t❤❛t ✐t ❣✐✈❡s ♥♦ ❡①♣•✐❝✐t ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ❇❱ ✐♥t❡❣r❛• ❛❣❛✐♥st A✳ ❚❤✐s ♠❛② ❜❡ ♣❛rt•② ❛❞❞r❡ss❡❞ t♦ ✐❢ ✇❡ ♠❛❦❡ ♦♥❡ ❡①tr❛ ❛ss✉♠♣t✐♦♥✱ ♥❛♠❡•② t❤❛t ❢♦r s♦♠❡ q > (p ′ ) * ✿ sup k A u k+1 -A u k S q,s ′ < ∞ ✭✹✳✺✶✮ ✇❤❡r❡ p ✐s ❣✐✈❡♥ ✐♥ ✭✹✳✸✮ ❛♥❞ t❤❡ s✉♣ ✐s t❛❦❡♥ ♦✈❡r ❛•• t❤❡ ♣❛rt✐t✐♦♥s ♦❢ [ǫ, 1]✳ ■♥❞❡❡❞ ✇❡ t❤❡♥ ❤❛✈❡ t❤❡✿ ▲❡♠♠❛ ✹✳✶✳✷✳✶✳ ❯♥❞❡r ❤②♣♦t❤❡s❡s ✭✹✳✸✮✱ ✭✹✳✹✽✮ ❛♥❞ ✭✹✳✺✶✮✱ t ǫ (∂ i T u ) • X u dA(u) = lim n→∞ t ǫ (∂ i (ρ n * T u )) (X u )dA(u) ✭✹✳✺✷✮ ✐s ❣✐✈❡♥ ❛s t❤❡ •✐♠✐t ♦❢ t❤❡ ❢♦••♦✇✐♥❣ |✐❡♠❛♥♥ s✉♠✿ k ((∂ i T u k ) • X u k ) • A u k+1 -A u k ✭✹✳✺✸✮ Pr♦♦❢✳ ❙❛♠❡ t❡❝❤♥✐q✉❡ ❛s ❢♦r t❤❡ ♣r♦♦❢ ♦❢ •❡♠♠❛ ✹✳✶✳✶✳✻✳ ❖❢ ❝♦✉rs❡✱ ❛s ✐♥ t❤❡ ♣r❡✈✐♦✉s ♣❛❛r❛❣r❛♣❤✱ ♦♥❡ ♠❛② ♥♦t✐❝❡ t❤❛t ✐t ✐s s✉✣❝✐❡♥t t♦ ♣r♦✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❛•• ❜✉t ♦♥❡ t❡r♠✱ ♦♥❡ ♠❛② ♦❜t❛✐♥ ♠♦r❡ ♣r❡❝✐s❡ r❡s✉•ts✳ ✾✵ ❈❍❆P❚❊| ✹✳ ❆ ❲❊❆❑ ■❚Ô ❋❖|▼❯▲❆ ✹✳✶✳✸ ❚❤❡ ❝❛s❡ ✇❤❡r❡ T t ✐s r❛♥❞♦♠ ▲❡t X ❜❡ ❛s ✐♥ s❡❝t✐♦♥ ✹✳✶✳✶ ❛♥❞ s✉♣♣♦s❡ t❤❛t T t ✐s ❛ S ′ ✲✈❛•✉❡❞ r❛♥❞♦♠ ♣r♦❝❡ss ✇✐t❤ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥s✱ ❛❞❛♣t❡❞ t♦ t❤❡ ✜•tr❛t✐♦♥ ♦❢ t❤❡ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ W ✱ s✉❝❤ t❤❛t ❢♦r ❡✈❡r② r > 1 ❛♥❞ ❡✈❡r② ✐♥t❡❣❡r k ≤ [s ′ ]✿ t 0 T s r D r,k (S ∞,s ′ -k ) ds < ∞ ✭✹✳✺✹✮ ❚❤❡♥ t❤❡ s❛♠❡ ✇❡❛❦ ■tô ❢♦r♠✉•❛ ❛s ✐♥ t❤❡♦r❡♠ ✹✳✶✳✶✳✶ ❤♦•❞s✳ ❚❤✐s ✐s ♣r♦✈❡❞ ❜② ❢♦••♦✇✐♥❣ t❤❡ ♣r♦♦❢ ♦❢ t❤❡♦r❡♠ ✹✳✶✳✶✳✶ ❜✉t ✉s✐♥❣ t❤❡♦r❡♠ ✸✳✺✳✵✳✷✵ ✐♥st❡❛❞ ♦❢ t❤❡♦r❡♠ ✸✳✷✳✵✳✶✺✳ ■♥❞❡❡❞ ❤②♣♦t❤❡s✐s ✹✳✺✹ ✇❛s ❞❡s✐❣♥❡❞ ❡①❛❝t•② s♦ t❤❛t •❡♠♠❛ ✸✳✺✳✵✳✷✵ ♠❛② ❜❡ ❛♣♣•✐❡❞✳ ✹✳✷ ❆♥ ■tô ✲ ❲❡♥t③❡•• ❢♦r♠✉•❛ ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ st✉❞② t❤❡ ❝❛s❡ ♦❢ T t • X t ✇❤❡r❡ X t ✐s ❛ r❡❣✉•❛r✱ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ♣r♦❝❡ss ♦❢ t❤❡ t②♣❡ ✭✹✳✶✮ ❛♥❞ T t ✐s ❛ S ′ ✲✈❛•✉❡❞ s❡♠✐✲♠❛rt✐♥❣❛•❡✳ ▼♦r❡ ♣r❡❝✐s❡•②✱ ✉s✐♥❣ t❤❡ r❡s✉•ts ❢r♦♠ s❡❝t✐♦♥ ✸✳✺✱ ✇❡ ✇✐•• ♣r♦✈❡ ❛♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ■tô✲❲❡♥t③❡•• ❢♦r♠✉•❛ ✭s❡❡✱ ❢♦r ❡①❛♠♣•❡✱ ❬✹✵(✮✳ ❲❡ st✐❝❦ t♦ t❤❡ ❝❛s❡ ✇❤❡r❡ t❤❡ ❞✐♠❡♥s✐♦♥ ✐s N = 1 ❢♦r s✐♠♣•❡r ♥♦t❛t✐♦♥ ❜✉t t❤❡r❡ ✐s ♥♦ ❞✐✣❝✉•t② ✐♥ ❡①t❡♥❞✐♥❣ t❤❡ r❡s✉•ts✳ ❲❡ s✉♣♣♦s❡ t❤❛t X ✈❡r✐✜❡s ❤②♣♦t❤❡s❡s ✭✹✳✺✮✱ ✭✹✳✻✮ ❛♥❞ ✭✹✳✻✮✳ ❲❡ ❛•s♦ ✐♥tr♦❞✉❝❡ t✇♦ t✐♠❡✲❝♦♥t✐♥♦✉s✱ ❞✐str✐❜✉t✐♦♥✲✈❛•✉❡❞ ♣r♦❝❡ss❡s✿ D t , V t ∈ k<[s] D ∞ -,k S ∞,-(s-k) ∩ D ∞ -,k W α,-(s-k-N α ) ✭✹✳✺✺✮ ❍❡r❡ ✇❡ s✉♣♣♦s❡ t❤❛t α ✐s s✉❝❤ t❤❛t {s} > N α ❀ ✐s s ✐s ❛♥ ✐♥t❡❣❡r ✇❡ ♦♥•② s✉♣♣♦s❡✿ D t , V t ∈ k<[s] D ∞ -,k S ∞,-(s-k) ✭✹✳✺✻✮ ❲❡ s✉♣♣♦s❡ t❤❛t t❤❡s❡ ♣r♦❝❡ss❡s ❛r❡ ❛❞❛♣t❡❞ t♦ t❤❡ ✜•tr❛t✐♦♥ ♦❢ W ✳ ❚❤❡♥ ✇❡ ♠❛② ❞❡✜♥❡ ❛♥ S ∞,-s ✲✈❛•✉❡❞ s❡♠✐♠❛rt✐♥❣❛•❡ ❜② ❛♥ ✐♥✐t✐❛• ✈❛•✉❡ T 0 ∈ S ∞,-s ❛♥❞ t❤❡ ❡q✉❛t✐♦♥✿ dT t = D t dt + V t dW t ✭✹✳✺✼✮ ✇❤❡r❡ t❤❡ ❛❜♦✈❡ ♠❡❛♥s t❤❛t ❢♦r ❛♥② φ ∈ S✿ T t , φ = T 0 , φ + t 0 D s , φ ds + t 0 V s , φ dW s ✭✹✳✺✽✮ ✹✳✸✳ ❆◆ ❆◆❚■❈■P❆❚■❱❊ ❱❊|❙■❖◆ ✾✶ ✇❤✐❝❤ ✐s ❛ r❡❣✉•❛r s❡♠✐♠❛rt✐♥❣❛•❡ ✐♥ R✳ ❋♦r ♠♦r❡ ❞❡t❛✐•s ❛❜♦✉t t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ s✉❝❤ ♦❜•❡❝ts✱ s❡❡✱ ❢♦r ❡①❛♠♣•❡✱ ❬✼✻( ♦r ❬✸✾(✳ ■t ✐s t❤❡♥ ❡❛s✐•② ✈❡r✐✜❡❞ t❤❛t✿ T t ∈ k<[s] D ∞ -,k S ∞,-(s-k) ✭✹✳✺✾✮ ❛♥❞ ❛•• t❤❡ t → T t D ∞ -,k (S∞,-(s-k)) ❛r❡ ❜♦✉♥❞❡❞✳ ❚❤❡r❡ ❛•s♦ ✐s ❛ s✐♠✐•❛r r❡s✉•t ❢♦r t❤❡ D ∞ -,k W α,-(s-k-N α ) s♣❛❝❡s✳ ❲❡ ♥♦✇ st❛t❡ ♦✉r r❡s✉•t✿ ❚❤❡♦r❡♠ ✹✳✷✳✵✳✶ ✭❲❡❛❦ ■tô✲❲❡♥t③❡•• ❢♦r♠✉•❛✮✳ ❯♥❞❡r t❤❡ ❤②♣♦t❤❡s❡s ✭✹✳✺✺✮✱ ✭✹✳✺✮✱ ✭✹✳✻✮ ❛♥❞ ✭✹✳✻✮✱ ❢♦r ❛♥② p ′ > p ❛♥❞ s ′ > s✱ ❢♦r ❛♥② ǫ > 0✱ t❤❡ ❢♦••♦✇✐♥❣ ❢♦r♠✉•❛ ❤♦•❞s ✐♥ D p ′ ∨2,-(s ′ +2) ✿ T (t) • X t -T (ǫ) • X ǫ = M (ǫ) t + t ǫ b(u) • (∂ x T u ) • X u du + 1 2 t ǫ σ(u) 2 • (∂ xx T u ) • X u du + t ǫ D u • X u du + t ǫ σ(u) • ∂ x V u • X u du ✭✹✳✻✵✮ ✇❤❡r❡ M (ǫ) t ǫ≤t≤1 ✐s ❛ ✇❡❛❦ ♠❛rt✐♥❣❛•❡ ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡✱ ❛♥❞ ✐♥ t❤❡ s❡♥s❡ ♦❢ ❞✐✈❡r❣❡♥❝❡s ✇❡ ❤❛✈❡✿ M (ǫ) t = t ǫ [b(u) • (∂ x T u ) • X u + V u • X u ] dW j u ✭✹✳✻✶✮ Pr♦♦❢✳ ❲❡ ✉s❡ t❤❡ s❛♠❡ ♠❡t❤♦❞ ❛s ❢♦r t❤❡ ♣r♦♦❢ ♦❢ t❤❡♦r❡♠ ✹✳✶✳✶✳✶✱ ❡①❝❡♣t t❤❛t ✇❡ ✉s❡ t❤❡♦r❡♠ ✸✳✺✳✵✳✷✵ ✐♥st❡❛❞ ♦❢ t❤❡♦r❡♠ ✸✳✷✳✵✳✶✺✳ ❚❤❡ ❝♦♥❞✐t✐♦♥ ✭✹✳✺✺✮ ❤❛s ❜❡❡♥ ✇r✐tt❡♥ s♦ t❤❛t t❤❡♦r❡♠ ✸✳✺✳✵✳✷✵ ❛•✇❛②s ❛♣♣•✐❡s ❞✐r❡❝t•②✱ s♦ t❤❡r❡ ✐s ♥♦ ❡①tr❛ ❞✐✣❝✉•t②✳ ✹✳✸ ❆♥ ❛♥t✐❝✐♣❛t✐✈❡ ✈❡rs✐♦♥ ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ st✐•• ❝♦♥s✐❞❡r ❛ S ′ ✲✈❛•✉❡❞ ♣❛t❤ T t ❛♥❞ ❛ ♣r♦❝❡ss✿ X t = X 0 + t 0 b u du + t 0 σ u dW u ✭✹✳✻✷✮ ✾✷ ❈❍❆P❚❊| ✹✳ ❆ ❲❊❆❑ ■❚Ô ❋❖|▼❯▲❆ ❜✉t ✇❡ ❛••♦✇ X 0 t♦ ❜❡ ❛ r❛♥❞♦♠ ✈❛r✐❛❜•❡ ❛♥❞ ✇❡ ❞♦ ♥♦t s✉♣♣♦s❡ t❤❛t b ♦r σ ❛r❡ ❛❞❛♣t❡❞ t♦ t❤❡ ❇r♦✇♥✐❛♥ ✜•tr❛t✐♦♥ ❛♥②♠♦r❡ ✭s♦ t❤❡ st♦❝❤❛st✐❝ ✐♥t❡❣r❛•s ❤❛✈❡ t♦ ❜❡ ✉♥❞❡rst♦♦❞ ❛s ❞✐✈❡r❣❡♥❝❡s✮✳ ❲❡ st✉❞② ✉♥❞❡r ✇❤✐❝❤ ❤②♣♦t❤❡s❡s ♦♥ t❤❡ ▼❛••✐❛✈✐♥ r❡❣✉•❛r✐t② ♦❢ X 0 ✱ b ❛♥❞ σ ✇❡ ♠❛② st✐•• ✇r✐t❡ ❛ ✇❡❛❦ ✭❛♥t✐❝✐♣❛t✐✈❡✮ ■tô ❢♦r♠✉•❛ ❛s ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✳ ❲❡ ✇✐•• ♥♦✇ r❡❝❛•• ❛ ❢❡✇ ♣♦✐♥ts ❛❜♦✉t ❛♥t✐❝✐♣❛t✐✈❡ st♦❝❤❛st✐❝ ❝❛•❝✉•✉s✳ ❲❡ ❞♦ ♥♦t ♣r❡t❡♥❞ t♦ ♣r♦✈✐❞❡ ❛ ❢✉•• ✐♥tr♦❞✉❝t✐♦♥ t♦ t❤❛t t♦♣✐❝❀ ✐♥st❡❛❞ ✇❡ ♦♥•② ✐♥tr♦❞✉❝❡ t❤❡ ♥♦t❛t✐♦♥ ✇❡ ✇✐•• ♥❡❡❞ ❛♥❞ ✇❡ r❡❢❡r t♦ ❬✺✸( ❛♥❞ t❤❡ r❡❢❡r❡♥❝❡s t❤❡r❡✐♥ ❢♦r ❞❡t❛✐•s✳ ❋✐rst✱ ❢♦r 1 < p < ∞ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ s♣❛❝❡ L p,1 ♦❢ t❤♦s❡ ♣r♦❝❡ss❡s s✉❝❤ t❤❛t✿ u p L 1,p := E 1 0 |u t | p dt + E s≤t |D s u t | p dsdt < ∞ ✭✹✳✻✸✮ ❚❤❡♥ ❢♦r 1 < q ≤ p ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ s♣❛❝❡s L q+ p,1 ✱ r❡s♣✳ L q- p,1 ♦❢ t❤♦s❡s ♣r♦❝❡ss❡s u ∈ L p,1 s✉❝❤ t❤❛t t❤❡r❡ ❡①✐sts ❛ ♣r♦❝❡ss D + u✱ r❡s♣✳ D -u s✉❝❤ t❤❛t ❢♦r ❡✈❡r② t✿ lim n→∞ 1 0 sup s<t≤(s+ 1 n )∧1 E |D s u t -(D + u) s | p ds = 0 ✭✹✳✻✹✮ r❡s♣✿ lim n→∞ 1 0 sup (s-1 n )∨0≤t<s E |D s u t -(D -u) s | p ds = 0 ✭✹✳✻✺✮ ❋✐♥❛••② ✇❡ ♥♦t❡✿ L q p,1 = L q+ 1,p ∩L q- p,1 ✳ ❚♦ ❛❝q✉✐r❡ ❛♥ ✐♥t✉✐t✐♦♥ ❛❜♦✉t t❤❛t ♥♦t✐♦♥✱ •❡t ✉s ♥♦t❡ t❤❛t ✐❢ u ✐s ❛ ♣r♦❝❡ss s✉❝❤ t❤❛t (s, t) → D s u t ✐s ❝♦♥t✐♥✉♦✉s ♦♥ ❛ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ t❤❡ ❞✐❛❣♦♥❛•✱ t❤❡♥✿ (D + u) t = D t u t = (D -u) t ✳ ❍♦✇❡✈❡r t❤❛t ❝♦♥t✐♥✉✐t② ♣r♦♣❡rt② ❞♦❡s ♥♦t ❤♦•❞ ❢♦r ♠❛♥② ♣r♦❝❡ss❡s ✇❤✐❝❤ ❛r❡ ♦❢ ✐♥t❡r❡st❀ ❢♦r ❡①❛♠♣•❡ t❤❡r❡ ✐s t❤❡✿ ❚❤❡♦r❡♠ ✹✳✸✳✵✳✷✳ ❈♦♥s✐❞❡r t❤❡ ♣r♦❝❡ss ✭✹✳✻✷✮ ❛♥❞ s✉♣♣♦s❡ t❤❛t✿ • X 0 ∈ D 2,1 ❀ • b ∈ D 2,2 (H)❀ • σ ∈ L 2,1 ✳ ❚❤❡♥ u ∈ L 2 2,1 ❛♥❞✿ (D -X) t = D t X 0 + t 0 D t b r dr + t 0 D t σ r dW r ✭✹✳✻✻✮ (D + X) t = (D -X) t + σ t ✭✹✳✻✼✮ ✹✳✸✳ ❆◆ ❆◆❚■❈■P❆❚■❱❊ ❱❊|❙■❖◆ ✾✸ ❆•s♦✱ ✐♥ ❬✺✸( ✭♦r ✐♥ t❤❡ ♦r✐❣✐♥❛• ❬✺✹(✮✱ ✐t ✐s ♣r♦✈❡❞ t❤❛t ✐❢ F ✐s ❛ C 2 ❢✉♥❝t✐♦♥ ❛♥❞ ✐❢ ♦♥❡ ❛ss✉♠❡s ❡♥♦✉❣❤ r❡❣✉•❛r✐t② ♦♥ b ❛♥❞ σ✱ t❤❡ ❢♦••♦✇✐♥❣ ■tô ❢♦r♠✉•❛ ❤♦•❞s✿ f (X t ) = f (X 0 ) + t 0 b s f ′ (X s ) + 1 2 σ 2 s f ′′ (X s ) ds + t 0 σ s f ′ (X s )dW s + t 0 (D -X) s σ s f ′′ (X s )ds ✭✹✳✻✽✮ ❍❡r❡ t❤❡ ❇r♦✇♥✐❛♥ t❡r♠ ✐s ❛ ❞✐✈❡r❣❡♥❝❡✳ ❲❡ ❛•s♦ ♥♦t❡ t❤❛t t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ✉s✉❛• ■tô ❢♦r♠✉•❛ ❛♥❞ t❤✐s ❛♥t✐❝✐♣❛t✐✈❡ ♦♥❡ •✐❡s ♦♥•② ✐♥ t❤❡ •❛st t❡r♠✳ ❋✐♥❛••② ✐t ✐s ✐♥t❡r❡st✐♥❣ t♦ ♥♦t✐❝❡ t❤❛t t❤✐s t❡r♠ ✐♥ t❤❡ ♦♥❡ ✇✐t❤ •❡❛st ▼❛••✐❛✈✐♥ r❡❣✉•❛r✐t②✳ ❚❤❡♥ ❜② t❤❡ s❛♠❡ ♠❡t❤♦❞s ❛s ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥ ✇❡ ♣r♦✈❡ t❤❡✿ ❚❤❡♦r❡♠ ✹✳✸✳✵✳✸ ✭❆♥t✐❝✐♣❛t✐✈❡ ✇❡❛❦ ■tô ❢♦r♠✉•❛✮✳ ▲❡t T ∈ S q,-δ ✳ ❈♦♥s✐❞❡r t❤❡ ♣r♦❝❡ss ✭✹✳✻✷✮✱ ✇✐t❤ t❤❡ ❤②♣♦t❤❡s❡s t❤❛t ❢♦r ❡✈❡r② p✿ s≤t E (Id + L) δ+3 2 D s σ t p dsdt < ∞ ✭✹✳✻✾✮ s≤t E (Id + L) δ+2 2 D s b t p dsdt < ∞ ✭✹✳✼✵✮ 1 0 1 Σ t p dt < ∞ ✭✹✳✼✶✮ ✇❤❡r❡ Σ ✐s t❤❡ ▼❛••✐❛✈✐♥ ♠❛tr✐① ♦❢ X t ✳ ❚❤❡♥ t❤❡ ❢♦••♦✇✐♥❣ ❤♦•❞s ✐♥ t❤❡ s♣❛❝❡ D q,-δ ✿ T • X t = T • X 0 + t 0 b s • T ′ • X s + 1 2 σ 2 s • T ′′ • X s ds + t 0 σ s • T ′ • X s dW s + t 0 (D -X) s σ s • T ′′ • X s ds ✭✹✳✼✷✮ Pr♦♦❢✳ ❚❤❡ ♠❡t❤♦❞ ✐s ❡①❛❝t•② t❤❡ s❛♠❡ ❛s ✐♥ t❤❡ ♣r❡✈✐♦✉s ♣❛r❛❣r❛♣❤ s♦ ✇❡ ♦♥•② ♠❡♥t✐♦♥ ❛ ❢❡✇ ♥♦t❡✇♦t❤② ❞✐✛❡r❡♥❝❡s✳ ❋✐rst s✐♥❝❡ ✇❡ ❛••♦✇ X 0 t♦ ❜❡ ❛ ✭♥♦♥❞❡❣❡♥❡r❛t❡✮ r❛♥❞♦♠ ✈❛r✐❛❜•❡ ♦✉r ❢♦r♠✉•❛ ♠❛② ✐♥✈♦•✈❡ ✐♥t❡❣r❛•s st❛rt✐♥❣ ❢r♦♠ 0 r❛t❤❡r t❤❛♥ s♦♠❡ ǫ✳ ✾✹ ❈❍❆P❚❊| ✹✳ ❆ ❲❊❆❑ ■❚Ô ❋❖|▼❯▲❆ ❙❡❝♦♥❞ ✇❡ ❝❛♥♥♦t •✐❢t t❤❡ ❇r♦✇♥✐❛♥ ✭❞✐✈❡r❣❡♥❝❡✮ t❡r♠ ✐♥t♦ ❛ ♠❛rt✐♥❣❛•❡ s✐♥❝❡ σ ✐s ♥♦t ❛❞❛♣t❡❞✳ ❚❤❡r❡❢♦r❡ ✐♥st❡❛❞ ♦❢ t❤❡ ❇✉r❦❤♦•❞❡r✲❉❛✈✐❡s✲•✉♥❞② ✐♥❡q✉•✐t✐❡s ✇❡ ✇✐•• ♥❡❡❞ t♦ ✉s❡ t❤❡ ❞✐✈❡r❣❡♥❝❡ ✐♥❡q✉❛✐•t②✿ δu Dp,s ≤ u D p,s+1 (H) ✭✹✳✼✸✮ ❋✐♥❛••② ✇❡ ♥♦t❡ t❤❛t ♦✉r ❤②♣♦t❤❡s❡s ❤❛✈❡ ❜❡❡♥ t❛✐•♦r❡❞ s♦ t❤❛t D -X ❤❛s ❡♥♦✉❣❤ ✭✐❡ δ + 3✮ ▼❛••✐❛✈✐♥ r❡❣✉•❛r✐t②✳ ✹✳✹ ❆♣♣•✐❝❛t✐♦♥ t♦ t❤❡ s♦•✉t✐♦♥ ♦❢ ❛ ❙❉❊ ■♥ t❤✐s ♣❛r❛❣r❛♣❤ ✇❡ ❣✐✈❡ ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ ❛ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥✲ t✐❛• ❡q✉❛t✐♦♥ ❢♦r ✐ts s♦•✉t✐♦♥ t♦ ❜❡ r❡❣✉•❛r ✐♥ t❤❡ s❡♥s❡ ♦❢ ▼❛••✐❛✈✐♥ ❝❛•❝✉•✉s ❛♥❞ ✉♥✐❢♦r♠•② ♥♦♥✲❞❡❣❡♥❡r❛t❡✳ ❚❤✐s ❛••♦✇s ✉s t♦ ❞❡r✐✈❡ ✉♥❞❡r ✇❤✐❝❤ ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ ❛♥ ❙❉❊ ✇❤✐❝❤ ❛r❡ s✉✣❝✐❡♥t t♦ ❛♣♣•② t❤❡ ✇❡❛❦ ■tô ❢♦r♠✉•❛ ✭♦r ✐ts ❡①t❡♥s✐♦♥s✮ t♦ ✐ts ✉♥✐q✉❡ str♦♥❣ s♦•✉t✐♦♥✳ ✹✳✹✳✶ ❋r❛❝t✐♦♥❛• r❡❣✉•❛r✐t② ♦❢ t❤❡ s♦•✉t✐♦♥ ♦❢ ❛♥ ❙❉❊ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ❍ö•❞❡r s♣❛❝❡s C s ✳ ■❢ s ✐s ❛♥ ✐♥t❡❣❡r✱ t❤❡♥ C s s✐♠♣•② ✐s t❤❡ s♣❛❝❡ ♦❢ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥s ✇✐t❤ ❜♦✉♥❞❡❞ ❞❡r✐✈❛t✐✈❡s ♦❢ t♦t❛• ♦r❞❡r •❡ss t❤❛♥ s✳ ❖t❤❡r✇✐s❡ ✇r✐t❡ s = [s] + {s}❀ t❤❡♥ C s ✐s t❤❡ s♣❛❝❡ ♦❢ t❤♦s❡ ❢✉♥❝t✐♦♥s ✐♥ C [s] ✇✐t❤ ❛•• ❞❡r✐✈❛t✐✈❡s ♦❢ t♦t❛• ♦r❞❡r [s] ❜❡✐♥❣ {s}✲❤♦•❞❡r✐❛♥✳ ❚❤❡s❡ s♣❛❝❡s ❛r❡ ❝•♦s❡•② r❡•❛t❡❞ t♦ t❤❡ P♦✐ss♦♥ s❡♠✐✲❣r♦✉♣ (Π t )✱ ❝❢ ❬✻✼(✳ ▲❡t ✉s r❡❝❛•• ❛ ❢❡✇ ❢❛❝ts ❛❜♦✉t t❤✐s s❡♠✐✲❣r♦✉♣✳ ❋✐rst ❢♦r t > 0 ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ❢✉♥❝t✐♦♥s✿ π t (x) = R N e -2iπy•x • e -2π|y|t dy = Γ( n+1 2 ) π n+1 2 • t (|x| 2 + t 2 ) n+1 2 ✭✹✳✼✹✮ ❛♥❞ ❢♦r f ∈ C 0 b ✇❡ s❡t✿ (Π t f )(x) = (π t * f )(x) ✭✹✳✼✺✮ ❚❤❡♥ Π t ❞❡✜♥❡s ❛ s❡♠✐❣r♦✉♣ ♦♥ C 0 b ✱ ✇❤✐❝❤ ❡♥•♦②s t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣❡rt②✿ ❚❤❡♦r❡♠ ✹✳✹✳✶✳✶✳ ▲❡t f ∈ L ∞ ❛♥❞ 0 < s < 1✳ ❚❤❡♥ t❤❡ ❢♦••♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛•❡♥t✿ • f ∈ C s ❀ • ∂(Πtf ) ∂t = O(t -1+s )❀ ✹✳✹✳ ❆PP▲■❈❆❚■❖◆ ❚❖ ❚❍❊ ❙❖▲❯❚■❖◆ ❖❋ ❆ ❙❉❊ ✾✺ • ❋♦r ❡✈❡r② 1 ≤ i ≤ N ✿ ∂(Πtf ) ∂x i = O(t -1+s ) ❲❡ ❛•s♦ ✐♥tr♦❞✉❝❡ t❤❡ s♣❛❝❡ C 0,s ♦❢ t❤♦s❡ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ [0, T ]×R N ✇❤✐❝❤ ❛r❡ C s ✐♥ s♣❛❝❡ ✉♥✐❢♦r♠•② ✐♥ t✐♠❡✱ ✇✐t❤ t❤❡ ♥♦r♠✿ f C 0,s = sup 0≤t≤T f (t, •) C s ✭✹✳✼✻✮ ❲❡ ✇✐•• ❝♦♥s✐❞❡r t❤❡ ❢♦••♦✇✐♥❣ N ✲❞✐♠❡♥s✐♦♥♥❛• st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛• ❡q✉❛t✐♦♥✿ dX t = b(t, X t )ds + σ(t, X t )dW t ✭✹✳✼✼✮ X 0 = x ✭✹✳✼✽✮ ✇✐t❤ b : R + × R N → R N ❛♥❞ σ : R + × R N → R N ⊗ R d ❛♥❞ W t❤❡ ❝❛♥♦♥✐❝❛• ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ♦♥ t❤❡ d✲❞✐♠❡♥s✐♦♥♥❛• ❲✐❡♥❡r s♣❛❝❡✳ ❚❤❡ ❢♦••♦✇✐♥❣ r❡s✉•t ✐s ✇❡•• ❦♥♦✇♥ ❛♥❞ ♠❛② ❜❡ ❢♦✉♥❞ ✐♥ ❬✻✻(✱ ❬✺✸( ♦r ❬✺✻(✿ ❚❤❡♦r❡♠ ✹✳✹✳✶✳✷✳ ❈♦♥s✐❞❡r t❤❡ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛• ❡q✉❛t✐♦♥ ✭✹✳✼✼✮ ❛♥❞ s✉♣✲ ♣♦s❡ t❤❛t b ❛♥❞ σ ❛r❡ ✐♥ C 0,k ❢♦r s♦♠❡ T ✳ ❚❤❡♥ ❢♦r ❛♥② p > 1 ❛♥❞ ❛♥② t ≥ 0✱ X t ∈ D p,k ❛♥❞ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t C ❞❡♣❡♥❞✐♥❣ ♦♥•② ♦♥ p✱ k ❛♥❞ t❤❡ C 0,k ♥♦r♠s ♦❢ b ❛♥❞ σ s✉❝❤ t❤❛t✿ sup 0≤t≤T |(Id + L) k/2 X t | L p ≤ e CT |❡♠❛r❦ ✹✳✹✳✶✳✶✳ ■♥ ❢❛❝t ✐t ✐s s✉✣❝✐❡♥t t❤❛t b ❛♥❞ σ ❜❡ C 0,k-1 ✇✐t❤ ✉♥✐❢♦r♠•② •✐♣s❝❤✐t③ ❞❡r✐✈❛t✐✈❡s ♦❢ ♦r❞❡r k -1✳ Pr♦♦❢✳ ❈•❛ss✐❝ ❛♥❞ ♦♠✐tt❡❞✱ ❝❢✱ ❢♦r ❡①❛♠♣•❡✱ ❬✺✸(✳ ❲❡ ✇✐•• ❣❡♥❡r❛•✐③❡ t❤✐s t♦ t❤❡ ❢♦••♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✹✳✹✳✶✳✸✳ ❈♦♥s✐❞❡r t❤❡ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛• ❡q✉❛t✐♦♥ ✭✹✳✼✼✮ ❛♥❞ s✉♣✲ ♣♦s❡ t❤❛t b ❛♥❞ σ ❛r❡ ✐♥ C 0,s ❢♦r s♦♠❡ T ❛♥❞ s♦♠❡ s ≥ 1✳ ❚❤❡♥ ❢♦r ❛♥② p > 1✱ s ′ < s ❛♥❞ ❛♥② t ≥ 0✱ X t ∈ D p,s ′ ❛♥❞ t❤❡r❡ ❡①✐st ❝♦♥st❛♥ts A ❛♥❞ B ❞❡♣❡♥❞✐♥❣ ♦♥•② ♦♥ p✱ s✱ s ′ ❛♥❞ t❤❡ C 0,s ♥♦r♠s ♦❢ b ❛♥❞ σ s✉❝❤ t❤❛t✿ X t D p,s ′ ≤ A exp(B( b C 0,s + σ C 0,s ) • t) ✭✹✳✼✾✮ Pr♦♦❢✳ ❲❡ st✐❝❦ t♦ t❤❡ ❝❛s❡ d = N = 1 ❞✉r✐♥❣ t❤❡ ♣r♦♦❢✱ ❜✉t t❤❡ ❣❡♥❡r❛• ❝❛s❡ ✐♥❝•✉❞❡s ♥♦ ♦t❤❡r ❞✐✣❝✉•t② t❤❛♥ t❡❞✐♦✉s ♥♦t❛t✐♦♥✳ ❲❡ ♥♦t❡ [s] = k ❛♥❞ {s} = α✳ ❲❡ s✉♣♣♦s❡ α > 0✱ ♦t❤❡r✇✐s❡ ♦✉r r❡s✉•t ✐s ✐♥❝•✉❞❡❞ ✐♥ t❤❡♦r❡♠ ✹✳✹✳✶✳✷✳ ❲❡ ♥♦t❡ t❤❛t t❤❡ ❢❛❝t t❤❛t s ≥ 1 ✐♠♣•✐❡s t❤❡ ❡①✐st❡♥❝❡ ❛♥❞ t❤❡ ✉♥✐❝✐t② ♦❢ ❛ str♦♥❣ s♦•✉t✐♦♥ t♦ t❤❡ ❙❉❊✳ ✾✻ ❈❍❆P❚❊| ✹✳ ❆ ❲❊❆❑ ■❚Ô ❋❖|▼❯▲❆ ❲❡ r❡❝❛•• t❤❛t Π ❞❡♥♦t❡s t❤❡ P♦✐ss♦♥ ❦❡r♥❡•❀ ❢♦r τ > 0 ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ❢✉♥❝✲ t✐♦♥s σ τ ❛♥❞ b τ ❞❡✜♥❡❞ ❜②✿ σ τ (t, •) = (Π τ σ)(t, •) ✭✹✳✽✵✮ ✇✐t❤ ❛♥ ❛♥❛•♦❣✉♦✉s ❞❡✜♥✐t✐♦♥ ❢♦r b τ ✳ ❋♦r τ > 0 t❤❡s❡ ❢✉♥❝t✐♦♥s ❤❛✈❡ ❜♦✉♥❞❡❞ ❞❡r✐✈❛t✐✈❡s ♦❢ ❛•• ♦r❞❡r❀ ✇❡ r❡❝❛•• t❤❛t ✇❡ ❤❛✈❡ t❤❡ ❡st✐♠❛t❡s✱ ❝❢ ❬✻✼(✿ σ τ C l ≤ σ C l ✭✹✳✽✶✮ ❢♦r l ≤ k ❛♥❞ σ τ C k+1 ≤ τ -1+α σ C s ✭✹✳✽✷✮ ❲❡ ❛•s♦ ❝♦♥s✐❞❡r X τ t t❤❡ s♦•✉t✐♦♥ ♦❢ t❤❡ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛• ❡q✉❛t✐♦♥ ❛ss♦❝✐❛t❡❞ t♦ t❤❡s❡ ❝♦❡✣❝✐❡♥ts✳ ❲❡ ♠❛② ❛♣♣•② t❤❡♦r❡♠ ✹✳✹✳✶✳✷ t♦ X t ❢♦r k ≤ s✱ ❛♥❞ t♦ t❤❡ X τ t ❢♦r ❛♥② ✐♥t❡❣❡r ✭❤❡r❡✿ k + 1✮ ✐❢ τ > 0✳ ▲❡t ✉s ✇r✐t❡✿ ∇X τ t = t 0 σ ′ τ (s, X τ s )∇X τ s dW s + t 0 b ′ τ (s, X τ s )∇X τ s dW s + min(t,•) 0 σ τ (s, X τ s )ds ✭✹✳✽✸✮ ❛♥❞ ❜② t❤❡ ▼❛••✐❛✈✐♥ ❝❛•❝✉•✉s ❛♥❛•♦❣✉❡ t♦ ❋❛❛ ❞✐ ❇r✉♥♦✬s ❢♦r♠✉•❛✿ ✐❢ k ≥ 1✿ ∇ k+1 X τ t = t 0 σ (k+1) τ (s, X τ s )∇ ⊗(k+1) X τ s dW s + t 0 σ ′ τ (s, X τ s )∇ k+1 X τ s dW s + t 0 b (k+1) τ (s, X τ s )∇ ⊗(k+1) X τ s ds + t 0 b ′ τ (s, X τ s )∇ k+1 X τ s ds + t 0 P (s, X)dW s + t 0 Q(s, X)ds + min(t,•) 0 R(s, X)ds ✭✹✳✽✹✮ ✇❤❡r❡ P ✱ Q ❛♥❞ R ❛r❡ ♣♦•②♥♦♠✐❛•s ✐♥ t❤❡ s♣❛❝❡ ❞❡r✐✈❛t✐✈❡s ♦❢ ♦r❞❡r •❡ss t❤❛♥ k ♦❢ b τ ❛♥❞ σ τ ❛♥❞ ✐♥ t❤❡ ∇ ⊗l X τ t ❢♦r l ≤ k ✐♥ t❤❡ s❡♥s❡ ♦❢ t❡♥s♦r ♣r♦❞✉❝ts✳ ❚❤❡r❡❢♦r❡✱ ❛♣♣•②✐♥❣ t❤❡ ❇✉r❦❤♦•❞❡r ❉❛✈✐s •✉♥❞② ✐♥❡q✉❛•✐t②✱ t❤❡♦r❡♠ ✹✳✹✳✶✳✷ ❛♥❞ ❍ö•❞❡r✬s t❤❡♦r❡♠ ♦♥❡ ❣❡ts✿ sup 0≤u≤t |∇ k+1 X τ u | p L p ≤ f (t)+τ -1+α g(t)+ t 0 ( b C 1 + σ C 1 ) p sup 0≤u≤s |∇ k+1 X τ u | p L p ds ✭✹✳✽✺✮ ✇❤❡r❡ f ❛♥❞ g ❛r❡ ❜♦✉♥❞❡❞ ❜② ❜♦✉♥❞s ♦❢ t❤❡ t②♣❡ exp(C( b C 0,s + σ C 0,s )✳ ◆♦✇ ❛♣♣•②✐♥❣ •r♦♥✇❛••✬s •❡♠♠❛ •❡❛❞s t♦ ❛ ♠❛•♦r✐③❛t✐♦♥ ♦❢ t❤❡ t②♣❡✿ sup 0≤u≤t |∇ k+1 X τ u | p L p ≤ (1 + τ (-1+α)p exp(C( b C 0,s + σ C 0,s ) • t) ✭✹✳✽✻✮ ✹✳✹✳ ❆PP▲■❈❆❚■❖◆ ❚❖ ❚❍❊ ❙❖▲❯❚■❖◆ ❖❋ ❆ ❙❉❊ ✾✼ ❛♥❞ ✐❢ 0 < α ′ < α✿ 1 0 τ -1+(1-α ′ )p sup 0≤u≤t |∇ k+1 X τ u | p L p dτ ≤ A • exp(B( b C 0,s + σ C 0,s ) • t) ✭✹✳✽✼✮ ◆♦✇ ✇❡ ♥❡❡❞ t♦ ❝♦♥tr♦• ∂ τ ∇ k X τ t ✳ ❲❡ ♥♦t❡ t❤❛t✱ s✐♥❝❡ s ≥ 1✱ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ P♦✐ss♦♥ ❦❡r♥❡• ❣✉❛r❛♥t❡❡ t❤❡ ❞✐✛❡r❡♥t✐❛❜✐•✐t② ♦❢ b τ ❛♥❞ σ τ ✐♥ τ t♦❣❡t❤❡r ✇✐t❤ ❛ ✉♥✐❢♦r♠ ❝♦♥tr♦• ♦♥ t❤❡s❡ ❞❡r✐✈❛t✐✈❡s✳ ❚❤❡r❡❢♦r❡ t❤❡ t❤❡♦r② ♦❢ st♦❝❤❛st✐❝ ✢♦✇s ✭❝❢ ❬✹✵( ♦r ❬✸✽(✱ t❤❡ r❡s✉•ts ❣❡♥❡r❛•✐③❡ ❡❛s✐•② t♦ ♣r♦❝❡ss❡s ✇✐t❤ ✈❛•✉❡s ✐♥ ❍✐•❜❡rt s♣❛❝❡s✮ ♣r♦✈❡s t❤❡ ❡①✐st❡♥❝❡ ♦❢ ∂ τ ∇ k X τ t ❛♥❞ t❤❛t t❤❡ ❝♦♠♣✉t❛t✐♦♥s ❜❡•♦✇ ❛r❡ ✈❛•✐❞✳ ❙t❛rt✐♥❣ ❢r♦♠ ✭✹✳✽✹✮ ❛♥❞ r❡♣•❛❝✐♥❣ k + 1 ❜② k ✇❡ ✜♥❞ ❛ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛• ❡q✉❛t✐♦♥ ✈❡r✐✜❡❞ ❜② ∂ τ ∇ k X τ t ✳ ❋✐rst✿ ∂ τ t 0 σ (k) τ (s, X τ s )∇ ⊗k X τ s dW s = t 0 ∂ τ σ (k) τ (s, X τ s ) + σ (k+1) τ (s, X τ s )∂ τ X τ s ∇ ⊗k X τ s dW s + k t 0 σ (k) τ (s, X τ s )∇ ⊗(k-1) X τ s ⊗ ∂ τ ∇X τ s dW s ✭✹✳✽✽✮ ❛♥❞ ∂ τ t 0 σ ′ τ (s, X τ s )∇ k X τ s dW s = t 0 [∂ τ σ ′ τ (s, X τ s ) + σ ′′ τ (s, X τ s )∂ τ X τ s ] ∇ k X τ s dW s + t 0 σ ′ τ (s, X τ s )∂ τ ∇ k X τ s dW s ✭✹✳✽✾✮ ❚❤❡r❡ ❛•s♦ ❛r❡ s✐♠✐•❛r ▲❡❜❡s❣✉❡ ✐♥t❡❣r❛•s ✐♥✈♦•✈✐♥❣ b✳ ❋✐♥❛••②✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❢♦r♠✿ ∂ τ X τ t = t 0 σ (k+1) τ (s, X τ s )∂ τ X τ s ∇ ⊗k X τ s + σ ′ τ (s, X τ s )∂ τ ∇ k X τ s dW s + t 0 b (k+1) τ (s, X τ s )∂ τ X τ s ∇ ⊗k X τ s + b ′ τ (s, X τ s )∂ τ ∇ k X τ s ds + t 0 P (s, X)dW s + t 0 Q(s, X)ds + min(t,•) 0 R(s, X)ds ✭✹✳✾✵✮ ✇❤❡r❡ P ✱ Q ❛♥❞ R ❛r❡ ♣♦•②♥♦♠✐❛•s ✐♥✈♦•✈✐♥❣ t❡r♠s ♦❢ ♦r❞❡r ❛t ♠♦st k -1 ✐♥ τ ✲❞❡r✐✈❛t✐✈❡s ❡♥❞ ❛t ♠♦st k ✐♥ s♣❛❝❡ ❛♥❞ ▼❛••✐❛✈✐♥ ❞❡r✐✈❛t✐✈❡s✱ s♦ ♦♥❝❡ ❛❣❛✐♥ ❜② ❛♣♣•②✐♥❣ t❤❡ ❇✉r❦❤♦•❞❡r ❉❛✈✐s •✉♥❞② ✐♥❡q✉❛•✐t②✱ t❤❡♦r❡♠ ✹✳✹✳✶✳✷✱ ❍ö•❞❡r✬s t❤❡♦r❡♠ ❛♥❞ ✜♥❛••② •r♦♥✇❛••✬s •❡♠♠❛ ✇❡ ♦❜t❛✐♥✿ sup 0≤u≤t |∂ τ ∇ k Xu τ | p L p ≤ (1 + τ (-1+α)p ) exp(C( σ C (0,s) + b C (0,s) )) ✭✹✳✾✶✮ ✾✽ ❈❍❆P❚❊| ✹✳ ❆ ❲❊❆❑ ■❚Ô ❋❖|▼❯▲❆ ❛♥❞ ❢♦r 0 < α ′ < α✿ 1 0 τ -1+(1-α ′ )p sup 0≤u≤t |∂ τ ∇ k Xu τ | p L p dτ ≤ A exp(B( b C 0,s + σ C 0,s ) • t) ✭✹✳✾✷✮ ❇② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ r❡❛• ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡s✱ ♥♦t✐♥❣ s ′ = k + α ′ t❤✐s ②✐❡•❞s✿ X t ∈ D p,s ′ ❛♥❞✿ X t D p,s ′ ≤ A exp(B( b C 0,s + σ C 0,s ) • t) ✭✹✳✾✸✮ ❚❤✐s ❡♥❞s t❤❡ ♣r♦♦❢✳ |❡♠❛r❦ ✹✳✹✳✶✳✷✳ ❍❡r❡ ❛s ♦♣♣♦s❡❞ t♦ t❤❡♦r❡♠ ✹✳✹✳✶✳✷ ✇❡ ❞♦ ♥♦t ♦❜t❛✐♥ ❛ ✉♥✐❢♦r♠ ♠❛•♦r✐③❛t✐♦♥✳ ❚❤✐s ✐s ❜❡❝❛✉s❡ ✇❡ ❝♦✉•❞ ♥♦t ❞✐✛❡r❡♥t✐❛t❡ t❤❡ s✉♣ ✇✐t❤ r❡❣❛r❞ t♦ τ ✳ ✹✳✹✳✷ ❆ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ♥♦♥✲❞❡❣❡♥❡r❛❝② ♦❢ t❤❡ s♦•✉t✐♦♥ ♦❢ ❛♥ ❙❉❊ ❲❡ ♥♦✇ st❛t❡ ❛ ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ♥♦♥✲❞❡❣❡♥❡r❛❝② ♦❢ t❤❡ s♦•✉t✐♦♥ ♦❢ t❤❡ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛• ❡q✉❛t✐♦♥ ✭✹✳✼✼✮✳ ❲❡ s✉♣♣♦s❡ t❤❛t σ ∈ C 0,s ❢♦r s♦♠❡ s ≥ 2 ❛♥❞ t❤❛t β ∈ C 0,1 ✳ ❚❤❡♥ ♦✉r st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛• s♦•✉t✐♦♥ ❤❛s ❛ ✉♥✐q✉❡ str♦♥❣ s♦•✉t✐♦♥✳ ❲❡ ♥♦t❡ k = [s]✳ ❋♦r 1 ≤ j ≤ d •❡t ✉s ❝♦♥s✐❞❡r t❤❡ ✈❡❝t♦r ✜❡•❞s ♦♥ R N ✇❤✐❝❤ ❛r❡ ❞❡✜♥❡❞ ❜②✿ V j = σ •,j ∂ x j ✳ ▲❡t L 0 ❜❡ t❤❡ s❡t ❝♦♥t❛✐♥✐♥❣ ❛•• t❤❡ V j ❀ ❞❡✜♥❡ ❜② r❡❝✉rr❡♥❝❡ L i+1 t❤❡ s❡t ❝♦♥t❛✐♥✐♥❣ t❤❡ [V j , Z] ✇❤❡r❡ Z ∈ L i ✳ ❍❡r❡ [•, •] ✐s t❤❡ ❝♦♠♠✉t❛t♦r ♦❢ t✇♦ ✈❡❝t♦r ✜❡•❞s✳ ❚❤❡ L i ❛r❡ ✇❡•• ❞❡✜♥❡❞ ❢♦r i ≤ k✳ ❆❞❛♣t✐♥❣ t❤❡ ♣r♦♦❢ ❢r♦♠ ❬✶✺( ✇❡ st❛t❡✿ ❚❤❡♦r❡♠ ✹✳✹✳✷✳✶✳ ❙✉♣♣♦s❡ t❤❛t ❢♦r s♦♠❡ n ≤ k -2 t❤❡r❡ ❡①✐sts c > 0 s✉❝❤ t❤❛t ❢♦r ❛•• ξ ∈ S N -1 ✿ n i=0 Z∈L k ξ, Z (0, x) > c ✭✹✳✾✹✮ ❙✉♣♣♦s❡ ❛•s♦ t❤❛t