Title: Analysis of Stochastic PDE
Abstract: Stochastic Partial Differential Equations are PDE involving random coefficients and/or random forcing terms and non-linearities. In applications, they give much more realistic models that can take into account noise, or lack of information. From a pure mathematics point of view, however, they present a range of challenges that are driving exciting developments in probability, PDE, and analysis. In this talk, I give an introduction to two of these challenges, and present some of my contributions to overcome them.
The first challenge arises from the study of infinitesimal generators of stochastic processes. These are partial differential operators that encode the probabilistic properties of solutions to SPDE. To understand why their analysis is so difficult, one should think of them as Laplace operators associated with the geometry of a probability space. Such geometries present analytic challenges that are substantially harder than their Riemannian counterparts. For instance, the lack of Hardy space theory over probability spaces had been considered a major roadblock since the 1980s. In the first part of the talk, I present some joint work with Maas (Vienna) and van Neerven (Delft) that lifts this roadblock.
The second challenge that I plan to discuss is arguably the most important difficulty in SPDE: the lack of smoothness. Because of the irregularity of even the simplest processes (such as the Brownian motion), solutions to SPDE have, in general, no regularity, even in parabolic situations. Consequently, even the basic deterministic estimates needed in SPDE require cutting edge harmonic analysis of differential operators with barely measurable coefficients. In the second part of the talk, I present some of my results on this topic, and their applications to SPDE. This includes joint works with Auscher (Paris 11-Orsay), Monniaux (Aix-Marseille), and van Neerven (Delft).