The large time profile for Hamilton--Jacobi--Bellman equations

TL;DR

This paper investigates the large-time behavior of viscosity solutions to second-order Hamilton--Jacobi--Bellman equations with convex Hamiltonians, providing a measure-based representation and characterizing the limit in terms of initial data and Mather measures.

Contribution

It introduces a novel measure-based representation for solutions and characterizes the large-time limit using generalized Mather measures, addressing solution selection issues.

Findings

Representation of viscosity solutions via generalized holonomic measures

Characterization of the large-time limit using initial data and Mather measures

New results on generalized Mather measures and duality theorems

Abstract

Here, we study the large-time limit of viscosity solutions of the Cauchy problem for second-order Hamilton--Jacobi--Bellman equations with convex Hamiltonians in the torus. This large-time limit solves the corresponding stationary problem, sometimes called the ergodic problem. This problem, however, has multiple viscosity solutions and, thus, a key question is which of these solutions is selected by the limit. Here, we provide a representation for the viscosity solution to the Cauchy problem in terms of generalized holonomic measures. Then, we use this representation to characterize the large-time limit in terms of the initial data and generalized Mather measures. In addition, we establish various results on generalized Mather measures and duality theorems that are of independent interest.