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Hitting Probabilities for Systems of Stochastic PDEs: An Overview

  • Robert C. DalangEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 229)

Abstract

We consider a d-dimensional random field that solves a possibly non-linear system of stochastic partial differential equations, such as stochastic heat or wave equations. We present results, obtained in joint works with Davar Khoshnevisan and Eulalia Nualart, and with Marta Sanz-Solé, on upper and lower bounds on the probabilities that the random field visits a deterministic subset of \(\mathbb {R}^d\), in terms, respectively, of Hausdorff measure and Newtonian capacity of the subset. These bounds determine the critical dimension above which points are polar, but do not, in general, determine whether points are polar in the critical dimension. For linear SPDEs, we discuss, based on joint work with Carl Mueller and Yimin Xiao, how the issue of polarity of points can be resolved in the critical dimension.

Keywords

Hitting probabilities Systems of stochastics PDEs Malliavin calculus Stochastic heat equation Stochastic wave equation Spatially homogeneous Gaussian noise Capacity Hausdorff measure 

2010 Mathematics Subject Classification

Primary: 60H15 Secondary: 60J45 60G15 60H07 60G60. 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut de Mathématiques, Ecole Polytechnique Fédérale de LausanneLausanneSwitzerland

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