The local time of a two-dimensional diffusion

Abstract

When a markovian random process taking values in a continuous state-space, such as R, visits a particular point repeatedly, it is natural to seek some quantity which records how long it spends there. Typically, however, the number of visits made to the point is uncountably infinite, and the (Lebesgue) length of time spent there is zero. One interesting object to consider is the local time, sometimes thought of as the occupation density of the process, which at each point is a random Cantor function that increases only when the process visits the point. The review article by Rogers (1989) contains a good introduction to the local time of a one-dimensional brownian motion and its relevance to the excursions of brownian motion from zero. In two dimensions, a typical diffusion, such as brownian motion in the plane, never revisits a point, so it does not have a local time. In this paper we shall construct the local times of some particular two-dimensional diffusions on a one-dimensional subspace, and show that they are jointly continuous in both time and space.