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

Haozhe Shan, Rubén Moreno-Bote, Jan Drugowitsch

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. 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
(
π
k
)
, where
k
∈
Z
+
∪
{
+
∞
}
. 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.