Time-Periodic Solutions of Stochastic Discounted Hamilton–Jacobi Equations

Abstract

We study time-periodic solutions to stochastic discounted Hamilton–Jacobi equations driven by a cylindrical Wiener process in the sense of most probable paths. We first analyze the existence and regularity of solutions. Then, within a spectral Galerkin limit framework, we derive the Onsager–Machlup functional using a stopping-time localization argument and $L^2$-type techniques. Finally, based on a minimization principle for this action functional, we characterize the system's most probable paths. By further combining weak KAM theory, we prove that there exist most probable paths with a time-periodic structure. This result extends the classical theory of time-periodic viscosity solutions for deterministic discounted Hamilton–Jacobi equations to the stochastic setting for the first time, and offers a new perspective on stochastic discounted Hamilton–Jacobi equations.