On regularization of formal Fourier--Wiener transform of the self-intersection local time of planar Gaussian process

TL;DR

This paper develops a regularization method for the Fourier-Wiener transform of the self-intersection local time of planar Gaussian processes, addressing divergence issues on diagonals and exploring geometric properties.

Contribution

It introduces a new regularization technique for divergent integrals related to self-intersection local times, utilizing strong local nondeterminism and geometric insights.

Findings

Regularization method successfully applied to planar Wiener process perturbations.

Analysis of strong local nondeterminism's role in regularization.

Explicit integral representation for the Fourier-Wiener transform.

Abstract

Fourier-Wiener transform of the formal expression for multiple self-intersection local time is described in terms of the integral, which is divergent on the diagonals. The method of regularization we use in this work related to regularization of functions with non-integrable singularities. The strong local nondeterminism property, which is more restrictive than the property of local nondeterminism introduced by S.Berman is considered. Its geometrical meaning in the construction of the regularization is investigated. As the example the problem of regularization is solved for the compact perturbation of the planar Wiener process.