Second order mean field games with degenerate diffusion and local coupling

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Second order mean field games with degenerate diffusion and local coupling

Pierre Cardaliaguet, J. Graber, Alessio Porretta, Daniela Tonon

To cite this version:

Pierre Cardaliaguet, J. Graber, Alessio Porretta, Daniela Tonon. Second order mean field games with degenerate diffusion and local coupling. Nonlinear Differential Equations and Applications, Springer Verlag, 2015, 22 (5), pp.1287-1317. �hal-01049834�

DIFFUSION AND LOCAL COUPLING

PIERRE CARDALIAGUET, P. JAMESON GRABER, ALESSIO PORRETTA, AND DANIELA TONON Version: July 25, 2014

Abstract. We analyze a (possibly degenerate) second order mean field games system of partial differential equations. The distinguishing features of the model considered are (1) that it is not uniformly parabolic, including the first order case as a possibility, and (2) the coupling is a local operator on the density. As a result we look for weak, not smooth, solutions. Our main result is the existence and uniqueness of suitably defined weak solutions, which are characterized as minimizers of two optimal control problems. We also show that such solutions are stable with respect to the data, so that in particular the degenerate case can be approximated by a uniformly parabolic (viscous) perturbation.

Introduction

This paper is devoted to the analysis of second order mean field games systems with a local coupling. The general form of these systems is:







(i) −∂_tφ−A_ij∂_ijφ+H(x, Dφ) =f(x, m(x, t)) (ii) ∂_tm−∂_ij(A_ijm)−div(mD_pH(x, Dφ)) = 0 (iii) m(0) =m₀, φ(x, T) =φ_T(x)

(1) whereA:R^d→R^d×dis symmetric and nonnegative, the HamiltonianH:R^d×R^d→Ris convex in the second variable, the couplingf :R^d×[0,+∞)→[0,+∞) is increasing with respect to the second variable,m₀ is a probability density and φ_T :R^d→Ris a given function. The functions H and f, and the matrix A, could as well depend on time, but since this does not give any additional difficulty, we will avoid it just to simplify notations.

Mean field game systems (MFG systems) have been introduced simultaneously by Lasry-Lions [17, 18, 19, 21] and Huang-Caines-Malham´e [15] to describe Nash equilibria in differential games with infinitely many players. The first unknownφ=φ(t, x) is the value function of an optimal control problem of a typical small player. In this control problem, the dynamics is given by the controlled stochastic differential equation

dXs=vsds+ Σ(Xs)dBs,

where (v_s) is the control, (B_s) is a Brownian motion and ΣΣ^T =A. The cost is given by E

Z T 0

H^∗(X_s,−v_s) +f(X_s, m(s, X_s))ds+φ_T(X_T)

For each timet∈[0, T] the quantitym(t, x) denotes the density of population of small players at positionx. In the control problem the term involvingf formalizes the fact that the cost of the player depends on this densitym. Asφis the value function of this control problem, the optimal control of a typical small player is formally given by the feedback (t, x)→ −DpH(x, Dφ(t, x)).

Hence the second equation (1)-(ii) is the Kolmogorov equation of the process (X_s) when the small player plays in an optimal way. By the mean field approach, this equation also describes the evolution of the whole population density as all players play in an optimal way.

1

MFG systems with uniformly parabolic diffusions—typically A_ij∂_ijφ = ∆φ—have been the object of several contributions, either by PDE methods (see, e.g., [8, 17, 18, 19, 21, 12, 13, 22]) or by stochastic techniques (see, e.g., [3, 15]): in this setting one often expects the solutions to be smooth, at least if the coupling is nonlocal and regularizing or if it has a “small growth”. The case of local couplings with an arbitrary growth has been discussed in [8] for purely quadratic hamiltonians (i.e. H=|Dφ|²), in which case solutions are proved to be smooth, and in [22] for general hamiltonians, by proving existence and uniqueness of weak solutions.

Here we concentrate on degenerate parabolic equations. In this case the usual fixed point techniques used to prove the existence of solutions in the uniformly parabolic setting break down by lack of regularity. One then has to rely on convex optimization methods: this idea, which goes back to the analysis of some optimal transport problems (see [2, 6]), has already been used to study first order MFG systems (i.e., A ≡0): see [4, 5, 14]. However it was not clear in these papers wether the weak solution was stable with respect to viscous approximation, i.e., if we could obtain weak solutions of the first order MFG systems by passing to the limit in uniformly parabolic ones. This issue has partially motivated our study.

In this paper we show the existence and uniqueness of a weak solution for the degenerate mean field game system (1) as well as the stability of solutions with respect to perturbation of the data: this includes of course stability by viscous approximation.

Concerning existence and uniqueness of solutions, the paper improves the existing results in two directions. First we consider non uniformly parabolic second order MFG systems, which have never been considered before. The introduction of second order derivatives induces several issues: in particular, in contrast with the first order equations, we do not expect the functionφ to be BV (as in [5, 14]), which obliges us to be very careful about trace properties. Secondly—

and this is new even for first order MFG systems—we drop a restriction between the growth condition ofH and the growth condition of f, restriction which was mandatory in the previous papers: see [4, 5]. To overcome the difficulty, we provide newintegral estimatesfor subsolutions of Hamilton-Jacobi equations with unbounded right-hand side (Theorems 2.1 and 2.3). We think that these results are of independent interest.

