Continuous Differentiability of Renormalized Intersection Local Times in R^{1}

TL;DR

This paper proves that the k-fold renormalized self-intersection local time for one-dimensional Brownian motion is almost surely continuously differentiable with respect to spatial variables, enhancing understanding of its regularity properties.

Contribution

It establishes the almost sure continuous differentiability of the renormalized intersection local times in one dimension, a novel regularity result for these stochastic processes.

Findings

Proves continuous differentiability of local times in R^1.

Shows regularity holds with probability 1.

Advances theoretical understanding of Brownian self-intersections.

Abstract

We study $\gamma_{k}(x_2,...,x_k;t)$, the k-fold renormalized self-intersection local time for Brownian motion in $R^1$. Our main result says that $\gamma_{k}(x_2,...,x_k;t)$ is continuously differentiable in the spatial variables, with probability 1.