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


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