Exponential ergodicity of stochastic heat equations with Hölder coefficients

Yi Han

Published 2022-11-15Version 1

We consider ergodicity of stochastic heat equations driven by space-time white noise in dimension one, whose drift and diffusion coefficients are merely H\"older continuous. We give a short proof that there exists a unique in law mild solution when the diffusion coefficient is $\beta$ - H\"older continuous for $\beta>\frac{3}{4}$ and uniformly nondegenerate, and that the drift is locally H\"older continuous. Due to non-smoothness of the coefficients, it is hard to obtain Bismut-Elworthy-Li type estimates that are central to the strong Feller property and unique ergodicity in the Lipschitz case, and transforming the coordinates via solving a Kolmogorov equation also seems intractable. Instead, we construct a distance function $d$ which is locally contracting and that bounded subsets of the state space are small in the sense of Harris' ergodic theorem. The construction utilizes a generalized coupling technique introduced by Kulik and Scheutzow in Ann.Probab vol.48 No.6: 3041-3076, 2020 in the setting of stochastic delay equations. Assuming the existence of a suitable Lyapunov function, we prove existence of a spectral gap and that transition probabilities converge exponentially fast to the unique invariant measure. The same method can be applied when the SPDE has a Burgers type nonlinearity $(-A)^{1/2}F(X_t)$ or $\partial_x F(X_t)$, where $F$ is continuous and has linear growth. We prove existence of a unique in law mild solution when $F$ is locally $\zeta$-H\"oloder continuous for $\zeta>\frac{1}{2}$, and the other assumptions are the same as before. We then extend spectral gap and exponential ergodicity results to this case once we have a suitable Lyapunov function.