zbMATH — the first resource for mathematics

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
MSC:
 60F10 Large deviations 60J25 Continuous-time Markov processes on general state spaces