Recurrence of the Brownian motion in multidimensional semi-selfsimilar environments and Gaussian environments

TL;DR

This paper investigates the recurrence properties of multidimensional Brownian motion in semi-selfsimilar and Gaussian environments, providing new sufficient conditions for recurrence in higher dimensions.

Contribution

It extends the analysis of Brownian motion recurrence from one dimension to multidimensional cases, introducing conditions based on environment properties.

Findings

Recurrence conditions are established for multidimensional Brownian motion.

Recurrence in Gaussian environments depends on correlation function properties.

Provides a framework for analyzing Brownian motion in complex random environments.

Abstract

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.