Paul Gassiat
email: gassiat@ceremade.dauphine.fr

Benjamin Gess
email: benjamin.gess@gmail.com

AND Pierre-Louis Lions
email: lions@ceremade.dauphine.fr

Panagiotis E Souganidis
email: souganidis@math.uchicago.edu

SPEED OF PROPAGATION FOR HAMILTON-JACOBI EQUATIONS WITH MULTIPLICATIVE ROUGH TIME DEPENDENCE AND CONVEX HAMILTONIANS

We show that the initial value problem for Hamilton-Jacobi equations with multiplicative rough time dependence, typically stochastic, and convex Hamiltonians satisfies finite speed of propagation. We prove that in general the range of dependence is bounded by a multiple of the length of the "skeleton" of the path, that is a piecewise linear path obtained by connecting the successive extrema of the original one. When the driving path is a Brownian motion, we prove that its skeleton has almost surely finite length. We also discuss the optimality of the estimate.

Introduction

We consider the initial value problem for Hamilton-Jacobi equations with multiplicative rough time dependence, that is where C 0 pr0, T sq denotes the space of continuous paths ξ : r0, T s Ñ R such that ξp0q " 0. When ξ is a C 1 or BV-path, (1.1) is the standard Hamilton-Jacobi equation that is studied using the Crandall-Lions theory of viscosity solutions. For such paths, in place of (1.1) we will often write (1.4) u t " HpDu, xq 9 ξ in R d ˆp0, T s up¨, 0q " u 0 in R d

When ξ is merely continuous, in (1.1) ¨simply denotes the way the path enters the equation. When ξ is a Brownian motion, then ¨dξ stands for the classical Stratonovich differential.

Lions and Souganidis introduced in [START_REF] Lions | Fully nonlinear stochastic partial differential equations[END_REF] the notion of stochastic or pathwise viscosity solutions for a general class of equations which contain (1.1) as a special case and studied its well-posedness; for this as well as further properties see Lions-Souganidis [START_REF] Lions | Fully nonlinear stochastic partial differential equations[END_REF][START_REF] Lions | Fully nonlinear stochastic partial differential equations: nonsmooth equations and applications[END_REF][START_REF] Lions | Stochastic viscosity solutions[END_REF][START_REF] Lions | Stochastic viscosity solutions of spatially dependent hamiltonjacobi equations with multiple paths[END_REF][START_REF] Panagiotis | Fully nonlinear first-and second-order stochastic partial differential equations[END_REF]. One of the questions raised in [START_REF] Panagiotis | Fully nonlinear first-and second-order stochastic partial differential equations[END_REF] was whether (1.1) has a finite speed of propagation, which is one of the important characteristics of the hyperbolic nature of the equations for regular paths. Roughly speaking, finite speed of propagation means that, if two solutions agree at some time in a ball, then they agree on a forward cone with a time dependent radius.

A partial result in this direction was shown in Lions and Souganidis [START_REF] Lions | Fully nonlinear stochastic partial differential equations: nonsmooth equations and applications[END_REF] (see also Souganidis [START_REF] Panagiotis | Fully nonlinear first-and second-order stochastic partial differential equations[END_REF]), while Gassiat showed in [START_REF] Gassiat | A stochastic Hamilton-Jacobi equation with infinite speed of propagation[END_REF] that, in general, when H is neither convex nor concave (1.1) does not have the finite speed of propagation property. In this work, assuming (1.2) and (1.3), we establish finite speed of propagation in the sense formulated precisely next.

Given T ą 0 and H : R d ˆRd Ñ R let ρ H pξ, T q :" sup ! R ě 0 : there exist solutions u 1 , u 2 of (1.1) and x P R d , (1.5) such that u 1 p¨, 0q " u 2 p¨, 0q in B R pxq and u 1 px, T q ‰ u 2 px, T q

) ,
where B R pxq is the ball in R d centered at x with radius R.

The classical theory for Hamilton-Jacobi equations (see Lions [START_REF] Lions | Generalized solutions of Hamilton-Jacobi equations[END_REF] and Crandall and Lions [START_REF] Michael | User's guide to viscosity solutions of second order partial differential equations[END_REF]) yields that, if ξ is a C 1 -or, more generally, a BV-path, then

(1.6) ρ H pξ, T q ď L}ξ} T V pr0,T sq , where }ξ} T V pr0,T sq :" sup 0"t0ď...ďtn"T n´1 ÿ i"0 |ξpt i`1 q ´ξpt i q| is the total variation semi-norm of ξ and L is the Lipschitz constant of H. It is easy to see that (1.6) is sharp when 9 ξ " 1. For general rough, that is only continuous, signal ξ it was shown in [START_REF] Lions | Fully nonlinear stochastic partial differential equations: nonsmooth equations and applications[END_REF], [START_REF] Panagiotis | Fully nonlinear first-and second-order stochastic partial differential equations[END_REF] that, if Hpp, xq " H 1 ppq ´H2 ppq, where H 1 , H 2 satisfy (1.2) with Lipschitz constant L and H 1 p0q " H 2 p0q " 0, then, for any constant A, if up0, ¨q " A on B R p0q, then upt, ¨q " A on B Rptq p0q, for Rptq :" R ´Lp max sPr0,T s ξpsq ´min sPr0,T s ξpsqq.

This does not, however, imply a finite domain of dependence. In fact, it was shown in [START_REF] Gassiat | A stochastic Hamilton-Jacobi equation with infinite speed of propagation[END_REF] that when Hppq " |p 1 |´|p 2 | equality is attained in (1.6) for all continuous ξ, a fact which implies that there is no finite domain of dependence if ξ R BV pr0, T sq. In other words, the counter-example in [START_REF] Gassiat | A stochastic Hamilton-Jacobi equation with infinite speed of propagation[END_REF] shows that for non-convex Hamiltonian H, all of the oscillations of ξ, measured in terms of the T V -norm, are relevant for the dynamics of (1.1). In contrast, in this paper we show that, if H is convex, there is an estimate, which is better than (1.6), and, in particular, implies that the rate of dependence ρ H pξ, T q is almost surely finite when ξ is a Brownian path. This new bound relies on a better understanding of which oscillations of the signal ξ are effectively relevant for the dynamics of (1.1).

In this spirit, we prove that, if H is convex, then ξ can be replaced by its skeleton. This is a reduced path R 0,T pξq which keeps track solely of the oscillations of ξ that are relevant for the dynamics of (1.1) without changing the solution to (1.1). Hence, in the convex case only the oscillations of ξ encoded in R 0,T pξq are relevant for (1.1). In the one-dimensional setting and for smooth, strictly convex, x-independent Hamiltonians a related result has been obtained independently and by different methods in Hoel, Karlsen, Risebro, and Storrøsten [START_REF] Hoel | Path-dependent convex conservation laws[END_REF].

We also establish that the reduced path of a Brownian motion has almost surely finite variation, a fact which implies that ρ H pξ, T q is almost surely finite.

τ 0 τ 1 T τ ´1 ξ R 0,T pξq τ ´3 τ ´2 R0,T pξq Figure 1.1.
The (fully) reduced path Given ξ P C 0 pr0, T sq, the sequence pτ i q iPZ of successive extrema of ξ is defined by (1.7) τ 0 :" sup

" t P r0, T s, ξptq " max 0ďsďT ξpsq or ξptq " min 0ďsďT ξpsq * ,
and, for all i ě 0,

(1.8) τ i`1 " # arg max rτi,T s ξ if ξpτ i q ă 0, arg min rτi,T s ξ if ξpτ i q ą 0,
and, for all i ď 0, (1.9) τ i´1 " # arg max r0,τis ξ if ξpτ i q ă 0, arg min r0,τis ξ if ξpτ i q ą 0.

The skeleton (resp. full skeleton) or reduced (resp. fully reduced) path R 0,T pξq (resp. R0,T pξq) of ξ P C 0 pr0, T sq is defined as follows (see Figure 1.1).

Definition 1.1. Let ξ P C 0 pr0, T sq.

(i) The reduced path R 0,T pξq is a piecewise linear function which agrees with ξ on pτ i q iPZ . (ii) The fully reduced path R0,T pξq is a piecewise linear function agreeing with ξ on pτ ´iq iPN Y tT u.

(iii) A path ξ P C 0 pr0, T sq is reduced (resp. fully reduced) if ξ " R 0,T pξq (resp. ξ " R0,T pξq).

Let u ξ be the solution to (1.1). We show in Theorem 2.9 in the next section that (1.10) u ξ p¨, T q " u R 0,T pξq p¨, T q, which immediately implies the following result.

Theorem 1.2. Assume (1.2). Then, for all ξ P C 0 pr0, T sq,

(1.11) ρ H pξ, T q ď L }R 0,T pξq} T V pr0,T sq .
The second main result of the paper, which is a probabilistic one and of independent interest, concerns the total variation of the reduced path of a Brownian motion. To state it, we introduce the random variable θ : r0, 8q Ñ r0, 8q given by (1.12) θpaq :" inftt ě 0 : max r0,ts B ´min r0,ts B " au.

