Traces of symmetric Markov processes and their characterizations

Details Traces of symmetric Markov processes and their characterizations Traces of symmetric Markov processes and their characterizations

2006-06-30

arxiv.org/abs/math/0606784v1

Annals of Probability 2006, Vol. 34, No. 3, 1052-1102

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117905000000657 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117905000000657

Time change is one of the most basic and very useful transformations for Markov processes. The time changed process can also be regarded as the trace of the original process on the support of the Revuz measure used in the time change. In this paper we give a complete characterization of time changed processes of an arbitrary symmetric Markov process, in terms of the Beurling--Deny decomposition of their associated Dirichlet forms and of Feller measures of the process. In particular, we determine the jumping and killing measure (or, equivalently, the L\'{e}vy system) for the time-changed process. We further discuss when the trace Dirichlet form for the time changed process can be characterized as the space of finite Douglas integrals defined by Feller measures. Finally, we give a probabilistic characterization of Feller measures in terms of the excursions of the base process.

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