# Limiting measure and stationarity of solutions to stochastic evolution equations with Volterra noise

## TL;DR

This paper investigates the long-term behavior of solutions to stochastic evolution equations driven by Volterra noise, establishing conditions for the existence of limiting measures and stationarity, with applications to heat equations driven by Rosenblatt processes.

## Contribution

It provides new sufficient conditions for limiting measure existence and stationarity of solutions to stochastic Volterra equations, including an example where these conditions are necessary.

## Key findings

- Conditions for limiting measure existence are established.
- Strict stationarity of solutions is characterized.
- Application to heat equations driven by Rosenblatt process is demonstrated.

## Abstract

Large-time behaviour of solutions to stochastic evolution equations driven by two-sided regular Volterra processes is studied. The solution is understood in the mild sense and takes values in a separable Hilbert space. Sufficient conditions for the existence of limiting measure and strict stationarity of the solution process are found and an example for which these conditions are also necessary is provided. The results are further applied to the heat equation driven by the two-sided Rosenblatt process.