Hitting Probability and Hausdorff Dimension Results for Gaussian Random Fields

Probabilités et Statistique

Salle séminaire M3-324
Yimin Xiao
Michigan State University
Mercredi, 30 Mai, 2018 - 10:30 - 11:30
Gaussian random fields generate many interesting random fractal sets whose fractal dimensions may be connected with various hitting probability problems. One of the open problems in the area is to determine when the trajectory of a Gaussian random field may intersecting a deterministic set.

We provide a solution to this problem for a special case, where the deterministic set is a singleton. More specifically, we show that for a wide class of Gaussian random fields, points are polar in the critical dimension. Examples of such random fields include solutions of systems of linear stochastic partial differential equations with deterministic coefficients, such as the stochastic heat equation or wave equation with space-time white noise, or colored noise in spatial dimensions $k \geq 1$. Our approach builds on a delicate covering argument developed by M. Talagrand (1995, 1998) for the study of fractional Brownian motion, and uses a harmonizable representation of the solutions of these stochastic pde's.  (This talk is based on a joint work with R. Dalang and C. Mueller.)