Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

In this talk, we shall discuss long-time behavior of parabolic stochastic partial differential equations (SPDEs) with singular nonlinear divergence-type drift subject to an additive stochastic perturbation by Gaussian noise. Examples include the stochastic singular $p$-Laplace equation, the multi-valued stochastic total variation flow and the stochastic curve shortening flow.
We shall present some improved pathwise regularity results and decay estimates. For additive Wiener noise, we prove the ergodicity of the associated Markovian semigroup and thus the uniqueness of the invariant measure, for which we shall also give new concentration results. The associated Kolmogorov operator is then seen to be maximal dissipative.
Finally, will discuss some new ergodicity results for SPDEs with locally monotone drift and additive Lévy noise, which include the stochastic incompressible 2D Navier-Stokes equation with Dirichlet boundary conditions on a bounded domain as an example.

The talk is based on joint works with Wei Liu (Xuzhou),
Benjamin Gess (Leipzig and Bielefeld),
Florian Seib (Berlin) and Wilhelm Stannat (Berlin),
as well as Gerardo Barrera (Lisbon).