Jacques Franchi

Yves Le Jan

Curvature Diffusions in General Relativity

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1 Introduction

It is known since Dudley's pioneer's work [Du] that a relativistic diffusion, i.e. a Lorentzcovariant Markov diffusion process, cannot exist on the base space, even in the Minkowski framework of special relativity, but possibly makes sense at the level of the tangent bundle. In this spirit, the general case of a Lorentz manifold was first investigated in [F-LJ], where a general relativistic diffusion was introduced. The quadratic variation of this diffusion is constant and does not vanish in the vacuum. In this article, we investigate diffusions whose quadratic variation is locally determined by the curvature of the space, and then vanishes in empty (or at least flat) regions.

The relativistic diffusion considered in [F-LJ] lives on the pseudo-unit tangent bundle T 1 M of the given generic Lorentz manifold (M, g). As is recalled in Section 3.1 below, it can be obtained by superposing, to the geodesic flow of T 1 M, random fluctuations of the velocity that are given by hyperbolic Brownian motion, if one identifies the tangent space T 1 ξ M with the hyperbolic space H d (at point ξ ∈ M, by means of the pseudo-metric g). These Brownian fluctuations of the velocity can be defined by the vertical Dirichlet form

T 1 M ∇ v ξ F (ξ, ξ)
2 µ(dξ, d ξ) , considered with respect to the Liouville measure µ .

The Dirichlet forms we investigate int his article depend only on the local geometry of (M, g), e.g. on the curvature tensor at the current point ξ , and on the velocity ξ . We consider several examples :

-If the scalar curvature R(ξ) is everywhere non-positive (which is physically relevant, see [L-L]), then the Dirichlet form can be : -

T 1 M ∇ v ξ F (ξ, ξ)
covariant Ξ-relativistic diffusions, or Ξ-diffusions, in Section 3 below.

-If the sectional curvatures of timelike planes are everywhere non-negative (sectional curvature has proved to be a natural tool in Lorentzian geometry, see for example [H], [H-R]), as this is often the case (at least in usual symmetrical examples), then it is possible to construct a covariant sectional relativistic diffusion which undergoes velocity fluctuations that are no longer isotropically Brownian, using the whole curvature tensor (not the Ricci tensor alone). See Section 6 below. This sectional relativistic diffusion depends on the curvature tensor in a canonical way. But its diffusion symbol vanishes in flat regions, but does not, in general, in empty regions.

Note that all these covariant diffusions are the projections on T 1 M of diffusions on the frame bundle G(M). Actually, they are constructed directly on G(M), as was done in [F-LJ]. These constructions are performed in Sections 3 and 6 below.

Note also that, while in the flat case the Dudley diffusion [Du] is the unique covariant diffusion, the above examples show that this is not at all the same for curved spaces.

In Sections 4 and 5, we study in more detail the case of warped products, and specify further some particularly symmetrical examples, namely Robertson-Walker manifolds, which are warped products with energy-momentum tensor of perfect fluid type.

We investigate more closely Einstein-de Sitter-like manifolds (Robertson-Walker manifolds for which the expansion rate is α(t) = t c for some positive c), reviewing in this simple class of examples, the relativistic diffusions we introduced, which appear to be distinct. We perform in this setting an asymptotic study of the R-diffusion, and of the minimal sub-diffusion (t s , ṫs ) relative to the energy E-diffusion.

Canonical vector fields and curvature

We present in this section the main notations and recall a few known facts (see [K-N]).

2.1 Isomorphism between Λ 2 R 1,d and so (1, d) On the Minkowski space-time R 1,d , we denote by η = ((η ij )) 0≤i,j≤d the Minkowski tensor      1 0 . . . 0 0 -1 . . . 0 . . . . . . . . .

0 . . . 0 -1     
. We also denote by ((η ij )) 0≤i,j≤d the inverse tensor, so that η ij η jk = δ k i (or equivalently :

η ij = η ij := 1 {i=j=0} -1 {1≤i=j≤d} ), and by •, • η the corresponding Minkowski pseudo-metric. For u, v, w ∈ R 1,d , we set u ∧ v (w) := u, w η v -v, w η u . (1)
In other terms, this is the interior product of u ∧ v by the dual of w with respect to η .

This defines an endomorphism of R 1,d which belongs to so(1, d), since for any w, w ′ ∈ R 1,d we have clearly u ∧ v(w), w ′ η + w, u ∧ v(w ′ ) η = 0 . It vanishes only if u and v are collinear, hence if and only if u ∧ v = 0 . We have thus an isomorphism between Λ 2 R 1,d and so(1, d).

Remark 2.1.1 The Lie bracket of so(1, d) can be expressed, for any a, b, u, v ∈ R 1,d , by :

[a ∧ b, u ∧ v] = a, u η b ∧ v + b, v η a ∧ u -a, v η b ∧ u -b, u η a ∧ v . The Minkowski pseudo-metric •, • η extends to Λ 2 R 1,d , by setting : u ∧ v, a ∧ b η := u, a η v, b η -u, b η v, a η = 1 2 u ∧ v(a), b η -u ∧ v(b), a η , (2)
so that, if (e 0 , . . . , e d ) is a Lorentz (i.e. pseudo-orthonormal) basis of (R 1,d , η), then

(e i ∧ e j | 0 ≤ i < j ≤ d) is an orthogonal basis of (Λ 2 R 1,d , η),
such that e i ∧ e j , e i ∧ e j η = η ii η jj .

Frame bundle G(M) over (M, g)

Let M be a time-oriented C ∞ (1+d)-dimensional Lorentz manifold, with pseudo-metric g having signature (+, -, . . . , -), and let T 1 M denote the positive half of the pseudounit tangent bundle. Let G(M) be the bundle of direct pseudo-orthonormal frames, with first element in T 1 M , which has its fibers modelled on the special Lorentz group. Let π 1 : u → π(u), e 0 (u) denote the canonical projection from G(M) onto the tangent bundle T 1 M , which to each frame e 0 (u), . . . , e d (u) associates its first vector e 0 .

We denote by T M π 2 -→ M the tangent bundle, by G(M) π -→ M the frame bundle, by u = π(u); e 0 (u), . . . , e d (u) the generic element of G(M). We extend (1) to a linear action of so(1, d) ≡ Λ 2 R 1,d on G(M), by setting :

e k ∧ e ℓ (e j (u)) := η jk e ℓ (u) -η jℓ e k (u) , for any 0 ≤ j, k, ℓ ≤ d ,
where (e 0 , . . . , e d ) denotes the canonical basis of R 1,d .

The action of SO(d) on (e 1 , . . . , e d ) induces the identification

T 1 M ≡ G(M)/SO(d).
The right action of so(1, d) on G(M) defines a linear map V from so(1, d) into vector fields on G(M) (i.e. sections of the canonical projection of T G(M) on G(M)), such that

[V a∧b , V α∧β ] = V [a∧b,α∧β] , for any a ∧ b , α ∧ β ∈ Λ 2 R 1,d . (3)
Vector fields V a∧b are called vertical.

Notation To abreviate the notations, we shall consider mostly the canonical vector fields :

V ij := V e i ∧e j , for 0 ≤ i, j ≤ d .
By (3) and (1), for 0 ≤ i, j, k, ℓ ≤ d we have :

[V ij , V kℓ ] = η ik V jℓ + η jℓ V ik -η iℓ V jk -η jk V iℓ . (4)
We shall often write V j for V 0j .

A connection σ , i.e. a section T M → T T M of T π 2 , is metric if it extends into a section : T M → T GM of T π . A metric connection σ defines the horizontal vector field H k on G(M), given by :

H k (u) := σ(e k (u)) , for any u ∈ G(M) and 0 ≤ k ≤ d . (5)
The canonical vectors

V ij , H k span T G(M) (the horizontal (resp. vertical) sub-bundle of T G(M) being spanned by H k 's (resp. V ij 's)).
Note that H 0 generates the geodesic flow, that V 1 , . . . , V d generate the boosts, and that the V ij (0 ≤ i, j ≤ d) generate rotations.

This allows to define the intrinsic torsion tensor ((T k ij )) and curvature tensor ((R ij kℓ )) (with 0 ≤ i, j, k, ℓ ≤ d) of the metric connection σ, by the assignment :

[H i , H j ] = d k=0 T k ij H k + 0≤k<ℓ≤d R ij kℓ V kℓ (6)
we can denote more simply

T k ij H k + 1 2 R ij kℓ V kℓ .
For any metric connection we have :

[V ij , H k ] = η ik H j -η jk H i , for 0 ≤ i, j, k ≤ d . (7)
There exists a unique metric connection with vanishing torsion, called the Levi-Civita connection. We shall henceforth consider this one.

Given two vector fields X, Y on M and ξ ∈ M, the covariant derivative

∇ X Y (ξ) ∈ T ξ M is defined by writing X(π(u)) = d k=0
X k (u)e k (u), and setting :

∇ X Y (ξ) := d k=0 X k (u) T π 1 (H k (u))Y (ξ) , for any u ∈ π -1 (ξ) . (8)
The curvature operator R ξ is defined on Λ 2 T ξ M by :

R ξ e i (u)∧e j (u) := 0≤k<ℓ≤d R ij kℓ e k (u)∧e ℓ (u), for any u ∈ π -1 (ξ) and 0 ≤ i, j ≤ d . (9)
The curvature operator is alternatively given by : for any

C 1 vector fields X, Y, Z, A , R (X ∧ Y ) , A ∧ Z η = [∇ X , ∇ Y ] -∇ [X,Y ] Z , A g . ( 10
)
The Ricci tensor and Ricci operator are defined, for 0 ≤ i, k ≤ d , by :

R k i := d j=0 R ij kj , and Ricci ξ e i (u) := d k=0 R k i e k (u) , for any u ∈ π -1 (ξ) . (11)
The scalar curvature is : R :=

d k=0 R k k .
The indexes of the curvature tensor ((R ij kℓ )) and of the Ricci tensor ((R k i )) are lowered or raised by means of the Minkowski tensor ((η ab )) and its inverse ((η ab )). For example, we have : R p jqr = R ij kℓ η ip η kq η ℓr , and

R ij = R k i η kj .
Remark 2.2.1 The curvature and Ricci operators and tensors are symmetrical :

R(a ∧ b), v ∧ w η = a ∧ b, R(v ∧ w) η , and Ricci(v), w η = v, Ricci(w) η , for any a, b, v, w ∈ R 1,d . Equivalently, for 0 ≤ i, j, k, ℓ ≤ d : R ij kℓ = R kℓ ij , and R ij = R ji .
The energy-momentum tensor ((T k j )) and operator T ξ are defined as :

T k j := R k j -1 2 R δ k j and T ξ := Ricci ξ -1 2 R . (12)
Note that

d j=0 T j j = -d-1 2 R . The energy at any line-element (ξ, ξ) ∈ T 1 M is E(ξ, ξ) := T ξ ( ξ), ξ g(ξ) = T 00 (ξ, ξ) . ( 13
)
The weak energy condition (see [H-E]) stipulates that E(ξ, ξ) ≥ 0 on the whole T 1 M.

