Brownian motion with variable drift: 0-1 laws, hitting probabilities and Hausdorff dimension

TL;DR

This paper investigates the properties of Brownian motion with variable drift functions outside the Dirichlet space, establishing 0-1 laws, analyzing Hausdorff dimensions, and examining the existence of double points in various dimensions.

Contribution

It introduces a general 0-1 law for Brownian motion with arbitrary drift, and characterizes the Hausdorff dimension and double points of such processes based on the regularity of the drift.

Findings

Hausdorff dimension of the image and graph are almost surely constant.

Brownian motion with Hölder(1/2) drift is intersection equivalent to standard Brownian motion.

Double points occur in dimensions ≤ 3 for Hölder(1/2) drifts, but not in higher dimensions.

Abstract

By the Cameron--Martin theorem, if a function $f$ is in the Dirichlet space $D$, then $B+f$ has the same a.s. properties as standard Brownian motion, $B$. In this paper we examine properties of $B+f$ when $f \notin D$. We start by establishing a general 0-1 law, which in particular implies that for any fixed $f$, the Hausdorff dimension of the image and the graph of $B+f$ are constants a.s. (This 0-1 law applies to any L\'evy process.) Then we show that if the function $f$ is H\"older$(1/2)$, then $B+f$ is intersection equivalent to $B$. Moreover, $B+f$ has double points a.s. in dimensions $d\le 3$, while in $d\ge 4$ it does not. We also give examples of functions which are H\"older with exponent less than $1/2$, that yield double points in dimensions greater than 4. Finally, we show that for $d \ge 2$, the Hausdorff dimension of the image of $B+f$ is a.s. at least the maximum of 2 and the dimension of the image of $f$.