Generalized Mather problem and homogenization of Hamilton-Jacobi equations

GENERALIZED MATHER PROBLEM AND HOMOGENIZATION OF HAMILTON-JACOBI EQUATIONS DIOGO A. GOMES & ENRICO VALDINOCI Abstract. We present a general approach to the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting through infinite dimensional linear programming and generalized Aubry-Mather theory.

Contents 1. Introduction 2. The main result 3. List of assumptions 4. Optimal control and linear programming 5. Moment estimates 6. Duality 7. Stationarity and Effective Lagrangian 8. Homogenized problems 9. Uniform estimates 10. Convergence 11. Improved Moment Estimates 12. Viscosity Solutions 13. Conclusion References

1 2 3 9 14 16 24 30 32 36 39 46 50 50

DG was supported in part by CAMGSD - FCT/POCTI/FEDER, POCI/FEDER/MAT/55745/2004 and EV was supported in part by MIUR Variational Methods and Nonlinear Differential Equations 1. Introduction The objective of this paper is to present a general approach to the homogenization of random stationary ergodic Hamilton-Jacobi equations arising in deterministic and stochastic optimal control. In our approach we will use infinite dimensional linear programming and the generalized Aubry-Mather theory. 1

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DIOGO A. GOMES & ENRICO VALDINOCI

Our setting is the following. Let Mat(n×n) be the space of (n×n) real matrices and Ω be a probability space. We consider a Hamiltonian H(M, p, x, X, ǫ, ω) : Mat(n × n) × Rn × Rn × Rn × R+ × Ω → R. We will study the homogenization properties of the Hamilton-Jacobi equation     − ∂ uǫ + H D 2 uǫ , D uǫ , x, x , ǫ, ω = 0 t x x ǫ (1)  uǫ (x, T, ω) = ψ(x) ,

as ǫ → 0. Here above, uǫ (x, t, ω) : Rn × [0, T ] × Ω −→ R and ǫ is a small, positive parameter and ψ(x, X, ω) : Rn × Rn × Ω → R is assumed to be a bounded smooth function of the variables (x, X), and to satisfy further additional stationarity hypotheses in ω. We will prove that, under suitable assumptions, there exists a non-random function u¯(x, t) such that uǫ → u¯. Furthermore, there is a function H(M, p, x), the so-called Effective Hamiltonian, non-random, such that u¯ is a viscosity solution of (
− ∂t u¯ + H Dx2 u¯, Dx u¯, x = 0 (2) u¯(x, T ) = ψ(x) .

Our strategy to study this homogenization problem is the following: first we establish some general properties of the original problem by converting it into a linear programming problem. Then, we introduce a homogenized problem. Finally, we prove the convergence of the sequence uǫ to a non-random limit and show that this limit agrees with the homogenized problem. To accomplish these goals, we will take a series of assumptions that will be put forward in section 3. Homogenization problems have been quite popular in recent years, and many steps have been made towards the understanding of these problems (see, e.g., [BLP76, LPV86, Eva92, Con96, Sou99, Alv99, Ish99, Ish00, RT00, CDI01, EG01, AB02, LS03, Gom03, LS05, KRV06, Kos06]). Several author, see [KS, CS02, MS98, GR06] have used linear programming techniques to study deterministic and stochastic optimal control problems. We should point out that there were some known connections between homogenization theory and Aubry-Mather theory (see, for instance, [LS03]), and that convex analysis and duality methods were used in [KRV06] to prove a homogenization result related to ours. However, to our knowledge, this is first paper in which this relation is worked out systematically, specially in what concerns second-order equations. 2. The main result We now state our main result. The assumptions under which it holds are very general and, for the convenience of the reader, their detailed statement will be postponed to section 3. Theorem 1. Let Assumptions 1–15 of section 3 hold and let uǫ be a viscosity solution of (1). Then:

GENERALIZED MATHER PROBLEM AND HOMOGENIZATION

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• The solution uǫ may be represented as Z ǫ u (x, t, ω) := inf L(x′ , X, v, ω) dµǫ(x′ , X, v, s; ω) Rn ×Rn ×U ×[t,T ] Z + ψ(x′ ) dνTǫ (x′ , X; ω) , Rn ×Rn

where the infimum is taken over all measures (µǫ , νTǫ ) which, for every ω ∈ Ω, satisfy: µǫ is a measure on Rn × Rn × U × [t, T ] with total mass T − t, νTǫ is a probability measure on Rn × Rn , and µǫ and νTǫ are such that Z ˜ ′ , X, s) dµǫ(x′ , X, v, s; ω) ∂t φ(x′ , X, s) + Aφ(x Rn ×Rn ×U ×[t,T ] Z Z ′ ǫ ′ = φ (x , X, T ) dνT (x , X; ω) − φ (x′ , X, t) dνtǫ (x′ , X) , Rn ×Rn

Rn ×Rn

for any φ ∈ C 2 (Rn × Rn × [t, T ]). • The norms kuǫt k∞ and kDuǫk∞ are bounded uniformly in ǫ. • Up to subsequences, uǫ (x, t, ω) converges to a suitable u¯(x, t) uniformly in K × [0, T ] × Ω, for any compact set K ⊂ Rn . The function u¯ is a viscosity solution of (2), for a suitable H.

During the proof of Theorem 1 several results of independent interest will show up. For instance: a “dual representation” of uǫ will be provided (see Proposition 6), a way to reconstruct uǫ from subsolutions is found (see (52) below), convergence issues of minimizing measures are discussed (see Proposition 11), a measure theoretic construction of u¯ is obtained (see (72) or (73), according to the proper scaling), and an Effective Lagrangian is constructed (see (59) below). Moreover, under an additional (still quite general) hypothesis (namely, Assumption 16 on page 43), the Effective Hamiltonian may be reconstructed from the Effective Lagrangian by duality (see Proposition 33). We now list the hypotheses needed for Theorem 1 to hold, that is Assumptions 1–15, collected in the next section (the additional hypothesis in Assumption 16 on page 43 is not needed for Theorem 1, but only for Proposition 33). 3. List of assumptions We now list the assumptions needed for our purposes. Some of these assumptions will be slightly redundant, but we prefer to list similar assumptions in a separate way any time they are needed in different parts of the paper, since some of the estimates that will be obtained may be of general interest. Our first assumption concerns the structure of the problem, and will be assumed without explicit mention throughout the paper. Assumption 1. We suppose that the Hamilton-Jacobi equation (1) originates from a stochastic control problem. The dynamics of this control problem is  x   x  (3) dx = f x, , v, ω dt + σ x, , ǫ, v, ω dWt , ǫ ǫ

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DIOGO A. GOMES & ENRICO VALDINOCI

where v is a bounded progressively measurable control, lying in a closed convex set U ⊆ Rm , which, without loss of generality, will be assumed to contain the origin 0, f : Rn × Rn × U × Ω → Rn is the drift coefficient, σ : Rn × Rn × R+ × U × Ω → Mat(n × p) is the diffusion coefficient, and Wt is an p-dimensional Brownian motion defined in a ˜ which is independent of Ω. The probability on Ω ˜ will be denoted probability space Ω by P , and E will denote the corresponding expected value. We assume enough regularity and growth conditions in f and σ such that the stochastic differential equation (3) is well defined for all bounded progressively measurable controls v. The running cost of this control problem is given by the Lagrangian L(x, X, v, ω) : Rn × Rn × U × Ω → R, where U ⊆ Rm is the control space, which is taken to be closed and convex. We suppose also that L is continuous in (x, X, v) and bounded from below. Without loss of generality, we will assume that inf L = 0. Furthermore, if U is unbounded we suppose that (4)

lim

R→+∞

inf

(x,X,ω) |v|≥R

L(x, X, v, ω) = +∞ . |v|

For simplicity, we assume that the terminal cost ψ is a bounded C ∞ function with bounded derivatives (less regular data may be treated as well). We suppose that, for T > 0, equation (1) possesses, for any fixed ω ∈ Ω, one and only one viscosity solution uǫ defined for all t ∈ [0, T ], which is the value function of the stochastic control problem, that is, it can be represented in the form (5)

uǫ (x, t, ω) = Z T    x(s) inf E L x(s), , v(s), ω ds + ψ (x(T )) , ǫ t

where the infimum is taken over all trajectories of the controlled dynamics (3) with initial condition x(t) = x. ◭ We refer to [FR75, Kry80, FS06, Eva06a] for further discussions about the stochastic version of Hamilton-Jacobi equations. The Hamiltonian H(M, p, z, x, X, ǫ, ω) : Mat(n × n) × Rn × Rn × Rn × R × Ω → R is related with the Lagrangian by duality using the generalized Legendre transform H(M, p, x, X, ǫ, ω) (6)

= sup −f (x, X, v, ω) · p − v∈U

σσ T (x, X, ǫ, v, ω) : M − L(x, X, v, ω) . 2

GENERALIZED MATHER PROBLEM AND HOMOGENIZATION

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Here and in the sequel, we use the following notation: given A, B ∈ Mat(n × n), we set A : B = tr AT B, if A and B are symmetric, and we will use the convention that for general matrices A : B = 14 tr (AT + A)(B + B T ). Note that (6) implies that H is decreasing in M, that is, if N ∈ Mat(n × n) is symmetric and non-negatively definite, then (7)

H(M + N, p, x, X, ǫ, ω) ≤ H(M, p, x, X, ǫ, ω) .

Assumption 2. We suppose that   x (8) sup sup sup H D 2 Ψ(x), DΨ(x), x − y, , ǫ, ω < ∞ , ǫ y ǫ>0 x,ω

for any Ψ ∈ C 2 (Rn ).

◭

We now suppose that the probability space Ω has nice slicing properties with respect to products with real spaces: ′

Assumption 3. If µ(r, ω, r ′) is a finite measure on Rd × Ω × Rd , there exist a finite ′ ′ measure Θ(r ′) on Rd and, for Θ-almost any r ′ ∈ Rd , a finite measure ζ(r, ω; r ′) on Rd × Ω in such a way that µ(r, ω, r ′) = ζ(r, ω; r ′)Θ(r ′ ) . Analogously, if µ(r, ω) is a finite measure on Rd × Ω, there exist a finite measure Θ(ω) on Ω and, for Θ-almost any ω ∈ Ω, a finite measure ζ(r; ω) on Rd in such a way that µ(r, ω) = ζ(r; ω)Θ(ω) . ◭ Assumption 3 is fulfilled, for instance, if Ω = Rm : see Theorem 10 on page 14 of [Eva90]. The next two assumptions will be used in section 4 and will be assumed from that point on. Assumption 4. Suppose that the Lagrangian L, the drift f and the diffusion coefficient σ satisfy   |σσ T (x′ , X, ǫ, v, ω)|p ′ |f (x′ , X, v, ω)|p + ≤ C 1 + L(x , X, v, ω) (1 + |x′ |2 )p/2

for any (x′ , X, v, ω) ∈ Rn × Rn × U × Ω and any ǫ ∈ R \ {0}, for suitable C > 0 and p > 1. ◭ Assumption 5. We assume that L(x, X, 0, ω) is uniformly bounded.

◭

As a notational remark, we point out that, in most of this paper, we think x to be the “frozen” initial datum of the controlled dynamics (and so the “frozen” variable of uǫ ), while x′ will denote the variable of the integration of the measure we will construct. The next assumption is quite natural and requires that the viscosity solutions can be approximated by smooth subsolutions. This assumption will be required from section 6 onwards.

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DIOGO A. GOMES & ENRICO VALDINOCI

Assumption 6. For any δ > 0 there exists vδǫ ∈ C 2 (Rn × [t, T ] × Ω) satisfying  x  ǫ 2 ǫ ǫ (9) ∂t vδ (x, t, ω) − H Dx vδ (x, t, ω), Dx vδ (x, t, ω), , ω + δ ≥ 0 , ǫ ǫ vδ (x, T, ω) = ψ(x), and lim vδǫ = uǫ

(10)

δ→0+

uniformly in (x, t, ω) ∈ Rn × [t, T ] × Ω. We further suppose that D 2 vδǫ is uniformly bounded in ǫ for fixed δ. ◭ In many applications, the smoothing process of section V.7 in [FS06] may be used to built semiconcave/convex sub/supersolutions. To obtain a non-random Effective Hamiltonian, one must impose certain ergodicity conditions on the data of the problem. So far, we have made no assumptions on the behavior of L, f or σ with respect to the random parameter ω. It turns out that a convenient hypothesis is stationarity, as we put forward next. For the purposes of section 7, the next assumption is enough, but later we will also require certain ergodicity hypothesis to obtain a non-random homogenized limit, see Assumption 13 below. Assumption 7. We assume that there is an action τ of Rn in the probability space Ω, denoted by τX ω, for X ∈ Rn , which preserves the measure in Ω. Furthermore, we suppose that the L, f and σ are stationary with respect to this action, that is, for all Y ∈ Rn (11)

L(x, X + Y, v, ω) = L(x, X, v, τY ω),

(12)

f (x, X + Y, v, ω) = f (x, X, v, τY ω),

and (13)

σ(x, X + Y, ǫ, v, ω) = σ(x, X, ǫ, v, τY ω). ◭

As an example, note that the periodic setting can be embedded in this framework in the following way: we let Ω = Tn , and consider in Ω the (normalized) Lebesgue measure. ˆ X, v) and σˆ (x, X, v) are periodic in X, we can write ˆ X, v), f(x, Then, if L(x, ˆ X + ω, v), L(x, X, v, ω) = L(x, with similar definitions for f and σ, and then we choose τY ω = ω + Y . Next, we state the concept of tight sequence of measures, by generalizing a standard definition in probability theory (see, e.g., page 336 of [Bil95]): Definition 1. A net of probability measures µα on a given space Z × Ω, with Z ⊂ Rd is said to be tight if for any ǫ > 0 there exists a compact set Kǫ in such a way that   lim sup µα (Z \ Kǫ ) × Ω ≤ ǫ . α

The following assumption is simply a compactness property on measures on Ω that is necessary to extract weak limits.

GENERALIZED MATHER PROBLEM AND HOMOGENIZATION

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Assumption 8. We assume that for each tight net of probability measures ηα (z, ω) on Z × Ω, with Z ⊂ Rd , there exists a subnet which converges to a probability measure η(z, ω), in the sense that: if ψ(X, z, ω) : Rd ×Z ×Ω → R is continuous in (X, z) and stationary (that is, ψ(X +Y, z, ω) = ψ(X, z, τY ω) for any (X, Y, z, ω) ∈ Rd ×Rd ×Z ×Ω), we have that Z Z lim ψ(z, 0, ω) dηα(z, ω) = ψ(z, 0, ω) dη(z, ω) , α

Z×Ω

Z×Ω

up to subnets.

◭

In section 7 we will use the following scaling hypothesis for the diffusion coefficient σ Assumption 9. We will assume that the viscosity coefficient σ is either A. σ = ǫσ0 B. σ = ǫ1/2 σ0 C. σ = σ0 , with σ0 = σ0 (x′ , X, v, ω) : Rn × Rn × U × Ω → Mat(n × m).

