## The averaging principle for diffusions with a small parameter in the case of a noncharacteristic boundary.(English)Zbl 0681.60078

Let $$D\subset R^ 2$$ be a region obtained from $$R_ 0<| x| <r$$ by depressing the circle of radius r inside in such a way that $$D\supset \{x|$$ $$R_ 0<| x| \leq r_ 0\}$$, $$r_ 0<r$$, and there are exactly two points $$(r_ 0,\theta_ 1)$$, $$(r_ 0,\theta_ 2)$$ that belong to the boundary $$\partial D$$. Consider the Dirichlet problem $$L_{\epsilon}u_{\epsilon}=0$$ on D and $$u_{\epsilon}=f$$ on $$\partial D$$, where $$L_{\epsilon}=\epsilon L_ 0+L_ 1$$, $$L_ 0$$ is elliptic, and $L_ 1=2^{-1}A(r,\theta)\partial^ 2/\partial \theta^ 2+B(r,\theta)\partial /\partial \theta$ has an invariant measure $$\nu_ r(d\theta)$$ on each circle with radius r. Denote by $$X^{\epsilon}(t)$$ the diffusion process associated with $$L_{\epsilon}$$. The main result states that the distribution of $$X^{\epsilon}(\tau_ D^{\epsilon})$$, where $$\tau_ D^{\epsilon}$$ is the first exit time of $$X^{\epsilon}(t)$$ from D, converges weakly, as $$\epsilon$$ $$\to 0$$, to $[1-p(| x|)]\delta_{R_ 0}(dr)\nu_{R_ 0}(\theta)d\theta +p(| x|)\delta_{r_ 0}(dr)\quad [\rho \delta_{\theta_ 1}(d\theta)+(1-\rho)\delta_{\theta_ 2}(d\theta)]$ with explicitly known p($$\cdot)$$, $$\nu_{R_ 0}(\cdot)$$, $$\rho$$, for $$x\in D\cap \{x|$$ $$| x| <r_ 0\}$$. The limit for $$x\in D\cap \{x| | x| \geq r_ 0\}$$ is also found. As a corollary, since $$u_{\epsilon}(x)=E_ xf(X^{\epsilon}(\tau_ D^{\epsilon})),$$ the limit of $$u_{\epsilon}(x)$$ as $$\epsilon$$ $$\to 0$$ is explicitly evaluated.
Reviewer: A.Korzeniowski

### MSC:

 60J60 Diffusion processes 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
Full Text: