Special Engagement Colloquium: Ergodic Theory of Singular Stochastic PDEs

The large-scale dynamic of many systems in statistical mechanics is described by analytically ill-posed stochastic partial differential equations. Recently, the theory of regularity structures has allowed to give a meaning to these equations and to construct their solutions. A natural question is that of the long-time behavior of these solutions. In particular, do they forget about their initial conditions? The speaker shows that a large class of singular SPDEs gives rise to Markov processes satisfying the strong Fellor property. By combining this with adequate support theorems, one can conclude that these processes are uniquely ergodic. Two prototypical examples covered by the results are the KPZ equation and the stochastic quantisation equation in dimension 2 and 3.