◭

We should remark that these are the relevant scalings, as many others could be studied with similar methods. From section 9 on, we will assume that our viscosity solutions are smooth, up to a small uniformly elliptic correction of the Hamiltonian. That is, we suppose: Assumption 10. For any ℓ ∈ N, there exist uǫℓ ∈ C 2 (Rn × (0, T )), ψℓ ∈ C 2 (Rn ) and a Hamiltonian Hℓ such that uǫℓ is a solution of the following Hamilton-Jacobi equation:     − ∂ uǫ + H D 2 uǫ , D uǫ , x, x , ǫ, ω = 0 t ℓ ℓ x ℓ x ℓ ǫ  uǫ (x, T, ω) = ψ (x) , ℓ

ℓ

and in such a way that

lim uǫℓ (x, t) = uǫ (x, t)

ℓ→+∞

lim ψℓ (x) = ψ(x) ,   x and lim sup Hℓ Dx2 Ψ(x), Dx Ψ(x), x, , ǫ, ω ℓ→+∞ x ǫ   x −H Dx2 Ψ(x), Dx Ψ(x), x, , ǫ, ω = 0 ǫ n 2 n for any x ∈ R , t ∈ [0, T ] and Ψ ∈ C (R ) with bounded C 2 -norm. Also, H ℓ satisfies ℓ→+∞

(14)

− DMij Hℓ ξi ξj ≥ cℓ |ξ|2 ,

for some cℓ > 0. We also assume that (15)

sup x∈Rn , t∈[0,T ]

|Duǫℓ | < +∞ . ◭

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DIOGO A. GOMES & ENRICO VALDINOCI

A natural possibility for such Hamiltonian Hℓ is of course H − tr M/ℓ, but other regularizations are possible (for general regularization results, see, e.g., [Eva82, Eva97, Wan90, Wan92a, Wan92b, Wan92c, CC95, LS05]). We remark that (15) is not assumed, a-priori, to be uniform in ǫ and ℓ (in fact, uniform estimates will be proved in Proposition 20 below). In many applications, bounds as in (15) are quite natural and may be obtained by comparing trajectories (see, e.g., page 555 of [Eva98] and Lemma 8.1 of [FS06]). The next three assumptions will be used in section 9 to prove various uniform estimates in ǫ. Assumption 11. We assume that H enjoys one of the following properties: P1) H(M, p, x, X, ǫ, ω) does not depend on M and satisfies the coercivity hypothesis (16)

lim H(p, x, X, ǫ, ω) = +∞,

|p|→∞

uniformly in (x, X, ǫ, ω) ∈ Rn × Rn × R+ × Ω. P2) There exists a function F (p, ζ) such that for all (M, p, x, X, ǫ, ω) ∈ Mat(n × n) × Rn × Rn × Rn × R+ × Ω and ζ ∈ R such that ζ + H(M, p, x, X, ǫ, ω) = 0 implies 1  1  1 − DMjk H M, p, x, X, ǫ, ω Mij Mik + ǫpi Dxi H M, p, x, X, ǫ, ω ǫ ǫ ǫ 1  +pi DXi H M, p, x, X, ǫ, ω ≥ F (p, ζ), ǫ and, furthermore, if K is a fixed compact set of Rn , (17)

inf F (p, ζ) → +∞,

ζ∈K

as |p| → +∞. ◭ Examples of Hamiltonians satisfying P2 are easily obtained, for instance, by considering the case σ := Idn , f := v, L := |v|2/2 − U(x, X, ω). In such a case, (6) yields that 1 1 (18) H = − tr M + |p|2 + U(x, X, ω) , 2 2 which is plainly seen to satisfy P2. The next assumption will be used in section 9 to prove that the limit is non-random. Assumption 12. We suppose that there exists a constant C independent of ǫ such that for any (x, z) ∈ Rn × Rn we have   x 2 ǫ ǫ Dx H D u (z, t), Du (z, t), z, , ǫ, ω ≤ C. ǫ ◭ Note that (18) provides an example of an Hamiltonian satisfying Assumption 12, due to Proposition 20 below.

GENERALIZED MATHER PROBLEM AND HOMOGENIZATION

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Assumption 13. We assume the following uniform ergodicity hypothesis: for any δ > 0 there exists M > 0, such that for any ω0 , ω1 ∈ Ω, x ∈ Rn , t ∈ [0, T ], there exists y ∈ Rn , |y| ≤ M, such that   x H D 2 uǫ (x, t, ω1 ), Duǫ(x, t, ω1 ), x, , ǫ, τy ω0 ǫ   x 2 ǫ ǫ −H D u (x, t, ω1 ), Du (x, t, ω1 ), x, , ǫ, ω1 ≤ δ. ǫ ◭

Heuristically, Assumption 13 means we are be able to get close to any point of Ω from any other by translating, the size of such translation depending only on how close we want to get (this is the case, for instance, of the geodesics with irrational slope on the flat torus). From section 11 on, we will also suppose the following: Assumption 14. We assume that there are constants C¯0 , C¯1 and C¯2 such that |L(x′ , X, 0, ω)| ≤ C¯0 , |f (x′ , X, 0, ω)| ≤ C¯1 , |σ(x′ , X, 0, ω)| ≤ C¯2 .

for any (x′ , X, ω) ∈ Rn × Rn × Ω.

◭

Assumption 15. Let p > 1 be as in Assumption 4. We suppose that |f (x′ , X, v, ω)|2 ≤ C(1 + L(x′ , X, v, ω)) ,

|σ(x′ , X, ǫ, v, ω)| ≤ C

and |σσ T (x′ , X, ǫ, v, ω)|p ≤ C(1 + L(x′ , X, v, ω)) , for any (x′ , X, v, ω) ∈ Rn × Rn × U × Ω.

◭

We now deal with the proof of Theorem 1. 4. Optimal control and linear programming In this section we are going to convert the original optimal control problem (1) into a linear programming problem. As the random parameter ω plays no role in most of this discussion we omit it until section 7. The controlled stochastic dynamics (3) has an infinitesimal generator (see, e.g., page 105 in [Eva06b]), given by σσ T : Dx2 ϕ. 2 To encode both the behavior at the slow scale x and fast scale X = ˜ given by introduce the extended generator A, (19)

Aϕ(x) = f · Dx ϕ +

x , ǫ

σσ T ˜ Aφ(x, X) = f · Dx φ + : Dx2 φ 2 1 1 σσ T 1 T 2 2 : DX φ. + f · DX φ + (σσ ) : DxX φ + 2 ǫ ǫ ǫ 2

we need to

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DIOGO A. GOMES & ENRICO VALDINOCI

This extended generator satisfies the following identity   x   x ˜ = A φ x, . (Aφ) x, ǫ ǫ Observe further that   x  ˜ (20) A ϕ X− = 0, ǫ

for all C 2 functions ϕ(x) : Rn → R. Before proceeding, we need some further definitions. If U is bounded, set γ = 1, otherwise, let γ : U → R+ be a function which satisfies lim inf

|v|→∞ x,X

L(x, X, v, ω) = +∞, γ(v)

and A˜v φ(x, X) = 0, |v|→∞ γ(v) lim

for all φ ∈ C 2 (Rn ×Rn ). Let p be the exponent for which Assumption 4 holds. Consider the spaces P and Q of signed1 measures, resp. µ on Rn × Rn × U × [t, T ] and ν on Rn × Rn , which satisfy Z Z p p γ(v) + |x| + ǫ|X| d|µ| < ∞, |x|p + ǫ|X|p d|ν| < ∞. We remark that P is the dual of the set C∗ (Rn ×Rn ×U ×[t, T ]) of continuous functions χ : Rn × Rn × U × [t, T ] → R, which satisfy lim

sup

|x|+|X|+|v|→∞ t′

χ(x, X, v, t′ ) = 0, γ(v) + |x|p + ǫ|X|p

and Q is the dual of the set C♯ (Rn × Rn ) of continuous functions on η : Rn × Rn → R, which satisfy η(x, X) = 0. lim |x|+|X|→∞ |x|p + ǫ|X|p We further consider the space C⋆ (Rn × Rn × [t, T ]) of continuous functions on φ : Rn × Rn → R, which satisfy lim

sup

|x|+|X|→∞ t′

φ(x, X, t′ ) = 0. |x|p + ǫ|X|p

In this section, we relax the original stochastic control problem by looking at generalized trajectories represented by certain measures, which will be elements of P and Q. To do this, we must consider measures that satisfy a generalized holonomy 1All

the measures will be tacitly assumed to be Radon. If not differently specified, they are also assumed to be positive.

GENERALIZED MATHER PROBLEM AND HOMOGENIZATION

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constraint. More precisely, let νtǫ be a given probability measure. We look for measures (µǫ (x′ , X, v, s), νTǫ (x′ , X)) ∈ P × Q which satisfy Z ˜ ′ , X, s) dµǫ(x′ , X, v, s; ω) (21) ∂t φ(x′ , X, s) + Aφ(x Rn ×Rn ×U ×[t,T ] Z Z ′ ǫ ′ = φ (x , X, T ) dνT (x , X) − φ (x′ , X, t) dνtǫ (x′ , X) , Rn ×Rn

Rn ×Rn

˜ ′ , X, s) ∈ C∗ (Rn × Rn × for any φ ∈ C 2 (Rn × Rn × [t, T ]) such that ∂t φ(x′ , X, s) + Aφ(x ′ ′ n n U × [t, T ]) and φ (x , X, t) , φ (x , X, T ) ∈ C♯ (R × R ). The measure νtǫ encodes the given initial condition x(t) = x, allowing for probability distributions of initial conditions, whereas the measure νTǫ , which is arbitrary, encodes the possible probability distribution of terminal conditions. We will always choose νtǫ = δx (x′ )δ xǫ (X)

(22)

and we will omit the dependence of νtǫ and νTǫ upon the prescribed initial point x. The motivation for this holonomy property is the following: let us consider a trajectory (x(·), v(·)) of the controlled dynamics (3); to this trajectory, one can associate measures µ and νT in the following way Z φ(x′ , X, v, s) dµǫ(x′ , X, v, s; ω)  (23) Z T  x(s) , v(s), s ds , := E φ x(s), ǫ t and (24)

Z

′

φ(x

, X) dνTǫ (x′ , X)

  x(T ) := Eφ x(T ), . ǫ

By Dynkin’s Formula (see, e.g., formula (3) on page 105 of [Eva06b]), the constraint in (21) is satisfied by these measures. Of course, by considering measures instead of trajectories, we are, in principle, enlarging the minimizing set. However, as we will see later, the value of the infimum is unchanged (see Propositions 4 and 9 below for further details on this). In the next proposition, we show that a measure µ satisfying (21) has always total mass T − t. Proposition 2. Suppose that µ is a measure on Rn ×Rn ×U ×[t, T ] which satisfies (21), for a suitable probability measure νTǫ on Rn × Rn . Then Z Z T (25) υ(s)dµ = υ(s)ds t

for any υ ∈ C([t, T ]). Also, (26)

µ(Rn × Rn × U × [t, T ]) = T − t.

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Proof. Choose Υ(t) such that υ(t) = Υ′ (t). Then, from (21), we have that Z Z Z Z T ǫ ǫ υ(s)dµ = Υ(T )dνT − Υ(t)dνt = Υ(T ) − Υ(t) = υ(s)ds, t

yielding (25). Then (26) follows by applying (25) to υ := 1. We now prove that both νTǫ and µǫ are supported on {X =

 x′ }, ǫ

as long as so is νtǫ .

′

Proposition 3. Suppose νtǫ is supported in the set X = xǫ , and that (µǫ , νTǫ ) sat′ isfy (21). Then both νTǫ and µǫ are supported in the set X = xǫ . 2 ′ ′ Proof. Let κ : Rn → R+ 0
be a C function such that κ(0) = 0 and κ(x ) > 0 if x 6= 0. ′ Assume that κ X − xǫ ∈ C♯ (Rn × Rn ). To prove the desired claim, we will show that if  Z  x′ κ X− dνtǫ (x′ , X) = 0, ǫ

then

Z

and

x′ κ X− ǫ 

Z

Set φ(x′ , X) := κ X − (27)

x′ ǫ

  x′ dνTǫ (x′ , X) = 0, κ X− ǫ






dµǫ (x′ , X, v, s; ω) = 0.

. Since, by (20), ˜ = 0, Aφ

the constraint in (21) implies that   Z  Z  x′ x′ ǫ dνT = κ X − dνtǫ = 0. (28) κ X− ǫ ǫ
′ Now choose φ(x′ , X, t) := tκ X − xǫ . Then, using (27), (28) and (21) once more, we gather that  Z  Z x′ ǫ ˜ dµǫ = 0 κ X− dµ = ∂t φ + Aφ ǫ as we wished.



Proposition 4. For t ∈ (0, T ), let Z ǫ u˜ (x, t, ω) := inf L(x′ , X, v, ω) dµǫ(x′ , X, v, s; ω) n n R ×R ×U ×[t,T ] Z (29) + ψ(x′ ) dνTǫ (x′ , X; ω) , Rn ×Rn

GENERALIZED MATHER PROBLEM AND HOMOGENIZATION

13

where the infimum is taken over all measures (µǫ , νTǫ ) which, for every ω ∈ Ω, satisfy the following: µǫ is a measure on Rn × Rn × U × [t, T ] with total mass T − t, νTǫ is a probability measure on Rn × Rn , and µǫ and νTǫ satisfy the following constraint: Z ˜ ′ , X, s) dµǫ(x′ , X, v, s; ω) ∂t φ(x′ , X, s) + Aφ(x Rn ×Rn ×U ×[t,T ] Z Z (30) ′ ǫ ′ = φ (x , X, T ) dνT (x , X; ω) − φ (x′ , X, t) dνtǫ (x′ , X) , Rn ×Rn

Rn ×Rn

for any φ ∈ C 2 (Rn × Rn × [t, T ]). Then, uǫ ≥ u˜ǫ .

Proof. Let uǫ be as in (5) and u˜ǫ given by (29). Fix δ > 0, and consider a bounded progressively measurable control vδ , and the corresponding trajectory xδ of (3) with the initial condition xδ (t) = x, such that uǫ (x, t, ω) + δ   Z T  xδ (s) , vδ (s), ω ds + ψ (xδ (T )) . ≥E L xδ (s), ǫ t

(31)

We introduce the probability measures µδ and νδ on Rn ×Rn ×U ×[t, T ] and on Rn ×Rn , respectively, defined by Z φ(x′ , X, v, s) dµδ (x′ , X, v, s) Rn ×Rn ×U ×[t,T ] Z T

:= E

t

and

  xδ (s) φ xδ (s), , vδ (s), s ds ǫ

  xδ (T ) , ϕ(x , X) dνδ (x , X) := Eϕ xδ (T ), ǫ Rn ×Rn for any φ ∈ C(Rn × Rn × U × [t, T ]) and ϕ ∈ C(Rn × Rn ). Then, if φ ∈ C 2 (Rn × Rn × [t, T ]), by Dynkin’s Formula (see, e.g., formula (3) on page 105 of [Eva06b]), Z ˜ ∂t φ + Aφdµ δ n n R ×R ×U ×[t,T ]      xδ (t) xδ (T ) , T − φ xδ (t), ,t = E φ xδ (T ), ǫ ǫ Z Z ′ ′ = φ(x , X, T )dνδ (x , X) − φ(x′ , X, t)dνtǫ (x′ , X), Z

′

Rn ×Rn

where

νtǫ

′

Rn ×Rn

is taken as

Z

 x  φ(x′ , X, t) dνtǫ (x′ , X) := φ x, , t , ǫ

which shows that µδ and νδ satisfy the constraint in (30). Accordingly, (29) and (31) give that uǫ + δ ≥ u˜ǫ , and so the claim follows by sending δ to zero. 

