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Comptes Rendus Mathématique
Volume 355, n° 7
pages 786-794 (juillet 2017)
Doi : 10.1016/j.crma.2017.06.001
Received : 25 April 2017 ;  accepted : 2 June 2017
On the existence of correctors for the stochastic homogenization of viscous Hamilton–Jacobi equations
Sur l'existence de correcteurs en homogénéisation stochastique d'équations de Hamilton–Jacobi
 

Pierre Cardaliaguet a , Panagiotis E. Souganidis b
a Ceremade, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16, France 
b Department of Mathematics, University of Chicago, Chicago, IL 60637, USA 

Abstract

We prove, under some assumptions, the existence of correctors for the stochastic homogenization of “viscous” possibly degenerate Hamilton–Jacobi equations in stationary ergodic media. The general claim is that, assuming knowledge of homogenization in probability, correctors exist for all extreme points of the convex hull of the sublevel sets of the effective Hamiltonian. Even when homogenization is not a priori known, the arguments imply the existence of correctors and, hence, homogenization in some new settings. These include positively homogeneous Hamiltonians and, hence, geometric-type equations including motion by mean curvature, in radially symmetric environments and for all directions. Correctors also exist and, hence, homogenization holds for many directions for nonconvex Hamiltonians and general stationary ergodic media.

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Résumé

Nous démontrons l'existence, sous certaines conditions, de correcteurs en homogénéisation stochastique d'équations de Hamilton–Jacobi et d'équations de Hamilton–Jacobi visqueuses. L'énoncé général est que, si l'on sait qu'il y a homogénéisation en probabilité, un correcteur existe pour toute direction étant un point extrémal de l'enveloppe convexe d'un ensemble de niveau du Hamiltonien effectif. Même lorsque que l'homogénéisation n'est pas connue a priori, les arguments développés dans cette note montrent l'existence d'un correcteur, et donc l'homogénéisation, dans certains contextes. Cela inclut les équations de type géométrique dans des environnements dont la loi est à symmétrie radiale. Dans le cas général stationnaire ergodique et sans hypothèse de convexité sur le hamiltonien, on montre que des correcteurs existent pour plusieurs directions.

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Introduction

The aim of this note is to show the existence of correctors for the stochastic homogenization of “viscous” Hamilton–Jacobi equations of the form
(1.1)utε−εtr(A(Duε,xε,ω)D2uε)+H(Duε,xε,ω)=0inRd×(0,∞). Here   is a small parameter that tends to zero,   is the Hamiltonian and   is a (possibly) degenerate diffusion matrix. Both A and H depend on a parameter  , where   is a probability space. We assume that   is stationary ergodic with respect to translations on   and that A and H are stationary.

The basic question in the stochastic homogenization of ((1.1)) is the existence of a deterministic effective Hamiltonian   such that the solutions   to ((1.1)) converge, as  , locally uniformly and with probability one, to the solution to the effective equation
(1.2)ut+H‾(Du)=0inRd×(0,∞). When H is convex with respect to the p variable and coercive, this was first proved independently by Souganidis [[17]] and Rezakhanlou and Tarver [[16]] for first-order Hamilton–Jacobi equations, and later extended to the viscous setting by Lions and Souganidis [[14]] and Kosygina, Rezakhanlou and Varadhan [[10]]. See also Armstrong and Souganidis [[1], [2]] and Armstrong and Tran [[4]] for generalizations and alternative arguments.

In periodic homogenization, convergence and, hence, homogenization rely on the existence of correctors (see Lions, Papanicolaou and Varadhan [[12]]). The random setting is, however, fundamentally different.

Following Lions and Souganidis [[13]], a corrector associated with a direction   is a solution χ to the corrector equation
(1.3)−tr(A(Dχ(x)+p,x,ω)D2χ(x))+H(Dχ(x)+p,x,ω)=H‾(p)in Rd which has a sublinear growth at infinity, that is, with probability one,(1.4)lim|x|→+∞⁡χ(x,ω)|x|=0.

