The packing measure of the support of super-Brownian motion
It is known that the value of super-Brownian motion in \(R^d\) is given for \(t > 0\) by a multiple of Hausdorff \(\varphi\)-measure restricted to its closed support where \(\varphi (s) = s^2 \log \log {1 \over s}\) for \(d \geq 3\) and \(s^2 \log {1 \over s} \log \log \log {1 \over s}\) for \(d = 2\). The purpose of this paper is to consider the support of super-Brownian motion from the point of view of packing measure. The packing measure [cf. the second author and \textit{C. Tricot}, Real Anal. Exch. 10, 58-67 (1985; Zbl 0578.28007) and Trans. Am. Math. Soc. 288, 679-699 (1985; Zbl 0537.28003)] of a set \(E\) uses small disjoint balls with centres in \(E\) and seeks to maximize the result. The main result is to show that, for \(d \geq 3\) there is no measure function \(\varphi\) such that the \(\varphi\)-packing measure of the closed support is finite and positive. An integral test is derived which distinguishes those \(\varphi\) which make the packing measure 0 or \(\infty\). Partial results are obtained for \(d = 2\). The results are obtained from estimates of lower densities with respect to a packing function using both the canonical representation and Le Gall's path-valued process.
scientific article; zbMATH DE number 828740
scientific article