Brownian motion with variable drift can be space filling

Antunović, Tonći; Peres, Yuval; Vermesi, Brigitta

doi:10.1090/S0002-9939-2011-10737-8

Brownian motion with variable drift can be space filling
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by Tonći Antunović, Yuval Peres and Brigitta Vermesi PDF

Proc. Amer. Math. Soc. 139 (2011), 3359-3373 Request permission

Abstract:

For $d \geq 2$ let $B$ be standard $d$-dimensional Brownian motion. For any $\alpha < 1/d$ we construct an $\alpha$-Hölder continuous function $f \colon [0,1] \to \mathbb {R}^d$ so that the range of $B-f$ covers an open set. This strengthens a result of Graversen (1982) and answers a question of Le Gall (1988).

References

Francesca Biagini, Yaozhong Hu, Bernt Øksendal, and Tusheng Zhang, Stochastic calculus for fractional Brownian motion and applications, Probability and its Applications (New York), Springer-Verlag London, Ltd., London, 2008. MR 2387368, DOI 10.1007/978-1-84628-797-8
Brahim Boufoussi, Marco Dozzi, and Raby Guerbaz, Path properties of a class of locally asymptotically self similar processes, Electron. J. Probab. 13 (2008), no. 29, 898–921. MR 2413288, DOI 10.1214/EJP.v13-505
A. R. Butz, Alternative algorithm for Hilbert’s space-filling curve, IEEE Trans. Comput. 20 (1971), no. 4, 424–426.
Arthur R. Butz, Convergence with Hilbert’s space filling curve, J. Comput. System Sci. 3 (1969), 128–146. MR 246631, DOI 10.1016/S0022-0000(69)80010-3
Kenneth Falconer, Fractal geometry, 2nd ed., John Wiley & Sons, Inc., Hoboken, NJ, 2003. Mathematical foundations and applications. MR 2118797, DOI 10.1002/0470013850
S. E. Graversen, “Polar”-functions for Brownian motion, Z. Wahrsch. Verw. Gebiete 61 (1982), no. 2, 261–270. MR 675615, DOI 10.1007/BF01844636
John Hawkes, On the Hausdorff dimension of the intersection of the range of a stable process with a Borel set, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 19 (1971), 90–102. MR 292165, DOI 10.1007/BF00536900
Jean-Pierre Kahane, Some random series of functions, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 5, Cambridge University Press, Cambridge, 1985. MR 833073
Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus, 2nd ed., Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991. MR 1121940, DOI 10.1007/978-1-4612-0949-2
Jean-François Le Gall, Sur les fonctions polaires pour le mouvement brownien, Séminaire de Probabilités, XXII, Lecture Notes in Math., vol. 1321, Springer, Berlin, 1988, pp. 186–189 (French). MR 960526, DOI 10.1007/BFb0084136
Paul Lévy, Sur certains processus stochastiques homogènes, Compositio Math. 7 (1939), 283–339 (French). MR 919
Peter Mörters and Yuval Peres, Brownian motion, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 30, Cambridge University Press, Cambridge, 2010. With an appendix by Oded Schramm and Wendelin Werner. MR 2604525, DOI 10.1017/CBO9780511750489
F. Przytycki and M. Urbański, On the Hausdorff dimension of some fractal sets, Studia Math. 93 (1989), no. 2, 155–186. MR 1002918, DOI 10.4064/sm-93-2-155-186
Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999. MR 1725357, DOI 10.1007/978-3-662-06400-9
Hans Sagan, Space-filling curves, Universitext, Springer-Verlag, New York, 1994. MR 1299533, DOI 10.1007/978-1-4612-0871-6