Raluca Balan, University of Ottawa

Recent advances on SPDEs using the random field approach - Part I

 Abstract: In a seminal article in 1944, Ito introduced the stochastic integral with respect to the Brownian motion, which turned out to be one of the most fruitful ideas in mathematics in the 20th century. This lead to the development of stochastic analysis, a field which includes the study of stochastic partial differential equations (SPDEs). One of the approaches for the study of SPDEs was initiated by Walsh (1986) and relies on the concept of random-field solution. This concept allows us to investigate the probabilistic behavior of the solution to an SPDE, simultaneously in time and space.

In these lectures, we will consider the stochastic heat equation and the stochastic wave equation on the entire space, perturbed by a Gaussian noise which is homogeneous in space (as introduced by Dalang in 1999) and is "colored" in time. This means that the noise behaves in time like a process with stationary increments, for instance the fractional Brownian motion (fBm). Since fBm is not a semi-martingale, Ito calculus techniques cannot be applied in this case. The methods that we will present are based on Malliavin calculus. Without going into technical details, the lectures will illustrate the dynamical interplay between the regularity of the noise and various properties of the solution (such as intermittency and Feyman-Kac representations).