Existence of generalized local times for Gaussian random fields

Abstract

We consider a Gaussian centered random field that has values in the Euclidean space. We investigate the existence of local time for the random field as a generalized functional, an element of the Sobolev space constructed for our random field. We give the sufficient condition for such an existence in terms of the field covariation and apply it in a few examples: the Brownian motion with additional weight and the intersection local time of two Brownian motions.