Local times of self-intersection and sample path properties of Volterra Gaussian processes

TL;DR

This paper investigates the sample path properties and local times of Volterra Gaussian processes, establishing laws of the iterated logarithm, local time existence, and self-intersection local times in both one and two dimensions.

Contribution

It provides new results on the law of the iterated logarithm, local time existence, and self-intersection local times for Volterra Gaussian processes, including the Rosen renormalized version.

Findings

Law of the iterated logarithm for one-dimensional processes

Existence of local times and their dependence on the kernel

Existence of Rosen renormalized self-intersection local times in planar processes

Abstract

We study a Volterra Gaussian process of the form $X(t)=\int^t_0K(t,s)d{W(s)},$ where $W$ is a Wiener process and $K$ is a continuous kernel. In dimension one, we prove a law of the iterated logarithm, discuss the existence of local times and verify a continuous dependence between the local time and the kernel that generates the process. Furthermore, we prove the existence of the Rosen renormalized self-intersection local times for a planar Gaussian Volterra process.