On a stiff problem in two-dimensional space

In this paper we will study a stiff problem in two-dimensional space and especially its probabilistic counterpart. Roughly speaking, the heat equation with a parameter $\epsilon >0$ is under consideration: $${\mathrm{\partial }}_t{u}^{\epsilon }(t,x)=\frac{1}{2}\mathrm{\nabla }\cdot ({\mathbf{A}}_{\epsilon }(x)\mathrm{\nabla }{u}^{\epsilon }(t,x)),t\ge 0,x\in {\mathbb{R}}^2,$$ where ${\mathbf{A}}_{\epsilon }(x)={\text{Id}}_2$, the identity matrix, for $x\notin \Omega_\varepsilon:={x=(x_1,x_2)\in $\mathbb{R}$^2: |x_2|<\varepsilon}$while$$$\mathbf{A}$\varepsilon(x):=\begin{pmatrix} a\varepsilon^- & 0 \ 0 & a^\shortmid_\varepsilon \end{pmatrix}$$withtwopositiveconstants$a^-\varepsilon, a^\shortmid\varepsilon$for$x\in \Omega_\varepsilon$. There exists a diffusion process $X^{\epsilon }$ on ${\mathbb{R}}^2$ associated to this heat equation in the sense that $u^{\epsilon }(t,x):={\mathbf{E}}^x{u}^{\epsilon }(0,X_t^{\epsilon })$ is its unique weak solution. Note that ${\mathrm{\Omega }}_{\epsilon }$ collapses to the $x_1$-axis, a barrier of zero volume, as $\epsilon \downarrow 0$. The main purpose of this paper is to derive all possible limiting process $X$ of $X^{\epsilon }$ as $\epsilon \downarrow 0$. In addition, the limiting flux $u$ of the solution $u^{\epsilon }$ as $\epsilon \downarrow 0$ and all possible boundary conditions satisfied by $u$ will be also characterized.