OCCUPATION TIMES OF BROWNIAN SEGMENTS AND THE σ-FINITE WIENER MEASURE

Abstract

We give an asymptotic result for the occupation of Borel sets of functions by the segments of recurrent Brownian motion on consecutive time intervals [n, n +1], n =0, 1, 2, …. This result provides new information on the behavior of Brownian motion, which is illustrated by examples. A formulation in terms of weak convergence of random measures on Polish space is also given. The proof is based on (a strengthened form of) the Darling–Kac occupation time theorem for Markov chains, and our result can be viewed as a "trajectorial" extension of that theorem. The main role in the occupation limit for Brownian segments is played by the σ-finite Wiener measure, which first appeared in a different context. An extension for segments of symmetric α-stable Lévy processes is also given.