Smoothness of Local Times and Self-Intersection Local Times of Gaussian Random Fields

TL;DR

This paper investigates the smoothness properties of local times for various Gaussian random fields, providing conditions for their existence and smoothness, including fractional Brownian motions, sheets, and stochastic heat equations.

Contribution

It establishes necessary and sufficient conditions for the existence and smoothness of local times and self-intersection local times for a broad class of Gaussian fields.

Findings

Conditions for existence of local times.

Criteria for smoothness in Meyer-Watanabe sense.

Applicability to fractional Brownian motions and stochastic heat equations.

Abstract

This paper is concerned with the smoothness (in the sense of Meyer-Watanabe) of the local times of Gaussian random fields. Sufficient and necessary conditions for the existence and smoothness of the local times, collision local times, and self-intersection local times are established for a large class of Gaussian random fields, including fractional Brownian motions, fractional Brownian sheets and solutions of stochastic heat equations driven by space-time Gaussian noise.