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Long time asymptotics of the ratio of measures of small tubes and a large deviation result. (English) Zbl 0656.60039
Probability theory and mathematical statistics, Proc. 5th Jap.-USSR Symp., Kyoto/Jap. 1986, Lect. Notes Math. 1299, 482-491 (1988).
[For the entire collection see Zbl 0626.00026.]
Asymptotics of \(T^{-1}\log \{\mu (B_ T(x,\delta))/m(B_ T(x,\delta))\}\) are discussed, for \(T\to \infty\) and then \(\delta\) \(\to 0\), where m denotes the law of an original Markov process on the path space X, \(\mu\) is the law of another Markov process and \[ B_ T(x,\delta)=\{y\in X:\quad d(F^ ty,F^ tx)\leq \delta \quad for\quad 0\leq t\leq T\}. \] Here \(F^ t\) is a compact dynamical system and d is the pseudo-metric on X induced from a metric on the state space. A meaning of the latter approach consists in reproducing the level three large deviation results of M. D. Donsker and S. R. S. Varadhan [see Commun. Pure Appl. Math. 36, 183-212 (1983; Zbl 0512.60068) and ibid. 28, 1-47 (1975; Zbl 0323.60069)] for Markov processes via these asymptotics of the above ratios.
Reviewer: J.Steinebach
60F10 Large deviations
60J25 Continuous-time Markov processes on general state spaces