On the Green's functions and Martin boundary structure of a planar diffusion in a discontinuous layered medium
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 $γ_+(x,y)$ and $γ_{-}(x,z)$. The introduction of independent auxiliary variables $y$ and $z$, associated with each half-plane, is a key idea.