  In this paper we consider a threshold surface absorption mechanism for a
particle diffusing in a domain containing a single target $\calU $. The target
boundary $\partial \calU$ is taken to be a reactive surface that modifies an
internal state $Z_t$ of the particle when in contact with the surface at time
$t$, with $Z_0=h$. The state $Z_t$ is taken to be a monotonically decreasing
function of the so-called boundary local time, and absorption occurs as soon as
$Z_t$ reaches zero. (The boundary local time is a Brownian functional that
determines the amount of time that the particle spends in a neighborhood of
$\partial \calU$.) We first show how to analyze threshold surface absorption in
terms of the joint probability density or generalized propagator
$P_0(\x,\ell,t|\x_0)$ 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 at time $t$, respectively, 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 $Z_t$ are reset to their initial values,
$\X_t\rightarrow \x_0$ and $Z_t\rightarrow h$, at a Poisson rate $r$. The
latter is mathematically equivalent to resetting the boundary local time,
$\ell_t\rightarrow 0$. 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 Laplace transform of $P_0(\x,\ell,t|\x_0)$
with respect to $t$. We contrast this with the case where only particle
position is reset, which is not governed by a renewal process. We illustrate
the theory using the simple examples of diffusion on the half-line and a
spherical target in a spherical domain.
