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\{[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 } .
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