Hitting Probability and Hausdorff Dimension Results for Gaussian Random Fields

Fractal Geometry and Dynamics

11 October 15:00 - 15:50

Yimin Xiao - Michigan State University

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 of 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 ≥ 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.)
Kenneth J. Falconer
University of St Andrews
Maarit Järvenpää
University of Oulu
Antti Kupiainen
University of Helsinki
Francois Ledrappier
University of Notre Dame
Pertti Mattila
University of Helsinki


Maarit Järvenpää


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