Branching explosion of Brownian Fleming-Viot systems
We resolve the long-standing non-extinction problem for Brownian Fleming-Viot systems in bounded domains. For every fixed N >= 3, we construct a bounded connected domain in R3 in which branching explosion occurs with positive probability before every prescribed positive time, from every interior initial configuration. On this event all particles converge to one boundary point. Together with two-particle non-extinction, this gives the sharp population threshold for bounded-domain Brownian explosion. The mechanism is directed transport generated by uniform copying in a periodic absorbing environment. We compute a positive cubic current for the same two-harmonic killing profile at every N >= 3, with an explicit population coefficient and a controlled fourth-order remainder. To realize this transport by spherical absorption, we develop a marked genealogical transfer theorem that retains the Euclidean displacement inherited at each copy. Common ancestry makes the integer winding reconstructible from phase records and yields convergence of the stationary signed current. A boundary restart law with P'olya weights supplies the continuity needed to turn that current into stable escape. Diffusion coefficients adapted to a common Kelvin clock preserve the joint Brownian law and uniform donor marks while compressing escape into finite lifetime. Disjoint spherical holes in a ball accumulate only at the explosion point. The boundary is smooth elsewhere, with finite area and perimeter. Boundary area within distance r of the singularity is of order r. For three particles, the transfer theorem covers every smooth positive periodic killing rate.