Recurrence of the Brownian motion in multidimensional semi-selfsimilar   environments and Gaussian environments
  Asymptotic behavior of the one-dimensional Brownian motion in general random environments has been investigated by many researchers. However, many of the methods used in the argument are available only for the one-dimensional case. In this paper the multidimensional case of the problem is considered, and we obtain some sufficient conditions for recurrence of the multi-dimensional Brownian motion in random environments. By using the sufficient conditions we show that the recurrence of the Brownian motion in Gaussian environments under some conditions on the correlation functions. 
Seiichiro Kusuoka, Hiroshi Takahashi and Yozo Tamura
math.PR