Hitting questions and multiple points for stochastic PDE (SPDE) in the critical case
New challenges in PDE: Deterministic dynamics and randomness in high and infinite dimensional systems October 19, 2015 - October 30, 2015
Location: MSRI: Simons Auditorium
60J70 - Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]
35R15 - Partial differential equations on infinite-dimensional (e.g. function) spaces (= PDE in infinitely many variables) [See also 46Gxx, 58D25]
60H15 - Stochastic partial differential equations [See also 35R60]
82C41 - Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]
60J45 - Probabilistic potential theory [See also 31Cxx, 31D05]
Hitting questions play a central role in the theory of Markov processes. For example, it is well known that Brownian motion hits points in one dimension, but not in higher dimensions. For a general Markov process, we can determine whether the process hits a given set in terms of potential theory. There has also been a huge amount of work on the related question of when a process has multiple points.
For SPDE, much less is known, but there have been a growing number of papers on the topic in recent years. Potential theory provides an answer in principle. But unfortunately, solutions to SPDE are infinite dimensional processes, and the potential theory is intractible. As usual, the critical case is the most difficult.
We will give a brief survey of known results, followed by a discussion of an ongoing project with R. Dalang, Y. Xiao, and S. Tindel which promises to answer questions about hitting points and the existence of multiple points in the critical case
Please report video problems to email@example.com.
See more of our Streaming videos on our main VMath Videos page.