On the Green's functions and Martin boundary structure of a planar diffusion in a discontinuous layered medium

TL;DR

This paper analyzes a two-dimensional diffusion in a layered medium, deriving explicit Green's functions, asymptotics, and Martin boundary structure, including the novel nonminimality phenomenon for diffusions with regular coefficients.

Contribution

It introduces a detailed theoretical framework for layered diffusions, deriving explicit Green's functions, asymptotics, and characterizing the Martin boundary, including the nonminimal boundary phenomenon.

Findings

Explicit Laplace transforms of Green's functions derived

Asymptotic behaviors of Green's functions computed

Full and minimal Martin boundaries identified

Abstract

We consider a two-dimensional diffusion process in a two-layered plane, governed by distinct covariance matrices in the upper and lower half-planes and by two drift vectors pointed away from the $x$-axis. We first analyze the case where the generator of the process is in divergence form, that is, when the flux is continuous across the interface. Then we extend the study to a broader class of processes whose behavior at the interface forms an oblique two-dimensional analogue of the skew Brownian motion. We provide a detailed theoretical analysis of this transient process. Our main results are as follows: (i) we derive explicit Laplace transforms of the Green's functions; (ii) we compute exact asymptotics of the Green's functions along all possible trajectories in the plane; (iii) We determine all positive harmonic functions, identifying the full and minimal Martin boundaries, which turn out to be distinct. The nonminimality of the Martin boundary is a noteworthy phenomenon for diffusions with regular coefficients. To obtain an analytical description of the process, we fully develop a three-variable version of the so-called kernel method by deriving and exploiting a functional equation involving unknown Laplace transforms of Green's functions and two known kernels $\gamma_+(x,y)$ and $\gamma_{-}(x,z)$. The introduction of independent auxiliary variables $y$ and $z$, associated with each half-plane, is a key idea.