With these estimates in hand, the structure of proof for the existence and uniqueness follows roughly the lines already developed in [4, 5, 6, 14]: basically it amounts to show that the MFG system can be viewed as an optimality condition for two convex problems, the first one being an optimal control of Hamilton-Jacobi equation, the second one an optimal control problem for the Fokker-Planck equation (see section 3 for details). A byproduct of this approach is the stability of weak solutions with respect to the data (Theorem 5.5), which can be obtained by Γ−convergence techniques.

The paper is organized as follows. First we introduce the notation and assumptions needed throughout the paper (section 1). Then (section 2) we give our new estimates for subsolutions of Hamilton-Jacobi equations with a superlinear growth in the gradient variable and an unbounded right-hand side. In section 3, we introduce the two optimal control problems and show that they are in duality while in section 4 we show that the optimal control problem for the Hamilton- Jacobi equation has a “relaxed solution.” Section 5 is devoted to the analysis of the MFG system (existence, uniqueness and characterization). In the last section we discuss the stability of solutions.

Acknowledgement: This work has been partially supported by the Commission of the European Communities under the 7-th Framework Programme Marie Curie Initial Training Net- works Project SADCO, FP7-PEOPLE-2010-ITN, No 264735, by the French National Research Agency ANR-10-BLAN 0112 and ANR-12-BS01-0008-01 and by the Italian Indam Gnampa project 2013 “Modelli di campo medio nelle dinamiche di popolazioni e giochi differenziali”.

1. Notations and assumptions

Notations : We denote byhx, yithe Euclidean scalar product of two vectors x, y∈R^dand by

|x|the Euclidean norm ofx. We use conventions on repeated indices: for instance, if a, b∈R^d, we often write a_ib_i for the scalar product ha, bi. More generally, if A and B are two square symmetric matrices of size d×d, we write A_ijB_ij for Tr(AB).

To avoid further difficulties arising from boundary issues, we work in the flatd−dimensional torus T^d = R^d\Z^d. We denote by P(T^d) the set of Borel probability measures over T^d. It is endowed with the weak convergence. For k, n∈N andT > 0, we denote byC^k([0, T]×T^d,Rⁿ) the space of maps φ= φ(t, x) of class C^k in time and space with values inRⁿ. For p∈ [1,∞]

and T >0, we denote by L^p(T^d) and L^p((0, T)×T^d) the set ofp−integrable maps over T^d and [0, T]×T^drespectively. We often abbreviateL^p(T^d) andL^p((0, T)×T^d) into L^p. We denote by kfkp theL^p−norm of a mapf ∈L^p.

Assumptions: We now collect the assumptions on the couplingf, the HamiltonianH and the initial and terminal conditionsm₀ and φ_T. These conditions are supposed to hold throughout the paper.

(H1) (Condition on the coupling) the coupling f :T^d×[0,+∞) → R is continuous in both variables, increasing with respect to the second variable m, and there exist q >1 and C₁ such that

1 C1

|m|^q−1−C₁ ≤f(x, m)≤C₁|m|^q−1+C₁ ∀m≥0. (2) Moreover we ask the following normalization condition to hold:

f(x,0) = 0 ∀x∈T^d. (3)

We denote by p the conjugate of q: 1/p+ 1/q = 1.

(H2) (Conditions on the Hamiltonian) The Hamiltonian H :T^d×R^d → R is continuous in both variables, convex and differentiable in the second variable, withDpHcontinuous in both variables, and has a superlinear growth in the gradient variable: there exist r >1 and C2>0 such that

1

rC₂|ξ|^r−C₂ ≤H(x, ξ)≤ C₂

r |ξ|^r+C₂ ∀(x, ξ)∈T^d×R^d. (4) We note for later use that the Fenchel conjugate H^∗ of H with respect to the second variable is continuous and satisfies similar inequalities

1

r^′C₂|ξ|^r′ −C₂≤H^∗(x, ξ)≤ C₂

r^′ |ξ|^r′ +C₂ ∀(x, ξ)∈T^d×R^d, (5) where r^′ is the conjugate ofr: 1

r + 1 r^′ = 1.

(H3) (Conditions on A) there exists a Lipschitz continuous map Σ : T^d → R^d×D such that ΣΣ^T =A: let C₃ be a constant such that

|Σ(x)−Σ(y)| ≤C₃|x−y| ∀x, y∈T^d, (6) Moreover we suppose that

eitherr ≥p or A≡0. (7)

We recall thatp is the conjugate of q.

(H4) (Conditions on the initial and terminal conditions) φ_T : T^d → R is of class C², while m₀:T^d→Ris aC¹ positive density (namelym₀ >0 and

Z

T^d

m₀dx= 1).

Condition (3) is just a normalization condition, which we may assume without loss of gen- erality. Indeed, if all the conditions (H1). . .(H4) but (3) hold, then one just needs to replace f(x, m) byf(x, m)−f(x,0) andH(x, p) byH(x, p)−f(x,0): the new H andf still satisfy the above conditions (H1). . .(H4) with (3).

