The Averaging Principle for Diffusions with a Small Parameter in the Case of a Noncharacteristic Boundary

Abstract

Let $L_\varepsilon = \varepsilon L_0 + L_1$, where $L_0$ is a nondegenerate elliptic operator on $R^2$ and $L_1 = \frac{1}{2}A(r, \theta)\partial^2/\partial\theta^2 + B(r, \theta)\partial/\partial\theta$. We assume that for fixed $r, L_1$ generates a positive recurrent diffusion on the circle with invariant measure $\mu_r(d\theta)$. Let $X^\varepsilon(t) = (r^\varepsilon(t), \theta^\varepsilon(t))$ denote the diffusion generated by $(1/\varepsilon)L_\varepsilon$ and let $u_\varepsilon$ be the solution to the Dirichlet problem $L_\varepsilon u = 0$ on $D$ and $u = f$ and $\partial D$, where $D = \{x: r_1 < |x| < r_2\}$ and $f$ is continuous. Then $u_\varepsilon(x) = E_x f(X^\varepsilon(\tau^\varepsilon_D))$, where $\tau^\varepsilon_D$ is the first exit time from $D$. By the averaging principle, the process $r^\varepsilon(t)$ converges weakly to the process $r^0(t)$ generated by $\bar{L}_0$, the operator obtained from $L_0$, by restricting to functions depending only on $r$ and averaging the coefficients with respect to $\mu_r(d\theta)$. Furthermore, $P_x(\theta^\varepsilon(\tau^\varepsilon_D \in d\theta\mid\tau^\varepsilon_{r_i} < \tau^\varepsilon_{r_j})$ converges weakly as $\varepsilon \rightarrow 0$, to a measure which can be calculated in terms of $\mu_{r_i}(\partial\theta)$ and the diffusion matrix of $L_0$, where $\tau^\varepsilon_r$ denotes the hitting time of the circle of radius $r$ and $(i, j) = (1, 2)$ or $(2, 1)$. The above information allows one to evaluate the limiting distribution of $(r^\varepsilon(\tau^\varepsilon_D), \theta^\varepsilon(\tau^\varepsilon_D))$ and thus also the asymptotics of $u_\varepsilon(x)$. Call $\theta$ the fast variable and $r$ the slow variable. In this paper we investigate what happens to the averaging principle in the case that the boundary of $D$ is no longer characteristic for the equation slow variable = constant.