Large Deviations for Stationary Probabilities of a Family of Continuous Time Markov Chains via Aubry–Mather Theory

Abstract

In the present paper, we consider a family of continuous time symmetric random walks indexed by \(k\in \mathbb {N}\), \(\{X_k(t),\,t\ge 0\}\). For each \(k\in \mathbb {N}\) the matching random walk take values in the finite set of states \(\Gamma _k=\frac{1}{k}(\mathbb {Z}/k\mathbb {Z})\); notice that \(\Gamma _k\) is a subset of \(\mathbb {S}^1\), where \(\mathbb {S}^1\) 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 \(k\rightarrow \infty \). Here we want to study other natural measures, obtained via a limit on \(k\rightarrow \infty \), that are concentrated on some points of \(\mathbb {S}^1\). We will disturb this process by a potential and study for each \(k\) the perturbed stationary measures of this new process when \(k\rightarrow \infty \). We disturb the system considering a fixed \(C^2\) potential \(V: \mathbb {S}^1 \rightarrow \mathbb {R}\) and we will denote by \(V_k\) the restriction of \(V\) to \(\Gamma _k\). Then, we define a non-stochastic semigroup generated by the matrix \(k\,\, L_k + k\,\, V_k\), where \(k\,\, L_k \) is the infinifesimal generator of \(\{X_k(t),\,t\ge 0\}\). 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 \(k\,V_k\), see (Lopes et al. in J Stat Phys 152:894–933, 2013). The stationary probability vector for such Markov Chain is denoted by \(\pi _{k,V}\). We assume that the maximum of \(V\) is attained in a unique point \(x_0\) of \(\mathbb {S}^1\), and from this will follow that \(\pi _{k,V}\rightarrow \delta _{x_0}\). Thus, here, our main goal is to analyze the large deviation principle for the family \(\pi _{k,V}\), when \(k \rightarrow \infty \). The deviation function \(I^V\), which is defined on \( \mathbb {S}^1\), 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 \(\{X_k(t),\,t\ge 0\}\). 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 \(k\rightarrow \infty \). Finally, we compute the entropy of the invariant probabilities on the Skorohod space associated to the Markov Chains we analyze.