On the Convergence of Diffusion Processes Conditioned to Remain in a Bounded Region for Large Time to Limiting Positive Recurrent Diffusion Processes

Abstract

Let $X(t)$ be a diffusion process on $R^d$ with generator $L = (1/2)\nabla \cdot a\nabla + b\nabla$ and let $\{P_x\}, x \in R^d$, be the corresponding measures on paths. Pick $0 < t < T < \infty$ and consider the process on the time interval $\lbrack 0, t\rbrack$ conditioned to remain in a certain open, connected bounded region $G$ up to time $T$. We obtain a new process $Y^T(s), 0 \leqq s \leqq t$. Let $\tau_G = \inf\{s: X(s) \not\in G\}$. With certain hypotheses on $P_x(\tau_G > s)$ (which are always satisfied if $a^{-1}b$ is a gradient function), we show that $Y^T(s)$ is an inhomogeneous diffusion process and that as $T \rightarrow \infty, Y^T(s), 0 \leq s \leq t$ converges to a limiting homogeneous positive recurrent diffusion $Y(s), 0 \leq s \leq t$, with state space $G$. Since $t$ is arbitrary, we actually obtain a limiting process $Y(s), 0 \leq s < \infty$. The generator of the limiting process may be written in the form $L_G = (1/2)\nabla \cdot a\nabla + b\nabla + a(\nabla g_0/g_0)\nabla - a\nabla h_{g_0} \nabla$ where $g_0$ is the square root of the density of a measure $\mu_0$ which minimizes the $I$-function for the process, over all $\gamma \in \mathscr{P}(\bar{G})$, the set of probability measures on $\bar{G}$. The function $h_{g_0}$ appears in the explicit calculation of $I(\mu_0)$ and solves a certain variational equation. The invariant measure for the process is $\mu_0$.