σ ∈ C (β,n+2) ❢♦r s♦♠❡ β > 0✳ ❚❤❡♥✱ ✐❢ Σ t ✐s t❤❡ ▼❛••✐❛✈✐♥ ♠❛tr✐① ♦❢ X t ✱ ❡st✐♠❛t❡s ♦❢ t❤❡ ❢♦••♦✇✐♥❣ t②♣❡ ❤♦•❞ ❢♦r t > 0✿ 1 Σ t L p ≤ C • 1 t ν • e Kt ✭✹✳✾✺✮ ❍❡r❡ C✱ ν ❛♥❞ K ❛r❡ ❝♦♥st❛♥ts ✇❤✐❝❤ ❞♦ ♥♦t ❞❡♣❡♥❞ ♦♥ t✳ ■♥ ♣❛rt✐❝✉•❛r✿ T ǫ 1 Σ t p L p dt < ∞ ✭✹✳✾✻✮ ❚❤❡ ♣r♦♦❢ ✐s ❡♥t✐r❡•② ❝♦♥t❛✐♥❡❞ ✐♥ ❬✶✺(✳ ❲❡ s✐♠♣•② ❛❝❝❡♣t t♦ ❝♦♣❡ ✇✐t❤ ❛ t❤❡✲ ♦r❡♠ ✇❤✐❝❤ ✐s ♠♦r❡ t❡❞✐♦✉s t♦ st❛t❡ ❛s ✇❡ ♣❛② ♠♦r❡ ❛tt❡♥t✐♦♥ t♦ t❤❡ ♦r❞❡rs ♦❢ ❞✐✛❡r❡♥t✐❛❜•✐t②❀ ✐♥ s♦ ❞♦✐♥❣ ✇❡ ❛✈♦✐❞ t♦ s✉♣♣♦s❡ t❤❛t t❤❡ ✈♦•❛t✐•✐t② ✐s C 0,∞ ✳ ✹✳✹✳ ❆PP▲■❈❆❚■❖◆ ❚❖ ❚❍❊ ❙❖▲❯❚■❖◆ ❖❋ ❆ ❙❉❊ ✾✾ ✹✳✹✳✸ ❲❡❛❦ ■tô ❢♦r♠✉•❛ ❢♦r t❤❡ s♦•✉t✐♦♥ ♦❢ ❛♥ ❙❉❊ ❲❡ ♥♦✇ ❝♦♠❜✐♥❡ t❤❡ r❡s✉•ts ✐♥ t❤❡ t✇♦ ♣r❡✈✐♦✉s ♣❛r❛❣r❛♣❤s t♦ ♦❜t❛✐♥ tr❛❝t❛❜•❡ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s ✇❤✐❝❤ •❡t ✉s ❛♣♣•② t❤❡ ✇❡❛❦ ■tô ❢♦r♠✉•❛ t♦ t❤❡ s♦•✉t✐♦♥ ♦❢ t❤❡ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛• ❡q✉❛t✐♦♥ ✭✹✳✼✼✮✳ ❚❤❡♦r❡♠ ✹✳✹✳✸✳✶✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛• ❡q✉❛✲ t✐♦♥ ✭✹✳✼✼✮ ❛r❡ ✐♥ C (0,s) ❢♦r s♦♠❡ s > 2 ❛♥❞ •❡t k = [s]✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ ❍♦r♠❛♥✲ ❞❡r ❝♦♥❞✐t✐♦♥ ✭✹✳✾✹✮ ❤♦•❞s ❢♦r s♦♠❡ n ≤ k -2 ❛♥❞ t❤❛t σ ∈ C (β,n+2) ✳ ❚❤❡♥ ✐❢ T ✐s ❛ ♣❛t❤ ✇✐t❤ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥s ✐♥ S p,-(s-1) t❤❡ ❢♦••♦✇✐♥❣ ■tô ❢♦r♠✉•❛ ❤♦•❞s✿ T t • X t -T ǫ • X ǫ = M ǫ t + i t ǫ b i (u, X u ) • (∂ i T u ) • X u du + 1 2 i,j t ǫ a ij (u, X u ) • (∂ ij T u ) • X u du + t ǫ T (du) • X u ✭✹✳✾✼✮ ❛♥❞ ✐♥ t❤❡ s❡♥s❡ ♦❢ ❞✐✈❡r❣❡♥❝❡s✿ M (ǫ) t = j t ǫ i [b i (u, X u ) • (∂ i T u ) • X u ] dW j u ✭✹✳✾✽✮ ❖❢ ❝♦✉rs❡✱ t❤❡♦r❡♠ ✹✳✶✳✶✳✷ ❛•s♦ ❡①t❡♥❞s ✐♥t♦ t❤❡ ❢♦••♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✹✳✹✳✸✳✷✳ ▲❡t X ❛♥❞ T ❜❡ ❛s ✐♥ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠✳ ❈♦♥s✐❞❡r t✇♦ ❢✉♥❝t✐♦♥s b Y , σ Y ∈ C (0,s) ❛♥❞ •❡t Y ❜❡ t❤❡ ♦♥•② str♦♥❣ s♦•✉t✐♦♥ ♦❢ t❤❡ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛• ❡q✉❛t✐♦♥✿ dY t = b(t, Y t )dt + σ(t, Y t )dW t ✭✹✳✾✾✮ Y 0 = y ✭✹✳✶✵✵✮ ❋✐♥❛••② •❡t f ∈ BV (C s )✳ ❚❤❡♥ t❤❡ ■tô ❢♦r♠✉•❛ ❤♦•❞s ❢♦r✿ f (t, Y t ) • T t • X t ✭✹✳✶✵✶✮ ▲❡t ✉s ♥♦t❡ t❤❛t ✇❡ ❞♦ ♥♦t ♥❡❡❞ t♦ ♠❛❦❡ ❛♥② ❛ss✉♠♣t✐♦♥ ♦♥ t❤❡ ♥♦♥❞❡❣❡♥❡r❛❝② ♦❢ Y ✳ ✶✵✵ ❈❍❆P❚❊| ✹✳ ❆ ❲❊❆❑ ■❚Ô ❋❖|▼❯▲❆ ✹✳✺ ❈♦♠♣❛r✐s♦♥ t♦ ♦t❤❡r ❡①t❡♥s✐♦♥s ♦❢ t❤❡ ■tô ❢♦r✲ ♠✉•❛ ✹✳✺✳✶ ❊①t❡♥s✐♦♥s ♦❢ t❤❡ ❋♦••♠❡r✲Pr♦tt❡r✲❙❤✐r②❛❡✈ t②♣❡ ❲❡ st❛rt ❜② s❤♦✇✐♥❣ t❤❛t ❢r♦♠ ♦✉r ■tô ❢♦r♠✉•❛✱ ✇❡ ❝❛♥ r❡❝♦✈❡r ❛ ❢♦r♠✉•❛ ♦❢ t❤❡ t②♣❡ ♦❢ t❤❡ ❋♦••♠❡r✲Pr♦tt❡r✲❙❤✐r②❛❡✈ ❢♦r♠✉•❛ ❣✐✈❡♥ ✐♥ ❬✷✵(✿ ❚❤❡♦r❡♠ ✹✳✺✳✶✳✶ ✭❋ö••♠❡r✲Pr♦tt❡r✲❙❤✐r②❛❡✈ ❢♦r♠✉•❛✮✳ ▲❡t f ❜❡ ❛ ❞✐✛❡r❡♥t✐❛❜•❡ ❢✉♥❝t✐♦♥ s✉❝❤ t❤❛t Df ∈ L 2 loc R N , R N ❛♥❞ •❡t W ❜❡ ❛ N ✲❞✐♠❡♥s✐♦♥♥❛• ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳ ❚❤❡♥✿ f (W t ) = f (0) + N i=1 ∂ i f (W s )dW s + 1 2 • [f ′ (W ), W ] t ✭✹✳✶✵✷✮ ✇❤❡r❡ t❤❡ q✉❛❞r❛t✐❝ ❝♦✈❛r✐❛❛t✐♦♥ [f ′ (W ), W ] ✐s ❞❡✜♥❡❞ ❛s t❤❡ •✐♠✐t ♦❢ |✐❡♠❛♥♥ s✉♠s✿ [f ′ (W ), W ] t = N i=1 lim n→∞ n k=1 f ′ (W k n ) -f ′ (W k-1 n ) • W k n -W k-1 n ✭✹✳✶✵✸✮ ◆♦✇✱ s✉♣♣♦s❡ t❤❛t ✇❡ ❛r❡ ✐♥ t❤❡ s❡tt✐♥❣ ♦❢ t❤❡♦r❡♠ ✹✳✹✳✸✳✶✳ ❙✉♣♣♦s❡✱ ✐♥ ❛❞❞✐✲ t✐♦♥✱ t❤❛t T t ❤❛s ✈❛•✉❡s✱ s❛②✱ ✐♥ S p,1 ♦r ✐♥ W p,1 ✳ ❚❤❡♥✱ t❤❡ T t ❛♥❞ ✐ts ✜rst ♦r❞❡r ❞❡r✐✈❛t✐✈❡s ❛r❡ ♣r♦♣❡r ❢✉♥❝t✐♦♥s ✇✐t❤ ❛t •❡❛st L p r❡❣✉•❛r✐t② s♦ t❤❡ ■tô ❢♦r♠✉•❛ r❡❛❞s✿ T (t, X t ) -T (ǫ, X ǫ ) = i t ǫ b i (u, X u ) • ∂ i T (u, X u )du + j t ǫ i [b i (u, X u ) • ∂ i T (u, X u )] dW j u + t ǫ T (du, X u ) + 1 2 i,j t ǫ a ij (u, X u ) • (∂ ij T u ) • X u du ✭✹✳✶✵✹✮ ■♥ t❤❡ ❡q✉❛t✐♦♥ ❛❜♦✈❡✱ ♦♥•② t❤❡ t❡r♠s ✐♥ t❤❡ •❛st •✐♥❡ ❛r❡ ❞✐str✐❜✉t✐♦♥s ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡✱ ✇❤✐•❡ ❛•• t❤❡ ♦t❤❡r ❛r❡ ♣r♦♣❡r r❛♥❞♦♠ ✈❛r✐❛❜•❡s✳ ❚❤❡r❡❢♦r❡✱ ♦♥❡ ❡①♣❡❝ts t❤❛t ✐t ✐s ♣♦ss✐❜•❡ t♦ •❡t ǫ t❡♥❞ t♦ ✵✱ ❛♥❞ t♦ r❡❝♦✈❡r ❛ ✉s✉❛• ✭❢✉♥❝t✐♦♥❛•✮ ■tô ❢♦r♠✉•❛ ✐♥ t❤✐s ❝❛s❡✳ ❆•s♦✱ ✐t s❤♦✉•❞ ❜❡ ♣♦ss✐❜•❡ t♦ ✐❞❡♥t✐❢② t❤❡ ❞✐str✐❜✉t✐♦♥ t❡r♠ ❛❜♦✈❡ ✇✐t❤ t❤❡ q✉❛❞r❛t✐❝ ❝♦✈❛r✐❛t✐♦♥ t❡r♠ ✐♥ t❤❡ ❋ö••♠❡r✲Pr♦tt❡r✲❙❤✐r②❛❡✈ ❢♦r♠✉•❛✳ ✹✳✺✳ ❈❖▼P❆|■❙❖◆ ❚❖ ❖❚❍❊| ❊❳❚❊◆❙■❖◆❙ ❖❋ ❚❍❊ ■❚Ô ❋❖|▼❯▲❆ ✶✵✶ ❋✐rst✱ ✇❡ ♣r♦✈❡ t❤❛t t❤✐s ❤♦•❞s ✐♥❞❡❡❞ ✐♥ t❤❡ ✉♥✐❢♦r♠•② ❡••✐♣t✐❝ ❝❛s❡✱ ✐❡ ✐❢ t❤❡ ❢✉♥❝t✐♦♥ a ✐s ✉♥✐❢♦r♠•② ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡•♦✇✳ ■♥❞❡❡❞✱ ✐♥ ❬✸✺( ✐t ✐s ♣r♦✈❡❞ t❤❛t ✉♥❞❡r t❤✐s ❝♦♥❞✐t✐♦♥✱ ❛♥❞ ✐❢ t❤❡ ❝♦❡✣❝✐❡♥ts b ❛♥❞ σ ❛r❡ C ∞ b ✉♥✐❢♦r♠•② ✐♥ t✐♠❡✱ t❤❡♥ t❤❡ ♣r♦❝❡ss X t ❤❛s ❛ s♠♦♦t❤ ❞❡♥s✐t② p Xt (y) ✇❤✐❝❤ ✈❡r✐✜❡s✿ c -1 t -N/2 exp - c y -x 2 t ≤ p Xt (y) ≤ C -1 t -N/2 exp - C y -x 2 t ✭✹✳✶✵✺✮ ❍❡r❡✱ c ❛♥❞ C ❛r❡ t✇♦ ❝♦♥st❛♥ts✱ ❛♥❞ ✇❡ r❡❝❛•• t❤❛t X 0 = x ❛✳s✳ ▼♦r❡♦✈❡r✱ ❜② t❛❦✐♥❣ ❛ ❝•♦s❡ •♦♦❦ ❛t ❑♦❤❛ts✉✲❍✐❣❛✬s ♣r♦♦❢✱ ♦♥❡ ♥♦t✐❝❡s t❤❛t t❤❡ ♠♦st ✐rr❡❣✉•❛r ❞✐str✐❜✉t✐♦♥s ♣•❛②✐♥❣ ❛ r♦•❡ ❛r❡ ♦❢ t❤❡ t②♣❡ δ ′ y • X t ✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ♣r♦♦❢ ✐s st✐•• ✈❛•✐❞ ✐❢ b ❛♥❞ σ ❛r❡ s✉❝❤ t❤❛t t❤❡s❡ ❞✐str✐❜✉t✐♦♥s ❛r❡ ✇❡•• ❞❡✜♥❡❞✳ |❡❝❛•• t❤❛t t❤❡ ❉✐r❛❝ ♠❛ss δ y ✐s ✐♥ t❤❡ s♣❛❝❡ S 1+ǫ,-N ǫ 1+ǫ ❢♦r ❛♥② ǫ✳ |❡❝❛•• ❛•s♦ t❤❛t t❤❡ ✉♥✐❢♦r♠ ❡••✐♣t✐❝✐t② ❤②♣♦t❤❡s✐s ✐♠♣•✐❡s ✉♥✐❢♦r♠ ♥♦♥✲❞❡❣❡♥❡r❛❝② ♦❢ t❤❡ ♣r♦❝❡s X✳ ❚❤❡r❡❢♦r❡✱ ❜② ❢♦••♦✇✐♥❣ t❤❡ ♣r♦♦❢ ❜② ❑♦❤❛ts✉✲❍✐❣❛ ❛♥❞ ✉s✐♥❣ t❤❡♦r❡♠ ✸✳✷✳✵✳✶✸ ✇❡ ❡①t❡♥❞ ❤✐s r❡s✉•t t♦ t❤❡ ❢♦••♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✹✳✺✳✶✳✷ ✭❉❡♥s✐t② ❝♦♥tr♦• ✉♥❞❡r ✉♥✐❢♦r♠ ❡••✐♣t✐❝✐t②✮✳ ▲❡t X t ❜❡ t❤❡ str♦♥❣ s♦•✉t✐♦♥ ♦❢ t❤❡ ❙❉❊ dX t = b(t, X t )dt + σ(t, X t )dW t ✭✹✳✶✵✻✮ X 0 = x ✭✹✳✶✵✼✮ ✇❤❡r❡ b, σ ∈ C 0,2+ǫ ❛♥❞ a ✐s ✉♥✐❢♦r♠•② ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡•♦✇✳ ❚❤❡♥ X t ❛❞♠✐ts ❛ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❜♦✉♥❞❡❞ ❞❡♥s✐t② p Xt ❛t ❡❛❝❤ t > 0 ❛♥❞ ♦♥❡ ❤❛s✿ c -1 t -N/2 exp - c y -x 2 t ≤ p Xt (y) ≤ C -1 t -N/2 exp - C y -x 2 t ✭✹✳✶✵✽✮ ❢♦r s♦♠❡ ❝♦♥st❛♥ts c, C✳ |❡♠❛r❦ ✹✳✺✳✶✳✶✳ ❇❡❝❛✉s❡ ♦❢ t❤❡ ✉♥✐❢♦r♠ ❡••✐♣t✐❝✐t② ✇❡ ❞♦ ♥♦t ♥❡❡❞ t♦ ❛ss✉♠❡ t❤❛t t❤❡ ❝♦❡✜❝✐❡♥ts ❛r❡ ❍ö•❞❡r ✐♥ t✐♠❡ ❛s ✇❡ ❞✐❞ ✐♥ t❤❡♦r❡♠ ✹✳✹✳✸✳✶ ❛s t❤✐s ❤②♣♦t❤❡s✐s ✇❛s ✉s❡❞ ♦♥•② t♦ ♣r♦✈❡ t❤❡ ♥♦♥✲❞❡❣❡♥❡r❛❝②✳ ❆•s♦✱ ✐❢ ✇❡ s✉♣♣♦s❡ ♠♦r❡ s♣❛❝❡ r❡❣✉✲ •❛r✐t② ♦♥ t❤❡ ❝♦❡✣❝✐❡♥ts✱ t❤❡♥ t❤❡ X t ❤❛✈❡ ♠♦r❡ ▼❛••✐❛✈✐♥ r❡❣✉•❛r✐t② ❛❝❝♦r❞✐♥❣ t♦ t❤❡♦r❡♠ ✹✳✹✳✶✳✸✱ ❛♥❞ ❤❡♥❝❡ p Xt ❤❛s ♠♦r❡ r❡❣✉•❛r✐t② ❜❡❝❛✉s❡ ♦❢ t❤❡♦r❡♠ ✸✳✹✳✵✳✶✽ |❡♠❛r❦ ✹✳✺✳✶✳✷✳ ❯♥❞❡r t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡♦r❡♠ ✹✳✺✳✶✳✷ ❢♦r ❛•• t > 0 X t ❤❛s ❢✉•• s✉♣♣♦rt✳ ❲❡ ❣✐✈❡ ❛♥♦t❤❡r r❡s✉•t ♦❢ t❤❡ s❛♠❡ t②♣❡ ✳ ■♥ ❬✹✷(✱ s❡❝t✐♦♥s ■❱✳✶✶ ❛♥❞ ❢♦••♦✇✐♥❣✱ t❤❡ ❢♦••♦✇✐♥❣ r❡s✉•t ✐s ❣✐✈❡♥✿ ✶✵✷ ❈❍❆P❚❊| ✹✳ ❆ ❲❊❆❑ ■❚Ô ❋❖|▼❯▲❆ ❚❤❡♦r❡♠ ✹✳✺✳✶✳✸ ✭❋✉♥❞❛♠❡♥t❛• s♦•✉t✐♦♥ t♦ ❛ ♣❛r❛❜♦•✐❝ ❈❛✉❝❤② ♣r♦❜•❡♠✮✳ ❈♦♥✲ s✐❞❡r ❛ s❡❝♦♥❞ ♦r❞❡r ♣❛r❛❜♦•✐❝ ♦♣❡r❛t♦r P(t, x, ∂ t , ∂ x ) ♦♥ ]0, T [×R N ✳ ❙✉♣♣♦s❡ t❤❛t✱ ❢♦r s♦♠❡ 0 < α < 1 t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ P ❛r❡ •♦✐♥t•② α 2 ✲❍ö•❞❡r ✐♥ t✐♠❡ ❛♥❞ α✲❍ö•❞❡r ✐♥ s♣❛❝❡✳ ▲❡t 0 < τ < t❀ ξ ∈ R N ✳ ❚❤❡♥ ❢♦r τ ≤ t ≤ T t❤❡r❡ ❡①✐sts ❛ ❢✉♥❞❛♠❡♥t❛• s♦•✉t✐♦♥ Z t♦ t❤❡ ❡q✉❛t✐♦♥✿ P(t, x, ∂ t , ∂ x )Z τ,ξ (t, x) = δ τ ⊗ δ ξ ✭✹✳✶✵✾✮ ✇❤✐❝❤ ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ❜♦✉♥❞❡❞ ✐♥ x ❢♦r ❛♥② t✳ ▼♦r❡♦✈❡r✱ t❤❡r❡ ❡①✐s✐ts ❛ ♥✉♠❜❡r C > 0 s✉❝❤ t❤❛t Z ✈❡r✐✜❡s✿ |Z τ,ξ (t, x)| ≤ C • 1 (t -τ ) -N 2 • exp -C |x -ξ| 2 t -τ ✭✹✳✶✶✵✮ ❚❤❡ s❛♠❡ r❡❢❡r❡♥❝❡ ❛•s♦ ♣r♦✈✐❞❡s ❡st✐♠❛t❡s ❢♦r s♠❛•• ✭❢r❛❝t✐♦♥❛•✮ ♦r❞❡r ❞❡r✐✈❛✲ t✐✈❡s ♦❢ Z✳ ❚❤❡ ✉♥✐❝✐t② ♦❢ t❤❡ ❈❛✉❝❤② ♣r♦❜•❡♠ ✐s ❛❞r❡ss❡❞ t♦ s❡♣❛r❛t❡•② ✐♥ t❤❡ s❛♠❡ ❜♦♦❦✳ ◆♦✇✱ ❝♦♥s✐❞❡r t❤❡ ❙❉❊ ✭✹✳✶✵✻✱ ✹✳✶✵✼✮ ❛♥❞ s✉♣♣♦s❡ t❤❛t ✐ts ❝♦❡✣❝✐❡♥ts ❛r❡ ▲✐♣s❝❤✐t③ ✐♥ s♣❛❝❡ ❛♥❞ α✲❍ö•❞❡r ✐♥ t✐♠❡ ❢♦r s♦♠❡ α > 0✳ ❚❤❡♥ ♦♥ ♦♥❡ ❤❛♥❞ ♦✉r ❙❉❊ ❛❞♠✐ts ❛ ✉♥✐q✉❡ str♦♥❣ s♦•✉t✐♦♥✱ ❛♥❞ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞ t❤❡ ♣❛r❛❜♦•✐❝ ♦♣✲ ❡r❛t♦r ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❙❉❊ ✈❡r✐✜❡s t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡♦r❡♠ ✹✳✺✳✶✳✸✳ ❚❤❡r❡❢♦r❡ X t ❤❛s t♦ ❤❛✈❡ ❛ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❜♦✉♥❞❡❞ ❞❡♥s✐t② ❛t ❛♥② t > 0 ✇❤✐❝❤ ✈❡r✐✜❡s ❛ ♠❛•♦r✐③❛t✐♦♥ ♦❢ t❤❡ ❛❜♦✈❡ t②♣❡ ❛♥❞ ✇❡ ❤❛✈❡✿ ❚❤❡♦r❡♠ ✹✳✺✳✶✳✹ ✭❉❡♥s✐t② ❡①✐st❡♥❝❡ ❛♥❞ ❝♦♥tr♦• ✇✐t❤♦✉t ✉♥✐❢♦r♠ ❡••✐♣t✐❝✐t②✮✳ ▲❡t X t ❜❡ t❤❡ str♦♥❣ s♦•✉t✐♦♥ ♦❢ t❤❡ ❙❉❊ ✭✹✳✶✵✻✱ ✹✳✶✵✼✮ ✇❤❡r❡ b, σ ❛r❡ α✲❍ö•❞❡r ✐♥ t✐♠❡ ❢♦r s♦♠❡ α > 0 ❛♥❞ ▲✐♣s❝❤✐t③ ✐♥ s♣❛❝❡✳ ❚❤❡♥ X t ❛❞♠✐ts ❛ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❜♦✉♥❞❡❞ ❞❡♥s✐t② p Xt ❛t ❡❛❝❤ t > 0 ❛♥❞ ♦♥❡ ❤❛s✿ p Xt (y) ≤ C • 1 (t -τ ) -N 2 • exp -C |x -ξ| 2 t -τ ✭✹✳✶✶✶✮ ❢♦r s♦♠❡ ❝♦♥st❛♥t C > 0✳ ❲❡ ♥♦t❡ t❤❛t ❛•t❤♦✉❣❤ t❤❡ s❡❝♦♥❞ t❤❡♦r❡♠ s❡❡♠s ❡❛s✐❡r t♦ ❛♣♣•②✱ ✐t ❞♦❡s ♥♦t ❜♦✉♥❞ t❤❡ ❞❡♥s✐t② ❢r♦♠ ❜❡•♦✇✳ ❲❡ ♥♦✇ ♣r♦✈❡ t❤❡ ❢♦••♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✹✳✺✳✶✳✺✳ ❙✉♣♣♦s❡ t❤❛t T t ✐s ❛ ❇❱ ♣❛t❤ t❛❦✐♥❣ ✈❛•✉❡s ✐♥ S p,1 ✭♦r W p,1 ✮ ❢♦r s♦♠❡ p ≥ 2 ❛♥❞ t❤❛t X ✈❡r✐✜❡s t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ ❡✐t❤❡r t❤❡♦r❡♠ ✹✳✺✳✶✳✸ ♦r ✹✳✺✳✶✳✷✳ ❚❤❡♥ ✐t ♠❛❦❡s s❡♥s❡ t♦ •❡t ǫ t❡♥❞ t♦ 0 ✐♥ ❢♦r♠✉•❛ ✭✹✳✶✵✹✮ Pr♦♦❢✳ ❲❡ ♦♥•② ♥❡❡❞ t♦ ♣r♦✈❡ t❤❛t ❡✈❡r② ✐♥t❡❣r❛• ✐s ✇❡•• ❞❡✜♥❡❞ ✐♥ L p ❛s ǫ t❡♥❞s t♦ 0✳ ▼♦r❡ ♣r❡❝✐s❡•②✱ ✇❡ ♣r♦✈❡ t❤❡ L p ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❡✈❡r② ✐♥t❡❣r❛• ❡①❝❡♣t t❤❡ ♦♥❡ ✹✳✺✳ ❈❖▼P❆|■❙❖◆ ❚❖ ❖❚❍❊| ❊❳❚❊◆❙■❖◆❙ ❖❋ ❚❍❊ ■❚Ô ❋❖|▼❯▲❆ ✶✵✸ ✐♥✈♦•✈✐♥❣ ❞✐str✐❜✉t✐♦♥s ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡✳ ❚❤❡♥✱ ❜② ❡•❡♠❡♥t❛r② ❛•❣❡❜r❛✱ t❤✐s ✐♥t❡❣r❛• ❤❛s t♦ ❝♦♥✈❡r❣❡✱ t♦♦✳ ❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❡✈❡r② t❡r♠ r❡•✐❡s ♦♥ ✐❞❡❛s s✐♠✐•❛r t♦ t❤♦s❡ ✇❡ ❞❡✈❡•♦♣❡❞ ✐♥ t❤❡ ♣r♦♦❢ ♦❢ t❤❡♦r❡♠ ✹✳✶✳✶✳✶✳ ❋♦r ❡①❛♠♣•❡✱ t 0 b(u, X u )∂ i T (u, X u )du L p ≤ b ∞ t 0 ∂ i T (u, X u ) L p du ✭✹✳✶✶✷✮ ❛♥❞ ∂ i T (u, X u ) p L p = R N |∂ i T (u, x)| p p Xu (x)dx ✭✹✳✶✶✸✮ s♦ ✜♥❛••② t 0 b(u, X u )∂ i T (u, X u )du L p ≤ b ∞ t 0 p Xu ∞ 1/p ∂ i (u, •) L p du ✭✹✳✶✶✹✮ ◆♦✇ u → ∂ i (u, •) L p ✐s ❝♦♥t✐♥✉♦✉s ❜❡❝❛✉s❡ ♦❢ t❤❡ ❤②♣♦t❤❡s❡s ♦♥ T ❛♥❞ p Xu ∞ ✐s ❝♦♥tr♦•❡❞ ❜② t❤❡♦r❡♠ ✹✳✺✳✶✳✷✱ s♦ ✜♥❛••② t❤✐s •❛st ✐♥t❡❣r❛• ✐s ✜♥✐t❡✳ ❚❤❡r❡❢♦r❡ ✇❡ ♠❛② ❛♣♣•② t❤❡ ❞♦♠✐♥❛t❡❞ ❝♦♥✈❡r❣❡♥❝❡ t❤❡♦r❡♠ t♦ ♦❜t❛✐♥ ♦✉r ❝♦♥✈❡r❣❡♥❝❡✳ ❚❤❡ ❇r♦✇♥✐❛♥ t❡r♠ ✐s ❞❡❛•t ✇✐t❤ ✐♥ ❛ s✐♠✐•❛r ✇❛②✱ ✉s✐♥❣ t❤❡ ❇✉r❦❤♦•❞❡r ❉❛✈✐❡s •✉♥❞② t❤❡♦r❡♠✳ ❋♦r t❤❡ ❇❱ ✐♥t❡❣r❛•✱ ♥♦t✐❝❡ t❤❛t✿ E t 0 T (du, X u ) ≤ t 0 |T |(du, p Xu ) ✭✹✳✶✶✺✮ |❡♠❛r❦ ✹✳✺✳✶✳✸✳ ■♥ ❬✹✼( ▼♦r❡t ❛♥❞ ◆✉❛•❛rt ❣✐✈❡ ❛ s✐♠✐•❛r r❡s✉•t ❢♦r ❛ ❣❡♥❡r❛• s❡♠✐♠❛rt✐♥❣❛•❡✳ ❲❡ ♦♥•② t❤❡ s♣❡❝✐❛•✐③❛t✐♦♥ ♦❢ t❤❡✐r r❡s✉•t t♦ s♦•✉t✐♦♥s ♦❢ ❙❉❊s✱ ❛s ♦❜t❛✐♥✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ❛♥❞ t❤❡ r❡❣✉•❛r✐t② ♦❢ ❛ ❞❡♥s✐t② ❢♦r ❛ ❣❡♥❡r❛• s❡♠✐♠❛rt✐♥❣❛•❡ ✐s ❛ ♠♦r❡ ❞✐✣❝✉•t ♣r♦❜•❡♠ t❤❛♥ ❞♦✐♥❣ ✐t •✉st ❢♦r t❤❡ s♦•✉t✐♦♥ ♦❢ ❛♥ ❙❉❊✳ |❡♠❛r❦ ✹✳✺✳✶✳✹✳ ■♥ ❬✷✵( t❤❡ ♦r❞❡r 2 t❡r♠ ✐s ❡①♣r❡ss❡❞ ❛s ❛ q✉❛❞r❛❞t✐❝ ❝♦✈❛r✐❛t✐♦♥ r❛t❤❡r t❤❛♥ ❛ D ′ ✲✈❛•✉❡❞ ✐♥t❡❣r❛•✱ s♦ ✇❡ ❤❛✈❡ t❤❡ ✐❞❡♥t✐✜❝❛t✐♦♥✿ 1 2 i,j t 0 a ij (u, X u ) • (∂ ij T u ) • X u du = 1 2 [(Df )(X), X] t ✭✹✳✶✶✻✮ |❡♠❛r❦ ✹✳✺✳✶✳✺✳ ❲❡ ♥❡❡❞ t❤❛t p ≥ 2 ❜❡❝❛✉s❡ ✇❡ ❛♣♣•② t❤❡ ❇✉r❦❤♦•❞❡r ❉❛✈✐❡s •✉♥❞② t❤❡♦r❡♠ t♦ t❤❡ ❇r♦✇♥✐❛♥ ✐♥t❡❣r❛•✳ ❙❡❡ t❤❡ ♣r♦♦❢ ♦❢ t❤❡♦r❡♠ ✹✳✶✳✶✳✶ ❢♦r ❞❡t❛✐•s✳ ❲❡ ♠❛② ❡①t❡♥❞ t❤❡ ❛❜♦✈❡ r❡s✉•t ❛ •✐tt•❡ ❜✐t ✐♥ t❤❡ s♣❡❝✐✜❝ ❝❛s❡ ✇❤❡r❡ N = 1 ❛♥❞ t❤❡ ❝♦❡✣❝✐❡♥ts b ❛♥❞ σ ❛r❡ t✐♠❡ ❤♦♠♦❣❡♥❡♦✉s✳ ■♥❞❡❡❞ ✐♥ ❬✼( t❤❡ ❢♦••♦✇✐♥❣ r❡s✉•t ✐s ♣r♦✈❡❞✿ ✶✵✹ ❈❍❆P❚❊| ✹✳ ❆ ❲❊❆❑ ■❚Ô ❋❖|▼❯▲❆ ❚❤❡♦r❡♠ ✹✳✺✳✶✳✻✳ ▲❡t b ∈ C 2 b (R) ❛♥❞ σ ∈ C n∨2 (R) ❢♦r s♦♠❡ n ∈ N ❛♥❞ x ∈ R s✉❝❤ t❤❛t✿ σ(x) = • • • = σ (n-1) (x) = 0 ✭✹✳✶✶✼✮ ❛♥❞ b(x)σ (n) (x) = 0 ✭✹✳✶✶✽✮ ▲❡t X t ❜❡ t❤❡ s♦•✉t✐♦♥ ♦❢ t❤❡ ❙❉❊✿ X t = x + t 0 b(X s )ds + t 0 σ(X s )dW s ✭✹✳✶✶✾✮ ❚❤❡♥ ❛t ❛•• t > 0✱ X t ❛❞♠✐ts ❛ ❞❡♥s✐t② p Xt ∈ C 0 b (R) s✉❝❤ t❤❛t✿ p Xt (x) ≤ C t n+ 1 2 ✭✹✳✶✷✵✮ |❡♠❛r❦ ✹✳✺✳✶✳✻✳ ❚❤❡ ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ ✈♦•❛t✐•✐t② ❡①❛❝t•② ❡①♣r❡ss ❤②♣♦❡••✐♣t✐❝✐t② ❛s ✐♥ ❞✐♠❡♥s✐♦♥ ✶ t❤❡ ▲✐❡ ❛•❣❡❜r❛ ❣❡♥❡r❛t❡❞ ❜② t❤❡ ✈♦•❛t✐•✐t② ❢✉♥❝t✐♦♥ ❛♥❞ ✐ts ❞❡r✐✈❛✲ t✐✈❡s ✐s ♦❢ ❞✐♠❡♥s✐♦♥ ❛t ♠♦st ✶✳ ❚❤❡ ❝❛s❡ ✇❤❡r❡ n = 0 ❝♦rr❡s♣♦♥❞s t♦ ❡••✐♣t✐❝✐t②✳ ■❢ a = σ 2 ✐s ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡•♦✇✱ ♦♥❡ ❤❛s ✉♥✐❢r♦♠ ❡••✐♣t✐❝✐t②✱ ✇❤✐❝❤ ✐♠♣•✐❡s t❤❛t ♦♥❡ ♠❛② t❛❦❡ n = 0 ✐ t❤❡ ❛❜♦✈❡ ❢r❛♠❡✇♦r❦✳ ❋✐♥❛••②✱ t❤❡ ❞✐✛❡r❡♥t✐❛❜✐•✐t② ❝♦♥❞✐t✐♦♥s ♦♥ b ❛♥❞ σ ❛r❡ s•✐❣❤t•② •❡ss ❞❡♠❛♥❞✐♥❣ ✐♥ t❤✐s ❝❛s❡ t❤❛♥ ✐♥ t❤❡♦r❡♠ ✹✳✹✳✸✳✶✱ ❛♥❞ ❡①tr❛ r❡❣✉•❛r✐t② ♠❛② ❜❡ ♣r♦✈❡♥ ❢♦r t❤❡ ❞❡♥s✐t② ❜② ✉s✐♥❣ t❤❡♦r❡♠ ✸✳✹✳✵✳✶✽✳ ❋r♦♠ t❤✐s r❡s✉•t ❛♥❞ ♦✉r ✇❡❛❦ ■tô ❢♦r✉•❛ ✇❡ r❡❝♦✈❡r t❤❡ ❢♦••♦✇✐♥❣ r❡s✉•t ✇❤✐❝❤ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❬✼(✿ ❚❤❡♦r❡♠ ✹✳✺✳✶✳✼✳ ▲❡t σ, b ∈ C 2 b (R)✱ •❡t x ∈ R s✉❝❤ t❤❛t b(x)σ(x) = 0✱ ❛♥❞ •❡t X t ❜❡ t❤❡ s♦•✉t✐♦♥ ♦❢✿ X t = x + t 0 b(X s )ds + t 0 σ(X s )dW s ✭✹✳✶✷✶✮ ❚❤❡♥✱ ❢♦r ❛♥② ❇❱ ♣❛t❤ T t ✇✐t❤ ✈❛•✉❡s ✐♥ S p,1 ✭r❡s♣ W p,1 ✮ ♦♥❡ ♠❛② •❡t ǫ t❡♥❞ t♦ ③❡r♦ ✐♥ ❢♦r♠✉•❛ ✭✹✳✶✵✹✮✳ Pr♦♦❢✳ ❙✐♠✐•❛r•② t♦ ❛❜♦✈❡ ✇❡ st❛rt ✇✐t❤ ♦✉r ✇❡❛❦ ■tô ❢♦r♠✉•❛ ❛♥❞ ✇❡ ♦♥•② ♥❡❡❞ t♦ s❤♦✇ t❤❛t ❛•• t❤❡ ✐♥t❡❣r❛•s st❛rt✐♥❣ ❛t 0 ❞♦ ❡①✐st ✐♥ L p ✳ ❲❡ ♣r♦❝❡❡❞ ❛s ✐♥ t❤❡ ♣r♦♦❢ ♦❢ t❤❡♦r❡♠ ✹✳✺✳✶✳✺ ❛♥❞ ✇❡ ✇r✐t❡✿ t 0 b(X u )∂ i T (u, X u )du L p ≤ b ∞ t 0 p Xu ∞ ∂ i (u, •) L p du ✭✹✳✶✷✷✮ ■♥ t❤❡ ❛❜♦✈❡ ✐♥❡q✉❛t✐♦♥✱ t❤❡ ❞❡♥s✐t② ❡①✐sts ❛♥❞ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ ✐♥t❡❣r❛• ✐s ✜♥✐t❡ ❜❡❝❛✉s❡ ♦❢ t❤❡♦r❡♠ ✹✳✺✳✶✳✻✳ ❚❤❡ ♦t❤❡r t❡r♠s ❛r❡ ❞❡❛•t ✇✐t❤ s✐♠✐•❛r•②✳ |❡♠❛r❦ ✹✳✺✳✶✳✼✳ ❖♥❝❡ ❛❣❛✐♥ ✇❡ ❡①♣r❡ss❡❞ t❤❡ ❝r♦❝❤❡t ✐♥ t❤❡ ❋♦••♠❡r✲Pr♦tt❡r✲ ❙❤✐r②❛❡✈ ❢♦r♠✉•❛ ❛s ❛♥ ✐♥t❡❣r❛• ✐♥ D ′ ✳ n = 0 ✐s ♥❡❡❞❡❞ t♦ ♦❜t❛✐♥ t❤❡ t✲✐♥t❡❣r❛❜✐•✐t② ♦❢ p Xt ❀ t❤✐s ✐s ❛ s•✐❣❤t ✐♠♣r♦✈❡♠❡♥t ♦✈❡r t❤❡ ❝♦♥❞✐t✐♦♥ ♦❢ ✉♥✐❢♦r♠ ❡••✐♣t✐❝✐t② r❡q✉✐r❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠✳ ✹✳✺✳ ❈❖▼P❆|■❙❖◆ ❚❖ ❖❚❍❊| ❊❳❚❊◆❙■❖◆❙ ❖❋ ❚❍❊ ■❚Ô ❋❖|▼❯▲❆ ✶✵✺ ✹✳✺✳✷ ❆r♦✉♥❞ t❤❡ •♦❝❛• t✐♠❡ |❡s✉•ts ✐♥ ❞✐♠❡♥s✐♦♥ ✶ ▲❡t (X t ) ❜❡ ❛s ✐♥ t❤❡♦r❡♠ ✹✳✹✳✸✳✶✱ ✇✐t❤ N = 1✳ ■❢ L K t (X) ✐s t❤❡ •♦❝❛• t✐♠❡ ♦❢ t❤❡ ♣r♦❝❡ss X ❛t t✐♠❡ t ❛♥❞ ♣♦✐♥t K✱ t❤❡♥ t❤❡ ❚❛♥❛❦❛ ❢♦r♠✉•❛ r❡❛❞s✿ |X t -K| -|x -K| = t 0 sgn(X s -K)b(s, X s )ds + t 0 sgn(X s -K)σ(s, X s )dW s + 1 2 L K t (X) ✭✹✳✶✷✸✮ ▼♦r❡♦✈❡r t❤❡ ❢✉♥❝t✐♦♥ x → |x -K| ✐s ✐♥ S p,-α ❢♦r ❛♥② α > 1 -1 p ❛♥❞ t❤❡ ❢✉♥❝t✐♦♥ x → sgn(x -K) ✐s ✐♥ S p,-α ❢♦r ❛♥② α > -1 p ✳ ❲❡ t❤❡r❡❢♦r❡ ♠❛② ❛♣♣•② t❤❡♦r❡♠ ✹✳✹✳✸✳✶ ❛♥❞ ✇❡ ♦❜t❛✐♥✿ |X t -K| -|X ǫ -K| = t ǫ sgn(X s -K)b(s, X s )ds + t ǫ sgn(X s -K)σ(s, X s )dW s + 1 2 t ǫ σ(s, X s ) 2 δ K • X s ds ✭✹✳✶✷✹✮ ❆s ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✱ ✐t ✐s t❡♠♣t✐♥❣ t♦ •❡t ǫ t❡♥❞ t♦ 0 ✐♥ t❤❡ ❛❜♦✈❡ ❛♥❞ t♦ ♦❜t❛✐♥ t❤❡ ✐❞❡♥t✐✜❝❛t✐♦♥✿ L K t (X) = t 0 σ(s, X s ) 2 δ K • X s ds ✭✹✳✶✷✺✮ ✇❤✐❝❤ ✐s ❛ t✐♠❡ ❞✐s✐♥t❡❣r❛t✐♦♥ ♦❢ t❤❡ •♦❝❛• t✐♠❡ ✐♥ ❛ ♥❡❣❛t✐✈❡ ♦r❞❡r ❙♦❜♦•❡✈ s♣❛❝❡✱ ✐♥ t❤❡ s❡♥s❡ ♦❢ P❡tt✐s ✐♥t❡❣r❛•s✳ ❚❤❡ r❡s✉•ts ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥ ❞♦ ♥♦t ❛♣♣•② ❞✐r❡❝t•② ❛s t❤❡ ❛❜s♦•✉t❡ ✈❛•✉❡ ❛♥❞ t❤❡ s✐❣♥ ❛r❡ ♥♦t r❡❣✉•❛r ❡♥♦✉❣❤ ❢✉♥❝t✐♦♥s❀ ✇❡ ✇✐•• ❤♦✇❡✈❡r ♣r♦✈❡ t❤❛t s✐♠✐•❛r ✭❛♥❞ ♠♦r❡ ♣r❡❝✐s❡✮ r❡s✉•ts ❞♦ ❤♦•❞✳ ❲❡ st❛rt ✇✐t❤ t❤❡✿ ▲❡♠♠❛ ✹✳✺✳✷✳✶✳ ❈♦♥s✐❞❡r ❛ ♥♦♥✲❞❡❣❡♥❡r❛t❡ X ∈ D ∞ -,2 ❛♥❞ f ∈ C 1 b ✳ ❚❤❡♥ t❤❡ q✉❛♥t✐t✐❡s ❜❡•♦✇ ❛r❡ ✇❡•• ❞❡✜♥❡❞ ❛♥❞✿ f (X) • δ K • X = f (K) • δ K • X ✭✹✳✶✷✻✮ Pr♦♦❢✳ ❋✐rst✱ s✐♥❝❡ ✇❡ ✇♦r❦ ✐♥ ❞✐♠❡♥s✐♦♥ 1✱ ✇❡ ♥♦t❡ t❤❛t δ K ∈ S p,-(1-1 p )-ǫ ❢♦r ❛♥② 1 < p < ∞ ❛♥❞ ǫ > 0✳ ■♥ ♣❛rt✐❝✉•❛r✱ •❡t ✉s ❝❤♦♦s❡ ǫ s✉❝❤ t❤❛t 1 -1 p ) -ǫ < 1✳ ❚❤❡♥ ❜② t❤❡♦r❡♠ ✸✳✷✳✵✳✶✺ δ K • X ✐s ✇❡•• ❞❡✜♥❡❞ ✐♥ D p,-(1-1 p )-ǫ ✳ ❆•s♦ ❜② t❤❡ ❝❤❛✐♥ ✶✵✻ ❈❍❆P❚❊| ✹✳ ❆ ❲❊❆❑ ■❚Ô ❋❖|▼❯▲❆ r✉•❡ f (X) ∈ D p * ,1 ⊂ D p * ,(1-1 p )+ǫ ✱ s♦ ❜② ❍ö•❞❡r✬s t❤❡♦r❡♠ t❤❡ •❡❢t ❤❛♥❞ s✐❞❡ ✐s ✇❡•• ❞❡✜♥❡❞✱ ❛t •❡❛st ✐♥ D 1,-(1-1 p )-ǫ ✳ ❆•s♦✱ ❜② ❡q✉❛t✐♦♥ ✭✸✳✾✹✮ ❢♦r ❛♥② φ ∈ D✱ p X,φ ✐s ✇❡•• ❞❡✜♥❡❞ ❛♥❞ t❤❡r❡ ❝♦♠❡s✿ f (X) • δ K • X, φ = E [f (X)φ|X = K] p X,φ (X) = E [f (K)φ|X = K] p X,φ (X) = f (K) • δ K • X, φ ✭✹✳✶✷✼✮ s♦ ✇❡ ❛r❡ ❞♦♥❡✳ ◆♦✇ ✇❡ ✇✐•• ♣r♦✈❡ t❤❡ ❢♦••♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✹✳✺✳✷✳✶✳ ▲❡t X ❜❡ ❛s ✐♥ t❤❡♦r❡♠ ✹✳✺✳✶✳✷✳ ❚❤❡♥✱ ❢♦r ❛♥② K✿ L K t (X) = t 0 σ(s, K) 2 δ K • X s ds ✭✹✳✶✷✽✮ ❛♥❞ ❢♦r ❛♥② r > 2✱ α < 1 r ✿ L K t (X) ∈ D r,α ✭✹✳✶✷✾✮ Pr♦♦❢✳ ❋✐rst ✇❡ s❤❛•• ♣r♦✈❡ t❤❛t ✉♥❞❡r t❤❡s❡ ❝♦♥❞✐t✐♦♥s ✐t ✐s •❡❣✐t✐♠❛t❡ t♦ •❡t ǫ t❡♥❞ t♦ ✵ ✐♥ ❡q✉❛t✐♦♥ ✭✹✳✶✷✹✮✳ ❇② t❤❡♦r❡♠ ✹✳✹✳✶✳✸✱ ❢♦r ❛♥② t > 0✱ X t ✐s ♥♦♥✲❞❡❣❡♥❡r❛t❡ ❛♥❞ X t ∈ D ∞ -,2 ✳ ❆•s♦✱ X t ❛❞♠✐ts ❛ ❞❡♥s✐t② p Xt ✇❤✐❝❤ ✐s ❛s ✐♥ t❤❡♦r❡♠ ✹✳✺✳✶✳✷✳ ■♥ ♣❛rt✐❝✉•❛r✱ t❤✐s ❞❡♥s✐t② ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ❜♦✉♥❞❡❞✳ ❲❡ t❤❡r❡❢♦r❡ ♠❛② ❛♣♣•② •❡♠♠❛ ✸✳✷✳✵✳✻ ❢♦r γ = 1 ❛♥❞ ✇❡ ♦❜t❛✐♥ t❤❛t 1 Xt>K ∈ D p,α ✱ ❛♥❞ ❢♦r s♦♠❡ ρ ✇❤✐❝❤ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t ❛♥❞ s♦♠❡ ❝♦♥st❛♥t C(p, α)✱ α < 1 r ✿ 1 Xt>K Dp,α ≤ C (1 + ∇X t L ρ ) p Xt ∞ ✭✹✳✶✸✵✮ ◆♦✇ s✐♥❝❡ p ≥ 2 ❜② t❤❡ ❇✉r❦❤♦•❞❡r ❉❛✈✐❡s •✉♥❞② t❤❡♦r❡♠ ❛♥❞ ❍ö•❞❡r✬s ✐♥❡q✉❛•✐t②✿ t 0 σ(s, X s )sgn(X s -K)dW s Dp,α ≤ C t 0 σ(s, X s ) Dq,α sgn(X s -K) Dr,α ✭✹✳✶✸✶✮ ✇✐t❤ 1 p = 1 q + 1 r ✳ ❈•❡❛r•②✿ σ(s, X s ) Dq,α ≤ σ(s, X s ) D q,1 ≤ σ ∞ X s D q,1 ✭✹✳✶✸✷✮ ❆•s♦✱ ✇r✐t✐♥❣ t❤❡ s✐❣♥ ❛s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t✇♦ ✐♥❞✐❝❛t♦rs✿ sgn(X s -K) Dr,α ≤ C (1 + ∇X s L ρ ) p Xs ∞ ✭✹✳✶✸✸✮ ✹✳✺✳ ❈❖▼P❆|■❙❖◆ ❚❖ ❖❚❍❊| ❊❳❚❊◆❙■❖◆❙ ❖❋ ❚❍❊ ■❚Ô ❋❖|▼❯▲❆ ✶✵✼ ◆♦✇ ♣✉tt✐♥❣ ❡✈❡r②t❤✐♥❣ t♦❣❡t❤❡r ❛♥❞ ❝♦♥tr♦••✐♥❣ t❤❡ ❞✐✛❡r❡♥t t❡r♠s t❤r♦✉❣❤ t❤❡✲ ♦r❡♠s ✹✳✹✳✶✳✸ ❛♥❞ ✹✳✺✳✶✳✷ t❤❡ ✐♥t❡❣r❛❜✐•✐t② ✐s ❝•❡❛r✳ ❚❤❡ ♦t❤❡r t❡r♠s ❛r❡ ❞❡❛•t ✇✐t❤ s✐♠✐•❛r•② s♦ ✇❡ ❤❛✈❡ ♦❜t❛✐♥❡❞✿ L K t (X) = t 0 σ(s, X s ) 2 δ K • X s ds ∈ D r,α ✭✹✳✶✸✹✮ ◆♦✇✱ ♦♥ ♦♥❡ ❤❛♥❞ t❤❡ ♣r❡✈✐♦✉s •❡♠♠❛ ❛♣♣•✐❡s✱ ❛♥❞ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞ ♦♥❡ ❝❛♥ t❛❦❡ r ❛s ❝•♦s❡ t♦ ✷ ❛s ♦♥❡ ❝❤♦♦s❡s✱ s♦ ✇❡ ❛r❡ ❞♦♥❡✳ |❡♠❛r❦ ✹✳✺✳✷✳✶✳ ❆ ♠♦r❡ ❣❡♥❡r❛• r❡s✉•t ♦❢ t❤❛t t②♣❡ ✇❛s ❣✐✈❡♥ ✐♥ ❬✸(❀ ✇❡ s✐♠♣•② ✇❛♥t t♦ s❤♦✇ ❤♦✇ t❤❡ ✐❞❡❛s ✐♥ t❤❡♦r❡♠ ✹✳✶✳✶✳✶ r❡•❛t❡ t♦ t❤❡ ❚❛♥❛❦❛ ❢♦r♠✉•❛✳ ▲❡t ✉s ❡s♣❡❝✐❛••② ♥♦t❡ ❤♦✇ t❤❡ t❡r♠✿ R L x t (X)T ′′ (dx) ✭✹✳✶✸✺✮ ✇❤✐❝❤ ❛♣♣❡❛rs ✐♥ t❤❡ ■tô✲❚❛♥❛❦❛ ❢♦r♠✉•❛ ✐s r❡•❛t❡❞ t♦ t❤❡ ❢♦••♦✇✐♥❣ t❡r♠ ❢r♦♠ ♦✉r ■tô ❢♦r♠✉•❛✿ t 0 σ(s, X s ) 2 • T ′′ • X s ds ✭✹✳✶✸✻✮ ❚❤❡s❡ t❡r♠s ❝♦✐♥❝✐❞❡ ✇❤❡♥ t❤❡② ❛r❡ ❜♦t❤ ❞❡✜♥❡❞ ❜✉t t❤❡ s❡❝♦♥❞ ♦♥❡ ♠❛② ❜❡ ❞❡✜♥❡❞ ❢♦r ♠♦r❡ ❣❡♥❡r❛• ❞✐str✐❜✉t✐♦♥s t❤❛♥ •✉st |❛❞♦♥ ♠❡❛s✉r❡s ✐❢ X ✐s r❡❣✉•❛r ❡♥♦✉❣❤✳ ❲❡ ♠❛② ❛•s♦ ❣✐✈❡ ❛♥♦t❤❡r r❡s✉•t r❡•❛t❡❞ t♦ t❤❡ ♦♥❡ ✐♥ ❬✸(✳ ▼♦r❡ ♣r❡❝✐s❡•②✱ ✇❡ ✇✐•• ♣r♦✈❡✿ ❚❤❡♦r❡♠ ✹✳✺✳✷✳✷✳ ▲❡t X ❜❡ ❛s ✐♥ t❤❡♦r❡♠ ✹✳✺✳✶✳✷✳ ▲❡t p > 2 ❛♥❞ α < 1 p ✳ ❚❤❡♥✱ ❢♦r ❛♥② t✱ K ∈ R → L K t (X) ∈ D p,α ✐s ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥✳ Pr♦♦❢✳ ❈♦♥s✐❞❡r K 1 < K 2 ∈ R✳ ❖♥❡ s❡❡s t❤❛t 1 ]K 1 ,K 2 [ ∈ W p,α ❢♦r ❛♥② α < 1 p ✳ ■♥❞❡❡❞✱ 1 ]K 1 ,K 2 [ ∈ L p ❛♥❞ ∇1 ]K 1 ,K 2 [ = δ K 1 -δ K 2 ∈ W p,α-1 ✳ ❚❤❡r❡❢♦r❡ ❜② •❡♠♠❛ ✸✳✷✳✵✳✺✱ ❢♦r s♦♠❡ q✿ 1 ]K 1 ,K 2 [ (X t ) Dp,α ≤ C p Xt ∞ (1 + ∇X L q ) 1 ]K 1 ,K 2 [ W p,α ≤ C p Xt ∞ (1 + ∇X L q ) • |K 1 -K 2 | + (Id -∆) -1-α 2 (δ K 2 -δ K 1 ) L p ✭✹✳✶✸✼✮ ◆♦✇✱ ❢♦r ❛♥② β > 1 -1 p ✱ K ∈ R → (Id -∆) -β δ K ∈ L p ✐s ❝♦♥t✐♥✉♦✉s✱ s♦ ✇❡ ♠❛② ❝♦♥tr♦• t❤❡ |❍❙✳ ❲❡ ♦❜t❛✐♥ ♦✉r r❡s✉•t ❜② ✇r✐t✐♥❣ L K 1 t (X) -L K 1 t (X) ✇✐t❤ t❤❡ ❚❛♥❛❦❛ ❢♦r♠✉•❛✱ ♠✐♠✐❝❦✐♥❣ t❤❡ ♣r♦♦❢ ♦❢ t❤❡♦r❡♠ ✹✳✺✳✷✳✶ ❛♥❞ ✉s✐♥❣ t❤❡ ❛❜♦✈❡ ❡st✐♠❛t❡✳ ✶✵✽ ❈❍❆P❚❊| ✹✳ ❆ ❲❊❆❑ ■❚Ô ❋❖|▼❯▲❆ ❆❝t✉❛••②✱ t❤❡ ♣r♦♦❢ ❛❜♦✈❡ ❣✐✈❡s ❛♥ ❡✈❡♥ ♠♦r❡ ♣r❡❝✐s❡ r❡s✉•t✱ ✇❤✐❝❤ ♠❡❛♥s t❤❛t ▼❛••✐❛✈✐♥ r❡❣✉•❛r✐t② ♠❛② ❜❡ tr❛❞❡❞ ♦✛ ❢♦r tr❛•❡❝t♦r② r❡❣✉•❛r✐t②✿ ❚❤❡♦r❡♠ ✹✳✺✳✷✳✸✳ ▲❡t X ❜❡ ❛s ✐♥ t❤❡♦r❡♠ ✹✳✺✳✶✳✷✳ ▲❡t p > 2✱ α < 1 p ❛♥❞ γ < α✳ ❚❤❡♥✱ ❢♦r ❛♥② t✱ K ∈ R → L K t (X) ∈ D p,α-γ ✐s ❛ γ✲❍ö•❞❡r ❢✉♥❝t✐♦♥✳ ❋r♦♠ t❤✐s✱ ✇❡ ♠❛② r❡❝♦✈❡r ❛♥ ❛•r❡❛❞② ✇❡••✲❦♥♦✇♥ r❡s✉•t✿ ❈♦r♦••❛r② ✹✳✺✳✷✳✶✳ ▲❡t X ❜❡ ❛s ✐♥ t❤❡♦r❡♠ ✹✳✺✳✶✳✷✳ ❚❤❡♥✱ ❢♦r ❛♥② t✱ t❤❡ ✭s♣❛❝❡✮ tr❛•❡❝t♦r✐❡s ♦❢ L • t (X) ❛r❡ ❛•♠♦st s✉r❡•② γ✲❍ö•❞❡r ❢♦r ❛♥② γ < 1 2 ✳ ❲❡ ❛r❡❛❞② ❦♥♦✇ t❤❛t t❤✐s ❝♦r♦••❛r② ✐s ♦♣t✐♠❛• ✭❝❢✱ ❢♦r ❡①❛♠♣•❡✱ t❤❡ ❝❛s❡ ♦❢ t❤❡ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✮✱ t❤❡r❡❢♦r❡ t❤❡r❡ ✐s •✐tt•❡ ❤♦♣❡ t♦ ❡①t❡♥❞ t❤❡♦r❡♠ ✹✳✺✳✷✳✶ t♦ t❤❡ ❝❛s❡ p < 2✳ ❆❝t✉❛••②✱ ✐♥ ❬✸( ✐t ✐s s❤♦✇♥ t❤❛t t❤❡ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ♣r♦✈✐❞❡s ❛ ❝♦✉♥t❡r ❡①❛♠♣•❡✳ ▲❡t ✉s ❛•s♦ ♥♦t❡ t❤❛t t❤❡ ❝❛s❡ p = 2 ❤❛s ❜❡❡♥ ❞❡❛•t ✇✐t❤ ❜② ♦t❤❡r ♠❡t❤♦❞s ✭❝❤❛♦s ❡①♣❛♥s✐♦♥✮ ❢♦r t❤❡ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ♦♥•②✱ s❡❡ ❬✺✶( ❛♥❞ ❬✺✵(✳ ❍❡r❡ ✐s ❛♥♦t❤❡r r❡s✉•t ♦❢ t❤❡ s❛♠❡ t②♣❡✿ ❚❤❡♦r❡♠ ✹✳✺✳✷✳✹✳ ▲❡t X ❜❡ ❛s ✐♥ t❤❡♦r❡♠ ✹✳✺✳✶✳✷✳ ❚❤❡♥ ∂ ∂K L K t (X) ✐s ✇❡•• ❞❡✜♥❡❞ ✐♥ D p,-(1-1 p )-ǫ ❢♦r ❛♥② ǫ > 0✱ ❛♥❞✿ ∂ ∂K L K t (X) = 2 t 0 σ(s, K) ∂ ∂K σ(s, K)δ K • X s ds - t 0 σ(s, K) 2 δ ′ K • X s ds ✭✹✳✶✸✽✮ ❋✐♥❛••②✱ K ∈ R → L K t (X) ∈ D p,-(1-1 p )-ǫ ✐s ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥✳ ■t ✐s ✐♥t❡r❡st✐♥❣ t♦ ❝♦♠♣❛r❡ t❤✐s ✇✐t❤ t❤❡ ❢♦••♦✇✐♥❣ r❡s✉•t ❢r♦♠ ❬✶✸(✿ ✐❢ W ✐s ❛ ❜r♦✇♥✐❛♥ ♠♦t✐♦♥✱ t❤❡♥ ❢♦r ❡✈❡r② t✱ x → L x t (W ) ✐s ❛ s❡♠✐♠❛rt✐♥❣❛•❡ ✐♥ ✐ts ♦✇♥ ✜•tr❛t✐♦♥ ❛♥❞ t❤❡r❡ ✐s t❤❡✿ ❚❤❡♦r❡♠ ✹✳✺✳✷✳✺ ✭❇♦✉•❡❛✉✲(♦r ❢♦r♠✉•❛✮✳ ▲❡t f ❜❡ ❛ ❞✐✛❡r❡♥t✐❛❜•❡ ❢✉♥❝t✐♦♥ ♦♥ R✱ ✇✐t❤ ❛ •♦❝❛••② ❜♦✉♥❞❡❞ ❞❡r✐✈❛t✐✈❡ ❛♥❞ •❡t W ❜❡ ❛ 1✲❞✐♠❡♥s✐♦♥♥❛• ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳ ❚❤❡♥✿ f (W t ) = f (0) + t 0 f ′ (W s )dW s - 1 2 • R f ′ (x)d x L x t (W ) ✭✹✳✶✸✾✮ ■t ✐s t❤❡♥ ♣♦ss✐❜•❡ t♦ s❤♦✇ t❤❛t t❤❡ t❡r♠✿ R f ′ (x)d x L x t (X) ✭✹✳✶✹✵✮ ❝♦✐♥❝✐❞❡s ✇✐t❤ ♦♥❡ ♦❢ t❤❡ t②♣❡✿ R f (x) ∂ ∂x L x t (W )dx ✭✹✳✶✹✶✮ ✇❤❡r❡ t❤❡ •❛tt❡r ✐s ❞❡✜♥❡❞ ❛s ❛ P❡tt✐s ✐♥t❡❣r❛• ✐♥ s♦♠❡ ♥❡❣❛t✐✈❡ ♦r❞❡r •r♦ss✲❙♦❜♦•❡✈ s♣❛❝❡✳ ❆•s♦✱ s✉❝❤ ❛ q✉❛♥t✐t② ❝♦✉•❞ ❜❡ ❞❡✜♥❡❞ ❢♦r ❛ ♠♦r❡ ❣❡♥❡r❛• st♦❝❤❛st✐❝ ♣r♦❝❡ss t❤❛♥ t❤❡ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✱ ✐❡ ♦♥❡ ✇❤✐❝❤ ✈❡r✐✜❡s t❤❡ ❤②♣♦t❤❡s❡s ✐♥ t❤❡♦r❡♠ ✹✳✺✳✶✳✷✳ ✹✳✺✳ ❈❖▼P❆|■❙❖◆ ❚❖ ❖❚❍❊| ❊❳❚❊◆❙■❖◆❙ ❖❋ ❚❍❊ ■❚Ô ❋❖|▼❯▲❆ ✶✵✾ |❡s✉•ts ✐♥ ♠✉•t✐♣•❡ ❞✐♠❡♥s✐♦♥s ❙♦♠❡ ❛✉t❤♦rs ❤❛✈❡ s❤♦✇♥ ❤♦✇ t♦ ❞❡✜♥❡ t❤❡ •♦❝❛• t✐♠❡ ♦❢ ❛ ♠✉•t✐✲❞✐♠❡♥s✐♦♥♥❛• ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ❛s ❛ ❞✐str✐❜✉t✐♦♥ ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡✱ s❡❡ ❢♦r ❡①❛♠♣•❡ ❬✷✼( ❛♥❞ t❤❡ r❡❢❡r❡♥❝❡s t❤❡r❡✐♥✳ ■♥ t❤✐s ♣❛r❛❣r❛♣❤ ✇❡ ✇✐•• s❤♦✇ ❤♦✇ t❤❡ ♠❡t❤♦❞ ✇❡ ❞❡✈❡•♦♣♣❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s ♣❛r❛❣r❛♣❤ ❛••♦✇s ✉s t♦ ❡①t❡♥❞ t❤✐s ✐❞❡❛ t♦ ♠♦r❡ ❣❡♥❡r❛• ❞✐✛✉s✐♦♥s✳ ❲❡ st❛rt ❜❡ r❡❝❛••✐♥❣ t❤❛t✱ ✐❢ w n ❞❡♥♦t❡s t❤❡ ✈♦•✉♠❡ ♦❢ t❤❡ ♥✲❞✐♠❡♥s✐♦♥♥❛• ✉♥✐t ❜❛••✱ t❤❡ ❢✉♥❞❛♠❡♥t❛• s♦•✉t✐♦♥ E n ♦❢ t❤❡ ▲❛♣•❛❝❡ ♦♣❡r❛t♦r ✐s ❞❡✜♥❡❞ ❛s ❢♦••♦✇s✿ E 2 (x) = 1 w 2 log(|x|) = 1 2π log(|x|) ✭✹✳✶✹✷✮ ❛♥❞ ✐❢ N > 2✿ E N (x) = - 1 N (N -2)w n • |x| 2-N ✭✹✳✶✹✸✮ ❇② ❞❡s✐❣♥✱ ∆E n = δ 0 ✳ ❆•s♦✱ ✐t ❝❛♥ ❜❡ ❝❤❡❝❦❡❞ t❤❛t E n ∈ S p,-N (1-1 p )-ǫ ❢♦r ❛♥② ǫ > 0✳ ◆♦✇ •❡t X ❜❡ t❤❡ ✉♥✐q✉❡ str♦♥❣ s♦•✉t✐♦♥ ♦❢ t❤❡ ❙❉❊✿ dX (i) t = b i (s, X s )ds + σ i (s, X s )dW (i) s ✭✹✳✶✹✹✮ X 0 = x ✭✹✳✶✹✺✮ ❙✉♣♣♦s❡ t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡♦r❡♠ ✹✳✹✳✸✳✶ ❤♦•❞ ❢♦r s♦♠❡ s > N (1 -1 p ) ✭✐❢ t❤❡ ❝♦❡✣❝✐❡♥ts ❛r❡ ❍ö•❞❡r ✐♥ t✐♠❡ ❛♥❞ 2 + ǫ ✐♥ s♣❛❝❡ t❤✐s ✐s ✈❡r✐✜❡❞ ❢♦r s♠❛•• ❡♥♦✉❣❤ p✮✳ ❚❤❡♥ ✇❡ ♠❛② ❛♣♣•② t❤❡ ✇❡❛❦ ■tô ❢♦r♠✉•❛ ❛♥❞ ✇❡ ♦❜t❛✐♥✿ E N (X t -K) -E N (X ǫ -K) = t ǫ b i (s, X s )∂ i E N (X s -K)ds + t ǫ sgn(X s -K)σ(s, X s )dW s + 1 2 t ǫ |σ(s, X s )| 2 δ K • X s ds ✭✹✳✶✹✻✮ ◆♦✇ ✇❡ ♠❛② ❛♣♣•② t❤❡ s❛♠❡ ♠❡t❤♦❞ ❛s ✐♥ t❤❡ ❞✐♠❡♥s✐♦♥ 1 ❝❛s❡ ✐♥ ♦r❞❡r t♦ ♣r♦✈❡ t❤❛t ✉♥❞❡r t❤❡s❡ ❝♦♥❞✐t✐♦♥s✿ ❚❤❡♦r❡♠ ✹✳✺✳✷✳✻✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡♦r❡♠ ✹✳✹✳✸✳✶ ❤♦•❞ ❢♦r s♦♠❡ s > N (1 -1 p ) ❛♥❞ p > 2✳ ❚❤❡♥ ❢♦r ❛♥② ǫ > 0 t❤❡ •♦❝❛• t✐♠❡ ♦❢ X ✐s ✇❡•• ❞❡✜♥❡❞ ✐♥ D p,1-N (1-1 p )-ǫ ❛s t❤❡ ❢♦••♦✇✐♥❣ ❇♦❝❤♥❡r ✐♥t❡❣r❛•✿ L K t (X) := t 0 |σ(s, K)| 2 δ K • X s ds ✭✹✳✶✹✼✮ ✶✶✵ ❈❍❆P❚❊| ✹✳ ❆ ❲❊❆❑ ■❚Ô ❋❖|▼❯▲❆ ▲❡t ✉s r❡♠❛r❦ t❤❛t s✐♥❝❡ t❤❡r❡ ❤❛s t♦ ❜❡ p > 2 ✐♥ t❤❡ t❤❡♦r❡♠ ❛❜♦✈❡✱ ✉♥•❡ss N = 1 t❤❡r❡ ❛•✇❛②s ✐s✿ 1 -N • 1 - 1 p < 0 ✭✹✳✶✹✽✮ ❚❤❡r❡❢♦r❡✱ t❤❡ •♦❝❛• t✐♠❡ ♦❢ ❛ ♠✉•t✐✲❞✐♠❡♥s✐♦♥♥❛• ♣r♦❝❡ss ✐s ♥♦t ❛ ♣r♦♣❡r r❛♥❞♦♠ ✈❛r✐❛❜•❡ ✐♥ ❣❡♥❡r❛•✳ ❚❤✐s ✐s t❤❡ ❝❛s❡ ❡✈❡♥ ❢♦r ✈❡r② r❡❣✉•❛r ♣r♦❝❡ss❡s✿ ❢♦r ❡①❛♠♣•❡ ✐t ✐s ✇❡•• ❦♥♦✇♥ t❤❛t t❤❡ ❜r♦✇♥✐❛♥ •♦❝❛• t✐♠❡ ✐s ♥♦t ❞❡✜♥❡❞ ❛s ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✐♥ ❞✐♠❡♥s✐♦♥ N > 2✳ ❲❡ ✜♥✐s❤ t❤✐s s❡❝t✐♦♥ ✇✐t❤ t❤❡ ❢♦••♦✇✐♥❣ r❡s✉•t✱ ✇❤✐❝❤ ✐s ♦❜t❛✐♥❡❞ ❜② ❛♣♣•②✐♥❣ t❤❡ ■tô ❢♦r♠✉•❛ t♦ K -δ E n r❛t❤❡r t❤❛♥ E n ✿ ❚❤❡♦r❡♠ ✹✳✺✳✷✳✼✳ ▲❡t X ❜❡ ❛s ✐♥ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠✱ p > 2 ❛♥❞ t > 0✳ α < 1 -N • 1 - 1 p < 0 ✭✹✳✶✹✾✮ ❚❤❡♥✱ ❢♦r ❛♥② γ > 0✱ t❤❡ ❢✉♥❝t✐♦♥✿ K ∈ R N → L t (K) ∈ D p,α-γ R N ✭✹✳✶✺✵✮ ✐s γ✲❤♦•❞❡r✐❛♥✳ ❚❤✐s ❛••♦✇s ✉s t♦ ✉♥❞❡rst❛♥❞ ♦✉r ✇❡❛❦ ■tô ❢♦r♠✉•❛ ❛s ❛ ♠✉•t✐✲❞✐♠❡♥s✐♦♥♥❛•✱ ✇❡❛❦ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❇♦✉•❡❛✉✲(♦r ❢♦r♠✉•❛✳ ❈❤❛♣t❡r ✺ ❆♣♣•✐❝❛t✐♦♥s t♦ s♦♠❡ ♣r♦❜•❡♠s ✐♥ ❛♥❛•②s✐s ✺✳✶ ❱❛r✐❛t✐♦♥❛• ✐♥❡q✉❛t✐♦♥s ✐♥ ❛ ♠❛r❦♦✈✐❛♥ ❢r❛♠❡✲ ✇♦r❦ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ st✐•• ❝♦♥s✐❞❡r X u t (x) t❤❡ ✉♥✐q✉❡ str♦♥❣ s♦•✉t✐♦♥ ♦♥ [u, T ] ♦❢ t❤❡ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛• ❡q✉❛t✐♦♥ ✭✹✳✼✼✮ s✉❝❤ t❤❛t X u = x✳ ❍❡r❡✱ ✇❡ s✉♣♣♦s❡ t❤❛t t❤❡ ❛ss✉♠♣t✐♦♥s ♦❢ t❤❡♦r❡♠ ✹✳✹✳✸✳✶ ❤♦•❞ ❢♦r s♦♠❡ s > 2✱ n ≤ k -2 ❛♥❞ β > 0✳ ❲❡ ✇✐•• ♥♦t❡ s = 1 + δ✳ ❲❡ ♥♦t❡ A t ❢♦r t❤❡ ❣❡♥❡r❛t♦r ♦❢ X✿ A t = a i,j (t, x)∂ i ∂ j + b i (t, x)∂ i ✭✺✳✶✮ ❛♥❞✿ K t = ∂ t + A t ✭✺✳✷✮ ❲❡ ♥♦t❡ Z t,T t❤❡ s❡t ♦❢ ❛•• st♦♣♣✐♥❣ t✐♠❡s τ s✉❝❤ t❤❛t t ≤ τ ≤ T ❛✳s✳ ◆♦✇ ✇❡ ❣✐✈❡ ♦✉rs❡•✈❡s s♦♠❡ ❝♦♥t✐♥✉♦✉s✱ ❜♦✉♥❞❡❞ ❛♥❞ ♣♦s✐t✐✈❡ ❢✉♥❝t✐♦♥s f ❛♥❞ g ❛♥❞ ❛ C (0,s) ✱ ♣♦s✐t✐✈❡ ❢✉♥❝t✐♦♥ r ❛♥❞ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❢♦••♦✇✐♥❣ ♣r♦❜•❡♠✿ K t u -ru ≤ -g ✭✺✳✸✮ u ≥ f ✭✺✳✹✮ (K t u -ru)(f -u) = 0 ✭✺✳✺✮ u(T, •) = f ✭✺✳✻✮ ❲❡ ✇✐•• ♥♦t❡ (A) ❢♦r t❤❡ s❡t ♦❢ ❛ss✉♠♣t✐♦♥s ✇❡ ♠❛❞❡ ♦♥ X✱ f ✱ g✱ ❡t❝✳ ❛♥❞ (V) ❢♦r t❤❡ ✈❛r✐❛t✐♦♥❛• ♣r♦❜•❡♠ ✇❡ ❝♦♥s✐❞❡r✳ ◆♦✇ ✇❡ ♣r♦✈❡✿ ✶✶✶ ✶✶✷ ❈❍❆P❚❊| ✺✳ ❆PP▲■❈❆❚■❖◆❙ ❚❖ ❙❖▼❊ P|❖❇▲❊▼❙ ■◆ ❆◆❆▲(❙■❙ ❚❤❡♦r❡♠ ✺✳✶✳✵✳✽✳ ❯♥❞❡r (A)✱ •❡t u ∈ BV (C 0 ) ❜❡ ❛ s♦•✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜•❡♠ (V)✳ ❚❤❡♥✿ u(t, x) = sup τ ∈Z t,T E f (X t τ (x)) exp - τ t r(u, X t u (x))du + τ t g(s, X t s (x)) exp - s t r(u, X t u (x))du ds ✭✺✳✼✮ Pr♦♦❢✳ ❲❡ ✇✐•• ♦♥•② ♣r♦✈❡ t❤❡ ❝❛s❡ ✇❤❡r❡ t = 0✱ t❤❡ ♦t❤❡r ❝❛s❡s ❜❡✐♥❣ s✐♠✐•❛r✳ ❲❡ ♥♦t❡ X t = X 0 t (x)✳ ❲❡ ❛•s♦ ✐♥tr♦❞✉❝❡✿ l t = exp - t 0 r(u, X u )du ✭✺✳✽✮ ❲❡ ♥♦t❡ t❤❛t ✇❡ ❤❛✈❡ t❤❡ ✐♥•❡❝t✐♦♥ C 0 ֒→ S p,-N/p-ǫ ❢♦r ❛♥② ǫ > 0✳ ❚❤❡r❡❢♦r❡✱ ❢♦r p ❜✐❣ ❡♥♦✉❣❤ ❛♥❞ ǫ s♠❛•• ❡♥♦✉❣❤ s♦ t❤❛t N/p + ǫ < s -1✱ ✇❡ ♠❛② ❛♣♣•② t❤❡ ❣❡♥❡r❛•✐③❡❞ ■tô ❢♦r♠✉•❛ ✐♥ t❤❡♦r❡♠ ✹✳✹✳✸✳✷ ❛♥❞ ❢♦r ❛♥② ǫ > 0 ✇❡ ♦❜t❛✐♥ t❤❡ ❢♦••♦✇✐♥❣✿ l t u(t, X t ) -l ǫ u(ǫ, X ǫ ) - t ǫ l s ((A s u)(s, X s ) -(ru)(s, X s )) ds - t ǫ l s u(ds, X s ) = M t ✭✺✳✾✮ ❢♦r s♦♠❡ ✇❡❛❦ ♠❛rt✐♥❣❛•❡ M ✳ ❆•s♦ K t u-ru+g ✐s ❛ ♥❡❣❛t✐✈❡ ❞✐str✐❜✉t✐♦♥✱ t❤❡r❡❢♦r❡ t❤❡ ❢♦••♦✇✐♥❣ ✐s ❛ ♣♦s✐t✐✈❡ ❞✐str✐❜✉t✐♦♥ ♦♥ t❤❡ ❲✐❡♥❡r s♣❛❝❡✿ D t := M t -l t u(t, x t ) + l ǫ u(ǫ, X ǫ ) - t ǫ l s g(s, X s )ds ✭✺✳✶✵✮ ◆♦✇ ❢♦r α > 0 ✐♥tr♦❞✉❝❡ t❤❡ ♣r♦❝❡ss❡s v α t = P α l t u(t, X t ) + t 0 l s g(s, X s )ds ✭✺✳✶✶✮ ❛♥❞ M α t = P α M t ✭✺✳✶✷✮ ❚❤❡s❡ ❤❛✈❡ ♠♦♠❡♥ts ♦❢ ❛•• ♦r❞❡rs✳ ❲❡ ♥♦t❡ t❤❛t t❤❡ ❢♦••♦✇✐♥❣ ✐s ❛ ♣♦s✐t✐✈❡ r❛♥❞♦♠ ✈❛r✐❛❜•❡✿ D α t := P α D t = M α t -v α t + P α l ǫ u(ǫ, X ǫ ) + ǫ 0 l s g(s, X s )ds ✭✺✳✶✸✮ ▲❡t τ ❜❡ ❛♥② st♦♣♣✐♥❣ t✐♠❡❀ ✇❡ ❡✈❛•✉❛t❡ ❛t t = τ ❛♥❞ t❛❦❡ ❡①♣❡❝t❛t✐♦♥s✳ M α ✐s ❛ ✉♥✐❢♦r♠•② ✐♥t❡❣r❛❜•❡ ♠❛rt✐♥❣❛•❡ st❛rt✐♥❣ ❢r♦♠ ✵✱ s♦✿ E[M α τ ] = 0 ✭✺✳✶✹✮ ✺✳✶✳ ❱❆|■❆❚■❖◆❆▲ ■◆❊◗❯❆❚■❖◆❙ ■◆ ❆ ▼❆|❑❖❱■❆◆ ❋|❆▼❊❲❖|❑ ✶✶✸ ❲❡ ❦♥♦✇ t❤❛t t❤❡ ❖r♥st❡✐♥✲❯❤•❡♥❜❡❝❦ s❡♠✐❣r♦✉♣ ♣r❡s❡r✈❡s t❤❡ ❡①♣❡❝t❛t✐♦♥s✱ s♦✿ E P α l ǫ u(ǫ, X ǫ ) + ǫ 0 l s g(s, X s )ds = E l ǫ u(ǫ, X ǫ ) + ǫ 0 l s g(s, X s )ds ✭✺✳✶✺✮ ❋✐♥❛••②✿ v α τ = P α l t u(t, X t ) + t 0 l s g(s, X s )ds |t=τ ✭✺✳✶✻✮ ❙✐♥❝❡ ❢♦r ❛♥② p l t u(t, X t ) + t 0 l s g(s, X s )ds ✐s ❜♦✉♥❞❡❞ ✐♥ L p ✭❜② u ∞ + T g ∞ ✮✱ t❤❡ ♣r♦❝❡ss l • u(•, X • ) + • 0 l s g(s, X s )ds ✐s ✉♥✐❢♦r♠•② ✐♥t❡❣r❛❜•❡✳ ❚❤❡r❡❢♦r❡✱ s♦ ✐s v α t ❛s ❛ ❜✐✈❛r✐❛t❡ ♣r♦❝❡ss✱ ❛♥❞ •❡tt✐♥❣ α t❡♥❞ t♦ 0 ✇❡ ♦❜t❛✐♥✿ E[v τ ] = E l τ u(τ, X τ ) + τ 0 l s g(s, X s )ds ✭✺✳✶✼✮ P✉tt✐♥❣ ❡✈❡r②t❤✐♥❣ t♦❣❡t❤❡r ♦♥❡ ♦❜t❛✐♥s✿ E l ǫ u(ǫ, X ǫ ) + ǫ 0 l s g(s, X s )ds ≥ E l τ u(τ, X τ ) + τ 0 l s g(s, X s )ds ✭✺✳✶✽✮ ❙✐♥❝❡ t❤✐s ✐s ❢♦r ❛♥② st♦♣♣✐♥❣ t✐♠❡ τ ❛♥❞ ❢♦r ❛♥② ǫ✱ •❡tt✐♥❣ ǫ t❡♥❞ t♦ 0 t❤❡r❡ ❝♦♠❡s✿ u(0, x) ≥ sup τ ∈Z 0,T E[l τ u(τ, X τ )] ✭✺✳✶✾✮ ◆♦✇ t♦ ✜♥✐s❤ t❤❡ ♣r♦♦❢ ✇❡ •✉st ♥❡❡❞ t♦ ✜♥❞ ♦♥❡ st♦♣♣✐♥❣ t✐♠❡ τ 0 ❢♦r ✇❤✐❝❤ u(0, x) = E[l τ 0 u(τ 0 , X τ 0 )] st❛♥❞s✳ ❈♦♥s✐❞❡r t❤❡ ♦♣❡♥ s❡t✿ D = {(t, y), u(t, y) = f (y)} ✭✺✳✷✵✮ ❛♥❞ ❞❡✜♥❡ ❛ st♦♣♣✐♥❣ t✐♠❡ ❛s✿ τ x = inf {t, (t, X t ) / ∈ D} ✭✺✳✷✶✮ ■❢ τ x = 0 ❛✳s✳ t❤❡♥ u(0, x) = f (x) ❛♥❞ ✐t ✐s ❝•❡❛r t❤❛t τ x r❡❛•✐③❡s t❤❡ s✉♣r❡♠✉♠ ✐♥ t❤✐s ❝❛s❡✳ ❖t❤❡r✇✐s❡✱ ❢r♦♠ t❤❡ 0 -1 •❛✇ ❢♦r t❤❡ ❇r♦✇♥✐❛♥ ✜•tr❛t✐♦♥✱ τ x > 0 ❛✳s✳ ◆♦✇ •❡t (u m , m ∈ N) ❜❡ ❛ s♠♦♦t❤ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ u✳ ❇② t❤❡ ✉s✉❛• ■tô ❢♦r♠✉•❛✱ ❢♦r ❡✈❡r② m ❛♥❞ ❢♦r ❡✈❡r② t > ǫ > 0✿ l t∧τx u m (t ∧ τ x , X t∧τx ) -l ǫ∧τx u m (ǫ ∧ τ x , X ǫ∧τx ) = t∧τx ǫ∧τx l s (∂ s + A s + r) u m (s, X s )ds + M (m,n,ǫ) t ✭✺✳✷✷✮ ✶✶✹ ❈❍❆P❚❊| ✺✳ ❆PP▲■❈❆❚■❖◆❙ ❚❖ ❙❖▼❊ P|❖❇▲❊▼❙ ■◆ ❆◆❆▲(❙■❙ ✇❤❡r❡ (M (m,n,ǫ) , t > ǫ) ✐s ❛ ♠❛rt✐♥❣❛•❡ ✐♥ t❤❡ ❇r♦✇♥✐❛♥ ✜•tr❛t✐♦♥ st❛rt✐♥❣ ❛t ǫ✳ ◆♦✇ •❡t φ ∈ D✳ ❖♥❡ ❤❛s✿ E φ • t∧τx ǫ∧τx l s (∂ s + A s + r) u m (s, X s )ds = E φ • t ǫ l s (∂ s + A s + r) u m (s, X s )1 (s,Xs)∈Dx ds = t ǫ E l s φ (∂ s + A s + r) u m (s, X s )1 (s,Xs)∈Dx ds = t ǫ (∂ s + A s + r) u m (s,
(x) = E (Id + L) δ/2 (l s • φ) • (Id + L) δ/2 (δ x • x s ) ✭✺✳✷✻✮ ♦♥❡ ❝❤❡❝❦s t❤❛t t❤❡ ❢♦••♦✇✐♥❣ ♣❛t❤ ✐s ❝♦♥t✐♥✉♦✉s ✭❢♦r ❛♥② ✈❛•✉❡ ♦❢ p✮✿ s ∈]ǫ, T [ → p Xs,lsφ ∈ S p,δ ✭✺✳✷✼✮ ◆♦✇ ✇❡ ❦♥♦✇ t❤❛t (∂ s + A s + r) u m -g ❝♦♥✈❡r❣❡s t♦ 0 ✐♥ t❤❡ s❡♥s❡ ♦❢ ❞✐str✐❜✉t✐♦♥s ♦♥ D❀ ♠♦r❡ ♣r❡❝✐s❡•② ♦♥ D ✇❡ ❤❛✈❡ t❤❡ ❡q✉❛•✐t②✿ Σσ ij ∂ ij u = -∂ t u -g + ru -Σ i b i ∂ i u ✭✺✳✷✽✮ ❛♥❞ u ∈ S p,-N p -ǫ ❢♦r ❛♥② p s♦ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ t❛❦❡s ♣•❛❝❡ ✐♥ S p,-N p -1-ǫ (D)❀ ✐♥ ♣❛rt✐❝✉•❛r ✐t t❛❦❡s ♣•❛❝❡ ✐♥ S p,-δ (D) ❢♦r s♠❛•• ❡♥♦✉❣❤ p✳ ❈❤♦♦s✐♥❣ s✉❝❤ ❛ s♠❛•• p ❛♥❞ p * ❢♦r t❤❡ r❡❣✉•❛r✐t② ♦❢ p Xs,lsφ ♦♥❡ ✜♥❛••② ❣❡ts✿ t∧τx ǫ∧τx l s [(∂ s + A s + r) u m (s, X s ) + g(s, X s )] ds → 0 ✭✺✳✷✾✮ ✺✳✶✳ ❱❆|■❆❚■❖◆❆▲ ■◆❊◗❯❆❚■❖◆❙ ■◆ ❆ ▼❆|❑❖❱■❆◆ ❋|❆▼❊❲❖|❑ ✶✶✺ ❛t •❡❛st ✐♥ D ′ ✳ ❲❡ t❤❡r❡❢♦r❡ ❤❛✈❡✿ l t∧τx u(t ∧ τ x , X t∧τx ) -l ǫ∧τx u(ǫ ∧ τ x , X ǫ∧τx ) + t∧τx ǫ∧τx l s g(s, X s )ds = M (ǫ) t ✭✺✳✸✵✮ ✇❤❡r❡ M (ǫ) ✐s ❛ ✇❡❛❦ ♠❛rt✐♥❣❛•❡✳ ◆♦✇ M (ǫ) ✐s ❛ ♣r♦♣❡r✱ ❜♦✉♥❞❡❞ ✐♥ L p ♣r♦❝❡ss ❜❡❝❛✉s❡ t❤❡ ▲❍❙ ✐s✱ s♦ ✐t ✐s str❛✐❣❤t❢♦r✇❛r❞ t♦ ❝❤❡❝❦ t❤❛t M (ǫ) ✐s ❛ ♠❛rt✐♥❣❛•❡ ✐♥ t❤❡ ✉s✉❛• s❡♥s❡✳ ■♥ ♣❛rt✐❝✉•❛r t❤❡ ❡①♣❡❝t❛t✐♦♥s ♦❢ t❤❡ ▲❍❙ ❛♥❞ t❤❡ |❍❙ ❜♦t❤ ❛r❡ 0 ❛t ❛♥② t✐♠❡✳ ❆•s♦✱ t❤❡ ▲❍❙ ✐s ❜♦✉♥❞❡❞ s✉r❡•② ❜② 2 u ∞ + T g ∞ s♦ ❜② t❛❦✐♥❣ ❡①♣❡❝t❛t✐♦♥s ❛♥❞ •❡tt✐♥❣ ǫ t❡♥❞ t♦ 0 ❛♥❞ t t❡♥❞ t♦ ✐♥✜♥✐t② ❜② t❤❡ ❞♦♠✐♥❛t❡❞ ❝♦♥✈❡r❣❡♥❝❡ t❤❡♦r❡♠ ♦♥❡ ❣❡ts✿ E l τx u(τ x , X τx ) + τx 0 l s g(s, X s )ds = u(0, x) ✭✺✳✸✶✮ s♦ ❜② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ τ x ✿ E l τx f (X τx ) + τx 0 l s g(s, X s )ds = u(0, x) ✭✺✳✸✷✮ ❛♥❞ t❤❡ ♣r♦♦❢ ✐s ✜♥✐s❤❡❞✳ |❡♠❛r❦ ✺✳✶✳✵✳✷ ✭■♠♣r♦✈❡♠❡♥ts ✇✳r✳t t❤❡ ♣r❡✈✐♦✉s r❡s✉•ts✮ ✳ ❚❤❡ ❝♦❡✣❝✐❡♥ts b✱ σ✱ r ♥❡❡❞ ✷ ♦r ✸ r❡❣✉•❛r✐t✐❡s✱ ♥♦t s♠♦♦t❤♥❡ss✳ f ♦♥•② ♥❡❡❞s t♦ ❜❡ ❝♦♥t✐♥✉♦✉s✳ ❚❤❡r❡ ✐s ♥♦ ♥❡❡❞ ❢♦r ❡••✐♣t✐❝✐t②✱ ♥♦r ❡✈❡♥ ❢♦r ❤②♣♦❡••✐♣t✐❝✐t②✿ ♦♥•② ▼❛••✐❛✈✐♥ ♥♦♥❞❡❣❡♥❡r❛❝② ❝♦♥❞✐t✐♦♥s ❛r❡ r❡q✉✐r❡❞✳ |❡♠❛r❦ ✺✳✶✳✵✳✸ ✭❆ s✐♠♣•❡r ♣r♦♦❢ ✉♥❞❡r str♦♥❣❡r ❤②♣♦t❤❡s❡s✿✮ ✳ ❙✉♣♣♦s❡ K t ✐s ❤②♣♦❡••✐♣t✐❝ ❛♥❞ r ✐s s♠♦♦t❤✳ ❙✐♥❝❡ K t u = ru ✐♥ t❤❡ s❡♥s❡ ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥s ♦♥ D✱ u ❤❛s t♦ ❜❡ ❛ s♠♦♦t❤ ❢✉♥❝t✐♦♥ ♦♥ D ❜❡❝❛✉s❡ r ✐s s♠♦♦t❤✳ ❚❤❡r❡❢♦r❡ t❤❡ ❝•❛ss✐❝❛• ■tô ❢♦r♠✉•❛ ♠❛② ❜❡ ❛♣♣•✐❡❞✱ ❛♥❞ ✐❢ τ n ❛♥♥♦✉♥❝❡s τ x ✿ l τn u(τ n , X τn ) -u(0, x) = τn 0 l s ( t σ(s, X s )∇u(s, X s ))dW s ✭✺✳✸✸✮ ❚❤❡r❡❢♦r❡ t❤❡ •❡❢t ❤❛♥❞ s✐❞❡ ❛❜♦✈❡ ✐s ❛ ✭✉s✉❛•✱ str♦♥❣✮ ❞✐s❝r❡t❡ t✐♠❡ ♠❛rt✐♥❣❛•❡ ❝♦♥✈❡r❣❡✐♥❣ ❛✳s✳ t♦ l τx u(τ x , X τx )-u(0, x) ❛♥❞ s✐♥❝❡ τ n ❛♥♥♦✉♥❝❡s τ x ♦✉r ♠❛rt✐♥❣❛•❡ ✐s ✉♥✐❢♦r♠•② ✐♥t❡❣r❛❜•❡✱ s♦ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ t❛❦❡s ♣•❛❝❡ ✐♥ L 1 ❛♥❞ t❤✐s ②✐❡•❞s✿ u(0, x) = E [l τx u(τ x , X τx )] ✭✺✳✸✹✮ ✇❤✐❝❤ ✜♥✐s❤❡s ♦✉r ♣r♦♦❢✳ ✶✶✻ ❈❍❆P❚❊| ✺✳ ❆PP▲■❈❆❚■❖◆❙ ❚❖ ❙❖▼❊ P|❖❇▲❊▼❙ ■◆ ❆◆❆▲(❙■❙ |❡♠❛r❦ ✺✳✶✳✵✳✹✳ ❆♥♦t❤❡r ♠❡t❤♦❞ ♠✐❣❤t ❤❛✈❡ ❜❡❡♥ t❤❡ ❢♦••♦✇✐♥❣✿ s✐♥❝❡ t∧τx ǫ∧τx l s (∂ s + A s + r) u m (s, X s )ds → 0 ✭✺✳✸✺✮ ✐♥ D ′ ✱ ♦♥❡ ♠❛② ✉s❡ t❤❡ ♣♦s✐t✐✈❡ ❞✐str✐❜✉t✐♦♥ ♠❡t❤♦❞s t♦ ❞❡❞✉❝❡ ❢r♦♠ t❤✐s t❤❛t t❤❡ ❝♦♥✈❡r❣❡♥❝❡ t❛❦❡s ♣•❛❝❡ ✐♥ •❛✇✳ ❚❤❡♥ ♦♥ ♠❛② ♣r♦✈❡ t❤❛t t❤❡ ▲❍❙ ✐s t❡♥s❡ ✭♣❡r❤❛♣s ♥❡❡❞ t♦ r❡str✐❝t t♦ ✐♥❝r❡❛s✐♥❣ ❝♦♠♣❛❝ts✮ ❜② t❤❡ ❆•❞♦✉s ❝r✐t❡r✐❛✳ ❋r♦♠ t❤✐s ✇❡ ❞❡❞✉❝❡ t❤❛t ♦✉r s❡q✉❡♥❝❡ ♦❢ ♠❛rt✐♥❣❛•❡s ✐s t❡♥s❡ ✭♦♥❡ t❡♥s❡ t❡r♠ ✰ ♦♥❡ t❡r♠ ❝♦♥✈❡r❣✐♥❣ t♦ ✵✮❀ ✇❡ ❛•s♦ ❦♥♦✇ t❤❛t ✐t ❝♦♥✈❡r❣❡s ✐♥ ✜♥✐t❡ ❞✐str✐❜✉t✐♦♥s✳ ❚❤❡r❡❢♦r❡ t❤❡ •✐♠✐t ❤❛s t♦ ❜❡ ❛ •♦❝❛• ♠❛rt✐♥❣❛•❡ ✭s❡❡ ❬✸✶( ❢♦r t❤❡ ♣r❡❝✐s❡ ❝r✐t❡r✐♦♥✮ ❛♥❞ s✐♥❝❡ ✐t ✐s ❜♦✉♥❞❡❞ ❛✳s✳ ❜② t❤❡ ❇✉r❦❤♦•❞❡r✲❉❛✈✐❡s✲•✉♥❞② t❤❡♦r❡♠ ❛♥❞ t❤❡ ◆♦✈✐❦♦✈ ❝r✐t❡r✐♦♥ ✐t ✐s ❛ ♠❛rt✐♥❣❛•❡✳ ❚❤✐s ✐s s✐♠✐•❛r ❜✉t ♠♦r❡ ❝♦♠♣•✐❝❛t❡❞❀ ✐t ✇❛s t❤❡ ✜rst ✐❞❡❛ ■ ❤❛❞✳ |❡♠❛r❦ ✺✳✶✳✵✳✺ ✭❖♥ t❤❡ ❝♦♥❞✐t✐♦♥✿ u ∈ BV (C 0 )✮✳ ■♥ ♦✉r t❤❡♦r❡♠ ✇❡ ✇♦r❦❡❞ ✉♥❞❡r t❤❡ ❤②♣♦t❤❡s✐s t❤❛t u ∈ BV (C 0 )✳ ■♥ ♣r❛❝t✐❝❡ t❤✐s ♠✐❣❤t ❜❡ ❤❛r❞ t♦ ✈❡r✐❢② ♦r ♦♥❡ ♠❛② ♦♥•② ❤❛✈❡ t❤❛t u ✐s ❇❱ ✐♥ t✐♠❡ ❢♦r ❡✈❡r② ♣♦✐♥t ✐♥ s♣❛❝❡ ❛♥❞ C 0 ✐♥ s♣❛❝❡ ❢♦r ❡✈❡r② ♣♦✐♥t ✐♥ t✐♠❡ ❜✉t ♥♦t ❤❛✈❡ •♦✐♥t r❡❣✉•❛r✐t②✳ ❚❤❡♥✱ ♦♥❡ ♠❛② st✐•• ♠❛❦❡ t❤❡ ❢♦••♦✇✐♥❣ ♦❜s❡r✈❛t✐♦♥✿ u ∈ BV (S ′ ) ✐❢ ❛♥❞ ♦♥•② ✐❢ ❢♦r ❡✈❡r② φ ∈ S✱ t → u(t, •), φ ✐s ❛ ❇❱ ❢✉♥❝t✐♦♥✳ ■♥ ♦✉r ♣r❡❝✐s❡ ❝❛s❡✱ t❤❡r❡ ✐s✿ ũ(t) := u(t, •), φ = u(t, x) • φ(x)dx ✭✺✳✸✻✮ ❛♥❞ ✐t ✐s t❤❡♥ ❡❛s② t♦ ❞❡r✐✈❡✿ V ar(ũ) ≤ V ar (u(•, x)) • φ(x)dx ✭✺✳✸✼✮ ❚❤❡♥ ✇❡ s❡❡ t❤❛t ✐❢ x → V aru(•, x) ✐♥ L p ✱ ❜② ❍ö•❞❡r✬s t❤❡♦r❡♠ u ∈ BV (L p )✳ ▼♦r❡ ❣❡♥❡r❛••②✱ ✐❢ t❤❡r❡ ❡①✐sts s♦♠❡ δ > 0 s✉❝❤ t❤❛t✿ x → V ar (u(•, x)) (1 + |x| 2 ) δ/2 ∈ L p ✭✺✳✸✽✮ t❤❡♥ u ∈ BV (S p,-δ )✳ ■♥❞❡❡❞ ✇❡ ❤❛✈❡ ❛•r❡❛❞② ♦❜s❡r✈❡❞ t❤❛t✿ x → 1 + |x| 2 δ/2 • φ(x) L p * ≤ C • φ S p * ,δ ✭✺✳✸✾✮ ❲❡ ♥♦✇ ♣r♦✈❡ t❤❡ ❢♦••♦✇✐♥❣ r❡❝✐♣r♦❝❛• t♦ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠✿ ❚❤❡♦r❡♠ ✺✳✶✳✵✳✾✳ ❯♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥s (A)✱ t❤❡ ❢✉♥❝t✐♦♥✿ u(t, x) = sup τ ∈Z t,T E f (X t τ (x)) exp - τ t r(u, X t u (x))du + τ t g(s, X t s (x)) exp - s t r(u, X t u (x))du ds ✭✺✳✹✵✮ ✐s ❛ s♦•✉t✐♦♥ t♦ t❤❡ ✈❛r✐❛t✐♦♥❛• ✐♥❡q✉❛t✐♦♥ (V)✳ ✺✳✶✳ ❱❆|■❆❚■❖◆❆▲ ■◆❊◗❯❆❚■❖◆❙ ■◆ ❆ ▼❆|❑❖❱■❆◆ ❋|❆▼❊❲❖|❑ ✶✶✼ Pr♦♦❢✳ ❚❤❛t u(t, x) ≥ f (x) ❛♥❞ u(T, x) = f (x) ❛r❡ ♦❜✈✐♦✉s✳ ❆•s♦✱ ✐t ✐s ❦♥♦✇♥ ✭❝❢ ❬✸✸(✮ t❤❛t u ✐s •♦✐♥t•② ❝♦♥t✐♥✉♦✉s✳ ❋✐♥❛••② ✐t ✐s ❞❡❝r❡❛s✐♥❣ ✐♥ t✐♠❡ ❜❡❝❛✉s❡ s♦ ✐s t❤❡ s❡t Z t,T ✱ ❤❡♥❝❡ u ✐s ❇❱ ✐♥ t✐♠❡ ❛♥❞✿ V ar (u(•, x)) = u(0, x) -u(T, x) ✭✺✳✹✶✮ ❚❤❡r❡❢♦r❡✿ sup x V ar (u(•, x)) ≤ 2 • f ∞ ✭✺✳✹✷✮ s♦ ❜② t❤❡ s❛♠❡ ❛r❣✉♠❡♥ts ❛s ✐♥ r❡♠❛r❦ ✺✳✶✳✵✳✺ ♦♥❡ ❣❡ts u ∈ BV S p,-N/p-ǫ ✳ ❲❡ ♥♦✇ ❝♦♥s✐❞❡r t❤❡ ❢♦••♦✇✐♥❣ ❛❞❛♣t❡❞ ♣r♦❝❡ss✿ S t := l t u(t, X t ) + t 0 l s g(s, X s )ds ✭✺✳✹✸✮ ❲❡ ✇✐•• ♣r♦✈❡ ✐t ✐s s✉♣❡r♠❛rt✐♥❣❛•❡✳ ■♥❞❡❡❞✱ ❢r♦♠ t❤❡ ❡①♣r❡ss✐♦♥ ♦❢ u ❛♥❞ t❤❡ str♦♥❣ ▼❛r❦♦✈ ♣r♦♣❡rt② X ❤❛s ❛s t❤❡ str♦♥❣ s♦•✉t✐♦♥ ♦❢ ❛♥ ❙❉❊✱ ♦♥❡ ❤❛s✿ u(t, X t ) = sup t≤τ ≤T E l τ l t f (X τ ) + τ t l s l t g(s, X s )ds|F t ✭✺✳✹✹✮ ❛♥❞ s✐♥❝❡ l • ❛♥❞ • 0 l s g(s, X s )ds ❛r❡ ❛❞❛♣t❡❞ ♣r♦❝❡ss❡s✿ S t = sup t≤τ ≤T E l τ f (X τ ) + τ 0 l s g(s, X s )ds|F t ✭✺✳✹✺✮ ❖♥❡ s❡❡s t❤❛t t❤✐s ✐s t❤❡ ❙♥❡•• ❡♥✈❡•♦♣❡ ❢♦r