Large Deviations for stationary probabilities of a family of continuous time Markov chains via Aubry-Mather theory

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 ${\mathrm{\Gamma }}_k=\frac{1}{k}(\mathbb{Z}\mathrm{/}k\mathbb{Z})$ which is a subset of the unitary circle. The stationary probability for such process converges to the uniform distribution on the circle, when $k\to \mathrm{\infty }$. We disturb the system considering a fixed $C^2$ potential $V: $\mathbb{S}$^1 \to $\mathbb{R}$$andwewilldenoteby$V_k$therestrictionof$V$to$\Gamma_k$\mathrm{.}Then,wedefineanon-stochasticsemigroupgeneratedbythematrix$k,, L_k + k,, V_k$,where$k,, L_k $istheinfinifesimalgeneratorof${X_k(t),,t\geq 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 ${\mathrm{\Gamma }}_k$. 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 ${\pi }_{k,V}\to {\delta }_{x_0}$. Our main goal is to analyze the large deviation principle for the family ${\pi }_{k,V}$, when $k\to \mathrm{\infty }$. 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.