We prove that the length of the reduced path is a random variable with almost Gaussian tails. We also show that if, instead of fixing the time horizon T , we fix the range, that is the maximum minus the minimum of B, then the length has Poissonian tails Theorem 1.3. Let B be a Brownian motion and fix T ą 0. Then, for each γ P p0, 2q, there exists C " Cpγ, T q ą 0 such that, for any x ě 2, (1.13) P ´}R 0,T pBq} T V pr0,T sq ě x ¯ď C exp p´Cx γ q , and

(1.14) lim xÑ8 ln P ´› › R 0,θp1q pBq › › T V pr0,θp1qsq ě x x lnpxq " ´1.
We also study the sharpness of the upper bound. For simplicity we only treat the case Hppq " |p|.

Theorem 1.4. Let Hppq " |p| on R d with d ě 1. Then, for all T ą 0 and ξ P C 0 pr0, T sq,

(1.15) ρ H pξ, T q ě } R0,T pξq} T V pr0,T sq . When d " 1, then ρ H pξ, T q " } R0,T pξq} T V pr0,T sq .
The paper is organized as follows. In section 2 we improve upon results of [START_REF] Lions | Fully nonlinear stochastic partial differential equations[END_REF][START_REF] Lions | Fully nonlinear stochastic partial differential equations: nonsmooth equations and applications[END_REF][START_REF] Lions | Stochastic viscosity solutions[END_REF] about representation formulae, the control of the oscillations in time and the domain of dependence of the solutions of (1.4) with piecewise linear paths. We then extend these estimates by density to general continuous paths. In order to avoid stating many assumptions on H, we introduce a new condition about solutions of (1.4) which is satisfied by the general class of Hamiltonians for which there is a well-posed theory of pathwise solutions as developed in [START_REF] Lions | Stochastic viscosity solutions of spatially dependent hamiltonjacobi equations with multiple paths[END_REF]. All these lead to the proof of Theorem 1.2. In section 3 we discuss the example which shows that the upper bound obtained in Theorem 1.2 is sharp. Section 4 is devoted to the study of "random" properties of the reduced path of the Brownian motion (Theorem 1.3).

Reduction to the skeleton path and domain of dependence

Notation and preliminaries. For all ξ P C 0 pr0, T sq and u 0 P BU CpR d q, let S ξ be the flow of solutions of (1.1). A simple rescaling shows that without loss of generality we may assume that

L " 1.
In view of ( We assume that H satisfies all assumptions needed (see [START_REF] Michael | User's guide to viscosity solutions of second order partial differential equations[END_REF]) for u t " HpDu, xq 9 ξ to be well posed when ξ is smooth and we denote by S ˘H ptq : BUCpR d q Ñ BUCpR d q the solution operator when 9 ξ " ˘1, that is, for u 0 P BUCpR d q, S ˘H ptqu 0 is the unique solution of (2.1)

u t " ˘HpDu, xq in R d ˆp0, T s up¨, 0q " u 0 in R d .
Moreover, for t ď 0, S H ptq :" S ´H p´tq.

Given S, S 1 : BUCpR d q Ñ BUCpR d q, we say that S ď S 1 if Su ď S 1 u for all u in BUCpR d q.

In the sequel we write ξ s,t :" ξ t ´ξs for the increments of ξ over the interval rs, ts.

Let ξ P Cpr0, T sq be a piecewise linear path, that is, for a partition 0 " t 0 ď . . . ď t N " T of r0, T s, and

a i , b i P R, i " 1, . . . N , ξptq " N ´1 ÿ i"0
1 rti,ti`1q pa i pt ´ti q `bi q.

We then set

S ξ H p0, T q :"S ξ H pt N ´1, t N q ˝¨¨¨˝S ξ H pt 0 , t 1 q and note that S ξ H p0, T q " S H pξ t N ´1,t N q ˝¨¨¨˝S H pξ t0,t1 q. We show later that ξ Þ Ñ S ξ
H p0, ¨q is uniformly continuous in sup-norm, which allows to extend S ξ H p0, T q to all continuous ξ. Monotonicity properties. The control representation of the solution u of (2.1) (see, for example, Lions [START_REF] Lions | Generalized solutions of Hamilton-Jacobi equations[END_REF]) with ξ t " t and u 0 P BUCpR d q is upx, tq " S H ptqu 0 pxq " sup qPA " u 0 pXptqq ´ż t 0 Lpqpsq, Xpsqqds : Xp0q " x, 9

Xpsq " qpsq for s P r0, ts * , and

S ´H ptqu 0 pyq " inf rPA " u 0 pY ptqq `ż t 0 Lprpsq, Y psqqds : Y p0q " y, 9 Y psq " ´rpsq for s P r0, ts * ,
where A " L 8 pR `; B 1 p0qq is the set of controls.

The next property is a refinement of an observation in [START_REF] Lions | Stochastic viscosity solutions[END_REF].

Lemma 2.1. Fix t ą 0 and u 0 P BU CpR d q. Then S H ptq ˝SH p´tqu 0 ď u 0 ď S H p´tq ˝SH ptqu 0 .

Proof. Since the arguments are identical we only show the proof of the inequality on the left.

We have

S H ptq ˝SH p´tqu 0 pxq " sup qPA inf rPA ! u 0 pY ptqq `ż t 0 Lprpsq, Y psqqds ´ż t 0 Lpqpsq, Xpsqqds : Y p0q " Xptq, 9 Y psq " ´rpsq, Xp0q " x, 9
Xpsq " qpsq for s P r0, ts

) . ξ t 1 T a 0 t 1 ξ 0,T ξ t 1 T a 0 t 1 ξ 0,T Figure 2.1. Reduction
Given q P A choose rpsq " qpt ´sq in the infimum above. Since Y psq " Xpt ´sq, it follows that

S H ptq ˝SH p´tqu 0 pxq ď sup qPA ! u 0 pXp0qq `ż t 0 Lpqpt ´sq, Xpt ´sqqds ´ż t 0
Lpqpsq, Xpsqqds :

Xp0q " x, 9
Xpsq " qpsq for s P r0, ts

) " u 0 pxq.
The next result is an easy consequence of Lemma 2.1 and the definition of S ξ H for piecewise linear paths.

Lemma 2.2. Let ξ t " 1 tPr0,t1s pa 0 tq `1tPrt1,T s pa 1 pt ´t1 q `a0 t 1 q. If a 0 ě 0 and a 1 ď 0 (resp. a 0 ď 0 and a 1 ě 0), then S ξ H p0, T q ě S H pξ 0,T q presp. S ξ H p0, T q ď S H pξ 0,T q.q Proof. Since the claim is immediate if a 0 " 0 or a 1 " 0, we assume next that a 0 ą 0 and a 1 ă 0 (see Figure 2.1).

If ξ 0,T ď 0, then S H pa 1 pT ´t1 qq " S ´H p´a 1 pT ´t1 qq " S ´H p´a 1 pT ´t1 q ´a0 t 1 q ˝S´H pa 0 t 1 q " S ´H p´ξ 0,T q ˝S´H pa 0 t 1 q " S H pξ 0,T q ˝SH p´a 0 t 1 q, and, hence, in view of Lemma 2.1, S ξ H p0, T q " S H pξ 0,T q ˝SH p´a 0 t 1 q ˝SH pa 0 t 1 q ě S H pξ 0,T q. If ξ 0,T ě 0 (see Figure 2.1), then, again using Lemma 2.1, we find S ξ H p0, T q " S H pa 1 pT ´t1 qq ˝SH p´a 1 pT ´t1 q `a0 t 1 `a1 pT ´t1 qq " S H pa 1 pT ´t1 qq ˝SH p´a 1 pT ´t1 qq ˝SH pa 0 t 1 `a1 pT ´t1 qq ď S H pξ 0,T q.

For the second inequality we note that S ´ξ ´H p0, T q " S ξ H p0, T q, S ´H p´tq " S H ptq. It then follows from the the first part that S ξ H p0, T q " S ´ξ ´H p0, T q ě S ´H p´ξ 0,T q " S H pξ 0,T q.

The next observation provides the first indication of the possible reduction encountered when using the max or min of a given path. For the statement, given piecewise linear path ξ, we set τ max :" suptt P r0, T s : ξ t " max sPr0,T s ξ s u and τ min :" suptt P r0, T s : ξ t " min sPr0,T s ξ s u.

Lemma 2.3. Fix a piecewise linear path ξ. Then S ξ H pτ max , T q ˝SH pξ 0,τmax q ď S ξ H p0, T q ď S H pξ τmin,T q ˝Sξ H p0, τ min q. Proof. Since the proofs of both inequalities are similar, we only show the details for the first. Note that without loss of generality we may assume that sgnpξ ti´1,ti q " ´sgnpξ ti,ti`1 q for all rt i´1 , t i`1 s Ď r0, τ max s. It follows that, if ξ |r0,τmaxs is linear, then S ξ H p0, τ max q " S H pξ 0,τmax q. If not, since ξ 0,τmax ě 0, there is an index j such that ξ tj´1,tj`1 ě 0 and ξ tj´1,tj ď 0. It then follows from Lemma 2.2 that S ξ H p0, τ max q ď S ξ H p0, τ max q, where ξ is piecewise linear and coincides with ξ for all t P tt i : i ‰ ju.

