Periodic solution of stochastic process in the distributional sense

TL;DR

This paper introduces the concept of periodic solutions for stochastic processes in a distributional sense, establishing their existence on bounded domains and for probability density functions using PDE techniques.

Contribution

It defines periodic solutions in the distributional sense for stochastic processes and proves their existence under various conditions, including the case with probability density functions.

Findings

Existence of periodic solutions on bounded domains.

Periodic solutions for probability density functions.

Application of PDE techniques to stochastic processes.

Abstract

In this paper, we aim to study a stochastic process from a macro point of view, and thus periodic solution of a stochastic process in distributional sense is introduced. We first give the definition and then establish the existence of periodic solution on bounded domain. Lastly, for the case that probability density function exists, we obtain the existence periodic solutions of the probability density function corresponding to the stochastic process by using the technique of deterministic partial differential equations.