A certain fractal 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
.
We define the
-parabolic
-Hausdorff outer measure for any set
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\}.}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/b3f301543c1a313342dcef55f14ffd11d7fdedda)
where the
-parabolic cylinders
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\}.}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/35d6f9b8078f5fc44557040d06dc6a2ed3b09fc4)
We define the
-parabolic Hausdorff dimension of
as
![{\displaystyle {\mathcal {P}}^{\alpha }-\dim A:=\inf \left\{\beta \geq 0:{\mathcal {P}}^{\alpha }-{\mathcal {H}}^{\beta }(A)=0\right\}.}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/f6b4a20edf9d3809d8ea2a894ce6c10c0a270da8)
The case
equals the genuine Hausdorff dimension
.
Let
. We can calculate the Hausdorff dimension of the fractional Brownian motion
of Hurst index
plus some measurable drift function
. 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}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/a0fe7c6cac4b1def98e17bc674572b045b7b7f68)
and
![{\displaystyle \dim {\mathcal {R}}_{T}\left(B^{H}+f\right)=\varphi _{\alpha }\wedge d.}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/5666aac4afe7096175d3e333b8b6eb5105d8dbba)
For an isotropic
-stable Lévy process
for
plus some measurable drift function
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}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/4de6bc39b894313734a1270abb6d9da6743982be)
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}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/719092f8d5e64edd310c1fc577fe3427691dd138)
Inequalities and identities
[edit]
For
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}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/eb5a96f566e6319172498f79a78acaf3f2e818dd)
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}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/2738626ae56f9e9a8fefd854ccf1abdbaa8d9026)
Further, for the fractional Brownian motion
of Hurst index
one has
![{\displaystyle {\mathcal {P}}^{\alpha }-\dim {\mathcal {G}}_{T}\left(B^{H}\right)=\alpha \cdot \dim T}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/21b357886dbf3cd8e12576e4a285d96961f8d845)
and for an isotropic
-stable Lévy process
for
one has
![{\displaystyle {\mathcal {P}}^{\alpha }-\dim {\mathcal {G}}_{T}\left(X\right)=(\alpha \vee 1)\cdot \dim T}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/5cbcfe3036ab1355ef5b7bb5c931a7a665e3ed75)
and
![{\displaystyle \dim {\mathcal {R}}_{T}(X)=\alpha \cdot \dim T\wedge d.}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/00d478231c8b68be3233fae78924bd7c6ef9cc8b)
For constant functions
we get
![{\displaystyle {\mathcal {P}}^{\alpha }-\dim {\mathcal {G}}_{T}\left(f_{C}\right)=(\alpha \vee 1)\cdot \dim T.}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/9fac9699c15312132b00faceeb610a684380a222)
If
, i. e.
is
-Hölder continuous, for
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}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/35f45601b74518258caf69e42b1694f55d198ff9)
hold.
Finally, for the Brownian motion
and
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}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/788f2993a60cb33b3007ff190f0dbc806f470d59)
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}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/17df00ed6a0e497d8148c1d8ea218bb3c88090e2)