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Many processes in cell biology involve diffusion in a domain $\Omega$ thatcontains a target $\calU$ whose boundary $\partial \calU$ is a chemicallyreactive surface. Such a target could represent a single reactive molecule, anintracellular compartment or a whole cell. Recently, a probabilistic frameworkfor studying diffusion-mediated surface reactions has been developed thatconsiders the joint probability density or propagator for the particle positionand the so-called boundary local time. The latter characterizes the amount oftime that a Brownian particle spends in the neighborhood of a point on atotally reflecting boundary. The effects of surface reactions are thenincorporated via an appropriate stopping condition for the boundary local time.In this paper we generalize the theory of diffusion-mediated surface reactionsto cases where the whole interior target domain $\calU$ acts as a partialabsorber rather than the target boundary $\partial \calU$. Now the particle canfreely enter and exit $\calU$, and is only able to react (be absorbed) within$\calU$. The appropriate Brownian functional is then the occupation time(accumulated time that the particle spends within $\calU$) rather than theboundary local time. We show that both cases can be considered within a unifiedframework by using a Feynman-Kac formula to derive a boundary value problem(BVP) for the propagator of the corresponding Brownian functional, andintroducing an associated stopping condition. We illustrate the theory bycalculating the mean first passage time (MFPT) for a spherical target $\calU$located at the center of a spherical domain $\Omega$. This is achieved bysolving the propagator BVP directly, rather than using spectral methods. Wefind that if the first moment of the stopping time density is infinite, thenthe MFPT is also infinite, that is, the spherical target is not sufficientlyabsorbing.

Diffusion-mediated surface reactions, Brownian functionals and the Feynman-Kac formula