A simple iteration yields S ξ H p0, τ max q ď S H pξ 0,τmax q, and, since S ξ H p0, T q " S ξ H pτ max , T q˝S ξ H p0, τ max q, this concludes the proof.

We combine the conclusions of the previous lemmata to establish the following monotonicity result.

Corollary 2.4. Let ξ, ζ be piecewise linear, ξp0q " ζp0q, ξpT q " ζpT q and ξ ď ζ on r0, T s. Then

(2.2)
S ξ H p0, T q ď S ζ H p0, T q. Proof. We assume that ξ and ζ are piecewise linear on each interval rt i , t i`1 s of a joint subdivision 0 " t 0 ď . . . ď t N " T of r0, T s. If N " 2, we show that, for all γ ě 0 and all a, b P R, The proof for N ą 2 follows by induction on N . Let ρ be piecewise linear on the same partition and coincide with ζ on t 0 , t 1 , and with ξ on t 2 , . . . , t N . The induction hypothesis then yields

(
S ξ H p0, t 2 q ď S ρ H p0, t 2 q and S ρ H pt 1 , T q ď S ζ H pt 1 , T q from which we deduce S ξ H p0, T q ď S ρ H p0, T q ď S ζ H p0, T q.
A uniform modulus of continuity. To extend the information obtained about the possible cancellations and oscillations from piecewise linear to arbitrary continuous paths, we need a wellposed theory for the pathwise viscosity solutions. Such a theory has been developed by the last two authors in [START_REF] Lions | Stochastic viscosity solutions[END_REF] and [START_REF] Lions | Stochastic viscosity solutions of spatially dependent hamiltonjacobi equations with multiple paths[END_REF]. The former reference imposes conditions on the joint dependence of the Hamiltonians in pp, xq but does not require convexity. A special (resp. a more general) class of convex or concave Hamiltonians, which do not require such conditions, is studied in Friz, Gassiat, Lions and Souganidis [START_REF] Peter K Friz | Eikonal equations and pathwise solutions to fully non-linear spdes[END_REF] (resp. Lions and Souganidis [START_REF] Lions | Stochastic viscosity solutions[END_REF]). An alternative, although less intrinsic, approach is to show that the solution operator has a unique extension from piecewise linear paths to arbitrary continuous ones.

To avoid stating additional conditions and since finding the optimal assumptions on the joint dependence on pp, xq of the Hamiltonians is not the main focus of this paper, we bypass this issue here. Instead, we formulate a general assumption that allows to have a unique extension of the solution operator to all continuous paths, which is enough to analyze the domain of dependence.

We only remark that this assumption is satisfied by the Hamiltonians considered in [START_REF] Lions | Stochastic viscosity solutions of spatially dependent hamiltonjacobi equations with multiple paths[END_REF] as well as some other ones that can be analyzed by the same methods. For t P p0, T q, the minimal action, also known as the fundamental solution, associated with Hamiltonians satisfying (1.2) is given by Lpx, y, tq :" inft ż t 0 Lp 9 γpsq, γpsqqds : γ P C 0,1 pr0, T sq such that γp0q " x, γptq " yu;

when we need to emphasize the dependence of L, we write L H . We recall (see, for example, [START_REF] Lions | Generalized solutions of Hamilton-Jacobi equations[END_REF]) that, for all t, s ě 0 and x, y, z P R d , (2.4) Lpx, z, t `sq ď Lpx, y, tq `Lpy, z, sq.

Moreover, for any u 0 P BUCpR d q, t ě 0 and x P R d , (2.5) upx, tq " S H ptqu 0 pxq " sup yPR d ru 0 pyq ´Lpx, y, tqs .

Finally, since ´S´H ptqu 0 " S | H p´u 0 q with q Hpp, xq " Hp´p, xq, we also have, for any u 0 P BUCpR d q, t ě 0 and x P R d ,

(2.6) S ´H ptqu 0 " inf yPR d " u 0 pyq `L| H px, y, tq ı .
We assume that, for all r ą 0, The claim above is a consequence of the following estimate.

Proposition 2.6. Let u " S ξ H p0, tqu 0 with ξ piecewise linear and u 0 P BU CpR d q. Then, for all t ě 0 and all x, y P R d , (2.9) upx, tq ´upy, tq ď inf δą0 ˜Lpy, x, δq `sup

x 1 ,y 1 PR d " u 0 px 1 q ´u0 py 1 q ´Lpy 1 , x 1 , δq ‰ ¸.

Proof. By induction it is enough to prove the estimate for u " S H ptqu 0 and u " S H p´tqu 0 .

We begin with the former and we fix x, y and Hence pu 0 px 1 q ´Lpx, x 1 , tqq ´sup y1PR d tu 0 py 1 q ´Lpy, y 1 , tqu ´Lpy, x, δq ď u 0 px 1 q ´u0 pỹ 1 q ´Lpx, x 1 , tq `Lpy, ỹ1 , tq ´Lpy, x, δq ď u 0 px 1 q ´u0 pỹ 1 q ´Lpỹ 1 , x 1 , δq ď sup

x 1 P R d .
x 1 ,y 1 PR d u 0 px 1 q ´u0 py 1 q ´Lpy 1 , x 1 , δq ( .

It follows that

upx, tq ´upy, tq ´Lpy, x, δq ď sup

x 1 ,y 1 u 0 px 1 q ´u0 py 1 q ´Lpy 1 , x 1 , δq ( and we conclude by taking the inf over δ.

In view of (2.6), a similar argument gives the estimate for u " S ´H ptqu 0 .

Proof of Proposition 2.5. Fix u 0 and let ηpδq :" sup

x 1 ,y 1 PR d `u0 px 1 q ´u0 py 1 q ´Lpy 1 , x H p0, T q ˝SH p´δq ď S ζ H p0, T q ď S H p´δp1 `ηqq ˝Sξ H p0, T q ˝SH pδq which implies that S ξ H p0, T q ´SH p´δp1 `ηqq ˝Sξ H p0, T q ˝SH pδq ď S ξ H p0, T q ´Sζ H p0, T q, and S ξ H p0, T q ´Sζ H p0, T q ď S ξ H p0, T q ´SH pδp1 ´ηqq ˝Sξ H p0, T q ˝SH p´δq. We now need to check that both sides of the above inequality go to 0 as δ Ñ 0. This follows if lim δÑ0 }S H pδq ˝Sξ H p0, T qu ´SH p´δq ˝Sξ H p0, T qu} 8,R d " 0 independently of ξ, which is a consequence of Proposition 2.5.

The next conclusion is an immediate consequence of Lemma 2.3 and Corollary 2.7.

Corollary 2.8. Let ξ be a continuous path such that ξ T " max r0,T s ξ and ξ 0 " min r0,T s ξ. Then, S ξ H p0, T q " S H pξ 0,T q. Similarly, if ξ T " min r0,T s ξ and ξ 0 " max r0,T s ξ, then S ξ H p0, T q " S ´H p´ξ 0,T q. It follows that we can have a general representation for the solution to (1.1) as a (countable) composition of the flows S H ptq, S H p´tq. Theorem 2.9. Let ξ be a continuous path. Then S ξ p0, T q " S R 0,T pξq p0, T q.

Proof. We apply Corollary 2.8 inductively to the successive extrema as defined in (1.7), (1.8), (1.9). It only remains to show that this procedure converges for i Ñ ˘8. This follows from the continuity of ξ in combination with Corollary 2.7.

The optimality of the domain of dependence

We consider the initial value problem (3.1) du " |Du| ¨dξ in R d ˆp0, T s up¨, 0q " u 0 p¨q in R d and prove Theorem 1.4. We remark that, in view of the geometric properties of (3.1), it is enough to consider the evolution of the level set P `ptq " tx P R : upx, tq ě 0u . Indeed, (3.1) is a level-set PDE, that is, if u is a solution, then also Φpuq is a solution. At this point the choice of the Stratonovich differential in (3.1) is important (see Souganidis [START_REF] Bardi | Viscosity solutions and applications[END_REF], [START_REF] Panagiotis | Fully nonlinear first-and second-order stochastic partial differential equations[END_REF] and Lions, Souganidis [START_REF] Lions | Stochastic viscosity solutions[END_REF]). It follows that P `ptq depends only on P `p0q and not on the particular form of u 0 . In fact, in the case of (3.1) this can be read off the explicit solution formula, for all δ ą 0,

S |¨| pδqupxq " sup |x´y| ďδ upyq, S |¨| p´δqupxq " inf |x´y| ďδ upyq.
In particular, in d " 1 and with the convention that rc, ds " H if c ą d, it follows that, for all δ P R,

(3.2) S |¨| pδqpra, bsq " ra ´δ, b `δs.
We notice that, informally, for general initial conditions, P `expands with speed |dξ| when dξ ą 0, and contracts with speed |dξ| when dξ ă 0.

