Properties of $\varphi$-sub-Gaussian stochastic processes related to the heat equation with random initial conditions

Abstract

In this paper, there are studied sample paths properties of stochastic processes representing solutions (in $L_2(\Omega)$ sense) of the heat equation with random initial conditions given by $\varphi$-sub-Gaussian stationary processes. The main results are the bounds for the distributions of the suprema for such stochastic processes considered over bounded and unbounded domains.