INTERSECTION LOCAL-TIMES OF ALL ORDERS FOR BROWNIAN AND STABLE DENSITY PROCESSES - CONSTRUCTION, RENORMALIZATION AND LIMIT LAWS

The Brownian and stable density processes are distribution valued proc esses 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-) inter section local times (ILT's) of various orders can be defined in a mann er somewhat akin to that used to define the self-intersection local ti mes of simple R(d)-valued processes; that is, via a limiting operation involving approximate delta functions. We obtain a full characterisat ion of this limiting procedure, determining precisely in which cases w e have convergence and deriving the appropriate renormalisation for ob taining weak convergence when only this is available. We also obtain r esults 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.