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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 $k\in \mathbb{N}$, $\{X_k(t),\,t\geq 0\}$. For each $k\in \mathbb{N}$ the matching random walk take values in the finite set of states $Γ_k=\frac{1}{k}(\mathbb{Z}/k\mathbb{Z})$ which is a subset of the unitary circle. The stationary probability for such process converges to the uniform distribution on the circle, when $k\to \infty$. We disturb the system considering a fixed $C^2$ potential $V: \mathbb{S}^1 \to \mathbb{R}$ and we will denote by $V_k$ the restriction of $V$ to $Γ_k$. 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 $\{X_k(t),\,t\geq 0\}$. 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 $Γ_k$. The stationary probability vector for such Markov Chain is denoted by $π_{k,V}$. 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 $π_{k,V}\to δ_{x_0}$. Our main goal is to analyze the large deviation principle for the family $π_{k,V}$, when $k \to\infty$. 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.