Sample path properties and small ball probabilities for stochastic fractional diffusion equations

TL;DR

This paper investigates the regularity, path properties, and small ball probabilities of solutions to a stochastic fractional diffusion equation driven by Gaussian noise, establishing existence, uniqueness, and detailed sample path behavior.

Contribution

It provides new results on the sample path regularity, moduli of continuity, and small ball probabilities for solutions to stochastic fractional diffusion equations, including Chung-type laws.

Findings

Established existence and uniqueness of solutions.

Derived exact moduli of continuity and laws of the iterated logarithm.

Analyzed small ball probabilities for the solutions.

Abstract

We consider the following stochastic space-time fractional diffusion equation with vanishing initial condition:$$ \partial^{\beta} u(t, x)=- \left(-\Delta\right)^{\alpha / 2} u(t, x)+ I_{0+}^{\gamma}\left[\dot{W}(t, x)\right],\quad t\in[0,T],\: x \in \mathbb{R}^d,$$ where $\alpha>0$, $\beta\in(0,2)$, $\gamma\in[0,1)$, $\left(-\Delta\right)^{\alpha/2}$ is the fractional/power of Laplacian and $\dot{W}$ is a fractional space-time Gaussian noise. We prove the existence and uniqueness of the solution and then focus on various sample path regularity properties of the solution. More specifically, we establish the exact uniform and local moduli of continuity and Chung-type laws of the iterated logarithm. The small ball probability is also studied.