Asymptotic properties of some multidimensional diffusions. (English) Zbl 0622.60062

Clark, Charles R.

Ann. Probab. 15, 985-1008 (1987).

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.

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60J25	Continuous-time Markov processes on general state spaces

Keywords:

transience; asymptotic behavior; stochastic differential equation