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 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.