Jump Processes with Self-Interactions: Large Deviation Asymptotics

Abstract: We consider a pure jump process ${X_t}{t\ge 0}$ with values in a finite state space $S= {1, \ldots, d}$ for which the jump rates at time instant $t$ depend on the occupation measure $L_t \doteq t^{-1} \int_0^t \delta{X_s}\,ds$. Such self-interacting chains arise in many contexts within statistical physics and applied probability. Under appropriate conditions, a large deviation principle is established for the pair $(L_t, R_t)$, as $t \to \infty$, where $R_t$ is the empirical flux process associated with the jump process. We show that the rate function takes a simple form that can be viewed as a dynamical generalization of the classical Donsker and Varadhan rate function for the analogous quantities in the setting of Markov processes, in particular, unlike the Markovian case, the rate function is not convex. Since the state process is non-Markovian, different techniques are needed than in the setting of Donsker and Varadhan and our proofs rely on variational representations for functionals of Poisson random measures and stochastic control methods.