Random Lax--Oleinik semigroups for Hamilton--Jacobi systems

TL;DR

This paper introduces a stochastic Lax--Oleinik formula for weakly coupled Hamilton--Jacobi systems, demonstrating that the associated value functions are viscosity solutions and exploring properties of minimizers and adjoint curves.

Contribution

It develops a novel stochastic Lax--Oleinik formula for Hamilton--Jacobi systems, extending scalar methods to coupled systems with new existence and differentiability results.

Findings

Value functions are viscosity solutions to the system.

Existence of minimal random curves under general conditions.

Differentiability and existence of adjoint random curves.

Abstract

Following the random approach of Mitake, Siconolfi,Tran and Yamada, we define a Lax--Oleinik formula adapted to evolutive weakly coupled systems of Hamilton--Jacobi equations. It is reminiscent of the corresponding scalar formula, with the relevant difference that it has a stochastic character since it involves, loosely speaking, random switchings between the various associated Lagrangians. We prove that the related value functions are viscosity solutions to the system, and establish existence of minimal random curves under fairly general hypotheses. Adding Tonelli like assumptions on the Hamiltonians, we show differentiability properties of such minimizers, and existence of adjoint random curves. Minimizers and adjoint curves are trajectories of a twisted generalized Hamiltonian dynamics.