Monotonic continuous-time random walks with drift and stochastic reset events

Abstract

In this paper we consider a stochastic process that may experience random reset events which suddenly bring the system to the starting value and analyze the relevant statistical magnitudes. We focus our attention on monotonic continuous-time random walks with a constant drift: The process increases between the reset events, either by the effect of the random jumps, or by the action of the deterministic drift. As a result of all these combined factors interesting properties emerge, like the existence (for any drift strength) of a stationary transition probability density function, or the faculty of the model to reproduce power-law-like behavior. General formulas for two extreme statistics, the survival probability, and the mean exit time are also derived. To corroborate in an independent way the results of the paper, Monte Carlo methods were used. These numerical estimations are in full agreement with the analytical predictions.

Slow Lévy flights

“…If π t (t ′ ) = δ t ′ ,0 , the site chosen for revisit is unique (the origin), a case which corresponds to the well-studied random walk with resetting to the origin [47][48][49][50]. For more general kernels, the walk is strongly path-dependent but still described by a master equation:…”

“…The exact equality (19) is general: it is valid at all t and for any kernel π t (t ′ ) (allowing to recover results on the resetting to the origin with bias [48]). Despite of being out-of-equilibrium, the FDR with constant bias in this system is the same as for ordinary random walks, where the response X t is entirely determined by the fluctuations at zero bias.…”

Optimal diffusive search: nonequilibrium resetting versus equilibrium dynamics

“…A continuous-time random walk model in the presence of a drift and resetting has also been studied recently [16]. Finally, in the context of search process, a related model has been studied by Gelenbe [17] where searchers are introduced stochastically into the system: there is a single searcher present at a given time with a random lifetime and when the searcher dies, a new searcher is introduced into the system at the initial starting point.…”

Directed random walk with random restarts: The Sisyphus random walk

“…The seminal work of Manrubia and Zanette [2] considering Markov chains where a reset mechanism operates has motivated new interest in the field, and presently the dynamics of systems with resets is being subjected to intense study. In [3,4] Brownian motion with resets was considered while in [5] resets were incorporated to a compound Poisson process with constant drift. Such intermittent strategies have been considered in general mathematical frameworks [3][4][5][6][7][8], but also in more specific contexts, like behavioral ecology where the browsing activity of living organisms (e.g., capuchin monkeys) may be suddenly interrupted to return to a preferred location [9,10], or econophysics where modifying Gibrat's law to include reset events has made it possible to account for the power law's distribution of a firm's growth [11].…”