In this paper we discuss a stochastic analog of Aubry-Mather theory in which a deterministic control problem is replaced by a controlled diffusion. We prove the existence of a minimizing measure (Mather measure) and discuss its main properties using viscosity solutions of Hamilton-Jacobi equations. Then we prove regularity estimates on viscosity solutions of Hamilton-Jacobi equation using the Mather measure. Finally we apply these results to prove asymptotic estimates on the trajectories of controlled diffusions and study the convergence of Mather measures as the rate of diffusion vanishes.
In this paper, we continue to develop Aubry-Mather and weak KAM theories for contact Hamiltonian systems $H(x,u,p)$ with certain dependence on the contact variable $u$. For the Lipschitz dependence case, we obtain some properties of the Mañé set. For the non-decreasing case, we provide some information on the Aubry set, such as the comparison property, graph property and a partially ordered relation for the collection of all projected Aubry sets with respect to backward weak KAM solutions. Moreover, we find a new flow-invariant set $\tilde{\mathcal{S}}_s$ consists of strongly static orbits, wh…
This paper is concerned with the study of Aubry-Mather and weak KAM theories for contact Hamiltonian systems with Hamiltonians $H(x,u,p)$ defined on $T^*M\times\mathbb{R}$, satisfying Tonelli conditions with respect to $p$ and $0<\frac{\partial H}{\partial u}\leqslant λ$ for some $λ>0$, where $M$ is a connected, closed and smooth manifold. First, we show the uniqueness of the backward weak KAM solutions of the corresponding Hamilton-Jacobi equation. Using the unique backward weak KAM solution $u_-$, we prove the existence of the maximal forward weak KAM solution $u_+$. Next, we analyse Aubry s…
The Vlasov equation models a group of particles moving under a potential $V$; moreover, each particle exerts a force, of potential $W$, on the other ones. We shall suppose that these particles move on the $p$-dimensional torus ${\bf T}^p$ and that the interaction potential $W$ is smooth. We are going to perturb this equation by a Brownian motion on ${\bf T}^p$; adapting to the viscous case methods of Gangbo, Nguyen, Tudorascu and Gomes, we study the existence of periodic solutions and the asymptotics of the Hopf-Lax semigroup.
The adjoint method introduced in [Eva] and [Tra] is used, to construct analogs to the Aubry-Mather measures for non convex Hamiltonians. More precisely, a general construction of probability measures, that in the convex setting agree with Mather measures, is provided. These measures may fail to be invariant under the Hamiltonian flow and a dissipation arises, which is described by a positive semidefinite matrix of Borel measures. However, in the important case of uniformly quasiconvex Hamiltonians the dissipation vanishes, and as a consequence the invariance is guaranteed.
Gomes and Valdinoci have introduced a time-step approximation scheme for a viscous version of Aubry-Mather theory; this scheme is a variant of that of Jordan, Kinderlehrer and Otto. Gangbo and Tudorascu have shown that the Vlasov equation can be seen as an extension of Aubry-Mather theory, in which the configuration space is the space of probability measures, i. e. the different distributions of infinitely many particles on a manifold. Putting the two things together, we show that Gomes and Valdinoci's theorem carries over to a viscous version of the Vlasov equation. In this way, we shall reco…
We present a large deviation principle for the entropy penalized Mather problem when the Lagrangian L is generic (in this case the Mather measure $μ$ is unique and the support of $μ$ is the Aubry set). Consider, for each value of $ε$ and h, the entropy penalized Mather problem $\min \{\int_{\tn\times\rn} L(x,v)dμ(x,v)+εS[μ]\},$ where the entropy S is given by $S[μ]=\int_{\tn\times\rn}μ(x,v)\ln\frac{μ(x,v)}{\int_{\rn}μ(x,w)dw}dxdv,$ and the minimization is performed over the space of probability densities $μ(x,v)$ that satisfy the holonomy constraint It follows from D. Gomes and E. Valdinoci th…
We study the connection between the Aubry-Mather theory and a mean-field game (MFG) price-formation model. We introduce a framework for Mather measures that is suited for constrained time-dependent problems in R. Then, we propose a variational problem on a space of measures, from which we obtain a duality relation involving the MFG problem examined in [36].
We introduce and study the class of linear transfers between probability distributions and the dual class of Kantorovich operators between function spaces. Linear transfers can be seen as an extension of convex lower semi-continuous energies on Wasserstein space, of cost minimizing mass transports, as well as many other couplings between probability measures to which Monge-Kantorovich theory does not readily apply. Basic examples include balayage of measures, martingale transports, optimal Skorokhod embeddings, and the weak mass transports of Talagrand, Marton, Gozlan and others. The class als…
In this article we develop an analogue of Aubry Mather theory for time periodic dissipative equation \[ \left\{ \begin{aligned} \dot x&=\partial_p H(x,p,t),\\ \dot p&=-\partial_x H(x,p,t)-f(t)p \end{aligned} \right. \] with $(x,p,t)\in T^*M\times\mathbb T$ (compact manifold $M$ without boundary). We discuss the asymptotic behaviors of viscosity solutions of associated Hamilton-Jacobi equation \[ \partial_t u+f(t)u+H(x,\partial_x u,t)=0,\quad(x,t)\in M\times\mathbb T \] w.r.t. certain parameters, and analyze the meanings in controlling the global dynamics. We also discuss the prospect of applyi…
Selection of critical solutions in non-monotone vanishing discount problems
Waseda University（早稻田大学）；Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）；School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）；Sorbonne Université, Université de Paris Cité, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche（索邦大学，巴黎西岱大学，法国国家科学研究中心，儒勒·让苏-巴黎左岸数学研究所）
We study the vanishing discount problem for a nonlinear monotone system of Hamilton-Jacobi equations. This continues the first author's investigation on the vanishing discount problem for a monotone system of Hamilton-Jacobi equations. As in Part 1, we introduce by the convex duality Mather measures and their analogues for the system, which we call respectively Mather and Green-Poisson measures, and prove a convergence theorem for the vanishing discount problem. Moreover, we establish an existence result for the ergodic problem.
Selection problems of Z^2-periodic entropy solutions and viscosity solutions
Unité de mathématiques pures et appliquées, CNRS UMR 5669 & École Normale Supérieure de Lyon（纯数学与应用数学单元，法国国家科学研究中心UMR 5669与里昂高等师范学院）；Department of Mathematics, Faculty of Science and Technology, Keio University（庆应义塾大学理工学部数学科）
