A stochastic analogue of Aubry-Mather theory*

A stochastic analogue of Aubry–Mather theory*

In this paper, we discuss a stochastic analogue 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 the Hamilton–Jacobi equation usin...

A Stochastic Analog of Aubry-mather Theory

In this paper we discuss a stochastic analog of AubryMather 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 HamiltonJacobi equation using the Mat...

Viscosity Solutions and Aubry-Mather theory

Wigner measures and quantum Aubry-Mather theory

In this paper we investigate the quantum action problem using Wigner measures on the torus. We prove existence, study its main properties, and prove convergence to Mather measures in the semiclassical limit. We also indicate how to extend these techniques to the study of stochastic Mather measures. (*) Supported in part by FCT/POCTI/FEDER (**) Supported in part by the Center for Mathematical An...

Aubry-mather Theory for Multi-dimensional Maps

In this paper we study a discrete multi-dimensional version of Aubry-Mather theory. We set this problem as an infinite dimensional linear programming problem and study its relations to Monge-Kantorowich optimal transport. The dual problem turns out to be a discrete analog of the Hamilton-Jacobi equations. As in optimal transport, Monge-Ampére equations also play a role in this problem. We prese...