14

DIOGO A. GOMES & ENRICO VALDINOCI

5. Moment estimates In this section, we discuss some a-priori moment estimates for minimizing measures (µǫ , νTǫ ), as given in (29), that will be essential to establish the tightness necessary to our scopes. More refined estimates, needed to prove that the homogenized limit is a viscosity solution of a Hamilton-Jacobi equation, will be developed in section 11. From this section on, we will assume Assumptions 4 and 5. By comparison with the control v := 0 in (3), recalling Assumption 5, we see that Z (32) L(x′ , X, v, ω)dµǫ ≤ C , with C > 0 independent of ǫ. Proposition 5. Let (µǫ , νTǫ ) be measures satisfying (30), with νtǫ given by (22). Let L satisfy (32). Let p be as in Assumption 4. Then, there exists C > 0, possibly depending on t and T , but independent of ǫ, in such a way that Z Z ′ p ǫ |x | dµ + |x′ |p dνTǫ ≤ C (1 + |x|p ) . Proof. In what follows, we will denote by Ci suitable positive quantities, which may depend on t, T , but that are independent of ǫ. We observe that |x′ |p ≤ (1 + |x′ |2 )p/2 ≤ 2p/2 (1 + |x′ |p ) .

(33)

We apply the constraint in (30) to the function φ(x′ ) := (1 + |x′ |2 )p/2 and we obtain that Z p(1 + |x′ |2 )(p−2)/2 f · x′ n   1X T ′ 2 (p−4)/2 ′ ′ ′ 2 (p−2)/2 + (σσ )ij p(p − 2)(1 + |x | ) xi xj + p(1 + |x | ) δij dµǫ 2 i,j=1 Z Z ′ 2 p/2 ǫ = (1 + |x | ) dνT − (1 + |x′ |2 )p/2 dνtǫ ,

and so, by (33), Z ≤ C2

′ p

|x | Z

T

dνTǫ

≤

Z

(1 + |x′ |2 )p/2 dνTǫ

|f | (1 + |x′ |2 )(p−1)/2 ′ 2 (p−2)/2

+|σσ | (1 + |x | )

ǫ

dµ +

Z

 (1 + |x′ |2 )p/2 dνtǫ .

GENERALIZED MATHER PROBLEM AND HOMOGENIZATION

15

Consequently, Young’s inequality, Assumption 4, estimates (32) and (33), and Proposition 2 give that Z |x′ |p dνTǫ  Z Z |σσ T |p ′ 2 p/2 ǫ ′ 2 p/2 ǫ p + (1 + |x | ) dµ + (1 + |x | ) dνt ≤ C3 |f | + (1 + |x′ |2 )p/2  Z Z (34) ′ p ǫ ′ p ǫ 1 + L + |x | dµ + 1 + |x | dνt ≤ C4   Z Z ′ p ǫ ′ p ǫ ≤ C5 1 + |x | dµ + |x | dνt .

To estimate this last term, we apply again (21) to the function φ(x′ , s) := (s − T )(1 + |x′ |2 )p/2 , to obtain Z h (1 + |x′ |2 )p/2 + (s − T ) p(1 + |x′ |2 )(p−2)/2 f · x′ n  i 1X T ′ 2 (p−4)/2 ′ ′ ′ 2 (p−2)/2 + (σσ )ij p(p − 2)(1 + |x | ) xi xj + p(1 + |x | ) δij dµǫ 2 i,j=1 Z = (T − t) (1 + |x′ |2 )p/2 dνtǫ .

Thus, using (33) and a scaled Young’s inequality, we obtain that Z (1 + |x′ |2 )p/2 dµǫ   Z Z |σσ T |p ǫ ′ p ǫ p dµ ≤ C6 1 + |x | dνt + |f | + (1 + |x′ |2 )p/2 Z 1 + (1 + |x′ |2 )p/2 dµǫ . 2 Thus, exploiting Assumption 4, estimate (32) and Proposition 2, we conclude that Z Z 1 1 ′ p ǫ |x | dµ ≤ (1 + |x′ |2 )p/2 dµǫ 2 2  Z Z ′ p ǫ 1 + Ldµ + |x | dνt ≤ C7   Z ′ p ǫ ≤ C8 1 + |x | dνt . This, (34) and the definition of νtǫ , see (22), yield the desired result.



As a consequence of this result, using (22), we have, using Proposition 3, that any ǫ minimizing sequence (µǫn , νT,n ) will satisfy the bound Z Z ′ p p p ǫ ǫ (35) |x | + ǫ |X| dµn ≤ C, |x′ |p + ǫp |X|p dνT,n ≤ C.

16

DIOGO A. GOMES & ENRICO VALDINOCI

The above tightness and Assumption 8 imply that there are minimizing measures µǫ and νTǫ attaining the infimum in (29).

6. Duality In addition to the Hamiltonian H as defined in (6), we need also the extended Hamiltonian operator (36)

˜ 2 φ(x, X), Dφ(x, X), x, X, ǫ, ω) := sup −Aφ ˜ − L, H(D v∈U

which is defined for φ ∈ C 2 (Rn × Rn ). It will be useful later the observation that ˜ 2 φλ (x, X), Dφλ(x, X), x, X, ǫ, ω), H(D with x φλ = φ(x, X) + λϕ(X − ), ǫ

(37)

is constant in λ, for any C 2 function ϕ, thanks to (20). Recalling the framework introduced on page 10, we define the set CΛ2 to be the set of functions φ ∈ C 2 (Rn × Rn × [t, T ]) such that φ(x′ , X, t), φ(x′ , X, T ) ∈ C♯ (Rn × Rn ), and ∂t φ + A˜v φ ∈ C∗ (Rn × Rn × U × [t, T ]). In order to prove the reverse inequality of the one in Proposition 4, we introduce the dual problem of (29): Proposition 6. In the notation of Proposition 4, we have that (38)

u˜ǫ (x, t, ω) = sup (T − t) φ(x′ ,X,s) ′



inf

(x′ ,X,s)∈Rn ×Rn ×(t,T )

˜ D 2 φ, Dφ, x′, X, ǫ, ω ∂t φ − H

′

+ inf [ψ(x , X, ω) − φ(x , X, T )] + ′ x ,X

Z

Rn ×Rn




φ(x′ , X, t) dνtǫ (x′ , X).

where the supremum is taken over any φ ∈ CΛ2 . To heuristically motivate the statement in (38), we apply the minimax principle, that is, we exchange formally the infimum with the supremum in the variational problem (29),

GENERALIZED MATHER PROBLEM AND HOMOGENIZATION

17

by using the constraint in (30), thus obtaining: u˜ǫ (x, t, ω) =

sup

inf

µǫ ,νTǫ φ(x′ ,X,t)

+

Z

Z

˜ + ∂t φ dµǫ L(x′ , X, v, ω) + Aφ Rn ×Rn ×U ×Ω×(t,T )

(ψ −

Rn ×Rn ×Ω

φ)dνTǫ

+

Z

Rn ×Rn ×Ω

φdνtǫ



(using the minimax principle) =

Z

sup φ(x′ ,X,s)

+ infǫ µ

+ inf ǫ νT

Z

Rn ×Rn ×Ω

φdνtǫ

˜ + ∂t φ dµǫ L(x′ , X, v, ω) + Aφ  ǫ (ψ − φ)dνT ,

Rn ×Rn ×U ×Ω×(t,T )

Z

Rn ×Rn ×Ω

and then, taking into account that both µǫ and νTǫ are measures, with νTǫ a probability ˜ we obtain (38). measure, and µǫ with mass T − t, the definition of A˜ and H Proof (of Proposition 6). The rigorous proof of Proposition 6 will be quite long (ending on page 20) and it will make use of the Legendre-Fenchel-Rockafellar Theorem. Let P and Q be defined as in page 10. Define the following sets:   Z Z M1 := (µ, ν) ∈ P × Q : dµ = T − t, dν = 1, µ, ν ≥ 0 ,

and

  Z Z Z ǫ 2 ˜ + ∂t φdµ = φdν − φdν , ∀φ ∈ C . M2 := (µ, ν) ∈ P × Q : Aφ t Λ

Given χ ∈ C∗ (Rn × Rn × U × [t, T ]) and φ ∈ C⋆ (Rn × Rn × [t, T ]), we set h(χ, φ) := (T − t) sup [−χ(x, X, v, s) − L(x, X, v, s, ω)] (x,X,v,s)

(39)

+ sup [ψ(x′ , X, ω) − φ(x′ , X, T )] . (x′ ,X)

Note that, for short, we have omitted the dependence of h on ω. We consider the set o n ˜ + ∂t φ ∈ C∗ , C = cl (χ, φ) : φ ∈ CΛ2 , χ = Aφ

where the closure is taken in the CΛ2 × C∗ topology. Let (µ∗ , ν∗ ) ∈ M1 ∩ M2 and define (R R χdµ∗ − φ(x, X, T )dν∗ g(χ, φ) := −∞

if (χ, φ) ∈ C otherwise.

18

DIOGO A. GOMES & ENRICO VALDINOCI

We will now compute the dual functions of h and g, that is Z Z ∗ − χdµ − φdν − h(χ, φ) , h (µ, ν) := sup (χ,φ)∈C∗ ×C⋆ (40) Z Z ∗ − χdµ − φdν − g(χ, φ) . g (µ, ν) := inf (χ,φ)∈C∗ ×C⋆

Lemma 7. We have (41)

∗

h (µ, ν) =

(R

Ldµ + +∞

R

ψdν

if (µ, ν) ∈ M1 otherwise

and (42)

( 0 g ∗ (µ, ν) = −∞

if (µ, ν) ∈ M2 otherwise.

Proof. We first claim that if either µ or ν are non-positive then h∗ (µ, ν) = +∞. To see this, we choose a sequence of non-negative functions (χj , φj ) such that Z Z − χj dµ − φj dν → +∞. Since L ≥ 0 we have that

sup −χj − L ≤ 0, and, furthermore, sup ψ − φj ≤ sup ψ . As a consequence, since ψ is bounded, h(χj , φj ) ≤ −(T − t) inf L + sup ψ is bounded from above uniformly in j, and so h∗ (µ, ν) = +∞. Now we claim that the following inequality holds: Z Z ∗ h (µ, ν) ≥ Ldµ + ψdν Z  Z  (43) ˜ ˜ + sup φdν − sup φ . χdµ ˜ − (T − t) sup χ˜ + sup χ ˜

φ˜

To establish (43), consider a sequence Lj ≥ inf L = 0 (see Assumption 1) of compactly supported functions, increasing pointwise to L. For any χ ∈ C∗ and φ ∈ C⋆ , define χ˜ ˜ Note that χ˜ ∈ C∗ and φ˜ ∈ C⋆ , so that and φ˜ by χ = −Lj − χ, ˜ and φ = −ψ − φ. Z Z Z Z ∗ ˜ h (µ, ν) ≥ sup Lj dµ + χdµ ˜ + ψdν + φdν χ, ˜ φ˜

i ˜ −(T − t) sup [Lj + χ ˜ − L] − sup φ .

GENERALIZED MATHER PROBLEM AND HOMOGENIZATION

19

Since Lj − L ≤ 0 we gather that Z Z ∗ h (µ, ν) ≥ Lj dµ + ψdν Z Z   ˜ − sup φ˜ . χdµ ˜ − (T − t) sup χ˜ + sup φdν + sup χ ˜

φ˜

By letting j → +∞ we obtain (43), due to the Monotone Convergence Theorem. Now we claim that if the mass of µ is not T −t or the mass of ν is not 1, then h∗ (µ, ν) = ˜ +∞. To see this just add constants to χ˜ or to φ. Finally, if the mass of µ is T − t and the mass of ν is 1, by choosing χ˜ := 0 and φ˜ := 0 in (43), we obtain Z Z h∗ (µ, ν) ≥

Lj dµ +

ψdν.

To obtain the opposite inequality, just use (39) and (40), and observe that, if the mass of µ is T − t and the mass of ν is 1, then, for each (χ, φ), we have Z −χ − Ldµ ≤ (T − t) sup(−χ − L), as well as

Z

so that − Z

Z

ψ − φdν ≤ sup(ψ − φ) ,

Z

χdµ − φdν − h(χ, φ) Z ≤ Ldµ + ψdν + (T − t) sup(−χ − L) + sup(ψ − φ) − h(χ, φ) Z Z = Ldµ + ψdν.

This completes the proof of (41). To compute the Legendre transform of g, observe that if (µ, ν) 6∈ M2 then there exist ˆ ∈ C such that functions (χ, ˆ φ) Z Z ˆ ∗ − ν) 6= 0 , (44) χd(µ ˆ ∗ − µ) − φd(ν

ˆ we may thanks to the fact that (µ∗ , ν∗ ) ∈ M2 . Possibly changing the signs of χˆ and ψ, ˆ We suppose that the quantity in (44) is positive. Then, take χˆj := j χˆ and φˆj := j φ. get that Z Z lim χ ˆj d(µ∗ − µ) − φˆj d(ν∗ − ν) = +∞ . j→+∞

The above considerations show that ( Z Z 0 g ∗ (µ, ν) = inf − χd(µ∗ − µ) + φd(ν∗ − ν) = χ,φ −∞

if (µ, ν) ∈ M2 otherwise,

20

DIOGO A. GOMES & ENRICO VALDINOCI

which gives (42).



We now complete the proof of Proposition 6 by arguing as follows. We note that the function g (resp., h), as defined above, is concave (resp., convex) and upper (resp., lower) semicontinuous (we remark indeed that C is a closed convex set). Then, by the Legendre-Fenchel-Rockafellar Theorem (see, e.g., [Vil03] or [Gom06]), sup(g − h) = inf(h∗ − g ∗ ) . Consequently, (29), (36) and Lemma 7 yield: Z Z ǫ u˜ = inf L dµ + ψ dνtǫ =

(µ,ν)∈M1 ∩M2 ∗ ∗

inf h − g

(µ,ν)

= sup g − h (χ,φ)

=

sup (t − T ) sup (−χ − L) − sup (ψ − φ) + (x′ ,X,v,s)

(χ,φ)∈C

=

φ dνtǫ

˜ − ∂t φ − L) − sup (ψ − φ) + sup (t − T ) sup (−Aφ (x′ ,X,v,s)

2 φ∈CΛ

=

(x′ ,X)

Z

(x′ ,X)

˜ − ∂t φ) − sup (ψ − φ) + sup (t − T ) sup (H (x′ ,X,s)

2 φ∈CΛ

(x′ ,X)

Z

Z

φ dνtǫ

φ dνtǫ ,

which ends the proof of of Proposition 6.



