Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-16T09:15:53.249Z Has data issue: false hasContentIssue false
  • English
  • Français

Assouad Dimension of Random Processes

Part of: Stochastic analysis Classical measure theory Stochastic processes Markov processes

Published online by Cambridge University Press:  16 November 2018

Douglas C. Howroyd  and
Han Yu
Douglas C. Howroyd*
Affiliation:
School of Mathematics & Statistics, University of St Andrews, St Andrews KY16 9SS, UK (dch8@st-andrews.ac.uk; hy25@st-andrews.ac.uk)
Han Yu
Affiliation:
School of Mathematics & Statistics, University of St Andrews, St Andrews KY16 9SS, UK (dch8@st-andrews.ac.uk; hy25@st-andrews.ac.uk)
*
*Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we study the Assouad dimension of graphs of certain Lévy processes and functions defined by stochastic integrals. We do this by introducing a convenient condition which guarantees a graph to have full Assouad dimension and then show that graphs of our studied processes satisfy this condition.

Keywords

Brownian motionWiener processstochastic integralAssouad dimensiongraph

MSC classification

Primary: 28A80: Fractals
Secondary: 60J65: Brownian motion 60G22: Fractional processes, including fractional Brownian motion 60H05: Stochastic integrals
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 1 , February 2019 , pp. 281 - 290
Copyright
Copyright © Edinburgh Mathematical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Adler, R. J., Hausdorff dimension and Gaussian fields, Ann. Probab. 5 (1977), 145–151.Google Scholar
2.Chen, C., Wu, M. and Wu, W., Accessible values of Assouad and the lower dimensions of subsets, Preprint, (2016), available at: https://arxiv.org/abs/1602.02180.Google Scholar
3.Cox, J. T. and Griffin, P. S., How porous is the graph of Brownian motion, Trans. Amer. Math. Soc. 325( 1) (1991), 19–140.Google Scholar
4.Falconer, K., From fractional Brownian motion to multifractional and multistable motion, In Benoit Mandelbrot – a life in many dimensions) (eds Frame, M. and Cohen, N.), Volume 4 ( World Scientific Publishing, 2015).Google Scholar
5.Fraser, J. M., Assouad type dimensions and homogeneity of fractals, Trans. Amer. Math. Soc. 366 (2014), 6687–6733.Google Scholar
6.Fraser, J. M. and Yu, H., Arithmetic patches, weak tangents, and dimension, Preprint, (2016), available at: http://arxiv.org/abs/1611.06960.Google Scholar
7.Kahane, J.-P., Some random series of functions, 2nd edn, Cambridge Studies in Advanced Mathematics ( Cambridge University Press, 1985).Google Scholar
8.Lévy, P., Sur les intégrales dont les éléments sont des variables aléatoires indépendentes, Annali della Scuola Normale Superiore di Pisa 3 (1934), 337–366.Google Scholar
9.Mandelbrot, B. B. and Van Ness, J., Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10(4) (1968), 422–437.Google Scholar
10.Möters, P. and Peres, Y., Brownian motion, Cambridge Series in Statistical and Probabilistic Mathematics (Cambridge University Press, 2010).Google Scholar
11.Protter, P., Stochastic integration and differential equations, 2nd edn, Stochastic Modelling and Applied Probability (Springer-Verlag, Berlin Heidelberg, 2005).Google Scholar
12.Robinson, J. C., Dimensions, embeddings and attractors (Cambridge University Press, Cambridge, 2011).Google Scholar
13.Taylor, S. J., The Hausdorff α-dimensional measure of Brownian paths in n-space, Math. Proc. Camb. Philos. Soc. 49(2) (1953), 31–39.Google Scholar