Ornstein-Uhlenbeck process in a random potential. (English) Zbl 0617.60078

Papanicolaou, G.; Varadhan, S. R. S.

Commun. Pure Appl. Math. 38, 819-834 (1985).

Let \(X_ t\) be an Ornstein-Uhlenbeck process in \({\mathbb{R}}^ d\times {\mathbb{R}}^ d\) with random infinitesimal generator of the form \[ \sum^{d}_{i=1}[\frac{\partial^ 2}{\partial v^ 2_ i}-v_ i\frac{\partial}{\partial v_ i}+q_ i(x)\frac{\partial}{\partial v_ i}+v_ i\frac{\partial}{\partial x_ i}] \] where \(q_ i\) are suitably random. Suppose X starts at 0 with initial velocity distribution \((2\pi)^{-d/2}\exp (2^{-1}| v|^ 2)dv\). Then the scaled process \(\epsilon X(t\epsilon^{-2})\) behaves, for small \(\epsilon\), like a d-dimensional Brownian motion with some non-singular matrix.

Reviewer: M.Rao

MSC:

60J65	Brownian motion
60G15	Gaussian processes

Keywords:

Ornstein-Uhlenbeck process; infinitesimal generator

References:

[1]	Kipnis, Comm. in Math. Physics.
[2]	A central limit theorem for Brownian motion in an external periodic force field, preprint, Heidelberg, 1985.
[3]	and , Martingale Limit Theorems and Applications, Academic Press, New York, 1980.