# Existence of Geometric Ergodic Periodic Measures of Stochastic Differential Equations

## TL;DR

This paper establishes conditions under which stochastic differential equations possess unique, geometrically convergent periodic measures, characterizing their long-term periodic behavior, with applications to physical systems and specific examples.

## Contribution

It provides general sufficient conditions for the existence, uniqueness, and geometric convergence of periodic measures in stochastic systems, extending previous results to broader classes of equations.

## Key findings

- Existence and uniqueness of periodic measures under certain conditions
- Periodic measures satisfy a specific Fokker-Planck equation
- Applications demonstrated in physical models with examples

## Abstract

Periodic measures are the time-periodic counterpart to invariant measures for dynamical systems and can be used to characterise the long-term periodic behaviour of stochastic systems. This paper gives sufficient conditions for the existence, uniqueness and geometric convergence of a periodic measure for time-periodic Markovian processes on a locally compact metric space in great generality. In particular, we apply these results in the context of time-periodic weakly dissipative stochastic differential equations, gradient stochastic differential equations as well as Langevin equations. We will establish the Fokker-Planck equation that the density of the periodic measure sufficiently and necessarily satisfies. Applications to physical problems shall be discussed with specific examples.