Parabolic Hausdorff dimension explained

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 usefull 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 .

Definitions

We define the

\alpha -parabolic \beta -Hausdorff outer measure for any set A \subseteq \R ^{d+1} asl{P} ^{\alpha-l{H}} ^{\beta} (A) : = \lim _{\delta} inf \left \{ \sum ^{inf ty} _{k =1} \left | P _{k} \right | ^{\beta:} A \subseteq c up ^{inf ty} _{k =1} P _{k,} P _{k} \in l{P} ^{\alpha,} \left | P _{k} \right | \leq \delta \right \}. where the \alpha -parabolic cylinders \left ( P _{k} \right ) _{k} are contained inl{P} ^{\alpha} : = \left \{ [t,t + c] x \prod ^{d} _{i =1} \left [ x _{i,} x _{i} + c ^{1/\alpha} \right ]; t, x _{i} \in R , c \in ( 0 , 1 ] \right \}. We define the \alpha -parabolic Hausdorff dimension of A asl{P} ^{\alpha-\dim} A : = inf \left \{ \beta \geq 0 : l{P} ^{\alpha-l{H}} ^{\beta} (A) = 0 \right \}. The case \alpha = 1 equals the genuine Hausdorff dimension \dim .

Application

Let

\varphi _{\alpha} : = l{P} ^{\alpha-\dim} l{G} _{T(f)} . We can calculate the Hausdorff dimension of the fractional Brownian motion B ^{H} of Hurst index 1 /\alpha = H \in ( 0 , 1 ] plus some measurable drift function f . We get\dim l{G} _{T} \left (B ^{H+f} \right ) = \varphi _{\alpha} \wedge (1)/(\alpha) ⋅ \varphi _{\alpha} + \left ( 1 - (1)/(\alpha) \right) ⋅ d and\dim l{R} _{T} \left (B ^{H} + f \right ) = \varphi _{\alpha} \wedge d. For an isotropic \alpha -stable Lévy process X for \alpha \in ( 0 , 2 ] plus some measurable drift function f we get\dim l{G} _{T(X+f)} = \begin{cases} \varphi _{1,} & \alpha \in ( 0 , 1 ], \\ \varphi _{\alpha} \wedge (1)/(\alpha) ⋅ \varphi _{\alpha} + \left ( 1 - (1)/(\alpha) \right ) ⋅ d, & \alpha \in [ 1 , 2 ] \end{cases} and\dim l{R} _{T} \left ( X + f \right ) = \begin{cases} \alpha ⋅ \varphi _{\alpha} \wedge d, & \alpha \in ( 0 , 1 ], \\ \varphi _{\alpha} \wedge d, & \alpha \in [ 1 , 2 ]. \end{cases}

Inequalities and identities

For

\phi _{\alpha} : = l{P} ^{\alpha-\dim} A one has\dim A \leq \begin{cases} \phi _{\alpha} \wedge \alpha ⋅ \phi _{\alpha} + 1 - \alpha, & \alpha \in ( 0 , 1 ], \\ \phi _{\alpha} \wedge (1)/(\alpha) ⋅ \alpha + \left ( 1 - (1)/(\alpha) \right ) ⋅ d, & \alpha \in [ 1 , inf ty) \end{cases} and\dim A \geq \begin{cases} \alpha ⋅ \phi _{\alpha} \vee \phi _{\alpha} + \left ( 1 - (1)/(\alpha) \right ) ⋅ d, & \alpha \in ( 0 , 1 ], \\ \phi _{\alpha} + 1 - \alpha, & \alpha \in [ 1 , inf ty). \end{cases} Further, for the fractional Brownian motion B ^{H} of Hurst index 1 /\alpha = H \in ( 0 , 1 ] one hasl{P} ^{\alpha-\dim} l{G} _{T} \left (B ^{H} \right ) = \alpha ⋅ \dim T and for an isotropic \alpha -stable Lévy process X for \alpha \in ( 0 , 2 ] one hasl{P} ^{\alpha-\dim} l{G} _{T} \left (X \right ) = (\alpha \vee 1 ) ⋅ \dim T and\dim l{R} _{T(X)} = \alpha ⋅ \dim T \wedge d. For constant functions f _{C} we getl{P} ^{\alpha-\dim} l{G} _{T} \left (f _{C} \right ) = (\alpha \vee 1 ) ⋅ \dim T. If f \in C ^{\beta(T,R} ^{d)} , i. e. f is \beta -Hölder continuous, for \varphi _{\alpha} = l{P} ^{\alpha-\dim} l{G} _{T(f)} the estimates\varphi _{\alpha} \leq \begin{cases} \dim T + \left ( (1)/(\alpha) - \beta \right ) ⋅ d \wedge (\dim T)/(\alpha ⋅ \beta) \wedge d + 1 , & \alpha \in ( 0 , 1 ], \\ \alpha ⋅ \dim T + ( 1 - \alpha ⋅ \beta) ⋅ d \wedge (\dim T)/(\beta) \wedge d + 1 , & \alpha \in \left [ 1 , (1)/(\beta) \right ],\\ \alpha ⋅ \dim T + (1)/(\beta) (\dim T -1 ) + \alpha \wedge d + 1 , & \alpha \in \left [ (1)/(\beta) , inf ty) \right ] \end{cases} hold.

Finally, for the Brownian motion

B and f \in C ^{\beta} \left (T, R ^{d} \right ) we get\dim l{G} _{T(B} + f) \leq \begin{cases} d + (1)/(2) , & \beta \leq (\dim T)/(d) - (1)/(2 d) ,\\ \dim T + ( 1 - \beta) ⋅ d, & (\dim T)/(d) - (1)/(2 d) \leq \beta \leq (\dim T)/(d) \wedge (1)/(,\\ 2) (\dim T)/(\beta) , & (\dim T)/(d) \leq \beta \leq (1)/(2) ,\\ 2 ⋅ \dim T \wedge \dim T + (d)/(2) , & else \end{cases} and\dim l{R} _{T(B} + f) \leq \begin{cases} (\dim T)/(\beta) , & (\dim T)/(d) \leq \beta \leq (1)/(2) ,\\ 2 ⋅ \dim T \wedge d, & (\dim T)/(d) \leq (1)/(2) \leq \beta,\\ d, & else . \end{cases}

Sources

• Kern . Peter. Pleschberger . Leonard. 2024. Parabolic Fractal Geometry of Stable Lévy Processes with Drift. math.PR. 2312.13800.
• Peres . Yuval Peres . Yuval . Sousi . Perla . 2016 . Dimension of fractional Brownian motion with variable drift. . . 165 . 3–4. 771–794. 10.1007/s00440-015-0645-5 . 1310.7002 .
• 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. 10.1017/S0305004100062873.