##
**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. \(J\).

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)

### Keywords:

Markov process; occupation time measure; large deviation principle; rate function; uniqueness theorem### Citations:

Zbl 0348.60032
PDFBibTeX
XMLCite

\textit{J. R. Baxter} et al., Commun. Pure Appl. Math. 44, No. 8--9, 911--923 (1991; Zbl 0749.60025)

Full Text:
DOI

### References:

[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) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.