It was shown in [[13]] that in general such solutions do not exist; see also the discussion by Davini and Siconolfi [[8]] in the 1-d case. Note that the main point is the existence of solutions satisfying ((1.4)).

Not knowing how to find correctors is the main reason that the theory of homogenization in random media is rather complicated and required the development of new arguments. General qualitative results in the references cited earlier required the quasiconvexity assumption. A more direct approach to prove homogenization (always in the convex setting), which is based on weak convergence methods and yields only convergence in probability, was put forward by Lions and Souganidis [[15]]. Our approach here is close in spirit to the one of [[15]]. With the exception of a case with Hamiltonians of a very special form (see Armstrong, Tran and Yu [[3], [5]]), the main results known in nonconvex settings are quantitative. That is it is necessary to make some strong assumptions on the environment (finite-range dependence) and to use sophisticated concentration inequalities to prove directly that the solutions to the oscillatory problems converge; see, for example, Armstrong and Cardaliaguet [[3]] and Feldman and Souganidis [[9]]. It should be noted that the counterexamples of Ziliotto [[18]] and [[9]] yield that in the setting of nonconvex homogenization in random media is not possible to prove the existence of correctors for all directions.

Our main result states that a corrector in the direction p exists provided p is an extreme point of the convex hull of the sub-level set  . For instance, this is the case if the law of the pair   under   is radially symmetric, and   satisfy some structure conditions.

This kind of result is already known in the context of first-passage percolation, where the correctors are known as Buseman function; see, for example, Licea and Newman [[11]]. The techniques we use here are strongly inspired by the arguments of Damron and Hanson [[7]]. There the authors build a type of weak solutions and prove that, when the time function is strictly convex, they are actually genuine Buseman functions.

The assumptions and the main result

The underlying probability space is denoted by  , where Ω is a Polish space,   is the Borel σ -field on Ω, and   is a Borel probability measure. We assume that there exists a one-parameter group   of measure preserving transformations on Ω, that is   preserves the measure   for any   and   for  . The maps  , the set of   real symmetric and nonnegative matrices, and   are supposed to be continuous in all variables and stationary, that is, for all   and  ,
(A,H)(p,x,τzω)=(A,H)(p,x+z,ω).

We also remark that the equations below, unless otherwise specified, are understood in the Crandall–Lions viscosity sense.

To avoid any unnecessary assumptions, in what follows we state a general condition, which we call assumption  , on the support of  .

Assumption  : We assume that, for any  , the approximate corrector equation
(2.1)δvδ,p−tr(A(Dvδ,p+p,x,ω)D2vδ,p)+H(Dvδ,p+p,x,ω)=0in Rd, has a comparison principle, and that, for any  , there exists   such that, if  , then the unique solution   to ((2.1)) satisfies‖δvδ,p‖∞+‖Dvδ,p‖∞≤CR. The conditions ensuring the comparison principle are well documented; see, for instance, the Crandall, Ishii, Lions “User's Guide” [[6]]. Given the comparison principle, it is well known that‖vδ,p(⋅,ω)‖∞≤supx∈Rd⁡|H(0,x,ω)|/δ, so that the  -assumption on   is not very restrictive. The Lipschitz bound, however, is more subtle and relies in general on a coercivity condition on the Hamiltonian. Such a structure condition is discussed, in particular, in [[14]].

Our main result is stated next.

Theorem 2.1

Assume   and, in addition, suppose that homogenization holds in probability, that is, for any  , the family   converges, as  , in probability to some constant  , where   is a continuous and coercive map. Let   be an extreme point of the convex hull of the sub level-set  . Then, for  -a.e.  , there exists a corrector   associated with p and ω, which is a Lipschitz continuous solution to ((1.3)) satisfying ((1.4)).

Some observations and remarks are in order here.

We begin noting that we do not know if the corrector χ has stationary increment, and we do not expect this to be true in general. Note that the corrector is not necessary unique, even up to an additive constant. However, by a measurable selection argument, we can assume that χ depends on ω in a measurable way.