The key behind the construction of the lower bound is the observation, already made in [START_REF] Lions | Stochastic viscosity solutions[END_REF], that there is some irreversibility in the dynamics. For example, once a hole is filled, that is two connected components of P `are joined by an increase in ξ, it cannot be recreated later when ξ decreases. Symmetrically, if a component of P `is destroyed by a decrease in ξ, it does not re-appear later. This intuition leads to the lower bound for ρ H pξ, T q derived below.

In what follows, to simplify the notation we omit the dependence of the solution operator and the speed of propagation on H, that is, we simply write S, S ξ and ρpξ, T q. We fix d " 1 and establish first the lower bound in Theorem 1.4, and then look at the upper bound. Note that considering initial conditions depending only on the first coordinate implies that the lower bound also holds for d ě 2.

Lower bound for the speed of propagation. The result is stated next.

Proposition 3.1. Let ξ be a continuous path. Then

(3.3) ρ H pξ, T q ě } Rξ p0, T q} T V pr0,T sq .
Proof. Without loss of generality we assume that ξ is a reduced path. Moreover, since the claim stays the same if we replace ξ by ´ξ, we further assume that ξpτ 0 q " max 0ďsďT ξpsq.

We first consider the case where N :" maxtn ď 0 : τ n " 0u is finite. Since ξ is constant if N " 0, we further assume N ď ´1 and fix a sequence x i , N ´1 ď i ď 1 such that x 1 " 0 and, for all

N ă k ď 0, (3.4) 2 ˇˇξ 0,τ k´2 ˇˇă x k`1 ´xk ă 2 |ξ 0,τ k | . ξ I 0 t " 0 t " 1 t " 2 x 1 " 0 x 0 x ´1 x 1 " 0 x 0 x ´1 x 1 " 0 x 0 x ´1 x 1 " 0 x 0 x ´1 x 1 " 0 x 0 x ´1 x 1 " 0 x 0 x ´1 x 1 " 0 x 0 x ´1 t " 3 x 1 " 0 x 0 x ´1 Figure 3.1. Lower bound Set (3.5) I k " " rx 2k´1 , x 2k s if 2k ´1 ą N H otherwise, and (3.6
) P 1 " ď kď0 I k Y r0, `8q, P 2 " p´8, x N s Y P 1 .
Since ξ is a reduced path and due to (3.2), the evolution of P 1 , P 2 can be easily described by induction on k as follows.

The component I k evolves individually, that is it does not intersect any other connected components, before τ 2k . This follows from the fact that |x 2k´1 ´x2k´2 | and |x 2k ´x2k`1 | are smaller than 2ξ 0,τ 2k .

Since ´2ξ 0,τ 2k´3 ă |x 2k ´x2k´1 | ă ´2ξ 0,τ 2k´1 , the component I k of P 1 disappears at time τ 2k´1 but not at any of the earlier τ i 's. Finally, given that x 2k´1 ´x2k´2 ă 2ξ 0,τ 2k´2 , the component I k of P 2 has joined the components I j with j ă k by the time τ 2k´2 .

It follows that

Spξ, τ 0 qpP 1 q " r´ξ 0,τ0 , `8q and S ξ p0, T qpP 1 q " r´ξ 0,T , `8q and S ξ p0, T qpP 2 q " R. Since P 1 and P 2 only differ for x ď x N , this implies ρ H pξ, T q ě p´ξ 0,T ´xN q

`.
Choosing the px k ´xk´1 q as large as possible in (3.4) we obtain ρpξ, T q ě ´ξ0,T `2

ÿ kď0 |ξ 0,τ k | " |ξ τ0,T | `ÿ kď0 |ξ 0,τ k | `ÿ kď´1 |ξ 0,τ k | .
Finally, using that, for k ď 0, ˇˇξ τ k´1 ,τ k ˇˇ" |ξ 0,τ k | `ˇξ 0,τ k´1 ˇˇ, we obtain

ρpξ, T q ě |ξ τ0,T | `ÿ kď0 |ξ τ k´1 ,τ k | " › › › R0,T pξq › › › T V
which concludes the proof in the case where τ N " 0 for some N .

Next we treat the general case. We fix N ď ´1 arbitrary, and as before we define x k , I k and P 1 , P 2 satisfying (3.4), (3.5) and (3.6). Now since (3.4) implies that

x 2k ´x2k´1 ą ´inf r0,τ N s ξ, x 2k`1 ´x2k ą sup r0,τ N s ξ,
the I k 's do not interact before time τ N . Let xk " x k `p´1q k ξ 0,τ N and

I k " " rx 2k´1 , x2k s if 2k ´1 ą N H otherwise.
Then

S ξ p0, τ N qpP 1 q " Y kď0 Ĩk Y r´ξ 0,τ N , `8q, S ξ pτ N , T qpP 2 q " p´8, xN s Y kď0 Ĩk . Note that xk`1 ´x k " x k`1 ´xk `p´1q k 2ξ 0,τ N is bounded from above by 2 | ξ 0,τ k | `p´1q k 2ξ 0,τ N " 2 | ξ τ N ,τ k | and, when k ě N `2, from below by 2 ˇˇξ τ N ,τ k´2 ˇˇ.
Hence, the evolution on rτ N , T s is then given as in the case τ N " 0, and we obtain again S ξ p0, T qpP 1 q " r´ξ 0,T , `8q and S ξ p0, T qpP 2 q " R.

Taking again tx k ´xk´1 as large as possible yields

ρpξ, T q ě › › › R0,T pξq › › › T V prτ N ,T sq
, and letting N Ñ ´8 finishes the proof.

Optimality in one space dimension. We assume d " 1 and consider x-independent Hamiltonians. In this case the representation obtained in Section 2 is even simpler, since only the fully reduced path is needed. Proposition 3.2. Let H : R Ñ R be continuous and convex. Then

(3.7)
up¨, T q " S ξ p0, T q " S R0,T pξq p0, T q, and (3.8) ρ H pξ, T q ď }H 1 } 8 } R0,T pξq} T V pr0,T sq .

Proof. The claim is shown for H smooth and strictly convex in [START_REF] Hoel | Path-dependent convex conservation laws[END_REF] using a regularization argument.

The result extends to convex H by approximation, since (1.1) is stable under the passage to the limit in H by standard viscosity theory; see [START_REF] Lions | Stochastic viscosity solutions[END_REF].

When d ě 2, (3.7) is not true in general. Indeed, this can be easily seen by the counter-example depicted in Figure 3. Note however that the claims in Proposition 3.2 hold in arbitrary dimension for Hppq " 1 2 |p| 2 (cf. Gassiat and Gess [START_REF] Gassiat | Regularization by noise for stochastic Hamilton-Jacobi equations[END_REF]) and more generally for a class of uniformly convex H (cf. Lions and Souganidis [10]). Since (3.7) is not valid in general in dimension d ě 2, the speed of propagation may depend on the total variation of the full reduced path R 0,T pξq. Note that we do not know, even for H " | ¨|, if one always has equality in (1.11) in that case. The following proposition gives an example of a situation where ρ H pξ, T q " }R 0,T pξq} T V pr0,T sq ą } R0,T pξq} T V pr0,T sq . Proposition 3.3. Let δ 1 ą δ 2 ą δ 3 ą 0 and ξ continuous on r0, 3s with

9 ξ " $ & % δ 1
on p0, 1q, ´δ2 on p1, 2q, `δ3 on p2, 3q.

Then

ρ |¨| pξ, 3q " δ 1 `δ2 `δ3 " }ξ} T V pr0,3sq .

Proof. Fix 0 ă L ă 2δ 2 and η ą 0, and consider initial conditions (see Figure 3.5 and Figure 3.4)

P 1 " r0, Ls ˆt´δ 1 , `δ1 u, P 2 " P 1 Y pt´δ 1 u ˆRq.
Then p´η, 0q, pL `η, 0q R S ξ |¨| p0, 1qP 1 , and, hence,

pB δ2 p´η, 0q Y B δ2 pL `η, 0qq X S ξ |¨| p0, 2qP 1 " H. Since L ă 2δ 2 ,
for η small enough, the interior of B δ2 p´η, 0q X B δ2 pL `η, 0q is non-empty. It follows that, for all η 1 P p0, ηq small enough,

B δ3 pL ´δ2 `δ3 ´η1 , 0q Ď B δ2 p´η, 0q Y B δ2 pL `η, 0q,

It follows that

B δ3 pL ´δ2 `δ3 ´η, 0q X S ξ |¨| p0, 2qP 2 ‰ H, and, hence, for each η ą 0, pL ´δ2 `δ3 ´η, 0q P S ξ |¨| p0, 3qP 2 . In conclusion, for all η ą 0 small enough, pL ´δ2 `δ3 ´η, 0q P S ξ |¨| p0, 3qpP 2 qzS ξ |¨| p0, 3qpP 1 q so that ρ |¨| pξ, 3q ě δ 1 `L ´δ2 `δ3 ´η. Letting L Ñ 2δ 2 and η Ñ 0 finishes the proof.

The Brownian case

We begin with some preliminary discussion and a few results that are needed for the proof of Theorem 1.3.

The key observation in the proof of Theorem 1.3 is that the length of R 0,T pBq on r0, τ 0 s, where τ 0 is given by (1.7), is the same as that of a left-continuous path obtained by removing all excursions of B between its minimum and maximum.

