  In this paper, we develop an encounter-based model of partial surface
adsorption for fractional diffusion in a bounded domain. We take the
probability of adsorption to depend on the amount of particle-surface contact
time, as specified by a Brownian functional known as the boundary local time
$\ell(t)$. If the rate of adsorption is state dependent, then the adsorption
process is non-Markovian, reflecting the fact that surface
activation/deactivation proceeds progressively by repeated particle encounters.
The generalized adsorption event is identified as the first time that the local
time crosses a randomly generated threshold. Different models of adsorption
(Markovian and non-Markovian) then correspond to different choices for the
random threshold probability density $\psi(\ell)$. The marginal probability
density for particle position $\X(t)$ prior to absorption depends on $\psi$ and
the joint probability density for the pair $(\X(t),\ell(t))$, also known as the
local time propagator. In the case of normal diffusion one can use a
Feynman-Kac formula to derive an evolution equation for the propagator. Here we
derive the local time propagator equation for fractional diffusion by taking a
continuum limit of a heavy-tailed continuous-time random walk (CTRW). We use
our encounter-based model to investigate the effects of subdiffusion and
non-Markovian adsorption on the long-time behavior of the first passage time
(FPT) density in a finite interval $[0,L]$ with a reflecting boundary at $x=L$.
In particular, we determine how the choice of function $\psi$ affects the
large-$t$ power law decay of the FPT density. Finally, we indicate how to
extend the model to higher spatial dimensions.
