15 research outputs found
Continuous-time random walks with reset events: Historical background and new perspectives
In this paper, we consider a stochastic process that may experience random
reset events which relocate the system to its starting position. We focus our
attention on a one-dimensional, monotonic continuous-time random walk with a
constant drift: the process moves in a fixed direction between the reset
events, either by the effect of the random jumps, or by the action of a
deterministic bias. However, the orientation of its motion is randomly
determined after each restart. As a result of these alternating dynamics,
interesting properties do emerge. General formulas for the propagator as well
as for two extreme statistics, the survival probability and the mean
first-passage time, are also derived. The rigor of these analytical results is
verified by numerical estimations, for particular but illuminating examples.Comment: 11 pages, 5 figure
Conditioned backward and forward times of diffusion with stochastic resetting: a renewal theory approach
Stochastic resetting can be naturally understood as a renewal process
governing the evolution of an underlying stochastic process. In this work, we
formally derive well-known results of diffusion with resets from a renewal
theory perspective. Parallel to the concepts from renewal theory, we introduce
the conditioned backward and forward times for stochastic processes with
resetting to be the times since the last and until the next reset, given that
the current state of the system is known. We focus on studying diffusion under
Markovian and non-Markovian resetting. For these cases, we find the conditioned
backward and forward time PDFs, comparing them with numerical simulations of
the process. In particular, we find that for power-law reset time PDFs with
asymptotic form , significant changes in the
properties of the conditioned backward and forward times happen at half-integer
values of . This is due to the composition between the long-time
scaling of diffusion and the reset time PDF.Comment: Submitted to Physical Review
Random Walks on Comb-like Structures under Stochastic Resetting
We study the long-time dynamics of the mean squared displacement of a random walker moving on a comb structure under the effect of stochastic resetting. We consider that the walker's motion along the backbone is diffusive and it performs short jumps separated by random resting periods along fingers. We take into account two different types of resetting acting separately: global resetting from any point in the comb to the initial position and resetting from a finger to the corresponding backbone. We analyze the interplay between the waiting process and Markovian and non-Markovian resetting processes on the overall mean squared displacement. The Markovian resetting from the fingers is found to induce normal diffusion, thereby minimizing the trapping effect of fingers. In contrast, for non-Markovian local resetting, an interesting crossover with three different regimes emerges, with two of them subdiffusive and one of them diffusive. Thus, an interesting interplay between the exponents characterizing the waiting time distributions of the subdiffusive random walk and resetting takes place. As for global resetting, its effect is even more drastic as it precludes normal diffusion. Specifically, such a resetting can induce a constant asymptotic mean squared displacement in the Markovian case or two distinct regimes of subdiffusive motion in the non-Markovian case