Exponential ergodicity of stochastic heat equations with Hölder coefficients

We investigate the stochastic heat equation driven by space-time white noise defined on an abstract Hilbert space, assuming that the drift and diffusion coefficients are both merely Hölder continuous. Random field SPDEs are covered as special examples. We give the first proof that there exists a unique in law mild solution when the diffusion coefficient is $\beta$ - Hölder continuous for $\beta >\frac{3}{4}$ and uniformly non-degenerate, and that the drift is locally Hölder continuous. Meanwhile, assuming the existence of a suitable Lyapunov function for the SPDE, we prove that the solution converges exponentially fast to the unique invariant measure with respect to a typical Wasserstein distance. Our technique generalizes when the SPDE has a Burgers type non-linearity $(-A{)}^{\vartheta }F(X_t)$ for any $\vartheta \in (0,1)$, where $F$ is $\vartheta +\epsilon$- Hölder continuous and has linear growth. For $\vartheta \in (\frac{1}{2},1)$ this result is new even in the case of additive noise.