Marta Sanz-Solé is a professor of Mathematics at the University of Barcelona. She is a member of the Institute of Catalan Studies and a Fellow of the IMS. Her research interests are in the field of stochastic analysis, specially Malliavin calculus and stochastic partial differential equations. She is, and has been, member of editorial boards of several journals, including the Annals of Probability. Marta Sanz-Solé has served on international scientific committees, advisory boards and evaluation panels. During the period 2011–2014, she held the position of President of the European Mathematical Society.
Marta’s Medallion lecture will be given at the 39th Conference on Stochastic Processes and their Applications (SPA) in Moscow (July 24–28, 2017). See http://www.spa2017.org/
Stochastic partial differential equations: trajectories and densities
The theory of stochastic partial differential equations (SPDEs) emerged about thirty years ago and since then, it has been undergoing a dramatic development. This new field of mathematics is at the crossroad of probability and analysis, combining tools from stochastic analysis and the classical theory of partial differential equations. Motivations for the study of SPDEs arise both within mathematics, as well as from applications to other scientific settings possessing an inherent component of randomness. This randomness can be, for example, in the initial conditions, in the environment, or as an external forcing. Fundamental problems in the theory of SPDEs are the existence and uniqueness of solutions, and the properties of their sample paths. In the relevant examples, these are non-smooth random functions.
A basic question in probabilistic potential theory is to determine whether a random field ever visits a fixed deterministic set A. This leads to a quantitative analysis of the hitting probabilities in terms of geometric measure notions, like the Hausdorff measure or the Bessel–Riesz capacity of the set A. For Gaussian (and also for Lévy) processes, the subject (initiated in the 40’s) has reached a state of maturity. Over the last decade, the study of hitting probabilities relative to sample paths of systems of SPDEs has been in the focus of interest. Substantial progress has been made, thereby contributing to the understanding of qualitative features of SPDEs. There are however many unsolved problems for further investigations.
In my lecture, I will describe the mathematical approach to obtaining upper and lower bounds for hitting probabilities of random fields in terms of Hausdorff measure and Bessel–Riesz capacity, respectively. The roles of the dimensions, the roughness of the sample paths, the one-point and two-point joint distributions and particularly, the structure of the covariance, will be highlighted. Then I will focus on a class of SPDEs defined through linear partial differential operators, with nonlinear Gaussian external forcing. The random field solutions to systems of such equations are random vectors on an abstract Wiener space. Except in very simple cases, they are not Gaussian stochastic processes.
Malliavin calculus provides a powerful toolbox to tackle many questions about probability laws on abstract Wiener spaces. The existence and properties of densities relative to our SPDEs can be proved using this calculus. In particular, the qualitative behaviour of the covariance of two-point joint distributions when these points get close to each other, can be achieved by a detailed analysis of the Malliavin matrix –an infinitesimal covariance type matrix on the Wiener space.
I will illustrate the implementation of this approach with two classical examples: the nonlinear stochastic heat and wave equations. Finally, I will mention some on-going work and open questions.