## 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)

### MSC:

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

### Citations:

Zbl 0679.60053; Zbl 0754.60062
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