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 $varepsilon>0$ is under consideration: [ partial_t u^varepsilon(t,x)=frac{1}{2} abla cdot left(mathbf{A}_varepsilon(x) abla u^varepsilon(t,x) right),quad tgeq 0, xin mathbb{R}^2, ] where $mathbf{A}_varepsilon(x)=text{Id}_2$, the identity matrix, for $x otin Omega_varepsilon:={x=(x_1,x_2)in mathbb{R}^2: |x_2|<varepsilon}$ while $$mathbf{A}_varepsilon(x):=begin{pmatrix} a_varepsilon^- & 0 0 & a^shortmid_varepsilon end{pmatrix}$$ with two positive constants $a^-_varepsilon, a^shortmid_varepsilon$ for $xin Omega_varepsilon$. There exists a diffusion process $X^varepsilon$ on $mathbb{R}^2$ associated to this heat equation in the sense that $u^varepsilon(t,x):=mathbf{E}^xu^varepsilon(0,X_t^varepsilon)$ is its unique weak solution. Note that $Omega_varepsilon$ collapses to the $x_1$-axis, a barrier of zero volume, as $varepsilondownarrow 0$. The main purpose of this paper is to derive all possible limiting process $X$ of $X^varepsilon$ as $varepsilondownarrow 0$. In addition, the limiting flux $u$ of the solution $u^varepsilon$ as $varepsilondownarrow 0 $ and all possible boundary conditions satisfied by $u$ will be also characterized.