Occupation Densities

Abstract

This is a survey article about occupation densities for both random and nonrandom vector fields $X: T \rightarrow \mathbb{R}^d$ where $T \subset \mathbb{R}^N$. For $N = d = 1$ this has previously been called the "local time" of $X$, and, in general, it is the Lebesgue density $\alpha(x)$ of the occupation measure $\mu(\Gamma) =$ Lebesgue measure $\{t\in T: X(t)\in \Gamma\}$. If we restrict $X$ to a subset $A$ of $T$ we get a corresponding density $\alpha(x, A)$ and we will be interested in its behavior both in the space variable $x$ and the set variable $A$. The first part of the paper deals entirely with nonrandom, nondifferentiable vector fields, focusing on the connection between the smoothness of the occupation density and the level sets and local growth of $X$. The other two parts are concerned, respectively, with Markov processes $(N = 1)$ and Gaussian random fields. Here the emphasis is on the interplay between the probabilistic and real-variable aspects of the subject. Special attention is given to Markov local times (in the sense of Blumenthal and Getoor) as occupation densities, and to the role of local nondeterminism in the Gaussian case.