In this talk, we shall discuss long time behavior of solutions to parabolic stochastic partial differential equations with singular nonlinear divergence-type diffusivity. As these kinds of equations usually lack good coercivity estimates in higher spatial dimensions, we choose to address the general well-posedness question by variational weak energy methods. Examples include the stochastic singular $p$-Laplace equation, and the stochastic curve shortening flow with additive Gaussian noise. We shall present improved pathwise regularity results and improved moment and decay estimates for a general class of singular divergence-type PDEs.

Based on joint works with Benjamin Gess (Leipzig and Bielefeld), Wei Liu (Xuzhou), Florian Seib (Berlin), and Wilhelm Stannat (Berlin).