Let us set

F(x, m) =





 Z _m

0

f(x, τ)dτ if m≥0

+∞ otherwise

Then F is continuous onT^d×(0,+∞), differentiable and strictly convex inm and satisfies 1

qC₁|m|^q−C₁≤F(x, m)≤ C₁

q |m|^q+C₁ ∀m≥0 (8)

(changing the constant C₁ if necessary). Let F^∗ be the Fenchel conjugate of F with respect to the second variable. Note thatF^∗(x, a) = 0 fora≤0 becauseF(x, m) is nonnegative and equal to +∞ form <0. Moreover,

1

pC₁|a|^p−C1 ≤F^∗(x, a)≤ C₁

p |a|^p+C1 ∀a≥0. (9) 2. Basic estimates on solutions of Hamilton-Jacobi equations

In this section we prove estimates in Lebesgue spaces for subsolutions of Hamilton-Jacobi equations of the form

(i) −∂_tφ−A_ij(x)∂_ijφ+H(x, Dφ)≤α(t, x)

(ii) φ(x, T)≤φT(x) (10)

in terms of Lebesgue norms ofαandφ_T. We assume that (4) and (6) hold, and (10) is understood in the sense of distributions. This means that Dφ∈L^r and, for any nonnegative test function ζ ∈C_c^∞((0, T]×T^d),

− Z

Td

ζ(T)φ_T + Z T

0

Z

Td

φ∂_tζ+hDζ, ADφi+ζ(∂_iA_ij∂_jφ+H(x, Dφ))≤ Z T

0

Z

Td

αζ.

The estimates will be a consequence of the divergence structure of second order terms.

Theorem 2.1. Assume that φ ∈ L^r((0, T);W^1,r(T^d)) is a nonnegative function satisfying, in distributional sense,

(i) −∂_tφ−∂_i(A_ij(x)∂_jφ) +c₀|Dφ|^r ≤α(t, x)

(ii) φ(x, T)≤φ_T(x) (11)

for some nonnegative, bounded Lipschitz matrixA_ij, and somer >1,c₀>0,α∈L^p((0, T)×T^d) andφ_T ∈L^∞(T^d). Then, there exists a constant C=C(p, d, r, c₀, T,kαk_Lp((0,T)×Td),kφ_Tk_Lη(Td)) such that

kφk_L^∞_((0,T),Lη(Td))+kφk_Lγ((0,T)×Td) ≤C where η= d(r(p−1)+1)

d−r(p−1) and γ = _d−r(p−1)^rp(1+d) ifp < 1 +^d_r andη =γ = +∞ if p >1 +^d_r. We note for later use thatγ > r.

Proof. Up to a rescaling, we may assume that c₀ = 1. We first claim that, for any real function g∈W^1,∞(R) which is nondecreasing, and nonnegative inR₊, we have

Z

Td

G(φ(τ))dx+ Z T

τ

Z

Td

|Dφ|^rg(φ)dxdt≤ Z T

τ

Z

Td

α g(φ)dxdt+ Z

Td

G(φT)dx (12) for a.e. τ ∈(0, T), whereG(r) =Rr

0 g(s)ds.

There are several possible ways to justify (12), one is to use regularization.

We first extend φ to (0, T + 1]×T^d by defining φ=φ_T on [T, T + 1]. Then it still holds in the sense of distributions

−∂_tφ−∂_i

A˜_ij(t, x)∂_jφ

+|Dφ|^rχ_(0,T)≤α(t, x)˜ (13) where ˜A_ij(t, x) =A_ij(x)χ_(0,T)(t) and ˜α(t, x) =α(t, x)χ_(0,T)(t). Let ξ be a standard convolution kernel in (t, x) defined on R^d+1 and ξ^ǫ(t, x) =ξ((t, x)/ǫ)/(ǫ)^d+1,ξ^ǫ ≥0, R

Rd+1ξ^ǫ(t, x)dtdx = 1, for all ǫ >0. Letφ_ǫ =ξ^ǫ⋆ φand α_ǫ=ξ^ǫ⋆α. Then˜ φ_ǫ, α_ǫ are C^∞ and converge toφ, α in their respective Lebesgue spaces. Convolving ξ^ǫ with (13) we obtain on (0, T + 1)×T^d:

−∂_tφ_ǫ−∂_i

A˜_ij(t, x)∂_jφ_ǫ

+|ξ_ǫ⋆(Dφ)|˜ ^r ≤α_ǫ(t, x) +R_ǫ

where Dφ˜ = Dφχ_(0,T) and we used the fact that Dφ 7→ |Dφ|^r is convex, and where R_ǫ =

−∂i

A˜ij(t, x)∂jφǫ

+ξǫ⋆ ∂i

A˜ij(t, x)∂jφ . Using the notation (cf. [10])

[ξ^ǫ, c](f) :=ξ^ǫ⋆(cf)−c(ξ^ǫ⋆ f)

we can rewriteR_ǫ asR_ǫ= [ξ_ǫ, ∂_iA˜_ij](∂_jφ) + [ξ_ǫ,A˜_ij∂_i](∂_jφ). Invoking [10, Lemma II.1], we have thatR_ǫ→0 in L^r, sinceDφ∈L^r and A_ij is Lipschitz.

Multiplying byg(φǫ) and integrating over [τ, T +ε]×T^d, for τ ∈(0, T), it follows Z

T^d

G(φ_ǫ(τ))dx− Z

T^d

G(φ_ǫ(T +ε))dx+ Z T

τ

Z

T^d

A_ij(x)∂_jφ_ǫg^′(φ_ǫ)∂_iφ_ǫdxdt +

Z T+ε τ

Z

T^d

|ξǫ⋆(Dφ)|˜ ^rg(φǫ)dxdt≤ Z T+ε

τ

Z

T^d

g(φǫ)αǫ(t, x)dxdt+ Z T+ε

τ

Z

T^d

g(φǫ)Rǫdxdt . Since RT

τ

R

TdA_ij(x)∂_jφ_ǫg^′(φ_ǫ)∂_iφ_ǫdxdt≥0, and since φ_ǫ(T +ε) =ξ^ǫ⋆ φ_T, we obtain Z

Td

G(φ_ǫ(τ))dx− Z

Td

G(ξ^ǫ⋆ φ_T)dx+ Z _T+ε

τ

Z

Td

|ξ_ǫ⋆(Dφ)|˜ ^rg(φ_ǫ)dxdt

≤ Z _T+ε

τ

Z

Td

g(φ_ǫ)α_ǫ(t, x)dxdt+ Z _T+ε

τ

Z

Td

g(φ_ǫ)R_ǫdxdt .

