A metric analysis of critical Hamilton--Jacobi equations in the stationary ergodic setting

TL;DR

This paper extends the metric approach to stationary ergodic Hamilton-Jacobi equations, defining an intrinsic random semidistance and analyzing the existence of solutions relative to a critical value.

Contribution

It introduces a new framework for analyzing critical Hamilton-Jacobi equations in a stochastic setting using a metric approach and admissible random solutions.

Findings

Existence of an asymptotic norm for levels above the critical value

Lax formula yields admissible subsolutions for certain random sets

Degeneracies of the stable norm relate to solution existence

Abstract

We adapt the metric approach to the study of stationary ergodic Hamilton-Jacobi equations, for which a notion of admissible random (sub)solution is defined. For any level of the Hamiltonian greater than or equal to a distinguished critical value, we define an intrinsic random semidistance and prove that an asymptotic norm does exist. Taking as source region a suitable class of closed random sets, we show that the Lax formula provides admissible subsolutions. This enables us to relate the degeneracies of the critical stable norm to the existence/nonexistence of exact or approximate critical admissible solutions.