Mather problem and viscosity solutions in the stationary setting Diogo

Mather problem and viscosity solutions in the stationary setting

In this paper we discuss the Mather problem for stationary Lagrangians, that is Lagrangians L : Rn ×Rn ×Ω → R, where Ω is a compact metric space on which Rn acts through an action which leaves L invariant. This setting allow us to generalize the standard Mather problem for quasi-periodic and almost-periodic Lagrangians. Our main result is the existence of stationary Mather measures invariant un...

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 Solution Methods and the Discrete Aubry-mather Problem

In this paper we study a discrete multi-dimensional version of AubryMather theory using mostly tools from the theory of viscosity solutions. We set this problem as an infinite dimensional linear programming problem. The dual problem turns out to be a discrete analog of the Hamilton-Jacobi equations. We present some applications to discretizations of Lagrangian systems.

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...

Viscosity Solutions and Aubry-Mather theory