Dynamical behavior of partial differential equations with multiplicative rough noise
A talk in the SPDEvent series by
Mazyar Ghani Varzaneh from Fern Universität Hagen
| Abstract: | Stochastic partial differential equations perturbed by the multiplicative noise are a family of interesting equations studied by researchers from different aspects. However, only a few results exist for the dynamical behavior of these equations. The main obstacle is that the Kolmogorov continuity theorem cannot be applied to these equations. Only in some exceptional cases can some transformations settle this problem. An alternative way to deal with this family of equations is to define the solution path-wise using the rough path theory. This approach has many advantages, including solving the problems path-wise and working with the equations perturbed by more irregular noises, including the fraction Brownian motions. The rough path theory can settle the problem of defining the random dynamical systems in the infinite dimension cases without using the Kolmogorov continuity theorem. Still, analyzing the priori bounds of the solutions is challenging due to the deterministic nature of this theory. In particular, to study the long-time behavior of the solutions of these equations, we must apply some techniques from the ergodic theory, which require certain integrability assumptions. In this talk, we show that the abovementioned challenges can be settled for the Gaussian measures when the corresponding Cameron–Martin space meets certain assumptions. These assumptions, in particular, can be verified for the interesting Gaussian processes, including the fraction Brownian and Brownian motions. We then show how we can prove the existence of the invariant manifolds (stable, center, and unstable) for this family of equations. In particular, this theory can easily yield several exponential stability results. This talk is based on joint work with Prof. Sebastian Riedel. |