Generalized Fleming-Viot processes with immigration via stochastic flows of partitions

The generalized Fleming-Viot processes were defined in 1999 by Donnelly and Kurtz using a particle model and by Bertoin and Le Gall in 2003 using stochastic flows of bridges. In both methods, the key argument used to characterize these processes is the duality between these processes and exchangeable coalescents. A larger class of coalescent processes, called distinguished coalescents, was set up recently to incorporate an immigration phenomenon in the underlying population. The purpose of this article is to define and characterize a class of probability-measure valued processes called the generalized Fleming-Viot processes with immigration. We consider some stochastic flows of partitions of Z_{+}, in the same spirit as Bertoin and Le Gall's flows, replacing roughly speaking, composition of bridges by coagulation of partitions. Identifying at any time a population with the integers $\mathbb{N}:=\{1,2,...\}$, the formalism of partitions is effective in the past as well as in the future especially when there are several simultaneous births. We show how a stochastic population may be directly embedded in the dual flow. An extra individual 0 will be viewed as an external generic immigrant ancestor, with a distinguished type, whose progeny represents the immigrants. The "modified" lookdown construction of Donnelly-Kurtz is recovered when no simultaneous multiple births nor immigration are taken into account. In the last part of the paper we give a sufficient criterion for the initial types extinction.Comment: typos and corrections in reference

Continuous-state branching processes with competition: duality and reflection at Infinity

The boundary behavior of continuous-state branching processes with quadratic competition is studied in whole generality. We first observe that despite competition, explosion can occur for certain branching mechanisms. We obtain a necessary and sufficient condition for $\infty$ to be accessible in terms of the branching mechanism and the competition parameter $c>0$. We show that when $\infty$ is inaccessible, it is always an entrance boundary. In the case where $\infty$ is accessible, explosion can occur either by a single jump to $\infty$ (the process at $z$ jumps to $\infty$ at rate $\lambda z$ for some $\lambda>0$) or by accumulation of large jumps over finite intervals. We construct a natural extension of the minimal process and show that when $\infty$ is accessible and $0\leq \frac{2\lambda}{c}<1$, the extended process is reflected at $\infty$. In the case $\frac{2\lambda}{c}\geq 1$, $\infty$ is an exit of the extended process. When the branching mechanism is not the Laplace exponent of a subordinator, we show that the process with reflection at $\infty$ get extinct almost-surely. Moreover absorption at $0$ is almost-sure if and only if Grey's condition is satisfied. When the branching mechanism is the Laplace exponent of a subordinator, necessary and sufficient conditions are given for a stationary distribution to exist. The Laplace transform of the latter is provided. The study is based on classical time-change arguments and on a new duality method relating logistic CSBPs with certain generalized Feller diffusions.Comment: minor modifications and new lemma 4.

A phase transition in the coming down from infinity of simple exchangeable fragmentation-coagulation processes

We consider the class of exchangeable fragmentation-coagulation (EFC) processes where coagulations are multiple and not simultaneous, as in a $\Lambda$-coalescent, and fragmentation dislocates at finite rate an individual block into sub-blocks of infinite size. We call these partition-valued processes, simple EFC processes, and study the question whether such a process, when started with infinitely many blocks, can visit partitions with a finite number of blocks or not. When this occurs, one says that the process comes down from infinity. We introduce two sharp parameters $\theta_{\star}\leq \theta^{\star}\in [0,\infty]$, so that if $\theta^{\star}<1$, the process comes down from infinity and if $\theta_\star>1$, then it stays infinite. We illustrate our result with regularly varying coagulation and fragmentation measures. In this case, the parameters $\theta_{\star},\theta^{\star}$ coincide and are explicit.Comment: 33 pages. Details and minor corrections added in Section

Evolution of linear warps in accretion discs and applications to protoplanetary discs in binaries

Warped accretion discs are expected in many protostellar binary systems. In this paper, we study the long-term evolution of disc warp and precession for discs with dimensionless thickness $H/r$ larger than their viscosity parameter $\alpha$, such that bending waves can propagate and dominate the warp evolution. For small warps, these discs undergo approximately rigid-body precession. We derive analytical expressions for the warp/twist profiles of the disc and the alignment timescale for a variety of models. Applying our results to circumbinary discs, we find that these discs align with the orbital plane of the binary on a timescale comparable to the global precession time of the disc, and typically much smaller than its viscous timescale. We discuss the implications of our finding for the observations of misaligned circumbinary discs (such as KH 15D) and circumbinary planetary systems (such as Kepler-413); these observed misalignments provide useful constraints on the uncertain aspects of the disc warp theory. On the other hand, we find that circumstellar discs can maintain large misalignments with respect to the plane of the binary companion over their entire lifetime. We estimate that inclination angles larger than $\sim 20^\circ$ can be maintained for typical disc parameters. Overall, our results suggest that while highly misaligned circumstellar discs in binaries are expected to be common, such misalignments should be rare for circumbinary discs. These expectations are consistent with current observations of protoplanetary discs and exoplanets in binaries, and can be tested with future observations.Comment: 15 pages, 10 figures, Accepted by MNRA