Intersection Local Times of All Orders for Brownian and Stable Density Processes--Construction, Renormalisation and Limit Laws

Abstract

The Brownian and stable density processes are distribution valued processes that arise both via limiting operations on infinite collections of Brownian motions and stable Levy processes and as the solutions of certain stochastic partial differential equations. Their (self-)intersection local times (ILT's) of various orders can be defined in a manner somewhat akin to that used to define the self-intersection local times of simple $\mathscr{R}^d$-valued processes; that is, via a limiting operation involving approximate delta functions. We obtain a full characterisation of this limiting procedure, determining precisely in which cases we have convergence and deriving the appropriate renormalisation for obtaining weak convergence when only this is available. We also obtain results of a fluctuation nature, that describe the rate of convergence in the former case. Our results cover all dimensions and all levels of self-intersection.