On a stiff problem in two-dimensional space

In this paper we study a stiff problem in two-dimensional space and especially characterize its probabilistic counterpart. More specifically, consider the heat equation with a parameter ε>0: ∂tuε(t,x)=1 2∇·(Aε(x)∇uε(t,x)),t≥0,x∈R2, where Aε(x):=Id2, the identity matrix, for x∉Ωε:={x=(x1,x2)∈R2:|x2|<ε} and Aε(x):=aε−00a ε|,x∈Ωε with two constants aε−,aε|>0. The solution uε is usually called a flux. Then the stiff problem is concerned with the existence and characterization of the limit u, called the limiting flux, of uε as ε↓0 in a certain sense. Note that there exists a diffusion process Xε on R2 associated to this heat equation in the sense that uε(t,x):=Exuε(0,Xtε) is its unique solution. The main result of this paper figures out the limiting process of Xε as ε↓0 for all possible cases. As a byproduct, the limiting flux u in an L2-sense and several boundary conditions on the x1-axis satisfied by u regarding various cases will be further obtained.