Distribution of the particles of a critical branching Wiener process
In this model, a single particle starts at the origin and executes a Wiener process \(W(t)\) in \(\mathbb{R}^d\) until time \(t=1\). It is then replaced by \(Y\geq 0\) offspring which independently perform Wiener processes, starting from the point \(W(1)\), and then after another time unit are each replaced by more offspring with the distribution of \(Y\), and so on. It is assumed that \(Y\) has mean \(1\). The main result concerns the standardized distribution function \(F_T({\mathbf x})\), \({\mathbf x}\in\mathbb{R}^d\), conditional on non-extinction, of the locations of particles living at time \(T=1,2,\dots\), and states that any absolutely continuous distribution function \(F({\mathbf x})\) with bounded density will be well approximated by \(F_T({\mathbf x})\) with positive probability, for all sufficiently large \(T\). The model has previously been treated by the author [``Random walks of infinitely many particles'' (1994; Zbl 0841.60053)].
empirical distribution of particle locations
scientific article; zbMATH DE number 887273