# Exponential mixing under controllability conditions for SDEs driven by a degenerate Poisson noise

## TL;DR

This paper establishes the existence, uniqueness, and exponential mixing of invariant measures for certain SDEs driven by degenerate Poisson noise, under controllability and dissipativity conditions, extending to PDEs and oscillator networks.

## Contribution

It introduces weaker controllability conditions than the parabolic Hörmander condition for proving exponential mixing in degenerate Poisson-driven SDEs.

## Key findings

- Proves exponential mixing for a class of degenerate Poisson-driven SDEs.
- Applies results to PDE Galerkin projections with polynomial nonlinearities.
- Extends to networks of oscillators connected to Poissonian baths.

## Abstract

We prove existence and uniqueness of the invariant measure and exponential mixing in the total-variation norm for a class of stochastic differential equations driven by degenerate compound Poisson processes. In addition to mild assumptions on the distribution of the jumps for the driving process, the hypotheses for our main result are that the corresponding control system is dissipative, approximately controllable and solidly controllable. The solid controllability assumption is weaker than the well-known parabolic H\"ormander condition and is only required from a single point to which the system is approximately controllable. Our analysis applies to Galerkin projections of stochastically forced parabolic partial differential equations with asymptotically polynomial nonlinearities and to networks of quasi-harmonic oscillators connected to different Poissonian baths.