For fixed r ą 0 set σ 0 prq :" r and, for all k P N, define the successive jump times by Using once again scaling and the fact that Bessel processes have the Gaussian tails, we also find that, for some C 0 ą 0 and all n P N, Proof of Theorem 1.3. We begin with (1.14) and note that on the interval r0, θp1qs we have τ 0 " θp1q.

It then follows from the definition of θp1q that Rpτ 0 q " Rpθp1qq " 1 and thus, using Lemma 4.1, we find › › R 0,θp1q pBq › › T V pr0,θp1qsq " LpRpθp1qqq " Lp1q. We further note that, in view of (4.3), we have that We present now the proof of (1.13). Throughout the argument below, C will denote a constant whose value may change from line to line. We first note that }R 0,1 pBq} T V pr0,1sq ď }R 0,1 pBq} T V pr0,τ0sq `}R 0,1 pBq} T V prτ´1,1sq .

Moreover, the symmetry of Brownian motion under time reversal gives }R 0,1 pBq} T V prτ´1,1sq pdq " }R 0,1 pBq} T V pr0,τ0sq .

It follows that it suffices to bound the tail probabilities of }R 0,1 pBq} T V pr0,τ0sqq and, hence, in view of Lemma 4.1, the tail of LpRp1qq.

Fix γ P p0, 2q and let α ă γ be such that γ " 2p1`αq

3

. For a fixed x ą 0, let r 0 :" x 4 and, for k P N, r k`1 :" r k p1 `e´x α q ´1. It is immediate that there exists an N such that N ď Ce Cx α and r N ď x ´1. Moreover, P pLpRp1qq ě xq ď N ´1 ÿ k"0 P pr k`1 ď Rp1q ď r k ; Lpr k q ě xq `P ´Rp1q ě x 4 ¯`P pLpr N q ě xq .

The second term on the right hand side is bounded by expp´Cx 2 q since Rp1q has Gaussian tails.

Moreover, Brownian scaling implies that Lpτ tq pdq " τ Lptq for all τ ą 0 and thus PpLpr N q ě xq ď P ˆL`1 x ˘ě x ˙ď PpLp1q ě x 2 q ď expp´Cx 2 q.

In addition, we note that, for k P t0, . . . , N ´1u, P pr k`1 ď Rp1q ď r k ; Lpr k q ě xq ď P ´rk`1 ď Rp1q; Lpr k`1 q ě x 2 ¯`P ´rk`1 ď Rp1q; Lpr k q ´Lpr k`1 q ě x 2 ¯.

Lemma 4.2 implies that P ´rk`1 ď Rp1q; Lpr k`1 q ě x 2 ¯ď P ´rk`1 ď Rp1q; Lpr k`1 q ´rk`1 ě x 4 ¯ď C expp´Cx 2 q.

Then, the Cauchy-Schwarz inequality and Lemma 4.3 give P ´rk`1 ď Rp1q; Lpr k q ´Lpr k`1 q ě x 2 ¯ď P pr k`1 ď Rp1qq 1{2 P ´Lpr k q ´Lpr k`1 q ě

x 2

¯1{2

ď C expp´Cr 2 k`1 q exp ˆ´C x

(1. 1 )

 1 du " HpDu, xq ¨dξ in R d ˆp0, T s up¨, 0q " u 0 in R d , with (1.2) H : R d ˆRd Ñ R convex and Lipschitz continuous in the first argument and (1.3) ξ P C 0 pr0, T sq,

2 and Figure 3 . 3 ,´1 2

 332 which correspond to the continuous, piece-wise linear path ξ with on p1, 3q; it is easy to observe that in this case S ξ |¨| p0, 3q ‰ S R0,3pξq |¨| p0, 3q.

Figure 3 . 2 . 3 Figure 3 . 3 .

 32333 Figure 3.2. Evolution of S ξ|¨| p0, tq at t " 0, t " 1, t " 2, t " 3

Figure 3 . 4 . 1 Figure 3 . 5 .

 34135 Figure 3.4. Evolution of S ξ |¨| p0, ¨qP 1

(4. 6 )

 6 P ´ζ ´1{2 n ě y ¯" P ˆsup 0ďtď1 Xn t ě y ˙ď expp´C 0 y 2 q.

Lp1qpdq" 1 `

 1 U0 `U0 U 1 `U0 U 1 U 2 `. . . , where the random variables U i are i.i.d. uniformly distributed on r0, 1s. It now follows from [6, Theorem 3.1] that Lp1q has Poissonian tails, that is, lim xÑ8 ln PpLp1q ě xq x ln x Ñ ´1.

  Proposition 2.5. If (2.7) and (2.8) hold, then, for each u 0 P BU CpR d q and T ě 0, the family

	)	lim sup δÑ0	inf rď|x´y|	Lpx, y, δq " `8,
	and
	(2.8)	lim δÑ0	lim rÑ0	|x´y|ďr sup	Lpx, y, δq " 0.
	Note that (2.8) is some sort of controllability assumption, while (2.7) follows from a uniform in x
	upper bound on H.
		! S ξ H p0, T qu 0 : ξ piecewise linear	)
	has a uniform modulus of continuity.

  Assuming in what follows the inf in the definition of L is attained, otherwise we work with approximate minimizers, we choose γ to be a minimizer for Lpy, x 1 , t `δq and set ỹ1 " γptq. It follows from (2.4) that Lpy, x, δq `Lpx, x 1 , tq ě Lpy, x 1 , t `δq "

	ż t	Lp 9 γpsq, γpsqqds	`ż t`δ	Lp 9 γpsq, γpsqqds
	0		t

ě Lpy, ỹ1 , tq `Lpỹ 1 , x 1 , δq.

  Extension and reduction. The extension result is stated next. In this subsection, we always assume that either H is independent of x or that (2.7) and (2.8) hold. In what follows we write } ¨}8,O for the L 8 -norm over O.Corollary 2.7. The map ξ Þ Ñ S H pξq is uniformly continuous in the sup-norm in the sense that, if pξ n q nPN is a sequence of piecewise-linear functions on r0, T s with lim n,mÑ8 }ξ n ´ξm } 8,r0,T s " 0, then, for all u P BU CpR d q, Fix δ ą 0 and let ξ, ζ be piecewise linear such that }ξ ´ζ} 8 ď δ on r0, T s. We extend ξ, ζ to all of R as constants on p´8, 0q and pT, `8q and choose η P r´1, 1s such that ξpT q " ζpT q `ηδ. Let ξ ˘δ be defined by ξ

	(2.10)	lim n,mÑ8 }S ξ n H p0, T qu ´Sξ m H p0, T qu} 8,R d " 0.
	It follows that we can extend ξ Þ Ñ S H pξq to all continuous paths. Indeed ξ n Ñ ξ in sup-norm as
	n Ñ 8, then
	(2.11)	S ξ H p0, T qu :" lim nÑ8 S ξ n H p0, T qu.
	Proof of Corollary 2.7
	If follows from Proposition 2.6 that, if v " S ξ H u 0 , then |vpxq ´vpyq| ď ωp|x ´y|q. On the other
	hand, in view of (2.7), lim δÑ0 ηpδq " 0, and, hence, using (2.8) we conclude that lim rÑ0 ωprq " 0.

1 , δq ȃnd νpx, yq :" inf δą0 pLpy, x, δq `ηpδqq , ωprq :" sup |x´y|ďr maxrνpx, yq, νpy, xqs. ˘δ :" # ξ ˘δ on r0, T s, ξ on p´8, ´δq Y pT `δ, `8q, and 9 ξ ˘δ " ˘1 on p´δ, 0q and 9 ξ ˘δ " ¯1 ´η on pT, T `δq. It follows that ξ ˘p´δq " ζp´δq, ξ ˘pT `δq " ζpT `δq, ξ ´δ ď ζ ď ξ δ , and, since S ξ ˘δ H p´δ, T `δq " S H p¯δ ´ηδq ˝Sξ H p0, T q ˝SH p˘δq, Corollary 2.4 yields S H pδp1 ´ηqq ˝Sξ

  σ k`1 prq " supts ă σ k prq : ∆Spsq ‰ 0u, prq `tq ^θn prqq ´Bpθ n`1 prqq| : t ě 0q ně0 pdq " pX n pt ^ζn prqq; t ě 0q ně0 where the processes X n are i.i.d. 3-dimensional Bessel processes, independent from S, starting from 0 andζ n prq :" inf tt ą 0 : X n " σ n prqu .At this point we recall that, in view of the scaling properties of the Brownian motion, it is enough to prove Theorem 1.3 with T " 1, that is to study the reduced path R 0,1 pBq.To prove (1.13) we need the following two lemmata.Since the i.i.d. Bessel random variables Xn are independent from the random variables U n , it follows that

	It follows from (4.2) that
	(4.3)		pσ n`1 prq{σ n prqq ně0	pdq " pU n q ně0 ,
	where pU n q ně0 is a sequence of i.i.d. random variables each with uniform distribution on r0, 1s, and
	pdq " denotes equality in the sense of distributions.
	Moreover, if
			θ n prq :" inf tt ě 0, Rptq " σ n prqu ,
	then
	(4.4) p|Bppθ n`1 It follows from (4.3) and (4.4) that there exist i.i.d. uniformly distributed in r0, 1s random variables
	U n such that
		Lprq " r	`ÿ ną1	σ n prq " r ˜1	n"0 `8 ÿ	U 1 ¨¨¨U n	¸.
	Moreover, in view of the scaling properties of the Bessel processes,
	θprq "	ÿ	pθ n prq ´θn`1 prqqq	pdq "	ÿ	ζ n prqq	pdq "	ÿ	pσ n prqq 2	ζn ,
		ně0							ně0	ně0
	where
				ζn " inf	! t ą 0 : Xn ptq " 1	)	.
	pθprq, Lprqq	pdq " ˜r2	8 ÿ	pU 1 . . . U n q 2		ζn , r ˜1	`8 ÿ
					n"0					n"0

Lemma 4.2. There exists C ą 0 such that, for all r, x ě 0, (4.5) P pRp1q ě r|Lprq ´r ě xq ď C exp `´Cx 2 ˘.

