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Zbl 0905.60031
Perkins, Edwin A.; Taylor, S.James
The multifractal structure of super-Brownian motion. (English)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 34, No.1, 97-138 (1998). ISSN 0246-0203

This paper deals with the multifractal structure of the the random Borel measure $\mu=X_t$ where $X$ is a super-Brownian motion on ${\Bbb R}^d$, for some $d\ge 3$. The behavior found in this particular case differs from previously studied examples and provides a counterexample to the multifractal formalism in the physics literature. If $\mu$ is a finite Borel measure on ${\Bbb R}^d$ and $B(x,r)$ denotes the Euclidean ball of radius $r$ centered in a point $x$, one can introduce the quantities $$\underline d(\mu,x)=\liminf_{r\downarrow 0}{\log\mu(B(x,r)) \over\log r},\quad\overline d(\mu,x)=\limsup_{r\downarrow 0}{\log\mu(B(x,r)) \over\log r}. $$ Known results yield that $\overline d(X_t,x)=\underline d(X_t,x)=2$ for $X_t$-a.a. $x$, a.s. on $\{X_t\neq 0\}$, for all $t>0$. While the result for $\underline d(X_t,x)$ extends to the pointwise statement $$\underline d(X_t,x)=2\text{ for all $x$ in the support $S(X_t)$ of $X_t$ a.s.,}$$ it is shown that there are exceptional points in $S(X_t)$ such that $\overline d(X_t,x)>2$. In fact, the Hausdorff dimension of all points $x\in S(X_t)$ for which $\overline d(X_t,x)=\alpha$ is shown to be equal to $8/\alpha-2$ for $2\le\alpha\le 4$. Furthermore, the mass exponents $B(q)$ and $b(q)$ for packing and Hausdorff measure, as defined by {\it L. Olsen} [Adv. Math. 116, No. 1, 82-196 (1995; Zbl 0841.28012)], are explicitly calculated. These phenomena are explained by introducing the notion of $\gamma$-thin sets.
[A.Schied (Berlin)]

MSC 2000:: *60G57 Random measures
28A78 Hausdorff measures
60J80 Branching processes
28A80 Fractals

Keywords: multifractal spectrum; mass exponents; super-Brownian motion; Hausdorff dimension

Citations: Zbl 0841.28012

Cited in: Zbl 1075.60104 Zbl 0978.60046

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