Derivatives of Self-intersection Local Times

Abstract

We show that the renormalized self-intersection local time $\gamma_t(x)$ for both the Brownian motion and symmetric stable process in R¹ is differentiable in the spatial variable and that $\gamma'_t(0)$ can be characterized as the continuous process of zero quadratic variation in the decomposition of a natural Dirichlet process. This Dirichlet process is the potential of a random Schwartz distribution. Analogous results for fractional derivatives of self-intersection local times in R¹ and R² are also discussed.

Jay Rosen: This research was supported, in part, by grants from the National Science Foundation and PSC-CUNY.