CENTRAL LIMIT THEOREM FOR A GAUSSIAN INCOMPRESSIBLE
FLOW WITH ADDITIONAL BROWNIAN NOISE
Abstract: We generalize the result of Komorowski and Papanicolaou published in [7]. We
consider the solution of stochastic differential equation
![dX(t) = V(t,X(t))dt+ V~ 2kdB(t),](files/23.2/HTML/23.2.13.abs0x.png)
where
![B(t)](files/23.2/HTML/23.2.13.abs1x.png)
is a standard
![d](files/23.2/HTML/23.2.13.abs2x.png)
-dimensional Brownian motion and
![V(t,x),(t,x) (- R ŚRd](files/23.2/HTML/23.2.13.abs3x.png)
,
is a
![d](files/23.2/HTML/23.2.13.abs4x.png)
-dimensional, incompressible, stationary, random Gaussian field decorrelating in finite
time. We prove that the weak limit as
![e |, 0](files/23.2/HTML/23.2.13.abs5x.png)
of the family of rescaled processes
![Xe(t) = eX(t/e2)](files/23.2/HTML/23.2.13.abs6x.png)
exists and may be identified as a certain Brownian motion.
2000 AMS Mathematics Subject Classification: 60F05 (60G15).
Key words and phrases: Weak convergence, random process, Gaussian field,
incompressible flow, diffusion.