Stochastic Analysis Seminar
Due to the presence of random noise, the long time behaviour of some stochastic partial differential equations (SPDEs) tends to be independent of the initial condition. More precisely, the dynamics are uniquely ergodic.
In the parabolic setting, a general strategy for proving unique ergodicity involves verifying the strong Feller property. However, for dispersive SPDEs, Tolomeo (2018) showed that this property fails and developed a new technique for proving unique ergodicity. In this talk, we will introduce these notions and methods for proving ergodicity, and will describe a result on the unique ergodicity for a hyperbolic SPDE, which does not fall within the scope of Tolomeo’s approach. Instead, our argument is inspired by the weaker asymptotic strong Feller property in this dispersive setting. This is a joint work in progress with Leonardo Tolomeo (University of Bonn).