The result to come shows that (38) is optimized when X =

x′ : ǫ

Proposition 8. Suppose that νtǫ is supported in the set X =

x′ . ǫ

Then

ǫ

u˜ (x, t, ω) =

(T − t)

sup

2 (Rn ×Rn ×[t,T ]) φ(x′ ,X,s)∈CΛ

′

2 (Rn ×[t,T ]) ϕ(x′ ,s)∈CΛ

′

(x ,s)∈R ×(t,T )

  ′ 2 ′ x −H Dx ϕ, Dx ϕ, x , , ǫ, ω ǫ ′ ′ + inf′ [ψ(x ) − ϕ(x , T )] Zx ϕ(x′ , t)dνtǫ + Rn ×Rn

inf

′ (x′ ,X= xǫ ,s)∈Rn ×Rn ×(t,T )

 x′ + inf′ ψ(x ) − φ(x , , T ) + x ǫ Z + φ(x′ , X, t)dνtǫ Rn ×Rn  (T − t) ′ infn sup 

=

"

∂t ϕ

˜ D 2 φ, Dφ, x′ , X, ǫ, ω ∂t φ − H

#


GENERALIZED MATHER PROBLEM AND HOMOGENIZATION

21

Proof. Let κ be as in the proof of Proposition 3, satisfying the additional uniformity hypothesis: for any δ > 0 there exists κ(δ) > 0 such that inf κ(y) ≥ κ(δ).

|y|≥δ

Then   x′ κ X− dνtǫ = 0. ǫ Rn ×Rn

Z

(45)

By contradiction, assume that the first equality claimed in Proposition 8 is false, hence, by (38), that
˜ D 2 φ, Dφ, x′ , X, ǫ, ω ∂t φ − H u˜ǫ (x, t, ω) < sup inf ′ φ(x′ ,X,s) (x′ ,X= xǫ ,s)∈Rn ×Rn ×(t,T )

  ′ ′ ′ x ψ(x ) − φ(x , , T ) + inf x′ ,X ǫ Z + φ(x′ , X, t)dνtǫ , Rn ×Rn

for some x ∈ Rn . This would mean that for some δ > 0, possibly depending on x, that u˜ǫ (x, t, ω) =

sup

inf

φ(x′ ,X,s) (x′ ,X,s)∈Rn ×Rn ×(t,T ),

(46)

+

inf

x′ ,X,

+

Z

′ |X− xǫ |>δ

Rn ×Rn

′ |X− xǫ |>δ

˜ D 2 φ, Dφ, x′, X, ǫ, ω ∂t φ − H

[ψ(x′ ) − φ(x′ , X, T )] +




φ(x′ , X, t)dνtǫ .

′
We replace φ(x′ , X, t) by φ(x′ , X, t) + λ(t − T )κ X − xǫ in (46), with λ > 0. Since, ˜ we have that the first due to the observation next to (37), this does not change H, infimum in (46) appears as  
x′ ˜ D 2 φ, Dφ, x′, X, ǫ, ω + ∂t φ − H λκ X − ǫ
˜ D 2 φ, Dφ, x′, X, ǫ, ω , ≥ λκ(δ) + ∂t φ − H

and the other infimum, as well as the integral are unchanged, due to (45). By sending λ → ∞ we would have a contradiction, and thus that the infimum in (38) must be ′ achieved when X = xǫ . This shows the first equality in Proposition 8. ′ The second one follows by taking ϕ(x′ , t) := φ(x′ , xǫ , t) and by observing that νtǫ is ′ supported on {X = xǫ }, thanks to (45), so that Z Z ′ ǫ φ(x , X, t) dνt = ϕ(x′ , t) dνtǫ .

22

DIOGO A. GOMES & ENRICO VALDINOCI

 From this point on, we will assume Assumption 6. With this, we are now in the position to improve Proposition 4: Proposition 9. u˜ǫ = uǫ in Rn × [t, T ] × Ω. Proof. By (22), νtǫ is supported in X = that u˜

ǫ

≥ (T − t) inf ′

x ,s

∂t vδǫ

x′ . ǫ

−H

Thus, by Proposition 8 and (9), we have 

Dx2 vδǫ , Dx vδǫ , x′ ,

+ inf′ [ψ(x′ ) − vδǫ (x′ , T )] + vδǫ (x, t)

x′ , ǫ, ω ǫ



x

≥ −δ + 0 + vδǫ (x, t) . Sending δ → 0, and using (10), we conclude that u˜ǫ ≥ uǫ . The converse inequality is assured by Proposition 4.  Finally, we present a result that converts the constrained minimization problem of (29) and (30) into an unconstrained problem. The idea, quite standard in linear programming, consists in using the approximate solution vδǫ of the dual problem as a Lagrange multiplier so to remove the constraints. Proposition 10 (Unconstrained minimization). Fix δ > 0, and let vδǫ be as in Assumption 6. Then,   Z Z ′ ǫ ′ x ǫ ǫ ǫ ǫ u (x, t, ω) − inf L(x , , v, ω) + Av + ∂ v dµ + v dν t δ δ δ t µ ǫ Rn ×U ×[t,T ]

(47)

≤ δ(T − t) + sup |uǫ (x, s, ω) − vδǫ (x, s, ω)| , s∈[t,T ]

where the above infimum is taken over all measures µ(x′ , v, s) on Rn × U × [t, T ], with total mass T − t. Proof. By (6), x′ x′ , v, ω) + Avδǫ ≥ −H(Dx2 vδǫ , Dx vδǫ , x′ , , ǫ, ω), ǫ ǫ and so, integrating with respect to any measure µ with total mass T − t, Z Z ′ ′ ′ x ǫ ǫ ǫ 2 ǫ ǫ ′ x L(x , , v, ω) + Avδ + ∂t vδ dµ ≥ ∂t vδ − H(Dx vδ , Dx vδ , x , , ǫ, ω) dµ ≥ −δ (T − t) , ǫ ǫ L(x′ ,

thanks to (9). This and the definition of νtǫ , equation (22), imply that   Z Z ′ ǫ ǫ ǫ ǫ ǫ ′ x u (x, t, ω) − inf L(x , , v, ω) + Avδ + ∂t vδ dµ + vδ dνt µ ǫ Rn ×U ×[t,T ] (48)

≤ δ(T − t) + uǫ (x, t, ω) − vδǫ (x, t, ω) .

GENERALIZED MATHER PROBLEM AND HOMOGENIZATION

23

On the other hand, let κ > 0, and suppose that (x(s), v(s)) is an almost optimal trajectory for (5), for s ∈ [t, T ], so that ZZ x′ ǫ (49) u ≥ L(x′ , , v, ω)dµǫo(x′ , v, s; ω) + Eψ (x(T )) − κ , ǫ where the measure µǫo on Rn × U × [t, T ] is defined by Z Z T ′ ǫ (50) φ(x , v, s) dµo := E φ(x(s), v(s), s) ds , t

n

for any φ ∈ C∗ (R × U × [t, T ]). Notice that µǫo(Rn × U × [t, T ]) = T − t, therefore Z x′ inf L(x′ , , v, ω) + Avδǫ + ∂t vδǫ dµ µ ǫ Z ′ x ≤ L(x′ , , v, ω) + Avδǫ + ∂t vδǫ dµǫo . ǫ

From this, (49) and (22), we get that   Z Z ′ ′ x ǫ ǫ ǫ ǫ ǫ L(x , , v, ω) + Avδ + ∂t vδ dµ + vδ dνt κ + u (x, t, ω) − inf µ ǫ Rn ×U ×[t,T ]   Z Z x(T ) ǫ ǫ ǫ (51) ≥ Eψ x(T ), , T − Avδ + ∂t vδ dµo − vδǫ dνtǫ ǫ  Z  x(T ) , T − Avδǫ + ∂t vδǫ dµǫo − vδǫ (x, t, ω) . = Eψ x(T ), ǫ Since, by (50) and Dynkin’s Formula, Z Avδǫ + ∂t vδǫ dµǫo + vδǫ (x, t, ω) = Evδǫ (x(T ), T ),

we deduce from (51) that

x′ L(x , , v, ω) + Avδǫ + ∂t vδǫ dµ + ǫ Rn ×U ×[t,T ]

 Z κ + u (x, t, ω) − inf ǫ

µ

′

Z

≥ E [ψ (x(T )) − vδǫ (x(T ), T, ω)] = E [uǫ (x(T ), T, ω) − vδǫ (x(T ), T ) ] . From this estimates, sending κ → 0, and (48), we obtain (47).

vδǫ dνtǫ

 

We remark that Proposition 10 and Assumption 6 imply that   Z Z ′ ǫ ǫ ′ x lim inf L x , , v, ω + Avδ + ∂t vδ dµ + vδǫ dνtǫ δ→0 µ ǫ n R ×U ×[t,T ] (52)

= uǫ (x, t, ω)

uniformly in (x, t, ω) ∈ Rn × [0, T ] × Ω, where the the infimum is taken over all measures µ(x′ , v, s) on Rn × U × [t, T ], with total mass T − t.

24

DIOGO A. GOMES & ENRICO VALDINOCI

7. Stationarity and Effective Lagrangian From this section on we will assume Assumption 7. We observe that, as a consequence of Assumption 7 and equation (6), the Hamiltonian is also stationary, that is (53)

H(M, p, x, X + Y, ǫ, ω) = H(M, p, x, X, ǫ, τY ω) .

Under reasonable hypothesis one can show that the minimizing measures µǫ are tight in the variable x, however, the same is false for the variable X. Therefore, we need a procedure to compactify the space in such a way that we can use weak convergence tools. The stationary setting provides a convenient way to achieve this, and for that we introduce the following construction: given a measure µǫ (x′ , X, v, s; ω) in Rn × Rn × U × [t, T ], we construct a measure µ ˆǫ (x′ , v, s, ω) in Rn × U × [t, T ] × Ω defined in the following way: Z ZZ ′ ǫ (54) φ(x , v, s, ω)dˆ µ := φ(x′ , v, s, τX ω)dµǫdω ,

for any bounded measurable function φ in Rn × U × [t, T ] × Ω. ¯ˆǫ (x′ , X, v, s, ω) Conversely, given a measure µ ˆ ǫ (x′ , v, s, ω), one can build a measure µ in Rn × Rn × U × [t, T ] × Ω in the following way: Z Z ′ ǫ ¯ˆ := (55) φ(x , X, v, s, ω)dµ φ(x′ , 0, v, s, ω)dˆ µǫ , for any bounded measurable φ in Rn × Rn × U × [t, T ] × Ω. Of course, with definitions similar to (54) and (55) (by omitting the dependence on v) one can construct from νTǫ measures νˆTǫ and ν¯ˆTǫ . Note that if φ is stationary, i.e., if φ(x′ , X, v, s, τY ω) = φ(x′ , X + Y, v, s, ω) ,

then

Z

¯ˆǫ = φ dµ

Z

φ dµǫdω .

Proposition 11. (i) Let µǫ and νTǫ be a sequence of minimizing measures for (29), corresponding to the initial trace νtǫ , according to (30). Then, the quantity in (29) is uniformly bounded. (ii) Let µǫ and νTǫ be a sequence of minimizing measures satisfying (30) for which the quantity in (29) is uniformly bounded. Let µ ˆ ǫ be as defined in (54), and define similarly νˆTǫ . Then, there exist subnets, which, by abuse of notation are still denoted as µ ˆǫ and νˆTǫ , and measures ρ(x′ , s) and θ(v, ω; x′, s) such that (56)

µ ˆǫ (x′ , v, s, ω) ⇀ θ(v, ω; x′, s) ρ(x′ , s) .

Furthermore, for each fixed (x′ , s), θ is a probability measure in U × Ω. Additionally, we have measures ν¯T and ν¯t on Rn , defined in the following way: Z Z ′ ′ (57) ϕ(x ) d¯ νT (x ) := lim ϕ(x′ ) dˆ νTǫ (x′ , X, ω) ǫ→0

GENERALIZED MATHER PROBLEM AND HOMOGENIZATION

25

and (58)

Z

′

′

ϕ(x ) d¯ νt (x ) := lim ǫ→0

for2 any ϕ ∈ Cc (Rn ).

Z

ϕ(x′ ) dˆ νtǫ (x′ , X, ω) ,

Proof. The claim in (i) easily follows by comparing with the control v := 0. We now prove the claim in (ii). Consider the control v := 0 and the corresponding trajectory x0 (t; ω). By comparing with the measures defined in (23) and (24), and by exploiting (4), we get that there exists a constant C such that   Z Z T  x0 (s; ω) C ≥ L x0 (s; ω), , 0, ω ds + 2 sup |ψ| dω ǫ t Z ≥ L(x, 0, v, ω) dˆ µǫ  i h ≥ (T − t) − µ ˆǫ Rn × (U \ BR ) × [t, T ] inf L   +µǫ Rn × (U \ BR ) × [t, T ] inf L. |v|≥R

From the growth conditions in v of L (see equation (4) in Assumption 1) we thus conclude that, as R → ∞,   n ǫ µ ˆ R × (U \ BR ) × [t, T ] → 0. Furthermore, we know, from Proposition 5 that   µ ˆ ǫ (Rn \ BR ) × U × [t, T ] → 0,

as R → ∞, which then implies that   n ǫ µ ˆ (R \ BR ) × (U \ BR ) × [t, T ] → 0,

as R → ∞, and then implies tightness. Therefore, the desired claim follows from Assumptions 8 and 3. From Proposition 5, it is also clear that the measure in Rn × Ω defined by Z E ϕ(x′ , ω)dνTǫ Rn

is tight, and therefore we have (57). The weak convergence in (58) is simply a consequence of (22), as then ν¯t = δx (x′ ). 

From this point on, we will also assume that the scaling hypotheses in Assumption 9 hold, that is σ = ǫα σ0 , with α = 1, 21 or 0 (resp. cases A, B or C). In each of these cases, we define the generalized Mather problem in the following way: given ̟ ∈ Rn 2A

standard observation is that the moment estimates in Proposition 5 imply that (57) and (58) also hold for any continuous ϕ(x′ ) growing at infinity less than |x′ |p once they hold for compactly supported ones.

