Family of closed-form solutions for two-dimensional correlated diffusion processes

Abstract

Diffusion processes with boundaries are models of transport phenomena with wide applicability across many fields. These processes are described by their probability density functions (PDFs), which often obey Fokker-Planck equations (FPEs). While obtaining analytical solutions is often possible in the absence of boundaries, obtaining closed-form solutions to the FPE is more challenging once absorbing boundaries are present. As a result, analyses of these processes have largely relied on approximations or direct simulations. In this paper, we studied two-dimensional, time-homogeneous, spatially-correlated diffusion with linear, axis-aligned, absorbing boundaries. Our main result is the explicit construction of a full family of closed-form solutions for their PDFs using the method of images (MoI). We found that such solutions can be built if and only if the correlation coefficient $ρ$ between the two diffusing processes takes one of a numerable set of values. Using a geometric argument, we derived the complete set of $ρ$'s where such solutions can be found. Solvable $ρ$'s are given by $ρ= - \cos \left( \fracπ{k} \right)$, where $k \in \mathbb{Z}^+ \cup \{ +\infty\}$. Solutions were validated in simulations. Qualitative behaviors of the process appear to vary smoothly over $ρ$, allowing extrapolation from our solutions to cases with unsolvable $ρ$'s.