Packing measure of the sample paths of fractional Brownian motion

Xiao, Yimin

doi:10.1090/S0002-9947-96-01712-6

Packing measure of the sample paths of fractional Brownian motion
HTML articles powered by AMS MathViewer

by Yimin Xiao

Trans. Amer. Math. Soc. 348 (1996), 3193-3213

DOI: https://doi.org/10.1090/S0002-9947-96-01712-6

PDF | Request permission

Abstract:

Let $X(t) (t \in \mathbf {R})$ be a fractional Brownian motion of index $\alpha$ in $\mathbf {R}^d.$ If $1 < \alpha d$, then there exists a positive finite constant $K$ such that with probability 1, \[ \phi -p(X([0,t])) = Kt\quad \text {for any $t > 0$}, \] where $\phi (s) = s^{\frac 1 { \alpha }}/ (\log \log \frac 1 s)^{\frac 1 {2 \alpha }}$ and $\phi$-$p (X([0,t]))$ is the $\phi$-packing measure of $X([0,t])$.

References

Z. Ciesielski and S. J. Taylor, First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path, Trans. Amer. Math. Soc. 103 (1962), 434–450. MR 143257, DOI 10.1090/S0002-9947-1962-0143257-8
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
Kenneth Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR 1102677
Bert E. Fristedt and S. James Taylor, The packing measure of a general subordinator, Probab. Theory Related Fields 92 (1992), no. 4, 493–510. MR 1169016, DOI 10.1007/BF01274265
Jean-Pierre Kahane, Some random series of functions, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 5, Cambridge University Press, Cambridge, 1985. MR 833073
P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
Michel Ledoux and Michel Talagrand, Probability in Banach spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 23, Springer-Verlag, Berlin, 1991. Isoperimetry and processes. MR 1102015, DOI 10.1007/978-3-642-20212-4
J.-F. Le Gall and S. James Taylor, The packing measure of planar Brownian motion, Seminar on stochastic processes, 1986 (Charlottesville, Va., 1986) Progr. Probab. Statist., vol. 13, Birkhäuser Boston, Boston, MA, 1987, pp. 139–147. MR 902430
M. B. Marcus, Hölder conditions for Gaussian processes with stationary increments, Trans. Amer. Math. Soc. 134 (1968), 29–52. MR 230368, DOI 10.1090/S0002-9947-1968-0230368-7
Loren D. Pitt, Local times for Gaussian vector fields, Indiana Univ. Math. J. 27 (1978), no. 2, 309–330. MR 471055, DOI 10.1512/iumj.1978.27.27024
Loren D. Pitt and Lanh Tat Tran, Local sample path properties of Gaussian fields, Ann. Probab. 7 (1979), no. 3, 477–493. MR 528325
Sidney C. Port and Charles J. Stone, Brownian motion and classical potential theory, Probability and Mathematical Statistics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0492329
Fraydoun Rezakhanlou and S. James Taylor, The packing measure of the graph of a stable process, Astérisque 157-158 (1988), 341–362. Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987). MR 976226
Xavier Saint Raymond and Claude Tricot, Packing regularity of sets in $n$-space, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 1, 133–145. MR 913458, DOI 10.1017/S0305004100064690
Keizo Takashima, Sample path properties of ergodic self-similar processes, Osaka J. Math. 26 (1989), no. 1, 159–189. MR 991287
Talagrand, M., Hausdorff measure of trajectories of multiparameter fractional Brownian motion. Ann. of Probab. 23 (1995) 767 - 775.
Talagrand, M., Lower classes for fractional Brownian motion. J. Theoret Probab. 9 (1996) 191–213.
S. James Taylor, The use of packing measure in the analysis of random sets, Stochastic processes and their applications (Nagoya, 1985) Lecture Notes in Math., vol. 1203, Springer, Berlin, 1986, pp. 214–222. MR 872112, DOI 10.1007/BFb0076883
S. James Taylor, The measure theory of random fractals, Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 3, 383–406. MR 857718, DOI 10.1017/S0305004100066160
S. James Taylor and Claude Tricot, Packing measure, and its evaluation for a Brownian path, Trans. Amer. Math. Soc. 288 (1985), no. 2, 679–699. MR 776398, DOI 10.1090/S0002-9947-1985-0776398-8