Discrete space-time resetting model: application to first-passage and transmission statistics

Abstract

We consider the dynamics of lattice random walks with resetting. The walker moving randomly on a lattice of arbitrary dimensions resets at every time step to a given site with a constant probability $r$. We construct a discrete renewal equation and present closed-form expressions for different quantities of the resetting dynamics in terms of the underlying reset-free propagator or Green's function. We apply our formalism to the biased random walk dynamics in one-dimensional unbounded space and show how one recovers in the continuous limits results for diffusion with resetting. The resetting dynamics of biased random walker in one-dimensional domain bounded with periodic and reflecting boundaries is also analyzed. Depending on the bias the first-passage probability in periodic domain shows multi-fold non-monotonicity as $r$ is varied. Finally, we apply our formalism to study the transmission dynamics of two lattice walkers with resetting in one-dimensional domain bounded by periodic and reflecting boundaries. The probability of a definite transmission between the walkers shows non-monotonic behavior as the resetting probabilities are varied.

Dynamics of lattice random walk within regions composed of different media and interfaces

“…It would be also interesting to study the effects of spatial heterogeneity on the biased random walk dynamics in the presence of stochastic resetting [44]. A discrete renewal equation in the presence of stochastic resetting has been derived in [34], where working formula are presented to compute different quantities in terms of the reset-free propagator. To study resetting in heterogeneous space, one may use these formula with the propagators derived here.…”

“…Here we derive a first principle procedure to represent the DSDT dynamics of a biased random walker in a heterogeneous space of two media separated by an interface and having different diffusivities. We extend the lattice random walk formalism developed in [32][33][34] to heterogeneous space and show that depending on the position of the interface on a lattice, there exist two exclusive ways to model it: (a) Type A interface placed between two lattice points belonging to different media, and (b) Type B interface placed exactly on a lattice point shared by both media. Very recently, an analytic framework based on the DSDT random walk approach has been developed to study diffusion with inert heterogeneities [35].…”

Stochastic dynamics with multiplicative dichotomic noise: Heterogeneous telegrapher’s equation, anomalous crossovers and resetting

“…Future research could be related to the investigation of ergodic properties of finite-velocity HDPs in absence and presence of resetting [35][36][37][94][95][96], including also corresponding higher-dimensional formulations [38,39]. Infiniteand finite-velocity HDPs in presence of time-dependent resetting [97], non-instantaneous [81,98] and space-time coupled returns [99], HDPs in presence of resetting in an interval [100,101] and bounded in complex potential [102], as well as discrete space-time resetting models [103] for HDPs, are other topics worth investigating.…”

Stochastic resetting in interacting particle systems: a review

“…Over the years, the effects of stochastic resetting have been investigated in a wide spectrum of dynamics, e.g. diffusion [2,[4][5][6][7][8][9][10], random walks [11,12], random walks on disordered lattices [13], Lévy flights [14], Bernoulli trials [15], discrete-time resets [16,17], active motion [18] and transport in cells [19], search problems [14,[20][21][22][23][24][25], RNA-polymerase dynamics [26,27], enzymatic reactions [28], dynamics of ecological systems [29,30], and in discussing Feynman-Kac path integral formalisms [31].…”