Some familiar examples for which the large deviation principle does not hold. (English) Zbl 0749.60025

Let \(X_ t\), \(t\in T=[0,\infty)\), denote a Markov process with state space \(E\), \(M(E)\) be the set of probability measures on the Borel subsets of \(E\), and \(L_ t(w,A)\) be the normalized occupation time measure. For a given \(x\in E\) and \(A\subset M(E)\) let \(\mu_ t(A)=P^ x(L_ t(w,.)\in A)\). M. D. Donsker and S. R. S. Varadhan [Commun. Pure Appl. Math. 29, 389-461 (1976; Zbl 0348.60032)] showed that for every \(x\in E\), \(\{\mu_ t,\;t\in T\}\) satisfies the restricted large deviation principle (LDP) with a certain rate function (r.f.) \(I\). A natural question arises whether another r.f. \(J: M(E)\to[0,\infty]\) can be found such that \(\{\mu_ t\}\) satisfies the full LDP.
In this paper the authors show that the answer to this question is negative. Their main result, the uniqueness theorem 2.1, implies that if a full LDP is to hold with a r.f. \(J\) and the lower bound holds with r.f. \(I\), then \(J=I\). Then the authors show that for some closed set \(C\subset M(E)\) the family \(\{\mu_ t\}\) does not satisfy the upper bound condition in LDP with r.f. \(I\). Hence, by the uniqueness theorem, \(\{\mu_ t\}\) cannot satisfy the full LDP with any r.f. \(J\).
Reviewer: E.Pancheva (Sofia)


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


Zbl 0348.60032
Full Text: DOI


[1] Boylan, III. Jour. Math. 8 pp 19– (1969)
[2] Donsker, Comm. Pure Appl. Math. 28 pp 1– (1975)
[3] Donsker, Comm. Pure Appl. Math. 29 pp 389– (1976)
[4] Stochastic Processes, John Wiley & Sons, New York, 1953.
[5] Kipnis, Comm. Math. Phy. 104 pp 1– (1986)
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