Asymptotic Properties of Some Multidimensional Diffusions

Abstract

Let $X_t \in \mathbf{R}^d$ be the solution to the stochastic differential equation $dX_t = \sigma(X_t) dB_t + b(X_t) dt, X_0 \in \mathbf{R}^d,$ where $B_t$ is a Brownian motion in $\mathbf{R}^d$. The aim of this paper is to make the following statement precise: "Let $x_t$ be a solution of $\dot{x} = b(x)$. If $|x_t| \rightarrow \infty$ as $t \rightarrow \infty$ and the drift vector field $b(x)$ is well behaved near $x_t$ then with positive probability, $X_t \rightarrow \infty$, and does so asymptotically like $x_t$." Examples are provided to illustrate the situations in which this theorem may be applied.