First passage of a diffusing particle under stochastic resetting in bounded domains with spherical symmetry

TL;DR

This paper analyzes how stochastic resetting affects the mean first passage time of a diffusing particle in spherical domains, revealing optimal resetting rates and transition behaviors depending on initial position and domain geometry.

Contribution

It derives explicit formulas for the mean time to absorption under resetting in spherical domains and uncovers phase transition phenomena in the optimal resetting rate behavior.

Findings

Existence of a nonzero optimal resetting rate for certain initial positions.

Continuous and discontinuous transitions in optimal resetting rate depending on domain ratios.

Resetting can significantly reduce mean first passage times in bounded spherical domains.

Abstract

We investigate the first passage properties of a Brownian particle diffusing freely inside a $d$-dimensional sphere with absorbing spherical surface subject to stochastic resetting. We derive the mean time to absorption (MTA) as functions of resetting rate $\gamma$ and initial distance $r$ of the particle to the center of the sphere. We find that when $r>r_c$ there exists a nonzero optimal resetting rate $\gamma_{{\rm opt}}$ at which the MTA is a minimum, where $r_c=\sqrt {d/\left( {d + 4} \right)} R$ and $R$ is the radius of sphere. As $r$ increases, $\gamma_{{\rm opt}}$ exhibits a continuous transition from zero to nonzero at $r=r_c$. Furthermore, we consider that the particle lies in between two two-dimensional or three-dimensional concentric spheres, and obtain the domain in which resetting expedites the MTA, which is $(R_1, r_{c_1}) \cup (r_{c_2},R_2)$, with $R_1$ and $R_2$ being the radius of inner and outer spheres, respectively. Interestingly, when $R_1/R_2$ is less than a critical value, $\gamma_{{\rm opt}}$ exhibits a discontinuous transition at $r=r_{c_1}$; otherwise, such a transition is continuous. However, at $r=r_{c_2}$, $\gamma_{{\rm opt}}$ always shows a continuous transition.