Local approximation and conditioning on Dawson-Watanabe superprocesses

We consider a critical, measure-valued branching diffusion ξ in Rd, where the branching is continuous and the spatial motion is given by the heat flow. For d ≥ 2 and fixed t > 0, ξt is known to be an a.s. singular random measure of Hasudorff dimension 2. We explain how it can be approximated by Lebesgue measure on ε-neighbourhoods of the support. Next we show how ξt can be approximated in total variation near n points, and how the associated Palm distributions arise in the limit from elementary conditioning. Finally we hope to explan the duality between moment and Palm measures, and to show how the latter can be described in terms of discrete “Palm trees.”