Sincegis bounded, whileR_ǫandα_εconverge inL^r((0, T)×T^d) and inL^p((0, T)×T^d) respectively, we can pass to the limit asǫgoes to zero and for almost everyτ we get (12).

Now we proceed with the desired estimate. First we observe that, up to replacingg(r) with g(r∧k), we can assume that φis bounded and that g may be any C¹ function. In particular, we takeg(φ) =φ^(σ−1)r forσ >1, obtaining

1 (σ−1)r+ 1

Z

Td

φ(τ)^(σ−1)r+1dx+ 1 σ^r

Z T τ

Z

Td

|Dφ^σ|^rdxdt

≤ Z T

τ

Z

Td

α φ^(σ−1)rdxdt+ 1 (σ−1)r+ 1

Z

Td

φ^(σ−1)r+1_T dx.

Let us denote henceforth bycpossibly different constants only depending onr,σ,dandT. By arbitrariness ofτ, the previous inequality implies

kφ^σk

(σ−1)r+1 σ

L^∞((0,T);L^{(σ−1)r+1σ} (T^d))

+kDφ^σk^r_Lr((0,T)×Td)

≤c Z T

0

Z

T^d

α φ^(σ−1)rdxdt+c Z

T^d

φ^(σ−1)r+1_T dx.

On the other hand, by interpolation we have (see e.g. [9, Proposition 3.1, Chapter 1]) kvk^q_{Lq((0,T)×Td)}≤ckvk

ηr d

L^∞((0,T);L^η(Td))kDvk^r_Lr((0,T)×T^d) whereq=r^d+η_d (14) for anyv∈L^r((0, T);W^1,r(T^d)) such thatR

Tdv(t)dx= 0 a.e. in (0, T). So we deduce that kφ^σk^q_Lq((0,T)×T^d)≤c

Z T 0

Z

Td

α φ^(σ−1)rdxdt+ Z

Td

φ^(σ−1)r+1_T dx 1+^r_d

+c Z _T

0

Z

T^d

φ(t)^σdx q

dt

forη= ^(σ−1)r+1_σ and q =r^η+d_d . We choose σ such that σq = (σ−1)rp^′ and therefore, by H¨older inequality, we conclude

kφ^σk^q_{Lq((0,T)×Td)}≤ckαk¹⁺

r d

L^p((0,T)×Td)kφ^σk

q p′(1+^r_d)

L^q((0,T)×Td)+c Z

Td

φ^(σ−1)r+1_T dx 1+^r_d

+c Z _T

0

Z

T^d

φ(t)^σdx q

dt . Since R

Tdφ(t)dx is estimated in terms of kαk_L1((0,T)×Td) and kφ_Tk_L1(Td), last term can be ab- sorbed into the left-hand side up to a constant C =C(kαk_L1((0,T)×T^d),kφ_Tk_L1(T^d)). Moreover, sincep <1 +^d_r, we have _p^q′(1 +_d^r)< q. Hence we end up with an estimate

kφ^σk^q_{Lq((0,T)×Td)}≤C(kαk_Lp((0,T)×Td),kφ_Tk_L(σ−1)r+1(Td)). Computing the value ofσ in terms ofr and p we get

qσ= rp(1 +d)

d−r(p−1) and (σ−1)r+ 1 = d(r(p−1) + 1) d−r(p−1) so the first part of the Theorem is proved.

Finally, we prove the L^∞ estimate by using a strategy which goes back to [23]. To this purpose, we replace φ withφ−k and use (12) with g(s) = (s+)^r′; for anyk ≥ kφTk_L^∞₍T^d) we obtain

Z

Td

[(φ−k)₊(τ)]^σ+1dx+ Z T

τ

Z

Td

|D(φ−k)^σ₊|^rdxdt≤ Z T

τ

Z

Td

α(φ−k)^σ₊dxdt withσ =r^′. Using as before the embedding (14) we get

k(φ−k)^σ₊k^q_Lq((0,T)×T^d)≤c Z T

0

Z

Td

α(φ−k)^σ₊dxdt 1+^r_d

+c Z _T

0

Z

Td

(φ−k)^σ₊dx q

dt

≤c Z _T

0

Z

T^d

α(φ−k)^σ₊dxdt 1+^r_d

+c Z _T

0

|{x :φ(t)> k}|^q−1 Z

T^d

(φ−k)^{σ q}₊ dxdt , where, using thatσ=r^′, we have

q=r

σ+1 σ +d

d = r

d(d+ 2−1

r). (15)

