Relative entropy and waiting times for continuous-time Markov processes
Jean-René Chazottes, CPht, CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, France.
Cristian Giardina, Eurandom, Postbus 513, 5600 MB Eindhoven, the Netherlands
Frank Redig, Mathematical Institute Leiden university
Abstract
For discrete-time stochastic processes, there is a close connection
between return (resp. waiting) times and entropy (resp. relative entropy). Such a connection cannot
be straightforwardly extended to the continuous-time
setting. Contrarily to the discrete-time case one needs a
reference measure on path space and so the natural object is relative entropy rather
than entropy. In this paper we elaborate on this in the case of
continuous-time Markov processes with finite state space. A reference
measure of special interest is the one associated to the time-reversed
process. In that case relative entropy is interpreted as the entropy
production rate. The main results of this paper are: almost-sure
convergence to relative entropy of the logarithm of
waiting-times ratios suitably normalized, and their
fluctuation properties (central limit theorem and large deviation principle).
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