Expediting Feller process with stochastic resetting.

Abstract

We explore the effect of stochastic resetting on the first-passage properties of the Feller process. The Feller process can be envisioned as space-dependent diffusion, with diffusion coefficient D(x)=x, in a potential U(x)=x(x/2-θ) that owns a minimum at θ. This restricts the process to the positive side of the origin and therefore, Feller diffusion can successfully model a vast array of phenomena in biological and social sciences, where realization of negative values is forbidden. In our analytically tractable model system, a particle that undergoes Feller diffusion is subject to Poissonian resetting, i.e., taken back to its initial position at a constant rate r, after random time epochs. We addressed the two distinct cases that arise when the relative position of the absorbing boundary (x_{a}) with respect to the initial position of the particle (x_{0}) differ, i.e., for (a) x_{0}<x_{a} and (b) x_{a}<x_{0}. Utilizing the Fokker-Planck description of the system, we obtained closed-form expressions for the Laplace transform of the survival probability and hence derived the exact expressions of the mean first-passage time 〈T_{r}〉. Performing a comprehensive analysis on the optimal resetting rate (r^{★}) that minimize 〈T_{r}〉 and the maximal speedup that r^{★} renders, we identify the phase space where Poissonian resetting facilitates first-passage for Feller diffusion. We observe that for x_{0}<x_{a}, resetting accelerates first-passage when θ<θ_{c}, where θ_{c} is a critical value of θ that decreases when x_{a} is moved away from the origin. In stark contrast, for x_{a}<x_{0}, resetting accelerates first-passage when θ>θ_{c}, where θ_{c} is a critical value of θ that increases when x_{0} is moved away from the origin. Our study opens up the possibility of a series of subsequent works with more case-specific models of Feller diffusion with resetting.