Sample path large deviations for a class of Markov chains related to disordered mean field models

We prove a large deviation principle on path space for a class of discrete time Markov processes whose state space is the intersection of a regular domain $\L\subset \R^d$ with some lattice of spacing $\e$. Transitions from $x$ to $y$ are allowed if $\e^{-1}(x-y)\in \D$ for some fixed set of vectors $\D$. The transition probabilities $p_\e(t,x,y)$, which themselves depend on $\e$, are allowed to depend on the starting point $x$ and the time $t$ in a sufficiently regular way, except near the boundaries, where some singular behaviour is allowed. The rate function is identified as an action functional which is given as the integral of a Lagrange function. %of time dependent relativistic classical mechanics. Markov processes of this type arise in the study of mean field dynamics of disordered mean field models.