Poisson stability of solutions for stochastic evolution equations driven by fractional Brownian motion

TL;DR

This paper investigates the Poisson stability of solutions to stochastic evolution equations driven by fractional Brownian motion, establishing existence, uniqueness, and stability of solutions with Poisson stable coefficients in a Hilbert space setting.

Contribution

It introduces conditions under which solutions to stochastic evolution equations driven by fractional Brownian motion are Poisson stable, extending stability analysis to this class of stochastic equations.

Findings

Existence and uniqueness of solutions established.

Solutions inherit Poisson stability from coefficients under smallness conditions.

Extended analysis on asymptotic stability of solutions.

Abstract

In this paper, we study the problem of Poisson stability of solutions for stochastic semi-linear evolution equation driven by fractional Brownian motion \mathrm{d} X(t)= \left( AX(t) + f(t, X(t)) \right) \mathrm{d}t + g\left(t, X(t)\right)\mathrm{d}B^H_{Q}(t), where A is an exponentially stable linear operator acting on a separable Hilbert space \mathbb{H}, coefficients f and g are Poisson stable in time, and B^H_Q (t) is a Q-cylindrical fBm with Hurst index H. First, we establish the existence and uniqueness of the solution for this equation. Then, we prove that under the condition where the functions f and g are sufficiently "small", the equation admits a solution that exhibits the same character of recurrence as f and g. The discussion is further extended to the asymptotic stability of these Poisson stable solutions. Finally, we include an example to validate our results.