Authors: Artur O Lopes Adriana Neumann
Publish Date: 2015/02/15
Volume: 159, Issue: 4, Pages: 797-822
Abstract
In the present paper we consider a family of continuous time symmetric random walks indexed by kin mathbb N X kttge 0 For each kin mathbb N the matching random walk take values in the finite set of states Gamma k=frac1kmathbb Z/kmathbb Z notice that Gamma k is a subset of mathbb S1 where mathbb S1 is the unitary circle The infinitesimal generator of such chain is denoted by L k The stationary probability for such process converges to the uniform distribution on the circle when krightarrow infty Here we want to study other natural measures obtained via a limit on krightarrow infty that are concentrated on some points of mathbb S1 We will disturb this process by a potential and study for each k the perturbed stationary measures of this new process when krightarrow infty We disturb the system considering a fixed C2 potential V mathbb S1 rightarrow mathbb R and we will denote by V k the restriction of V to Gamma k Then we define a nonstochastic semigroup generated by the matrix k L k + k V k where k L k is the infinifesimal generator of X kttge 0 From the continuous time Perron’s Theorem one can normalized such semigroup and then we get another stochastic semigroup which generates a continuous time Markov Chain taking values on Gamma k This new chain is called the continuous time Gibbs state associated to the potential kV k see Lopes et al in J Stat Phys 152894–933 2013 The stationary probability vector for such Markov Chain is denoted by pi kV We assume that the maximum of V is attained in a unique point x 0 of mathbb S1 and from this will follow that pi kVrightarrow delta x 0 Thus here our main goal is to analyze the large deviation principle for the family pi kV when k rightarrow infty The deviation function IV which is defined on mathbb S1 will be obtained from a procedure based on fixed points of the Lax–Oleinik operator and Aubry–Mather theory In order to obtain the associated Lax–Oleinik operator we use the Varadhan’s Lemma for the process X kttge 0 For a careful analysis of the problem we present full details of the proof of the Large Deviation Principle in the Skorohod space for such family of Markov Chains when krightarrow infty Finally we compute the entropy of the invariant probabilities on the Skorohod space associated to the Markov Chains we analyze
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