The existence of a corrector yields that, in fact, the  's converge to   for  -a.e.  ; see Proposition 1.2 in [[13]]. In the rest of the paper, we will use this fact repeatedly. Note also that convexity plays absolutely no role here.

Our result readily applies to the case where (H) holds, the law of the pair   under   is radially symmetric, and   and   are homogeneous in p of degrees 0 and 1, respectively; this is stated in Corollary 3.9. Moreover, since   for some positive  , Theorem 2.1 implies the existence of a corrector for any direction p . Note that this case covers the homogenization of equations of mean curvature type and the result is new. Other known results for such equations are quantitative.

This result also extends to the case where H satisfies, for all  ,   and  ,
0≤H(λp,x,ω)≤λH(p,x,ω). Then there exists a corrector for any direction p such that   is positive. Indeed, following Corollary 3.9, homogenization holds in probability for any direction p and   for some map   which is increasing when positive.

If H is convex in p and A is independent of p , our proof implies that, for any  , the limit   exists in probability; see Proposition 3.10. This result and its the proof are very much in the favor of [[15]].

Finally, we note that our arguments also yield the existence of a corrector in some directions and, thus, homogenization, for nonconvex Hamiltonians and p -dependent A . More precisely, for any direction p , there exists a constant   such that p belongs to the convex hull of directions   for which a corrector exists with an associated homogenized constant equal to  ; see Corollary 3.8.

The Proof of Theorem 2.1

Throughout the section, we assume that condition   is satisfied, but do not suppose that homogenization holds (even in probability): this condition is added only at the very end of the section.

Fix  , let   be as in   and define the metric space
Θ:={θ∈C0,1(Rd):θ(0)=0and‖Dθ‖∞≤CR} with distance, for all  ,d(θ1,θ2):=supx∈Rd⁡|θ1(x)−θ2(x)|1+|x|2. It is immediate that Θ is a compact.

Next we enlarge the probability space to  , which is endowed with the one-parameter group of transformations   defined, for  , by
τ˜x(ω,θ,s)=(τxω,θ(⋅+x)−θ(x),s); below, by an abuse of notation, we write  .

Fix   with  , let   be the solution to ((2.1)), define the map   by
Φδ,p(ω)=(ω,vδ,p(⋅,ω)−vδ,p(0,ω),−δvδ,p(0,ω)), which is clearly measurable, and consider the push-forward measureμδ,p=Φδ,p♯P, which is a Borel probability measure on  .

Note that, since the first marginal of   is   and Ω is a Polish space while   is compact, the family of measures   is tight.

Let μ be a limit, up to a subsequence  , of the  's.

Lemma 3.1

For each  , the transformation   preserves the measure μ.

Proof

Fix a continuous and bounded map  . Since the map   is continuous and bounded and   converges weakly to μ , we have
∫Ω˜ξ(ω˜)τx♯μ(dω˜)=∫Ω˜ξ(τx(ω˜))μ(dω˜)=limn⁡∫Ω˜ξ(τx(ω˜))μδn,p(dω˜). In view of the definition of   and  , we get∫Ω˜ξ(τx(ω˜))μδn,p(dω˜)=∫Ωξ(τxω,vδn,p(x+⋅,ω)−vδn,p(x,ω),−δnvδn,p(0,ω))dP(ω)=∫Ωξ(τxω,vδn,p(⋅,τxω)−vδn,p(0,τxω),−δnvδn,p(−x,τxω))dP(ω)=∫Ωξ(ω,vδn,p(⋅,ω)−vδn,p(0,ω),−δnvδn,p(−x,ω))dP(ω), the last line being a consequence of the stationarity of  .

Using that   is Lipschitz continuous uniformly in δ and ξ is continuous on the set  , we find
∫Ω˜ξ(τx(ω˜))μδn,p(dω˜)=∫Ωξ(ω,vδn,p(⋅,ω)−vδn,p(0,ω),−δnvδn,p(0,ω)+O(δn))dP(ω)=∫Ω˜ξ(ω˜)dμδn,p(ω˜)+o(1). Letting  , we finally get∫Ω˜ξ(ω˜)τx♯μ(dω˜)=∫Ω˜ξ(ω˜)dμ(ω˜), and, hence, the claim. □

The next lemma asserts that there exists some   such that the restriction of μ to the last component is just a Dirac mass. If we know that homogenization holds, then   is of course nothing but  . Note that in what follows, abusing once again the notation, we denote by μ the restriction of μ to the first two components  .