26

DIOGO A. GOMES & ENRICO VALDINOCI

and Ξ ∈ Mat(n × n) we look, for each fixed x, at probability measures η(v, ω; x, ̟, Ξ) on U × Ω, which minimize Z L(x, ̟, Ξ) := L(x, 0, v, ω)dη(v, ω; x, ̟, Ξ) (59)

under the constraints, which depend on the scaling hypothesis put forward in Assumption 9, given by: A. In this first case, the Effective Lagrangian L(x, ̟) will not depend on Ξ, and we take as constraints the rotation vector Z (60) f (x, 0, v, ω)dη = ̟, (61)

and the holonomy constraint Z f (x, 0, v, ω) · DX φ(0, ω)dη = 0,

for all stationary functions φ(X, ω), which are C 1 in the first variable. Note that in this scaling the contribution of the diffusion vanishes. B. In this intermediate scaling, the Effective Lagrangian will also not depend on Ξ, and we require, similarly, the rotation vector Z (62) f (x, 0, v, ω)dη = ̟, and the holonomy constraint, which in this case has a contribution from the diffusion Z σ0 σ0T (x, 0, v, ω) 2 (63) f (x, 0, v, ω) · DX φ(0, ω) + : DX φ(0, ω)dη = 0, 2 for all C 2 (in the first variable) stationary functions φ(X, ω). C. In this scaling, in which diffusion dominates, we require both an average diffusion coefficient Ξ, a rotation vector ̟ Z 1 σ0 σ0T dη = Ξ, 2 Z (64) f (x, 0, v, ω)dη = ̟,

and the holonomy constraint, which does not have a drift term since the diffusion dominates Z σ0 σ0T (x, 0, v, ω) 2 (65) : DX φ(0, ω)dη = 0, 2 for all C 2 in X stationary functions φ(X, ω). Given the limiting measures ρ and θ constructed in Proposition 11, we define, for each (x′ , s) ∈ Rn × [t, T ], Z ′ (66) ̟(x ¯ , s) := f (x′ , 0, v, ω) dθ(ω, v; x′, s) .

GENERALIZED MATHER PROBLEM AND HOMOGENIZATION

27

Also, in case C, we define ¯ ′ , s) := 1 Ξ(x 2

(67)

Z

σ0 σ0T (x′ , 0, v, ω) dθ(ω, v; x′, s).

It turns out that ρ(x′ , s) almost everywhere, the measure θ(v, ω; x′ , s) constructed in Proposition 11 satisfies the holonomy constraints corresponding to cases A-C. More precisely: Proposition 12. For almost any (x′ , s) ∈ Rn ×[t, T ] with respect to the measure ρ(x′ , s), the measure θ(v, ω; x′ , s) satisfies conditions (61), (63) or (65), and conditions (60), (62) ¯ ′ , s)), according to cases A-C. or (64) with ̟ (resp., Ξ) replaced by ̟(x ¯ ′ , s) (resp., Ξ(x Moreover, in cases A and B, if we define the measure ̺(x′ , ̟, s) on Rn × Rn × [t, T ] by ̺(x′ , ̟, s) := δ̟(x′ ,s) (̟)ρ(x′ , s), we have that Z Z Z ′ ′ ′ (68) ̟ · Dx ϕ(x , s) + ∂t ϕ(x , s)d̺ = ϕ(x , T )d¯ νT − ϕ(x′ , t)d¯ νt , for all C 1 functions ϕ. Analogously, in case C, if we define the measure ̺(x′ , ̟, Ξ, s) on Rn × Rn × Mat(n × n) × [t, T ] by ̺(x′ , ̟, Ξ, s) := δ̟(x′ ,s)(̟)δΞ(x′ ,s) (Ξ)ρ(x′ , s), we have that Z (69) ̟ · Dx ϕ(x′ , s) + Ξ : Dx2 ϕ(x′ , s) + ∂t ϕ(x′ , s)d̺ Z Z ′ = ϕ(x , T )d¯ νT − ϕ(x′ , t)d¯ νt , for all C 2 functions ϕ. Proof. As usual, given a, b ∈ Rn , we define a ⊗ b ∈ Mat(n × n) by (a ⊗ b)ij := ai bj . Let φ(X, ω) : Rn × Ω → R be C 2 in X and stationary, let α(x′ , s) ∈ C(Rn × [t, T ]) and consider the function (x′ , X, s, ω) 7→ α(x′ , s) φ(X, ω) . By (21), (12) and (13), we see that (70) Z 

 1 1 ′ T ′ 2 f (x , 0, v, ω) · DX φ(0, ω) + 2 (σσ )(x , 0, ǫ, v, ω) : DX φ(0, ω) α(x′ , s) dˆ µǫ ǫ 2ǫ  Z  1 T ′ 2 ′ µǫ + ∂t α + f (x , 0, v, ω) · Dx α + (σσ )(x , 0, ǫ, v, ω) : Dx α φ(0, ω) dˆ 2 Z 1 + (σσ T )(x′ , 0, ǫ, v, ω) : (Dx α ⊗ DX φ(0, ω)) dˆ µǫ ǫ ZZ ZZ ǫ = αφ dνT dω − αφ dνtǫ dω .

28

DIOGO A. GOMES & ENRICO VALDINOCI

We first consider cases A and B. We now take a := 1 in case A or a := 1/2 in case B and we multiply (70) by ǫ, thus obtaining  Z  ǫ2a−1 T ′ 2 ′ (σ0 σ0 )(x 0, v, ω) : DX φ(0, ω) α(x′ , s) dˆ µǫ f (x , 0, v, ω) · DX φ(0, ω) + 2  Z  1 ′ T ′ 2 +ǫ ∂t α + f (x , 0, v, ω) · Dx α + (σσ )(x , 0, ǫ, v, ω) : Dx α φ(0, ω) dˆ µǫ 2 Z µǫ + ǫ2a (σ0 σ0T )(x′ 0, v, ω) : (Dx α ⊗ DX φ(0, ω)) dˆ Z Z  ZZ ǫ ǫ =ǫ αφ dνT dω − αφ dνt dω .

By sending ǫ → 0 via Proposition 11, it follows that ZZ h i c T ′ 2 ′ f (x , 0, v, ω) · DX φ(0, ω) + (σ0 σ0 )(x 0, v, ω) : DX φ(0, ω) 2 ′ ′ dθ(v, ω; x , s) α(x , s) dρ(x′ , s) = 0 ,

where c = 0 in case A, and c = 1 in case B. Since α is arbitrary, we conclude that Z h i c ′ T ′ 2 f (x , 0, v, ω) · DX φ(0, ω) + (σ0 σ0 )(x 0, v, ω) : DX φ(0, ω) dθ(v, ω; x′, s) = 0 2 ′ for almost any (x , s) with respect to ρ, thence conditions (61) or (63) are fulfilled, according to cases A and B. Multiplying (70) by ǫ2 and arguing as above, one shows that (65) holds in case C. The fact that θ satisfies (60), (62) or (64) with ̟ replaced by ̟(x ¯ ′ , s) follows at once by (66) and (67). To prove (68) and (69), it is sufficient to plug φ := 1 in (70) and send ǫ → 0.  Proposition 13. In the notation of Proposition 12, we have that Z ZZ (71) L d̺ ≤ L dθ dρ .

Proof. We use the notation of case C, since cases A and B are analogous and easier, to compute that Z L(x′ , ̟, Ξ) d̺ ZZ = L(x′ , 0, v, ω) dη(v, ω; x′, ̟, Ξ) d̺(x′, ̟, Ξ, s) ZZ ¯ ′ , s)) dρ(x′ , s) = L(x′ , 0, v, ω) dη(v, ω; x′, ̟(x ¯ ′ , s), Ξ(x ZZ ≤ L(x′ , 0, v, ω) dθ(v, ω; x′, s) dρ(x′ , s) , as we wished.



GENERALIZED MATHER PROBLEM AND HOMOGENIZATION

29

We remark that, for some choices of ̟ and Ξ, the constrained minimization problem given in (59) may be ill posed, since there might be no measure fulfilling conditions (61), (63) or (65), and conditions (60), (62) or (64), according to cases A-C. However, one of the consequences of Proposition 12 is that such a problem is always well defined for some particular choice of ̟ and Ξ, i.e., at least for ̟ := ̟(x ¯ ′ , s) ′ ∗ ∗ ∗ ¯ , s). In case, for some (x , ̟ , Ξ ), there are no measures fulfilling condiand Ξ := Ξ(x tions (61), (63) or (65), and conditions (60), (62) or (64), according to cases A-C, we define L(x∗ , ̟ ∗, Ξ∗ ) to be +∞. Proposition 14. The Effective Lagrangian L is lower semicontinuous. Note that this result, via Yosida regularization (see, e.g., Theorem 2.64 in [Att84]), gives that L may be approximated monotonically from below by continuous functions. Proof (of Proposition 14). Let (x′n , ̟n , Ξn ) → (x′ , ̟, Ξ) and consider the corresponding optimal measures ηn on U × Ω. Let Z ′ Ln := L(xn , ̟n , Ξn ) = L(x′n , 0, v, ω)dηn(v, ω; x′n , ̟n , Ξn ) .

Without loss of generality, we may suppose that Ln is bounded by above, say Ln ≤ K (otherwise, either lim inf Ln = ∞, in which case there is nothing to show, or lim inf Ln < ∞ and by extracting a subsequence we are in the previous case). But then ηn is tight by (4) and so the result follows from Assumption 8.  We now observe some growth conditions of L: Proposition 15. In cases A and B, we have that |̟|2 ≤ C(1 + L(x′ , ̟)). In case C, we have that |̟|2 + |Ξ|p ≤ C(1 + L(x′ , ̟, Ξ)). Proof. We will make use of Assumptions 4 and 15. By (60), (62) or (64) (according to the cases), 1/2 Z Z ′ 2 ′ |f (x , 0, v, ω)| dη |̟| ≤ |f (x , 0, v, ω)|dη ≤ 1 · ≤ C

Z

′

1 + L(x , 0, v, ω)dη

= C(1 + L)1/2 .

1/2

In case C, an analogous argument applied to (64) gives Z 1/p Z T ′ T ′ p 2|Ξ| ≤ |σ0 σ0 (x , 0, v, ω)|dη ≤ 1 · |σ0 σ0 (x , 0, v, ω)| dη ≤ C

Z

1 + L(x , 0, v, ω)dη

= C(1 + L)1/p , as desired.

′

1/p



30

DIOGO A. GOMES & ENRICO VALDINOCI

8. Homogenized problems In this section, we discuss the homogenized problem, which is built using the Effective Lagrangian constructed in the previous section. Let νt be as in (22) and let L(x′ , ̟, Ξ) or L(x′ , ̟) be defined as in the previous section. We consider the following problems: A or B. We look for measures ̺(x′ , ̟, s) on Rn × Rn × [t, T ] and ν¯T (x′ ) on Rn which minimize Z Z ′ νT , (72) u¯(x, t) := L(x , ̟)d̺ + ψd¯

(73)

under the constraint in (68). C. We look for measures ̺(x′ , ̟, Ξ, s) on Rn × Rn × Mat(n × n) × [t, T ] and ν¯T (x′ ) on Rn which minimize Z Z u¯(x, t) :=

L(x′ , ̟, Ξ)d̺ +

ψd¯ νT ,

under the constraint in (69). This variational problem is well posed, according to the result to come:

Proposition 16. There exist minimizing measures ̺(x′ , ̟, s) on Rn × Rn × [t, T ] and ν¯T (x′ ) on Rn , as requested above. Moreover, (74)

|¯ u(x, t)| ≤ C(1 + kψk∞ + T − t)

for any x ∈ Rn and t ≤ T Proof. The idea of the proof is that the measures built in Propositions 11 and 12 give an upper bound on the action and a natural candidate for the minimization process (though maybe not the minimum itself). The minimization process then converges due to the moment estimates of Proposition 5 and the lower semicontinuity of L given by Proposition 14. Following are the details of the proof. First, we consider the process   xǫ   xǫ ǫ ǫ ǫ dx = f x , , 0, ω dt + σ x , , ǫ, 0, ω dWt ǫ ǫ ǫ with x (t) = x and let µǫ and νǫ be as in (23) and (24) with v(s) := 0. Then, by Assumption 14, Z Z Ldµǫ + ψdνǫ Z T    xǫ  xǫ (s) (75) ≤ C(1 + kψk∞ + T − t) = E L xǫ (s), , 0, ω ds + ψ xǫ , ǫ ǫ t and so the second claim in Proposition 11 holds true. Consequently, the claim in Proposition 12 holds true. In particular, the measure µǫ induces in the limit a measure ̺ satisfying (68) and (69). By (71) and (75), we get that Z Z (76) Ld̺ + ψd¯ νT ≤ C(1 + kψk∞ + T − t)

GENERALIZED MATHER PROBLEM AND HOMOGENIZATION

31

that is, the quantities in (72) and (73) are not always infinite. This allow us to use the direct method in the calculus of variations, that is, we take a minimizing sequence of measures, say ̺n and ν¯T,n . We apply to these measures the moment estimates of Proposition 5 (with respect to the notation in Proposition 5, we replace what there is called µǫ , νTǫ , f , σσ T and L with the quantities denoted here by ̺n , ν¯T,n , ̟, Ξ and L). Note that the measures ̺n and ν¯T,n satisfy a constraint of the type in (30), thanks to (68) or (69), that the uniform bound on the action of the type in (32) is assured here by (76) and that the growth condition in Assumption 4 is given here by Proposition 15. Then, the tightness given by Proposition 5 and Assumption 8 gives the existence of the desired minimal measures, since, by Proposition 14, L is lower semicontinuous. Also, the uniform bound in (76) easily implies (74).  Proposition 17. The measure ̺, as introduced in Proposition 12, satisfies (68) in cases A and B, and (69) in case C. Proof. Let ϕ(x′ , t) be a smooth function. We use (21), (12) and (13) to gather that Z σσ T (x′ , 0, ǫ, v, ω) ∂t ϕ(x′ , s) + f (x′ , 0, v, ω) · Dx ϕ(x′ , s) + : Dx2 ϕ(x′ , s) dˆ µǫ 2 ZZ ZZ = ϕ(x′ , T ) dνTǫ dω − ϕ(x′ , t) dνtǫ dω .

Therefore, we use Proposition 11 to send ǫ → 0, and we obtain ZZ ∂t ϕ(x′ , s) + f (x′ , 0, v, ω) · Dx ϕ(x′ , s) (77) =

σ ′ σ ′ T (x′ , 0, v, ω) : Dx2 ϕ(x′ , s) dθ(v, ω; x′, s) dρ(x′ , s) + 2 Z Z ϕ(x′ , T ) d¯ νTǫ −

ϕ(x′ , t) d¯ νtǫ ,

where σ ′ := 0 in cases A and B, and σ ′ := σ0 (x′ , 0, v, ω) in case C. By means of Proposition 12, we know that Z 1 ¯ ′ , s) σ0 σ0T dθ(v, ω; x′s) = Ξ(x 2 Z and f (x, 0, v, ω) dθ(v, ω; x′s) = ̟(x ¯ ′ , s) .

Accordingly, the left hand side of (77) is equal to Z ¯ ′ , s) : D 2 ϕ(x′ , s) dρ(x′ , s) , ∂t ϕ(x′ , s) + ̟(x ¯ ′ , s) · Dx ϕ(x′ , s) + cΞ(x x where c := 0 in cases A and B, and c := 1 in case C. The latter quantity equals Z ∂t ϕ + ̟ · Dx ϕ(x′ , s) + cΞ(x′ , s) : Dx2 ϕ(x′ , s) d̺ ,

32

DIOGO A. GOMES & ENRICO VALDINOCI

thus yielding the desired result.



