The existence and smoothness of self-intersection local time for a class of Gaussian processes

Abstract

In this paper sufficient conditions for the existence and smoothness of the self-intersection local time of a class of Gaussian processes are given in the sense of Meyer–Watanabe through $L^2$ convergence and Wiener chaos expansion. Let $X$ be a centered Gaussian process, whose canonical metric $E\left[(X\left(t\right)-X{\left(s\right)}^2)\right]$ is commensurate with ${\sigma }^2\left(|t-s|\right)$, where $\sigma \left(\cdot \right)$ is continuous, increasing and concave. If ${\int }_0^T\frac{1}{\sigma \left(\gamma \right)}d\gamma <\infty$, then the self-intersection local time of the Gaussian process exists, and if ${\int }_0^T{\left(\sigma \left(\gamma \right)\right)}^{-\frac{3}{2}}d\gamma <\infty$, the self-intersection local time of the Gaussian process is smooth in the sense of Meyer–Watanabe.