Proof. Note that Rp1q ě r is equivalent to θprq ď 1.

U 1 ¨¨¨U n ¸¸.

  Ce Cx α e ´Cx γ ď Ce ´Cx γ .

			1`α	ď
			r k`1
	C exp ´´Cx	3 2p1`αq	¯.
	It follows that
	P pLpRp1qq ě xq ď

Acknowledgements

Gassiat was partially supported by the ANR via the project ANR-16-CE40-0020-01. Souganidis was partially supported by the National Science Foundation grant DMS-1600129 and the Office for Naval Research grant N000141712095. Gess was partially supported by the DFG through CRC 1283.

We fix an arbitrary continuous path B : r0, `8q Ñ R with Bp0q " 0. Let M ptq :" sup sďt Bpsq, mptq :" inf sďt Bpsq, Rptq :" M ptq ´mptq, recall that for a ě 0 θpaq " inf tt ě 0, Rptq " au , and, r ě 0, define Sprq :" Bpθprqq.

Then S is a left continuous process with right limits, the dynamics of which are simple to describe (see the proof of Lemma 4.1 below): Away from the jumps, S has a drift given by signpSprqq, and the jumps are given by ∆Sprq :" Spr `q ´Sprq " ´signpSprqqr. In particular, (4.1) Lprq :" }S} T V pr0,rsq " r `ÿ 0ďsăr s1 ∆Spsq‰0 .

The following lemma relates L and R 0,T pBq.

Lemma 4.1. For all T ě 0, }R 0,T pBq} T V pr0,τ0sq " LpRpT qq.

Proof. Recall (1.7), (1.8) and (1.9) and note that }R 0,T pBq} T V pr0,τ0sq "

|Bpτ i´1 q ´Bpτ i q| .

Fix i ď 0 and assume that Bpτ i´1 q " mpτ i´1 q ă 0. If r P pRpτ i´1 q, Rpτ i qs, then monotonicity of θ and the definition of τ i give θprq ď θpRpτ i qq " τ i , and, hence, mpθprqq " mpτ i´1 q.

Moreover, for r P pRpτ i´1 q, Rpτ i qs, we have Bpθprqq " M pθprqq and thus r " Rpθprqq " M pθprqq ´mpτ i´1 q " Bpθprqq ´Bpτ i´1 q.

In conclusion, for all r P pRpτ i´1 q, Rpτ i qs,

Sprq " Bpθprqq " Bpτ i´1 q `r " SpRpτ i´1 qq `r, that is S has a jump of size Rpτ i´1 q at r " Rpτ i´1 q and is affine with slope 1 on pRpτ i´1 q, Rpτ i qs.

If Bpτ i´1 q " M pτ i´1 q ą 0, the same reasoning shows that S has a jump of size ´Rpτ i´1 q at r " Rpτ i´1 q and is affine with slope ´1 on pRpτ i´1 q, Rpτ i qs.

Finally we get

LpRpτ 0 qq " }S} T V pr0,Rpτ0qsq "

In view of the definition of τ 0 and R we have Rpτ 0 q " RpT q and thus LpRpT qq " LpRpτ 0 qq, which finishes the proof.

Next we assume that B is a linear Brownian motion on a probability space pΩ, F, Pq and we describe some of the properties of the time change S that we will use below. Then, Jensen's inequality yields that, for any nonnegative sequence α n ,

and thus

On the other hand, (4.6), the independence of the ζn and a straightforward Chernoff bound yield, for all y ě 0 and some C ą 0, We are now ready for the proof of Theorem 1.3.


François Laudenbach
email: francois.laudenbach@univ-nantes.fr

Moraga Carlos

Ferrándiz

A geometric Morse-Novikov complex with infinite series coefficients

Let M be a closed n-dimensional manifold, n > 2, whose first real cohomology group H 1 (M ; R) is non-zero. We present a general method for constructing a Morse 1-form α on M , closed but non-exact, and a pseudo-gradient X such that the differential ∂ X of the Novikov complex of the pair (α, X) has at least one incidence coefficient which is an infinite series. This is an application of our previous study of the homoclinic bifurcation of pseudogradients of multivalued Morse functions.

Introduction

Let us consider an n-dimensional closed smooth manifold M with a non-zero de Rham cohomology in degree one. Let u ∈ H 1 dR (M), u = 0, and let α be a differential closed form in the class u. By Thom's transversality theorem with constraints [START_REF] Thom | Un lemme sur les applications différentiables[END_REF][START_REF] Thom | Les singularités des applications différentiables[END_REF], generically in the class u, the local primitives of a closed 1-form α are Morse functions (such an α is named a Morse 1-form). The finite set Z(α) made of its zeroes is graded by the Morse index: Z(α) = ∪ n k=0 Z k (α). Taking the point of view that a Morse 1-form is nothing but a multivalued function (that is, a function well-defined up to an additive constant), S.P. Novikov developed a theory analogous to Morse theory [START_REF] Novikov | Multivalued functions and functionals. An analogue of Morse theory[END_REF][START_REF] Novikov | The Hamiltonian formalism and a multivalued analogue of Morse theory[END_REF] (today referred to as Morse-Novikov theory) which leads to some homology denoted by H * (M; u). Here, we summarize it with the point of view adopted by J.-C. Sikorav [START_REF] Sikorav | Points fixes de difféomorphismes symplectiques, intersections de sous-variétés lagrangiennes, et singularités de 1-formes fermées[END_REF].

The class u is seen as the morphism π 1 (M) → R defined by u(γ) = γ α for every oriented loop γ. Let ũ : Zπ 1 (M) → R be the associate valuation: for λ ∈ Zπ 1 (M), that is a finite sum i n i g i with n i ∈ Z * and g i ∈ π 1 (M), one defines ũ(λ) = max u(g i ). The universal Novikov ring Λ u is the completion of Zπ 1 (M) for ũ(λ) → -∞. It is worth noticing that we get the same completion by replacing π 1 (M) with the fundamental groupoid of M. This choice is made in what follows.

The Novikov complex N * (α, ∂ X ) is the free Λ u -module based on Z * (α); the differential ∂ X is detailed right below. For defining ∂ X , one chooses some (descending) pseudo-gradient X adapted to α, meaning that X fulfils the following:

1) α(x), X(x) < 0 for every x ∈ M apart from Z(α); 2) for every p ∈ Z k (α), the field -X is the Euclidean gradient of a local primitive h p of α in Morse coordinates (x 1 , . . . , x n ) about p where h p reads

h p (x 1 , . . . , x n ) = -x 2 1 -. . . -x 2 k + x 2 k+1 . . . + x 2 n .
The zeroes of X, which coincide with the zeroes of α, are hyperbolic. Therefore, for each p ∈ Z k (α), we have a stable manifold W s (p, X) of dimension nk and an unstable manifold W u (p, X) of dimension k. According to Kupka-Smale's theorem (see [START_REF] Palis | Geometric Theory of Dynamical Systems, An introduction[END_REF]), one can approximate X so that all stable and unstable manifolds are mutually transverse. For p ∈ Z k (α), the value of ∂ X in p reads:

∂ X p = q∈Z k-1 (α)
p, q X q where the incidence coefficient p, q X belongs to the Novikov ring Λ u .

As far as we know, there are no published example1 where some of the incidence coefficients are infinite series as the Novikov ring allows it. By contrast, it is known that if f : M → R is a Morse function, then for every c ≫ 0, the differential of the Novikov complex N * (α + c df, X) has all its incidence coefficients in Zπ 1 (M). Here, X denotes a pseudo-gradient adapted to α + c df which is C 1 -close to a pseudo-gradient adapted to f [3, Lemma 3.7]. When n > 2, we present a general method for constructing some pairs (α, X) such that the differential ∂ X of N * (α) have some incidence coefficients which are infinite series. Our method is based on our study [START_REF] Laudenbach | Homoclinic bifurcation in Morse-Novikov theory, a doubling phenomenon[END_REF] of the homoclinic bifurcations of the pseudo-gradient X.