Notice that |{x : φ(t) > k}| is uniformly small provided k is large, only depending on kαk_L¹ and kφ_Tk_L1. Therefore, absorbing last term in the left-hand side we deduce

k(φ−k)^σ₊k^q_Lq((0,T)×T^d) ≤c Z T

0

Z

Td

α(φ−k)^σ₊dxdt ¹⁺

r d

for some c = c(kαk_L1((0,T)×Td),kφ_Tk_L1(Td)). One can check that, since r > 1, (15) implies q >1 +_d^r and, in particular, ¹_q+ ¹_p <1. Thus, by H¨older inequality we get

k(φ−k)^σ₊k^q_Lq((0,T)×T^d) ≤ck(φ−k)^σ₊k¹⁺

r d

L^q((0,T)×T^d)kαk¹⁺

r d

L^p((0,T)×T^d)|A_k|^{(1−1q−1p)(1+rd)}, whereA_k :={(t, x) :φ(t, x)> k}. Since, for anyh > k we have

Z T 0

Z

Td

(φ−k)^σq₊ dxdt≥ |A_h|(h−k)^σq, we end up with the inequality

|A_h|^1−1q(1+rd)(h−k)^{σ q−σ(1+rd)}≤ k(φ−k)^σ₊k^q−(1+

r d)

L^q((0,T)×Td)≤ckαk¹⁺

r d

L^p((0,T)×Td)|A_k|^{(1−1q−1p)(1+rd)} which means that

|A_h| ≤C |A_k|^β

(h−k)^δ ∀h > k≥ kφ_Tk_L^∞₍T^d)

for someC =C(kαk_Lp((0,T)×Td)), some δ >0 and with β = ⁽¹⁻

1

q−¹_p)(1+^r_d)

1−¹_q(1+_d^r) . One can check that β >1 since p >1 +^d_r. Therefore, by a classical iteration lemma (see e.g. [23]), it follows that

|A_k0|= 0 for some (explicit) k₀ >0, which in particular implies the desired bound in terms of

kαk_Lp((0,T)×Td) and kφ_Tk_L^∞₍Td).

Remark 2.2. The assumption that φ is nonnegative can be dropped and in this case the estimates are given onφ₊; indeed, if φsatisfies (11), then φ₊also does. This can be seen in the previous proof by takingg=g(r₊), withg(0) = 0.

Let us also stress that the Lipschitz continuity of the matrixA was only used to recover the estimate from the distributional formulation (namely, to be sure that φis limit of solutions of smooth approximating problems). The constant C of the estimate, however, does not depend on Ain any way; in particular, the estimate will hold uniformly for any viscous approximation to possibly less regular matrices.

As a corollary, we deduce the following result for problem (10).

Theorem 2.3. Assume that (4)and (6)hold true and letφsatisfy (10)withα∈L^p((0, T)×T^d), φ_T ∈L^∞(T^d). Then, φ₊ satisfies the estimates of Theorem 2.1. In particular, if φ is bounded below, we have

kφk_L^∞_((0,T),Lη(Td))+kφk_Lγ((0,T)×Td) ≤C whereη= d(r(p−1)+1)

d−r(p−1) andγ = _d−r(p−1)^rp(1+d) ifp <1+^d_r andη =γ = +∞ifp >1+^d_r, with a constant C depending on T, p, d, r, C₂, C₃ (appearing in (4) and (6)) and on kαk_Lp((0,T)×T^d),kφ_Tk_Lη(T^d)

and kφ₋k_L^∞₍Td).

3. Two optimization problems

Mean field games systems with local coupling can be studied as an optimality condition between two problems in duality.

The first optimization problem is described as follows: let us denote by K₀ the set of maps φ∈ C²([0, T]×T^d) such thatφ(T, x) =φ_T(x) and define, on K₀, the functional

A(φ) = Z _T

0

Z

T^d

F^∗(x,−∂_tφ(t, x)−A_ij∂_ijφ+H(x, Dφ(t, x))) dxdt− Z

T^d

φ(0, x)dm₀(x). (16) Then the problem consists in optimizing

φ∈Kinf0

A(φ). (17)

For the second optimization problem, let K₁ be the set of pairs (m, w) ∈ L¹((0, T)×T^d)× L¹((0, T)×T^d,R^d) such that m(t, x) ≥0 a.e., with

Z

Td

m(t, x)dx = 1 for a.e. t ∈ (0, T), and which satisfy in the sense of distributions the continuity equation

∂_tm−∂_ij(A_ij(x)m) + div(w) = 0 in (0, T)×T^d, m(0) =m₀. (18) On the set K₁, let us define the following functional

B(m, w) = Z T

0

Z

T^d

m(t, x)H^∗

x,−w(t, x) m(t, x)

+F(x, m(t, x))dxdt+ Z

T^d

φ_T(x)m(T, x)dx where, form(t, x) = 0, we impose that

m(t, x)H^∗

x,−w(t, x) m(t, x)

=

+∞ if w(t, x)6= 0 0 if w(t, x) = 0 .

Since H^∗ and F are bounded from below and m ≥0 a.e., the first integral in B(m, w) is well defined in R∪ {+∞}. In order to give a meaning to the last integral R

TdφT(x)m(T, x)dx we proceed as follows: let us definev(t, x) =−w(t, x)

m(t, x) if m(t, x)>0 and v(t, x) = 0 otherwise.

Thanks to the growth ofH^∗ (implied by (5), which follows from (H2)), B(m, w) is infinite if m|v|^r′ ∈/ L¹(dxdt). Therefore, we can assume without loss of generality thatm|v|^r′ ∈L¹(dxdt), or, equivalently, that v ∈ L^r′(m dxdt). In this case equation (18) can be rewritten as a Kol- mogorov equation

∂tm−∂ij(Aij(x)m)−div(mv) = 0 in (0, T)×T^d, m(0) =m0. (19) Lemma 3.1. The map t7→m(t) is H¨older continuous a.e. for the weak* topology of P(T^d).

