We study the pathwise description of a (sub-)critical continuous-state branching process (CSBP) conditioned to be never extinct, as solution stochastic differential equation driven by Brownian motion and Poisson point measures. The interest our approach, which relies on applying Girsanov theorem SDE that describes unconditioned CSBP, is it points out an explicit mechanism build immigration term appearing in process, randomly selecting jumps original one. These techniques should also useful...

In this survey, we explore the connections between two areas of probability: percolation theory and population genetic models. Our first goal is to highlight a construction on Galton-Watson trees, which has been described in different ways: Bernoulli bond neutral mutations. Next, introduce novel connection Divide-and-Color model particular multi-type tree. We provide gentle introduction these topics while presenting an overview results that connect them.

Motivated by the stochastic Lotka-Volterra model, we introduce discrete-state interacting multitype branching processes. We show that they can be obtained as sum of a multidimensional random walk with Lamperti-type change proportional to population size; and Poisson process time-change pairwise interactions. define analogous continuous-state unique strong solution SDE. prove scaling limits correspond its continuous counterpart. In addition, model constructed generalized transformation L\'evy

We introduce flows of branching processes with competition, which describe the evolution general continuous state populations in interactions between individuals give rise to a negative density dependence term. This generalizes logistic studied by Lambert. Following approach developed Dawson and Li, we first construct such as solutions certain flow stochastic differential equations. then propose novel genealogical description for competition based on interactive pruning L\'evy-trees,...

We present a lookdown construction for Moran seed-bank model with variable active and inactive population sizes we show that the empirical measure of our coincides Seed-Bank-Moran Model latency Greven, den Hollander Oomen, 2022. Furthermore, prove time to most recent common ancestor, starting from $N$ individuals stationary distribution over its state (active or inactive), has same asymptotic order as largest inactivity period. then obtain an TMRCA, use this result find first fixation single...

We consider a system of N queues with decentralized load balancing such as power-of-D strategies(where D may depend on N) and generic scheduling disciplines. To measure the dependence queues, we use clan ancestors, technique coming from interacting particle systems. Relying in that analysis prove quantitative estimates correlations implying propagation chaos for systems Markovian arrivals general service time distribution. This solves conjecture posed by Bramsom et. al. [*] concerning...