9. Uniform estimates This section is dedicated to proving certain uniform estimates that allow us to obtain convergence of uǫ to a non-random limit. For that we will assume, in addition to the previous ones, Assumptions 11-13. We will prove two types of estimates: the first type concerns a-priori estimates for the modulus of continuity of uǫ which are uniform in ǫ – these estimates apply fairly standard viscosity solutions and optimal control techniques and are used to establish that the solutions uǫ converge to a limit pointwise in ω. The second type of estimates show that the dependence on ω decreases as ǫ → 0, and therefore establish that any limit of uǫ is non-random. Proposition 18. We have (78)

  x kut k∞ ≤ lim sup sup H D 2 ψ, Dψ, x, , ǫ, ω . ǫ ǫ→0 x

Proof. By Assumption 10, we may assume that both the terminal data ψ and the solution uǫ are smooth. Define w(x, t) = ut (x, T − t). Then, w satisfies the parabolic equation (79)

wt + Lw = 0,

where L is the elliptic operator Lw = DM H : D 2 w + Dp H · Dw, and w(x, 0) = H(D 2 ψ, Dψ, x, xǫ ). Note that the fact that L is elliptic (thence (79) is parabolic) is a consequence of (14). Since equation (79) has maximum principle (see, for instance, Proposition 4.1 and Theorem 4.2 in [DiB95] or page 165 in [Shu92]), we obtain the required estimate.  Remark 19. We observe that the right hand side of (78) is finite, due to (8). Proposition 20. kDuǫ k∞ is uniformly bounded in ǫ. Proof. We will prove separately the cases in which H satisfies P1 or P2 of Assumption 11. Case 1. If H satisfies P1 then we have, from Proposition 18 H(Duǫ, x, X) ≤ C, which by (16) implies that kDuǫ k∞ is bounded. Case 2. By Assumption 10, we may suppose that uǫ and ψ are smooth. Consider the following rescaled version of the solution 1 U(τ, y) = uǫ (ǫτ, ǫy). ǫ Then (80)

kUk∞ < +∞

GENERALIZED MATHER PROBLEM AND HOMOGENIZATION

33

because of Proposition 18 and Remark 19, and U satisfies the equation 1  −Uτ + H D 2 U, DU, ǫy, y, ǫ, ω = 0, ǫ

with terminal data 1ǫ ψ(ǫy) at τ = Tǫ . A simple argument shows that uǫ is Lipschitz if and only if so is U, with the same Lipschitz bound. According to our smoothness hypothesis, we must now show that DU is bounded. Let w = |DU|2 /2. A simple computation yields (81) wτ = DU · DUτ 1 3 = Di UDMjk HDijk U + Di UDpj HDij2 U + ǫDi UDxi H + Di UDXi H ǫ 1 1 2 2 w − DMjk HDij2 UDik U + Dpj HDj w + ǫDi UDxi H + Di UDXi H. = DMjk HDjk ǫ ǫ If w admits an interior maximum, we have wt = 0 and Dw = 0 there, and 2 DMjk HDjk w ≥ 0,

due to (14). Thus, 1 2 − DMjk HDij2 UDik U + ǫDi UDxi H + Di UDXi H ≤ 0. ǫ But then P2 of Assumption 11 and (80) imply that DU is uniformly bounded in this case. We now modify the above argument in case w does not attain its maximum. For this, we observe that kwk∞ < +∞, thanks to (15). Hence, if, for a small q (even smaller than ǫ), we define w (q) (y) := w(y) − q ln(1 + |y|2) we have that w (q) has an interior (q) (q) (q) 2 maximum, say at yq . Then, wt (yq ) = 0, Dwt (yq ) = 0 and DMjk HDjk wt (yq ) ≥ 0, (q) 2 by (14). As a result, wt (yq ) = 0, |Dwt (yq )| ≤ Cq and DMjk HDjk wt (yq ) ≥ −Cq, for some C > 0. Substituting in (81), we obtain that 1 2 − DMjk HDij2 UDik U + ǫDi UDxi H + Di UDXi H ≤ Cǫ q ǫ at the point yq , for some Cǫ > 0, due to (8). Then, P2 of Assumption 11 and (80) yield that 1 ≥ Cǫ q ≥ F (DU(yq ), Uτ (yq )) ≥

inf

|ζ|≤kUτ k∞

F (DU(yq ), ζ) ,

as long as q is small enough. From this and (17), we infer that |DU(yq )| is uniformly bounded for small q. But then, w ≤ w(yq ) is uniformly bounded, and so is DU, as desired.  As a consequence, we have

34

DIOGO A. GOMES & ENRICO VALDINOCI

Proposition 21. Fixed any ω ∈ Ω, through an appropriate subsequence, we have that uǫ (x, t, ω) converges to u♯ (x, t, ω) locally uniformly in (x, t) ∈ Rn × [0, T ], for some u♯ : Rn × [0, T ] → R. Also, Z lim uǫ (x, t, ω) dω = u0 (x, t) , ǫ→0

0

Ω

n

for some u : R × [0, T ] → R.

R Proof. This follows from the Ascoli-Arzel`a Theorem, since Ω uǫ (x, t, ω) dω is equicontinuous, thanks to Propositions 18 and 20, and equibounded, due to the terminal condition.  Proposition 22. We have that |uǫ (x, t, τy ω) − uǫ (x, t, ω)| ≤ Cǫ|y| .

(82) Moreover, (83)

|

Z

ǫ

u (x − y, t, ω) dω −

Proof. Notice that −uǫt (x, t, τy ω) By (53),

Z

h i uǫ (x, t, ω) dω| ≤ |y| kDψk∞ + C(T − t) .

  x 2 ǫ ǫ + H D u (x, t, τy ω), Du (x, t, τy ω), x, , ǫ, τy ω = 0. ǫ

x −uǫt (x, t, τy ω) + H(D 2uǫ (x, t, τy ω), Duǫ(x, t, τy ω), x, y + , ǫ, ω) = 0. ǫ ǫ Set v(x, y, t; ω) = u (x − ǫy, t, τy ω). Then, v solves x −vt + H(D 2 v, Dv, x − ǫy, , ǫ, ω) = 0. ǫ As before, by Assumption 10, we may assume that v is smooth. Then set ∂v . w= ∂yk Thus, −wt + DM H : D 2 w + Dp H · Dw − ǫDxk H = 0. By Assumption 12, the ellipticity condition in (14) and the maximum principle (see, e.g., Proposition 4.1 and Theorem 4.2 in [DiB95]), we conclude that kw(x, t)k∞ ≤ ǫkDψk∞ + ǫC(T − t), which implies (84)

|uǫ (x − ǫy, t, τy ω) − uǫ (x, t, ω)| ≤ |y| [ǫkDψk∞ + ǫC(T − t)] .

Furthermore, the uniform Lipschitz continuity of uǫ (recall Proposition 20) implies that |uǫ (x − ǫy, t, τy ω) − uǫ (x, t, τy ω)| ≤ Cǫ|y|. The latter two estimates yield the claim in (82). Then, (83) follows from (84) by replacing ǫy with y and recalling that τy is measure preserving, according to Assumption 7. 

GENERALIZED MATHER PROBLEM AND HOMOGENIZATION

35

Proposition 23. We have lim sup sup |uǫ (x, t, ω0 ) − uǫ (x, t, ω1 )| → 0. ǫ→0

x∈Rn t∈[0,T ] ω0 ,ω1 ∈Ω

Proof. To prove the above estimate, observe that |uǫ (x, t, ω0 ) − uǫ (x, t, ω1 )| ≤ |uǫ (x, t, ω0 ) − uǫ (x, t, τy ω0 )| + |uǫ (x, t, τy ω0 ) − uǫ (x, t, ω1 )|. Let δ1 > 0 be fixed, and suppose that for every ǫ there exists y = y(x, t, ǫ, δ1 ) such that |y| ≤ M, with M depending only on δ1 (and so independent of ǫ), for which (85)

sup |uǫ(x, t, τy ω0 ) − uǫ (x, t, ω1 )| ≤

x∈Rn t∈[0,T ] ω0 ,ω1 ∈Ω

δ1 . 2

Then, from Proposition 22, for ǫ sufficiently small, we have sup |uǫ (x, t, ω0 ) − uǫ (x, t, τy ω0 )| ≤

x∈Rn t∈[0,T ] ω0 Ω |y|≤M

δ1 , 2

which then, by sending δ1 → 0+ yields the estimate. Therefore our task now consists in establishing (85). To this extent, fix δ > 0 and choose y as in Assumption 13. Then u˜ǫ (x, t) := uǫ (x, t, ω1 ) + δ(T − t) satisfies

that is

  x 0 = −uǫt (x, t, ω1 ) + H D 2 uǫ (x, t, ω1 ), D˜ uǫ (x, t, ω1 ), x, , ǫ, ω1 ǫ   x ǫ 2 = −δ − u˜t + H D u˜, D˜ u, x, , ǫ, ω1 ǫ   x ǫ 2 ≤ −˜ ut + H D u˜, D˜ u, x, , ǫ, τy ω0 , ǫ

  x −˜ uǫt + H D 2 u˜, D˜ u, x, , ǫ, τy ω0 ≥ 0 . ǫ ǫ ǫ Then, u − u˜ is a subsolution of a linear parabolic equation, due to (14), which then implies, by the comparison principle for viscosity solutions (see Proposition 4.1 and Theorem 4.2 in [DiB95]) that uǫ (x, t, τy ω0 ) ≤ u˜ǫ (x, t). Accordingly, we obtain (85) by possibly exchanging the roles of τy ω0 and ω1 , as desired. 

By collecting the results in this section, we get the following result:

36

DIOGO A. GOMES & ENRICO VALDINOCI

Proposition 24. Through an appropriate subsequence, lim uǫ (x, t, ω) = u0 (x, t) , ǫ→0

uniformly in K × [0, T ] × Ω, for any compact set K ⊂ Rn . Proof. Fix ω0 ∈ Ω. Then, by Proposition 21, uǫj (x, t, ω0 ) converges to u♯ (x, t, ω0 ) locally uniformly in (x, t). Let u0 (x, t) := u♯ (x, t, ω0 ). Then, lim

sup

|uǫj (x, t, ω) − u0 (x, t)|

lim

sup

|uǫj (x, t, ω) − uǫj (x, t, ω0 )|

j→+∞ (x,t,ω)∈K×[0,T ]×Ω

≤ +

j→+∞ (x,t,ω)∈K×[0,T ]×Ω

lim

sup

j→+∞ (x,t)∈K×[0,T ]

|uǫj (x, t, ω0 ) − u♯ (x, t, ω0 )|

≤ 0 +0, where Proposition 23 has been exploited.



10. Convergence In this section, we establish that the limit of the viscosity solutions uǫ agrees with u¯, which was defined in (72). We divide the proof into two parts, a lower bound and an upper bound. Proposition 25 (Lower Bound). u0 ≥ u¯. Proof. Consider the measures ̺ and θ constructed in section 7. Let also µǫ and νTǫ be the measures optimizing (5). Then, we use Proposition 9, (29) (11), (54), Proposition 11, (71), Proposition 17, (72) and (73) to obtain that Z 0 u (x, t) = lim uǫ (x, t, ω) dω ǫ→0 Ω Z Z  ZZ ′ ǫ ′ ǫ = lim L(x , X, v, s, ω)dµ dω + ψ(x )dνT dω ǫ→0  Z  Z ′ ǫ νT µ + ψd¯ = lim L(x , 0, v, s, ω)dˆ ǫ→0 ZZ Z ′ = L(x , 0, v, s, ω)dθ dρ + ψd¯ νT Z Z ′ ≥ L(x , ̟, Ξ)d̺ + ψd¯ νT = u¯(x, t). 

In order to obtain the opposite inequality to Proposition 25, we need the result in Proposition 24. Proposition 26 (Upper bound). u0 = u¯.

GENERALIZED MATHER PROBLEM AND HOMOGENIZATION

37

Proof. Let δ > 0 and vδǫ be as in Assumption 6. We denote by oδ quantities (possibly depending on (x, t, ω)) which tend to zero as δ → 0. Let κ > 0. We consider almost optimal measures (̺κ (x′ , ̟, Ξ, s), ν¯T,κ(x′ )) for (72) or (73), and an almost optimal family of probability measure ηκ (v, ω; x′, s, ̟, Ξ) for (59), that is, we suppose that ̺κ (x′ , ̟, Ξ, s) almost everywhere Z ′ L(x , ̟, Ξ) ≥ L(x, 0, v, ω)dηκ(v, ω; x, ̟, Ξ) − κ, and that, in case A or B,

u¯(x, t) ≥ whereas, in case C, u¯(x, t) ≥

Z

Z

′

L(x , ̟)d̺κ +

′

Z

L(x , ̟, Ξ)d̺κ +

Z

ψd¯ νT,κ − κ,

ψd¯ νT,κ − κ.

We construct a trial measure µ ˜κ (x′ , v, s, ω) on Rn ×U ×[t, T ]×Ω defined in the following way: Z ZZ ′ (86) φ(x , v, s, ω)d˜ µκ := φ(x′ , v, s, τ− x′ ω)dηκ d̺κ , ǫ

Rn ×U ×[t,T ]×Ω

for any φ ∈ Cc (Rn × U × [t, T ] × Ω). Note that, by taking ϕ(s) := (s − T ) in (68) or (69), we obtain that Z d̺κ = T − t ,

and so, since ηκ is a probability measure, it follows that µ ˜κ has total mass T − t. In particular, the total mass of µ ˜κ is finite and so, recalling Assumption 3, we can slice µ ˜κ as (87)

µ ˜κ (x′ , v, s, ω) =: ζκ (x′ , v, s; ω) Θκ(ω) ,

where Θκ (ω) is a probability measure, and ζκ is, for Θκ -almost any ω ∈ Ω fixed, a measure of finite total mass. Let m(ω) be the total mass of ζκ for a fixed ω. By construction, Z ZZ Z 1 1 m(ω) (88) 1= d˜ µκ = dζκ dΘκ = dΘκ (ω) . T −t T −t T −t Let also

T −t ζˆκ := ζκ , m(ω)

and note that ζˆ has total mass T − t for any fixed ω. Then, by Proposition 10, Z Z ′ ǫ ǫ ˆ ǫ ′ x oδ + u ≤ L(x , , v, ω) + Avδ + ∂t vδ dζκ + vδǫ dνtǫ , ǫ

38

DIOGO A. GOMES & ENRICO VALDINOCI

and so, by (87) and (88), Z m(ω) ǫ u (x, t, ω)dΘκ(ω) oδ + T −t Z Z ZZ ′ m(ω) ǫ ǫ ′ x ǫ ǫ ≤ L(x , , v, ω)d˜ µκ + Avδ + ∂t vδ d˜ µ+ v dν dΘκ . ǫ T −t δ t

Making use of (11) and (59), we get that Z Z ′ x′ ′ x L(x , , v, ω)d˜ µκ = L(x′ , , v, τ− x′ ω)dηκ d̺κ ǫ ǫ ǫ ZZ = L(x′ , 0, v, ω)dηκd̺κ Z ≤ L(x′ , ̟, Ξ)d̺κ + Cκ, .

for some suitable constant C > 0. Similarly, by (19), (86), (12), (13), (60), (62) and (64), Z Z σσ T ′ x′ x′ (x , , v, ω) : Dx2 ϕ d˜ µκ Aϕ d˜ µκ = f (x′ , , v, ω) · Dx ϕ + ǫ 2 ǫ ZZ x′ = f (x′ , , v, τ− x′ ω) · Dx ϕ ǫ ǫ ′ T x σσ (x′ , , v, τ− x′ ω) : Dx2 ϕ dηκ d̺κ + ǫ 2 ǫ ZZ σσ T ′ = f (x′ , 0, v, ω) · Dx ϕ + (x , 0, v, ω) : Dx2 ϕ dηκ d̺κ 2 Z = ̟ · Dx ϕ + Ξǫ : D 2 ϕ d̺κ ,

for any ϕ(x′ ) ∈ C 2 (Rn ), with

1 Ξ (x ) := 2 ǫ

′

Z

σσ T (x′ , 0, v, ω) dηκ .

