Space-dependent diffusion with stochastic resetting: A first-passage study.

Abstract

We explore the effect of stochastic resetting on the first-passage properties of space-dependent diffusion in the presence of a constant bias. In our analytically tractable model system, a particle diffusing in a linear potential U(x) ∝ μ|x| with a spatially varying diffusion coefficient D(x) = D0|x| undergoes stochastic resetting, i.e., returns to its initial position x0 at random intervals of time, with a constant rate r. Considering an absorbing boundary placed at xa < x0, we first derive an exact expression of the survival probability of the diffusing particle in the Laplace space and then explore its first-passage to the origin as a limiting case of that general result. In the limit xa → 0, we derive an exact analytic expression for the first-passage time distribution of the underlying process. Once resetting is introduced, the system is observed to exhibit a series of dynamical transitions in terms of a sole parameter, ν≔(1+μD0 -1), that captures the interplay of the drift and the diffusion. Constructing a full phase diagram in terms of ν, we show that for ν < 0, i.e., when the potential is strongly repulsive, the particle can never reach the origin. In contrast, for weakly repulsive or attractive potential (ν > 0), it eventually reaches the origin. Resetting accelerates such first-passage when ν < 3 but hinders its completion for ν > 3. A resetting transition is therefore observed at ν = 3, and we provide a comprehensive analysis of the same. The present study paves the way for an array of theoretical and experimental works that combine stochastic resetting with inhomogeneous diffusion in a conservative force field.