Invariant Measure for Linear Stochastic PDEs in the space of Tempered distributions

TL;DR

This paper establishes the existence and uniqueness of invariant measures for linear stochastic PDEs with potential in the space of tempered distributions, utilizing monotonicity inequalities to analyze stability and measure properties.

Contribution

It introduces a novel approach using monotonicity inequalities to prove invariant measure existence and uniqueness for a class of linear stochastic PDEs in tempered distributions.

Findings

Proves exponential stability of solutions.

Establishes existence of invariant measures.

Demonstrates uniqueness of invariant measures.

Abstract

In this paper, we first explore exponential stability by using Monotonicity inequality and use this information to obtain the existence of Invariant measure for linear Stochastic PDEs with potential in the space of tempered distributions. The uniqueness of Invariant Measure follows from Monotonicity inequality.