Non-linear diffusion with stochastic resetting

Abstract

Resetting or restart, when applied to a stochastic process, usually brings its dynamics to a time-independent stationary state. In turn, the optimal resetting rate makes the mean time to reach a target to be finite and the shortest one. These and other innovative problems have been intensively studied over the last decade mainly in the case of ordinary diffusive processes. Intrigued by this fact we consider here the influence of stochastic resetting on the non-linear diffusion analysing its fundamental properties. We derive the exact formula for the mean squared displacement and demonstrate how it attains the steady-state value under the influence of the exponential resetting. This mechanism brings also about that the spatial support of the probability density function, which for the free non-linear diffusion is confined to the domain of a finite size, tends to span the entire set of real numbers. In addition, the first-passage properties for the non-linear diffusion intermittent by the exponential resetting are investigated. We find analytical expressions for the mean first-passage time and determine by means of the numerical method the optimal resetting rate which minimizes the mean time needed for a particle to reach a pre-determined target. Finally, we test and confirm the universal property that the relative fluctuation in the mean first-passage time of optimally restarted non-linear diffusion is equal to unity.