Partially Reflected Diffusion

Source abstract

The radiation (reactive or Robin) boundary condition for the diffusion equation is widely used in chemical and biological applications to express reactive boundaries. The underlying trajectories of the diffusing particles are believed to be partially absorbed and partially reflected at the reactive boundary; however, the relation between the reaction constant in the Robin boundary condition and the reflection probability is not well defined. In this paper we define the partially reflected process as a limit of the Markovian jump process generated by the Euler scheme for the underlying Itô dynamics with partial boundary reflection. Trajectories that cross the boundary are terminated with probability $P\sqrt{\Delta t}$ and otherwise are reflected in a normal or oblique direction. We use boundary layer analysis of the corresponding master equation to resolve the nonuniform convergence of the probability density function of the numerical scheme to the solution of the Fokker–Planck equation in a half-space, with the Robin constant $\kappa$. The boundary layer equation is of the Wiener–Hopf type. We show that the Robin boundary condition is recovered if and only if trajectories are reflected in the conormal direction $\boldsymbol{\sigma}\boldsymbol{n}$, where $\boldsymbol{\sigma}$ is the (possibly anisotropic) constant diffusion matrix and $\boldsymbol{n}$ is the unit normal to the boundary. Otherwise, the density satisfies an oblique derivative boundary condition. The constant $\kappa$ is related to P by $\kappa= rP\sqrt{\sigma_n}$, where $r=1/\sqrt{\pi}$ and $\sigma_n=\boldsymbol{n}^T\boldsymbol{\sigma}\n$. The reflection law and the relation are new for diffusion in higher dimensions.