t❤❡ ❝♦st ❢✉♥❝t✐♦♥ Y u := l u f (X u ) + u 0 l s g(s, X s )ds ✭✺✳✹✻✮ t❤❡r❡❢♦r❡ ❜② t❤❡ r❡s✉•ts ✐♥ ❬✸✸( ✇❡ ❦♥♦✇ t❤❛t ✐t ✐s ❛ s✉♣❡r♠❛rt✐♥❣❛•❡✳ ❆•s♦✱ S ✐s ✭❛•♠♦st✮ s✉r❡•② ❜♦✉♥❞❡❞✿ |S t | ≤ u ∞ + (T -t) g ∞ ≤ f ∞ + T g ∞ ✭✺✳✹✼✮ ■♥ ♣❛rt✐❝✉•❛r✱ S ✐s ❛ ❝•❛ss ❉ s✉♣❡r♠❛rt✐♥❣❛•❡✱ t❤❡r❡❢♦r❡ ✐t ❛❞♠✐ts ❛ ❉♦♦❜✲▼❡②❡r ❞❡❝♦♠♣♦s✐t✐♦♥✱ ❝❢ ❬✶✼(✿ S t -u(0, x) = M t + D t ✭✺✳✹✽✮ ✇❤❡r❡ M ✐s ❛ ♠❛rt✐♥❣❛•❡ ❛♥❞ D ✐s ❛ ❞❡❝r❡❛s✐♥❣ ♣r♦❝❡ss✱ ❛♥❞ M 0 = D 0 = 0✳ ◆♦✇ s✐♥❝❡ u ∈ BV S p,-N/p-ǫ ✱ ❛s ✐♥ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠✱ ♦♥❡ ♠❛② ✇r✐t❡ t❤❡ ✇❡❛❦ ■tô ❢♦r♠✉•❛✿ l t u(t, X t )-l ǫ u(t, X ǫ ) = t ǫ l s (∂ s + A s + r) u(s, X s )ds+ t ǫ l s σ(s, X s )∂ x σ(s, X s )dW s ✭✺✳✹✾✮ ✶✶✽ ❈❍❆P❚❊| ✺✳ ❆PP▲■❈❆❚■❖◆❙ ❚❖ ❙❖▼❊ P|❖❇▲❊▼❙ ■◆ ❆◆❆▲(❙■❙ ❛♥❞ t❤❡r❡❢♦r❡✿ S t -S ǫ = t ǫ l s ((∂ s + A s + r) u(s, X s ) + g(s, X s )) ds + t ǫ l s σ(s, X s )∂ x σ(s, X s )dW s ✭✺✳✺✵✮ ❚❤❡ ✜rst t❡r♠ ♦❢ t❤❡ |❍❙ ✐s ❛ ✇❡❛❦ ❇❱ ♣r♦❝❡ss✱ ❛♥❞ ✐t s❡❝♦♥❞ t❡r♠ ✐s ❛ ✇❡❛❦ ♠❛rt✐♥❣❛•❡✳ ◆♦✇ ❛♣♣•② t❤❡ ♦♣❡r❛t♦r (Id + L) -k 2 ❢♦r ❜✐❣ ❡♥♦✉❣❤ k s♦ t❤❡② ❜❡❝♦♠❡ ❛ ❇❱ ♣r♦❝❡ss ❛♥❞ ❛ ♠❛rt✐♥❣❛•❡✱ r❡s♣❡❝t✐✈❡•②✳ ❆•s♦ ❜② ✭✺✳✹✽✮✿ l t u(t, X t ) -u(ǫ, X ǫ ) = (M t -M ǫ ) + (D t -D ǫ ) ✭✺✳✺✶✮ s♦ ❜② t❤❡ ✉♥✐❝✐t② ♦❢ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛ s❡♠✐♠❛rt✐♥❣❛•❡ ❛s ❛ •♦❝❛• ♠❛rt✐♥❣❛•❡ ♣•✉s ❛ ❇❱ ♣r♦❝❡ss ♦♥❡ ♠❛② ✐❞❡♥t✐❢②✿ (Id + L) -k 2 t ǫ l s σ(s, X s )∂ x σ(s, X s )dW s = (Id + L) -k 2 (M t -M ǫ ) ✭✺✳✺✷✮ (Id + L) -k 2 t ǫ (l s (∂ s + A s + r) u(s, X s ) + g(s, X s )) ds = (Id + L) -k 2 (D t -D ǫ ) ✭✺✳✺✸✮ s♦ ✐♥ ♣❛rt✐❝✉•❛r✿ t ǫ l s ((∂ s + A s + r) u(s, X s ) + g(s, X s )) ds = D t -D ǫ ✭✺✳✺✹✮ t❤❡r❡❢♦r❡ t❤✐s ✇❡❛❦•② ❇❱ ♣r♦❝❡ss ✐s ❛ ♣r♦♣❡r ♣r♦❝❡ss✱ ❛♥❞ ✐t ✐s ❞❡❝r❛s✐♥❣✳ ◆♦✇ •❡t φ ❜❡ s♠♦♦t❤ ❛♥❞ ❛✳s✳ ♣♦s✐t✐✈❡✳ ❋♦r ❛♥② 0 < ǫ < t < u✿ E φ u ǫ l s ((∂ s + A s + r) u(s, X s ) + g(s, X s )) ds ≤ E φ t ǫ ((∂ s + A s + r) u(s, X s ) + g(s, X s )) ds ✭✺✳✺✺✮ s♦✿ E φ u t l s ((∂ s + A s + r) u(s, X s ) + g(s, X s )) ds = u t E [φl s ((∂ s + A s + r) u(s, X s ) + g(s, X s ))] ds ≤ 0 ✭✺✳✺✻✮ ❚❤❡ ❢✉♥❝t✐♦♥ ✉♥❞❡r t❤❡ ✐♥t❡❣r❛• ✐s✿ s → E [φl s ((∂ s + A s + r) u(s, X s ) + g(s, X s ))] = ((∂ s + A s + r) u(s, y) + g(s, y)) p Xs,lsφ (y)dy ✭✺✳✺✼✮ ✺✳✶✳ ❱❆|■❆❚■❖◆❆▲ ■◆❊◗❯❆❚■❖◆❙ ■◆ ❆ ▼❆|❑❖❱■❆◆ ❋|❆▼❊❲❖|❑ ✶✶✾ ✇❤✐❝❤ ✐s ❜♦✉♥❞❡❞ ♦♥ ]ǫ, T [✱ t❤❡r❡❢♦r❡ ✐t ✐s •✐❝✐t t♦ t❛❦❡ •✐♠✐ts ✐♥ t❤❡ ◆❡✇t♦♥ r❛t✐♦ ❛♥❞ t♦ ❞✐✛❡r❡♥t✐❛t❡ ❛t •❡❛st ✐♥ t❤❡ ❛❜s♦•✉t❡•② ❝♦♥t✐♥✉♦✉s s❡♥s❡ t♦ ♦❜t❛✐♥✱ ❢♦r ❛•• s > ǫ✿ E [φl s ((∂ s + A s + r) u(s, X s ) + g(s, X s ))] ≤ 0 ✭✺✳✺✽✮ ❆♣♣•② t❤✐s t♦ φ = h(X s ) ❢♦r s♦♠❡ r❡❣✉•❛r ❡♥♦✉❣❤✱ ♣♦s✐t✐✈❡ ❢✉♥❝t✐♦♥ h✳ ❚❤❡♥ t❤❡r❡ ❝♦♠❡s✿ (∂ s + A s + r) u(s, •) + g, p Xs,ls h ≤ 0 ✭✺✳✺✾✮ ❆•s♦ ❜② ♣r♦♣♦s✐t✐♦♥ ✸✳✹✳✵✳✺ supp(p Xs ) = supp(p Xs,ls ) ❜❡❝❛✉s❡ l s ✐s ❛✳s✳ str✐❝t•② ♣♦s✐t✐✈❡✳ ❋r♦♠ t❤✐s ✇❡ ❞❡❞✉❝❡ t❤❛t (∂ s + A s + r) u + g ≤ 0 ✐♥ t❤❡ s❡♥s❡ ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥s ♦♥ supp(p Xs )✳ ❚♦ ✜♥✐s❤ t❤❡ ♣r♦♦❢✱ ✇❡ ♦♥•② ♥❡❡❞ t♦ ✈❡r✐❢② t❤❛t (∂ s + A s + r) u + g = 0 ✐♥ t❤❡ s❡♥s❡ ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥s ♦♥ t❤❡ ♦♣❡♥ s❡t supp(p Xs ) ∩ {u(s, •) = f }✳ ❆s ✐♥ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠✱ ✇❡ ✐♥tr♦❞✉❝❡✿ D = {(t, y), u(t, y) = f (y)} ✭✺✳✻✵✮ ❛♥❞ ❞❡✜♥❡ ❛ st♦♣♣✐♥❣ t✐♠❡ ❛s✿ τ x = inf {t, (t, X t ) / ∈ D} ✭✺✳✻✶✮ ❲❡ ♥♦t❡ t❤❛t τ x < ∞ ❛✳s✳ ❜❡❝❛✉s❡ ✇❡ ❝♦♥s✐❞❡r ❛ ✜♥✐t❡ ❤♦r✐③♦♥ ♣r♦❜•❡♠❀ ❛•s♦ τ x ✐s t❤❡ s♠❛••❡st ♦♣t✐♠❛• st♦♣♣✐♥❣ t✐♠❡ ❢♦r ♦✉r ♦♣t✐♠❛• st♦♣♣✐♥❣ ♣r♦❜•❡♠✳ ❇② t❤❡ ❉♦♦❜ st♦♣♣✐♥❣ t❤❡♦r❡♠ ❛♥❞ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ τ x ♦♥❡ s❡❡s t❤❛t✿ E[l τx f (X τx ) + τx 0 l s g(s, X s )ds] = E[S τx ] = u(0, x) + E[D τx ] ✭✺✳✻✷✮ ❛♥❞ t❤❡r❡❢♦r❡ E[D τx ] = 0✳ ◆♦✇ •❡t (u m ) ❜❡ ❛ s♠♦♦t❤ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ u✳ ❇② ♠❡t❤♦❞s s✐♠✐•❛r t♦ t❤♦s❡s ❞❡✈❡•♦♣❡❞ ✐♥ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠✱ t❤❡r❡ ❝♦♠❡s✿ E t∧τx ǫ∧τx l s ((∂ s + A s + r) u m (s, X s ) + g(s, X s )) ds = ]ǫ,t[×R N ∩U ((∂ s + A s + r) u m (s, y) + g(s, y)) p Xs,ls (y)dyds ✭✺✳✻✸✮ s♦ ❜② t❛❦✐♥❣ •✐♠✐ts ❛♥❞ ❝♦♠♣❛r✐♥❣ ✇✐t❤ E[B τx ] ♦♥❡ s❡❡s t❤❛t ❢♦r ❛♥② s✿ (∂ s + A s + r) u + g, p Xs,ls U = 0 ✭✺✳✻✹✮ ❛♥❞ s✐♥❝❡ ✇❡ ❛•r❡❛❞② ❦♥♦✇ t❤❛t (∂ s + A s + r) u + g ✐s ❛ ♥❡❣❛t✐✈❡ ❞✐str✐❜✉t✐♦♥ ✇❡ ❛r❡ ❞♦♥❡✳ ✶✷✵ ❈❍❆P❚❊| ✺✳ ❆PP▲■❈❆❚■❖◆❙ ❚❖ ❙❖▼❊ P|❖❇▲❊▼❙ ■◆ ❆◆❆▲(❙■❙ |❡♠❛r❦ ✺✳✶✳✵✳✻✳ ❬✺✷( ❛♥❞ s♦♠❡ r❡❢❡r❡♥❝❡s t❤❡r❡✐♥ ♣r♦✈✐❞❡ r❡s✉•ts ❛❜♦✉t t❤❡ s✉♣✲ ♣♦rt ♦❢ p Xt ✳ ■♥ ♣❛rt✐❝✉•❛r✱ ✇✐t❤✐♥ t❤❡ ❤②♣♦t❤❡s❡s ✇❡ ♠❛❞❡ ♦♥ b ❛♥❞ σ✱ t❤❡ r❡s✉•ts ✐♥ ❬✺✷( ✐♠♣•② t❤❛t {p Xt > 0} ✐s ❛♥ ♦♣❡♥✱ ❝♦♥♥❡❝t❡❞ s❡t✱ s♦ ✇❡ ♠❛② ✇♦r❦ ✇✐t❤ t❤❡ ❞✐str✐❜✉t✐♦♥s ♦♥ supp(p Xt )✳ ❆•s♦✱ ✐♥ ♠❛♥② ❝❛s❡s✱ ❢♦r ❡①❛♠♣•❡ ✇❤❡♥ A ✐s ❡••✐♣t✐❝✱ ✇❡ ❤❛✈❡ •♦✇❡r ❜♦✉♥❞s ❢♦r p Xt ✱ s♦ p Xt ❤❛s ❢✉•• s✉♣♣♦rt✳ |❡♠❛r❦ ✺✳✶✳✵✳✼✳ ❇② ✉s✐♥❣ r❡s✉•ts ♦♥ ♣♦s✐t✐✈❡ ❞✐str✐❜✉t✐♦♥s ✐t s❤♦✉•❞ ❜❡ ♣♦ss✐❜•❡ t♦ ❣❡t r✐❞ ♦❢ t❤❡ ♥❡❝❡ss✐t② t❤❛t t❤❡ ❞✐s❝♦✉♥t ❢❛❝t♦r l t ✐s r❡❣✉•❛r✳ ❚❤❡♥ t❤❡ r❡s✉•ts ✇♦✉•❞ ❤♦•❞ s✐♠♣•② ✇✐t❤ ❛ ❝♦♥t✐♥✉♦✉s ✐♥t❡r❡st r❛t❡✳ |❡♠❛r❦ ✺✳✶✳✵✳✽✳ ❚♦ ❞♦✿ ❡①t❡♥❞ t❤❡ r❡s✉•ts t♦ t❤❡ ❝❛s❡ ✇❤❡r❡ t❤❡ ❤♦r✐③♦♥ ✐s ♥♦t ✜♥✐t❡ ❛♥❞ t♦ t❤❡ ❝❛s❡ ✇❤❡r❡ t❤❡ ❞♦♠❛✐♥ ✐s ♥♦t R N ✳ ✺✳✷ ❖♥ ❛ q✉❛s✐ ✈❛r✐❛t✐♦♥❛• ✐♥❡q✉❛t✐♦♥ ❈♦♥❝•✉s✐♦♥ ✶✷✶ ✶✷✷ ❈❍❆P❚❊| ✺✳ ❆PP▲■❈❆❚■❖◆❙ ❚❖ ❙❖▼❊ P|❖❇▲❊▼❙ ■◆ ❆◆❆▲(❙■❙ ❆♣♣❡♥❞✐① ❆ ❙♦♠❡ r❡s✉•ts ✐♥ t♦♣♦•♦❣② ■♥ t❤✐s ❛♣♣❡♥❞✐① ✇❡ ❣✐✈❡ s♦♠❡ r❡s✉•ts ♦♥ t❤❡ t❤❡♦r② ♦❢ t♦♣♦•♦❣✐❝❛• ✈❡❝t♦r s♣❛❝❡s ✇❤✐❝❤ ❛r❡ r❡q✉✐r❡❞ ❢♦r ♦✉r st✉❞②✳ ❊s♣❡❝✐❛••②✱ ✇❡ ✇✐•• ♠❛❦❡ ✉s❡ ♦❢ t❤❡ str✉❝t✉r❡ ♦❢ S ❛♥❞ S ′ ❛s ♥✉❝•❡❛r s♣❛❝❡s✱ ❜❛rr❡•❡❞ s♣❛❝❡s ❛♥❞ ▼♦♥t❡• s♣❛❝❡s✱ s♦ ✇❡ r❡❝❛•• t❤❡ r❡s✉•ts ❛❜♦✉t t❤♦s❡ s♣❛❝❡s ✇❤✐❝❤ ❛r❡ r❡•❡✈❛♥t t♦ ♦✉r ♣✉r♣♦s❡✳ ❲❡ ✇✐•• ♠♦st•② ❢♦••♦✇ ❬✻✾(✱ ❬✷✹(✱ ❬✻✹(✱ ❬✸✻( ❛♥❞ ❬✺✼(✳ ▲❡t ✉s ❝♦♥s✐❞❡r ❛ ✜❡•❞ K = R ♦r C ✇❤✐❝❤ ✇❡ ❡q✉✐♣ ✇✐t❤ ✐ts ❛r❝❤✐♠❡❞❡❛♥ t♦♣♦•♦❣②✳ ▲❡t E ❜❡ s♦♠❡ ✈❡❝t♦r s♣❛❝❡ ♦♥ K❀ ✇❡ ❡q✉✐♣ E ✇✐t❤ ❛ t♦♣♦•♦❣② Θ✳ ❲❡ r❡❝❛•• t❤❡✿ ❉❡✜♥✐t✐♦♥ ❆✳✵✳✵✳✶ ✭❚♦♣•♦❣✐❝❛• ✈❡❝t♦r s♣❛❝❡✮✳ (E, Θ) ✐s ❛ t♦♣♦•♦❣✐❝❛• ✈❡❝t♦r s♣❛❝❡✱ ❚❱❙ ❢♦r s❤♦rt✱ ✐❢ t❤❡ ❛♣♣•✐❝❛t✐♦♥s (x, y) → x + y ❛♥❞ (λ, x) → λ • x ❛r❡ ❝♦♥t✐♥✉♦✉s ♦♥ E × E ❛♥❞ K × E r❡s♣❡❝t✐✈❡•②✳ ❲❡ r❡❝❛•• t❤❛t t❤❡ t♦♣♦•♦❣② ♦❢ ❛ ❚❱❙ ✐s ❢✉••② ❞❡✜♥❡❞ ❜② ❛ ❜❛s✐s ♦❢ ♥❡✐❣❤❜♦✉r✲ ❤♦♦❞s ♦❢ 0✳ ■♥ t❤❡ s❡q✉❡• ♦❢ t❤✐s ❛♣♣❡♥❞✐①✱ E, F, . . . ✇✐•• ❞❡♥♦t❡ t♦♣♦•♦❣✐❝❛• ✈❡❝t♦r s♣❛❝❡s ✭✇❡ ♦♠✐t t♦ ♥♦t❡ t❤❡✐r r❡s♣❡❝t✐✈❡ t♦♣♦•♦❣✐❡s✮✳ ❆✳✶ ▲♦❝❛••② ❝♦♥✈❡① t♦♣♦•♦❣✐❝❛• ✈❡❝t♦r s♣❛❝❡s ❆✳✶✳✶ ❋✐rst ❞❡✜♥✐t✐♦♥s ❉❡✜♥✐t✐♦♥ ❆✳✶✳✶✳✶ ✭▲♦❝❛••② ❝♦♥✈❡① t♦♣♦•♦❣✐❝❛• ✈❡❝t♦r s♣❛❝❡✮✳ ❆ ❚❱❙ E ✐s s❛✐❞ t♦ ❜❡ •♦❝❛••② ❝♦♥✈❡① ✐❢ 0 ∈ E ❤❛s ❛ ❜❛s✐s ♦❢ ♥❡✐❣❤❜♦✉r❤♦♦❞s ❝♦♥s✐st✐♥❣ ✐♥ ❝♦♥✈❡① s❡ts✳ ❆ ❝r✉❝✐❛• ♥♦t✐♦♥ ✇✐•• ❜❡ t❤❛t ♦❢✿ ❉❡✜♥✐t✐♦♥ ❆✳✶✳✶✳✷ ✭❇❛rr❡•✮✳ ❆ ♣❛rt A ⊂ E ✐s ❛ ❜❛rr❡• ✐❢ ✐t ✈❡r✐✜❡s✿ ✶✷✸ ✶✷✹ ❆PP❊◆❉■❳ ❆✳ ❙❖▼❊ |❊❙❯▲❚❙ ■◆ ❚❖P❖▲❖•( • A ✐s ❛❜s♦r❜✐♥❣✿ ❢♦r ❡✈❡r② x ∈ E t❤❡r❡ ❡①✐sts c > 0 s✳t✳ |λ| < c ⇒ λ • x ∈ A❀ • A ✐s ❜❛•❛♥❝❡❞✿ ❢♦r ❡✈❡r② x ∈ A✱ ❢♦r ❡✈❡r② λ s✳t✳ |λ| < 1✱ λ • x ∈ A❀ • A ✐s ❝•♦s❡❞❀ • A ✐s ❝♦♥✈❡①✳ ❲❡ ❤❛✈❡ t❤❡✿ Pr♦♣♦s✐t✐♦♥ ❆✳✶✳✶✳✶✳ ■♥ ❛ ❚❱❙✱ 0 ❤❛s ❛ ❜❛s✐s ♦❢ ♥❡✐❣❤❜♦✉r❤♦♦❞s ❝♦♥s✐st✐♥❣ ✐♥ ❜❛rr❡•s✳ ❆✳✶✳✷ ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ❜② s❡♠✐♥♦r♠s ❉❡✜♥✐t✐♦♥ ❆✳✶✳✷✳✶ ✭❙❡♠✐♥♦r♠✮✳ p : E → R + ✐s ❛ s❡♠✐♥♦r♠ ♦♥ E ✐❢ ✐t ✈❡r✐✜❡s✿ • ❋♦r ❡✈❡r② x, y ∈ E p(x + y) ≤ p(x) + p(y)❀ • ❋♦r ❡✈❡r② λ ∈ K✱ x ∈ E✿ p(λ • x) = |λ| • p(x) ❲❡ ♥♦t❡ t❤❛t ✐❢ p ✐s ❛ s❡♠✐♥♦r♠ t❤❡r❡ ❤❛s t♦ ❜❡ p(0) = 0✳ ❚❤❡ ❝♦♥t✐♥✉✐t② ♦❢ s❡♠✐♥♦r♠s ✐s ❡❛s✐•② ❝❤❛r❛❝t❡r✐③❡❞✿ Pr♦♣♦s✐t✐♦♥ ❆✳✶✳✷✳✶✳ ▲❡t p ❜❡ ❛ s❡♠✐♥♦r♠ ♦♥ E✳ ❚❤❡♥ t❤❡ t❤r❡❡ ❢♦••♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛•❡♥t✿ • ❚❤❡ ♦♣❡♥ ✉♥✐t ❜❛•• {x|p(x) < 1} ✐s ❛♥ ♦♣❡♥ s❡t❀ • p ✐s ❝♦♥t✐♥✉♦✉s ❛t 0❀ • p ✐s ❝♦♥t✐♥✉♦✉s✳ ❆•s♦✱ ✐♥ t❤❛t ❝❛s❡✱ t❤❡ ♦♣❡♥ ✉♥✐t ❜❛•• ♦❢ p ✐s ❛ ❜❛rr❡•✳ ❈♦♥✈❡rs❡•② t❤❡r❡ ✐s t❤❡✿ Pr♦♣♦s✐t✐♦♥ ❆✳✶✳✷✳✷✳ ▲❡t T ❜❡ ❛ ❜❛rr❡• ✐♥ E✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ s❡♠✐✲ ♥♦r♠ p s✉❝❤ t❤❛t T ✐s t❤❡ ♦♣❡♥ ✉♥✐t ❜❛•• ♦❢ p ❛♥❞ p ✐s ❝♦♥t✐♥✉♦✉s ✐❢ ❛♥❞ ♦♥•② ✐❢ T ✐s ❛ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ 0✳ p ✐s t❤❡ ❣❛✉❣❡ ♦❢ t❤❡ ❜❛rr❡• T ✿ p(x) = inf {λ ≥ 0|x ∈ λ • T } ✭❆✳✶✮ ❚❤❡s❡ t✇♦ r❡s✉•ts ❝♦♠❜✐♥❡ ✐♥t♦ t❤❡✿ ❚❤❡♦r❡♠ ❆✳✶✳✷✳✶✳ ■♥ ❛ •♦❝❛••② ❝♦♥✈❡① t♦♣♦•♦❣✐❝❛• ✈❡❝t♦r s♣❛❝❡✱ t❤❡ ♦♣❡♥ ✉♥✐t ❜❛••s ♦❢ t❤❡ ❝♦♥t✐♥✉♦✉s s❡♠✐♥♦r♠s ❢♦r♠ ❛ ❜❛s✐s ♦❢ ♥❡✐❣❤❜♦✉r❤♦♦❞s ♦❢ 0✳ ❆✳✶✳ ▲❖❈❆▲▲( ❈❖◆❱❊❳ ❚❖P❖▲❖•■❈❆▲ ❱❊❈❚❖| ❙P❆❈❊❙ ✶✷✺ ◆♦✇ ✐❢ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡✿ ❉❡✜♥✐t✐♦♥ ❆✳✶✳✷✳✷ ✭❇❛s✐s ♦❢ ❝♦♥t✐♥✉♦✉s s❡♠✐♥♦r♠s✮✳ ❆ ❢❛♠✐•② P ♦❢ ❝♦♥t✐♥✉♦✉s s❡♠✐♥♦r♠s ♦♥ ❛ •♦❝❛••② ❝♦♥✈❡① ❚❱❙ ✐s ❛ ❜❛s✐s ♦❢ ❝♦♥t✐♥✉♦✉s s❡♠✐♥♦r♠s ✐❢ ❢♦r ❡✈❡r② p ∈ P✱ t❤❡r❡ ❡①✐sts q ∈ P s✉❝❤ t❤❛t p = O(q)✳ t❤❡ t✇♦ ♥♦t✐♦♥s ❝♦♠❜✐♥❡ ✐♥ t❤❡✿ ❚❤❡♦r❡♠ ❆✳✶✳✷✳✷✳ ■♥ ❛ •♦❝❛••② ❝♦♥✈❡① ❚❱❙✿ • ▲❡t P ❜❡ ❛ ❜❛s✐s ♦❢ ❝♦♥t✐♥✉♦✉s s❡♠✐♥♦r♠s❀ ✐❢ p ∈ P ✇❡ ♥♦t❡ U p ❢♦r ✐ts ♦♣❡♥ ✉♥✐t ❜❛••✳ ❚❤❡♥ t❤❡ λU p ✱ λ > 0✱ p ∈ P ❛r❡ ❛ ❜❛s✐s ♦❢ ♥❡✐❣❤❜♦✉r❤♦♦❞s ♦❢ 0✳ • ❈♦♥✈❡rs❡•② ✐❢ ♦♥❡ ❣✐✈❡s ♦♥❡s❡•❢ ❛ ❜❛s✐s ♦❢ ♥❡✐❣❤❜♦✉r❤♦♦❞s ♦❢ 0 ❝♦♥s✐st✐♥❣ ✐♥ ❜❛rr❡•s✱ t❤❡♥ t❤❡ ❛ss♦❝✐❛t❡❞ s❡♠✐♥♦r♠s ❛r❡ ❛ ❜❛s✐s ♦❢ ❝♦♥t✐♥✉♦✉s s❡♠✐♥♦r♠s✳ ❚❤❡ r❡s✉•t ✇❡ ✇✐•• ✉s❡ ✐♥ ♣r❛❝t✐❝❡ ✐s t❤❡ ❢♦••♦✇✐♥❣✿ ❈♦r♦••❛r② ❆✳✶✳✷✳✶✳ ▲❡t E✱ F ❜❡ t✇♦ •♦❝❛••② ❝♦♥✈❡① ❚❱❙✳ ❚❤❡♥ ❛ •✐♥❡❛r ♠❛♣♣✐♥❣ f : E → F ✐s ❝♦♥t✐♥✉♦✉s ✐❢ ❛♥❞ ♦♥•② ✐❢ ❢♦r ❡✈❡r② ❝♦♥t✐♥✉♦✉s s❡♠✐♥♦r♠ q ♦♥ F t❤❡r❡ ❡①✐sts ❛ ❝♦♥t✐♥✉♦✉s s❡♠✐♥♦r♠ p ♦♥ E s✉❝❤ t❤❛t ❢♦r ❡✈❡r② x ∈ E✿ q(f (x)) ≤ p(x) ✭❆✳✷✮ ❊s♣❡❝✐❛••② ❛ •✐♥❡❛r ❢♦r♠ ♦♥ E ✐s ❝♦♥t✐♥✉♦✉s ✐❢ ❛♥❞ ♦♥•② ✐❢ ✐t ✐s ❝♦♥t✐♥✉♦✉s ❢♦r ❛t •❡❛st ♦♥❡ ❝♦♥t✐♥✉♦✉s s❡♠✐♥♦r♠ ♦♥ E✳ ❆✳✶✳✸ ▼❡tr✐③❛❜•❡ •♦❝❛••② ❝♦♥✈❡① ❚❱❙ ❋✐rst ✇❡ r❡❝❛•• t❤❡ ❣❡♥❡r❛•✿ ❉❡✜♥✐t✐♦♥ ❆✳✶✳✸✳✶ ✭▼❡tr✐③❛❜•❡ ❚❱❙✮✳ ❚❤❡ ❚❱❙ E ✐s s❛✐❞ t♦ ❜❡ ♠❡tr✐③❛❜•❡ ✐❢ ✐t ✐s ❍❛✉ss❞♦r❢ ❛♥❞ ✐❢ 0 ∈ E ❛❞♠✐ts ❛ ❝♦✉♥t❛❜•❡ ❜❛s✐s ♦❢ ♥❡✐❣❤❜♦✉r❤♦♦❞s✳ ❛♥❞✿ ❚❤❡♦r❡♠ ❆✳✶✳✸✳✶✳ ❆ ❚❱❙ ✐s ♠❡tr✐③❛❜•❡ ✐❢ ❛♥❞ ♦♥•② ✐❢ ✐ts t♦♣♦•♦❣② ✐s ❞❡✜♥❡❞ ❜② ❛ tr❛♥s•❛t✐♦♥✲✐♥✈❛r✐❛♥t ❞✐st❛♥❝❡✳ ■♥ t❤❡ ❝❛s❡ ♦❢ •♦❝❛••② ❝♦♥✈❡① ❚❱❙✱ ✇❡ ❤❛✈❡ t❤❡ ♠♦r❡ ♣r❡❝✐s❡✿ ❚❤❡♦r❡♠ ❆✳✶✳✸✳✷✳ ▲❡t E ❜❡ ❛ ♠❡tr✐③❛❜•❡✱ •♦❝❛••② ❝♦♥✈❡① ❚❱❙❀ t❤❡r❡ ❡①✐sts ❛ ❝♦✉♥t❛❜•❡ ❜❛s✐s ♦❢ ❝♦♥t✐♥✉♦✉s s❡♠✐♥♦r♠s (p j ) j∈N ♦♥ E✳ ▲❡t ❛•s♦ (a j ) ∈ R + ❜❡ s✉❝❤ t❤❛t a j < ∞ ❛♥❞ ❞❡✜♥❡✿ d(x, y) = ∞ j=0 a j • p j (x -y) 1 + p j (x -y) ✭❆✳✸✮ ❚❤❡♥ d ✐s ❛ tr❛♥s•❛t✐♦♥✲✐♥✈❛r✐❛♥t ❞✐st❛♥❝❡ ♦♥ E ✇❤✐❝❤ ❞❡✜♥❡s t❤❡ t♦♣•♦❣② ♦♥ E✳ ✶✷✻ ❆PP❊◆❉■❳ ❆✳ ❙❖▼❊ |❊❙❯▲❚❙ ■◆ ❚❖P❖▲❖•( ❲❡ ❛•s♦ ✐♥tr♦❞✉❝❡ t❤❡✿ ❉❡✜♥✐t✐♦♥ ❆✳✶✳✸✳✷ ✭❋ré❝❤❡t s♣❛❝❡✮✳ ❆ ❋ré❝❤❡t s♣❛❝❡ ✐s ❛ ♠❡tr✐③❛❜•❡✱ •♦❝❛••② ❝♦♥✲ ✈❡① ❚❱❙ ✇❤✐❝❤ ✐s ❝♦♠♣•❡t❡✳ ❆✳✷ ❉✉❛•✐t② ▲❡t E ❜❡ ❛ ❚❱❙ ✭E ✐s ♥♦t ♥❡❝❡ss❛r✐•② •♦❝❛••② ❝♦♥✈❡①✮❀ ✇❡ ♥♦t❡ E ′ ❢♦r ✐ts t♦♣♦•♦❣✐❝❛• ❞✉❛•✱ ✐❡ t❤❡ ✈❡❝t♦r s♣❛❝❡ ♦❢ t❤❡ ❝♦♥t✐♥✉♦✉s •✐♥❡❛r ❢♦r♠s ♦♥ E✳ ❲❡ r❡❝❛•• t❤❛t ❛ s✉❜s❡t A ♦❢ ❛ ❣❡♥❡r❛• ❚❱❙ E ✐s ❝❛••❡❞ ❜♦✉♥❞❡❞ ✐❢ ❢♦r ❡✈❡r② ♥❡✐❣❤❜♦✉r❤♦♦❞ U ♦❢ 0 t❤❡r❡ ❡①✐sts ❛ ♥✉♠❜❡r λ s✉❝❤ t❤❛t U ⊂ λ • A ❛♥❞ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ❢♦••♦✇✐♥❣✿ ❉❡✜♥✐t✐♦♥ ❆✳✷✳✵✳✸ ✭P♦•❛r ♦❢ ❛ s✉❜s❡t✮✳ ▲❡t A ⊂ E✳ ❲❡ ❞❡✜♥❡ t❤❡ ♣♦•❛r s❡t ♦❢ A ❛s✿ A • = x ′ ∈ E ′ sup x∈A |x ′ (x)| < 1 ✭❆✳✹✮ ❲❡ ♥♦t❡ t❤❛t A • ✐s ❛ ❝♦♥✈❡① ❛♥❞ ❜❛•❛♥❝❡❞ s❡t ❛♥❞ t❤❛t t❤❡r❡ ✐s✿ (A ∪ B) • = A • ∩ B • ✭❆✳✺✮ ❛♥❞ (λ • A) • = (1/λ) • A • ✭❆✳✻✮ ▼♦r❡♦✈❡r ✐❢ A ✐s ❜♦✉♥❞❡❞ t❤❡♥ A • ✐s ❛❜s♦r❜✐♥❣✳ ❖♥❡ ♠❛② t❤❡r❡❢♦r❡ ♣r♦✈❡ t❤❡✿ ❚❤❡♦r❡♠ ❆✳✷✳✵✳✸✳ ▲❡t S ❜❡ ❛ ♥♦♥❡♠♣t② ❢❛♠✐•② ♦❢ ❜♦✉♥❞❡❞ s❡ts ✐♥ A s✉❝❤ t❤❛t✿ • ❋♦r ❡✈❡r② A, B ∈ S t❤❡r❡ ❡①✐sts C ∈ S s✉❝❤ t❤❛t A ∪ B ⊂ C❀ • ❋♦r ❡✈❡r② A ∈ S ❛♥❞ λ ∈ K t❤❡r❡ ❡①✐sts B ∈ S s✉❝❤ t❤❛t λ • A ⊂ B✳ ❚❤❡♥✱ S • := {A • , A ∈ S} ❞❡✜♥❡s ❛ ❜❛s✐s ♦❢ ♥❡✐❣❤❜♦✉r❤♦♦❞s ♦❢ 0 ❢♦r ❛ ❝❡rt❛✐♥ t♦♣♦•♦❣② ♦♥ E ′ ✳ |❡•❡✈❛♥t ❡①❛♠♣•❡s ❛r❡✿ • ❚❤❡ ✇❡❛❦ t♦♣♦•♦❣②✱ ♦❜t❛✐♥❡❞ ❜② t❛❦✐♥❣ t❤❡ ✜♥✐t❡ s✉❜s❡ts ♦❢ E ❢♦r S❀ • ❚❤❡ ❝♦♥✈❡①✲❝♦♠♣❛❝t t♦♣♦•♦❣②✱ ♦❜t❛✐♥❡❞ ❜② t❛❦✐♥❣ t❤❡ ❝♦♥✈❡① ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡ts ♦❢ E ❢♦r S❀ • ❚❤❡ ❝♦♠♣❛❝t t♦♣♦•♦❣②✱ ♦❜t❛✐♥❡❞ ❜② t❛❦✐♥❣ t❤❡ ❝♦♠♣❛❝t s✉❜s❡ts ♦❢ E ❢♦r S❀ • ❚❤❡ str♦♥❣ t♦♣♦•♦❣②✱ ♦❜t❛✐♥❡❞ ❜② t❛❦✐♥❣ ❛•• ❜♦✉♥❞❡❞ s✉❜s❡ts ♦❢ E ❢♦r S✳ ❆✳✸✳ ❇❆||❊▲❊❉ ❙P❆❈❊❙ ❆◆❉ ▼❖◆❚❊▲ ❙P❆❈❊❙ ✶✷✼ ❆✳✸ ❇❛rr❡•❡❞ s♣❛❝❡s ❛♥❞ ▼♦♥t❡• s♣❛❝❡s ❆✳✸✳✶ ❇❛rr❡•❡❞ s♣❛❝❡s ❉❡✜♥✐t✐♦♥ ❆✳✸✳✶✳✶ ✭❇❛rr❡•❡❞ s♣❛❝❡✮✳ ❆ t♦♣♦•♦❣✐❝❛• ✈❡❝t♦r s♣❛❝❡ E ✐s s❛✐❞ t♦ ❜❡ ❜❛rr❡•❡❞ ✐❢ ❡✈❡r② ❜❛rr❡• ✐♥ E ✐s ❛ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ 0 ✐♥ E✳ ■t ♠❛② ❜❡ ♣r♦✈❡❞ t❤❛t ❛♥② ❇❛✐r❡ s♣❛❝❡ ✐s ❜❛rr❡•❡❞❀ ✇❡ ✇✐•• ♦♥•② ♥❡❡❞ t❤❡ ❢♦••♦✇✐♥❣✿ ❚❤❡♦r❡♠ ❆✳✸✳✶✳✶✳ ❆♥② ❋ré❝❤❡t s♣❛❝❡ ✐s ❜❛rr❡•❡❞✳ ❇❛rr❡•❡❞ s♣❛❝❡s ❤❛✈❡ ♠❛♥② ♣r♦♣❡rt✐❡s ✇❤✐❝❤ ❛r❡ ✉s❡❢✉• ✐♥ ❛♥❛•②s✐s✳ ❚❤❡ ♦♥❡ ✇❡ ✇✐•• ✉s❡ ✐s t❤❡ ❢♦••♦✇✐♥❣✿ ❚❤❡♦r❡♠ ❆✳✸✳✶✳✷ ✭❈•♦s❡❞ ❣r❛♣❤ t❤❡♦r❡♠✮✳ ▲❡t E ❛♥❞ F ❜❡ t✇♦ ❚❱❙❀ s✉♣♣♦s❡ t❤❛t E ✐s ❜❛rr❡•❡❞ ❛♥❞ t❤❛t F ✐s ❝♦♠♣•❡t❡✳ ❚❤❡♥✱ ✐❢ u : E → F ✐s ❛ •✐♥❡❛r ♠❛♣ ✇✐t❤ ❝•♦s❡❞ ❣r❛♣❤✱ t❤❡♥ u ✐s ❝♦♥t✐♥✉♦✉s✳ ❆✳✸✳✷ ▼♦♥t❡• s♣❛❝❡s ❉❡✜♥✐t✐♦♥ ❆✳✸✳✷✳✶ ✭▼♦♥t❡• s♣❛❝❡✮✳ ❆ ❚❱❙ E ✐s s❛✐❞ t♦ ❜❡ ❛ ▼♦♥t❡• s♣❛❝❡ ✐❢ ✐t ✈❡r✐✜❡s t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿ • E ✐s ❍❛✉ss❞♦r❢❀ • E ✐s •♦❝❛••② ❝♦♥✈❡①❀ • E ✐s ❜❛rr❡•❡❞❀ • ❊✈❡r② ❝•♦s❡❞ ❛♥❞ ❜♦✉♥❞❡❞ s♣❛❝❡ ✐♥ E ✐s ❝♦♠♣❛❝t✳ ❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣❡rt② ♦❢ ▼♦♥t❡• s♣❛❝❡s✿ ❚❤❡♦r❡♠ ❆✳✸✳✷✳✶✳ ■♥ t❤❡ ❞✉❛• ♦❢ ❛ ▼♦♥t❡• s♣❛❝❡✱ ❡✈❡r② ✇❡❛❦•② ❝♦♥✈❡r❣❡♥t s❡✲ q✉❡♥❝❡ ✐s str♦♥❣•② ❝♦♥✈❡r❣❡♥t✳ ▲❡t ✉s ❛•s♦ ♠❡♥t✐♦♥✿ ❚❤❡♦r❡♠ ❆✳✸✳✷✳✷✳ ❚❤❡ str♦♥❣ ❞✉❛• ♦❢ ❛ ▼♦♥t❡• s♣❛❝❡ ✐s ❛ ▼♦♥t❡• s♣❛❝❡✳ ✶✷✽ ❆PP❊◆❉■❳ ❆✳ ❙❖▼❊ |❊❙❯▲❚❙ ■◆ ❚❖P❖▲❖•( ❆✳✹ ◆✉❝•❡❛r s♣❛❝❡s ❆✳✹✳✶ ❚♦♣♦•♦❣✐❝❛• t❡♥s♦r ♣r♦❞✉❝ts ▲❡t E ❛♥❞ F ❜❡ t✇♦ ❚❱❙ ❛♥❞ •❡t B(E, F ) ❜❡ t❤❡ ✈❡❝t♦r s♣❛❝❡ ♦❢ ❝♦♥t✐♥✉♦✉s ❜✐•✐♥❡❛r ❢♦r♠s ♦♥ E × F ✳ ❋♦r (x, y) ∈ E × F ✇❡ ❝♦♥s✐❞❡r t❤❡ ❡✈❛•✉❛t✐♦♥ ♠❛♣♣✐♥❣✿ u x,y = f ∈ B(E, F ) → f (x, y) ∈ K ✭❆✳✼✮ ❈•❡❛r•②✱ u x,y ∈ B(E, F ) ′ ✱ t❤❡ ❞✉❛• s♣❛❝❡ ♦❢ B(E, F )✳ ◆♦✇ •❡t ✉s ❝♦♥s✐❞❡r✿ χ = (x, y) ∈ E × F → u x,y ∈ B(E, F ) ′ ✭❆✳✽✮ ❲❡ ♥♦t❡ E ⊗ F ❢♦r t❤❡ •✐♥❡❛r ❤✉•• ♦❢ χ(E × F ) ✐♥ B(E, F ) ′ ❀ ✐t ✇✐•• s♦♠❡t✐♠❡s ♠❛❦❡ ✐♥t✉✐t✐✈❡ s❡♥s❡ t♦ ♥♦t❡ x ⊗ y ❢♦r u x,y ✳ ❉❡✜♥✐t✐♦♥ ❆✳✹✳✶✳✶ ✭❚❡♥s♦r ♣r♦❞✉❝t✮✳ E ⊗ F ✐s t❤❡ t♦♣♦•♦❣✐❝❛• t❡♥s♦r ♣r♦❞✉❝t ♦❢ E ❛♥❞ F ✳ ❉❡✜♥✐t✐♦♥ ❆✳✹✳✶✳✷ ✭❈❛♥♦♥✐❝❛• ❜✐•✐♥❡❛r ♠❛♣✮✳ χ ✐s t❤❡ ❝❛♥♦♥✐❝❛• ❜✐•✐♥❡❛r ♠❛♣ ♦❢ E × F ✐♥t♦ E ⊗ F ✳ ❲❡ ❤❛✈❡ t❤❡✿ ❚❤❡♦r❡♠ ❆✳✹✳✶✳✶✳ ▲❡t E✱ F ❛♥❞ G ❜❡ t❤r❡❡ t♦♣♦•♦❣✐❝❛• ✈❡❝t♦r s♣❛❝❡s✳ ❚❤❡♥ t❤❡ ❢♦••♦✇✐♥❣ ❛♣♣•✐❝❛t✐♦♥✿ u ∈ L(E ⊗ F, G) → u • χ ∈ B(E, F ; G) ✭❆✳✾✮ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠✳ ❊s♣❡❝✐❛••②✱ t❤❡r❡ ✐s✿ (E ⊗ F ) ′ ≈ B(E, F ) ✭❆✳✶✵✮ ❲❡ ♠❛② ❝❤❛r❛❝t❡r✐③❡ t❤❡ t♦♣♦•♦❣② ♦♥ E ⊗ F t❤r♦✉❣❤ t❤❡ ❢♦••♦✇✐♥❣✿ ❚❤❡♦r❡♠ ❆✳✹✳✶✳✷✳ ▲❡t p ❛♥❞ q ❜❡ t✇♦ s❡♠✐♥♦r♠s ♦♥ E ❛♥❞ F r❡s♣❡❝t✐✈❡•②✱ s✉❝❤ t❤❛t p ❛♥❞ q ❛r❡ t❤❡ ❣❛✉❣❡s ♦❢ t❤❡ ❜❛rr❡•s T p ❛♥❞ T q r❡s♣❡❝t✐✈❡•②✳ ❚❤❡♥✱ t❤❡ ❢♦••♦✇✐♥❣ ❞❡✜♥❡s ❛ s❡♠✐♥♦r♠ ♦♥ E ⊗ F ✿ r(z) = inf p(x i ) • q(y i ) u = x i ⊗ y i ✭❆✳✶✶✮ ▼♦r❡♦✈❡r t❤❡r❡ ✐s✿ r(x ⊗ y) = p(x) • q(y) ✭❆✳✶✷✮ ❛♥❞ r ✐s t❤❡ ❣❛✉❣❡ ♦❢ t❤❡ s♠❛••❡st ❜❛rr❡• ❝♦♥t❛✐♥✐♥❣ χ(T 1 × T 2 )✳ ❲❡ ♥♦t❡✿ r = p ⊗ q ✭❆✳✶✸✮ ❆✳✹✳ ◆❯❈▲❊❆| ❙P❆❈❊❙ ✶✷✾ ■❢ E ❛♥❞ F ❛r❡ ♠❡tr✐③❛❜•❡ •♦❝❛••② ❝♦♥✈❡① ❚❱❙ ✇✐t❤ r❡s♣❡❝t✐✈❡ ❝♦✉♥t❛❜•❡ ❜❛s✐s ♦❢ s❡♠✐♥♦r♠s (p i ) ❛♥❞ (q j ) ✇❡ ✇✐•• ✉s✉❛••② ❡q✉✐♣ E ⊗ F ✇✐t❤ t❤❡ ♠❡tr✐③❛❜•❡ •♦❝❛••② ❝♦♥✈❡① ❚❱❙ t♦♣♦•♦❣② ✇✐t❤ ❜❛s✐s ♦❢ s❡♠✐♥♦r♠s (p i ⊗ q j )✳ ■♥ t❤❡ ❝❛s❡ ✇❤❡r❡ E ❛♥❞ F ❛r❡ ❇❛♥❛❝❤ s♣❛❝❡s✱ ✇❡ s✐♠♣•② ♦❜t❛✐♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡ str✉❝t✉r❡ ♦♥ E ⊗ F ✱ ✇✐t❤ ♥♦r♠ • E ⊗ • F ✳ ❚❤✐s s♣❡❝✐❛• ❝❛s❡ ✐s r❡•❡✈❛♥t t♦ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ s♣❛❝❡s D p,k ✐♥ ▼❛••✐❛✈✐♥ ❝❛•❝✉•✉s✳ ❲❡ ❝♦♥❝•✉❞❡ t❤✐s ♣❛r❛❣r❛♣❤ ✇✐t❤ t❤❡✿ ❚❤❡♦r❡♠ ❆✳✹✳✶✳✸✳ ▲❡t E ❛♥❞ F ❜❡ t✇♦ ♠❡tr✐③❛❜•❡✱ •♦❝❛••② ❝♦♥✈❡① ❚❱❙✳ ❚❤❡♥ ❛♥② ❡•❡♠❡♥t ♦❢ E ⊗ F ♠❛② ❜❡ ✇r✐tt❡♥ ✐♥ t❤❡ ❢♦r♠✿ ∞ i=1 λ i • x i ⊗ y i ✭❆✳✶✹✮ ✇❤❡r❡ |λ i | < ∞✳ ❆✳✹✳✷ ◆✉❝•❡❛r ♠❛♣♣✐♥❣s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ▲❡t E ❛♥❞ F ❜❡ t✇♦ ❇❛♥❛❝❤ s♣❛❝❡s❀ ✇❡ ❡q✉✐♣ t❤❡ str♦♥❣ ❞✉❛• E ′ ♦❢ E ✇✐t❤ ✐ts ♥❛t✉r❛• ❇❛♥❛❝❤ s♣❛❝❡ str✉❝t✉r❡✳ ■❢✿ v ∈ E ′ ⊗ F = r i=1 f i ⊗ y i ✭❆✳✶✺✮ ✇❡ ❞❡✜♥❡ ❛♥ ❡•❡♠❡♥t u ∈ L(E, F ) ❛s✿ u(x) = r i=1 f i (x) • y i ✭❆✳✶✻✮ ❚❤❡♥ t❤❡ •✐♥❡❛r ♠❛♣♣✐♥❣✿ v ∈ E ′ ⊗ F → u ∈ L(E, F ) ✭❆✳✶✼✮ ✐s ❛♥ ✭❛•❣❡❜r❛✐❝✮ ✐s♦♠♦r♣❤✐s♠❀ •❡t ✉s ♥♦t❡ τ ❢♦r t❤❡ ❡①t❡♥s✐♦♥ ♦❢ t❤✐s ✐s♦♠♦r♣❤✐s♠ t♦ t❤❡ ❝♦♠♣•❡t✐♦♥ ♦❢ E ′ ⊗ F ✳ ❚❤❡♥ t❤❡ ♥✉❝•❡❛r •✐♥❡❛r ♠❛♣♣✐♥❣s ✐♥ L(E, F ) ❛r❡ t❤♦s❡ •✐♥❡❛r ❛♣♣•✐❝❛t✐♦♥s ✇❤✐❝❤ ❛r❡ ✐♥ t❤❡ ✐♠❛❣❡ ♦❢ τ ✳ ❆✳✹✳✸ ◆✉❝•❡❛r ♠❛♣♣✐♥❣s ✐♥ ❛ •♦❝❛••② ❝♦♥✈❡① ❚❱❙ ❲❡ st❛rt ✇✐t❤ s♦♠❡ ♥♦t❛t✐♦♥✳ ▲❡t T ❜❡ ❛ ❜❛rr❡• ✐♥ t❤❡ ✈❡❝t♦r E✳ ❚❤❡♥✱ t❤❡ ❢❛♠✐••② (n -1 • T, n ∈ N -{0}) ✐s ❛ ♥❡✐❣❤❜♦✉r❤♦♦❞ ❜❛s✐s ♦❢ 0 ❢♦r s♦♠❡ ♠❡tr✐③❛❜•❡ •♦❝❛••② ❝♦♥✈❡① ❚❱❙ t♦♣♦•♦❣② ✶✸✵ ❆PP❊◆❉■❳ ❆✳ ❙❖▼❊ |❊❙❯▲❚❙ ■◆ ❚❖P❖▲❖•( ♦♥ E✳ ▲❡t ❛•s♦ p ❜❡ t❤❡ ❣❛✉❣❡ ♦❢ T ❀ t❤❡♥ ❛ ♥♦r♠ ✐s ❞❡✜♥❡❞ ♦♥ t❤❡ ✈❡❝t♦r s♣❛❝❡ E T := E/p -1 (0) ❜②✿ x = p(x) ✭❆✳✶✽✮ ❢♦r ❛♥② x ∈ x✳ ❲❡ ♥♦t❡ ẼT t❤❡ ❝♦♠♣•❡t✐♦♥ ♦❢ E T ❢♦
( a 0 A 0 + t • a 1 A 1 ) ✭❉✳✼✮ ■❢ a ∈ A 0 ∩ A 1 ✇❡ ❞❡✜♥❡ ❛ ❜♦r❡•✐❛♥ ❢✉♥❝t✐♦♥ ♦♥ R + ❜② t❤❡ ❡q✉❛t✐♦♥✿ J(t, a) = max ( a A 0 , t • a A 1 ) ✭❉✳✽✮ ❋✐♥❛••② ✐❢ φ ✐s ❛ ♣♦s✐t✐✈❡ ❜♦r❡•✐❛♥ ❢✉♥❝t✐♦♥ ♦♥ R + ✱ ❢♦r 0 ≤ θ ≤ 1 ❛♥❞ q ∈ [1, ∞] ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ q✉❛♥t✐t② ✭✇❤✐❝❤ ✐s ♣♦ss✐❜•② +∞✮✿ Φ θ,q (φ) = ∞ 0 t -θ φ(t) q dt t 1 q
✭❉✳✾✮ ❖♥ ♦♥❡ ❤❛♥❞ ✇❡ ❞❡✜♥❡ ❛ s♣❛❝❡ t❤r♦✉❣❤ K ❜② s❡tt✐♥❣✿ a θ,q,K = Φ θ,q (K(•, a)) ✭❉✳✶✵✮ ❉✳✷✳ ❚❍❊ |❊❆▲ ▼❊❚❍❖❉ ❖❋ ■◆❚❊|P❖▲❆❚■❖◆ ✶✹✸ ❛♥❞✿ Āθ,q,K = a ∈ 0 + A 1 a θ,q,K < ∞ ✭❉✳✶✶✮ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞ ✇❡ ❞❡✜♥❡ ❛ s❡t Āθ,q,J ❜❛s❡❞ ♦♥ J ❛s t❤❡ s❡t ♦❢ t❤♦s❡ a ∈ A 0 ∩ A 1 ✇❤✐❝❤ ♠❛② ❜❡ r❡♣r❡s❡♥t❡❞ ❛s✿ • Āθ,q,K , a θ,q,K ❛♥❞ Āθ,q,J , a θ,q,J ❛r❡ t✇♦ ♥♦r♠❡❞ ✈❡❝t♦r s♣❛❝❡s✱ t♦ ✇❤✐❝❤ ✇❡r❡❢❡r ❛s t❤❡ ❑✲♠❡t❤♦❞ ✭r❡s♣✳ t❤❡ ❏✲♠❡t❤♦❞✮ r❡❛• ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡s✳

