On the local time of the multiparameter wiener process and the asymptotic behaviour of an associated integral

Abstract

Let W be the p-parameter Wiener process with values in . Let f be a continuous function with compact support in , and be given by . We show that Xt, normalized by the factors , converges in law (resp. a.s.) as t↑∞. This can be applied to describe the behaviour of the local time of W when x→0, and to prove an ergodic property of Xt. Finally we give a stochastic integral representation of the local time of the two-parameter real-valued Wiener process