Continuous flows driving branching processes and their nonlinear evolution equations

Abstract

Abstract We consider on M(ℝ d ) (the set of all finite measures on ℝ d ) the evolution equation associated with the nonlinear operator F ↦ Δ F ′ + ∑ k ⩾ 1 b k F k F \mapsto \Delta F' + \sum\nolimits_{k \geqslant 1} b_k F^k , where F′ is the variational derivative of F and we show that it has a solution represented by means of the distribution of the d-dimensional Brownian motion and the non-local branching process on the finite configurations of M(ℝ d ), induced by the sequence (bk ) k ⩾1 of positive numbers such that ∑ k ⩾ 1 b k ⩽ 1 \sum\nolimits_{k \geqslant 1} b_k \leqslant 1 . It turns out that the representation also holds with the same branching process for the solution to the equation obtained replacing the Laplace operator by the generator of a Markov process on ℝ d instead of the d-dimensional Brownian motion; more general, we can take the generator of a right Markov process on a Lusin topological space. We first investigate continuous flows driving branching processes. We show that if the branching mechanism of a superprocess is independent of the spatial variable, then the superprocess is obtained by introducing the branching in the time evolution of the right continuous flow on measures, canonically induced by a right continuous flow as spatial motion. A corresponding result holds for non-local branching processes on the set of all finite configurations of the state space of the spatial motion.