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Parabolic Hausdorff dimension

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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 f {\displaystyle f} .

Definitions

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\{\times \prod _{i=1}^{d}\left;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 } .

Application

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 \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 .\end{cases}}}

Inequalities and identities

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,\\\alpha \cdot \dim T+{\frac {1}{\beta }}(\dim T-1)+\alpha \wedge d+1,&\alpha \in \left\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}}}

References

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

Sources

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