  We study regularity properties for invariant measures of semilinear
diffusions in a separable Hilbert space. Based on a pathwise estimate for the
underlying stochastic convolution, we prove a priori estimates on such
invariant measures. As an application, we combine such estimates with a new
technique to prove the $L^1$-uniqueness of the induced Kolmogorov operator,
defined on a space of cylindrical functions. Finally, examples of stochastic
Burgers equations and thin-film growth models are given to illustrate our
abstract result.