We shall need the following general computation rule.

Lemma 2.2.2 For 0 ≤ i, j, k, ℓ, p, q ≤ d , we have :

V qp R ij kℓ = η qi R pj kℓ -η ip R qj kℓ +η qj R ip kℓ -η jp R iq kℓ +δ k q R ijp ℓ -δ k p R ijq ℓ -δ ℓ q R ijp k +δ ℓ p R ijq k .
Proof Using ( 9) and (1), we have indeed :

V qp R ij kℓ = V qp R(e i ∧ e j
), e n ∧ e m η η kn η ℓm = R (η qi e p -η ip e q ) ∧ e j , e n ∧ e m η + R e i ∧ (η qj e p -η jp e q ) , e n ∧ e m η η kn η ℓm + R(e i ∧ e j ), (η qn e p -η np e q ) ∧ e m η + R(e i ∧ e j ), e n ∧ (η qm e p -η mp e q ) η η kn η ℓm

= η qi R pj kℓ -η ip R qj kℓ + η qj R ip kℓ -η jp R iq kℓ + δ k q R ijp ℓ -δ k p R ijq ℓ -δ ℓ q R ijp k + δ ℓ p R ijq k . ⋄

Expressions in local coordinates

Consider local coordinates (ξ i , e k j ) for u = (ξ, e 0 , . . . , e d ) ∈ G(M), with e j = e k j ∂ ∂ξ k . Then the horizontal and vertical vector fields V ij , V j , H k , which satify the commutation relations (4),( 6),(7) of the preceding section 2.2, read as follow. First, denoting by Γ ℓ kj = Γ ℓ jk the Christoffel coefficients of the Levi-Civita connexion ∇, we have for 0 ≤ i, j ≤ d :

∇ ∂ ∂ξ i ∂ ∂ξ j = Γ k ij (ξ) ∂ ∂ξ k and H j = e k j ∂ ∂ξ k -e k j e m i Γ ℓ km (ξ) ∂ ∂e ℓ i . ( 14
)
Recall that the Christoffel coefficients (of ∇) are computed by :

Γ k ij = 1 2 g kℓ ∂g ℓj ∂ξ i + ∂g iℓ ∂ξ j - ∂g ij ∂ξ ℓ ,
or equivalently, by the fact that geodesics solve ξk + Γ k ij ξi ξj = 0 .

Then for 0 ≤ i, j, k ≤ d :

V ij e k = η ik e j -η jk e i = (η ik e ℓ j -η jk e ℓ i ) ∂ ∂ξ ℓ = (η iq e m j -η jq e m i ) ∂ ∂e m q e ℓ k ∂ ∂ξ ℓ ,
whence for 0 ≤ i, j ≤ d :

V ij = (η iq e m j -η jq e m i ) ∂ ∂e m q , ( 15
)
that is :

V ij = e k i ∂ ∂e k j -e k j ∂ ∂e k i and V j = e k 0 ∂ ∂e k j + e k j ∂ ∂e k 0 , for 1 ≤ i, j ≤ d .
The curvature operator expresses in a local chart as : for 0 ≤ m, n, p, q ≤ d ,

R mnpq := R ∂ ∂ξ m ∧ ∂ ∂ξ n , ∂ ∂ξ p ∧ ∂ ∂ξ q g = g mr Γ r ps Γ s nq -Γ r qs Γ s np + ∂Γ r nq ∂ξ p - ∂Γ r np ∂ξ q . (16)
Then, the Ricci operator can be computed similarly, as : for 0 ≤ m, p ≤ d ,

Rmp := Ricci( ∂ ∂ξ m ) , ∂ ∂ξ p g = R mnpq g nq = Γ n nq Γ q mp -Γ n pq Γ q mn + ∂Γ n mp ∂ξ n - ∂Γ n mn ∂ξ p . ( 17
)
The scalar curvature and the energy-momentum operator can be computed by :

R = Rij g ij and Tℓm = Rℓm -1 2 R g ℓm . ( 18
)
To summarize, the Riemann curvature tensor ((R ij kℓ )) is made of the coordinates of the curvature operator R in an orthonormal moving frame, and its indexes are lowered or raised by means of the Minkowski tensor ((η ab )), while the curvature tensor (( Rmnpq )) is made of the coordinates of the curvature operator in a local chart, and its indexes are lowered or raised by means of the metric tensor ((g ab )).

To go from one tensor to the other, note that by ( 16) and ( 9) we have

R ∂ ∂ξ m ∧ ∂ ∂ξ n = 1 2 R mn ab ∂
∂ξ a ∧ ∂ ∂ξ b , whence : e k i e ℓ j R kℓ pq = R ij mn e p m e q n , or equivalently :

R ijab = R kℓrs e k i e ℓ j e r a e s b , or as well : R rspq = R abmn e r a e s b e p m e q n . (19)

Case of a perfect fluid

The energy-momentum tensor T (of ( 12), or equivalently T , recall ( 18)) is associated to a perfect fluid (see [H-E]) if it has the form :

Tkℓ = q U k U ℓ -p g kℓ , (20)
for some C 1 field U in T 1 M , and some C 1 functions p, q on M .

By Einstein Equations ( 18), ( 20) is equivalent to :

Rkℓ = q U k U ℓ + p g kℓ , with p = (2p -q)/(d -1), (21)
or as well, by ( 17), to :

Ricci(W ), W η = q × g(U, W ) 2 + p × g(W, W ) , for any W ∈ T M . (22)
The quantity U (ξ s ), ξs , (which is the hyperbolic cosine of the distance, on the unit hyperboloid at ξ s identified with the hyperbolic space between the space-time velocities of the fluid and the path) will be denoted by A s or A(ξ s , ξs ).

Note that necessarily A s ≥ 1 if U, U = 1 . By Formulas ( 13) and (20), the energy equals :

E(ξ, ξ) = q(ξ) A(ξ, ξ) 2 -p(ξ). (23)
Remark 2.4.1 (i) The energy of the fluid is simply : Tkℓ U k U ℓ = q -p . and the scalar curvature equals R = 2 [(d + 1) p -q]/(d -1).

(ii) By ( 23), the weak energy condition reads here : q ≥ p + .

Covariant Ξ-relativistic diffusions

Let Ξ denote a non-negative smooth function on G(M), invariant under the right action of SO(d) (so that it identifies with a function on T 1 M).

Our examples will be Ξ = -̺ 2 R and Ξ = ̺ 2 E (for a positive constant ̺).

We call Ξ-relativistic diffusion or Ξ-diffusion the G(M)-valued diffusion process (Φ s ) associated to Ξ we will construct in Section 3.2 below, as well as its T 1 M-valued projection π 1 (Φ s ). Let us first recall our previous construction, which corresponds to a constant Ξ .

The basic relativistic diffusion

The relativistic diffusion process (ξ s , ξs ) was defined in [F-LJ], as the projection under π 1 of the G(M)-valued diffusion (Ψ s ) solving the following Stratonovitch stochastic differential equation (for a given R d -valued Brownian motion (w j s ) and some fixed ̺ > 0) :

dΨ s = H 0 (Ψ s ) ds + ̺ d j=1 V j (Ψ s ) • dw j s . (24)
The infinitesimal generator of the G(M)-valued relativistic diffusion (Ψ s ) is

H := H 0 + ̺ 2 2 d j=1 V 2 j , (25)
and the infinitesimal generator of the relativistic diffusion (ξ s , ξs ) := π 1 (Ψ s ) is the relativistic operator :

H 1 := L 0 + ̺ 2 2 ∆ v = ξk ∂ ∂ξ k + d ̺ 2 2 ξk -ξi ξj Γ k ij (ξ) ∂ ∂ ξk + ̺ 2 2 ( ξk ξℓ -g kl (ξ)) ∂ 2 ∂ ξk ∂ ξℓ , ( 26
)
where L 0 denotes the vector field on T 1 M generating the geodesic flow, and ∆ v denotes the vertical Laplacian, i.e. the Laplacian on T 1 ξ M equipped with the hyperbolic metric induced by g(ξ). The relativistic diffusion (ξ s , ξs ) is parametrized by proper time s ≥ 0 , possibly till some positive explosion time.

In local coordinates (ξ i , e k j ), setting Ψ s = (ξ i s , e k j (s)), Equation (24) becomes locally equivalent to the following system of Itô equations :

dξ k s = ξk s ds = e k 0 (s) ds ; d ξk s = -Γ k iℓ (ξ s ) ξi s ξℓ s ds + ̺ d i=1 e k i (s) dw i s + d ̺ 2
2 ξk s ds , and

de k j (s) = -Γ k iℓ (ξ s ) e ℓ j (s) ξi s ds + ̺ ξk s dw j s + ̺ 2 2 e k j (s) ds , for 1 ≤ j ≤ d , 0 ≤ k ≤ d . Remark 3.1.1 We have on T 1 M : d j=1 V 2 j E = 2(d + 1) E -2 Tr(T ) = 2(d + 1) E + (d -1) R . (27)
Indeed, since for each 1 ≤ j ≤ d V j exchanges the basis vectors e 0 = ξ and e j (recall (15)) we get :

V 2 j E = 2 V j ( Tℓm ξℓ e m j ) = 2 Tℓm ( ξℓ ξm + e ℓ j e m j ), whence d j=1 V 2 j E = 2d E + 2 Tℓm ( ξℓ ξm -g ℓm ) = 2(d + 1) E -2 Tr( T ) ,
and Formulas ( 27) follow at once by ( 12).

As an application, a direct computation yields the following evolution of the energy.

Remark 3.1.2 The random energy process E s = E(ξ s , ξs ) associated to the basic relativistic diffusion π 1 (Ψ s ) = (ξ s , ξs ) satisfies the following equation (where ∇ v := v j ∇ j ) :

dE s = ∇ ξs E s ds + ̺ 2 (d + 1)E s + d-1 2 R(ξ s ) ds + dM E s ,
with the quadratic variation of its martingale part dM E s given by :

[dE s , dE s ] = [dM E s , dM E s ] = 4̺ 2 [E 2 s -T ξs , T ξs ] ds .
We have here in particular

∇ ξs E s = ∂ ξ k Tij (ξ s ) -2 Tiℓ (ξ s ) Γ ℓ jk (ξ s ) ξi s ξj s ξk s .
Note that the energy E s is not, in general, a Markov process.

