Diffusion-mediated surface reactions and stochastic resetting

In this paper, we investigate the effects of stochastic resetting on diffusion in $\R^d\backslash \calU$, where $\calU$ is a bounded obstacle with a partially absorbing surface $\partial \calU$. We begin by considering a Robin boundary condition with a constant reactivity $\kappa_0$, and show how previous results are recovered in the limits $\kappa_0\rightarrow 0,\infty$. 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 propagator $P(\x,\ell,t|\x_0)$ for the pair $(\X_t,\ell_t)$ in the case of a perfectly reflecting surface, where $\X_t$ and $\ell_t$ 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 (BVP) satisfied by the propagator in the presence of 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 allows us to explore the dependence of the MFPT on the resetting rate $r$ and the type of surface reactions. The theory is illustrated using the example of a spherically symmetric surface.