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 epsilon > 0: partial derivative(t)u(epsilon)(t,x) = 1/2 del <middle dot> (A(epsilon) (x)del u(epsilon)(t,x)), t >= 0 , x is an element of R-2, where A(epsilon) (x) := Id(2), the identity matrix, for x is not an element of Omega(epsilon) := { x = (x(1),x(2)) is an element of R-2: |x(2) | < epsilon } and A(epsilon) (x) := (a(epsilon)(-) 0, 0 a(epsilon)(-)) x is an element of Omega(epsilon) with two constants a(epsilon)(-), a(epsilon)(1) > 0. The solution u epsilon 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 u epsilon as epsilon down arrow 0 in a certain sense. Note that there exists a diffusion process X-epsilon on R-2 associated to this heat equation in the sense that u(epsilon)(t, x) := E(x)u(epsilon) (0, X-t(epsilon) ) is its unique solution. The main result of this paper figures out the limiting process of X-epsilon as epsilon down arrow 0 for all possible cases. As a by product, the limiting flux u in an L-2-sense and several boundary conditions on the x(1)-axis satisfied by u regarding various cases will be further obtained.