### A characterization of Markov solutions for stochastic differential equations with jumps

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In this paper, we prove a Donsker theorem for one-dimensional processes generated by an operator with measurable coefficients. We construct a random walk on any grid on the state space, using the transition probabilities of the approximated process, and the conditional average times it spends on each cell of the grid. Indeed we can compute these quantities by solving some suitable elliptic PDE problems.

We consider the stochastic equation $${X}_{t}={x}_{0}+{\int}_{0}^{t}b(u,{X}_{u})\mathrm{d}{B}_{u},\phantom{\rule{1.0em}{0ex}}t\ge 0,$$ where $B$ is a one-dimensional Brownian motion, ${x}_{0}\in \mathbb{R}$ is the initial value, and $b\phantom{\rule{0.222222em}{0ex}}[0,\infty )\times \mathbb{R}\to \mathbb{R}$ is a time-dependent diffusion coefficient. While the existence of solutions is well-studied for only measurable diffusion coefficients $b$, beyond the homogeneous case there is no general result on the uniqueness in law of the solution. The purpose of the present note is to give conditions on $b$ ensuring the existence as well as the uniqueness in law of the solution.

We consider a generic diffusion on the 1D torus and give a simple representation formula for the large deviation rate functional of its invariant probability measure, in the limit of vanishing noise. Previously, this rate functional had been characterized by M. I. Freidlin and A. D. Wentzell as solution of a rather complex optimization problem. We discuss this last problem in full generality and show that it leads to our formula. We express the rate functional by means of a geometric transformation...