Construction of the Ξ-diffusion

Let us start with the following Stratonovitch stochastic differential equation on G(M) (for a given R d -valued Brownian motion (w j s )) :

dΦ s = H 0 (Φ s ) ds + 1 4 d j=1 V j Ξ (Φ s )V j (Φ s ) ds + d j=1 Ξ(Φ s ) V j (Φ s ) • dw j s . (28)
Note that all coefficients in this equation are clearly smooth, except √ Ξ on its vanishing set Ξ -1 (0). However, √ Ξ is a locally Lipschitz function ; see ([I-W], Proposition IV.6.2). Hence, Equation (28) does define a unique G(M)-valued diffusion (Φ s ).

We have the following proposition, defining the Ξ-relativistic diffusion (or Ξ-diffusion) (Φ s ) on G(M), and (ξ s , ξs ) on T 1 M, possibly till some positive explosion time.

Proposition 3.2.1 The Stratonovitch stochastic differential equation ( 28) has a unique solution (Φ s ) = (ξ s ; ξs , e 1 (s), . . . , e d (s)), possibly defined till some positive explosion time. This is a G(M)-valued covariant diffusion process, with generator

H Ξ := H 0 + 1 2 d j=1 V j Ξ V j . ( 29
)
Its projection π 1 (Φ s ) = (ξ s , ξs ) defines a covariant diffusion on T 1 M, with SO(d)-invariant generator H 1 Ξ := L 0 + 1 2 ∇ v Ξ ∇ v , (30)
∇ v denoting the gradient on T 1 ξ M equipped with the hyperbolic metric induced by g(ξ). Moreover, the adjoint of H Ξ with respect to the Liouville measure of G(M) is

H * Ξ := -H 0 + 1 2 d j=1 V j Ξ V j .
In particular, if there is no explosion, then the Liouville measure is invariant. Furthermore, if Ξ does not depend on ξ , i.e. is a function on M, then the Liouville measure is preserved by the stochastic flow defined by Equation ( 28).

We specify at once how this looks in a local chart, before giving a proof for both statements.

Corollary 3.2.2

The T 1 M-valued Ξ-diffusion (ξ s , ξs ) satisfies dξ s = ξs ds , and in any local chart, the following Itô stochastic differential equations : for 0 ≤ k ≤ d , (denoting

Ξ s = Ξ(ξ s , ξs )) d ξk s = dM k s -Γ k ij (ξ s ) ξi s ξj s ds + d 2 Ξ s ξk s ds + 1 2 [ ξk s ξℓ s -g kℓ (ξ s )] ∂Ξ ∂ ξℓ (ξ s , ξs ) ds , (31)
with the quadratic covariation matrix of the martingale term (dM s ) given by :

[d ξk s , d ξℓ s ] = [ ξk s ξℓ s -g kℓ (ξ s )] Ξ s ds , for 0 ≤ k, ℓ ≤ d .
Proof In local coordinates (ξ i , e k j ), Φ = (ξ ; e 0 , e 1 , . . . , e d ), using Section 2.3, Equation (28) reads : for any 0

≤ k ≤ d , dξ k s = ξk s ds = e k 0 (s) ds , d ξk s = -Γ k iℓ (ξ s ) ξi s ξℓ s ds + 1 4 d j=1 V j Ξ (ξ s , ξs ) e k j (s) ds + d j=1 Ξ(ξ s , ξs ) e k j (s) • dw j s ;
and for

1 ≤ j ≤ d , 0 ≤ k ≤ d , de k j (s) = -Γ k iℓ (ξ s ) e ℓ j (s) ξi s ds + 1 4 V j Ξ (ξ s , ξs ) ξk s ds + Ξ(ξ s , ξs ) ξk s • dw j s .
We compute now the Itô corrections, which involve the partial derivatives of Ξ(ξ, ξ) with respect to ξ . We get successively, for 1

≤ j ≤ d , 0 ≤ k ≤ d : Ξ(ξ s , ξs ) d Ξ(ξ s , ξs ) e k j (s) , dw j s = Ξ(ξ s , ξs ) de k j (s), dw j s + 1 2 e k j (s) d Ξ(ξ s , ξs ), dw j s = Ξ (ξ s , ξs ) 3/2 ξk s ds + 1 2 e k j (s) ∂ Ξ ∂ ξℓ (ξ s , ξs ) Ξ(ξ s , ξs ) e ℓ j (s) ds , hence, d Ξ(ξ s , ξs ) e k j (s) , dw j s = Ξ (ξ s , ξs ) ξk s ds + 1 2 V j Ξ (ξ s , ξs ) e k j (s) ds ; Ξ(ξ s , ξs ) d Ξ(ξ s , ξs ) ξk s , dw j s = Ξ(ξ s , ξs ) d ξk s , dw j s + 1 2 ξk s d Ξ(ξ s , ξs ), dw j s = Ξ (ξ s , ξs ) 3/2 e k j (s) ds + 1 2 ξk s ∂ Ξ ∂ ξℓ (ξ s , ξs ) Ξ(ξ s , ξs ) e ℓ j (s) ds , hence, d Ξ(ξ s , ξs ) ξk s , dw j s = Ξ (ξ s , ξs ) e k j (s) ds + 1 2 V j Ξ (ξ s , ξs ) ξk s ds .
Note that the simplification by Ξ(ξ s , ξs ) is allowed, since on Ξ -1 (0) both sides of the simplified formula vanish identically.

Hence, in local coordinates and in Itô form, Equation ( 28) reads : for any 0 ≤ k ≤ d , dξ k s = ξk s ds = e k 0 (s) ds , and setting dM k s :=

d j=1 Ξ(ξ s , ξs ) e k j (s) dw j s , d ξk s = dM k s -Γ k iℓ (ξ s ) ξi s ξℓ s ds + d 2 Ξ (ξ s , ξs ) ξk s ds + 1 2 [ ξk s ξℓ s -g kℓ (ξ s )] ∂Ξ ∂ ξℓ (ξ s , ξs ) ds ; de k j (s) = Ξ(ξ s , ξs ) ξk s dw j s -Γ k iℓ (ξ s ) e ℓ j (s) ξi s ds + 1 2 Ξ (ξ s , ξs ) e k j (s) ds + 1 2 V j Ξ (ξ s , ξs ) ξk s ds .
Note that we used the formula

d j=1 e k j (s)e ℓ j (s) = ξk s ξℓ s -g kℓ (ξ s ) , which expresses that Φ ∈ G(M).
Using again this formula, we get the quadratic covariation matrix of the martingale term (dM s ), displayed in the above proposition, which shows that (π 1 (Φ s )) is indeed a diffusion, and proves Corollary 3.2.2.

On the other hand, comparing the above equations with Equations ( 24) and ( 25), which correspond to Ξ ≡ ̺ 2 , we get precisely the wanted form (29) for the generator of (Φ s ). 26) (for which Ξ ≡ ̺ 2 ), we see that establishing Formula (30) giving the projected generator H 1 Ξ reduces now to proving that

Then, since ∇ v Ξ ∇ v = Ξ ∆ v + (∇ v Ξ) ∇ v ,
(∇ v Ξ) ∇ v ≡ [ ξk s ξℓ s -g kℓ ] ∂Ξ ∂ ξℓ ∂ ∂ ξk . Now this becomes clear, noting that ∇ v j = e k j ∂ ∂ ξk , for each 1 ≤ j ≤ d .
Finally, the assertions relative to the Liouville measure are direct consequences of the fact that the vectors H 0 and V j are antisymmetric with respect to the Liouville measure of G(M). (The invariance with respect to H 0 , i.e. to the geodesic flow, is proved in the same way as in the Riemannian case, and the invariance with respect to V j is straightforward.) ⋄ Remark 3.2.3 (i) The vertical terms could be seen as an effect of the matter or the radiation present in the space-time M. The Ξ-diffusion (Φ s ) reduces to the geodesic flow in the regions of the space where Ξ vanishes, which happens in particular for empty spacetimes M in the cases

Ξ = -̺ 2 R(ξ), or Ξ = ̺ 2 E(ξ, ξ), or also Ξ = -̺ 2 R(ξ) e κ E(ξ, ξ)/R(ξ)
(for positive constant κ) for example.

(ii) As well as for the basic relativistic diffusion, the law of the Ξ-relativistic diffusion is covariant with any isometry of (M, g). The basic relativistic diffusion corresponds to Ξ ≡ ̺ 2 > 0 , and the geodesic flow to Ξ ≡ 0 .

(iii) In [B] is considered a general model for relativistic diffusions, which may be covariant or not. Up to enlarge it slightly, by allowing the "rest frame" (denoted by z in [B]) to have space vectors of non-unit norm, this model includes the generic Ξ-diffusion (compare the above equation ( 29) to (2.5) in [B]).

The R-diffusion

We assume here that the scalar curvature R = R(ξ) is everywhere non-positive on M, which is physically relevant (see [L-L]), and consider the particular case of Section 3.2 corresponding to Ξ = -̺ 2 R(ξ), with a constant positive parameter ̺ .

In this case, as its central term clearly vanishes, Equation (28) takes on the simple form :

dΦ s = H 0 (Φ s ) ds + ̺ d j=1 -R(Φ s ) V j (Φ s ) • dw j s .

The E-diffusion

We assume that the Weak Energy Condition (recall Section 2.3) holds (everywhere on T 1 M), which is physically relevant (see [L-L], [H-E]), and consider the particular case of Section 3.

2 corresponding to Ξ = ̺ 2 E = ̺ 2 E(ξ, ξ) = ̺ 2 T 00 .
We call energy relativistic diffusion or E-diffusion the G(M)-valued diffusion process (Φ s ) we get in this way, as well as its T 1 M-valued projection π 1 (Φ s ).

The following is easily derived from Lemma 2.2.2 and Formula (11). As a consequence, the central drift term in Equation ( 28) is a function of the Ricci tensor alone when Ξ is.

Lemma 3.4.1 We have V j R k i = δ 0i R k j -η ij R k 0 + δ k 0 R ij -δ k j R 0i , for 0 ≤ i, k ≤ d and 1 ≤ j ≤ d .
In particular, V j R = 0 , and V j E = V j T 00 = V j R 00 = 2R 0j . Another expression is got by using Einstein equation ( 18), or equivalently, by computing directly from ( 31) and ( 13) :

By

[ ξk ξℓ -g kℓ ] ∂E ∂ ξℓ = 2[ ξk ξℓ -g kℓ ] Tℓm ξm = 2[E ξ -T ξ] k , where the notation ( T ξ) k ≡ T k m
ξm has the meaning of a matrix product.

