Diffusion-mediated surface reactions and stochastic resetting

Abstract

Abstract In this paper, we investigate the effects of stochastic resetting on diffusion in R d \ U , where U is a bounded obstacle with a partially absorbing surface ∂ U . We begin by considering a Robin boundary condition with a constant reactivity κ 0, and show how previous results are recovered in the limits κ 0 → 0, ∞. We then generalize the Robin boundary condition to a more general probabilistic model of diffusion-mediated surface reactions using an encounter-based approach. The latter considers the joint probability density or generalized propagator P(x, ℓ, t|x 0) for the pair (X t , ℓ t ) in the case of a perfectly reflecting surface, where X t and ℓ t denote the particle position and local time, respectively. The local time determines the amount of time that a Brownian particle spends in a neighborhood of the boundary. The effects of surface reactions are then incorporated via an appropriate stopping condition for the boundary local time. We construct the boundary value problem satisfied by the propagator in the presence of position resetting, and use this to derive implicit equations for the marginal density of particle position and the survival probability. We highlight the fact that these equations are difficult to solve in the case of non-constant reactivities, since resetting is not governed by a renewal process. We then consider a simpler problem in which both the position and local time are reset. In this case, the survival probability with resetting can be expressed in terms of the survival probability without resetting, which considerably simplifies the analysis. We illustrate the theory using the example of a spherically symmetric surface. In particular, we show that the effects of a partially absorbing surface on the mean first passage time (MFPT) for total absorption differs significantly if local time resetting is included. That is, the MFPT for a totally absorbing surface is increased by a multiplicative factor when the local time is reset, whereas the MFPT is increased additively when only particle position is reset.