Lookdown-Constructions of Symbiotic Branching Processes

Abstract

A symbiotic diffusion is a bivariate diffusion that models the masses of two branching populations. The branching rate of one population is proportional to the mass of the other population and vice versa. The driving Brownian motions are correlated with a constant correlation coefficient. We are concerned with the construction of so-called lookdown representations for symbiotic diffusions and their discrete mass analoga. A lookdown representation is a particle model where the particles, representing families or lineages, are assigned levels that evolve in time and govern the reproductive dynamics. Lookdown representations carry genealogical information. We study models, where the levels take non discrete values. This kind of lookdown construction was introduced for the Dawson-Watanabe process by Kurtz and Rodrigues in 2011.

The construction of Kurtz and Rodrigues relies on a deterministic evolution of the levels. We modify their approach insofar as the level motion is random or deterministic only given the level configuration of the partner population. We explore possible birth and death mechanisms that allow for coupling of the branching events in both populations. In the discrete mass setting, we construct lookdown representations for the whole range [-1,1] of possible correlation coefficients. In contrast to the Kurtz-Rodrigues model, continuity of the level paths is lost and, in general, only right continuity remains. In the diffusive limit however, the discontinuous paths converge to conditional geometric Brownian motions plus additional drift. We construct lookdown representations of symbiotic diffusions for nonnegative correlation coefficients in [0,1) as weak limits of discrete mass models. For the uncorrelated, mutually catalytic case we also give an explicit construction.