Consider evolution of density of a mass or a population, geographically situated in a compact region of space, assuming random creation-annihilation and migration, or dispersion of mass, so the evolution is a random measure. When the creation-annihilation and dispersion are diffusions the situation is described formally by a stochastic partial differential equation; ignoring dispersion make approximations to the initial density by atomic measures and if the corresponding discrete random measures converge "in law" to a unique random measure call it a solution. To account for dispersion Trotter's product formula is applied to semiflows corresponding to dispersion and creation-annihilation. Existence of solutions has been a conjecture for several years despite a claim in ([2], J. Multivariate Anal. 5, 1-52). We show that solutions exist and that non-deterministic solutions are "smeared" continuous-state branching diffusions.
Measure-valued process stochastic differential equation transition probability vague topology narrow topology branching Markov process immigration nonlinear semiflow on Banach space Trotter's product formula infinitely decomposable continuous-state branching test function
