Expediting Feller process with stochastic resetting

Abstract

We explore the effect of stochastic resetting on the first-passage properties of Feller process. The Feller process can be envisioned as space-dependent diffusion, with diffusion coefficient $D(x)=x$, in a potential $U(x)=x\left(\frac{x}{2}-\theta \right)$ that owns a minimum at $\theta$. 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) $x_a<x_0$. We observe that for $x_0<x_a$, resetting accelerates first-passage when $\theta<\theta_c$, where $\theta_c$ is a critical value of $\theta$ that decreases when $x_a$ is moved away from the origin. In stark contrast, for $x_a<x_0$, resetting accelerates first-passage when $\theta>\theta_c$, where $\theta_c$ is a critical value of $\theta$ 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.