Hence, Formula (31) of Proposition 3.2.2 expressing d ξs reads here :

d ξk s = dM k s -Γ k ij (ξ s ) ξi s ξj s ds + ̺ 2 d 2 E s ξk s ds + ̺ 2 Rmn (ξ s ) ξm [ ξk s ξn s -g kn (ξ s )] ds , (32)
or equivalently :

d ξs = dM s -Γ • ij (ξ s ) ξi s ξj s ds + ̺ 2 ( d 2 + 1) E s ξs ds -̺ 2 T ξs ds . ( 33
)
We can then compute the equation satisfied by the random energy E s . In particular, the drift term discussed above for d ξk s contributes now for :

2̺ 2 Tkm ξm [E ξ -T ξ] k = 2̺ 2 (E Tkm ξm ξk -Tkm ξm [ T ξ] k ) = 2̺ 2 (E 2 -[ T ξ] k [ T ξ] k ).
This leads to the following, to be compared with Corollary 3.1.2.

Remark 3.4.2 The random energy E s := E(ξ s , ξs ) associated to the E-diffusion (Φ s ) satisfies the following equation (where ∇ v := v j ∇ j ) :

dE s = ∇ ξs E(ξ s , ξs ) ds + (d + 2) ̺ 2 E 2 s ds -2̺ 2 g( T ξs , T ξs ) ds + 2̺ dM E s ,
with the quadratic variation of its martingale part dM E s given by :

[dE s , dE s ] = 4̺ 2 [dM E s , dM E s ] = 4̺ 2 [E 2 s -g( T ξs , T ξs )] E s ds .
Note that the diffusion coefficient [E 2 s -g( T ξs , T ξs )] appearing in the quadratic variation [dE s , dE s ] is necessarily non-negative, which can of course be checked directly.

Remark 3.4.3 Case of Einstein Lorentz manifolds.

The Lorentz manifold M is said to be Einstein if its Ricci tensor is proportional to its metric tensor. Bianchi's contracted identities (see [H-E] or [K-N]), which entail the conservation equations ∇ k T jk = 0 , force the proportionality coefficient p to be constant on M. Hence : Rℓm (ξ) = p g ℓm (ξ) , for any ξ in M and 0 ≤ ℓ, m ≤ d .

Then the scalar curvature is R(ξ) = (d + 1)p , and by Einstein Equations (18) we have :

Tℓm (ξ) = (Λ -d-1
2 p) g ℓm (ξ) =: -p g ℓm (ξ) . Hence Equation (20) holds, with q = 0 : we are in a limiting case of perfect fluid. Moreover, R and E are constant, so that on an Einstein Lorentz manifold, the R-diffusion and the E-diffusion coincide with the basic relativistic diffusion (of Section 3.1).

Warped (or skew) products

Let us consider here a Lorentz manifold (M, g) having the warped product form : M = I × M , where I is an open interval of R + and (M, h) is a Riemannian manifold, is endowed with the Lorentzian pseudo-norm g given by :

ds 2 := dt 2 -α(t) 2 |dx| 2 h , (34)
or equivalently

g 0k := δ 0k and g ij := -α(t) 2 h ij (x) , for 0 ≤ k ≤ d , 1 ≤ i, j ≤ d . (35)
Here ξ ≡ (t, x) ∈ I × M denotes the generic point of M, and the expansion factor α is a positive C 2 function on I . The so-called Hubble function is :

H(t) := α ′ (t)/α(t) . ( 36
)
This structure is considered in [B-E], which contains most of the following proposition.

Proposition 4.1 Consider a Lorentz manifold (M, g) having the warped product form.

(i) Its curvature operator R is given by : for u, v, w, a

∈ C 0 (I) and X, Y, Z, A ∈ T M , R ((u∂ t + X) ∧ (v∂ t + Y )) , (a∂ t + A) ∧ (w∂ t + Z) g = -α 2 K (X ∧ Y ) , A ∧ Z + αα ′′ h(u Y -v X, a Z -w A) + (αα ′ ) 2 [h(X, Z) h(Y, A) -h(X, A) h(Y, Z)] , (37)
K denoting the curvature operator of (M, h).

(ii) Denoting by Ric the Ricci operator of (M, h) and by •,• the standard canonical inner product of R d , we have :

Ricci(v∂ t + Y ), w∂ t + Z η = Ric(Y ), Z + [(d -1) |α ′ | 2 + αα ′′ ] h(Y, Z) -d α ′′ α vw. (38)
(iii) If ∇ g and ∇ h denote the Levi-Civita connections of (M, g) and (M, h) respectively :

∇ g (u∂t+X) (v∂ t + Y ) = [uv ′ + α(t)α ′ (t) h(X, Y )] ∂ t + ∇ h X Y + H(t) (u Y + v X) . ( 39
)
Remark 4.2 In any warped local chart (t, x j ) : for 1 ≤ m, n, p, q ≤ d (K mnpq denoting the curvature tensor of (M, h), and K mp its Ricci tensor) and 0 ≤ k ≤ d , we have

Γ • 00 (g) = 0 ; Γ p 0n (g) = H δ p n ; Γ 0 mn (g) = αα ′ h mn ; Γ p mn (g) = Γ p mn (h) , (40)
R 0nkq = δ 0k α(t)α ′′ (t) h nq ; R mnpq = (αα ′ ) 2 (t) [h mq h np -h mp h nq ] -α 2 (t) K mnpq , (41) R0k = -δ 0k d × α ′′ (t)/α(t) ; Rmp = Kmp + [(d -1) |α ′ | 2 + αα ′′ ] h mp . ( 42
)
Let us outline the proof for the convenience of the reader.

Proof We get (39), and then (40), by using Koszul formula :

2 h(∇ h X Y, Z) = Xh(Y, Z)+Y h(X, Z)-Zh(X, Y )+h([X, Y ], Z)-h([X, Z], Y )+h([Z, Y ], X),
for both ∇ h and ∇ g . Hence,

∇ g (u∂t+X) ∇ g (v∂t+Y ) (w∂ t + Z) = ∇ g (u∂t+X) [vw ′ + αα ′ h(Y, Z)]∂ t + ∇ h Y Z + H [vZ + wY ] = u[v ′ w ′ +vw ′′ +(αα ′ ) ′ h(Y, Z)]+αα ′ Xh(Y, Z)+h(X, ∇ h Y Z)+H[vh(X, Z)+wh(X, Y )] ∂ t +∇ h X ∇ h Y Z + H [v ∇ h X Z + w ∇ h X Y ] + uH ′ [v Z + w Y ] + uH [v ′ Z + w ′ Y ] , +H u ∇ h Y Z + u H [vZ + wY ] + [vw ′ + αα ′ h(Y, Z)] X . Therefore ∇ g (u∂t+X) , ∇ g (v∂t+Y ) (w∂ t + Z) = [uv ′ -u ′ v]w ′ + (αα ′ ) ′ [u h(Y, Z) -v h(X, Z)] + αα ′ [Xh(Y, Z) -Y h(X, Z)] ∂ t + αα ′ [h(X, ∇ h Y Z) -h(Y, ∇ h X Z)] + α 2 H 2 [v h(X, Z) -u h(Y, Z)] ∂ t +[∇ h X , ∇ h Y ]Z + H (v ∇ h X Z -u ∇ h Y Z + w [X, Y ]) + (Hw) ′ [u Y -v X] + H [uv ′ -u ′ v] Z +H u ∇ h Y Z -v ∇ h X Z + (Hw -w ′ )(u Y -v X) + αα ′ [h(Y, Z) X -h(X, Z) Y ] .
And since ∇ g [(u∂t+X),(v∂t+Y )] (w∂

t + Z) = ∇ g ([uv ′ -u ′ v] ∂t+[X,Y ]) (w∂ t + Z) = [uv ′ -u ′ v]w ′ + αα ′ h([X, Y ], Z) ∂ t + ∇ h [X,Y ] Z + H ([uv ′ -u ′ v] Z + w [X, Y ]) , we get ∇ g (u∂t+X) , ∇ g (v∂t+Y ) -∇ g [(u∂t+X),(v∂t+Y )] (w∂ t + Z) = (αα ′ ) ′ [u h(Y, Z) -v h(X, Z)] + αα ′ [Xh(Y, Z) -Y h(X, Z)] ∂ t + αα ′ [h(X, ∇ h Y Z) -h(Y, ∇ h X Z) -h([X, Y ], Z)] + α 2 H 2 [v h(X, Z) -u h(Y, Z)] ∂ t + [∇ h X , ∇ h Y ] -∇ h [X,Y ] Z + H (v ∇ h X Z -u ∇ h Y Z) + (Hw) ′ [u Y -v X] +H ∇ h u Y -v X Z + H (Hw -w ′ )(u Y -v X) + α 2 H 2 [h(Y, Z) X -h(X, Z) Y ] = αα ′′ h(uY -vX, Z)∂ t +([∇ h X , ∇ h Y ]-∇ h [X,Y ] )Z + α ′′ α w(uY -vX)+α ′2 [h(Y, Z)X -h(X, Z)Y ].
By Formula (10), this entails Formula (37), which is equivalent to (41), by ( 16).

Then, denoting by (e 1 , . . . , e d ) an orthonormal basis of (M, h) :

Ricci(v∂ t + Y ), w∂ t + Z η = R (∂ t ∧ (v∂ t + Y )) , ∂ t ∧ (w∂ t + Z) η -α(t) -2 d j=1
R (e j ∧ (v∂ t + Y )) , e j ∧ (w∂ t + Z) η , which yields (38) ; which is in turn equivalent to (42), by ( 17). ⋄ Corollary 4.3 A Lorentz manifold (M, g) having the warped product form is of perfect fluid type (recall Section 2.4) if and only if the Ricci operator of its Riemannian factor M is conform to the identical map : Ric = Ω × Id , for some Ω ∈ C 0 (M ).