Thus, exploiting (68) and (69), Z Z ǫ ǫ Avδ + ∂t vδ d˜ µκ = ̟ · Dx vδǫ + Ξǫ : Dx2 vδǫ + ∂t vδǫ d̺κ Z Z Z ǫ ǫ = vδ d¯ νT − vδ d¯ νt + (Ξǫ − Ξ∗ ) : Dx2 vδǫ d̺κ ,

with Ξ∗ := 0 in cases A and B and Ξ∗ = Ξ = Ξǫ in case C. Note that (89)

lim Ξǫ − Ξ∗ = 0 ǫ→0

uniformly in (x, v, ω) in all the cases A-C.

GENERALIZED MATHER PROBLEM AND HOMOGENIZATION

39

Also, by (72) and (73), u¯(x, t) ≥

Z

′

L(x , ̟, Ξ)d̺κ +

Z

ψd¯ νT,κ − κ .

By collecting the above estimates, we conclude that Z m(ω) ǫ oδ + u (x, t, ω)dΘ(ω) T −t Z Z ZZ m(ω) ǫ ǫ ǫ v dν dΘ − vδǫ d¯ νt ≤ u¯(x, t) + (vδ − ψ)d¯ νT + T −t δ t Z + (Ξǫ − Ξ∗ ) : Dx2 vδǫ d̺κ + Cκ . That is, by Assumption 6,

m(ω) ǫ oδ + u dΘ T −t Z ZZ m(ω) ǫ ǫ ǫ ¯ d¯ ≤ u¯ + (u − ψ) νT + u dνt dΘ T −t Z Z ǫ − u d¯ νt + (Ξǫ − Ξ∗ ) : Dx2 vδǫ d̺ + Cκ . Z

We now send ǫ → 0 and we obtain

oδ + u0 ≤ u¯ + Cκ , thanks to Proposition 24 and formulas (58), (88) and (89). We now send δ and κ to zero, getting u0 ≤ u¯. This and Proposition 25 yield the desired claim.  11. Improved Moment Estimates In this section we improve the moment estimates of section 5 in order to show that for small time “optimal trajectories don’t go too far”. These estimates will be essential in establishing that the homogenized limit is a viscosity solution of a suitable equation. Here, we will take a small time step h ∈ (0, T ). We start with an auxiliary result: Lemma 27. Consider the diffusion     0 0 0 0 x 0 x dx = f x , , 0, ω dt + σ x , , 0, ω dWt , ǫ ǫ 0 with initial condition x (T − h) = x. Then, we have |E[x0 (T − h) − x0 (T )]| ≤ Ch, and E|x0 (T − h) − x0 (T )|2 ≤ Ch, where the constant C can be chosen independent of ǫ, h and x.

40

DIOGO A. GOMES & ENRICO VALDINOCI

Proof. Fix h > 0, let T − h ≤ t ≤ T . By Dynkin’s Formula we have   0 d 0 0 0 x E[x (T − h) − x (t)] = −Ef x , , 0, ω , (90) dt ǫ which, by Assumption 14, yields the first part of the claim. Similarly, we have   0 d 0 0 2 0 0 0 x E|x (T − h) − x (t)| =2E (x (T − h) − x (t))f (x , , 0, ω) dt ǫ 0 0 x x (91) + tr E[σ(x0 , , 0, ω)σ T (x0 , , 0, ω)]. ǫ ǫ Thus, if we set A(t) = E|x0 (T − h) − x0 (t)|2 we have d A ≤ A + C, A(T − h) = 0. dt A Gronwall estimate yields |A(t)| ≤ Ch.



Lemma 28. Consider an optimal diffusion x(t), for T − h ≤ t ≤ T , with initial condition x(T − h) = x. Let Z T   x M := E L x, , v, ω dt. ǫ T −h Then, we have |E[x(T − h) − x(T )]| ≤ M 1/2 Ch1/2 + Ch, and E|x(T − h) − x(T )|2 ≤ MCh + Ch, where C can be chosen independently of ǫ, h and x. Proof. Using again (90) and Assumption 15 we obtain Z T   x E[x(T − h) − x(T )] = E f x, , v, ω dt ǫ T −h  Z T   2 1/2 x 1/2 E ≤h f x, , v, ω dt ǫ T −h  Z T  1/2  x 1/2 E ≤ Ch L x, , v, ω + Cdt ǫ T −h ≤ CM 1/2 h1/2 + Ch.

Similarly, by (91), we have h  x i d E|x(T − h) − x(t)|2 =2E (x(T − h) − x(t))f x, , v, ω dt h  x   ǫx i + tr E σ x, , v, ω σ T x, , v, ω ǫ ǫ 

 x ≤Ch E|x(T − h) − x(t)| + ChEL x, , v, ω + C. ǫ −1

2

GENERALIZED MATHER PROBLEM AND HOMOGENIZATION

41

Thus, if we set A(t) = E|x(T −h)−x(t)|2 we have, by a Gronwall estimate, that |A(t)| ≤ CMh + Ch.  Now we use the two previous results to prove that the integral of the Lagrangian along an optimal trajectory is not too big: Proposition 29. Consider an optimal diffusion x for T − h ≤ t ≤ T , minimizing Z T   x E L x, , v, ω dt + ψ(x(T )), ǫ T −h

with initial condition x(T − h) = x. Then there exists a constant C independent of h, ǫ and x such that Z T   x E L x, , v, ω dt ≤ Ch. ǫ T −h Proof. Consider the diffusion     x0 x0 dx0 = f x0 , , 0, ω dt + σ x0 , , 0, ω dWt , ǫ ǫ with x(T − h) = x. We have, by the optimality of x, that  Z T   x E L x, , v, ω dt + ψ(x(T )) ǫ T −h  Z T   0 0 0 x ≤E L x , , 0, ω dt + ψ(x (T )) . ǫ T −h

Moreover, the fact that x(T − h) = x0 (T − h) = x, via a second order Taylor expansion of ψ, gives that ψ(x0 (T )) − ψ(x(T )) ≤ Dψ(x) · (x0 (T ) − x0 (T − h)) − Dψ(x) · (x(T ) − x(T − h))   +C |x0 (T ) − x0 (T − h)|2 + |x(T ) − x(T − h)|2 .

By the previous estimates and Lemmata 27 and 28, Z T  Z T      0 x 0 0 x M =E L x, , v, ω dt ≤E L x , , 0, ω dt + ψ(x (T )) − ψ(x(T )) ǫ ǫ T −h T −h ≤ C(1 + M)h + CM 1/2 h1/2 ,

which implies the desired bound. We will also need the following estimate: Proposition 30. Fix r > 0. Consider the optimal diffusion  x   x  dx = f x, , v, ω dt + σ x, , v, ω dWt , ǫ ǫ + with x(T − h) = x. Then, as h → 0 , we have P (|x(T ) − x| > r) = o(h).



42

DIOGO A. GOMES & ENRICO VALDINOCI

Proof. Write x = y + z, where

  x   x dz = σ x, , v, ω dWt , dy = f x, , v, ω dt, ǫ ǫ and y(T − h) = x, z(T − h) = 0. We have  x  d 2 E|y − x| = 2E(y − x)f x, , v, ω dt ǫ

1/2 2 1/2 ≤ 2 E|y − x| E|f |2 ,

which implies


1/2
1/2 d E|y − x|2 ≤ C E|f |2 . dt Thus, using again Assumption 15, Z T

1/2 2 1/2 E|y(t) − x| ≤C E|f |2 dt T −h

1/2

≤ Ch

Z

T

1/2 E|f | dt 2

T −h

 Z 1/2 E ≤ Ch

T

T −h

due to Proposition 29, and so

1/2 L + Cdt ≤ Ch,

E|y(t) − x|2 ≤ Ch2 . Accordingly, by Chebychev’s inequality, we have  h2 r ≤ C 2. (92) P |y(T ) − x| > 2 r To handle the second term, we use Itˆo’s formula (see, e.g., page 105 in [Eva06b]) and we get λ|z|2 d Ee t+2h−T dt λ|z|2   λ|z|2 λ|z|2 λ2 z ⊗ z 1 2λI λ|z|2 e t+2h−T T t+2h−T t+2h−T + + Eσσ : e e = −E (t + 2h − T )2 2 t + 2h − T (t + 2h − T )2   λ|z|2 λI T ≤ Eσσ : e t+2h−T t + 2h − T λ|z|2 C ≤ Ee t+2h−T , h if λ is sufficiently small and t ∈ [T − h, T ]. Thus, recalling that z(T − h) = 0, we obtain by a Gronwall estimate that λ|z|2

Ee t+2h−T ≤ C,

GENERALIZED MATHER PROBLEM AND HOMOGENIZATION

43

for all T − h ≤ t ≤ T . Again by Chebychev’s inequality,  λr 2 r ≤ Ce− 2h , (93) P |z(T )| ≥ 2 which is exponentially small as h → 0. The desired result then follows from (92) and (93).  We will need a final moment estimate for the optimal measures of the homogenized problem. Such moment estimate will only be in use in the proof of Proposition 33 and it requires the following additional condition: Assumption 16. We assume that for any fixed (x0 , ̟0 , σ0 ) ∈ Rn × Rn × Mat(n × n) there exists a map x 7→ Sx0 ,̟0 ,σ0 (x) = (Sx̟0 ,̟0 ,σ0 (x), Sxσ0 ,̟0 ,σ0 (x)) ∈ Rn × Mat(n × n) with the properties listed below.
T Let Ξ0 = 21 σ0 σ0T , and SxΞ0 ,̟0,Ξ0 (x) = 21 Sxσ0 ,̟0 ,σ0 (x) Sxσ0 ,̟0 ,σ0 (x) . Then (1) Sx0 ,̟0 ,σ0 (x0 ) = (̟0 , σ0 ); (2) the mapping x 7→ Sx0 ,̟0,σ0 (x) is globally bounded and globally Lipschitz; (3) the mapping x 7→ L(Sx̟0 ,̟0 ,σ0 (x), SxΞ0 ,̟0 ,Ξ0 (x), x) is globally Lipschitz. ◭ Assumption 16 is obviously fulfilled, by choosing Sx0 ,̟0 ,σ0 (x) to be identically equal to (̟0 , σ0 ), when L is globally Lipschitz in x and, in particular, if L does not depend on x at all – this is the case, for instance, if L does not depend on x (though it may depend on X = x/ǫ). Proposition 31. Suppose that T ∈ (0, 1] is conveniently small. Let ̺ and νT be optimal measures for (72) under the constraint in (68) in cases A and B, or optimal measures for (73) under the constraint in (69) in case C, with t = 0. Let also Assumption 16 hold. Then, Z ′ (x − x)dνT ≤ CT, (94) (95) and (96) Furthermore,

Z

|x′ − x|2 d̺ ≤ CT 2

Z

|x′ − x|2 dνT ≤ CT.

(97) Proof. First of all, we observe that, (98)

Z Ld̺ ≤ CT.

|̟|2 + |Ξ| + |Ξ|p ≤ C(1 + L(x′ , ̟, Ξ))

thanks to Proposition 15.

44

DIOGO A. GOMES & ENRICO VALDINOCI

Also, by taking ϕ(x′ , s) = x′j − xj in the holonomy constraint (68) or (69) (according to the case), with j = 1, . . . , n, we obtain the identity Z Z (99) ̟d̺ = x′ − xdνT .

By taking ϕ = 12 (T − s)|x′ − x|2 , we obtain Z 1 (100) (T − s)(̟ · (x′ − x) + tr Ξ) − |x′ − x|2 d̺ = 0. 2 Similarly, take ϕ = 21 |x′ − x|2 to get Z Z ′ (101) ̟ · (x − x) + tr Ξ d̺ = |x′ − x|2 dνT .

By taking ϕ = 1 and ϕ = T − s we also see that the total masses of νT and ̺ are 1 and T , respectively. Thus, using (99), the H¨older Inequality and (98), we conclude that Z Z 1/2 Z 1/2 2 1/2 1/2 ′ (x − x)dνT ≤ CT |̟| d̺ ≤ CT + CT (102) Ld̺ . Also, from (100) and a scaled H¨older Inequality, Z Z Z Z 1 ′ 2 ′ ′ 2 |x − x| d̺ ≤ CT |̟||x − x| + |Ξ|d̺ ≤ |x − x| d̺ + CT |̟|2 + |Ξ|d̺ 2 and so, by (98), Z    Z Z ′ 2 2 (103) |x − x| d̺ ≤ CT |̟| + |Ξ|d̺ ≤ CT T + Ld̺ .

We now consider an additional small parameter d > 0. From (101), combined with (98),  Z Z |x′ − x|2 2 p −1/(p−1) ′ 2 + d|̟| + d|Ξ| + d d̺ |x − x| dνT ≤ C d   Z Z C −1/(p−1) ′ 2 ≤ Cd T+ |x − x| d̺ + Cd T + Ld̺ . d Hence, exploiting (103), Z Z ′ 2 (104) |x − x| dνT ≤ Cd T + C(d + T ) Ld̺ ,

where Cd > 0 may depend on d. R To get the desired estimates, it remains to estimate Ld̺. Fixed x ∈ Rn , we take ̟0 = ̟0 (x), σ0 = σ0 (x) and Ξ0 = Ξ0 (x) = σ0 σ0T in such a way that sup −Dx ψ(x) · ̟ − Ξ : Dx2 ψ(x) − L(x, ̟, Ξ) ̟,Ξ

= −Dx ψ(x) · ̟0 − Ξ0 : Dx2 ψ(x) − L(x, ̟0 , Ξ0 ) .

