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

We explore the effect of stochastic resetting on the first-passage properties of space-dependent diffusion in presence of a constant bias. In our analytically tractable model system, a particle diffusing in a linear potential $U(x)\propto\mu |x|$ with a spatially varying diffusion coefficient $D(x)=D_0|x|$ undergoes stochastic resetting, i.e., returns to its initial position $x_0$ at random intervals of time, with a constant rate $r$. Considering an absorbing boundary placed at $x_a<x_0$, 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 $x_a\to0$, 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, $\nu=(1+\mu D_0^{-1})$, that captures the interplay of the drift and the diffusion. Constructing a full phase diagram in terms of $\nu$, we show that for $\nu<0$, i.e., when the potential is strongly repulsive, the particle can never reach the origin. In contrast, for weakly repulsive or attractive potential ($\nu>0$), it eventually reaches the origin. Resetting accelerates such first-passage when $\nu<3$, but hinders its completion for $\nu>3$. A resetting transition is therefore observed at $\nu=3$, 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.