If this holds, we must have :

U = ∂ t , p + q = -d (α ′′ /α), and q = Ω α -2 -(d -1)H ′ .
Proof By ( 22) and ( 38), this happens if and only if for any v ∈ C 0 (I), Y ∈ T M :

Ric(Y ), Y + [(d -1)α ′ 2 + αα ′′ ]h(Y, Y ) -d(α ′′ /α)v 2 = q g(U, v∂ t + Y ) 2 + p [v 2 -α 2 h(Y, Y )] ,
which (seen as a polynomial in v) forces U = ∂ t , and then splits into

p + q = -d (α ′′ /α) and Ric(Y ), Y = -[(d -1)α ′ 2 + αα ′′ + p α 2 ] h(Y, Y ). This latter equation is equivalent to p = -[(d -1)H 2 + (α ′′ /α) + Ω α -2 ]
, and then, using

p + q = -d (α ′′ /α), to q = Ω α -2 -(d -1)H ′ . ⋄
Corollary 4.4 Consider a Lorentz manifold (M, g) having the warped product form. The energy (13) at ξ ≡ ( ṫ, ẋ) ∈ T 1 ξ M ≡ T 1 (t,x) (I × M ) equals :

E(ξ, ξ) = Ric( ẋ), ẋ -(d -1) H ′ (t) ( ṫ2 -1) + 1 2 d (d -1) H 2 (t) + R M 2 α 2 (t) . ( 43
)
Then the weak energy condition is equivalent to the following lower bounds for the Ricci operator and the scalar curvature R M of the Riemannian factor (M, h) :

inf w∈T M Ric(w), w h(w, w) ≥ (d -1) sup I {α α ′′ -α ′ 2 } ; R M ≥ -d (d -1) inf I α ′ 2 . ( 44
)
And the scalar curvature R of (M, g) equals :

R = -α -2 R M -d (d -1) | α ′ α | 2 + 2 α ′′ α , (45)
so that its non-positivity is equivalent to the lower bound on the scalar curvature R M :

R M ≥ -d × inf I {(d -1) α ′ 2 + 2 α α ′′ } . ( 46
)
Proof From (38), we compute the scalar curvature of M :

R = Ricci(∂ t ), ∂ t η -α -2 d j=1
Ricci(e j ),

e j η = -α -2 R M -d (d -1) | α ′ α | 2 + 2 α ′′ α .
On the other hand, by ( 38) we have at ξ ≡ ( ṫ, ẋ)

∈ T 1 ξ M ≡ T 1 (t,x) (I × M ) : Ricci( ξ), ξ η = Ric( ẋ), ẋ + [(d -1) |α ′ | 2 + αα ′′ ] ṫ2 -1 α 2 -d α ′′ α ṫ2 = Ric( ẋ), ẋ -(d -1) H ′ (t) ( ṫ2 -1) -d α ′′ α (t)
. Thence Formula (43). Then, the weak energy condition holds if and only if for any t ∈ I and w ∈ T M :

Ric(w), w ≥ (d -1) α 2 (t)H ′ (t) h(w, w) -1 2 d (d -1) H 2 (t) - R M 2 α 2 (t)
.

By homogeneity with respect to w , this can be split into the following lower bound for the Ricci operator of the Riemannian factor M :

inf w∈T M Ric(w), w h(w, w) ≥ (d -1) sup I {α α ′′ -α ′ 2 } ,
together with the condition particularised to w = 0 , which yields the following lower bound for the scalar curvature of the Riemannian factor M :

d(d -1) H 2 (t) + α(t) -2 R M ≥ 0 , or R M ≥ -d (d -1) inf I α ′ 2 . ⋄

Example of Robertson-Walker (R-W) manifolds

These important manifolds are particular cases of warped product : they can be written M = I × M , where I is an open interval of R + and M ∈ {S 3 , R 3 , H 3 }, with canonical global spherical coordinates ξ ≡ (t, r, ϕ, ψ), and are endowed with the pseudo-norm :

g( ξ, ξ) := ṫ2 -α(t) 2 ṙ2 1 -kr 2 + r 2 φ2 + r 2 sin 2 ϕ ψ2 , ( 47
)
where the constant scalar spatial curvature k belongs to {-1, 0, 1} (note that r ∈ [0, 1] for k = 1 and r ∈ R + for k = 0, -1), and the expansion factor α is as in the previous section 4. Note that we have necessarily ṫ ≥ 1 everywhere on T 1 M .

By (37), we have the curvature operator given by :

R ((u∂ t + X) ∧ (v∂ t + Y )) , (a∂ t + A) ∧ (w∂ t + Z) η = αα ′′ h(uY -vX, aZ -wA) -α 2 (α ′ 2 + k) [h(X, A)h(Y, Z) -h(X, Z)h(Y, A)] .
By (38), the Ricci tensor (( Rℓm )) is diagonal, with diagonal entries :

-3 α ′′ (t) α(t) , A(t) 1 -kr 2 , A(t) r 2 , A(t) r 2 sin 2 ϕ , where A(t) := α(t) α ′′ (t) + 2 α ′ (t) 2 + 2k ,
and the scalar curvature is

R = -6 [α(t) α ′′ (t) + α ′ (t) 2 + k] α(t) -2 .
The Einstein energy-momentum tensor Rℓm -1 2 R g ℓm = Tℓm is diagonal as well, with diagonal entries :

3 α ′ (t) 2 + k α(t) 2 , -Ã(t) 1 -kr 2 , -Ã(t) r 2 , -Ã(t) r 2 sin 2 ϕ , with Ã(t) := 2α(t)α ′′ (t) + α ′ (t) 2 + k .
Hence, we have

Tℓm -α(t) -2 Ã(t) g ℓm = 2 [k α(t) -2 -H ′ (t)] 1 {ℓ=m=0} .
Thus, in accordance with Corollary 4.3, we have here an example of perfect fluid : Equation ( 20) holds, with

U j ≡ δ 0 j , -p(ξ) = k α(t) -2 + 2H ′ (t) + 3H 2 (t) , q(ξ) = 2 [k α(t) -2 -H ′ (t)] , (48)
p(ξ) = -2 [2k α(t) -2 + H ′ (t) + 3H 2 (t)]/(d -1) .

Note that

A s = U i (ξ s ) ξi s = ṫs and E s = 2 [k α(t s ) -2 -H ′ (t s )] ṫ2 s -p(ξ s ) . ( 49
)
By Corollary 4.4 (or by Remark 2.4.1(ii) as well), the weak energy condition is equivalent to :

α ′ 2 + k ≥ (α α ′′ ) + .

Case of an expanding R-W manifold

We consider an eternal Robertson-Walker space-time, which has its future-directed half-geodesics complete. This amounts to

I = R * + , together with ∞ α √ 1 + α 2 = ∞ .
In the case of the basic relativistic diffusion (solving Equation ( 24) in this same

Robertson-Walker model), we have in particular :

d ṫs = ̺ ṫ2 s -1 dw s + 3̺ 2 2 ṫs ds -H(t s )[ ṫ2 s -1] ds . ( 50
)

Ξ-relativistic diffusions in an Einstein-de Sitter-like manifold

We consider henceforth the particular case I = ]0, ∞[ , k = 0 , and α(t) = t c , with exponent c > 0 . Note that such expansion functions α can be obtained by solving a proportionality relation between p and q (see [H-E] or [L-L]).

Thus q = 2c t -2 , p = (2 -3c)c t -2 , R = -6 c (2c -1) t -2 , E = c t -2 (2 ṫ2 + 3c -2) .
Note that the weak energy condition holds. The scalar curvature is non-positive if and only if c ≥ 1/2 , and the pressure p is non-negative if and only if c ≤ 2/3 .

Note that the particular case c = 2 3 corresponds to a vanishing pressure p , and is precisely known as that of Einstein-de Sitter universe (see for example [H-E]). And the analysis of [L-L] shows up precisely both limiting cases c = 2 3 and c = 1 2 .

Basic relativistic diffusion in an Einstein-de Sitter-like manifold

In order to compare with the other relativistic diffusions, we mention first for the basic relativistic diffusion (of Section 3.1), the stochastic differential equations satisfied by the main coordinates ṫs and ṙs , appearing in the 4-dimensional sub-diffusion (t s , ṫs , r s , ṙs ). By (50), we have, for independent standard real Brownian motions w, w :

d ṫs = ̺ ṫ2 s -1 dw s + 3̺ 2 2 ṫs ds - c t s ( ṫ2 s -1) ds ; (51)
d ṙs = ̺ ṫs ṙs ṫ2 s -1 dw s + ̺ 1 t 2c s - ṙ2 s ṫ2 s -1 d ws + 3̺ 2 2 ṙs ds + ṫ2 s -1 t 2c s -ṙ2 s ds r s - 2c t s ṫs ṙs ds . ( 52
)
Almost surely (see [A]), lim s→∞ ṫs = ∞ , and x s /r s ∼ ẋs /| ẋs | converges in S 2 .

R-diffusion in an Einstein-de Sitter-like manifold

With the above, Section 3.3 reads here, for the R-relativistic diffusion, when c ≥ 1/2 :

d ξs = ̺ dM s + 9c (2c -1)̺ 2 t -2 s ξs ds -Γ • ij (ξ s ) ξi s ξj s ds , (53)
with the quadratic covariation matrix of the martingale part dM s given by :

̺ -2 [d ξk s , d ξℓ s ] = 6c (2c -1) [ ξk s ξℓ s -g kℓ (ξ s )] t -2 s ds , for 0 ≤ k, ℓ ≤ d .
In particular, we have for independent standard real Brownian motions w, w :

d ṫs = ̺ t s 6c (2c -1)( ṫ2 s -1) dw s + 9̺ 2 c (2c -1) t 2 s ṫs ds - c t s ( ṫ2 s -1) ds ; ( 54
)
d ṙs = ̺ 6c (2c -1) t s ṫs ṙs ṫ2 s -1 dw s + 1 t 2c s - ṙ2 s ṫ2 s -1 d ws (55) + 9̺ 2 c (2c -1) t 2 s ṙs ds + ṫ2 s -1 t 2c s -ṙ2 s ds r s - 2c t s ṫs ṙs ds .
As the scalar curvature R s = 6c (1 -2c)/t 2 s vanishes asymptotically, we expect that almost surely the R-diffusion behaves eventually as a timelike geodesic, and in particular that lim s→∞ ṫs = 1 .

