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 useful 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 ${\displaystyle f}$.

Definitions

We define the ${\displaystyle \alpha }$-parabolic ${\displaystyle \beta }$-Hausdorff outer measure for any set ${\displaystyle A\subseteq \mathbb {R} ^{d+1}}$ as

${\displaystyle {\mathcal {P}}^{\alpha }-{\mathcal {H}}^{\beta }(A):=\lim _{\delta \downarrow 0}\inf \left\{\sum _{k=1}^{\infty }\left|P_{k}\right|^{\beta }:A\subseteq \bigcup _{k=1}^{\infty }P_{k},P_{k}\in {\mathcal {P}}^{\alpha },\left|P_{k}\right|\leq \delta \right\}.}$

where the ${\displaystyle \alpha }$-parabolic cylinders ${\displaystyle \left(P_{k}\right)_{k\in \mathbb {N} }}$ are contained in

${\displaystyle {\mathcal {P}}^{\alpha }:=\left\{[t,t+c]\times \prod _{i=1}^{d}\left[x_{i},x_{i}+c^{1/\alpha }\right];t,x_{i}\in \mathbb {R} ,c\in (0,1]\right\}.}$

We define the ${\displaystyle \alpha }$-parabolic Hausdorff dimension of ${\displaystyle A}$ as

${\displaystyle {\mathcal {P}}^{\alpha }-\dim A:=\inf \left\{\beta \geq 0:{\mathcal {P}}^{\alpha }-{\mathcal {H}}^{\beta }(A)=0\right\}.}$

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

Application

Let ${\displaystyle \varphi _{\alpha }:={\mathcal {P}}^{\alpha }-\dim {\mathcal {G}}_{T}(f)}$. We can calculate the Hausdorff dimension of the fractional Brownian motion ${\displaystyle B^{H}}$ of Hurst index ${\displaystyle 1/\alpha =H\in (0,1]}$ plus some measurable drift function ${\displaystyle f}$. We get

${\displaystyle \dim {\mathcal {G}}_{T}\left(B^{H}+f\right)=\varphi _{\alpha }\wedge {\frac {1}{\alpha }}\cdot \varphi _{\alpha }+\left(1-{\frac {1}{\alpha }}\right)\cdot d}$

and

${\displaystyle \dim {\mathcal {R}}_{T}\left(B^{H}+f\right)=\varphi _{\alpha }\wedge d.}$

For an isotropic ${\displaystyle \alpha }$-stable Lévy process ${\displaystyle X}$ for ${\displaystyle \alpha \in (0,2]}$ plus some measurable drift function ${\displaystyle f}$ we get

${\displaystyle \dim {\mathcal {G}}_{T}(X+f)={\begin{cases}\varphi _{1},&\alpha \in (0,1],\\\varphi _{\alpha }\wedge {\frac {1}{\alpha }}\cdot \varphi _{\alpha }+\left(1-{\frac {1}{\alpha }}\right)\cdot d,&\alpha \in [1,2]\end{cases}}}$

and

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

Inequalities and identities

For ${\displaystyle \phi _{\alpha }:={\mathcal {P}}^{\alpha }-\dim A}$ one has

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

and

${\displaystyle \dim A\geq {\begin{cases}\alpha \cdot \phi _{\alpha }\vee \phi _{\alpha }+\left(1-{\frac {1}{\alpha }}\right)\cdot d,&\alpha \in (0,1],\\\phi _{\alpha }+1-\alpha ,&\alpha \in [1,\infty ).\end{cases}}}$

Further, for the fractional Brownian motion ${\displaystyle B^{H}}$ of Hurst index ${\displaystyle 1/\alpha =H\in (0,1]}$ one has

${\displaystyle {\mathcal {P}}^{\alpha }-\dim {\mathcal {G}}_{T}\left(B^{H}\right)=\alpha \cdot \dim T}$

and for an isotropic ${\displaystyle \alpha }$-stable Lévy process ${\displaystyle X}$ for ${\displaystyle \alpha \in (0,2]}$ one has

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

and

${\displaystyle \dim {\mathcal {R}}_{T}(X)=\alpha \cdot \dim T\wedge d.}$

For constant functions ${\displaystyle f_{C}}$ we get

${\displaystyle {\mathcal {P}}^{\alpha }-\dim {\mathcal {G}}_{T}\left(f_{C}\right)=(\alpha \vee 1)\cdot \dim T.}$

If ${\displaystyle f\in C^{\beta }(T,\mathbb {R} ^{d})}$, i. e. ${\displaystyle f}$ is ${\displaystyle \beta }$-Hölder continuous, for ${\displaystyle \varphi _{\alpha }={\mathcal {P}}^{\alpha }-\dim {\mathcal {G}}_{T}(f)}$ the estimates

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

hold.

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

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

and

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

References

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

Sources

• Kern, Peter; Pleschberger, Leonard (2024). "Parabolic Fractal Geometry of Stable Lévy Processes with Drift". arXiv: 2312.13800 [math.PR].
• Peres, Yuval; Sousi, Perla (2016). "Dimension of fractional Brownian motion with variable drift". Probability Theory and Related Fields . 165 (3–4): 771–794. arXiv: 1310.7002 . doi:10.1007/s00440-015-0645-5.
• Taylor, S. J.; Watson, N. A. (1985). "A Hausdorff measure classification of polar sets for the heat equation". Mathematical Proceedings of the Cambridge Philosophical Society . 97 (2): 325–344. doi:10.1017/S0305004100062873.