Draft: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 usefull 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 $$A \subseteq \R^{d+1}$$ as
 * $$ \mathcal{P}^\alpha-\mathcal{H}^\beta (A) := \lim_{\delta \downarrow 0} \inf \left \{ \sum_{k=1}^\infty \left | P_k \right |^\beta: A \subseteq \bigcup_{k=1}^\infty P_k, P_k \in \mathcal{P}^\alpha, \left | P_k \right | \leq \delta \right \}.$$

where the $$\alpha$$-parabolic cylinders $$\left ( P_k \right )_{k \in \mathbb{N}}$$ are contained in
 * $$ \mathcal{P}^\alpha := \left \{ [t,t+c] \times \prod_{i=1}^d \left [ x_i, x_i + c^{1/\alpha} \right ]; t, x_i \in \mathbb{R}, c \in (0,1] \right \}.$$

We define the $$\alpha$$-parabolic Hausdorff dimension of $$A$$ as
 * $$\mathcal{P}^\alpha-\dim A := \inf \left \{ \beta \geq 0: \mathcal{P}^\alpha-\mathcal{H}^\beta (A) = 0 \right \}.$$

The case $$\alpha = 1$$ equals the genuine Hausdorff dimension $$\dim$$.

Application
Let $$\varphi_\alpha := \mathcal{P}^\alpha-\dim \mathcal{G}_T(f)$$. 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
 * $$ \dim \mathcal{G}_T \left (B^H+f \right ) = \varphi_\alpha \wedge \frac{1}{\alpha} \cdot \varphi_{\alpha} + \left (1 - \frac{1}{\alpha} \right) \cdot d $$

and
 * $$ \dim \mathcal{R}_T \left (B^H +f \right ) = \varphi_\alpha \wedge d. $$

For an isotropic $$\alpha$$-stable Lévy process $$X$$ for $$\alpha \in (0,2]$$ plus some measurable drift function $$f$$ we get
 * $$ \dim \mathcal{G}_T(X+f) =

\begin{cases} \varphi_1, & \alpha \in (0,1], \\ \varphi_\alpha \wedge \frac{1}{\alpha} \cdot \varphi_\alpha + \left ( 1 - \frac{1}{\alpha} \right ) \cdot d, & \alpha \in [1,2] \end{cases} $$ and

\dim \mathcal{R}_T \left ( X + f \right ) = \begin{cases} \alpha \cdot \varphi_\alpha \wedge d, & \alpha \in (0,1], \\ \varphi_\alpha \wedge d, & \alpha \in [1,2]. \end{cases} $$

Inequalities and identities
For $$\phi_\alpha := \mathcal{P}^\alpha-\dim A$$ one has

\dim A \leq \begin{cases} \phi_\alpha \wedge \alpha \cdot \phi_\alpha + 1 - \alpha, & \alpha \in (0,1], \\ \phi_\alpha \wedge \frac{1}{\alpha} \cdot \alpha + \left ( 1 - \frac{1}{\alpha} \right ) \cdot d, & \alpha \in [1,\infty) \end{cases} $$ and

\dim A \geq \begin{cases} \alpha \cdot \phi_\alpha \vee \phi_\alpha + \left ( 1 - \frac{1}{\alpha} \right ) \cdot d, & \alpha \in (0,1], \\ \phi_\alpha + 1 - \alpha, & \alpha \in [1,\infty). \end{cases} $$ Further, for the fractional Brownian motion $$B^H$$ of Hurst index $$1/\alpha = H \in (0,1]$$ one has

\mathcal{P}^\alpha-\dim \mathcal{G}_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{P}^\alpha-\dim \mathcal{G}_T \left (X \right ) = (\alpha \vee 1) \cdot \dim T $$ and

\dim \mathcal{R}_T(X) = \alpha \cdot \dim T \wedge d. $$ For constant functions $$ f_C $$ we get

\mathcal{P}^\alpha-\dim \mathcal{G}_T \left (f_C \right ) = (\alpha \vee 1) \cdot \dim T. $$ If $$f \in C^\beta(T,\mathbb{R}^d) $$, i. e. $$f $$ is $$\beta$$-Hölder continuous, for $$\varphi_\alpha = \mathcal{P}^\alpha-\dim \mathcal{G}_T(f)$$ the estimates

\varphi_\alpha \leq \begin{cases} \dim T + \left ( \frac{1}{\alpha} - \beta \right ) \cdot d \wedge \frac{\dim T}{\alpha \cdot \beta} \wedge d + 1, & \alpha \in (0,1], \\ \alpha \cdot \dim T + (1 - \alpha \cdot \beta) \cdot d \wedge \frac{\dim T}{\beta} \wedge d + 1, & \alpha \in \left [1,\frac{1}{\beta} \right ],\\ \alpha \cdot \dim T + \frac{1}{\beta}(\dim T -1) + \alpha \wedge d + 1, & \alpha \in \left [\frac{1}{\beta}, \infty) \right ] \end{cases} $$ hold.

Finally, for the Brownian motion $$B$$ and $$f \in C^\beta \left (T,\mathbb{R}^d \right )$$ we get

\dim \mathcal{G}_T(B + f) \leq \begin{cases} d + \frac{1}{2}, & \beta \leq \frac{\dim T}{d} - \frac{1}{2d},\\ \dim T + (1 - \beta) \cdot d, & \frac{\dim T}{d} - \frac{1}{2d} \leq \beta \leq \frac{\dim T}{d} \wedge \frac{1}{2},\\ \frac{\dim T}{\beta}, & \frac{\dim T}{d} \leq \beta \leq \frac{1}{2},\\ 2 \cdot \dim T \wedge \dim T + \frac{d}{2}, & \text{ else} \end{cases} $$ and

\dim \mathcal{R}_T(B + f) \leq \begin{cases} \frac{\dim T}{\beta}, & \frac{\dim T}{d} \leq \beta \leq \frac{1}{2},\\ 2 \cdot \dim T \wedge d, & \frac{\dim T}{d} \leq \frac{1}{2} \leq \beta,\\ d, & \text{ else}. \end{cases} $$