This Lemma implies, in particular, that the measurem(t) is defined for any t, therefore the second integral term in the definition ofB(m, w) is well defined.

For the sake of completeness, we give the proof here.

Proof. We first extend the pairs (m, w) to [−1, T]×T^d by defining m = m₀ on [−1,0] and w(s, x) = 0 for (s, x)∈(−1,0)×T^d. Note that ∂_tm−∂_ij( ˜A_ij(t, x)m) + div(w) = 0 holds in the sense of distributions on (−1, T)×T^d, where ˜A_ij(t, x) =A_ij(x) if t ∈ (0, T) and ˜A_ij(t, x) = 0 otherwise. Let ξ be a standard convolution kernel in (t, x), a support compact on R^d+1 and ξ^ǫ(t, x) = ξ((t, x)/ǫ)/(ǫ)^d+1, ξ^ǫ ≥ 0, R

Rd+1ξ^ǫ(t, x)dtdx = 1, for all ǫ > 0. Letm_ǫ =ξ^ǫ⋆ m and w_ǫ = ξ^ǫ⋆ w. Then m_ǫ, w_ǫ are C^∞ and R

T^dm_ǫ(t, x)dx = 1 for all t ∈ (0, T) and ǫ > 0 small enough.

Convolvingξ^ǫ with (18), we obtain

∂_tm_ǫ−∂_ij(ξ^ǫ∗( ˜A_ij(t, x)m)) + div(w_ǫ) = 0 in (−1/2, T)×T^d, with

m_ǫ(−1/2, x) = Z

R

Z

Td

ξ^ǫ(s, x−y)m₀(y)dyds.

The equation can be rewritten as

∂_tm_ǫ−∂_ij( ˜A^ǫ_ij(t, x)m_ǫ))−div(m_ǫv_ǫ) = 0 in (−1/2, T)×T^d (20) where ˜A^ǫ_ij = ^{ξǫ⋆( ˜}_m^Aijm)

ǫ and v_ǫ =−_m^wǫ

ǫ.

Let us consider the following stochastic differential equations defined for allǫ >0 dX_t^ǫ =v_ǫ(t, X_t^ǫ)dt+ Σ_ǫ(X_t^ǫ)dB_t^ǫ t∈[−1/2, T]

X_−1/2^ǫ =Z_−1/2^ǫ , (21)

wheredB_t^ǫ is a standardd-dimensional Brownian motion over some probability space (Ω,A,P), Σ_ǫΣ^T_ǫ = ˜A^ǫ, and the initial condition Z_−1/2^ǫ ∈L¹(T^d) is random, independent of (B_t^ǫ) and with lawm_ǫ(−1/2,·).

For all ǫ > 0, the vector field v_ǫ is continuous, uniformly Lipschitz continuous in space and bounded. Therefore, there exists a unique solution to (21). Moreover, as a consequence of Ito’s formula, we have that, if the density L(Z₀^ǫ) = ξ^ǫ⋆ m₀, then m_ǫ(t) = L(X_t^ǫ) solves (20) in the sense of distributions.

Let d₁ be the Kantorovich-Rubinstein distance on P(T^d) and γ_ǫ ∈ Π(m_ǫ(t), m_ǫ(s)) the law of the pair (X_t^ǫ, X_s^ǫ) for 0 ≤ s < t ≤ T, where Π(m_ǫ(t), m_ǫ(s)) is the set of Borel probability measuresµonT^d×T^dsuch that µ(A×T^d) =mǫ(t, A) andµ(T^d×A) =mǫ(s, A) for any Borel setA∈T^d. We have

d₁(m_ǫ(t), m_ǫ(s))≤ Z

T^d×T^d

|x−y|dγ_ǫ(x, y) =E[|X_t^ǫ−X_s^ǫ|].

Moreover,

E[|X_t^ǫ−X_s^ǫ|]≤E[ Z t

s

|v_ǫ(τ, X_τ^ǫ)|dτ] +E

Z t s

Σ_ǫ(X_τ^ǫ)dB_τ

≤ Z t

s

Z

Td

|vǫ(τ, x)|mǫ(τ, x)dxdτ +

E Z t

s

ΣǫΣ^∗_ǫ(X_τ^ǫ)dτ 1/2

≤ Z t

s

Z

Td

|v_ǫ(τ, x)|m_ǫ(τ, x)dxdτ +kAk_∞C|t−s|¹². Recalling the definition of vǫ, we have that mǫ|vǫ|^r′ = ^|wǫ|r

′

m^{r′ −}ǫ ¹

belongs to L¹([0, T]×T^d) for all ǫ >0. Indeed, the function (m, w)7→ ^|w|r

′

m^{r′ −1} is convex and ^|w|r

′

m^{r′ −1} belongs toL¹([0, T]×T^d). Thus Z T

0

Z

T^d

|ξ^ǫ⋆ w|^r′

(ξ^ǫ⋆ m)^r′−1dxdτ ≤ Z T

0

Z

T^d

ξ^ǫ⋆ |w|^r′ m^r′−1

!

dxdτ ≤

|w|^r′ m^r′−1

1

.

Therefore, using H¨older inequality, d₁(m_ǫ(t), m_ǫ(s))≤

Z _t

s

Z

T^d

|v_ǫ(τ, x)|^r′m_ǫ(τ, x)dxdτ

_r¹′ Z _t

s

Z

T^d

m_ǫ(τ, x)dxdτ ¹_r

+kAk_∞C|t−s|¹²

≤

|w|^r′ m^r′−1

1 r′

1

|t−s|^1r +kAk_∞|t−s|²¹.