Lemma 3.2

There exits a constant   such that, for any Borel measurable set  ,
μ(E×[−Mp,Mp])=μ(E×{c‾}). In particular, the sequence   converges in probability to  .

Proof

Let   large and  , set   and
Ek:={ω∈Ω:∃θ∈Θand∃s∈[tk,tk+1]such that(ω,θ,s)∈sppt(μ)}. Since the first marginal of μ is   and  , there exists   such that  .

It turns out that   is translation invariant, that is, for each  ,  . Indeed, if  , there exists   and   such that   and, hence,   belongs to  . Since μ is invariant under  , so is its support. Hence   and ω belongs to  . The opposite implication follows in the same way.

The ergodicity of   yields that  , which means that μ is concentrated in some  . Thus μ is also concentrated on  . Letting   implies that there exists   such that μ is concentrated of the set  .

It remains to check that   converges in probability to  . This is a consequence of the classical Porte-Manteau Theorem, since, for any  ,
limsupn→∞P[|δnvδn,p(0,⋅)+c‾|≥ε]=limsupμδn,p[Ω×Θ×([−Mp,Mp]\(c‾−ε,c‾+ε))]≤μ[Ω×Θ×([−Mp,Mp]\(c‾−ε,c‾+ε))]=0. □

The next lemma is the first step in finding a corrector and possibly identifying   and  , when the latter exists.

Lemma 3.3

Let   be defined by Lemma 3.2. For μ-a.e.  , θ is a solution to
(3.1)−tr(A(Dθ+p,x,ω)D2θ)+H(Dθ+p,x,ω)=c‾in Rd.

Proof

Fix   and let   be the set of   such that θ such that, in the open ball  ,
−tr(A(Dθ+p,x,ω)D2θ)+H(Dθ+p,x,ω)≥c‾−ε and−tr(A(Dθ+p,x,ω)D2θ)+H(Dθ+p,x,ω)≤c‾+ε. Recall that Lemma 3.2 gives that   converge in probability to  . Since   solves ((2.1)) and is uniformly Lipschitz continuous, it follows that, as  ,  .

Finally observing that   is closed in  , we infer, using again the Porte-Manteau Theorem, that  .

As R and ε are arbitrary, we conclude that the set  , for which the fact that the equation is satisfied in the viscosity sense is of full probability. □

Next we investigate some properties of θ .

Lemma 3.4

For any  ,  .

Proof

Since the map   is continuous on   and   is stationary, we have
Eμ[θ(x)]=lim⁡Eμδn,p[θ(x)]=lim⁡EP[vδn,p(x)−vδn,p(0)]=0. □

Lemma 3.5

For μ-a.e.   and any direction  , the (random) limit
ρω˜(q):=limt→∞⁡θ(tq)t exists. Moreover,   is invariant under   for  , that is, ρτ˜x(ω˜)(q)=ρω˜(q)μ−a.e.

Proof

We first show that, for any  , the limit
limt→+∞⁡1t(∫Br(0)θ(tq+y)dy−∫Br(0)θ(y)dy) exists  -a.s.

Since the uniform converge of uniformly Lipschitz continuous maps implies the  -weak ⋆ convergence of their gradients, the map   defined by
ξ((ω,θ)):=∫Br(0)Dθ(y)⋅qdy is continuous and bounded on  .

Moreover,
1t(∫Br(0)θ(tq+y)dy−∫Br(0)θ(y)dy)=1t∫0t∫Br(0)Dθ(sq+y)⋅qdyds=1t∫0tξ(τ˜sq(ω˜))ds. It follows from the ergodic theorem that the above expression has, as   and μ -a.s. a limit  .

