Accumulation time of stochastic processes with resetting

Abstract

One of the characteristic features of a stochastic process under resetting is that the probability density converges to a nonequilibrium stationary state (NESS). In addition, the approach to the stationary state exhibits a dynamical phase transition, which can be interpreted as a traveling front separating spatial regions for which the probability density has relaxed to the NESS from those where it has not. One can establish the existence of the phase transition by carrying out an asymptotic expansion of the exact solution. In this paper we develop an alternative, direct method for characterizing the approach to the NESS of a stochastic process with resetting that is based on the calculation of the so-called accumulation time. The latter is the analog of the mean first passage time of a search process, in which the survival probability density is replaced by an accumulation fraction density. In the case of one-dimensional Brownian motion with Poissonian resetting, we derive the asymptotic formula $|x-x_0|\approx \sqrt{4rD}T(x)$ for $|x-x_0|\gg \sqrt{D/r}$, where $T(x)$ is the accumulation time at $x$, $r$ is the constant resetting rate, $D$ is the diffusivity and $x_0$ is the reset point. This is identical in form to the traveling front condition for the dynamical phase transition. We also derive an analogous result for diffusion in higher spatial dimensions and for non-Poissonian resetting. We then consider the effects of delays such as refractory periods and finite return times. In both cases we establish that the asymptotic behavior of $T(x)$ is independent of the delays. Finally, we extend the analysis to a run-and-tumble particle with resetting. We thus establish the accumulation time of a stochastic process with resetting as a useful quantity for characterizing the approach to an NESS (if it exists) that is relatively straightforward to calculate.

arXiv comment14 pages

References in corpus (19)

• Diffusion with Stochastic Resetting
• Stochastic Resetting and Applications
• Diffusion with Optimal Resetting
• First order transition for the optimal search time of Lévy flights with resetting
• Diffusion under time-dependent resetting
• Run and tumble particle under resetting: a renewal approach
• Diffusion in a potential landscape with stochastic resetting
• Diffusion with resetting in arbitrary spatial dimension
• Dynamical transition in the temporal relaxation of stochastic processes under resetting
• Diffusion with stochastic resetting at power-law times
• Non-equilibrium steady states of stochastic processes with intermittent resetting
• Search with home returns provides advantage under high uncertainty
• Resetting processes with non-instanteneous return
• Effects of refractory period on stochastic resetting
• Diffusion with stochastic resetting is invariant to return speed
• Transport properties of random walks under stochastic non-instantaneous resetting
• Invariants of motion with stochastic resetting and space-time coupled returns
• Occupation time of a run-and-tumble particle with resetting
• Stochastic processes subject to a reset-and-residence mechanism: transport properties and first arrival statistics

Cited by in corpus (9)

• The inspection paradox in stochastic resetting
• The cost of stochastic resetting
• Diffusion processes with Gamma-distributed resetting and non-instantaneous returns
• A probabilistic model of diffusion through a semi-permeable barrier
• Search with stochastic home-returns can expedite classical first passage under resetting
• Accumulation time of diffusion in a 3D singularly perturbed domain
• Accumulation times for diffusion-mediated surface reactions
• Local accumulation time for diffusion in cells with gap junction coupling
• Accumulation time of diffusion in a 2D singularly perturbed domain