The Hausdorff measure of the support of two-dimensional super-Brownian motion. (English) Zbl 0856.60055

Let \(\mathbb{Q}^\gamma_\mu\) denote the law of two-dimensional super-Brownian motion \(Y_t\) \((t\geq 0)\) having branching rate \(\gamma> 0\) and starting from some \(\mu\) in the space \(M_F(\mathbb{R}^2)\) of positive finite Borel measures on \(\mathbb{R}^2\). The main result states that \(Y_t\) can be obtained as deterministic multiple of the \(h\)-Hausdorff measure \(h- m\) restricted to the closed support \(S(Y_t)\) of \(Y_t\), where \[ h(r)= r^2\log^+(1/r) \log^+ \log^+ \log^+(1/r). \] More precisely, there exists a universal constant \(c_0\in (0, \infty)\) such that \[ Y_t(A)= c_0 \gamma h- m(A\cap S(Y_t))\quad \forall\text{ Borel sets }A\subset \mathbb{R}^2,\quad \mathbb{Q}^\gamma_\mu\text{-a.s.}, \] for all \(t> 0\), \(\gamma> 0\) and \(\mu\in M_F(\mathbb{R}^2)\). This theorem is obtained by moment estimates for the excursion measure of Le Gall’s Brownian snake process. It complements results of the second author [Ann. Inst. Henri Poincaré, Probab. Stat. 25, No. 2, 205-224 (1989; Zbl 0679.60053)] and D. Dawson and the second author [Mem. Am. Math. Soc. 454 (1991; Zbl 0754.60062)] for higher dimensions. There it has been shown that an analogous statement holds on \(\mathbb{R}^d\) with \(d\geq 3\), but with \(h\) replaced by \(\varphi(r)= r^2\log^+ \log^+(1/r)\).
Reviewer: A.Schied (Berlin)


60G57 Random measures
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
Full Text: DOI