Asymptotic properties of some multidimensional diffusions (Q1091041)

Let \(X_ t\in {\mathbb{R}}^ d\) be the solution to the stochastic differential equation \[ dX_ t=\sigma (X_ t)dB_ t+b(X_ t)dt,\quad X_ 0\in {\mathbb{R}}^ d, \] where \(B_ t\) is a Brownian motion in \({\mathbb{R}}^ d\). The aim of the paper is to make the following statement precise: ''Let \(x_ t\) be a solution of \(\dot x=b(x)\). If \(| x_ t| \to \infty\) as \(t\to \infty\) and the drift vector field b(x) is well behaved near \(x_ t\) then with positive probability, \(X_ t\to \infty\), and does so asymptotically like \(x_ t.''\) Examples are provided to illustrate the situations in which this theorem may be applied.