Exponential ergodicity of stochastic heat equations with H\"older 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 H\"older 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 $\beta$ - H\"older continuous for $\beta>\frac{3}{4}$ and uniformly non-degenerate, and that the drift is locally H\"older 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 $\vartheta+\epsilon$- H\"older continuous and has linear growth. For $\vartheta\in(\frac{1}{2},1)$ this result is new even in the case of additive noise.