• ❚❤❡s❡ s♣❛❝❡s ❛r❡ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡s ♦❢ ❡①♣♦♥❡♥t θ✳ • ■❢ 0 < θ < 1 ❛♥❞ q ∈ [1, ∞] t❤❡ ❑ ❛♥❞ ❏ s♣❛❝❡s ❝♦✐♥❝✐❞❡ ✇✐t❤ ❡q✉✐✈❛•❡♥❝❡ ♦❢ ♥♦r♠s✳ ❚❤❡♥ ✇❡ s✐♠♣•② ♥♦t❡ Āθ,q ♦r ❡q✉✐✈❛•❡♥t•② (A 0 , A 1 ) θ,q ❢♦r t❤❛t s♣❛❝❡✳

• ■❢ A 0 ❛♥❞ A 1 ❛r❡ ❇❛♥❛❝❤ s♣❛❝❡s t❤❡♥ s♦ ✐s Āθ,q ✳ • ■❢ q < ∞ t❤❡♥ A 0 ∩ A 1 ✐s ❞❡♥s❡ ✐♥ Āθ,q ✳ ❲❡ ✇✐• ❛•s♦ ♥❡❡❞ t❤❡ t✇♦ ❢♦••♦✇✐♥❣ r❡s✉•ts✿ ❚❤❡♦r❡♠ ❉✳✷✳✵✳✶✼ ✭|❡✐t❡r❛t✐♦♥ t❤❡♦r❡♠✮✳ ■❢ A 0 ✱ A 1 ❛r❡ t✇♦ ❝♦♠♣❛t✐❜•❡ ❇❛♥❛❝❤ s♣❛❝❡s ❛♥❞ ✐❢ 0 < θ, θ 0 , θ 1 , η < 1 ✈❡r✐❢②✿

