On unbounded solutions of ergodic problems in r^m for viscous hamilton-jacobi equations

TL;DR

This paper investigates the existence, uniqueness, and properties of solutions to ergodic viscous Hamilton-Jacobi equations in the entire space, identifying a critical value that determines solution behavior without extra assumptions in certain cases.

Contribution

It establishes new PDE-based proofs for the existence and uniqueness of solutions to ergodic problems in R^m, generalizing previous probabilistic and PDE results.

Findings

Existence of a critical value for solution existence

Solutions are unique up to an additive constant

Results hold under less restrictive conditions in superquadratic case

Abstract

In this article we study ergodic problems in the whole space R m for viscous Hamilton-Jacobi Equations in the case of locally Lips-chitz continuous and coercive right-hand sides. We prove in particular the existence of a critical value $\lambda$ * for which (i) the ergodic problem has solutions for all $\lambda$ $\le$ $\lambda$ * , (ii) bounded from below solutions exist and are associated to $\lambda$ * , (iii) such solutions are unique (up to an additive constant). We obtain these properties without additional assumptions in the superquadratic case, while, in the subquadratic one, we assume the right-hand side to behave like a power. These results are slight generalizations of analogous results by N. Ichihara but they are proved in the present paper by partial differential equations methods, contrarily to N. Ichihara who is using a combination of pde technics with probabilistic arguments.