GENERALIZED MATHER PROBLEM AND HOMOGENIZATION

45

Indeed, note that, since L is coercive, by (98), and lower semicontinuous, by Proposition 14, we have that such ̟0 and Ξ0 exist, although they may not be unique. Consider now the diffusion dx0 = Sx̟0 ,̟0,σ0 (x0 )dt + SxΞ0 ,̟0 ,Ξ0 (x0 )dWt ,

(105)

with x0 (0) = x and let the measures d̺0 (x′ , ω, Ξ, s) be defined by Z φ(x′ , ̟, Ξ, s)d̺0 Rn ×Rn ×Mat(n×n)

:= E

Z

0

T

φ(x0 (s), Sx̟0 ,̟0,σ0 (x0 (s)), SxΞ0 ,̟0 ,Ξ0 (x0 (s)), s)ds

and ν0,T (x′ ) by Z

ϕ(x′ )dν0,T (x′ ) := Eϕ(x0 (T )). Rn

Then, by Dynkin’s Formula and the minimality of ̺ and νT , Z Z Z Z (106) Ld̺ + ψdνT ≤ Ld̺0 + ψdν0,T . By Lemma 27, we have Z (x′ − x)dν0,T ≤ CT

Z

2

|(x′ − x)| dν0,T ≤ CT.

Thus

(107)

Z ψ(x′ ) − ψ(x) dν0,T (x′ ) Z Z 2 ′ ≤ |Dψ(x)| (x − x)dν0,T + C |(x′ − x)| dν0,T ≤ CT .

On the other hand, by (102) and (104), Z ′ ′ ψ(x ) − ψ(x) dνT (x ) Z Z 2 ′ ≤ |Dψ(x)| (x − x)dνT + C |(x′ − x)| dνT (108) Z 1/2 Z ≤ CT + CT 1/2

Ld̺ + Cd T + C(d + T ) Z ≤ Cd T + C(d + T ) Ld̺ .

Ld̺

46

DIOGO A. GOMES & ENRICO VALDINOCI

Therefore, putting together (106), (107) and (108), Z Z Z Ld̺ ≤ ψ(x′ ) − ψ(x)dν0,T + ψ(x) − ψ(x′ )dνT Z + Ld̺0 Z ≤ Cd T + C(d + T ) Ld̺.

This yields the estimate in (97). Then, combining together (97), (102), (103) and (104), we obtain (94), (95) and (96).  12. Viscosity Solutions This last section is dedicated to the proof of the main result of the paper, that is that the homogenized limit is a viscosity solution to an Effective Hamilton-Jacobi equation. Proposition 32. There exists a function continuous function H : Mat(n × n) × Rn × Rn → R such that u¯ is a viscosity solution of
(109) − u¯t + H Dx2 u¯, Dx u¯, x = 0.

Proof. To prove that u¯ is a viscosity solution, we are going to use the results from [Bit01]. To do so, we first define an operator: Z Z (110) Tt1 ,t2 ψ(x) := inf Ld̺ + ψdνt2 , ̺,νt

where the infimum is taken over all measures ̺ on Rn × Rn × [t1 , t2 ] (or on Rn × Rn × Mat(n × n) × [t1 , t2 ]), and νt on Rn that satisfy (68) (or (69), according to case A, B or C) with T replaced by t2 and t replaced by t1 , and νt1 = δ(x). We also set (111)

Tt := T0,t .

Let Y be the set of all ψ ∈ C 2 (Rn ) such that   x 2 (112) sup sup H D ψ, Dψ, x, , ǫ, ω < ∞. ǫ ǫ>0 x,ω

Then, by (72) and Propositions 20, 24 and 26, we have that Tt maps Y into Lipschitz functions, for all t ≥ 0. Define X := ∪t≥0 Tt Y . If we prove that Tt1 ,t2 satisfies the semigroup property: Tt1 ,t2 ◦ Tt2 ,t3 = Tt1 ,t3 , for all 0 ≤ t1 < t2 < t3 , it is then clear that X is an invariant set for Tt . Note that Y contains Cc∞ (Rn ), thanks to (8), thus (H1) in [Bit01] holds. To address the semigroup property, it suffices to observe that for ǫ > 0 the evolution associated to the Hamilton-Jacobi equation Ttǫ1 ,t2 is a semigroup, which converges uniformly, and therefore the limit also satisfies the semigroup property. In further detail: one considers Z Z ǫ ǫ Tt1 ,t2 ψ := inf Ldµ + ψdνtǫ2 ,

GENERALIZED MATHER PROBLEM AND HOMOGENIZATION

47

where the infimum is taken on the measures ρ on Rn × Rn × U × [t1 , t2 ] with total mass t2 − t1 and νt probability measures on Rn satisfying (30). From Proposition 9 and (5),  Z t2  x(s) ǫ L x(s), Tt1 ,t2 ψ = inf E , v(s), ω ds + ψ (x(t2 )) , ǫ t1 where the infimum above is taken over all trajectories of the controlled dynamics (3) with x(t1 ) = x. Then, Tt1ǫ ,t2 is a semigroup, by the optimality of the latter dynamic programming principle, thence so is Tt1 ,t2 , due to Propositions 24 and 26, as desired. Let now the translation operator τ be defined by τy ψ(x) = ψ(x+y), for any ψ : Rn → R. The hypothesis (H2) in [Bit01] requires that for all f ∈ X and all y ∈ Rn , τy f ∈ X. However, this is not really necessary, and we may replace it by requiring for all y ∈ Rn and ψ ∈ Y , τy ψ ∈ Y . This condition is then verified, thanks to (8). Hypothesis (H3) of [Bit01] is obviously fulfilled here too. Moreover, clearly, Tt1 ,t2 is monotone, that is, if ψ¯1 ≤ ψ¯2 then Tt1 ,t2 ψ¯1 ≤ Tt1 ,t2 ψ¯2 . Therefore, to establish the result it suffices to check that the continuity and regularity hypothesis I-IV that we spell next. We will need the following notation: for any sequence d = (dk ) of positive reals we define Qd = {η ∈ Cc∞ (Rn ), kD αηk∞ ≤ dk , |α| ≤ k}. I Continuity: for every ψ ∈ X the function (t, x) 7→ Tt [ψ](x) is continuous and for all b > a ≥ 0 there exists C = C(a, b, ψ) such that |Tt ψ| ≤ C, for any t ∈ [a, b]. II Locality: for every ψ1 , ψ2 ∈ C ∞ (Rn ) ∩ X and any fixed x ∈ Rn , and r > 0, such that ψ1 = ψ2 in the ball B(x, r) then Tt−h,t ψ1 − Tt−h,t ψ2 = o(h), as h → 0+ . III Regularity: for any sequence of positive numbers d = (dk ), any compact set K ⊂ Rn and for every ψ ∈ C ∞ (Rn ) ∩ X there exists a function mK,f,d(·) : R+ → R+ such that mK,f,d (0+ ) = 0, |Tt [ψ + λη] − Tt [ψ] − λη(x)| ≤ mK,f,d (λ)t, for any (x, η) ∈ K × Qd and any λ, t ≥ 0. IV Translation: for any compact subset K ⊂ Rn and every ψ ∈ Cc∞ (K), there exists a function nK,ψ : R+ → R+ , with nK,ψ (0+ ) = 0 such that |τy Tt [ψ](x) − Tt [τy ψ](x)| ≤ nK,ψ (|y|)t, for any x ∈ K and t ≥ 0. Property [I] holds by (74). Properties [II] and [III] follow from improved moment estimates of the previous section, as we explain next. Property [IV] follows from (72) and (83). Let us now give the details about how to check Properties [II] and [III]. First, we will use Proposition 30 to prove that [II] holds. Assume that ψ1 and ψ2 satisfy the

48

DIOGO A. GOMES & ENRICO VALDINOCI

conditions of [II], and let x be an optimal trajectory for ψ2 , that is x optimizes (5) for ψ := ψ2 and x(t − h) = x. Then, by (72) and Propositions 24 and 26,   Z T  x(s) , v(s), ω ds + ψ2 (x(T )) Tt−h,t ψ2 = lim E L x(s), ǫ→0 ǫ t

while

Tt−h,t ψ1 ≤ lim E ǫ→0

Z

t

T

   x(s) , v(s), ω ds + ψ2 (x(T )) , L x(s), ǫ

via a suitable subsequence. Consequently, using that ψ1 and ψ2 agree on B(x, r), we gather that

Tt−h,t ψ1 − Tt−h,t ψ2 ≤ (kψ1 k∞ + kψ2 k∞ ) P (|x − x| > r), which is o(h) by Proposition 30. To prove [III], we argue in an analogous way. Namely, we take an optimal trajectory for ψ and we take limit for ǫ → 0, as above. In this case, we also exploit Lemma 28 to conclude that TT −h,T (ψ + λη) − TT −h,T ψ − λη ≤ lim Eψ(x(T )) + λη(x(T )) − ψ(x(T )) − λη(x(T − h)) ǫ→0

≤ lim λE(η(x(T )) − η(x(T − h))) ≤ CkDηk∞λh ≤ Cd1 λh , ǫ→0

as required. The desired result then follows from Theorem 3.1 of [Bit01].



We remark that Theorem 3.1 of [Bit01] explicitly gives that − H(D 2 ψ(x), Dψ(x), x) = lim

(113)

T →0

T0,T ψ − ψ , T

in the notation of (110) and (111). Therefore, under the additional Assumption 16, we can characterize completely H as the Legendre transform of L. Proposition 33. If also Assumption 16 holds, then H is given by (114)

H(M, p, x) =

sup

−p · ̟ − Ξ : M − L(x, ̟, Ξ) .

Ξ=σσT ,σ∈Mat(n×n),̟∈Rn

Proof. We will be using formula (113) to establish the result. We only consider case C, since cases A and B are analogous, with just less variables involved. By taking ϕ(x, t) := t and ϕ(x, t) := 1 in (69), we see that if ̺ and ν¯T satisfy (69), then they have total mass T and 1, respectively. Also, the supremum in (114) is finite and attained, due to Propositions 14 and 15. Thus, take ̟0 = ̟0 (x), σ0 = σ0 (x) and Ξ0 = Ξ0 (x) = σ0 σ0T that satisfy sup −Dx ψ(x) · ̟ − Ξ : Dx2 ψ(x) − L(x, ̟, Ξ) ̟,Ξ

= −Dx ψ(x) · ̟0 − Ξ0 : Dx2 ψ(x) − L(x, ̟0 , Ξ0 ) .

GENERALIZED MATHER PROBLEM AND HOMOGENIZATION

We also consider the dynamics of the associated control problem dx0 = Sx̟0 ,̟0,σ0 (x0 )dt + SxΞ0 ,̟0 ,Ξ0 (x0 )dWt , and corresponding measures ̺0 (x′ , ω, Ξ, s) and ν0,T (x′ ), given by Z φ(x′ , ̟, Ξ, s)d̺0 Rn ×Rn ×Mat(n×n)

:= E

Z

T

φ(x0 (s), Sx̟0 ,̟0,σ0 (x0 (s)), SxΞ0 ,̟0 ,Ξ0 (x0 (s)), s)ds

0

and

Z

ϕ(x′ )dν0,T (x′ ) := Eϕ(x0 (T )). Rn

We have that (69) holds, by Dynkin’s Formula, and so Z Z ψ − TT ψ ≥ ψ − ψd¯ ν0,T − Ld̺0 Z = −Dx ψ · ̟ − Ξ : Dx2 ψ − Ld̺0 Z = −Dx ψ(x) · ̟0 − Ξ0 : Dx2 ψ(x) − L(x, ̟0 , Ξ0 )d̺0 Z + Dx ψ · ̟ + Ξ : Dx2 ψ +L − Dx ψ(x) · ̟0 − Ξ0 :

Dx2 ψ(x)

− L(x, ̟0 , Ξ0 ) d̺0 .

≥ T sup −Dx ψ(x) · ̟ − Ξ : Dx2 ψ(x) − L(x, ̟, Ξ) ̟,Ξ

−C

Z

T

E|x0 (s) − x| ds,

0

thanks to Assumption 16. That is, ψ − TT ψ ≥ sup −Dx ψ(x) · ̟ − Ξ : Dx2 ψ(x) − L(x, ̟, Ξ) T ̟,Ξ (115) Z T C − E|x0 (s) − x| ds . T 0 We now observe that, by Lemma 27, Z T Z T E|x0 (s) − x| ds ≤ (E|x0 (s) − x|2 )1/2 ds ≤ CT 3/2 , 0

0

and so (116)

as T → 0.

1 T

Z

0

T

E|x0 (s) − x| ds → 0,

49

50

DIOGO A. GOMES & ENRICO VALDINOCI

Consequently, making use of (113), (115) and (116), we thus deduce that H(D 2 ψ(x), Dψ(x), x) ≥ sup −Dx ψ(x) · ̟ − Ξ : Dx2 ψ(x) − L(x, ̟, Ξ).

(117)

̟,Ξ

Now we will establish the reverse inequality and complete the proof of (114). For that matter, let ̺(x′ , ̟, Ξ, s) and νT (x′ ) be minimizing measures for (73) under the constraint in (69), with t = 0. Then, T sup −Dx φ(x) · ̟ − Ξ : Dx2 φ(x) − L(x, ̟, Ξ) ̟,Ξ Z ≥ −Dx φ(x) · Sx̟′ ,̟,Ξ(x) − SxΞ′ ,̟,Ξ(x) : Dx2 φ(x) − L(x, Sx̟′ ,̟,Ξ(x), SxΞ′ ,̟,Ξ(x))d̺ Z Z 2 ≥ −Dx φ(x) · ̟ − Ξ : Dx φ(x) − L(x, ̟, Ξ) d̺ − C |x′ − x|d̺ ,

thanks to Assumption 16. The above estimate, the holonomy constraint and Proposition 31 then give sup −Dx φ(x) · ̟ − Ξ : Dx2 φ(x) − L(x, ̟, Ξ) ̟,Ξ

R  Z 1/2 ψd¯ νT − Ld̺ C ′ 2 ≥ T |x − x| d̺ − T T ψ − TT ψ − CT 1/2 ≥ T as long as T is small enough. Therefore, by sending T to zero and recalling (113), ψ−

R

sup −Dx ψ(x) · ̟ − Ξ : Dx2 ψ(x) − L(x, ̟, Ξ) ≥ H(D 2 ψ(x), Dψ(x), x) − 0. ̟,Ξ

This and (117) complete the proof of (114).



13. Conclusion Proof of Theorem 1. Propositions 4 and 9 imply the first claim of Theorem 1. The second claim follows from Propositions 18 and 20 and Remark 19. By collecting the results in Propositions 24, 26 and 32, we obtain the last claim of Theorem 1.  References [AB02]

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Diogo A. Gomes Departamento de Matem´ atica, Instituto Superior T´ecnico, Lisboa, 1049-001, Portugal e-mail: [email protected]

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Enrico Valdinoci Dipartimento di Matematica, Universit`a di Roma Tor Vergata, Roma, I-00133, Italy e-mail: [email protected]

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