Effect of stochastic resetting on Brownian motion with stochastic diffusion coefficient

Abstract

We study the dynamics of a Brownian motion with a diffusion coefficient which evolves stochastically. We first study this process in arbitrary dimensions and find the scaling form and the corresponding scaling function of the position distribution. We find that the tails of the distribution have exponential tails with a ballistic scaling. We then introduce the resetting dynamics where, at a constant rate, both the position and the diffusion coefficient are reset to zero. This eventually leads to a nonequilibrium stationary state, which we study in arbitrary dimensions. In stark contrast to ordinary Brownian motion under resetting, the stationary position distribution in one dimension has a logarithmic divergence at the origin. For higher dimensions, however, the divergence disappears and the distribution attains a dimension-dependent constant value at the origin, which we compute exactly. The distribution has a generic stretched exponential tail in all dimensions. We also study the approach to the stationary state and find that, as time increases, an inner core region around the origin attains the stationary state, while the outside region still has a transient distribution -- this inner stationary region grows $\sim t^2$, i.e., with a constant acceleration, much faster than ordinary Brownian motion.