PDE and applications
- Lecturer: Lauri Viitasaari
- Contact person: Kaj Nyström
Title: Stochastic fractional heat equation - existence, uniqueness, and quantitative normal approximations
Abstract: Stochastic PDEs arise naturally as unpredictable cousins of more classical deterministic PDEs by allowing to take into account random shocks representing e.g., some external random force affecting the system or possibly some measurement errors. However, typical random noises behave rather badly which leads to some extra difficulties to an already complicated mathematical problems. In particular, even the existence and uniqueness results for the solutions are far from obvious. Moreover, as there is a random force affecting the system, the solution is also a random object. Taming this randomness explicitly is typically a challenging task, and hence one wishes to derive some approximation results that would allow to analyse statistical properties of the solution. In this talk we consider a non-linear stochastic version of the fractional heat equation driven by a Gaussian noise that, while being white in time, admits rather general spatial covariance structure. We provide existence and uniqueness result for the solution under mild assumptions on the spatial covariance R. Under some further restrictions on R, we also provide quantitative approximation results for the spatial averages of the solution. Such approximations can, in particular, be extremely handy in practical applications where one needs to assess statistical features of the solution.
The talk is based on a joint work with Obayda Assaad (University of Lille), David Nualart (University of Kansas), and Ciprian Tudor (University of Lille).