Letting ǫ → 0 we have m_ǫ → m in L¹([0, T]×T^d) and for a.e. τ ∈ [0, T], m_ǫ(τ) → m(τ) in L¹(T^d), moreover for a.e. 0≤s < t≤T

ǫ→0limd₁(m_ǫ(t), m_ǫ(s)) =d₁(m(t), m(s)).

Thus for a.e. 0≤s < t≤T

d₁(m(t), m(s))≤C|t−s|^1r +kAk∞|t−s|¹².

The second optimal control problem is the following:

(m,w)∈Kinf 1

B(m, w). (22)

Lemma 3.2. We have

φ∈Kinf0

A(φ) =− min

(m,w)∈K1

B(m, w).

Moreover, the minimum in the right-hand side is achieved by a unique pair(m, w)∈ K₁satisfying (m, w)∈L^q((0, T)×T^d)×L

r′ q

r′+q−1((0, T)×T^d).

Remark 3.3. Note that _r^′_+q−1^r′q >1 becauser^′>1 andq >1.

Proof. The strategy of proof—which is very close to the corresponding one in [4, 5, 6]—consists in applying the Fenchel-Rockafellar duality theorem (cf. e.g., [11]). In order to do so, it is better to reformulate the first optimization problem (17) in a more suitable form. LetE₀=C²([0, T]×T^d) and E₁=C⁰([0, T]×T^d,R)× C⁰([0, T]×T^d,R^d). We define onE₀ the functional

F(φ) =− Z

T^d

m₀(x)φ(0, x)dx+χ_S(φ),

whereχ_S is the characteristic function of the setS ={φ∈E₀, φ(T,·) =φ_T}, i.e., χ_S(φ) = 0 if φ∈S and +∞otherwise. For (a, b)∈E₁, we define

G(a, b) = Z _T

0

Z

T^d

F^∗(x,−a(t, x) +H(x, b(t, x)))dxdt .

The functional F is convex and lower semi-continuous on E₀ while G is convex and continuous onE₁. Let Λ :E₀ →E₁ be the bounded linear operator defined by Λ(φ) = (∂_tφ+A_ij∂_ijφ, Dφ).

We can observe that

φ∈Kinf0

A(φ) = inf

φ∈E0

{F(φ) +G(Λ(φ))}.

It is easy to verify that the qualification hypothesis, that ensures the stability of the above optimization problem, holds. Indeed, there is a map φ such that F(φ) <+∞ and such that G is continuous at Λ(φ): it is enough to take φ(t, x) =φ_T(x).

Therefore we can apply the Fenchel-Rockafellar duality theorem, which states that

φ∈Einf0

{F(φ) +G(Λ(φ))}= max

(m,w)∈E^′₁{−F^∗(Λ^∗(m, w))− G^∗(−(m, w))}

where E₁^′ is the dual space of E1, i.e., the set of vector valued Radon measures (m, w) over [0, T]×T^d with values inR×R^d,E₀^′ is the dual space ofE0, Λ^∗ :E₁^′ →E₀^′ is the dual operator of Λ andF^∗ andG^∗are the convex conjugates ofF andG respectively. By a direct computation we have

F^∗(Λ^∗(m, w)) =





 Z

Td

φ_T(x)dm(T, x) if∂_tm−∂_ij(A_ijm) + div(w) = 0, m(0) =m₀

+∞ otherwise

where the equation ∂_tm−∂_ij(A_ijm)+div(w) = 0, m(0) =m₀ holds in the sense of distributions.

Following [4], we haveG^∗(m, w) = +∞ if (m, w)∈/ L¹ and, if (m, w)∈L¹, G^∗(m, w) =

Z T 0

Z

Td

K^∗(x, m(t, x), w(t, x))dtdx, where

K^∗(x, m, w) =







F(x,−m)−mH^∗(x,−_m^w) if m <0

0 if m= 0, w= 0

+∞ otherwhise

is the convex conjugate of

K(x, a, b) =F^∗(x,−a+H(x, b)) ∀(x, a, b)∈T^d×R×R^d.

Therefore

(m,w)∈Emax ₁^′{−F^∗(Λ^∗(m, w))− G^∗(−(m, w))}

= max Z T

0

Z

T^d

−F(x, m)−mH^∗(x,−w

m) dtdx− Z

T^d

φ_T(x)m(T, x)dx

where the last maximum is taken over theL¹ maps (m, w) such that m≥0 a.e. and

∂_tm−∂_ij(A_ijm) + div(w) = 0, m(0) =m₀ holds in the sense of distributions. Since

Z

Td

m₀ = 1 , it follows that Z

Td

m(t) = 1 for any t ∈ [0, T]. Thus the pair (m, w) belongs to the set K₁ and the first part of the statement is proved.

Take now an optimal (m, w) ∈ K1 in the above system. Observe that due to optimality we havew(t, x) = 0 for all (t, x)∈[0, T]×T^dsuch that m(t, x) = 0. The growth conditions (4) and (8) imply

C ≥ Z _T

0

Z

T^d

F(x, m) +mH^∗(x,−w

m)dtdx+ Z

T^d

φ_T(x)m(T, x) dx

≥ Z T

0

Z

Td

1

C|m|^q+ m C

w m

r^′

−C(m+ 1)

dxdt− kφ_Tk_∞. Thereforem∈L^q. Moreover, by H¨older inequality, we also have

Z T 0

Z

Td

|w| ^r

′q r′+q−1 =

Z Z

{m>0}

|w| ^r

′q

r′+q−1 ≤ kmk

r′

−1 r′+q−1

q

Z Z

{m>0}

|w|^r′ m^r′−1

! ^q

r′+q−1

≤C so thatw∈L ^r

′q

r′+q−1. Finally, a minimizer to (22) should be unique, because the setK₁ is convex and the mapsF(x,·) andH^∗(x,·) are strictly convex: thusmis unique and so is w

m in{m >0}.