θ = (1 -η) • θ 0 + η • θ 1 ✭❉✳✶✹✮ t❤❡♥ t❤❡ ❢♦••♦✇✐♥❣ ❤♦•❞s✿
Āθ 0 ,q , Āθ 1 ,q η,q = Āθ,q ✭❉✳✶✺✮ ❚❤❡♦r❡♠ ❉✳✷✳✵✳✶✽ ✭❉✉❛•✐t② t❤❡♦r❡♠✮✳ ■❢ A 0 ❛♥❞ A 1 ❛r❡ t✇♦ ❝♦♠♣❛t✐❜•❡ ❇❛♥❛❝❤ s♣❛❝❡s s✉❝❤ t❤❛t A 0 ∩ A 1 ✐s ❞❡♥s❡ ✐♥ ❜♦t❤ A 0 ❛♥❞ A 1 t❤❡♥✿ ❚❤❡♦r❡♠ ❉✳✹✳✵✳✷✹✳ ❖♥ X α,r,q ✱ t❤❡ ❢♦••♦✇✐♥❣ q✉❛♥t✐t②✿

x := x X + ∞ 0 (T t -Id) r x q X t -αq dt t

1 q ✭❉✳✷✺✮ ❞❡✜♥❡s ❛ ♥♦r♠ ✇❤✐❝❤ ✐s ❡q✉✐✈❛•❡♥t t♦ t❤❡ ✉s✉❛• r❡❛• ✐♥t❡r♣♦•❛t✐♦♥ ♥♦r♠s✳ ❖♥❡ ♠❛② ❛•s♦ ♣r♦✈❡✿ ❚❤❡♦r❡♠ ❉✳✹✳✵✳✷✺✳ ▲❡t ✉s ✇r✐t❡ [α] = k ❛♥❞ {α} = β✳ ❚❤❡♥✱ x ∈ X α,r,q ✐❢ ❛♥❞ ♦♥•② ✐❢ x ∈ D(A k ) ❛♥❞ A k x ∈ X β,1,q ✳ ❆•s♦✱ t❤❡ ❢♦••♦✇✐♥❣ ❞❡✜♥❡s ❛ ♥♦r♠ ♦♥ X α,r,q
✇❤✐❝❤ ✐s ❡q✉✐✈❛•❡♥t t♦ t❤❡ ♣r❡✈✐♦✉s ♦♥❡✿ x = x A k + ∞ 0 (T t -Id)x q A k t -αq dt t

  ét❛❜•✐r♦♥s ❞❡s •✐❡♥s ❞✐r❡❝ts ❡♥tr❡ ❝❡s tr♦✐s ❞❡r♥✐❡rs rés✉•t❛ts ✭❢♦r♠✉•❡s ❞❡ ❚❛♥❛❦❛✱ ❞❡ ❇♦✉•❡❛✉✲(♦r ❡t ❞❡ ❋ö••♠❡r✲Pr♦tt❡r✲❙❤✐r②❛❡✈✮ ❡t •❡s ♥ôtr❡s❀ ♥♦✉s r❡✈✐❡♥❞r♦♥s s✉r ❝❡ ♣♦✐♥t ♣•✉s t❛r❞✳ ▼❡♥t✐♦♥♥♦♥s é❣❛•❡♠❡♥t ✉♥ ❞❡r♥✐❡r rés✉t❛t ❞✉ ♠ê♠❡ t②♣❡✱ ✈♦✐r ❬✷✷( ♦✉ ❬✶✷(✱ ❜✐❡♥ q✉✬✐• s♦✐t ♠♦✐♥s ❞✐r❡❝t❡♠❡♥t •✐é à ♥♦tr❡ ét✉❞❡✿ ❚❤é

  ✈❡rs❛♥t✧ •❡s ✐♥té❣r❛t✐♦♥s ♣❛r ♣❛rt✐❡s✳ ✵✳✸ ❆♣♣♦rts ❞❡ ❝❡ tr❛✈❛✐• ❙✐ •✬✐♥térêt t❤é♦r✐q✉❡ ❞❡s rés✉•t❛ts ❞✉ ♣❛r❛❣r❛♣❤❡ ♣ré❝é❞❡♥t ❡st ❝•❛✐r✱ ♦♥ ✈♦✐t é❣❛•❡✲ ♠❡♥t q✉❡ •❡s ❤②♣♦t❤ès❡s ❢❛✐t❡s ✭•❡s ✈❛r✐❛❜•❡s ❛•é❛t♦✐r❡s ❞♦✐✈❡♥t êtr❡ ❞❛♥s D✮ s♦♥t tr♦♣ ❢♦rt❡s ♣♦✉r •❡s ❛♣♣•✐❝❛t✐♦♥s ♣r❛t✐q✉❡s✳ ❆✉ ❝♦✉rs ❞❡ ❝❡ tr❛✈❛✐•✱ ♦♥ s✬❡st ❞♦♥❝ ❛t✲ t❛❝❤é à •❡s ❛✛❛✐❜•✐r✳ ▲❡s ❞❡✉① ♣r❡♠✐❡rs ❝❤❛♣✐tr❡s ❞❡ ❝❡tt❡ t❤ès❡ ❢♦✉r♥✐ss❡♥t •❡s ♦✉t✐•s ♥é❝❡ss❛✐r❡s à •❛ ♣♦✉rs✉✐t❡ ❞❡ ❝❡t ♦❜•❡❝t✐❢✳ ❆✉ ❝❤❛♣✐tr❡ ✶✱ ♦♥ ❢❛✐t ❞❡s r❛♣♣❡•s ❞❡ ❝❛•❝✉• ❞❡ ▼❛••✐❛✈✐♥✱ ❝❡tt❡ t❤é♦r✐❡ ✐♥t❡r✈❡♥❛♥t ❞❛♥s •❛ ♠❛•♦r✐té ❞❡ ♥♦s ♣r❡✉✈❡s✳ ❉✬❛✉tr❡s ✵✳✸✳ ❆PP❖|❚❙ ❉❊ ❈❊ ❚|❆❱❆■▲ ①✐① ♣❛♥s ❞❡s ♠❛t❤é♠❛t✐q✉❡s ✐♥t❡r✈✐❡♥♥❡♥t é❣❛•❡♠❡♥t✿ ❡s♣❛❝❡s ✈❡❝t♦r✐❡•s t♦♣♦•♦❣✐q✉❡s✱ ✐♥té❣r❛t✐♦♥ ❞❡ ❢♦♥❝t✐♦♥s à ✈❛•❡✉rs ❇❛♥❛❝❤✱ ✐♥t❡r♣♦•❛t✐♦♥✱ t❤é♦r✐❡ ❞❡s s❡♠✐✲❣r♦✉♣❡s✱ éq✉❛t✐♦♥s ❛✉ ❞ér✐✈é❡s ♣❛rt✐❡••❡s✳✳✳ ❖♥ ❢❛✐t •❡s r❛♣♣❡•s ♥é❝❡ss❛✐r❡s à ♥♦tr❡ ét✉❞❡ ❞❛♥s •❡s ❛♣♣❡♥❞✐❝❡s ❝♦rr❡s♣♦♥❞❛♥ts✱ ♦ù •✬♦♥ ❢♦✉r♥✐t é❣❛•❡♠❡♥t ❞❡s ré❢ér❡♥❝❡s✳ ❉❛♥s •❡ ❝❤❛♣✐tr❡ ✷✱ ♦♥ ét✉❞✐❡ •❛ str✉❝t✉r❡ ❞❡ •❛ ❝•❛ss❡ ❞❡ ❙❝❤✇❛rt③ ❡t ❞❡ •✬❡s♣❛❝❡ ❞❡s ❞✐s✲ tr✐❜✉t✐♦♥s t❡♠♣éré❡s ❡♥ t❛♥t q✉✬❡s♣❛❝❡s ✈❡❝t♦r✐❡•s t♦♣♦•♦❣✐q✉❡s ✭✈♦✐r ♣❛r ❡①❡♠♣•❡ ❬✻✺(✱ ❬✷✸(✱ ❬✻✶(✱ ❬✺✾(✱ ❬✻✾(✳✳✳✮ ❊♥ ♣❛rt✐❝✉•✐❡r✱ ♦♥ ❝♦♥str✉✐t •❡s ♣✉✐ss❛♥❝❡s ❢r❛❝t✐♦♥♥❛✐r❡s ❞❡ •✬♦♣ér❛t❡✉r✿ K := Id + X 2 -∆ ✭✸✷✮ ♣❛r ❞❡s ♠ét❤♦❞❡s ❞✬✐♥t❡r♣♦•❛t✐♦♥ ❞❡ s❡♠✐✲❣r♦✉♣❡s ❡t ♦♥ ✐♥tr♦❞✉✐t •❡s ❡s♣❛❝❡s✿ S p,s = dom L p K s/2 ✭✸✸✮ ▲❡✉r ✐♥térêt ❡st q✉❡ •✬♦♥ ❛✿ S = ✐♦♥s ❡♥s❡♠❜•✐st❡s ❞é✜♥✐ss❡♥t é❣❛•❡♠❡♥t ✉♥❡ t♦♣♦•♦❣✐❡ ❞✬❡s♣❛❝❡ ❞❡ ❋ré❝❤❡t s✉r S ❡t ❞❡ ❞✉❛• ❞✬❡s♣❛❝❡ ❞❡ ❋ré❝❤❡t s✉r S ′ ❝♦♠♠❡ ✐• ❡st ❡①♣•✐q✉é ❞❛♥s •❡ t❤é♦rè♠❡ ✷✳✸✳✵✳✸✳ ❈❡ s♦♥t s✉r •❡s é•é♠❡♥ts ❞é✈❡•♦♣♣és ❞❛♥s •❡s ❞❡✉① ♣r❡♠✐❡rs ❝❤❛♣✐tr❡s ❡t •❡s ❛♣♣❡♥❞✐❝❡s q✉❡ ♥♦✉s ♥♦✉s ❛♣♣✉②♦♥s ♣♦✉r ét❛❜•✐r✱ ❛✉ ❝♦✉rs ❞✉ ❝❤❛♣✐tr❡ ✸✱ ❞❡ ♥♦♠✲ ❜r❡✉① •❡♠♠❡s t❡❝❤♥✐q✉❡s q✉✐ ♦♥t ✉♥ ✐♥térêt ♣r♦♣r❡✳ ◆♦✉s ♣♦✉✈♦♥s ❛✐♥s✐ ét❛❜•✐r ❞❡s ❡①t❡♥s✐♦♥s ❞❡ •❛ ❢♦r♠✉•❡ ❞✬■tô ❞❛♥s •❡ ❝❤❛♣✐tr❡ ✹ ❡t ❡♥ ❞é✈❡•♦♣♣❡r ✉♥❡ ❛♣♣•✐❝❛t✐♦♥ à ✉♥ ♣r♦❜•è♠❡ ✈❛r✐❛t✐♦♥♥❡• ❛✉ ❝❤❛♣✐tr❡ ✺✳ ✵✳✸✳✶ |és✉•t❛ts ❞✬✐♥té❣r❛t✐♦♥ ♣❛r ♣❛rt✐❡s ▲❡s ♦✉t✐•s ✉❡ ♥♦✉s ✈❡♥♦♥s ❞❡ ♣rés❡♥t❡r ♣❡r♠❡tt❡♥t ❞❡ ♣r♦✉✈❡r •❡ t❤é♦rè♠❡ ❝✐✲ ❞❡ss♦✉s✱ q✉✐ ❡st •❡ rés✉•t❛t ♣r✐♥❝✐♣❛• ❞✉ ❝❤❛♣✐tr❡ ✸❀ ✐• s✬❛❣✐t ❞✉ t❤é♦rè♠❡ ✸✳✷✳✵✳✶✸ q✉❡ ♥♦✉s r❡♣r♦❞✉✐s♦♥s ✐❝✐✿ ❚❤é