Choosing   and letting  , we also find that, as  ,   has μ -a.s., a limit   because θ is  -Lipschitz continuous.

Fix   and   for which   and   are well defined; recall that this holds for μ -a.e.  .

Then, in view of the Lipschitz continuity of θ , we have
ρτ˜x(ω˜)(q)=limt→+∞⁡τ˜x(θ)(tq)t=limt→+∞⁡1t(θ(x+tq)−θ(x))=ρω˜(q). □

Lemma 3.6

There exists a random vector   such that, μ-a.s. and for any direction  ,
limt→+∞⁡θ(tv)t=rω˜⋅v.

Proof

Since θ is  -Lipschitz continuous, it is enough to check that the map   is linear on   for μ -a.e.  .

Let   be a set of μ -full probability in Ω such that the limit   in Lemma 3.5 exists for any  .

Restricting further the set   if necessary; we may also assume (see, for instance, the proof of Lemma 4.1 in [[1]]) that, for any   and  , there exists   such that, for all   with  , all   and  ,
|θ(x+tq)−θ(x)t−ρω˜(q)|≤η(|x|+1). Fix  ,   with  ,   and  , and let T be associated with   as above. Then, for any  , we haveθ(t(q1+q2))=θ(t(q1+q2))−θ(tq2)+θ(tq2). Thus|θ(t(q1+q2))t−ρω˜(q1)−ρω˜(q2)|≤|θ(t(q1+q2))−θ(tq2)t−ρω˜(q1)|+|θ(tq2)t−ρω˜(q2)|≤η(|q2|+t−1)+η Letting   and   yields the claim, since η and M are arbitrary. □

Lemma 3.7

Let r be defined as in Lemma 3.6. Then  .

Proof

Lemma 3.4 yields that, for any  ,
0=limt→+∞⁡Eμ[θ(tv)t]=Eμ[limt→+∞⁡θ(tv)t]=Eμ[r⋅v]=Eμ[r]⋅v. □

As a straightforward consequence of the previous results, we have the existence of a corrector and, hence, homogenization for at least one vector  .

Corollary 3.8

For μ-a.e.  ,   exists for   and is given by  . Moreover,   is a corrector for  , in the sense that
−tr(A(Dθ′+p′,x,ω)D2θ′)+H(Dθ′+p′,x,ω)=c‾in Rdwithlim|x|→+∞⁡θ′(x)/|x|=0.

Another consequence of the above results is that homogenization holds if the law of   under   is a radially symmetric. By this we mean that, for any rotation matrix R , the law of   is the same as the law of the pair   given by
(A˜,H˜)(p,x,ω):=(RTAR,H)(Rp,Rx,ω). Note that this implies that   has the same law as  .

Corollary 3.9

Assume that,  -a.s.,   is 0-homogeneous in p, H satisfies, for all  ,
(3.2)0≤H(λp,x,ω)≤λH(p,x,ω) and suppose that the law of   under   is radially symmetric. Then homogenization holds in probability, that is, for any  ,   in probability. Moreover, the map   satisfies, for any  , 0≤c‾(s1)/s1≤c‾(s2)/s2.

Note that the map   is increasing as soon as it is positive. Moreover, one easily checks that, if, in addition, H is 1-homogeneous in p and coercive, then   for some positive constant  .

Proof

It follows from the assumed bounds and the stationarity, that there exists a set   with   such that, for any   and  ,   and   exist and are deterministic. The radial symmetry assumption and as well as (Corollary 3.9) imply that   and, in addition, for all  ,
0≤c±(λs)≤λc±(s). Also note that the maps   are nondecreasing. Indeed given  , choosing   and  , we findc±(s1)/s1≤c±(s2)/s2≤c±(s2)/s1. It follows that   is increasing as soon as it is positive.

On the other hand, for any  , we can find a subsequence of   converging to some measure   as δ tends to 0. By a diagonal argument, we can assume that this is the same subsequence for any  .

Let   be associated with the limit measure   as above. It follows from (H) that the map   is uniformly continuous and thus can be continuously extended to  . Moreover, the assumed radial symmetry yields that  . Finally, note that, for all  ,
(3.3)0≤c−(s)≤c‾(s)≤c+(s). Let  . Then ((3.3)) implies   on  .