Asw= 0 in {m= 0}, uniqueness ofw follows as well.

4. Analysis of the optimal control of the HJ equation

In general, we do not expect problem (17) to have a solution. In this section we exhibit a relaxation for (17) (Proposition 4.2) and show that this relaxed problem has at least one solution (Proposition 4.4).

4.1. The relaxed problem. Recall that the exponentsη >1 andγ >1 are defined in Theorem 2.3. Let K be the set of pairs (φ, α) ∈ L^γ((0, T) ×T^d) ×L^p((0, T)×T^d) such that Dφ ∈ L^r((0, T)×T^d) and which satisfy in the sense of distributions

−∂tφ−Aij(x)∂ijφ+H(x, Dφ)≤α, φ(T,·)≤φT (23) (for the precise meaning of the inequality, see the beginning of Section 2). The following state- ment explains thatφhas a “trace” in a weak sense.

Lemma 4.1. Let(φ, α)∈ K. Then, for any Lipschitz continuous map ζ:T^d→R, the mapt→ Z

Td

ζ(x)φ(t, x)dxhas a BV representative on[0, T]. Moreover, if we denote by Z

Td

ζ(x)φ(t⁺, x)dx its right limit att∈[0, T), then the map ζ →

Z

T^d

ζ(x)φ(t⁺, x)dx is continuous in L^η′(T^d).

As a consequence, for any nonnegative C¹ map ϑ : [0, T]×T^d → R, one can write the integration by parts formula: for any 0≤t1 ≤t2≤T,

− Z

Td

ϑφ t2

t1

+ Z t2

t1

Z

Td

φ∂tϑ+hDϑ, ADφi+ϑ(∂iAij∂jφ+H(x, Dφ))≤ Z t2

t1

Z

Td

αϑ.

Proof of Lemma 4.1. One easily checks that, for any Lipschitz continuous, nonnegative map ζ :T^d→R,

−d dt

Z

Td

ζφ(t) + Z

Td

hDζ, ADφ(t)i+ζ(∂iAij∂jφ+H(x, Dφ)−α)≤0, holds in the sense of distributions. As the second integral is inL¹((0, T)), the mapt→R

Tdζφ(t) is BV. If now ζ is Lipschitz continuous and changes sign, one can write ζ =ζ⁺−ζ⁻ and the mapt→R

T^dζφ(t) =R

T^dζ⁺φ(t)−R

T^dζ⁻φ(t) is still BV. The continuity with respect toζ comes from the L^∞((0, T), L^η(T^d)) estimate on φgiven in Theorem 2.3.

We extend the functionalA to K by setting A(φ, α) =

Z _T

0

Z

T^d

F^∗(x, α(x, t))dxdt− Z

T^d

φ(x,0)m₀(x) dx ∀(φ, α)∈ K.

The next proposition explains that the problem

(φ,α)∈Kinf A(φ, α) (24)

is the relaxed problem of (17). For this we first note that

(φ,α)∈Kinf A(φ, α) = inf

(φ,α)∈K, α≥0 a.e.A(φ, α) (25)

because one can always replaceα by α∨0 sinceF^∗(x, α) = 0 for α≤0.

Proposition 4.2. We have

φ∈Kinf0

A(φ) = inf

(φ,α)∈KA(φ, α).

The proof requires the following inequality:

Lemma 4.3. Let (φ, α) ∈ K and (m, w) ∈ K₁. Assume that mH^∗(·,−w/m)∈L¹((0, T)×T^d) and m∈L^q((0, T)×T^d). Then

Z

T^d

mφ T

t

+ Z T

t

Z

T^d

m

α+H^∗(x,−w m)

≥ 0 (26)

and

Z

Td

mφ t

0

+ Z t

0

Z

Td

m

α+H^∗(x,−w m)

≥ 0.

Moreover, if equality holds in the inequality (26) for t= 0, then w=−mD_pH(x, Dφ) a.e.

Proof. We first extend the pairs (m, w) to [−1, T + 1]×T^d by defining m = m₀ on [−1,0], m = m(T) on [T, T + 1] and w(s, x) = 0 for (s, x) ∈ (−1,0)∪(T, T + 1)×T^d. Note that

∂tm−∂ij( ˜Aij(t, x)m) + div(w) = 0 on (−1, T + 1)×T^d, where ˜Aij(t, x) =Aij(x) if t ∈(0, T) and ˜A_ij(t, x) = 0 otherwise. Let ξ^ǫ = ξ^ǫ(t, x) be a smooth convolution kernel with support in B_ǫ; we smoothen the pair (m, w) in a standard way into (m_ǫ, w_ǫ). Then (m_ǫ, w_ǫ) solves

∂_tm_ǫ−∂_ij( ˜A_ijm_ǫ) + div(w_ǫ) =∂_iR_ǫ in (−1/2, T + 1/2) (27) in the sense of distributions, where

R_ǫ:= [ξ^ǫ, ∂_jA˜_ij](m) + [ξ^ǫ,A˜_ij∂_j](m). (28)