VISCOSITY SOLUTION METHODS AND THE DISCRETE AUBRY-MATHER PROBLEM

. In this paper we study a discrete multi-dimensional version of Aubry- Mather theory using mostly tools from the theory of viscosity solutions. We set this problem as an inﬁnite dimensional linear programming problem. The dual problem turns out to be a discrete analog of the Hamilton-Jacobi equations. We present some applications to discretizations of Lagrangian systems. theory such as the existence of invariant sets and measures, the graph theorem, and asymptotic behaviour. The other one is to show that in the continuous limit one can recover the corresponding objects such as Mather’s measures and viscosity solutions. We would like to point out that our results concerning Mather sets and measures are not new, the main novelty consists in the formulation and the methods used to obtain them, as well as, in pointing out the connection between certain diﬀerence equations and the corresponding Hamilton-Jacobi equations in the continuous setting.

1. Introduction. Certain variational problems which arise in applications such as Lagrangian mechanics, Mather's problem [Mn92,Mn96], Monge-Kantorowich optimal transport problems [Eva99], or stationary stochastic optimal control [Gom02a], can be written as infinite dimensional linear programming problems in spaces of measures. The standard strategy to study such problems is to compute the dual problem. In general, the dual yields important information about the original (primal) problem. This dual may be of interest itself, and therefore the primal problem may provide useful insights about the dual.
In the continuous case, the dual of Mather's problem is related to viscosity solutions of Hamilton-Jacobi equations [Fat97a,Fat97b,Fat98a,Fat98b]. For stationary stochastic optimal control, it turns out to be related with second order nonlinear elliptic equations.
For discrete problems, such as optimal transport and the problem discussed in this paper, the dual is not a partial differential equation but a difference equation. These equations can be seen as discretizations of the corresponding partial differential equations in the appropriate limit.
The objective of this paper is twofold, one is to show that viscosity solutions methods can be adapted and used to study certain discrete dynamical system, and can be used to prove many well known facts about Mather's theory such as the existence of invariant sets and measures, the graph theorem, and asymptotic behaviour. The other one is to show that in the continuous limit one can recover the corresponding objects such as Mather's measures and viscosity solutions. We would like to point out that our results concerning Mather sets and measures are not new, the main novelty consists in the formulation and the methods used to obtain them, as well as, in pointing out the connection between certain difference equations and the corresponding Hamilton-Jacobi equations in the continuous setting.

DIOGO A. GOMES
Before stating the main problem, we need some notation. To that effect, let T n be the n dimensional torus T n = R n /Z n , Ω = T n × R n , and consider a Lagrangian L(x, v) : Ω → R, smooth, periodic in x (which is implicit by assuming x ∈ T n ), coercive, and strictly convex in v. We look for positive probability measures µ (x, v) in Ω, and that minimize the average action This is a linear programming problem on a space of measures, and so it admits a dual problem, which, as we will see in section 2, is given by: (1.4) In analogy with the continuous case we will study the discrete stationary Hamilton-Jacobi equation associated with (1.4): The solution to (1.5) contains information about properties of minimizing measures such as the support. The analog of the Euler-Lagrange equations that arise in continuous time, are the discrete maps that we study in this paper. It is instructive to observe several facts about the invariant measure that differ from the continuous case. In the continuous case, we look for probability measures that minimize (1.3) with the constraint Since L is strictly convex in v, and the constraints (1.1) and (1.6) are linear in v, it is immediate that the minimizing measure is supported on a graph v = v(x). This is, however, an issue for the discrete version since (1.2) is non-linear in v. Nevertheless, if this measure were supported on a graph v = v(x), and its projection on T n had a density θ(x) then it would satisfy the Monge-Ampere equation: (1.7) We will prove that, indeed, µ is supported on a graph and therefore if this graph is smooth the minimizing measure satisfies (1.7) as consequence of the change of coordinates formula. Furthermore, as we will see in the last section the constraint (1.7) should be regarded as the discrete version of which, together with the minimality condition, implies the invariance of the Mather measure in the continuous case.
2. Duality. As it was mentioned in the introduction, the minimization problem (1.3) with the constraints (1.1) and (1.2) is a linear programming problem in a space of Radon measures. The Fenchel-Rockafellar duality theorem [Roc66] is the natural tool to study such problems. In particular, it identifies the dual problem that, as in the continuous Aubry-Mather theory, contains important information concerning the original minimization. The setting is the following: let E be a Banach space with dual E and dual pairing denoted by (·, ·). Let h 1 (x) be a convex lower semicontinuous function in Similarly, if h 2 is a concave upper semicontinuous function in E assuming values in Theorem 2.1 (Rockafellar). Let h 1 and h 2 be as above. Then provided either h 1 or h 2 is continuous at some point where both functions are finite.
and observe that the dual of C 0 * is the space M of Radon measures µ in Ω with Ω |v|d|µ| < ∞.
and set We should observe that the functions φ ∈ C are discrete versions of exact one forms.
, as in the continuous case the integral along a closed curve of an exact one-form is zero. Define and M 1 = µ ∈ M : µ ≥ 0 and Ω dµ = 1 .
If µ is non-positive then we can choose a sequence of non-negative functions Letting n → ∞, and using monotone convergence theorem proves the lemma.