Theorem. Let M be an n-dimensional closed manifold with n > 2. Let u ∈ H 1 dR (M) be a non-zero cohomology class and let α be a Morse 1-form in the class u without centers, that is, with no zeroes of extremal index. For 1 ≤ k ≤ n -2, let α ′ be any Morse 1-form obtained from α by creating the birth of a pair of zeroes (q, p) of respective indices (k, k + 1). Then, it is possible to build a pseudo-gradient X ′ adapted to α ′ such that the incidence coefficient p, q X ′ is an infinite series in the Novikov ring Λ u .

Remark. Observe that, when α is a non-exact Morse 1-form, it is possible to cancel all zeroes of extremal indices [1, Remarque section 1.2]. Therefore, the assumption of having no centers is irrelevant.

Proof

2.1. Making a tube of periodic orbits. Let X be a Kupka-Smale pseudo-gradient adapted to α. After the no-center assumption, almost no orbit of X goes to (or comes from) one of its zeroes. A C 0 -approximation of X near an accumulation point of such an orbit allows us to create not only one periodic orbit but a tube T ∼ = D n-1 × S 1 made of periodic orbits {pt} × S 1 (still name X the pseudo-gradient after this approximation). The (free) homotopy class of these periodic orbits is denoted by g. Note u(g) < 0.

Birth of a pair of zeroes in cancelling position

. Let B ⊂ T , B ∼ = D n-1 × D 1 ,
be a polydisc bi-foliated by leaves of α and orbits of X. Let h be a primitive of α| B . Let us insert a birth model into B to create a pair (p, q) of critical points of respective indices k + 1 and k (see [2, chap. III] and also [START_REF] Whitney | On singularities of mappings of Euclidean spaces. I. Mappings of the plane into the plane[END_REF]). We impose this one-parameter deformation starting from h to be supported in B. Denote h ′ the function ending this deformation; set α ′ = α + dh ′dh. Let X 1 be a pseudo-gradient adapted to α ′ near p and q and coinciding with X outside B. There is exactly one orbit of X 1 descending from p to q inside B. However, the pseudo-gradient X 1 is not convenient for our theorem as some extra connecting orbit exists in T from p to q. Thus, we change X 1 into X ′ as follows: make ∂T an attractor of X ′ | T (while keeping X ′ | ∂T = X 1 | ∂T ) and one of the periodic orbits in T B a repeller in order that any orbit of X ′ emanating from the bottom of B, namely ∂ -B := {h ′ = h ′ (q) -ε}, 0 < ε ≪ 1, never returns to B in positive time. The outcome of this construction is that the Novikov complex N * (α ′ , X ′ ) fulfils the following: ∂ X ′ q = 0 and ∂ X ′ p = q for suitable orientations of the unstable manifolds. Moreover, as the boundary of T is invariant by X ′ , no zero of α ′ other than p has a connecting orbit to q; and no zero is connected to p.

Let S g be the stratum of pseudo-gradients adapted to α ′ which are equal to X ′ outside T and have one homoclinic connecting orbit from p to p in T in the class g with the minimal defect of transversality. This is a codimension-one stratum which is canonically co-oriented.

Notation. Denote by

∂ + B := {h ′ = h ′ (p) + ε} the top of B. It contains the so-called belt sphere Σ + of p, that is, Σ + := W s (p, X ′ ) ∩ ∂ + B.
This sphere is (nk -2)-dimensional and co-oriented by the chosen orientation of W u (p, X ′ ). Moreover, it is the boundary of the closed (nk -1)-disc ∆ + whose interior is made of points in ∂ + B whose positive X ′ -flow goes to q. Similarly, set

∂ -B := {h ′ = h ′ (q) -ε}. Denote by ∆ -the closure of W u (p, X ′ ) ∩ ∂ -B (a k-disc) and set Σ -:= W u (q, X ′ ) ∩ ∂ -B. The sphere Σ -bounds ∆ -. Let L ± denote the leaf of α ′ in T which contains ∂ ± B.
The ball ∂ + B shall be regarded as the union of a normal tube N(Σ + ) to Σ + and a normal tube N(∆ + ) to ∆ + in L + . Moreover, there exists a larger tube N ′ (Σ + ) which contains all fibres of N(∆ + ) which meet N(Σ + ).

Let ρ : L -→ L + be the first holonomy diffeomorphism of the positive flow of X ′ . Denote by Desc :

L + ∆ + → L -∆ -the gradient descent by X ′ . Set D := ρ(∆ -); it is a k-disc in L + disjoint from ∂ + B whose boundary lies in W u (q, X ′ ).
Finally, let µ a be the fibre in a ∈ Σ + of the tube N ′ (Σ + ). By descent, the (k + 1)-disc µ a descends to Γ - a , a cylinder over ∆ -, pinched along ∂∆ -. The cylinder Γ - a is foliated by rays which are traced by the normal fibres to ∆ -in L -. The holonomy diffeomorphism ρ carries Γ - a and its foliation to a foliated pinched cylinder Γ + a whose leaves are still named rays.

2.4. Some isolated self-slide of p; homoclinic bifurcation. In this setup, a positive self-slide of p consists in making an isotopy of D in L + , relative to ∂D, which crosses Σ + exactly once and positively with respect to its co-orientation [START_REF] Laudenbach | Homoclinic bifurcation in Morse-Novikov theory, a doubling phenomenon[END_REF]Remark 3.2]. This isotopy lifts to a deformation

(X ′ s ) s∈[0,1] of X ′ 0 = X ′ .
At the time of crossing (let us say s = 1 2 ), the homoclinic connecting orbit from p to p lies in the homotopy class g. The path (X ′ s ) s is said to cross the codimension-one stratum S g positively in the space of pseudo-gradients adapted to α ′ .

Up to homotopy, the path (X ′ s ) s is well-defined by the following data: -a simple smooth path γ : [0, 1 2 ] → L + , starting non-tangentially to D from any point γ(0) ∈ int(D), ending in a + := γ( 12 ) normally to Σ + and which, for every s ∈ (0, 1 2 ), avoids both of W u (p, X ′ ) ∪ ∆ + and the repelling periodic orbit.

-a transverse framing τ = (τ 1 , . . . , τ n-2 ) to γ in L + such that τ k

1 := τ 1 ∧ • • • ∧ τ k is tangent to D in γ(0) and such that ( γ( 1 2 ), τ 1 , . . . , τ n-2
) is an orthogonal framing positively normal to Σ + in a + .

Note that any point a + ∈ Σ + is reachable from D by such a path. In what follows, we will specify γ more precisely near s ∈ {0, 1 2 }. For further use, we introduce a -:= ρ -1 (γ(0)), a point of ∆ -. Let Σ-:= W u (p, X ′ ) ∩ {h ′ = h ′ (p) -ε} be the attaching k-sphere associated with p and denote by ã-∈ Σthe point which descends to a -through the flow of

X ′ | B .
Let Y be a vector field on L + , tangent to ∆ + , whose support S is contained in a neighbourhood of γ([0, 1 2 ]) ∪ ∂ + B and whose flow is noted Y s , s ∈ R. The triple (γ, τ, Y ) may be chosen so that the following holds.

(i) For every s ∈ [0, 1 2 ], we have Y s (γ(0)) = γ(s) and τ k 1 is tangent to Y s (D) in γ(s); moreover, we impose the vector γ( 12 ) to be tangent to ∆ + pointing inwards. (ii) Denoting by T the closure of ∪ +∞ s=0 Y s (D), named the tongue, the intersection T ∩ ∂ + B is a corned disc νa + contained in the fibre µ a + of the tube N ′ (Σ + ). Denote by ν a + the corresponding fibre of N(Σ + ). (iii) For every s ∈ [ 1 2 , +∞), the intersection Y s (D) ∩ ∂ + B is one fibre of N(∆ + ) which meets N(Σ + ). The distance of this fibre to Σ + is noted r(s); set η = max r(s). A suitable choice of Y makes it as small as desired. (iv) The vector field Y is required to be tangent to the foliation of Γ + a + by rays. This requirement forces the path γ to be tangent to a ray of Γ + a + and to follow it entirely. (v) The support S satisfies the following condition:

(S \ ∂ + B) is contained in a small neighbourhood of γ([0, 1 2 ]
). The last requirement needs some preparation. By (i), the vector γ( 12 ) is tangent to ∆ + and points in the direction marked by the point ∆ + ∩ ∂ν a + . This vector determines a co-oriented equator E in the sphere ∂ν a + (see [4, subsection 2.4]). Recall that the Morse model of p produces a canonical diffeomorphism from the meridian sphere ∂ν a + to Σ-. It maps E to an equator Ẽ ⊂ Σwhich avoids the connecting orbit from p to q by the choice of γ( 12 ). Thus, Ẽ descends through the flow of X ′ | B to a (k -1)-sphere E -contained in the interior of ∆ -. This E -makes a decomposition ∆ -= δ -∪ E -δ + where δ -is a k-disc and δ + is diffeomorphic to S k-1 × [0, 1]. The signs of δ ± reflect the co-orientation of E in ∂ν a + , and hence, of E -in ∆ -. By moving γ(0) in D, we can fulfill the last requirement below.

