Diffusion in a partially absorbing medium with position and occupation time resetting

Abstract

AbstractIn this paper we consider diffusion in a domain Ω containing a partially absorbing targetMwith position and occupation time resetting. The occupation timeAtis a Brownian functional that determines the amount of time that the particle spends inMover the time interval [0,t]. We assume that there exists some internal stateUtof the particle at timetwhich is modified whenever the particle is diffusing withinM. The stateUtis taken to be a monotonically increasing function ofAt, and absorption occurs as soon asUtcrosses some fixed threshold. We first show how to analyze threshold absorption in terms of the joint probability density or generalized propagatorP(x,a,t|x0) for the pair (Xt,At) in the case of a non-absorbing substrateM, whereXtis the particle position at timetandx0is the initial position. We then introduce a generalized stochastic resetting protocol in which both the positionXtand the internal stateUtare reset to their initial values,Xt→x0andUt→0, at a Poisson rater. The latter is mathematically equivalent to resetting the occupation time,At→ 0. Since resetting is governed by a renewal process, the survival probability with resetting can be expressed in terms of the survival probability without resetting, which means that the statistics of absorption can be determined by calculating the double Laplace transform ofP(x,a,t|x0) with respect totanda. In order to develop the basic theory, we focus on one-dimensional diffusion withMgiven by a finite or semi-infinite interval, and explore how the mean first passage time with resetting depends on various model parameters. We also compare the threshold mechanism with the classical case of a constant absorption rate.