A certain fractal dimension
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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
f
{\displaystyle f}
.
We define the
α
{\displaystyle \alpha }
-parabolic
β
{\displaystyle \beta }
-Hausdorff outer measure for any set
A
⊆
R
d
+
1
{\displaystyle A\subseteq \mathbb {R} ^{d+1}}
as
P
α
−
H
β
(
A
)
:=
lim
δ
↓
0
inf
{
∑
k
=
1
∞
|
P
k
|
β
:
A
⊆
⋃
k
=
1
∞
P
k
,
P
k
∈
P
α
,
|
P
k
|
≤
δ
}
.
{\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
α
{\displaystyle \alpha }
-parabolic cylinders
(
P
k
)
k
∈
N
{\displaystyle \left(P_{k}\right)_{k\in \mathbb {N} }}
are contained in
P
α
:=
{
[
t
,
t
+
c
]
×
∏
i
=
1
d
[
x
i
,
x
i
+
c
1
/
α
]
;
t
,
x
i
∈
R
,
c
∈
(
0
,
1
]
}
.
{\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
α
{\displaystyle \alpha }
-parabolic Hausdorff dimension of
A
{\displaystyle A}
as
P
α
−
dim
A
:=
inf
{
β
≥
0
:
P
α
−
H
β
(
A
)
=
0
}
.
{\displaystyle {\mathcal {P}}^{\alpha }-\dim A:=\inf \left\{\beta \geq 0:{\mathcal {P}}^{\alpha }-{\mathcal {H}}^{\beta }(A)=0\right\}.}
The case
α
=
1
{\displaystyle \alpha =1}
equals the genuine Hausdorff dimension
dim
{\displaystyle \dim }
.
Let
φ
α
:=
P
α
−
dim
G
T
(
f
)
{\displaystyle \varphi _{\alpha }:={\mathcal {P}}^{\alpha }-\dim {\mathcal {G}}_{T}(f)}
. We can calculate the Hausdorff dimension of the fractional Brownian motion
B
H
{\displaystyle B^{H}}
of Hurst index
1
/
α
=
H
∈
(
0
,
1
]
{\displaystyle 1/\alpha =H\in (0,1]}
plus some measurable drift function
f
{\displaystyle f}
. We get
dim
G
T
(
B
H
+
f
)
=
φ
α
∧
1
α
⋅
φ
α
+
(
1
−
1
α
)
⋅
d
{\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
dim
R
T
(
B
H
+
f
)
=
φ
α
∧
d
.
{\displaystyle \dim {\mathcal {R}}_{T}\left(B^{H}+f\right)=\varphi _{\alpha }\wedge d.}
For an isotropic
α
{\displaystyle \alpha }
-stable Lévy process
X
{\displaystyle X}
for
α
∈
(
0
,
2
]
{\displaystyle \alpha \in (0,2]}
plus some measurable drift function
f
{\displaystyle f}
we get
dim
G
T
(
X
+
f
)
=
{
φ
1
,
α
∈
(
0
,
1
]
,
φ
α
∧
1
α
⋅
φ
α
+
(
1
−
1
α
)
⋅
d
,
α
∈
[
1
,
2
]
{\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
dim
R
T
(
X
+
f
)
=
{
α
⋅
φ
α
∧
d
,
α
∈
(
0
,
1
]
,
φ
α
∧
d
,
α
∈
[
1
,
2
]
.
{\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
edit
For
ϕ
α
:=
P
α
−
dim
A
{\displaystyle \phi _{\alpha }:={\mathcal {P}}^{\alpha }-\dim A}
one has
dim
A
≤
{
ϕ
α
∧
α
⋅
ϕ
α
+
1
−
α
,
α
∈
(
0
,
1
]
,
ϕ
α
∧
1
α
⋅
α
+
(
1
−
1
α
)
⋅
d
,
α
∈
[
1
,
∞
)
{\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
dim
A
≥
{
α
⋅
ϕ
α
∨
ϕ
α
+
(
1
−
1
α
)
⋅
d
,
α
∈
(
0
,
1
]
,
ϕ
α
+
1
−
α
,
α
∈
[
1
,
∞
)
.