Fix   with  , and let  ,   and r be associated with p . For  -a.e.   with  ,   is a corrector for   with associated ergodic constant  . It follows that  , and, hence,
c+(|p|)≥c‾(|p|)=c+(|p+r|)μp−a.s. Since   is increasing on  , this inequality implies that   a.s. Then   gives    -a.s., so that, for  -a.e.  , θ is a corrector for p with associated ergodic constant  . It follows that  . In conclusion,   for any  , and thus, by continuity, for any  , which, in turn, proves that homogenization holds. □

Another application of the previous results is the convergence in law of the random variable   when H is convex in the gradient variable. The argument is a variant of [[15]]. Of course, the result is much weaker than the a.s. convergence is established in [[14]]; see also [[1], [2]]. The proof is, however, rather simple.

Proposition 3.10

Assume that,  -a.e.,   is convex in the p variable and that   does not depend on p. Then, for any  , homogenization holds in probability, that is there exists   such that   in probability.

Proof

Let μ be a measure built as in the beginning of the section. It follows that there exists a random family of measures   on Θ such that, for any continuous map  , one has
∫Ω×Θϕ(ω,θ)dμ(ω,θ)=∫Ω[∫Θϕ(ω,θ)dμω(θ)]dP(ω). Set  . Since   and μ are invariant with respect to   and   respectively, for any bounded measurable map   and any  , we have∫Ωϕ(ω)(θˆ(x+z,ω)−θˆ(z))dP(ω)=∫Ω×Θϕ(ω)(θ(x+z)−θ(z))dμ(ω,θ)=∫Ω×Θϕ(τ−zω)θ(x)τ˜z♯dμ(ω,θ)=∫Ωϕ(τ−zω)θˆ(x,ω)dP(ω)=∫Ωϕ(ω)θˆ(x,τzω)dP(ω). This shows that   has stationary increments. Moreover, in view of Lemma 3.4,   has mean zero, and, hence,   is stationary with average 0. In particular,   is  -a.s. strictly sublinear at infinity. Since, for μ -a.e.  , θ is a solution to (Lemma 3.3) and H is convex in the gradient variable,   is a subsolution to (Lemma 3.3) and, thus a subcorrector. Following [[15]], this implies thatliminfδ→0δvδ,p(0,ω)≥−c‾. In particular, for any sequence   that tends to 0 such that   and   converge respectively to a measure   and a constant  , we have  . Exchanging the roles of   and   leads to the equality  . The conclusion now follows. □

We are now ready to prove the main result.

Proof of Theorem 2.1

We assume that homogenization holds in probability and   is an extreme point of the convex hull of the set  .

Let μ be a measure built as in the beginning of the section and r be defined by Lemma 3.6.

Then   μ -a.s., that is   belongs to S μ -a.s. Indeed Lemma 3.3 gives   and
−tr(A(Dθ+p,x,ω)D2θ)+H(Dθ+p,x,ω)=c‾in Rd, while, in view of Lemma 3.6, for all  ,limt→+∞⁡θ(tx)t=r⋅x. Thus   is a corrector for  , that is it satisfies−tr(A(Dθ˜+p+r,x,ω)D2θ˜)+H(Dθ˜+p+r,x,ω)=c‾in Rdandlim|x|→+∞⁡θ˜(x)/|x|=0. It follows that   μ -a.s.

Next we recall (Lemma 3.7) that  . Since   μ -a.s. and p is an extreme point of the convex hull of S , the equality   implies that   μ -a.s. Therefore   μ -a.s., which, together with the fact that θ solves the corrector equation for p , implies that θ is a corrector for p itself. □


 P. Cardaliaguet was partially supported by the ANR (Agence nationale de la recherche) project ANR-12-BS01-0008-01. P. E. Souganidis was partially supported by the National Science Foundation Grants DMS-1266383 and DMS-1600129 and the Office of Naval Research Grant N000141712095.

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