Expected Exit Time for Time-Periodic Stochastic Differential Equations and Applications to Stochastic Resonance

TL;DR

This paper develops a PDE-based method to compute the expected exit time for time-periodic stochastic differential equations, enabling numerical analysis of stochastic resonance in various physical systems.

Contribution

It establishes a Feynman-Kac duality for time-periodic SDEs and proposes a fixed point and convex optimization approach for solving the associated PDEs.

Findings

Method is implementable with standard numerical schemes.

Applicable to systems with weakly dissipative coefficients.

Provides a framework for analyzing stochastic resonance.

Abstract

In this paper, we derive a parabolic partial differential equation for the expected exit time of non-autonomous time-periodic non-degenerate stochastic differential equations. This establishes a Feynman-Kac duality between expected exit time of time-periodic stochastic differential equations and time-periodic solutions of parabolic partial differential equations. Casting the time-periodic solution of the parabolic partial differential equation as a fixed point problem and a convex optimisationproblem, we give sufficient conditions in which the partial differential equation is well-posed in a weak and classical sense. With no known closed formulae for the expected exit time, we show our method can be readily implemented by standard numerical schemes. With relatively weak conditions (e.g. locally Lipschitz coefficients), the method in this paper is applicable to wide range of physical systems including weakly dissipative systems. Particular applications towards stochastic resonance will be discussed.