E-diffusion in an Einstein-de Sitter-like manifold

Similarly, using ( 33), ( 48), ( 49), we have here E ξ -T ξ = 2(0 -H ′ )( ṫ2 ξṫ U ) , so that Section 3.4 reads here, for the E-diffusion :

d ξs = ̺ dM s + 3̺ 2 c 2 t -2 s (2 ṫ2 s + 3c -2) ξs ds + 2̺ 2 c t -2 s ( ṫs ξs -U s ) ṫs ds -Γ • ij (ξ s ) ξi s ξj s ds , ( 56
)
with the quadratic covariation matrix of the martingale part dM s given by :

̺ -2 [d ξk s , d ξℓ s ] = c [ ξk s ξℓ s -g kℓ (ξ s )] (2 ṫ2 s + 3c -2) t -2 s ds , for 0 ≤ k, ℓ ≤ d .
In particular, we have for some standard real Brownian motion w :

d ṫs = ̺ √ c t s (2 ṫ2 s -2 + 3c)( ṫ2 s -1) dw s + c 5̺ 2 ( ṫ2 s -1 + 9c 10 ) ṫs t 2 s - ṫ2 s -1 t s ds ; (57)
d ṙs = ̺ √ c t s 2 ṫ2 s -2 + 3c ṫs ṙs ṫ2 s -1 dw s + 1 t 2c s - ṙ2 s ṫ2 s -1 d ws (58) + ̺ 2 c (5 ṫ2 s -3 + 9c 2 ) ṙs t 2 s ds - 2c t s ṫs ṙs ds + ṫ2 s -1 t 2c s -ṙ2 s ds r s . Remark 5.2.4 Comparison of Ξ-diffusions in an E.-d.S.
-like manifold Along the preceding sections 5.2.1, 5.2.2, 5.2.3, we specified to an Einstein-de Sitter-like manifold the various Ξ-diffusions we considered successively in Sections 3.1, 3.3, 3.4. Restricting to the only equation relating to the hyperbolic angle A s = ṫs , or in other words, to the simplest sub-diffusion (t s , ṫs ), this yields Equations ( 51), ( 54), ( 57) respectively. We observe that even in this simple case, all these covariant relativistic diffusions differ notably, having pairwise distinct minimal sub-diffusions (with 3 non-proportional diffusion factors).

Asymptotic behavior of the R-diffusion in an

Einstein-de Sitter-like manifold

We present here the asymptotic study of the R-diffusion of an Einstein-de Sitter-like manifold (recall Sections 5.2, 5.2.2). We will focus our attention on the simplest subdiffusion (t s , ṫs ), and on the space component x s ∈ R 3 . Recall from (49) that ṫs = A s equals the hyperbolic angle, measuring the gap between the ambient fluid and the velocity of the diffusing particle. Recall also that, by the unit pseudo-norm relation, ṫs controls the behavior of the whole velocity ξs . We get as a consequence the asymptotic behavior of the energy E s . As quoted in Section 5.2.2, we must have here c ≥ 1 2 . Note that for c = 1 2 , the scalar curvature vanishes, and the R-diffusion reduces to the geodesic flow, whose equations are easily solved and whose time coordinate satisfies (for constants a and s 0 ) :

s -s 0 = t s (t s + a 2 ) -a 2 log[ √ t s + √ t s + a 2 ] , whence t s ∼ s .
The proofs of this section (and of the following one) will use several times the elementary fact that almost surely a continuous local martingale cannot go to infinity.

The following confirms a conjecture stated in Section 5.1.

Proposition 5.3.1 The process ṫs goes almost surely to 1, and E s → 0 , as s → ∞ .

Proof By Equation ( 54),

log ṫs ṫ1 -3̺ 2 c(2c -1) s 1 (2 + ṫ-2 τ ) dτ t 2 τ + c s 1 (1 -ṫ-2 τ ) ṫτ t τ dτ
is a continuous martingale with quadratic variation 6̺ 2 c (2c -1)

s 1 (1 -ṫ-2 τ ) dτ t 2 τ
. Hence, since ṫτ ≥ 1 and therefore t τ ≥ τ , the non-negative process

log ṫs + c s 1 (1 -ṫ-2 τ ) ṫτ t τ dτ
converges almost surely as s → ∞ . This forces the almost sure convergence of the integral :

∞ 1 (1 -ṫ-2 τ ) ṫτ t τ dτ < ∞ , and of ṫs , towards some ṫ∞ ∈ [1, ∞[ . This implies in turn t τ = O(τ ), hence ∞ 1 (1 -ṫ-2 τ ) dτ τ < ∞ , whence finally ṫ∞ = 1 . ⋄
Consider now the functional a := t c ṫ2 -1 , which is constant along any geodesic.

Lemma 5.3.2 For c > 1 2 , the process a s := t c s ṫ2 s -1 goes almost surely to infinity, and cannot vanish. Moreover, for any ε > 0 we have almost surely :

∞ 1 t 2c-2 s ds a 2+ε s < ∞ .
Proof We get from Equation (54) :

da s = ̺ t s 6c (2c -1)(a 2 s + t 2c s ) dw s + 3̺ 2 c (2c -1) 3 a 2 s + 2 t 2c s t 2 s a s ds ,
and then for any ε ∈ ]0, 1] and for some continuous local martingale M :

0 ≤ a -ε s = a -ε 1 -M s -3ε̺ 2 c (2c -1) s 1 [2 -ε] a 2 τ + [1 -ε] t 2c τ t 2 τ a 2+ε τ dτ .
The signs in this last formula, and the fact that almost surely a continuous local martingale cannot go to infinity, imply the the convergence of the last integral and of the martingale term M s , entailing the almost sure existence of a finite limit a -1 ∞ , hence of a ∞ ∈ ]0, ∞], and the almost sure convergence of the integral

∞ 1 dτ a 2+2ε τ t 2-2c τ < ∞ . Now, by Proposition 5.3.1, this implies ∞ 1 dτ a 2+2ε τ τ ≤ ∞ 1 dτ a 2+2ε τ τ 2-2c < ∞ , hence a ∞ = ∞ .
Finally, the equation for a -ε s forbids also the existence of a finite zero s 0 for a s . Indeed, s ր s 0 would force the martingale term of this equation to go to -∞, which is impossible. ⋄

The following reveals the asymptotic behavior of the space component (x s ) for c > 1 2 .

Proposition 5.3.3 For c > 1 2 , the space component converges almost surely (as s → ∞) :

x s → x ∞ ∈ R 3 .
Proof (i) Let us consider the non-negative process u s := t s ( ṫ2 s -1), which is constant for c = 1 2 , and cannot vanish for c > 1 2 , by Lemma 5.3.2. By Equation (54),

u s + (2c -1) s 1 u τ ṫτ t τ dτ -6̺ 2 c (2c -1) s 1 [4 ṫ2 τ -1] dτ t τ
is a continuous local martingale. Then, for any ε > 0 ,

u s t ε s -6̺ 2 c [2c -1] s 1 [4 ṫ2 τ -1] dτ t 1+ε τ + [2c -1 + ε] s 1 u τ dt τ t 1+ε
:= c -1 2 , to get : ∞ 1 | ẋs | ds 2 ≤ ∞ 1 t ε-2c s ds × ∞ 1 t 2c-ε s | ẋs | 2 ds ≤ 2 2c-1 ∞ 1 u s t 1+ε s ds < ∞ .
This proves that

x s = x 1 + s 1 ẋτ dτ → x 1 + ∞ 1 ẋs ds ∈ R 3 , almost surely as s → ∞ . ⋄
In the case c = 1 2 of the R-diffusion being the geodesic flow, we have

r s = b 2 /a 2 + (a + o(1)) log s ∼ a log s as s → ∞ ,
which shows that Proposition 5.3.3 does not hold for the limiting case c = 1 2 . To compare the R-diffusion with geodesics, note that (as is easily seen ; see for example [A]) along any timelike geodesic, we have

x s = x 1 + ẋ1 | ẋ1 | s 1 a dτ t 2c τ and ẋs | ẋs| = ẋ1 | ẋ1 |
, which converges precisely for c > 1 2 ; and along any lightlike geodesic, we have On the other hand, for c ≤ 1 , the behavior of the basic relativistic diffusion proves to satisfy (see [A]) :

x s = x 1 + ẋ1 | ẋ1 | ts t 1 dτ τ c ∼ V × s 1-c 1+c
r s ∼ s→∞ s 1 a τ dτ t 2c
τ -→ ∞ (exponentially fast, at least for c < 1).

Hence, the R-diffusion behaves asymptotically more like a (timelike) geodesic than like the basic relativistic diffusion. However, owing to Lemma 5.3.2, the asymptotic behavior of the R-diffusion seems to be somehow intermediate between those of the geodesic flow and of the basic relativistic diffusion.

Asymptotic energy of the E-diffusion in an E.-d.S. manifold

We consider here the case of Section 5.2.3, dealing with the energy diffusion in an Einstein-de Sitter-like manifold, and more precisely, with its absolute-time minimal subdiffusion (t s , ṫs ) satisfying Equation (57), and with the resulting random energy : Proof By Equation ( 57),

E s = c t -2 s (2 ṫ2 s + 3c -2) =
1 ṫs∧ζ -c s∧ζ s 0 1 -1 ṫ2 τ dτ t τ + ̺ 2 c s∧ζ s 0 3 + 3c+2 2 ṫ2 τ + 3c-2 ṫ4 τ dt τ t 2 τ is a continuous martingale with quadratic variation 2̺ 2 c s∧ζ 1 1+ 3c-2 2 ṫ2 τ 1-1 ṫ2 τ dτ t 2 τ
. Hence, since ṫτ ≥ 1 and therefore t τ ≥ τ , the process

ṫ-1 s∧ζ -c s∧ζ s 0 (1 -ṫ-2 τ ) dτ t τ
converges almost surely as s → ∞ . As 0 < ṫ-1 s∧ζ ≤ 1 , this forces the almost sure convergence of the integral :

ζ s 0 1 -1 ṫ2 τ dτ t τ
< ∞ , and of ṫs∧ζ , towards some ṫζ ∈ [1, ∞].

Moreover, the convergence of the integral forces either ζ < ∞ and then ṫζ = ∞ , or ζ = ∞ and then ṫζ ∈ {1, ∞}. ⋄

The asymptotic behavior can, with positive probability, be partly opposite to that of the preceding R-diffusion : Proposition 5.4.2 From any starting point (t s 0 , ṫs 0 ), there is a positive probability that both A s = ṫs and the energy E s explode. This happens with arbitrary large probability, starting with ṫs 0 /t s 0 sufficiently large and t 0 bounded away from zero.

On the other hand, there is also a positive probability that the hyperbolic angle A s = ṫs does not explode and goes to 1, and then that the random energy E s goes to 0. This happens actually with arbitrary large probability, starting with sufficiently large t s 0 / ṫs 0 .

