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

## TL;DR

This paper derives a family of explicit closed-form solutions for the probability density functions of two-dimensional correlated diffusion processes with boundaries, valid for specific correlation values, advancing analytical understanding of such models.

## Contribution

It introduces a complete set of closed-form solutions for correlated diffusion with boundaries, valid for specific correlation coefficients, using the method of images and geometric analysis.

## Key findings

- Solutions valid for specific correlation coefficients $
ho = - ext{cos}(rac{ extpi}{k})$
- Validated solutions through simulations
- Qualitative behaviors vary smoothly over $
ho$

## 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 $\rho$ between the two diffusing processes takes one of a numerable set of values. Using a geometric argument, we derived the complete set of $\rho$'s where such solutions can be found. Solvable $\rho$'s are given by $\rho = - \cos \left( \frac{\pi}{k} \right)$, where $k \in \mathbb{Z}^+ \cup \{ +\infty\}$. Solutions were validated in simulations. Qualitative behaviors of the process appear to vary smoothly over $\rho$, allowing extrapolation from our solutions to cases with unsolvable $\rho$'s.