VISCOSITY SOLUTION METHODS AND DISCRETE AUBRY-MATHER PROBLEM 107
3. Solution to the discrete Hamilton-Jacobi equation. The previous section motivates the study of the equation: which should be seen as a discrete analog of the Hamilton-Jacobi equation: In this section we discuss the existence of solutions to (3.8), and, as we will see, the techniques used for the continuous case extend easily to this problem.
To construct a solution to (3.8) we are going to consider an approximate problem and then pass to the limit. To that effect, we define an operator T α acting on bounded periodic functions u by for α > 0. We look for fixed points u α of T α .
First observe that for any u and w bounded and periodic. Therefore it is a strict contraction, and so there is a unique fixed point. Without loss of generality we may assume L ≥ 0, by adding a constant to L. Therefore this fixed point is non-negative since T α maps non-negative functions into non-negative functions. Furthermore, we have some a-priori bounds for the fixed point: Lemma 3.1. T α u is uniformly (with respect to α) semiconcave and therefore, since it is periodic, Lipschitz. Proof.
Let v * such that Then The last remark in the statement follows from the fact that for periodic functions, semiconcavity implies Lipschitz continuity.
Note also that we have an a-priori bound on any fixed point: Remark. The assumption L(x, 0) ≥ 0 is not critical (and of course implied by the assumption L ≥ 0) but simplifies the bound. The other bound on the Lagrangian L is just a consequence of the compactness of T n . Proof. It suffices to observe that Finally, the main result is  u solves (3.8).
Proof. Since u α − min u α is uniformly Lipschitz, with respect to α, and consequently bounded by periodicity, it converges uniformly, possibly through some subsequence, to a limit u.
The uniform bounds on the fixed point imply that (1 − e −α ) min u α is bounded and its Lipschitz constant tends to zero. Therefore, through a subsequence, it converges to a constant which define to be H.
To check that (3.8) holds, it suffices to observe that therefore in the limit we have Proposition 3.4. The number H is unique.
Proof. By contradiction, assume that there is H 1 > H 2 and corresponding functions u 1 , u 2 that satisfy Then it is possible to find a sequence of points (x j , v j ) with x j+1 = x j + v j and 1 ≤ j ≤ n such that Then, since u 1 and u 2 are bounded we obtain Proof. We have To prove the reverse inequality, suppose that ψ is such that Then we would have for all x for all x and some > 0 sufficiently small. The difference ψ − u has an absolute maximum at some point x 0 . At this point we have

u is differentiable at x + v(x), and
Proof. The first statement is a consequence that u is bounded and To prove the second part, observe that for any x with equality for y = 0. Since the right hand side is differentiable in y, u has non-empty super-differential at any point (this also follows from the semiconcavity of u, but this proof is instructive).
which implies that at x + v(x) u has non-empty sub-differential. When a function has non-empty sub and super differentials it is differentiable. From (4.13), by differentiating in y we obtain (4.14) Finally observe that from (4.12), by differentiating in y, we get An easy consequence of this theorem is recorded in the next corollary. Let H, the Hamiltonian be defined by

