Jump-Drift and Jump-Diffusion Processes : Large Deviations for the density, the current and the jump-flow and for the excursions between jumps

Abstract

For one-dimensional Jump-Drift and Jump-Diffusion processes converging towards some steady state, the large deviations of a long dynamical trajectory are described from two perspectives. Firstly, the joint probability of the empirical time-averaged density, of the empirical time-averaged current and of the empirical time-averaged jump-flow are studied via the large deviations at Level 2.5. Secondly, the joint probability of the empirical jumps and of the empirical excursions between consecutive jumps are analyzed via the large deviations at Level 2.5 for the alternate Markov chain that governs the series of all the jump events of a long trajectory. These two general frameworks are then applied to three examples of positive jump-drift processes without diffusion, and to two examples of jump-diffusion processes, in order to illustrate various simplifications that may occur in rate functions and in contraction procedures. I. II. JUMP-DIFFUSION PROCESS CONVERGING TOWARDS SOME LOCALIZED STEADY-STATEA.Diffusion with drift v(x) and diffusion coefficient D(x) ; jumps with rate λ(x) and probability Π(x |x)We consider the following one-dimensional jump-diffusion dynamics : when at position x, the particle experiences the drift v(x) and diffuses with the diffusion coefficient D(x), but with the jump rate λ(x) per unit time, it can also

Large deviations for metastable states of Markov processes with absorbing states with applications to population models in stable or randomly switching environment

“…As a consequence, as in many other Markov jump processes [31,38,39,41,45], the rate function obtained from Eq. 120…”

“…measure the density of jumps from one configuration y to another configuration x. The joint probability to see the empirical density ρ(x) and the empirical flows q(x, y) follows the large deviation form for large T [25,[28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46]]…”

“…Indeed, the initial classification of large deviations into three nested levels (see the reviews [22][23][24] and references therein), with Level 1 for empirical observables, Level 2 for the empirical density, and Level 3 for the empirical process, has been more recently supplemented by the Level 2.5 concerning the joint distribution of the empirical density and of the empirical flows. The rate functions at Level 2.5 are explicit for various types of Markov processes, including discrete-time Markov chains [24][25][26][27][28][29], continuous-time Markov jump processes [25,[28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46] and Diffusion processes [28,29,33,34,37,[47][48][49]. As a consequence, the explicit Level 2.5 can be considered as a starting point from which many other large deviations properties can be derived via contraction.…”