The multifractal structure of super-Brownian motion
This paper deals with the multifractal structure of the the random Borel measure \(\mu=X_t\) where \(X\) is a super-Brownian motion on \({\mathbb R}^d\), for some \(d\geq 3\). 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 \({\mathbb R}^d\) 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 \(2\leq\alpha\leq 4\). Furthermore, the mass exponents \(B(q)\) and \(b(q)\) for packing and Hausdorff measure, as defined by \textit{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.
scientific article; zbMATH DE number 1156746