Proof Let us set λ s := t s / ṫs ≥ 0 . From the above proof of Lemma 5.4.1, we get directly : (i) Let us first start the time sub-diffusion from (t s 0 , ṫs 0 ) such that t s 0 ≥ s 0 ≥ 1 and ṫs 0 ≥ n m t s 0 , with fixed n ≥ 2 and m ≥ 2 + 1+c 3̺ 2 c , and consider T := ζ ∧ inf{s > s 0 | m λ s > 1}. Thus, we have on [s 0 , T ] : λ -1 τ ≥ m , and then

λ s∧ζ -λ s 0 = M s∧ζ + [1 + c] s∧ζ s 0 1 -c [1+c] ṫ2 τ dτ -̺ 2 c s∧ζ s 0 3 + 3c-2 2 ṫ2 τ + 3c-2 ṫ4 τ dτ λ τ (59) ≤ M s∧ζ + [1 + c](s ∧ ζ -s 0 ) -̺ 2 c s∧ζ s 0 3 λ τ - 1 + 2 ṫ-2 τ t τ ṫτ dτ , (60)
3 λτ -1+2 ṫ-2 τ tτ ṫτ ≥ 3m -3 ≥ 3 + 1+c ̺ 2 c
. Therefore, by inequality 60, we have almost surely for any s ≥ s 0 :

0 ≤ λ s∧T ≤ λ s 0 -3̺ 2 c (s ∧ T -s 0 ) + M s∧T .
Integrating this inequality and letting s ր ∞ yields :

1 m P (ts 0 , ṫs 0 ) [T < ζ] ≤ lim inf s→∞ E[λ s∧T ] ≤ λ s 0 ≤ 1 nm , whence P (ts 0 , ṫs 0 ) [T = ζ] ≥ 1 -1 n .
Moreover, almost surely on the event {T = ζ}, the above inequality This proves that we have P (ts 0 , ṫs

0 ≤ λ s∧ζ ≤ λ s 0 -3̺ 2 c (s ∧ ζ -s 0 ) + M s∧ζ
0 ) ζ < ∞ , λ ζ = 0 ≥ 1 -n -1 .
Since (by the support theorem of Stroock and Varadhan, see for example Theorem 8.1 in [I-W]) from any starting point the sub-diffusion (t s , ṫs ) hits with a positive probability some (t s 0 , ṫs 0 ) as above, we find there is always a positive probability that ṫs and E s explode (together).

(ii) Let us now start the time sub-diffusion from (t s 1 , ṫs 1 ) such that n m ′ ṫs 1 ≤ t s 1 , with fixed m ′ ≥ 9̺ 2 c (1 + c) and n ≥ 2 + c , and consider T

′ := ζ ∧ inf{s > s 1 | λ s < m ′ }.
By Equation ( 57) we have at once : almost surely, for any s ≥ s 1 ,

λ -1 s∧T ′ = λ -1 s 1 + 5̺ 2 c s∧T ′ s 1 1 + 9c-10 10 ṫ2 τ λ -3 τ dτ -[1 + c] s∧T ′ s 1 λ -2 τ dτ + c s∧ζ s 1 t -2 τ dτ + M ′ s∧T ′ , (M ′ s ) denoting a martingale having quadratic variation 2̺ 2 c s s 1 1 + 3c-2 2 ṫ2 τ 1 -1 ṫ2 τ λ -4 τ dτ . Since on [s 1 , T ′ ] we have λ -1 τ ≤ 1/m ′ , and then 5̺ 2 c 1 + 9c-10 10 ṫ2 τ λ -1 τ ≤ 5̺ 2 c (1+c) m ′
≤ 5/9 , we get :

0 ≤ λ -1 s∧T ′ ≤ λ -1 s 1 -c s∧T ′ s 1 λ -2 τ dτ + c t s 1 + M ′ s∧T ′ .
This entails

T ′ s 1 λ -2 τ dτ < ∞ and λ -1 s∧T ′ → λ -1 T ′ ∈ R + (which implies moreover λ -1 T ′ = 0 almost surely on {T ′ = ∞} ), and 1 m ′ P (ts 1 , ṫs 1 ) [T ′ < ζ] ≤ E (ts 1 , ṫs 1 ) λ -1 T ′ ≤ λ -1 s 1 + c ts 1 ≤ 1+c n m ′ . Hence, we get P (ts 1 , ṫs 1 ) [T ′ = ζ] ≥ 1 -1+c n > 0 . Furthermore, as in (i) above, ṫT ′ = ∞ and λ T ′ > 0 for finite T ′ is impossible, which excludes T ′ = ζ < ∞ . Therefore P (ts 1 , ṫs 1 ) [T ′ = ζ = ∞] ≥ 1 -1+c n > 0 . Then from the equation for λ , using that 3 + 9c 2 ̺ 2 c λ -1 τ ≤ (6+9c)̺ 2 c 2 m ′ < 1 2 on [s 1 , T ′ ],
we get almost surely :

λ s∧T ′ -λ s 1 ≥ (s ∧ T ′ -s 1 ) -3 + 9c 2 ̺ 2 c s∧T ′ s 1 dτ λ τ + M s∧T ′ ≥ 1 2 (s ∧ T ′ -s 1 ) + M s∧T ′ , which shows (since [M s , M s ] = O(s)) that almost surely {T ′ = ζ = ∞} ⊂ {λ s → ∞}.
On this same event, by Equation (57) we have almost surely for any s ≥ s 1 :

ṫs -ṫs ′ 1 = ̺ s s ′ 1 2c 1 + 3c-2 2 ṫ2 τ 1 -1 ṫ2 τ ṫτ λ τ dw τ + c s s ′ 1 5̺ 2 1 + 9c-10 10 ṫ2 τ λ 2 τ - 1 -ṫ-2 τ λ τ dt τ ,
which shows that ṫs cannot go to infinity, since this would forbid the last integral, and then the right hand side, to go to infinity. Hence, by Lemma 5.4.1, we obtain that almost surely {T ′ = ζ = ∞} ⊂ { ṫs → 1}. The proof is ended as in (i) above, by applying the support theorem of Stroock and Varadhan, and by taking n arbitrary large. ⋄ Proof As for the proof of Lemma 6.1.3, we use the commutation relations ( 4) and ( 6), that V kℓ = 0 on C 2 (T 1 M) for 1 ≤ k, ℓ ≤ d , (11), Lemmas 2.2.2 and 6.1.2, and the symmetries of the Riemann tensor. We have thus on one hand and on C 2 (T 1 M) :

[[H p , H j ], V ij ] = 1 2 [R pj kℓ V kℓ , V mn ] η im η jn = 1 2 R pj kℓ [V kℓ , V mn ]η im η jn -1 2 (V mn R pj kℓ )η im η jn V kℓ = 1 2 R pj kℓ η km V ℓn + η ℓn V km -η ℓm V kn -η kn V ℓm η im η jn -1 2 × η mp R nj kℓ -η np R mj kℓ +η mj R np kℓ -η jn R mp kℓ +δ k m R pjn ℓ -δ k n R pjm ℓ -δ ℓ m R pjn k +δ ℓ n R pjm k η im η jn V kℓ = 1 2 R p nkℓ δ i k V ℓn + η ℓn V k i -δ i ℓ V kn -η kn V ℓ i -1 2 -R i p kℓ + R i p kℓ -(d + 1)R i p kℓ + η ik R pj jℓ -η jk R pj iℓ -η iℓ R pj jk + η jℓ R pj ik V kℓ = 1 2 (R p niℓ V ℓn + R k p V k i -R p nki V kn + R ℓ p V ℓ i ) + 1 2 [(d + 1)R i p kℓ + η ik R ℓ p + R p kiℓ -η iℓ R k p -R p ℓik ]V kℓ = 1 2 -R p kiℓ + η iℓ R k p -R p ℓki -η ik R ℓ p + (d + 1)R i p kℓ + η ik R ℓ p + R p kiℓ -η iℓ R k p -R p ℓik V kℓ = ( d+1 2 ) R i p kℓ V kℓ .
In particular, we get

[ [H i , H j ], V ij ] = ( d+1 2 ) R i i kℓ V kℓ = 0 .
And on the other hand, on C 2 (T 1 M) again :

[H i , H j ]V ij = 1 2 η im η jn R ij kℓ V kℓ V mn = η i0 η jn R ij kℓ V kℓ V n = R 0 jkℓ V kℓ V j = R 0 jkℓ ([V kℓ , V j ] + V j V kℓ ) = R 0 jkℓ (η jk V ℓ -η jℓ V k + η 0ℓ V jk -η 0k V jℓ ) + R 0 jkℓ V j V kℓ = -2 R k 0 V k + 2 R 0 jk 0 V jk + 2 R 0 j0ℓ V j V ℓ = 2 (R 0 j0k V j V k -R k 0 V k ) = -2 H o curv .
⋄ The final assertion relating to the Liouville measure is proved as in Theorem 3.2.1. Proposition 6.1.5 In local coordinates, the second order operator H 1 curv defined on T 1 M by Proposition 6.1.1 reads :

H 1 curv = ξj ∂ ∂ξ j -ξi ξj Γ k ij ∂ ∂ ξk + ̺ 2 2 ξn Rk n ∂ ∂ ξk -̺ 2 2 ξp ξq R p k q ℓ ∂ 2 ∂ ξk ∂ ξℓ = ξj ∂ ∂ξ j -ξi ξj Γ k ij ∂ ∂ ξk + ̺ 2 2 ξm R mnpq g nq g pk ∂ ∂ ξk -ξp g nk g qℓ ∂ 2 ∂ ξk ∂ ξℓ .
Proof By Section 2.3, we have on C 2 (T 1 M) : (including now j = 0) .

V j V k = e n j e
On the other hand, by Formula (19) we have :

R 0 j0k = R 0abc η aj η b0 η ck = e m 0 e n a R mnpℓ e p b e ℓ c η aj η b0 η ck = e m 0 e p 0 e r a R mrpℓ e ℓ c η aj η ck , whence R 0 j0k e n j e q k = e m 0 e p 0 e r a R mrpℓ e ℓ c η aj η ck e n j e q k = e m 0 e p 0 R mrpℓ g rn g qℓ = e m 0 e p 0 R m n p q .

And in a similar way, by ( 17) :

R j 0 e n j = e m 0 e q i Rmq η ij e n j = e m 0 Rn m = e m 0 R mℓpq g ℓq g pn . This and ( 14),( 15) yield the wanted formula, whose coefficients depend only on (ξ, ξ) ∈ T 1 M, as it must be by the SO(d)-invariance underlined in Proposition 6.1.1. ⋄

Sign condition on timelike sectional curvatures

The generator H 1 curv defined on T 1 M by Proposition 6.1.1 is covariant with any Lorentz isometry of (M, g). Hence, it is a candidate to generate a covariant "sectional" relativistic diffusion on T 1 M, provided it be semi-elliptic.

As a consequence of Section 6.1, the intrinsic sectional generator we are led to consider on

T 1 M is H 1 curv , restriction of H 0 + ̺ 2 2 R j 0 V j -R 0 j0k V j V k .
Now, a necessary and sufficient condition, in order that such an operator be the generator of a well defined diffusion, is that it be subelliptic.

We are thus led to consider the following negativity condition on the curvature : R(u ∧ v), u ∧ v η ≤ 0 , for any timelike u and any spacelike v .

