Large deviations for unbounded additive functionals of a Markov process with discrete time (noncompact case).

Gulinsky, O.V.; Liptser, R.S.; Lototsky, S.V.

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Let $Ω$ be a bounded open subset of $ℝ^{n}$ , $n > 2$ . In $Ω$ we deduce the global differentiability result $u \in H^{2} (Ω, ℝ^{N})$ for the solutions $u \in H^{1} (Ω, ℝ^{n})$ of the Dirichlet problem $u - g \in H_{0}^{1} (Ω, ℝ^{N}), - \sum_{i} D_{i} a^{i} (x, u, D u) = B_{0} (x, u, D u)$ with controlled growth and nonlinearity $q = 2$ . The result was obtained by first extending the interior differentiability result near the boundary and then proving the global differentiability result making use of a covering procedure.

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A Large Deviation Principle (LDP) is proved for the family $\frac{1}{n} \sum_{1}^{n} 𝐟 (x_{i}^{n}) \cdot Z_{i}^{n}$ where the deterministic probability measure $\frac{1}{n} \sum_{1}^{n} δ_{x_{i}^{n}}$ converges weakly to a probability measure $R$ and ${(Z_{i}^{n})}_{i \in ℕ}$ are $ℝ^{d}$ -valued independent random variables whose distribution depends on $x_{i}^{n}$ and satisfies the following exponential moments condition: $sup_{i, n} 𝔼 e^{α^{*} | Z_{i}^{n} |} < + \infty forsome 0 < α^{*} < + \infty .$ In this context, the identification of the rate function is non-trivial due to the absence of equidistribution. We rely on fine convex analysis to address this issue. Among...

On Lebesgue points of functions in the Sobolev class $W^{1, n}$ .

Latvala, Visa (2002)

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