Super-Brownian motion: Path properties and hitting probabilities. (English) Zbl 0692.60063

Dawson, D. A.; Iscoe, I.; Perkins, E. A.

Probab. Theory Relat. Fields 83, No. 1, 135-205 (1989).

The model under consideration is a super-Brownian motion \(X=\{X_ t;t\geq 0\}\) in \({\mathbb{R}}^ d\), i.e. a measure-valued diffusion process related to the equation \(\partial u/\partial t=2^{-1}(\Delta u-u^ 2)\) via the Laplace transition functional. The basic goal is the derivation of path properties of \(\{S(X_ t)\); \(t\geq 0\}\) where \(S(X_ t)\) is the closed support of the measure state \(X_ t\) at time t. This includes a modulus of continuity (d\(\geq 1)\), a study of “polar sets” and “multiple points”, and the derivation of an exact Hausdorff measure function for the “range” of this support process (d\(\geq 4).\)
The super-Brownian motion exhibits a richer family of behaviors than Brownian motion. For instance, points are hit in dimensions \(d\leq 3\), in \(d=4\) multiple points of all orders exist, and for \(d\leq 7\) (and only for these dimensions) there are double points. Proofs are much more complicated as for analogous Brownian motion sample path properties. They include analytic estimates for the hitting of balls by the X process. Also nonstandard analysis arguments are involved.

Reviewer: K.Fleischmann

MSC:

60J80

Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords:

Hausdorff measure; closed support; multiple points; polar sets; measure- valued diffusion process; Laplace transition functional; modulus of continuity

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