# Invariant measure for the stochastic Cauchy problem driven by a cylindrical L\'evy process

## TL;DR

This paper establishes necessary and sufficient conditions for the existence and uniqueness of invariant measures for stochastic Cauchy problems driven by cylindrical Lévy processes, with applications to heat equations.

## Contribution

It provides a comprehensive characterization of invariant measures for such stochastic equations, including simplified conditions for common cases like the heat equation.

## Key findings

- Necessary and sufficient conditions for invariant measures
- Simplified conditions for heat equation scenarios
- Application examples demonstrating the theory

## Abstract

In this work, we present sufficient conditions for the existence of a stationary solution of an abstract stochastic Cauchy problem driven by an arbitrary cylindrical L\'evy process, and show that these conditions are also necessary if the semigroup is stable, in which case the invariant measure is unique. For typical situations such as the heat equation, we significantly simplify these conditions without assuming any further restrictions on the driving cylindrical L\'evy process and demonstrate their application in some examples.