Draft: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

edit

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

edit

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

edit

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

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

Sources

edit
  • 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.