# Mixing via controllability for randomly forced nonlinear dissipative PDEs

## TL;DR

This paper extends the understanding of mixing in randomly forced nonlinear dissipative PDEs by showing that global controllability ensures unique stationary measures and convergence, even without exponential rates.

## Contribution

It introduces a new approach linking stability to conditional random walks, relaxing previous assumptions of a unique equilibrium.

## Key findings

- Unique stationary measure under global controllability
- Convergence to the stationary measure without exponential rate
- Applicable to a broad class of parabolic PDEs

## Abstract

We continue our study of the problem of mixing for a class of PDEs with very degenerate noise. As we established earlier, the uniqueness of stationary measure and its exponential stability in the dual-Lipschitz metric holds under the hypothesis that the unperturbed equation has exactly one globally stable equilibrium point. In this paper, we relax that condition, assuming only global controllability to a given point. It is proved that the uniqueness of a stationary measure and convergence to it are still valid, whereas the rate of convergence is not necessarily exponential. The result is applicable to randomly forced parabolic-type PDEs, provided that the deterministic part of the external force is in general position, ensuring a regular structure for the attractor of the unperturbed problem. The proof uses a new idea that reduces the verification of a stability property to the investigation of a conditional random walk.