540 research outputs found
Traces of symmetric Markov processes and their characterizations
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.Comment: Published at http://dx.doi.org/10.1214/009117905000000657 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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