Parabolic Hausdorff Dimension

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

Definitions

We define the $\alpha$-parabolic $\beta$-Hausdorff outer measure for any set $A \subseteq \R^$ as
:$\mathcal^\alpha-\mathcal^\beta (A) := \lim_ \inf \left \.$
where the $\alpha$-parabolic cylinders $\left ( P_k \right )_$ are contained in
:$\mathcal^\alpha := \left \.$
We define the $\alpha$-parabolic Hausdorff dimension of $A$ as
:$\mathcal^\alpha-\dim A := \inf \left \.$
The case $\alpha = 1$ equals the genuine Hausdorff dimension $\dim$.

Application

Let $\varphi_\alpha := \mathcal^\alpha-\dim \mathcal_T(f)$. We can calculate the Hausdorff dimension of the fractional Brownian motion $B^H$ of Hurst index $1/\alpha = H \in (0,1]$ plus some measurable drift function $f$. We get
:$\dim \mathcal_T \left (B^H+f \right ) = \varphi_\alpha \wedge \frac \cdot \varphi_ + \left (1 - \frac \right) \cdot d$
and
:$\dim \mathcal_T \left (B^H +f \right ) = \varphi_\alpha \wedge d.$
For an isotropic $\alpha$-stable Lévy process $X$ for $\alpha \in (0,2]$ plus some measurable drift function $f$ we get
: \dim \mathcal_T(X+f) =
\begin
\varphi_1, & \alpha \in (0,1], \\
\varphi_\alpha \wedge \frac \cdot \varphi_\alpha + \left ( 1 - \frac \right ) \cdot d, & \alpha \in ,2\end

and
:
\dim \mathcal_T \left ( X + f \right ) =
\begin
\alpha \cdot \varphi_\alpha \wedge d, & \alpha \in (0,1], \\
\varphi_\alpha \wedge d, & \alpha \in ,2
\end

Inequalities and identities

For $\phi_\alpha := \mathcal^\alpha-\dim A$ one has
:
\dim A \leq
\begin
\phi_\alpha \wedge \alpha \cdot \phi_\alpha + 1 - \alpha, & \alpha \in (0,1], \\
\phi_\alpha \wedge \frac \cdot \alpha + \left ( 1 - \frac \right ) \cdot d, & \alpha \in ,\infty)
\end

and
:
\dim A \geq
\begin
\alpha \cdot \phi_\alpha \vee \phi_\alpha + \left ( 1 - \frac \right ) \cdot d, & \alpha \in (0,1 \\
\phi_\alpha + 1 - \alpha, & \alpha \in ,\infty).
\end

Further, for the fractional Brownian motion $B^H$ of Hurst index 1/\alpha = H \in (0,1/math> one has
:$\mathcal^\alpha-\dim \mathcal_T \left (B^H \right ) = \alpha \cdot \dim T$
and for an isotropic $\alpha$-stable Lévy process $X$ for $\alpha \in (0,2]$ one has
:$\mathcal^\alpha-\dim \mathcal_T \left (X \right ) = (\alpha \vee 1) \cdot \dim T$
and
:$\dim \mathcal_T(X) = \alpha \cdot \dim T \wedge d.$
For constant functions $f_C$ we get
:$\mathcal^\alpha-\dim \mathcal_T \left (f_C \right ) = (\alpha \vee 1) \cdot \dim T.$
If $f \in C^\beta(T,\mathbb^d)$, i. e. $f$ is $\beta$-Hölder continuous, for $\varphi_\alpha = \mathcal^\alpha-\dim \mathcal_T(f)$ the estimates
:
\varphi_\alpha \leq
\begin
\dim T + \left ( \frac - \beta \right ) \cdot d \wedge \frac \wedge d + 1, & \alpha \in (0,1], \\
\alpha \cdot \dim T + (1 - \alpha \cdot \beta) \cdot d \wedge \frac \wedge d + 1, & \alpha \in \left ,\frac \right \\
\alpha \cdot \dim T + \frac(\dim T -1) + \alpha \wedge d + 1, & \alpha \in \left frac, \infty) \right \end

hold.

Finally, for the Brownian motion $B$ and $f \in C^\beta \left (T,\mathbb^d \right )$ we get
:$\dim \mathcal_T(B + f) \leq \begin d + \frac, & \beta \leq \frac - \frac,\\ \dim T + (1 - \beta) \cdot d, & \frac - \frac \leq \beta \leq \frac \wedge \frac,\\ \frac, & \frac \leq \beta \leq \frac,\\ 2 \cdot \dim T \wedge \dim T + \frac, & \text \end$
and
:$\dim \mathcal_T(B + f) \leq \begin \frac, & \frac \leq \beta \leq \frac,\\ 2 \cdot \dim T \wedge d, & \frac \leq \frac \leq \beta,\\ d, & \text. \end$

References

Sources

*
*
*{{cite journal
, last1=Taylor , first1=S. J.
, last2=Watson , first2=N. A.
, date=1985
, title=A Hausdorff measure classification of polar sets for the heat equation
, journal=Mathematical Proceedings of the Cambridge Philosophical Society
, volume=97
, issue=2
, pages=325–344
, doi=10.1017/S0305004100062873

Dimension theory
Fractals
Metric geometry