Sample Paths of the Solution to the Fractional-colored Stochastic Heat Equation

TL;DR

This paper investigates the path properties of solutions to a fractional-colored stochastic heat equation, deriving precise continuity moduli and laws of the iterated logarithm for the process in both time and space.

Contribution

It provides new exact results on the uniform and local moduli of continuity and Chung-type laws for the solution to the fractional-colored stochastic heat equation.

Findings

Derived exact uniform and local moduli of continuity.

Established Chung-type laws of the iterated logarithm.

Analyzed path properties in both time and space variables.

Abstract

Let u = {u(t, x), t $\in$ [0, T ], x $\in$ R d } be the solution to the linear stochastic heat equation driven by a fractional noise in time with correlated spatial structure. We study various path properties of the process u with respect to the time and space variable, respectively. In particular, we derive their exact uniform and local moduli of continuity and Chung-type laws of the iterated logarithm.