Parabolic Hausdorff dimension

In fractal geometry, the parabolic Hausdorff dimension is a restricted version of the genuine Hausdorff dimension.[1] Only parabolic cylinders, i. e. rectangles with a distinct non-linear scaling between time and space are permitted as covering sets. It is usefull to determine the Hausdorff dimension of self-similar stochastic processes, such as the geometric Brownian motion[2] or stable Lévy processes[3] plus Borel measurable drift function .

Definitions

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We define the  -parabolic  -Hausdorff outer measure for any set   as

 

where the  -parabolic cylinders   are contained in

 

We define the  -parabolic Hausdorff dimension of   as

 

The case   equals the genuine Hausdorff dimension  .

Application

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Let  . We can calculate the Hausdorff dimension of the fractional Brownian motion   of Hurst index   plus some measurable drift function  . We get

 

and

 

For an isotropic  -stable Lévy process   for   plus some measurable drift function   we get

 

and

 

Inequalities and identities

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For   one has

 

and

 

Further, for the fractional Brownian motion   of Hurst index   one has

 

and for an isotropic  -stable Lévy process   for   one has

 

and

 

For constant functions   we get

 

If  , i. e.   is  -Hölder continuous, for   the estimates

 

hold.

Finally, for the Brownian motion   and   we get

 

and

 

References

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  1. ^ Taylor & Watson, 1985.
  2. ^ Peres & Sousi, 2016.
  3. ^ Kern & Pleschberger, 2024.

Sources

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  • Kern, Peter; Pleschberger, Leonard (2024). "Parabolic Fractal Geometry of Stable Lévy Processes with Drift". arXiv:2312.13800 [math.PR].{{cite arXiv}}: CS1 maint: multiple names: authors list (link)
  • Peres, Yuval; Sousi, Perla (2016). "Dimension of fractional Brownian motion with variable drift". Probab. Theory Relat. Fields. 165 (3–4): 771–794. arXiv:1310.7002. doi:10.1007/s00440-015-0645-5.
  • Taylor, S.; Watson, N. (1985). "A Hausdorff measure classification of polar sets for the heat equation", Math. Proc. Camb. Phil. Soc. 97: 325–344.