On sample paths properties of sub-Gaussian type random fields and applications to stochastic heat equations

Abstract

Abstract. We study sample paths properties of a class of sub-Gaussian type random fields X(t), t ∈ T, focusing on the case where a parameter set T is endowed with an anisotropic metric and imposing some kind of Hölder continuity condition on the field X. Our aim is to establish upper bounds for the distribution of the supremum P

sup t∈T |X(t)| > u for bounded T and to evaluate a rate of growth of X over an unbounded domain V by considering upper bounds for P

sup t∈V |X(t)| f(t) > u for a properly chosen continuous function f. The study is motivated by applications to random fields related to stochastic heat equations. Extensive recent investigations of such equations resulted, in particular, in establishing the Hölder continuity of solutions in various settings. It is quite natural and appealing to consider different functionals of solutions. We evaluate the distribution of suprema of solutions and their asymptotic rate of growth.