  ✐q✉❡r q✉❡ •❡s rés✉•t❛ts ❡①✐st❛♥ts❀ ❡♥ ♣❛rt✐❝✉•❡r ♦♥ ♥✬❛ ♣•✉s ❜❡s♦✐♥ ❞❡ s✉♣♣♦s❡r X t ∈ D✳ ✵✳✸✳✸ ▲❡ ❝❛s ❞✬✉♥❡ ❊❉❙ ◆♦✉s ❛✈♦♥s s♦✉❤❛✐té s♣é❝✐❛•✐s❡r ♥♦tr❡ ❢♦r♠✉•❡ ❞✬■tô ❛✉ ❝❛s ❞❡s s❡♠✐♠❛rt✐♥❣❛•❡s q✉✐ s♦♥t s♦•✉t✐♦♥ ❞✬✉♥❡ éq✉❛t✐♦♥ ❞✐✛ér❡♥t✐❡••❡ st♦❝❤❛st✐q✉❡ q✉✐ ❡st s❛♥s ❞♦✉t❡ •❡ ♣•✉s ✉t✐•❡ ♣♦✉r •❡s ❛♣♣•✐❝❛t✐♦♥s✳ ❈✬❡st •✬♦❜•❡t ❞✉ t❤é♦rè♠❡ ✹✳✹✳✸✳✶✳ P♦✉r •✬♦❜t❡♥✐r✱ ♥♦✉s ❛✈♦♥s ❞✬✉♥❡ ♣❛rt ♣r♦✉✈é ✉♥ rés✉•t❛t ❞❡ ❞ér✐✈❛❜✐•✐té ✲ ❡ss❡♥t✐❡••❡♠❡♥t✱ s✐ •❡s ❝♦❡✣❝✐❡♥ts ❞❡ •✬❊❉❙ ❛♣♣❛rt✐❡♥♥❡♥t à •✬❡s♣❛❝❡ ❞❡ ❍ö•❞❡r Λ s ❛•♦rs s❛ s♦•✉t✐♦♥ ❛ s ❞ér✐✈é❡s ❢r❛❝t✐♦♥♥❛✐r❡s ❛✉ s❡♥s ❞❡ ▼❛••✐❛✈✐♥ ✲ ❡t ❞✬❛✉tr❡ ♣❛rt ♥♦✉s ❛✈♦♥s ❡♠♣r✉♥té ✉♥ rés✉•t❛t ❞❡ ❬✶✺( ♣♦✉r •❛ ♥♦♥✲❞é❣é♥ér❡s❝❡♥❝❡✳ ◆♦✉s r❡♣r♦❞✉✐s♦♥s ❝❡s rés✉•❛ts ❝✐✲❞❡ss♦✉s✳ ❖♥ ❝♦♥s✐❞èr❡ •✬éq✉❛t✐♦♥ ❞✐✛ér❡♥t✐❡

  ét❡♥❞✉ ♥♦s rés✉•t❛ts ❞❛♥s ❞✐✈❡rs❡s ❞✐r❡❝t✐♦♥s✳ ❚♦✉t ❞✬❛❜♦r❞✱ ♥♦✉s ❛✈♦♥s ❞é•à s✐❣♥❛•é q✉❡ ♥♦♦✉s ❞♦♥♥♦♥s ❛✉ ❝❤❛♣✐tr❡ ✷ ❞❡s rés✉•t❛ts s✉r •❡s ❡s♣❛❝❡s S ∞,s q✉✐ ♥♦✉s ♣❡r♠❡tt❡♥t ❞✬♦❜t❡♥✐r✱ ❛✉ ❝❤❛♣✐tr❡ ✸✱ ✉♥❡ ✈❡rs✐♦♥ ❞✉ t❤é♦rè♠❡ ✸✳✷✳✵✳✶✸ ❢❛✐s❛♥t ✐♥t❡r✈❡♥✐r ❝❡s ❡s♣❛❝❡s ❛✉ •✐❡✉ ❞❡s S p,s ✿ ❝✬❡st •❡ t❤é♦rè♠❡ ✸✳✷✳✵✳✶✺✳ ❈❡ rés✉•t❛t ❛ ✉♥ ✐♥térêt ❞♦✉❜•❡✳ ❉✬❛❜♦r❞✱ ❝♦♠♠❡ ♥♦✉s •✬❛✈♦♥s ❞é•à ❞✐t✱ ✐• ♣❡r♠❡t ❞❡ ♥❡ ♣❛s ❢❛✐r❡ ❞✬❤②♣♦t❤ès❡ ❛ ♣r✐♦r✐ s✉r •✬❡①✐st❡♥❝❡ ❞✬✉♥❡ ❞❡♥s✐té ♣♦✉r •❛ ✈❛r✐❛❜•❡ ❛•é❛t♦✐r❡ ♣❛r •❛q✉❡••❡ ♦♥ r❡•è✈❡✳ ❊♥s✉✐t❡✱ ✐• ♥♦✉s ❛ ♣❡r♠✐s ❞✬ét❡♥❞r❡ •❡ t❤é♦rè♠❡ ✸✳✷✳✵✳✶✺ ❛✉ ❝❛s ♦ù •❛ ❞✐str✐❜✉t✐♦♥ q✉❡ •✬♦♥ r❡•è✈❡ ❡st ❡••❡✲♠ê♠❡ ❛•é❛t♦✐r❡✳ ◆♦✉s ❞♦♥♥♦♥s ❝❡s rés✉•t❛ts ❞❛♥s •❡ ♣❛r❛❣r❛♣❤❡ ✸✳✺❀ ♣♦✉r •❡s ét❛❜•✐r✱ ♥♦✉s ❛✈♦♥s ét❡♥❞✉ •❛ ❢♦r♠✉•❡ ❞✬✐♥té❣r❛t✐♦♥ ♣❛r ♣❛rt✐❡s ✵✳✷✳✷✳✶ ❛✉ ❝❛s ♦ù •❛ ❢♦♥❝t✐♦♥ ❡st ❛•é❛t♦✐r❡✳ ❉❛♥s ❝❡ ❝❛s •❛ ❢♦r♠✉•❡ ❞✬✐♥té❣r❛t✐♦♥ ♣❛r ♣❛rt✐❡s ❞❡✈✐❡♥t✿

T(

  r t❤✐s ♥♦r♠ ❛♥❞ ✇❡ ♥♦t❡ t❤❛t t❤❡ ❝❛♥♦♥✐❝❛• ✐♥•❡❝t✐♦♥✿ φ T : E → ẼT ✭❆✳✶✾✮ ✐s ❝♦♥t✐♥✉♦✉s✳ ◆♦✇ •❡t B ❜❡ ❛ ❜♦✉♥❞❡❞ ❛♥❞ ♥♦♥✲❡♠♣t② ❜❛rr❡• ✐♥ E❀ ✇❡ ❞❡✜♥❡ ❛ s✉❜s♣❛❝❡ ♦❢ E ❛s✿ E 1 := ∞ n=1 n • B ✭❆✳✷✵✮ ❚❤❡ ❣❛✉❣❡ p B ♦❢ B ✐s ❛ ♥♦r♠ ♦♥ E 1 ❛♥❞ ✇❡ ♥♦t❡ E B ❢♦r t❤❡ ♥♦r♠❡❞ s♣❛❝❡ (E 1 , p B )✳ ■❢ B ✐s ❝♦♠♣•❡t❡ E B ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ❢♦••♦✇✐♥❣ ♥♦t❛t✐♦♥ ❢♦r t❤❡ ❝❛♥♦♥✐❝❛• ❡♠❜❡❞❞✐♥❣✿ ψ B = E B ֒→ E ✭❆✳✷✶✮ ❛♥❞ ✇❡ ♥♦t❡ t❤❛t ψ B ✐s ❝♦♥t✐♥✉♦✉s✳ ❋✐♥❛••②✱ ✇❡ ♥♦t❡ t❤❛t ✐❢ T ❛♥❞ B ❝♦✐♥❝✐❞❡ t❤❡♥ t❤❡ s♣❛❝❡s E T ❛♥❞ E B ❝♦✐♥❝✐❞❡ ❛s ✇❡••✳ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ❢♦••♦✇✐♥❣✿ ❉❡✜♥✐t✐♦♥ ❆✳✹✳✸✳✶ ✭❇♦✉♥❞❡❞ •✐♥❡❛r ♠❛♣✮✳ ▲❡t E ❛♥❞ F ❜❡ t✇♦ •♦❝❛••② ❝♦♥✈❡① ❚❱❙✳ u ∈ L(E, F ) ✐s s❛✐❞ t♦ ❜❡ ❛ ❜♦✉♥❞❡❞ •✐♥❡❛r ♠❛♣ ✐❢ t❤❡r❡ ❡①✐sts ❛ ♥❡✐❣❤❜♦✉r❤♦♦❞ U ♦❢ 0 ∈ E s✉❝❤ t❤❛t u(U ) ✐s ❛ ❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ F ✳ ▲❡t ✉s ♥♦t❡ t❤❛t ✐❢ u ✐s ❛ ❜♦✉♥❞❡❞ ♠❛♣♣✐♥❣✱ ♦♥❡ ♠❛② t❛❦❡ U t♦ ❜❡ ❛ ❜❛rr❡•❀ t❤❡♥ u(U ) ✐s ✐♥❝•✉❞❡❞ ✐♥ ❛ ❜♦✉♥❞❡❞ ❜❛rr❡• B ⊂ F ❛♥❞ ♦♥❡ ♠❛② ✇r✐t❡✿ u = ψ B • u 0 • φ U ✭❆✳✷✷✮ ✇❤❡r❡ u 0 ∈ L(E U , F B )✳ ❆•s♦✱ ✐❢ F B ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡✱ u 0 ❤❛s ❛ ❝♦♥t✐♥✉♦✉s ❡①t❡♥s✐♦♥ t♦ L( ẼU , F B ) ✇❤✐❝❤ ✇❡ st✐•• ❞❡♥♦t❡ u 0 ✳ ❚❤✐s •❡❛❞s ✉s t♦ t❤❡ ❢♦••♦✇✐♥❣✿ ❉❡✜♥✐t✐♦♥ ❆✳✹✳✸✳✷ ✭◆✉❝•❡❛r ♠❛♣♣✐♥❣✮✳ ❆ ♠❛♣♣✐♥❣ ❜❡t✇❡❡♥ t✇♦ •♦❝❛••② ❝♦♥✈❡① ❚❱❙ E ❛♥❞ F ✐s s❛✐❞ t♦ ❜❡ ♥✉❝•❡❛r ✐❢ ✐t ♠❛② ❜❡ ✇r✐tt❡♥ ❛s ✐♥ ❡q✉❛t✐♦♥ ✭❆✳✷✷✮✱ ✇✐t❤ U ❛ ❜❛rr❡•✱ B ❛ ❜♦✉♥❞❡❞ ❜❛rr❡• s✉❝❤ t❤❛t F B ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❛♥❞ u 0 ❛ ♥✉❝•❡❛r ♠❛♣♣✐♥❣ ✭✐♥ t❤❡ s❡♥s❡ ♦❢ ❇❛♥❛❝❤ s♣❛❝❡s✮✳ ■t ✐s ✐♥t❡r❡st✐♥❣ t♦ ♥♦t❡ t❤❛t ❢♦r ❡✈❡r② S✲t♦♣♦•♦❣② ♦♥ L(E, F )✱ t❤❡ ♥✉❝•❡❛r ♠❛♣s ❛r❡ ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ ❝•♦s✉r❡ ♦❢ E ′ ⊗F ✭✈✐❡✇❡❞ ❛s ❛ s✉❜s♣❛❝❡ ♦❢ L(E, F )✮✳ ❚❤❡r❡❢♦r❡ ❢r♦♠ t❤❡♦r❡♠ ❆✳✹✳✶✳✸ ♦♥❡ ♦❜t❛✐♥s t❤❡✿ ❆✳✹✳ ◆❯❈▲❊❆| ❙P❆❈❊❙ ✶✸✶ ❚❤❡♦r❡♠ ❆✳✹✳✸✳✶ ✭❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ♥✉❝•❡❛r ♠❛♣s✮✳ ▲❡t E ❛♥❞ F ❜❡ t✇♦ •♦❝❛••② ❝♦♥✈❡① ❚❱❙✳ ❆ •✐♥❡❛r ♠❛♣ u ∈ L(E, F ) ✐s ♥✉❝•❡❛r ✐❢ ❛♥❞ ♦♥•② ✐❢ ✐t ♠❛② ❜❡ ✇r✐tt❡♥✿ u(x) = ∞ n=1 λ n • f n (x) • y n ✭❆✳✷✸✮ ✇❤❡r❡ (λ n ) ∈ l 1 (K)✱ (f n ) ✐s ❛♥ ❡q✉✐❝♦♥t✐♥✉♦✉s s❡q✉❡♥❝❡ ✐♥ E ′ ❛♥❞ (y n ) ✐s ❛ s❡q✉❡♥❝❡ ✐♥❝•✉❞❡❞ ✐♥ s♦♠❡ ❜♦✉♥❞❡❞ ❜❛rr❡• B ⊂ F ❢♦r ✇❤✐❝❤ F B ✐s ❝♦♠♣•❡t❡✳ ❙♦♠❡t✐♠❡s ✇❡ ✇✐•• ❛❜✉s✐✈❡•② ♥♦t❡✿ u = ∞ n=1 λ n • f n ⊗ y n ✭❆✳✷✹✮ ❆✳✹✳✹ ◆✉❝•❡❛r s♣❛❝❡s ❍❛✈✐♥❣ ❞❡✜♥❡❞ t❤❡ ♥♦t✐♦♥ ♦❢ ♥✉❝•❡❛r ♠❛♣♣✐♥❣ ✇❡ ❛r❡ ❛❜•❡ t♦ ✐♥tr♦❞✉❝❡ t❤❡✿ ❉❡✜♥✐t✐♦♥ ❆✳✹✳✹✳✶ ✭◆✉❝•❡❛r s♣❛❝❡✮✳ ❆ •♦❝❛••② ❝♦♥✈❡① ❚❱❙ ✐s s❛✐❞ t♦ ❜❡ ♥✉❝•❡❛r ✐❢ ✐t ❛❞♠✐ts ❛ ❜❛s❡ B ♦❢ ♥❡✐❣❤❜♦✉r❤♦♦❞s ♦❢ 0 s✉❝❤ t❤❛t✿ • ❊✈❡r② V ∈ B ✐s ❛ ❜❛rr❡•❀ • ❋♦r ❡✈❡r② V ∈ B✱ t❤❡ ❝❛♥♦♥✐❝❛• ♠❛♣♣✐♥❣ E → ẼB ✐s ♥✉❝•❡❛r✳ ❚❤❡ ❢♦••♦✇✐♥❣ ❝❤❛r❛❝t❡r✐③❛③t✐♦♥ ✐s ♠♦r❡ tr❛❝t❛❜•❡✿ ❚❤❡♦r❡♠ ❆✳✹✳✹✳✶✳ ▲❡t E ❜❡ ❛ •♦❝❛••② ❝♦♥✈❡① ❚❱❙❀ t❤❡ t❤r❡❡ ❢♦••♦✇✐♥❣ ❛r❡ ❡q✉✐✈✲ ❛•❡♥t✿ • E ✐s ♥✉❝•❡❛r❀ • ❆♥② ❝♦♥t✐♥✉♦✉s •✐♥❡❛r ♠❛♣ ❢r♦♠ E ✐♥t♦ ❛♥② ❇❛♥❛❝❤ s♣❛❝❡ B ✐s ♥✉❝•❡❛r❀ • ▲❡t B ❜❡ ❛ ❜❛s❡ ♦❢ ♥❡✐❣❤❜♦✉r❤♦♦❞s ♦❢ 0 ✇❤✐❝❤ ❛r❡ ❛•• ❜❛rr❡•s❀ t❤❡♥ ❢♦r ❡✈❡r② T 1 ∈ B t❤❡r❡ ❡①✐sts T 1 ∈ B s✉❝❤ t❤❛t T 2 ⊂ T 1 ❛♥❞ t❤❡ ❝❛♥♦♥✐❝❛• ♠❛♣ ẼT 2 → ẼT 1 ✐s ♥✉❝•❡❛r✳ ❋♦r ❋ré❝❤❡t s♣❛❝❡s✱ t❤❡ ❢♦••♦✇✐♥❣ ✐s ❡✈❡♥ s✐♠♣•❡r✿ ❚❤❡♦r❡♠ ❆✳✹✳✹✳✷✳ ❆ ❋ré❝❤❡t E s♣❛❝❡ ✐s ♥✉❝•❡❛r ✐❢ ❛♥❞ ♦♥•② ✐❢ ✐t ✐s t❤❡ ♣r♦•❡❝✲ t✐✈❡ •✐♠✐t ♦❢ ❛ s❡q✉❡♥❝❡ ♦❢ ❍✐•❜❡rt s♣❛❝❡s ✇✐t❤ ♥✉❝•❡❛r ✐♥•❡❝t✐♦♥s✱ ✐❡ t❤❡r❡ ❡①✐sts ❛ ❞❡❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ ❍•✐❜❡rt s♣❛❝❡s (H n ) s✉❝❤ t❤❛t✿ ❢♦r ❡✈❡r② m < n✿ H n ֒→ H m ✭❆✳✷✻✮ ✐s ♥✉❝•❡❛r✳ ❲❡ ♠❡♥t✐♦♥ ❛ ❝♦✉♣•❡ ♠♦r❡ r❡s✉•ts ❛❜♦✉t ♥✉❝•❡❛r s♣❛❝❡s✿ ❚❤❡♦r❡♠ ❆✳✹✳✹✳✸✳ ❆ ❋ré❝❤❡t s♣❛❝❡ ✐s ♥✉❝•❡❛r ✐❢ ❛♥❞ ♦♥•② ✐❢ ✐ts str♦♥❣ ❞✉❛• ✐s ♥✉❝•❡❛r✳ ❚❤❡♦r❡♠ ❆✳✹✳✹✳✹✳ ❆ ❝♦♠♣•❡t❡ ❜❛rr❡•❡❞ s♣❛❝❡ ✇❤✐❝❤ ✐s ♥✉❝•❡❛r ✐s ❛ ▼♦♥t❡• s♣❛❝❡✳ ❆♣♣❡♥❞✐① ❇ ❚❤❡ ❇♦❝❤♥❡r ❛♥❞ P❡tt✐s ✐♥t❡❣r❛•s ❋♦r t❤✐s ❛♣♣❡♥❞✐① ♦✉r s❡tt✐♥❣ ✇✐•• ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡ X ❛♥❞ ❛ ♠❡❛s✉r❡❞ s♣❛❝❡ (Ω, F, ν)❀ ✇❡ ✇✐•• ❛•s♦ ♥♦t❡ X ′ ❢♦r t❤❡ str♦♥❣ t♦♣♦•♦❣✐❝❛• ❞✉❛• s♣❛❝❡ ♦❢ X✳ ❚❤❡ ❛✐♠ ✐s t♦ ❞❡✜♥❡ ❛♥❞ st✉❞② t❤❡ ♠❡❛s✉r❛❜✐•✐t② ❛♥❞ t❤❡ ✐♥t❡❣r❛❜✐•✐t② ♦❢ X✲✈❛•✉❡❞ ❜♦r❡•✐❛♥ ❢✉♥❝t✐♦♥s ♦♥ Ω✳ ❲❡ st✐❝❦ t♦ t❤❡ ❝❛s❡ ♦❢ ❛ ♣♦s✐t✐✈❡ ♠❡❛s✉r❡ ❢♦r s✐♠♣•✐❝✐t②✳ ❖✉r r❡❢❡r❡♥❝❡ ✐s ❬✶✽(✱ ✇❤❡r❡ ❛•• t❤❡ ♣r♦♦❢s ♠❛② ❜❡ ❢♦✉♥❞✳ ❇✳✶ ▼❡❛s✉r❛❜✐•✐t② ❉❡✜♥✐t✐♦♥ ❇✳✶✳✵✳✷ ✭❙✐♠♣•❡ ❢✉♥❝t✐♦♥✮✳ f : Ω → X ✐s ❝❛••❡❞ ❛ s✐♠♣•❡ ❢✉♥❝t✐♦♥ ✐❢ ♦♥❡ ♠❛② ✇r✐t❡ ❛ ✜♥✐t❡ •✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥✿ f = i 1 E i α i ✭❇✳✶✮ ✇❤❡r❡ E i ∈ F ❛♥❞ α i ∈ X✳ ❉❡✜♥✐t✐♦♥ ❇✳✶✳✵✳✸ ✭❙tr♦♥❣ ♠❡❛s✉r❛❜✐•✐t②✮✳ f : Ω → X ✐s ✭str♦♥❣•②✮ ♠❡❛s✉r❛❜•❡ ✐❢ t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ (f n ) ♦❢ s✐♠♣•❡ ❢✉♥❝t✐♦♥s s✉❝❤ t❤❛t t❤❡ ❢♦••♦✇✐♥❣ ❤♦•❞s ✐♥ t❤❡ ν✲❛✳s✳ s❡♥s❡✿ lim n→∞ ff n X = 0 ✭❇✳✷✮ ❉❡✜♥✐t✐♦♥ ❇✳✶✳✵✳✹ ✭❲❡❛❦ ♠❡❛s✉r❛❜✐•✐t②✮✳ f : Ω → X ✐s ✇❡❛❦•② ♠❡❛s✉r❛❜•❡ ✐❢ ❢♦r ❡✈❡r② x * ∈ X ′ t❤❡ r❡❛•✲✈❛•✉❡❞ ❢✉♥❝t✐♦♥ x * , f ✐s ♠❡❛s✉r❛❜•❡ ✐♥ t❤❡ ✉s✉❛• ✭✜♥✐t❡✲ ❞✐♠❡♥s✐♦♥♥❛•✮ s❡♥s❡✳ ❖♥❡ ♠❛② ✈❡r✐❢② t❤❛t ✐♥ t❤❡ ❝❛s❡ ✇❤❡r❡ X ✐s ♦❢ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥✱ t❤❡s❡ t✇♦ ♥♦t✐♦♥s ♦❢ ♠❡❛s✉r❛❜✐•✐t② ❝♦✐♥❝✐❞❡ ✇✐t❤ t❤❡ ✉s✉❛• ♦♥❡✳ ❆•s♦ t❤❡r❡ ✐s t❤❡✿ ❚❤❡♦r❡♠ ❇✳✶✳✵✳✺✳ f : Ω → X ✐s ν✲♠❡❛s✉r❛❜•❡ ✐❢ ❛♥❞ ♦♥•② ✐❢ t❤❡ t✇♦ ❢♦••♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❤♦•❞✿ ✶✸✸ ✶✸✹ ❆PP❊◆❉■❳ ❇✳ ❚❍❊ ❇❖❈❍◆❊| ❆◆❉ P❊❚❚■❙ ■◆❚❊•|❆▲❙ • f ✐s ν✲✇❡❛❦•② ♠❡❛s✉r❛❜•❡❀ • ❚❤❡r❡ ❡①✐sts ❛ s❡♣❛r❛❜•❡ s✉❜s♣❛❝❡ Y ♦❢ X s✉❝❤ t❤❛t f ✐s ν✲❛✳s✳ Y ✲✈❛•✉❡❞✳ ■♥ ♣❛rt✐❝✉•❛r✱ ✐❢ X ✐s ❛ s❡♣❛r❛❜•❡ ❇❛♥❛❝❤ s♣❛❝❡✱ ❛s ✇✐•• ❜❡ t❤❡ ❝❛s❡ ✐♥ ♦✉r ♣r❛❝t✐❝❛• ❛♣♣•✐❝❛t✐♦♥s✱ t❤❡ ♥♦t✐♦♥s ♦❢ ✇❡❛❦ ❛♥❞ str♦♥❣ ♠❡❛s✉r❛❜✐•✐t② ❝♦✐♥❝✐❞❡✳ ❇✳✷ ❚❤❡ ❇♦❝❤♥❡r ✐♥t❡❣r❛• ■❢ f ✐s ❛ s✐♠♣•❡ ❢✉♥❝t✐♦♥✱ f = i 1 E i α i ✱ ✇❡ ❞❡✜♥❡ ✐ts ✭❇♦❝❤♥❡r✮ ✐♥t❡❣r❛• ❛s✿ Ω f dν = i ν(E i )α i ✭❇✳✸✮ ❚❤❡♥ •❡t ✉s ✐♥tr♦❞✉❝❡ t❤❡✿ ❉❡✜♥✐t✐♦♥ ❇✳✷✳✵✳✺ ✭❇♦❝❤♥❡r ✐♥t❡❣r❛❜✐•✐t②✮✳ ❲❡ s❛② t❤❛t ❛ ✭str♦♥❣•②✮ ♠❡❛s✉r❛❜•❡ ❢✉♥❝t✐♦♥ f : Ω → X ✐s ν✲❇♦❝❤♥❡r ✐♥t❡❣r❛❜•❡ ✐❢ t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ ♦❢ s✐♠♣✇❡ st❛t❡ t❤❡ ♠♦r❡ ❝♦♥✈❡♥✐❡♥t ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ✇❤✐❝❤ ✇❡ ✇✐•• ✉s❡ ✐♥ ♣r❛❝✲ t✐❝❡✿ ❚❤❡♦r❡♠ ❇✳✷✳✵✳✻✳ f : Ω → X ✐s ν✲❇♦❝❤♥❡r ✐♥t❡❣r❛❜•❡ ✐❢ ❛♥❞ ♦♥•② ✐❢✿ Ω f X dν < ∞ ✭❇✳✻✮ ❲❡ ✇✐•• ❛•s♦ ♠❛❦❡ ✉s❡ ♦❢ t❤❡ ❢♦••♦✇✐♥❣ r❡s✉•ts✿ ❚❤❡♦r❡♠ ❇✳✷✳✵✳✼ ✭❉♦♠✐♥❛t❡❞ ❝♦♥✈❡r❣❡♥❝❡ t❤❡♦r❡♠✮✳ ▲❡t (f n ) ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ ❇♦❝❤♥❡r✲✐♥t❡❣r❛❜•❡ ❢✉♥❝t✐♦♥s✳ ❲❡ s✉♣♣♦s❡ t❤❛t✿ • lim n→∞ ν ({ ff n X > ǫ}) = 0 ❢♦r ❡✈❡r② ǫ > 0❀ • ❚❤❡r❡ ❡①✐sts ❛ ν✲✐♥t❡❣r❛❜•❡✱ R + ✲✈❛•✉❡❞ ❢✉♥❝t✐♦♥ g s✉❝❤ t❤❛t f n ≤ g ❢♦r ❡✈❡r② n✳ ❇✳✸✳ ❚❍❊ P❊❚❚■❙ ■◆❚❊•|❆▲ ✶✸✺ ❚❤❡♥ f ✐s ✭str♦♥❣•② ♠❡❛s✉r❛❜•❡ ❛♥❞✮ ❇♦❝❤♥❡r ✐♥t❡❣r❛❜•❡ ❛♥❞✿ ♠ ❇✳✷✳✵✳✽ ✭❆❜s♦•✉t❡ ❝♦♥t✐♥✉✐t② ♦❢ t❤❡ ❇♦❝❤♥❡r ✐♥t❡❣r❛•✮✳ ▲❡t f ❜❡ ❛ ❇♦❝❤♥❡r ✐♥t❡❣r❛❜•❡ ❢✉♥❝t✐♦♥❀ t❤❡♥♠ ❇✳✷✳✵✳✾ ✭❚r✐❛♥❣•❡ ✐♥❡q✉❛•✐t②✮✳ ■❢ f ✐s ❛ ❇♦❝❤♥❡r ✐♥t❡❣r❛❜•❡ ❢✉♥❝t✐♦♠ ❇✳✷✳✵✳✶✵ ✭❉✐✛❡r❡♥t✐❛t✐♦♥✮✳ ■❢ f ❛♥❞ g ❛r❡ t✇♦ ❇♦❝❤♥❡r ✐♥t❡❣r❛❜•❡ ❢✉♥❝✲ t✐♦♥s s✉❝❤ t❤❛t ❢♦r ❡✈❡r② E ∈ Σ✿ E f dν = E gdν ✭❇✳✶✶✮ t❤❡♥ f = g ν✲❛✳s✳ ❚❤❡♦r❡♠ ❇✳✷✳✵✳✶✶ ✭❍✐••❡✮✳ ▲❡t X ❛♥❞ Y ❜❡ t✇♦ s❡♣❛r❛❜•❡ ❇❛♥❛❝❤ s♣❛❝❡s ❛♥❞ T ∈ L c (X, Y )✳ ■❢ ❜♦t❤ f ❛♥❞ T • f ❛r❡ ❇♦❝❤♥❡r✲✐♥t❡❣r❛❜•❡ t❤❡♥✿ ❛ ❇✳✸✳✵✳✶✳ ❈♦♥s✐❞❡r ❛ ✇❡❛❦•② ♠❡❛s✉r❛❜•❡ f : Ω → X s✉❝❤ t❤❛t ❢♦r ❡✈❡r② x * ∈ X ′ ✿ x * , f ∈ L 1 (ν) ✭❇✳✶✸✮ ❚❤❡♥ ❢♦r ❡✈❡r② E ∈ Σ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ❡•❡♠❡♥t I E ∈ X * * s✉❝❤ t❤❛t ❢♦r ❡✈❡r② x * ∈ X ′ ✿ I E (x * ) = E x * , f dν ✭❇✳✶✹✮ I E ✐s t❤❡ ❉✉♥❢♦r❞ ✐♥t❡❣r❛• ♦❢ f ♦♥ t❤❡ ♠❡❛s✉r❛❜•❡ s❡t E✳ ■❢ X ✐s r❡✢❡①✐✈❡✱ ✇❤✐❝❤ ✇✐•• ❜❡ t❤❡ ❝❛s❡ ✐♥ ♦✉r ❛♣♣•✐❝❛t✐♦♥s✱ ✇❡ ♠❛② ❝♦♥s✐❞❡r t❤❛t I E ∈ X ❛♥❞ ✇❡ ❝❛•• ✐t ❛ P❡tt✐s ✐♥t❡❣r❛•✳ ❚❤❡♥ t❤❡ P❡tt✐s ✐♥t❡❣r❛• ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ❇♦❝❤♥❡r ✐♥t❡❣r❛• ❢♦r str♦♥❣•② ♠❡❛s✉r❛❜•❡ ❢✉♥❝t✐♦♥s✳ ✶✸✻ ❆PP❊◆❉■❳ ❇✳ ❚❍❊ ❇❖❈❍◆❊| ❆◆❉ P❊❚❚■❙ ■◆❚❊•|❆▲❙ ❆♣♣❡♥❞✐① ❈ ❙♦♠❡ r❡s✉•ts ✐♥ s❡♠✐❣r♦✉♣ t❤❡♦r② ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ❣✐✈❡ ❛ ❜r✐❡❢ ✐♥tr♦❞✉❝t✐♦♥ t♦ t❤❡ t❤❡♦r② ♦❢ s❡♠✐❣r♦✉♣s ❛♥❞ ❣❡♥✲ ❡r❛t♦rs✳ ❲❡ ♠♦st•② ❢♦••♦✇ ❬✽✹( ❛♥❞ ❬✶✹(❀ ❬✻✵( ♣r♦✈✐❞❡s ❛♥ ✐♥tr♦❞✉❝t✐♦♥ t♦ ❤♦✇ t❤❡ t❤❡♦r② ❛♣♣•✐❡s t♦ ♣r♦❜❛❜✐•✐t② ✇❤✐•❡ ❬✶(✱ ❬✼✵( ♦r ❬✼✶( r❡•❛t❡s ✐t t♦ t❤❡ ❝♦♥str✉❝t✐♦♥ ❛♥❞ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ s♣❛❝❡s ✇❤✐❝❤ ❛r❡ ♦❢ ❝♦♠♠♦♥ ✉s❡ ✐♥ ❛♥❛•②s✐s✳ ❈✳✶ ❉❡✜♥✐t✐♦♥ ▲❡t X ❜❡ s♦♠❡ ❇❛♥❛❝❤ s♣❛❝❡✳ ❲❡ st❛rt ❜② ✐♥tr♦❞✉❝✐♥❣ t❤❡ ♥♦t✐♦♥ ✇✐t❤ ✇❤✐❝❤ t❤✐s ❛♣♣❡♥❞✐① ❞❡❛•s✿ ❉❡✜♥✐t✐♦♥ ❈✳✶✳✵✳✻ ✭❙❡♠✐❣r♦✉♣ ♦♥ X✮✳ (T t ) t≥0 ✱ ♦r T t ❢♦r s❤♦rt✱ ✐s s❛✐❞ t♦ ❜❡ ❛ s❡♠✐❣r♦✉♣ ♦♥ X ✐❢ ✐t ✈❡r✐✜❡s t❤❡ t❤r❡❡ ❢♦••♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿ • ∀t ≥ 0✿ T t ∈ L c (X) • ∀t 1 , t 2 ≥ 0✿ T t 1 +t 2 = T t 1 • T t 2 • T 0 = Id ❉❡✜♥✐t✐♦♥ ❈✳✶✳✵✳✼ ✭❙❡♠✐❣r♦✉♣ ♦❢ ❝•❛ss C 0 ✮✳ ❆ s❡♠✐❣r♦✉♣ T t ✐s s❛✐❞ t♦ ❜❡ ♦❢ ❝•❛ss C 0 ✐❢ ❢♦r ❡✈❡r② x ∈ X t❤❡ ❢♦••♦✇✐♥❣ •✐♠✐t ❤♦•❞s ✐♥ t❤❡ str♦♥❣ s❡♥s❡✿ lim t→0 T t x = x ✭❈✳✶✮ ❉❡✜♥✐t✐♦♥ ❈✳✶✳✵✳✽ ✭❈♦♥tr❛❝t✐♦♥ s❡♠✐❣r♦✉♣✮✳ ■❢ ❛ s❡♠✐❣r♦✉♣ T t ✐s s✉❝❤ t❤❛t ❢♦r ❡✈❡r② t ≥ 0✱ T t Lc(X) ≤ 1✱ ✭r❡s♣✳ < 1✮ ✇❡ ❝❛•• T t ❛ ❝♦♥tr❛❝t✐♦♥ s❡♠✐❣r♦✉♣ ✭r❡s♣✳ ❛ str✐❝t ❝♦♥tr❛❝t✐♦♥ s❡♠✐❣r♦✉♣✮✳ ✶✸✼ ✶✸✽ ❆PP❊◆❉■❳ ❈✳ ❙❖▼❊ |❊❙❯▲❚❙ ■◆ ❙❊▼■•|❖❯P ❚❍❊❖|( ❈✳✷ ❚❤❡ ✐♥✜♥✐t❡s✐♠❛• ❣❡♥❡r❛t♦r ▲❡t T t ❜❡ s♦♠❡ s❡♠✐❣r♦✉♣ ♦❢ ❝•❛ss C 0 ♦♥ X✳ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ❢♦••♦✇✐♥❣ str♦♥❣ •✐♠✐t✱ ✇❤❡♥❡✈❡r ✐t ❡①✐sts✿ Ax = lim t→0 T t xx t ✭❈✳✷✮ ❛♥❞ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡✿ ❉❡✜♥✐t✐♦♥ ❈✳✷✳✵✳✾ ✭■♥✜♥✐t❡s✐♠❛• ❣❡♥❡r❛t♦r ♦❢ ❛ s❡♠✐❣r♦✉♣✮✳ A ✐s t❤❡ ✐♥✜♥✐t❡s✐♠❛• ❣❡♥❡r❛t♦r ♦❢ T t ✳ ❉❡✜♥✐t✐♦♥ ❈✳✷✳✵✳✶✵ ✭❉♦♠❛✐♥ ♦❢ t❤❡ ✐♥✜♥✐t❡s✐♠❛• ❣❡♥❡r❛t♦r✮✳ ❚❤❡ s❡t D(A) ♦❢ t❤♦s❡ x ∈ X s✉❝❤ t❤❛t Ax ✐s ✇❡•• ❞❡✜♥❡❞ ✐s ❝❛••❡❞ t❤❡ ❞♦♠❛✐♥ ♦❢ A✳ ❚❤❡ ♣r♦♣❡rt✐❡s ♦❢ ❛♥ ✐♥✜♥✐t❡s✐♠❛• ❣❡♥❡r❛t♦r t❤❛t ✇❡ ✇✐•• ✉s❡ ❛r❡ s✉♠♠❛r✐③❡❞ ✐♥ t❤❡✿ ❚❤❡♦r❡♠ ❈✳✷✳✵✳✶✷✳ ❖♥❡ ❤❛s✿ • D(A) ✐s ❛ s✉❜s♣❛❝❡ ♦❢ X ❛♥❞ A ∈ L (D(A), X) • D(A) ✐s ❞❡♥s❡ ✐♥ X ✭❢♦r t❤❡ t♦♣♦•♦❣② ♦❢ • X ) • x A := x X + Ax X ❞❡✜♥❡s ❛ ♥♦r♠ ♦♥ D(A) ❛♥❞ D(A) ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❛t ♥♦r♠ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ • ■❢ x ∈ X ❛♥❞ t > 0 t❤❡♥ T t x ∈ D(A) ❛♥❞ t❤❡ ❢♦••♦✇✐♥❣ r❡•❛t✐♦♥s ❤♦•❞✿ AT t x = d dt (T t x) ✭❈✳✸✮ T t xx = t 0 (T s Ax) ds ✭❈✳✹✮ ✇❤❡r❡ t❤❡ ✐♥t❡❣r❛• ✐s ❞❡✜♥❡❞ ✐♥ t❤❡ ❇♦❝❤♥❡r s❡♥s❡✱ s❡❡ ❛♣♣❡♥❞✐① ❇✳ • ■❢ x ∈ D(A)✿ AT t x = T t Ax ✭❈✳✺✮ ❙✐♠✐•❛r•②✱ ♦♥❡ ♠❛② ❞❡✜♥❡ ♣♦✇❡rs A r ❢♦r r ∈ N✱ D(A r+1 ) ❜❡✐♥❣ ❛ s✉❜s♣❛❝❡ ♦❢ D(A r )✳ ❍❡r❡ ❛r❡ t❤❡ ♣r♦♣❡rt✐❡s ✇❡ ✇✐•• ✉s❡ ❛❜♦✉t t❤❡ ♣♦✇❡rs ♦❢ A✿ ❚❤❡♦r❡♠ ❈✳✷✳✵✳✶✸✳ ❖♥❡ ❤❛s✿ • r∈N D(A r ) ✐s ❞❡♥s❡ ✐♥ X ❛♥❞ ❝♦♥t❛✐♥s ❛•• t❤❡ T t x✱ x ∈ X✳ ❈✳✸✳ |❊❙❖▲❱❊◆❚ ❖❋ ❆ ❙❊▼■•|❖❯P ✶✸✾ • ■❢ x ∈ D(A r ) t❤❡ ❢♦••♦✇✐♥❣ ❤♦•❞✿ d r dt r (T t x) = A r T t x = T t A r x ✭❈✳✻✮ ❛♥❞ ❛s ❛ ❇♦❝❤♥❡r ✐♥t❡❣r❛•✿ T t xs) r-1 • T s A r x ds ✭❈✳✼✮ • ❋♦r ❡✈❡r② x ∈ X✿ (T t -Id) r x = s 1 +•••+sr x ds 1 . . . ds r ✭❈✳✽✮ ❊q✉❛t✐♦♥ ✭❈✳✼✮ •✉st✐✜❡s t❤❛t ✇❡ ✇✐•• s♦♠❡t✐♠❡s ❛❜✉s✐✈❡•② ♥♦t❡✿ T t = e t•A ✭❈✳✾✮ ❈✳✸ |❡s♦•✈❡♥t ♦❢ ❛ s❡♠✐❣r♦✉♣ ❲❡ st❛rt ✇✐t❤ t❤❡✿ ❚❤❡♦r❡♠ ❈✳✸✳✵✳✶✹ ✭❍✐••❡✮✳ w 0 := sup t∈R + 1 t log T t Lc(X) < ∞ ✭❈✳✶✵✮ ❚❤❡♥✱ ❢♦r λ > w 0 ✱ ♦♥❡ ♠❛② ❞❡✜♥❡ t❤❡ ❢♦••♦✇✐♥❣ ✐♥t❡❣r❛• ✐♥ t❤❡ s❡♥s❡ ♦❢ ❇♦❝❤♥❡r✿ R λ (A)x := ∞ 0 e -λt T t x dt ✭❈✳✶✶✮ ❉❡✜♥✐t✐♦♥ ❈✳✸✳✵✳✶✶ ✭|❡s♦•✈❡♥t ♦❢ ❛ s❡♠✐❣r♦✉♣✮✳ R λ (A) ✐s t❤❡ r❡s♦•✈❡♥t ♦❢ t❤❡ s❡♠✐❣r♦✉♣ T t ✳ ❚❤✐s ✐s ✐♥t❡r❡st✐♥❣ ❜❡❝❛✉s❡ ♦❢ t❤❡✿ Pr♦♣♦s✐t✐♦♥ ❈✳✸✳✵✳✶✳ ❋♦r λ > w 0 ✱ ❢♦r ❡✈❡r② x ∈ X✱ R λ (A)x ∈ D(A) ❛♥❞✿ R λ A (λ • Id -A) x = x ✭❈✳✶✷✮ ❚❤❡r❡❢♦r❡ ✇❡ ✇✐•• s♦♠❡t✐♠❡s t❤✐♥❦ ♦❢ R λ (A) ❛s t❤❡ ✐♥✈❡rs❡ ♦❢ λ • Id -A✳ ✶✹✵ ❆PP❊◆❉■❳ ❈✳ ❙❖▼❊ |❊❙❯▲❚❙ ■◆ ❙❊▼■•|❖❯P ❚❍❊❖|( ❈✳✹ ■s ❛♥ ♦♣❡r❛t♦r t❤❡ ❣❡♥❡r❛t♦r ♦❢ ❛ s❡♠✐❣r♦✉♣❄ ❈✳✺ ❋r❛❝t✐♦♥❛• ♣♦✇❡rs ♦❢ ❛ ❝♦♥tr❛❝t✐♦♥ s❡♠✐❣r♦✉♣ ❛♥❞ s✉❜♦r❞✐♥❛t✐♦♥ ❙✉♣♣♦s❡ t❤❛t T t ✐s ❛ str✐❝t ❝♦♥tr❛❝t✐♦♥ s❡♠✐❣r♦✉♣✳ ❚❤❡♥✱ ✉s✐♥❣ t❤❡ ♥♦t❛t✐♦♥ ♦❢ t❤❡♦r❡♠ ❈✳✸✳✵✳✶✹✱ ♦♥❡ ❤❛s w 0 < 0 s♦ ❢♦r 0 < β < 1 ✐t ✐s ♣♦ss✐❜•❡ t♦ ❞❡✜♥❡✱ ✐♥ t❤❡ s❡♥s❡ ♦❢ ❇♦❝❤♥❡r✿ -1 (T λ xx) dλ ✭❈✳✶✹✮ ❲❡ ❝❛•• t❤❡s❡ ❢r❛❝t✐♦♥❛• ♣♦✇❡rs ♦❢ A❀ t❤✐s ♠❛❦❡s ✐♥t✉✐t✐✈❡ s❡♥s❡ ❜❡❝❛✉s❡ ✇✐t❤✐♥ ❛ ❤✐•❜❡rt✐❛♥ s❡tt✐♥❣ ♦♥❡ ♠❛② ❞✐❛❣♦♥❛•✐③❡ A ❛♥❞ t❤❡r❡ ✐s✿ x β = x • sin(βπ) ♦t❤❡r ❤❛♥❞✱ ✐t ✐s ♣♦ss✐❜•❡ t♦ ❝♦♥str✉❝t t❤❡ ❢❛♠✐•② T β t ♦❢ t❤❡ s♦✲❝❛••❡❞ s✉❜♦r❞✐♥❛t❡❞ s❡♠✐❣r♦✉♣s ❢r♦♠ T t ❀ ✐♥❞❡❡❞✱ ✐♥ t❤❡ s❡♥s❡ ♦❢ ❇♦❝❤♥❡r✱ ❞❡✜♥❡✿ β t ❛r❡ ❞❡✜♥❡❞ t❤r♦✉❣❤ t❤❡✐r ▲❛♣•❛❝❡ tr❛♥s❢♦r♠s ❛s✿ ∞ 0 e -γs λ β t (ds) = e -γ β t ✭❈✳✶✽✮ ❚❤❡♥✱ t❤❡ ❣❡♥❡r❛t♦r ♦❢ T β t ✐s -(-A) β ✳ ❈♦♠❜✐♥✐♥❣ t❤❡ ♥♦t✐♦♥s ♦❢ r❡s♦•✈❡♥t ❛♥❞ ♦❢ ❢r❛❝t✐♦♥❛• ♣♦✇❡rs✱ ♦♥❡ ❛•s♦ ♠❛② ❞❡✜♥❡ (λ • Id -A) -s ❢♦r ❛ ❣❡♥❡r❛• s > 0✳ ■♥❞❡❡❞✱ ✐♥ t❤❡ s❡♥s❡ ♦❢ ❇♦❝❤♥❡r ✇❡ s❡t✿ (λ • Id -A) -s x 1 e -λt T t x dt ✭❈✳✶✾✮ ❆♣♣❡♥❞✐① ❉ ■♥t❡r♣♦•❛t✐♦♥ t❤❡♦r② ■♥ t❤✐s ❛♣♣❡♥❞✐① ✇❡ ❣✐✈❡ ❛ ❜r✐❡❢ r❡✈✐❡✇ t❤❡ r❡❛• ❛♥❞ ❝♦♠♣•❡① ♠❡t❤♦❞s ♦❢ ✐♥✲ t❡r♣♦•❛t✐♦♥ ❢♦r ❇❛♥❛❝❤ s♣❛❝❡s✳ ■♥ ♦✉r ❛♣♣•✐❝❛t✐♦♥s✱ ✇❡ ✇✐•• ♠♦st•② r❡❢❡r t♦ t❤❡ ❝❛s❡ ✇❤❡r❡ t❤❡ s♣❛❝❡s ✇❡ ✐♥t❡r♣♦•❛t❡ ❛r❡ ❞❡✜♥❡❞ ❛s t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ✐♥✜♥✐t❡s✐♠❛• ❣❡♥❡r❛t♦r ♦❢ s♦♠❡ s❡♠✐❣r♦✉♣✱ s♦ ✇❡ ❛•s♦ ❣✐✈❡ ♠♦r❡ s♣❡❝✐✜❝ r❡s✉•ts ❢♦r t❤❡ ✐♥t❡r✲ ♣♦•❛t✐♦♥ ♦❢ s❡♠✐❣r♦✉♣s✳ ❲❡ ♠❛✐♥•② ❢♦••♦✇ ❬✶✵( ❛♥❞ ❬✶✹(❀ ❝•❛ss✐❝ ❡①❛♠♣•❡s ✭❙♦❜♦•❡✈ ❛♥❞ ❇❡s♦✈ s♣❛❝❡s✱ ❍ö•❞❡r s♣❛❝❡s✱ ❡t❝✳✮ ❛r❡ ❡①♣•❛✐♥❡❞ ✐♥ ❞❡t❛✐• ✐♥ ❬✶(✱ ❬✻✼(✱ ❬✼✵( ❛♥❞ ❬✼✶( ❢♦r ❡①❛♠♣•❡✳ ❲❡ ❣✐✈❡ ♥♦ ♣r♦♦❢s ❛♥❞ r❡❢❡r t♦ t❤❡s❡ r❡❢❡r❡♥❝❡s ✐♥st❡❛❞✳ ❉✳✶ •❡♥❡r❛• ❞❡✜♥✐t✐♦♥s ■♥ t❤✐s ❝❤❛♣t❡r ✇❡ ✇✐•• ❝♦♥s✐❞❡r t✇♦ ♥♦r♠❡❞ s♣❛❝❡s A 0 ❛♥❞ A 1 ❀ ✇❡ ✇✐•• ❛•s♦ ♥♦t❡✿ Ā = (A 0 , A 1 ) ✭❉✳✶✮ ❉❡✜♥✐t✐♦♥ ❉✳✶✳✵✳✶✷ ✭❈♦♠♣❛t✐❜•❡ ♥♦r♠❡❞ s♣❛❝❡s✮✳ ❚❤❡ ♥♦r♠❡❞ ✈❡❝t♦r s♣❛❝❡s A 0 ❛♥❞ A 1 ❛r❡ s❛✐❞ t♦ ❜❡ ❝♦♠♣❛t✐❜•❡ ✐❢ t❤❡r❡ ❡①✐sts ❛ ❍❛✉ss❞♦r❢ s♣❛❝❡ A s✉❝❤ t❤❛t t❤❡ ❢♦••♦✇✐♥❣ ✭❝♦♥t✐♥✉♦✉s✮ ✐♥•❡❝t✐♦♥s ❤♦•❞✿ A 0 , A 1 ֒→ A ✭❉✳✷✮ ❚❤❡♥ ✐t ✐s ♣♦ss✐❜•❡ t♦ ❞❡✜♥❡ t❤❡ ✈❡❝t♦r s♣❛❝❡s A 0 ∩ A 1 ❛♥❞ A 0 + A 1 ❛♥❞ t♦ ❡q✉✐♣ t❤❡♠ ✇✐t❤ t❤❡ ♥♦r♠s✿ a A 0 ∩A 1 = max ( a A 0 , a A 1 ) ✭❉✳✸✮ a A 0 +A 1 = inf a 0 +a 1 =a ( a A 0 + a A 1 ) ✭❉✳✹✮ ❲❡ ❤❛✈❡ t❤❡ ❡❛s②✿ Pr♦♣♦s✐t✐♦♥ ❉✳✶✳✵✳✷✳ ■❢ A 0 ❛♥❞ A 1 ❛r❡ ❝♦♠♣❛t✐❜•❡ ❇❛♥❛❝❤ s♣❛❝❡s✱ t❤❡♥ A 0 ∩ A 1 ❛♥❞ A 0 + A 1 ❛r❡❇❛♥❛❝❤ s♣❛❝❡s✳ ✶✹✶ ✶✹✷ ❆PP❊◆❉■❳ ❉✳ ■◆❚❊|P❖▲❆❚■❖◆ ❚❍❊❖|( ■♥ ♣r❛❝t✐❝❡ ✇❡ ✇✐•• ♠♦st•② t❤❡ ❝❛s❡ ✇❤❡r❡ A 0 ⊂ A 1 = A ❜✉t ✇❡ st✐•• st❛t❡ t❤❡ ❣❡♥❡r❛• r❡s✉•ts ❛s t❤✐s ❞♦❡s ♥♦t ✐♥✈♦•✈❡ ❛♥② ❡①tr❛ ❞✐✣❝✉•t②✳ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡✿ ❉❡✜♥✐t✐♦♥ ❉✳✶✳✵✳✶✸ ✭■♥t❡r♠❡❞✐❛t❡ s♣❛❝❡✮✳ A ✐s s❛✐❞ t♦ ❜❡ ❛♥ ✐♥t❡r♠❡❞✐❛t❡ s♣❛❝❡ ✐♥ Ā ✐❢ t❤❡ ❢♦••♦✇✐♥❣ ✭❝♦♥t✐♥✉♦✉s✮ ✐♥•❡❝t✐♦♥s ❤♦•❞✿ A 0 ∩ A 1 ֒→ A ֒→ A 0 + A 1 ✭❉✳✺✮ ❉❡✜♥✐t✐♦♥ ❉✳✶✳✵✳✶✹ ✭■♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡✮✳ ❚❤❡ ✐♥t❡r♠❡❞✐❛t❡ s♣❛❝❡ A ✐♥ Ā ✐s s❛✐❞ t♦ ❜❡ ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ✐♥ Ā ✐❢ ❢♦r ❡✈❡r② T ∈ L c ( Ā) ♦♥❡ ❛•s♦ ❤❛s T ∈ L c (A)✳ ❚❤❡ ❦❡② r❡s✉•t ✐s t❤❡✿ ❚❤❡♦r❡♠ ❉✳✶✳✵✳✶✺✳ ■❢ A ✐s ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ✐♥ Ā t❤❡r❡ ❡①✐sts ❛ ♥✉♠❜❡r C > 0 ❛♥❞ ❛ ♥✉♠❜❡r θ ∈ [0, 1] s✉❝❤ t❤❛t ❢♦r ❡✈❡r② T ∈ L c ( Ā)✿ T Lc( Ā) ≤ C • T 1-θ Lc(A 0 ) • T θ Lc(A 1 ) ✭❉✳✻✮ ❉❡✜♥✐t✐♦♥ ❉✳✶✳✵✳✶✺✳ ■♥ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠✱ A ✐s s❛✐❞ t♦ ❜❡ ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ♦❢ ❡①♣♦♥❡♥t θ✳ ❉✳✷ ❚❤❡ r❡❛• ♠❡t❤♦❞ ♦❢ ✐♥t❡r♣♦•❛t✐♦♥ ❲❡ st❛rt ✇✐t❤ s♦♠❡ ♥♦t❛t✐♦♥✳ ▲❡t a ∈ A 0 + A 1 ❀ ✇❡ ❞❡✜♥❡ ❛ ❜♦r❡•✐❛♥ ❢✉♥❝t✐♦♥ ♦♥ R + ❜② t❤❡ ❡q✉❛t✐♦♥✿ K(t, a) = inf a=a 0 +a 1

