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 L2 convergence and Wiener chaos expansion. Let X be a centered Gaussian process, whose canonical metric E[(X(t)−X(s)2)] is commensurate with σ2(|t−s|), where σ(⋅) is continuous, increasing and concave. If ∫0T1σ(γ)dγ<∞, then the self-intersection local time of the Gaussian process exists, and if ∫0T(σ(γ))−32dγ<∞, the self-intersection local time of the Gaussian process is smooth in the sense of Meyer–Watanabe.