Exponential Mixing for a Stochastic PDE Driven by Degenerate Noise

TL;DR

This paper proves that certain reaction-diffusion stochastic PDEs with highly degenerate noise still exhibit exponential convergence to equilibrium, using a weighted variation norm and a Doeblin condition.

Contribution

It demonstrates exponential mixing for reaction-diffusion SPDEs driven by degenerate noise, extending previous results to cases with non-full rank forcing.

Findings

Exponential convergence to invariant measure established.

Convergence proven in weighted variation norm.

Applicable to reaction-diffusion equations with degenerate noise.

Abstract

We study stochastic partial differential equations of the reaction-diffusion type. We show that, even if the forcing is very degenerate (i.e. has not full rank), one has exponential convergence towards the invariant measure. The convergence takes place in the topology induced by a weighted variation norm and uses a kind of (uniform) Doeblin condition.