{\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
B
H
{\displaystyle B^{H}}
of Hurst index
1
/
α
=
H
∈
(
0
,
1
]
{\displaystyle 1/\alpha =H\in (0,1]}
one has
P
α
−
dim
G
T
(
B
H
)
=
α
⋅
dim
T
{\displaystyle {\mathcal {P}}^{\alpha }-\dim {\mathcal {G}}_{T}\left(B^{H}\right)=\alpha \cdot \dim T}
and for an isotropic
α
{\displaystyle \alpha }
-stable Lévy process
X
{\displaystyle X}
for
α
∈
(
0
,
2
]
{\displaystyle \alpha \in (0,2]}
one has
P
α
−
dim
G
T
(
X
)
=
(
α
∨
1
)
⋅
dim
T
{\displaystyle {\mathcal {P}}^{\alpha }-\dim {\mathcal {G}}_{T}\left(X\right)=(\alpha \vee 1)\cdot \dim T}
and
dim
R
T
(
X
)
=
α
⋅
dim
T
∧
d
.
{\displaystyle \dim {\mathcal {R}}_{T}(X)=\alpha \cdot \dim T\wedge d.}
For constant functions
f
C
{\displaystyle f_{C}}
we get
P
α
−
dim
G
T
(
f
C
)
=
(
α
∨
1
)
⋅
dim
T
.
{\displaystyle {\mathcal {P}}^{\alpha }-\dim {\mathcal {G}}_{T}\left(f_{C}\right)=(\alpha \vee 1)\cdot \dim T.}
If
f
∈
C
β
(
T
,
R
d
)
{\displaystyle f\in C^{\beta }(T,\mathbb {R} ^{d})}
, i. e.
f
{\displaystyle f}
is
β
{\displaystyle \beta }
-Hölder continuous , for
φ
α
=
P
α
−
dim
G
T
(
f
)
{\displaystyle \varphi _{\alpha }={\mathcal {P}}^{\alpha }-\dim {\mathcal {G}}_{T}(f)}
the estimates
φ
α
≤
{
dim
T
+
(
1
α
−
β
)
⋅
d
∧
dim
T
α
⋅
β
∧
d
+
1
,
α
∈
(
0
,
1
]
,
α
⋅
dim
T
+
(
1
−
α
⋅
β
)
⋅
d
∧
dim
T
β
∧
d
+
1
,
α
∈
[
1
,
1
β
]
,
α
⋅
dim
T
+
1
β
(
dim
T
−
1
)
+
α
∧
d
+
1
,
α
∈
[
1
β
,
∞
)
]
{\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
B
{\displaystyle B}
and
f
∈
C
β
(
T
,
R
d
)
{\displaystyle f\in C^{\beta }\left(T,\mathbb {R} ^{d}\right)}
we get
dim
G
T
(
B
+
f
)
≤
{
d
+
1
2
,
β
≤
dim
T
d
−
1
2
d
,
dim
T
+
(
1
−
β
)
⋅
d
,
dim
T
d
−
1
2
d
≤
β
≤
dim
T
d
∧
1
2
,
dim
T
β
,
dim
T
d
≤
β
≤
1
2
,
2
⋅
dim
T
∧
dim
T
+
d
2
,
else
{\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
dim
R
T
(
B
+
f
)
≤
{
dim
T
β
,
dim
T
d
≤
β
≤
1
2
,
2
⋅
dim
T
∧
d
,
dim
T
d
≤
1
2
≤
β
,
d
,
else
.
{\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}}}
^ Taylor & Watson, 1985.
^ Peres & Sousi, 2016.
^ Kern & Pleschberger, 2024.
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.