We consider a class of Markov processes with resettings, where at random times, the Markov processes are restarted from a predetermined point or a region. These processes are frequently applied in physics, chemistry, biology, economics, and in population dynamics. In this paper we establish the local large deviation principle (LLDP) for the Wiener processes with random resettings, where the resettings occur at the arrival time of a Poisson process. Here, at each resetting time, a new resetting point is selected at random, according to a conditional distribution.

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The large deviation principle on phase space is proved for a class of Markov processes known as random population dynamics with catastrophes. In the paper we study the process which corresponds to the random population dynamics with linear growth and uniform catastrophes, where an eliminating portion of the population is chosen uniformly. The large deviation result provides an optimal trajectory of large fluctuation: it shows how the large fluctuations occur for this class of processes.

АннотацияThe paper considers a continuous-time birth-death process where the jump rate has an asymptotically polynomial dependence on the process position. We obtain a rough exponential asymptotics for the probability of excursions of a re-scaled process contained within a neighborhood of a given continuous non-negative function.

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