Continuous differentiability of renormalized intersection local times in R1

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.