Polarity of points for Gaussian random fields. (English) Zbl 1390.60135

Dalang, Robert C.; Mueller, Carl; Xiao, Yimin

Ann. Probab. 45, No. 6B, 4700-4751 (2017).

The authors study polarity of points of Gaussian random fields \(v(x), x\in \mathbb R^k,\) in the case of critical dimensions \(Q.\) For various random fields, it is shown that \(P\left(\exists x\in J: v(x)=z\right)=0\) for any closed box \(J\subset \mathbb R^k\) and \(z\in \mathbb R^Q.\) The authors develop some Talagrand approaches for fractional Brownian motion to the case of stochastic PDEs using harmonizable representations of their solutions. In particular, random fields that are solutions of systems of linear stochastic partial differential equations with deterministic coefficients are studied. Fractional Brownian fields, the stochastic heat equations and wave equations with space-time white noise and colored noise in spatial dimensions are investigated in detail.

Reviewer: Andriy Olenko (Victoria)

MSC:

60G15	Gaussian processes
60J45	Probabilistic potential theory
60G60	Random fields
60G22	Fractional processes, including fractional Brownian motion

Keywords:

hitting probabilities; polarity of points; critical dimension; harmonizable representation; stochastic partial differential equations