A generalised telegraph process with resetting driven by random trials

Iuliano Antonella, University of Basilicata

We analyse a stochastic process X(t) that describes the random motion on the real line characterized by two alternating velocities c and −v (c, v > 0) and subject to resets. Specifically, at the end of each (forward or backward) displacement a Bernoulli trial is performed, so that the particle is subject to an instantaneous reset at the origin with probability p, or it begins a new displacement along the other direction with probability 1 − p, for a given p ∈ [0, 1]. We assume also that the resetting mechanism is based on independent Bernoulli trials, that are independent from the random displacements. After each reset the motion restarts with the same initial velocity V (0) = y ∈ {−v, c} and proceeds adopting the same rules, independently from the previous history of the motion. Hence, the sequence of resets constitutes the renewal instants of a renewal process, say K(t). Moreover, our approach is based on the Markov property of the resetting times and on the knowledge of the distribution of the intertimes between consecutive velocity changes. Under these assumptions we obtain the probability laws of the process when (i) the random times have identical exponential distribution with parameter λ, and (ii) the random times have Erlang distribution with equal parameters (n, λ). Some results on moments and related limiting behaviour are also disclosed.

Area: CS32 - Stochastic processes with random resetting (Mario Abundo and Antonella Iuliano)

Keywords: Bernoulli trials, random intertimes, random velocity, resetting, telegraph process

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