(

) 61
This condition is equivalent to the lower bound on sectional curvatures of timelike planes Ru + Rv : R(u ∧ v), u ∧ v η g(u ∧ v, u ∧ v) ≥ 0 , since g(u ∧ v, u ∧ v) := g(u, u)g(v, v) -g(u, v) 2 < 0 for such planes.

Note that Sectional Curvature has proved to be a natural tool in Lorentzian geometry, see for example [H], [H-R].

We test this negativity condition on warped products, in Corollary 6.2.1 below. When this negativity condition is fulfilled, we call the resulting covariant diffusion on T 1 M, which has generator H 1 curv given by Propositions 6.1.1, 6.1.5, the sectional relativistic diffusion.

for any non-collinear X, Y ∈ T M such that α 2 h(X ∧Y, X ∧Y )+h(Y, Y )-h(X, X) = α -2 and h(Y, Y ) ≥ α -2 . We shall have proved that this condition is equivalent to the wanted inequality (62), if we show now that for any given α > 0 , any given plane P in T M is generated by pairs X, Y such that α 2 h(X ∧ Y, X ∧ Y ) + h(Y, Y ) -h(X, X) = α -2 , h(Y, Y ) ≥ α -2 , and such that h(X ∧ Y, X ∧ Y ) is arbitrary large. Now, starting from an arbitrary {X 0 , Y 0 } generating P , take Y :=

Y 0 α √ h(Y 0 ,Y 0 )
and X := q X 0 -h(X 0 ,Y 0 ) h(Y 0 ,Y 0 ) Y 0 . Then α 2 h(X ∧ Y, X ∧ Y ) + h(Y, Y ) -h(X, X) = α -2 = h(Y, Y ), and h(X ∧ Y, X ∧ Y ) = q 2 h(X 0 ∧Y 0 ,X 0 ∧Y 0 )

α 2 h(Y 0 ,Y 0 )
is indeed arbitrary large, for arbitrary large q . ⋄ In particular, in an Einstein-de Sitter-like manifold, the sign condition (61) holds if and only if α ′′ ≤ 0 , i.e. if and only if c ≤ 1 .

Sectional diffusion in an Einstein-de Sitter-like manifold

We must have here c ≤ 1 . By (41), we have : for 0 ≤ k ≤ d and 1 ≤ m, n, p, q ≤ d, R 0nkq = δ 0k c (c -1) t 2c-2 h nq and R mnpq = c 2 t 4c-2 [h mq h np -h mp h nq ] -t 2c

Kmnpq .

Using Cartesian coordinates (x j ) instead of spherical coordinates (r, ϕ, ψ) for the Euclidian factor M = R 3 , we have merely :

R 0nkq = c (c -1) t 2c-2 δ 0k δ nq and R mnpq = c 2 t 4c-2 [δ mq δ np -δ mp δ nq ] .

Thence, for 0 ≤ k ≤ d and 1 ≤ m, n, p, q ≤ d : R 0 n k q = c(c-1) t -2c-2 δ 0k δ nq and R m n p q = c(c-1) t 2c-2 δ mp δ tn δ tq +c 2 t -2 [δ q m δ n p -δ mp δ nq ].

And Rij = δ ij 3c (1 -c) t -2 δ 0j + c (3c -1) t 2c-2 [1 -δ 0j ] , whence Rk n = -δ k n c t -2 3(c -1)δ 0n + (3c -1)[1 -δ 0n ] . And by ( 40), the non-vanishing Christoffel coefficients are :

Γ k 0j = Γ k j0 = c t -1 δ k j
, and Γ 0 ij = c t 2c-1 δ ij , for 1 ≤ i, j, k ≤ d . Therefore, by Proposition 6.1.5, we have :

H 1 curv = ξj ∂ ∂ξ j -ξi ξj Γ k ij ∂ ∂ ξk + ̺ 2 2 ξn Rk n ∂ ∂ ξk -̺ 2 2 ξp ξq R p k q ℓ ∂ 2 ∂ ξk ∂ ξℓ = ξj ∂ ∂ξ j - c t ( ṫ2 -1) ∂ ∂ ṫ - 2c t ṫ ẋj ∂ ∂ ẋj - 3̺ 2 c 2 t 2 (c -1) ṫ ∂ ∂ ṫ - ̺ 2 c 2 t 2 (3c -1) ẋj ∂ ∂ ẋj - ̺ 2 c (c -1) 2 t 2 ( ṫ2 -1) ∂ 2 ∂ ṫ2 - ̺ 2 c (c -1) 2 t 2c+2 ṫ2 ∆ x + ̺ 2 c 2 2 t 2 t -2c ( ṫ2 -1)∆ x -ẋi ẋj ∂ 2 ∂ ẋi ∂ ẋj = ξj ∂ ∂ξ j - c t ( ṫ2 -1) ∂ ∂ ṫ - 3̺ 2 c 2 t 2 (c -1) ṫ ∂ ∂ ṫ - 2c t ṫ ẋj ∂ ∂ ẋj - ̺ 2 c (3c -1) 2 t 2 ẋj ∂ ∂ ẋj + ̺ 2 c (1 -c) 2 t 2 ( ṫ2 -1) ∂ 2 ∂ ṫ2 + ̺ 2 c 2 t 2c+2 ( ṫ2 -c) ∆ x - ̺ 2 c 2 2 t 2 ẋi ẋj ∂ 2 ∂ ẋi ∂ ẋj .
We see that even in this simple case, the sectional and curvature diffusion differ tangibly, apart from the fact that the range of values of c for which they are defined are different.

  Isomorphism between Λ 2 R 1,d and so(1, d) page 2.2) Frame bundle G(M) over (M, Ξ-relativistic diffusions in an Einstein-de Sitter-like manifold page 5.3) Asymptotic behavior of the R-diffusion in an E.-d.S.-like manifold page 5.4) Asymptotic energy of the E-diffusion in an E.-d.S.-like manifold page 6

  comparing the above equation (31) for ξk s with Equation (

  Lemma 3.4.1, the drift term of Equation (31) which involves the derivatives of Ξ equals here : ̺ 2 d j=1 R 0j (ξ s ) e k j (s) ds . As we have R 0j = Rmn e m 0 e n j by (17), we get the alternative expression : ̺ 2 d j=1 R 0j (ξ s ) e k j (s) ds = ̺ 2 Rmn (ξ s ) ξm [ ξk s ξn s -g kn (ξ s )] ds .

  and ẋs| ẋs| = ẋ1 | ẋ1 | , which converges only for c > 1 .

  2c ( ṫs /t s ) 2 + O(s -2 ). Let us denote by ζ the explosion time : ζ := sup{s > 0 | ṫs < ∞} ∈ ]0, ∞] . Lemma 5.4.1 We have almost surely : either lim s→ζ ṫs = 1 and ζ = ∞ , or lim s→ζ ṫs = ∞ .

(

  M s ) denoting a martingale having quadratic variation 2̺ 2 c

  implies clearly (using that a continuous martingale almost surely cannot go to infinity) ζ < ∞ , and by the previous lemma that ṫζ = ∞ . Then 59 implies the convergence of λ s∧ζ to some λ ζ ∈ R + . Furthermore, ṫζ = ∞ and λ ζ > 0 for finite ζ would imply trivially t ζ = ∞ , whence its logarithmic derivative should explode, which leads to a contradiction.

Sectional relativistic diffusion

We turn now our attention towards a different class of intrinsic relativistic generators on G(M), whose expressions derive directly from the commutation relations of Section 2.2, on canonical vector fields of T G(M). They all project on the unit tangent bundle T 1 M onto a unique relativistic generator H 1 curv , whose expression involves the curvature tensor. Semi-ellipticity of H 1 curv requires the assumption of non-negativity of timelike sectional curvatures. Note that in general H 1 curv does not induce the geodesic flow in an empty space.

Intrinsic relativistic generators on G(M)

We shall actually consider among these generators, those which are invariant under the action of SO(d) on G(M). To this aim, we introduce the following dual vertical operators, by lifting indexes :

We consider again a positive parameter ̺ . Proposition 6.1.1 The following four SO(d)-invariant differential operators define the same operator H 1 curv on T 1 M :

Note that (H 1 curv -L 0 ) is self-adjoint with respect to the Liouville measure of T 1 M.

The proof will be broken in several lemmas. We begin with the following general and useful computation rules, derived from Sections 2.1 and 2.2.

Lemma 6.1.2 For 0 ≤ j, k, ℓ ≤ d , we have :

Proof We get the first formula by multiplying by η ip the formula of Lemma 2.2.2, and particularising to q = 0 . As to the second one, by particularising the latter to j = 0 and changing sign, we get :

Finally, the last formula derives from the second one and from the commutation relations ( 4) and ( 6), as follows :

We get then first the following.

V j ] = 0 , and

Proof Using the commutation relations ( 4) and ( 6), that and (11), we have on one hand :

On the other hand, using this first part of proof and Lemma 6.1.2, we get :

We get then the following.

Corollary 6.2.1 Consider a Lorentz manifold (M, g) having the warped product form.

Then the sign condition ( 61) is equivalent to : α ′′ ≤ 0 on I, together with the following lower bound on sectional curvatures of the Riemannian factor (M, h) :

Proof Let us denote by SR(U ∧V ) the sectional curvature of the timelike plane associated with U ∧ V . Recall from Section 6.2 that the negativity condition (61) reads simply SR(U ∧ V ) ≥ 0 , for any timelike U and spacelike V . By choosing a pseudo-orthonormal basis of such given timelike plane, we can moreover restrict to g(U, U ) = 1 = -g(V, V ) and g(U, V ) = 0 . Setting U = u∂ t + X and V = v∂ t + Y , with u, v ∈ C 0 (I) and X, Y ∈ T M , we can thus suppose that : Now,by (37), this entails :

Reciprocally, given any X, Y ∈ T M such that

, and thence a basis (U, V ) as above.

Hence, the negativity condition ( 61) is equivalent to :

Distinguishing between collinear and non-collinear pairs X, Y , and denoting in the latter case by SK(X ∧ Y ) the sectional curvature of the plane associated with X ∧ Y , this condition splits into both : (α ′′ /α) ≤ 0 , together with :

A.M.S. Classification : Primary 58J65, secondary 53C50, 83C10, 60J65.

Key Words : Covariant relativistic diffusions, General relativity, Lorentz manifold, Riemann curvature tensor, Stochastic flow, Robertson-Walker manifolds.

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• Jacques FRANCHI : jacques.franchi@math.unistra.fr bénéficiaire d'une aide de l'Agence Nationale de la Recherche, n o ANR-09-BLAN-0364-01.

• Yves LE JAN : yves.lejan@math.u-psud.fr Université Paris Sud 11, Mathématiques, Bâtiment 425, 91405 Orsay, France.

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