Sample Path Properties of the Local Times of Strongly Symmetric Markov Processes Via Gaussian Processes

Abstract

Necessary and sufficient conditions are obtained for the almost sure joint continuity of the local time of a strongly symmetric standard Markov process $X$. Necessary and sufficient conditions are also obtained for the almost sure global boundedness and unboundedness of the local time and for the almost sure continuity, boundedness and unboundedness of the local time in the neighborhood of a point in the state space. The conditions are given in terms of the 1-potential density of $X$. The proofs rely on an isomorphism theorem of Dynkin which relates the local times of Markov processes related to $X$ to a mean zero Gaussian process with covariance equal to the 1-potential density of $X$. By showing the equivalence of sample path properties of Gaussian processes with the related local times, known necessary and sufficient conditions for various sample path properties of Gaussian processes are carried over to the local times. The results are used to obtain examples of local times with interesting sample path behavior.