(vi) The point a -lies in δ + . Moreover, we make the support S of Y disjoint from ρ(δ -).

Finally, the isotopy (Y s ) s∈[0,1] lifts to the desired one-parameter deformation (X ′ s ) of X ′ realizing a self-slide of p in the homotopy class g. Lemma 1. With the above requirements (i-vi), the self-slide at s = 1 2 is isolated. More precisely, for s = 1 2 , the pseudo-gradient X ′ s has no homoclinic orbit.

Proof. The dynamics of ρ which moves ∂D away from S makes clear that W u (q, X ′ s ) never meet ∆ + . This implies that no s ∈ [0, 1] is a time of self-slide of q.

Concerning the self-slides of p, we argue as follows. By (ii), the corned disc νa + descends to a corned cylinder C -⊂ Γ - a + , also based on ∆ -and pinched along ∂∆ -. Then, the holonomy diffeomorphism ρ carries C -to a pinched corned cylinder C + ⊂ Γ + a + . By (iv-v), the intersection C + ∩ S is foliated by (truncated) rays of C + starting from D ∩ S. In particular, S ∩ T may be viewed as a sub-tongue of T .

We are now ready to prove Lemma 1. We limit ourselves to prove that X ′ s is never a self-slide of p in the class g 2 when s = 1 2 , that is:

( * ) ∀s = 1 2 , [(Y s ρ) • Desc • 1 (Y s ρ)(∆ -)] ∩ Σ + = ∅ .
Here, the operator • 1 stands for Desc where it is defined, that is in L + ∆ + . The proof concerning the classes g k , k > 2, is similar with the same choices (i) -(vi). When s < 1 2 and as long as the disc Y s (D) does not touch the ball ∂ + B, the dynamics of ρ • Desc implies ( * ). When Y s (D) ∩ N(Σ + ) is non-empty, we define D - s := Desc • 1 Y s (D). When s < 1 2 , one checks that D - s is contained in the union of thin tube Desc(S ∂ + B) and the part of the cylinder C -over δ -(in the sense of the rays). Thus by (vi), ρ(D - s ) does not approach the support S of Y and ( * ) holds for every s ∈ [0, 1 2 ). For s > 1 2 , D - s is now an annulus whose boundary is made of two (k -1)-spheres, one of which is Desc(∂D) and the other is ∂∆ -. By [START_REF] Laudenbach | Homoclinic bifurcation in Morse-Novikov theory, a doubling phenomenon[END_REF]Lemma 3.8], when s goes to 1 2 from above, it approaches any compact domain given in the interior of the annulus δ + in the C 1 -topology. As a consequence of (iv), when s - g (resp. S - g ) the non-empty open set in S g where χ > 0 (resp. χ < 0). When crossing S g transversely, the change of the differential in the Morse-Novikov complex is different according to the crossing: it can be positive or negative; through S + g or S - g . A positive crossing through S + g implies a change by multpiplication with the infinite series 1 + g + g2 + • • • .

In our setting, the considered pseudo-gradient is X ′ 1/2 and the involved homoclinic orbit ℓ is the orbit of X ′ 1/2 which leaves the box B at a -and enters B at a + . We wish χ(X ′ 1/2 ) > 0. By the construction, the character reads χ(X ′ 1/2 ) = λω + + ω -with λ > 0 and ω ± ∈ [-1, +1]. Each term 2 in this formula depends on h only. The term ω -is positive by the choice of γ( 12 ) (which fixes the decomposition of ∆ -= δ -∪ E -δ + ) and by taking a -∈ δ + . The term ω + is not estimated but the factor λ may be made as small as desired by the choice of the holonomy of Y along γ([0, 1 2 ]) once γ(0) = ρ(a -) and γ( 12 ) = a + are fixed. And this choice is free. As a consequence, by [4, Theorem 3.6], we have:

( * ) ∂ X ′ 1 q = 0, ∂ X ′ 1 p = (1 + g + g 2 + g 3 + • • • ) (∂ X ′ 0 p) = (1 + g + g 2 + g 3 + • • • ) q .
Some comment about ( * ) is necessary here. In the theorem which is referred to, ( * ) is stated only for the germ of (X ′ s ) s at s = 1 2 with truncation of the incidence coefficients. The support of a representative of the germ depends on the order of the truncation, the higher the order of truncation the smaller the domain of the representative. Here, we are in a very particular situation: by Lemma 1 the self-slide is isolated. As a consequence, the truncation in question is not needed. This finishes the proof of our theorem.

1 2 2 . 5 .

 125 is small enough the intersection of ρ(D - s ) with S is the graph G us in the cylinder C + of some positive function u s : ρ(δ + ) ∩ S → R >0 . Since the flow of Y preserves the order on each orbit, [(Y s ρ)(D - s )] ∩ S (which lies in the tongue) is in front position with respect to [Y 1/2 (D)] ∩ S on each flow line of Y . This claim holds true when s > 1 2 is very close to 1 2 and it remains true up to 1 by the flow rule. Therefore, [(Y s ρ)(D - s )] ∩ S is still beyond [Y 1/2 (D)] ∩ S in the tongue, and hence, it cannot meet Σ + . The character function. This function is a natural continuous function χ : S g → R which takes positive and negative values on each connected component of S g [4, Theorem 1.1]. For a pseudo-gradient Y ∈ S g , the value χ(Y ) only depends on the 1-jet of the holonomy h of Y along the unique homoclinic orbit ℓ Y from a zero of Y to itself in the homotopy class g. The construction and properties of χ(Y ) are given in [4, Section 2]. Denote by S +

A 3-dimensional example where u is a rational class has been given by A. Pajitnov in [7,Section 3].

The terms ω + and ω -are named the latitudes in[START_REF] Laudenbach | Homoclinic bifurcation in Morse-Novikov theory, a doubling phenomenon[END_REF] and the factor λ is called the holonomy factor.


Mustapha Alaoui

Hattab Maker

Azeddine Mouhsen

Interleaved Pv Emulator

Con- Verter Buck

PV Emulator based on Interleaved Buck Converter using Adaptive PI Controller

controlled by an adaptive PI controller, this system is generally used for the purpose of reproducing the electrical behavior of a PV module without needing the use of real PV panels which are very cost and require a large space besides their dependence on weather parameters of irradiance and temperature. For those reasons, the PV emulator has been proposed to overcome these limitations and allow researchers, engineers and manufacturers to carry out tests and experimentation on PV systems such as PV MPPT algorithms and connected PV inverters irrespective of environmental conditions change [2]. The reason behind using interleaved topology is its ability to give a little amount of ripples and reduce conduction losses; in addition, it allows downsizing LC filters, which improves the transient response speed [1]. Simulation results are given and discussed in order to verify the effectiveness of PV emulator system as well as evaluate its performances in terms of accuracy, speed, and overshoot limitation.

The whole PV emulator circuit, as shows in figure 1, contains the power part which concerns the interleaved buck converter, and the control part which consists on the current reference generator where the I-V curve is implemented and the PI controller which tracks the I-V characteristic and ensures the robustness in terms of parameters variation and external disturbances. The subsystem below is used to compare the PV emulator current and voltage outputs with the PV model 'ones.

IMAT3E'18

International Meeting on Advanced Technologies in Energy and Electrical Engineering The adaptive PI controller parameters are defined using the following equations:

(1)

(2) ( 3
)
Where τ is the desired closed loop time constant .The PI parameters are adjusted for any value of resistance (R) in order to maintain the performances of the closed loop system, besides, those parameters are tweaked to make the system over-damped in the voltage region of I-V curve so as to ensure the stability of the PV emulator whereas the current reference changes excessively [START_REF] Prasanth | Analysis on solar PV emulators[END_REF].

IMAT3E'18

International Meeting on Advanced Technologies in Energy and Electrical

Engineering

It has to be noted that Kd is very low in the order of 1e-6, so it can be neglected. As shown in figure 3 above, the designed PV emulator can generate accurately the voltage and current outputs of PV module for any operating point in the I-V curve, so it can efficiently mimic the electrical behavior of an actual PV module.

Références

  Advanced Technologies in Energy and Electrical Engineering PV Emulator based on Interleaved Buck Converter using Adaptive PI Controller 1 *ALAOUI Mustapha, 1 MAKER Hattab, 2 MOUHSEN Azeddine. 1 mu.alaoui@uhp.ac.ma, High School of Technologies, Berrechid, Morocco 2 Faculty of Sciences and Technologies, Settat, Morocco This paper presents a modeling, design and simulation of PV array emulator based on Switched Mode Power Supply which consists on an Interleaved DC-DC Buck Converter

Figure 1

 1 Figure 1: PV Array Emulator Simulink Model

Figure 2 :

 2 Figure 2: Voltage and Current outputs of the proposed PV emulator for R = 15 Ω under STC conditions As illustrated in figure 2, the proposed PV emulator is able to track the I-V curve of PV module with accuracy and fast dynamic response; the settling time doesn't exceed 100 µs and the output ripples are very small.

Figure 3 :

 3 Figure 3: Comparison between I-V curve characteristic of PV model, real PV module and PV emulator under STC conditions.