Abstract
AbstractBy 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 fD. 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 Lévy process.) Then we show that if the function f is Hölder(1/2), then B + f is intersection equivalent to B. Moreover, B + f has double points a.s. in dimensions d ≤ 3, while in d ≥ 4 it does not. We also give examples of functions which are Hölder with exponent less than 1/2, that yield double points in dimensions greater than 4. Finally, we show that for d ≥ 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.
Publisher
Cambridge University Press (CUP)
Reference12 articles.
1. Double points of a Gaussian sample path
2. Brownian sheet images and Bessel-Riesz capacity
3. Sur les fonctions polaires pour le mouvement brownien
4. T. Antunović , Y. Peres and B. Vermesi Brownian motion with variable drift can be space-filling, (2010). Available at http://arxiv.org/abs/1003.0228.
5. T. Antunović , K. Burdzy , Y. Peres and J. Ruscher Isolated zeros for Brownian motion with variable drift (2010). Avalaible at http://arxiv.org/abs/1009.3603.
Cited by
6 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献