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

In this paper we consider diffusion in a domain $\Omega$ containing a partially absorbing target $\calM$ with position and occupation time resetting. The occupation time $A_t$ is a Brownian functional that determines the amount of time that the particle spends in $\calM$ over the time interval $[0,t]$. We assume that there exists some internal state $\calU_t$ of the particle at time $t$ which is modified whenever the particle is diffusing within $\calM$. The state $\calU_t$ is taken to be a monotonically increasing function of $A_t$, and absorption occurs as soon as $\calU_t$ crosses some fixed threshold. We first show how to analyze threshold absorption in terms of the joint probability density or generalized propagator $P(\x,a,t|\x_0)$ for the pair $(\X_t,A_t)$ in the case of a non-absorbing substrate $\calM$, where $\X_t$ is the particle position at time $t$ and $\x_0$ is the initial position. We then introduce a generalized stochastic resetting protocol in which both the position $\X_t$ and the internal state $\calU_t$ are reset to their initial values, $\X_t\rightarrow \x_0$ and $\calU_t\rightarrow 0$, at a Poisson rate $r$. The latter is mathematically equivalent to resetting the occupation time, $A_t\rightarrow 0$. 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 of $P(\x,a,t|\x_0)$ with respect to $t$ and $a$. In order to develop the basic theory, we focus on one-dimensional (1D) diffusion with $\calM$ given by a finite or semi-infinite interval, and explore how the MFPT with resetting depends on various model parameters. We also compare the threshold mechanism with the classical case of a constant absorption rate.