Corollary 4.2 (Discrete Hamilton equations). Set
Then Proof. Observe that from (4.10) we have and (4.11) reads . Subtracting these two equations we obtain the first identity in (4.15), since The second line follows from the fact that v(x n ) = −D p H(x n , p n+1 ). 5. Regularity and the graph property. In this section, we study the regularity of the solution of the discrete Hamilton-Jacobi equation, and prove the graph property, that is, that a minimizing measure lies on a Lipschitz graph.
The first two propositions deal with the semiconcavity of the solution of the discrete Hamilton-Jacobi equation (4.9) (proposition 5.1), as well as the local semiconcavity along a minimizing trajectory (proposition 5.2). Then, following the ideas of [EG01], we prove that in the minimizing trajectory the solution of (4.9) is Lipschitz. The graph theorem, which characterizes the support of the minimizing measure is proven in theorem 5.5.
Proposition 5.1. Let u be any solution of (4.9). Then u is semiconcave, that is Proof. Let v * be the optimal velocity for x, that is Proposition 5.2. Let u be a solution of (4.9), x 0 an arbitrary point in T n , and v 0 = v(x 0 ) the optimal solution to (4.9). Then, at x = x 0 + v 0 , u is locally semiconvex, that is, Note that this proposition does not hold at all points x, otherwise u would be both semiconcave and semiconvex and thus differentiable everywhere. Now we borrow the ideas from [EG01], to show that along a minimizing trajectory, Du is Lipschitz.
Theorem 5.3. Let x 0 be any point in T n and v 0 = v(x 0 ) the optimal solution to (4.9). Let x = x 0 + v 0 . There exists a constant C such that Proof. For any h semiconcavity yields: and (5.17) Also, by local semiconvexity at x, Combining this with (5.17) we get which, together with (5.16), proves the first estimate. Let now z be such that |z| ≤ 2|h|. We have, . Furthermore, the semiconcavity of u implies Therefore by combining these estimates we have which implies the second part. Now we should observe that to a solution of (4.9) we can associate a dynamics by (4.15), and the initial conditions are determined by D x u(x 0 ), provided that at the initial point x 0 u is differentiable. To this dynamics corresponds a measure on T n × R n which is supported in the graph (x, v(x)). Furthermore u is differentiable in the support of this measure. Observe that has a unique solution if L is strictly convex in v and D x L has a small Lipschitz constant in v, these hypothesis are quite natural as we will see in the last section.
Theorem 5.4. Let µ be any minimizing measure and u any solution of (4.9). Then µ-almost everywhere Proof. By (4.9) we have for all (x, v), namely those in supp µ. Therefore it suffices to prove the reverse inequality. By contradiction, assume that there is > 0 such that

Recall that
which contradicts Theorem 3.5. The last part of the statement is simply the optimality condition from the previous section.
Theorem 5.5. If the equation has a unique differentiable solution v(x, p) for every p and x then any minimizing measure is supported on a Lipschitz graph.
Proof. By the previous theorem almost every v in the support of µ is optimal, and thus u is differentiable at x + v with φ(x + v)dµ < , defines v uniquely, and since D x u is Lipschitz at x then v is Lipschitz.
6. Asymptotic behavior. By replacing the Lagrangian L with for P ∈ R n we obtain a family of solutions u(x, P ) and a function H(P ) that satisfy Proof. It suffices to observe that H(P ) can be written as the supremum of a family of convex functions: The next theorem should be compared with the corresponding result for Hamilton-Jacobi equations [Gom02b].
Theorem 6.2. Let x n be an optimal trajectory for the problem corresponding to P . Then lim provided H is differentiable at P .
Proof. We have and, for any ∆ By subtraction we obtain thus sending n → ∞ and ∆ → 0 we obtain the result.
7. Convergence in the continuous limit. In this section we show that these discretizations are approximations to the continuous Mather problem. The Euler method for an ODEẋ = v(t) yields the discrete dynamics x n+1 = x n + hv n . To study the limit h → 0, it is convenient to consider a rescaled problem in which the constraint (1.2) is replaced by 1 h Ω w(x + vh) − w(x)dµ h = 0.
The corresponding discrete Hamilton-Jacobi is then There are several questions that we would like to address The first point, the convergence of H h can be addressed in two parts -it is clear that the sequence H h is uniformly bounded in h, therefore through some subsequence it converges to a limit. To see that this number is indeed H 0 we need to address the second point. The main problem in dealing with these convergence issues is that the proof of proposition 5.1 for the semiconcavity constant for a solution of (7.18) is O( 1 h ), therefore we need a slightly improved proof: Proposition 7.1. Let u be a solution of (7.18). Then u is semiconcave, that is in which the constant is independent of h.
Proof. To simplify the proof, suppose h = 1/n for some integer n > 0. Let Then u(x) = u(w) + 1