  s♦♠❡ ❜♦r❡•✐❛♥ u : R + → A 0 ∩ A 1 s✉❝❤ t❤❛t Φ (J(•, u(•))) < ∞ ❛♥❞ ✇❡ ✐♥tr♦❞✉❝❡✿ a θ,q,J = inf u Φ (J(•, u(•))) ✭❉✳✶✸✮ ◆♦✇ ✇❡ ♠❛② st❛t❡ t❤❡ ♠❛✐♥ r❡s✉•t ✐♥ t❤✐s ♣❛r❛❣r❛♣❤✿ ❚❤❡♦r❡♠ ❉✳✷✳✵✳✶✻ ✭|❡❛• ✐♥t❡r♣♦•❛t✐♦♥✮✳ ❖♥❡ ❤❛s✿

  ♦♠♣•❡① ♠❡t❤♦❞ ♦❢ ❲❡ ✐♥tr♦❞✉❝❡✿ S := z ∈ C 0 ≤ ℜz ≤ 1 ✭❉✳✶✼✮ ❚❤❡♥ ✐❢ A 0 ❛♥❞ A 1 ❛r❡ t✇♦ ❝♦♠♣❛✐❜•❡ ❇❛♥❛❝❤ s♣❛❝❡s✱ ✇❡ ❞❡✜♥❡ ❛ ✈❡❝t♦r s♣❛❝❡ F( Ā) ❛s t❤❡ s❡t ♦❢ t❤♦s❡ ❢✉♥❝t✐♦♥s f s✉❝❤ t❤❛t✿ • f ✐s C 0 b ♦♥ S❀ • f ✐s ❤♦•♦♠♦r♣❤✐❝ ♦♥ S • ❀ • ❋♦r j ∈ {0, 1}✿ t ∈ R + → f (j + it) ∈ A j ✐s ❝♦♥t✐♥✉♦✉s❀ • ❋♦r j ∈ {0, 1}✿ lim |t|→∞ f (j + it) = 0 ❛♥❞ ✇❡ st❛t❡ t❤❡✿ ❚❤❡♦r❡♠ ❉✳✸✳✵✳✶✾✳ F( Ā) ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡ ✇❤❡♥ ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ♥♦r♠✿ f F ( Ā) = max sup t f (it) A 0 , sup t f (1 + it) A 1 ✭❉✳✶✽✮ ◆♦✇ ❢♦r 0 ≤ θ ≤ 1 ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ s❡t✿ [ Ā] θ = a ∈ A 0 + A 1 ∃f ∈ F( Ā), f (θ) = a ✭❉✳✶✾✮ ❛♥❞ ✇❡ st❛t❡✿ ❚❤❡♦r❡♠ ❉✳✸✳✵✳✷✵ ✭❈♦♠♣•❡① ✐♥t❡r♣♦•❛t✐♦♥✮✳ [ Ā] θ ✐s ❛ ✈❡❝t♦r s♣❛❝❡❀ ✐t ❛•s♦ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡ ✐❢ ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ♥♦r♠✿ a θ = inf f F ( Ā) f (θ) = a ✭❉✳✷✵✮ ▼♦r❡♦✈❡r [ Ā] θ ✐s ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ♦❢ ♦r❞❡r θ ❛♥❞ A 0 ∩ A 1 ✐s ❞❡♥s❡ ✐♥ [ Ā] θ ✳ |❡♠❛r❦ ❉✳✸✳✵✳✶✳ ❬✶✵( ✐♥tr♦❞✉❝❡s ❛♥♦t❤❡r ❝♦♠♣•❡① ✐♥t❡r♣♦•❛t✐♦♥ ♠❡t❤♦❞✳ ❍♦✇✲ ❡✈❡r✱ ✐t ✐s s❤♦✇♥ t❤❛t t❤❡ t✇♦ ♠❡t❤♦❞s ❛r❡ ❡q✉✐✈❛•❡♥ts ✇❤❡♥ ❛t •❡❛st ♦♥❡ ♦❢ t❤❡ s♣❛❝❡s A 0 ♦r A 1 ✐s r❡✢❡①✐✈❡✱ ✇❤✐❝❤ ✇✐•• ❛•✇❛②s ❜❡ t❤❡ ❝❛s❡ ✐♥ ♦✉r ❛♣♣•✐❝❛t✐♦♥s✳ ❆•s♦✱ ✇❡ ✇✐•• ♥♦t ♠❛❦❡ ❛♥② ❞✐r❡❝t ✉s❡ ♦❢ t❤❡ ♦t❤❡r ♠❡t❤♦❞✱ t❤❡r❡❢♦r❡ ✇❡ ❞♦ ♥♦t ❞❡✈❡•♦♣ ✐t ❤❡r❡✳ ❲❡ ✇✐•• ❛•s♦ ♠❛❦❡ ✉s❡ ♦❢ t❤❡ ❢♦••♦✇✐♥❣ r❡s✉•ts✿ ❉✳✹✳ ❙❊▼■•|❖❯P ■◆❚❊|P❖▲❆❚■❖◆ ✶✹✺ ❚❤❡♦r❡♠ ❉✳✸✳✵✳✷✶ ✭❉✉❛•✐t② t❤❡♦r❡♠✮✳ ■❢ ❛t •❡❛st ♦♥❡ t❤❡ ❇❛♥❛❝❤ s♣❛❝❡s A 0 ♦r A 1 ✐s r❡✢❡①✐✈❡✱ t❤❡♥✿ [ Ā] ′ θ = (A ′ 0 , A ′ 1 ) [θ] ✭❉✳✷✶✮ ❚❤❡♦r❡♠ ❉✳✸✳✵✳✷✷ ✭|❡✐t❡r❛t✐♦♥ t❤❡♦r❡♠✮✳ ■❢ A 0 ✱ A 1 ❛r❡ t✇♦ ❝♦♠♣❛t✐❜•❡ ❇❛♥❛❝❤ s♣❛❝❡s ❛♥❞ ✐❢ 0 < θ, θ 0 , θ 1 , η < 1 ✈❡r✐❢②✿ θ = (1η) • θ 0 + η • θ 1 ✭❉✳✷✷✮ t❤❡♥ t❤❡ ❢♦••♦✇✐♥❣ ❤♦•❞s✿ [ Ā] θ 1 , [ Ā] θ 0 η = [ Ā] θ ✭❉✳✷✸✮ ❚❤❡♦r❡♠ ❉✳✸✳✵✳✷✸ ✭|❡•❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ t❤❡ ❝♦♠♣•❡① ❛♥❞ r❡❛• ✐♥t❡r♣♦•❛t✐♦♥ ♠❡t❤♦❞s✮✳ ❖♥❡ ❤❛s✿ • Āθ,1 ⊂ [ Ā] θ ⊂ Āθ,∞ • [ Ā] θ 0 , [ Ā] θ 1 η,p = Āθ,p • Āθ 0 ,p 0 , Āθ 1 ,p 1 η = Āθ,p ✇❤❡r❡ θ = (1η) • θ 0 + η • θ 1 ❛♥❞ 1 p = 1-η p 0 + η p 1 ✳ ❉✳✹ ❙❡♠✐❣r♦✉♣ ✐♥t❡r♣♦•❛t✐♦♥ ■❢ T t ✐s ❛ s❡♠✐❣r♦✉♣ ♦❢ ❝•❛ss C 0 ✇✐t❤ ❣❡♥❡r❛t♦r G ♦♥ s♦♠❡ ❇❛♥❛❝❤ s♣❛❝❡ X✱ ✐t✐s ✐♥t❡r❡st✐♥❣ t♦ ❝♦♥s✐❞❡r t❤❡ ❋❛✈❛r❞ ❝•❛ss X α,r,q ✱ ✇❤✐❝❤ ✐s ❞❡✜♥❡❞✱ ❢♦r r ∈ N✱ 0 < α < r ❛♥❞ q ∈ [1, ∞] ❛s t❤❡ ❢♦••♦✇✐♥❣ r❡❛• ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡✿ X α,r,q = (X, D(A r ))

  t❤❛t • A k ❤❛s ❜❡❡♥ ❞❡✜♥❡❞ ✐♥ ❛♣♣❡♥❞✐① ❈✳ ✶✹✻ ❆PP❊◆❉■❳ ❉✳ ■◆❚❊|P❖▲❆❚■❖◆ ❚❍❊❖|( ❆♣♣❡♥❞✐① ❊ ❆ r❡s✉•t ❢r♦♠ P❉❊ t❤❡♦r② ❲❡ ♠❛❦❡ ✉s❡ ♦❢ t❤❡ ❢♦••♦✇✐♥❣ r❡s✉•ts ✇❤✐❝❤ ✇❡ ❝♦♣② ❢r♦♠ ❬✼✵(✳ ❲❡ st❛rt ✇✐t❤ t❤❡✿ ❉❡✜♥✐t✐♦♥ ❊✳✵✳✵✳✶✻ ✭•♦♦❞ ✇❡✐❣❤t ❢✉♥❝t✐♦♥✮✳ ρ ✐s s❛✐❞ t♦ ❜❡ ❛ ❣♦♦❞ ✇❡✐❣❤t ❢✉♥❝✲ t✐♦♥ ♦♥ R N ✐❢ ✐t ✐s ❛ ♣♦s✐t✐✈❡✱ C ∞ ❢✉♥❝t✐♦♥ s✉❝❤ t❤❛t✿ • ❢♦r ❛♥② ♠✉•t✐✲✐♥❞❡① γ✱ |D γ ρ| = O ρ 1+|γ| ❤♦•❞s❀ • ❢♦r s♦♠❡ a > 0✱ ρ -a ∈ L 1 ✳ ◆♦✇✱ ❜❡✐♥❣ ❣✐✈❡♥ ❛ ❣♦♦❞ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ρ✱ ❛ ♥♦♥✲③❡r♦ ✐♥t❡❣❡r m✱ ❛ r❡❛• ♥✉♠❜❡r µ ❛♥❞ ❛ r❡❛• ♥✉♠❜❡r ν s✉❝❤ t❤❛t ν > µ + 2m ❛♥❞ ν ≥ 0✱ ✇❡ ♠❛② ✐♥tr♦❞✉❝❡✿ ❉❡✜♥✐t✐♦♥ ❊✳✵✳✵✳✶✼ ✭❋❛♠✐•② A m µ,ν R N ♦❢ ❞✐✛❡r❡♥t✐❛• ♦♣❡r❛t♦rs✮✳ ❚❤❡s❡ ❛r❡ t❤❡ ❞✐✛❡r❡♥t✐❛• ♦♣❡r❛t♦rs Au = m l=0 |α|=2l ρ χ 2l b α D α u + |β|<2m a β D β u ✭❊✳✶✮ ✇❤❡r❡ t❤❡ b α ❛♥❞ t❤❡ a β ❛r❡ C ∞ ❢✉♥❝t✐♦♥s ❛♥❞✿ • χ l = 1 2m (ν (2ml) + µl) • b 0 ≥ C > 0 • (-1) m |α|=2m b α (x)ξ α ≥ C|ξ| 2m • ❢♦r 1 ≤ l ≤ m -1✿ (-1) l |α|=2l b α (x)ξ α ≥ 0 • ❢♦r ❛•• ♠✉•t✐✲✐♥❞❡①❡s β ❛♥❞ γ✿ D γ a β = o ρ χ |β| +|γ| ■♥ t❤❡ ❛❜♦✈❡ ❢r❛♠❡✇♦r❦✱ ❬✼✵( ♣r♦✈✐❞❡s t❤❡ ❢♦••♦✇✐♥❣✿ ✶✹✼ ✶✹✽ ❆PP❊◆❉■❳ ❊✳ ❆ |❊❙❯▲❚ ❋|❖▼ P❉❊ ❚❍❊❖|( ❚❤❡♦r❡♠ ❊✳✵✳✵✳✷✻ ✭❊st✐♠❛t❡s ❢♦r ❛♥ ♦♣❡r❛t♦r ✐♥ A m µ,ν R N ✮✳ ▲❡t A ∈ A m µ,ν R N ✱ χ ∈ R ❛♥❞ 1 < p < ∞✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ r❡❛• ♥✉♠❜❡r C 1 ❛♥❞ t✇♦ ♣♦s✐t✐✈❡ ♥✉♠❜❡rs C 2,3 ❞❡♣❡♥❞✐♥❣ ♦♥•② ♦♥ A✱ p ❛♥❞ χ s✉❝❤ t❤❛t✱ ❢♦r ❛♥② ❝♦♠♣•❡① ♥✉♠❜❡r λ s✉❝❤ t❤❛t Reλ ≤ C 1 ✱ t❤❡ ❢♦••♦✇✐♥❣ ❤♦•❞s✿ C 2 u W 2mp (ρ χ+pµ ,ρ χ+pν ) ≥ Auλu L p (ρ χ ) ≥ C 3 u W 2m p (ρ χ+pµ ,ρ χ+pν ) ✭❊✳✷✮

  ••❡ st♦❝❤❛st✐q✉❡✿ dX t = b(t, X t )dt + σ(t, X t )dW t ✭✺✸✮ ❖♥ ✐♥tr♦❞✉✐t •✬❡s♣❛❝❡ C 0,s ❞❡s ❢♦♥❝t✐♦♥s ❝♦♥t✐♥✉❡s ❡♥ t❡♠♣s✱ ❛❞♠❡tt❛♥t [s] ❞ér✐✈é❡s ❡♥ ❡s♣❛❝❡✱ ❞♦♥t •❛ ❞ér✐✈é❡ ❞✬♦r❞r❡ [s] ❡♥ ❡s♣❛❝❡ ❡st {s}✲❤♦•❞❡r✐❡♥♥❡✳ ❖♥ ❛ ❛•♦rs •❡✿ ①①✐✈ ■◆❚|❖❉❯❈❚■❖◆ ❚❤é♦rè♠❡ ✵✳✸✳✸✳✶✳ ❖♥ ❝♦♥s✐❞èr❡ •✬❊❉❙ ❝✐✲❞❡ss✉s ❡t ♦♥ s✉♣♣♦s❡ q✉❡ b ❡t σ s♦♥t ❞❛♥s C 0,s •✉sq✉✬à ✉♥ ❝❡rt❛✐♥ t❡♠♣s T ✱ ♣♦✉r ✉♥ ❝❡rt❛✐♥ s ≥ 1✳ ❆•♦rs ♣♦✉r t♦✉s p > 1✱ s ′ < s ❡t ♣♦✉r t♦✉t t ≥ 0✱ ♦♥ ❛ X t ∈ D p,s ′ ❡t ✐• ❡①✐st❡ ❞❡s ❝♦♥st❛♥t❡s A ❡t B ♥❡ ❞é♣❡♥❞❛♥tq✉❡ ❞❡ p✱ s✱ s ′ ❡t ❞❡s ♥♦r♠❡s C 0,s ❞❡ b ❡t σ t❡••❡s q✉❡✿