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Register a new userCoalescent processes arising in a study of diffusive clustering
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This paper studies the spatial coalescent on $Z^2$. In our setting, the partition elements are located at the sites of $Z^2$ and undergo local delayed coalescence and migration. That is, pairs of partition elements located at the same site coalesce into one partition element after exponential waiting times. In addition, the partition elements perform independent random walks. The system starts in either locally finite configurations or in configurations containing countably many partition elements per site. These two situations are relevant if the coalescent is used to study the scaling limits for genealogies in Moran models respectively interacting Fisher-Wright diffusions (or Fleming-Viot processes), which is the key application of the present work. Our goal is to determine the longtime behavior with an initial population of countably many individuals per site restricted to a box $[-t^{alpha/2}, t^{alpha/2}]^2 cap Z^2$ and observed at time $t^beta$ with $1 geq beta geq alphage 0$. We study both asymptotics, as $ttoinfty$, for a fixed value of $alpha$ as the parameter $betain[alpha,1]$ varies, and for a fixed $beta$, as the parameter $alphain [0,beta]$ varies. This exhibits the genealogical structure of the mono-type clusters arising in 2-dimensional Moran and Fisher-Wright systems. (... for more see the actual preprint)
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We revisit the discrete additive and multiplicative coalescents, starting with $n$ particles with unit mass. These cases are known to be related to some combinatorial coalescent processes: a time reversal of a fragmentation of Cayley trees or a parking scheme in the additive case, and the random graph process $(G(n,p))_p$ in the multiplicative case. Time being fixed, encoding these combinatorial objects in real-valued processes indexed by the line is the key to describing the asymptotic behaviour of the masses as $nto +infty$. We propose to use the Prim order on the vertices instead of the classical breadth-first (or depth-first) traversal to encode the combinatorial coalescent processes. In the additive case, this yields interesting connections between the different representations of the process. In the multiplicative case, it allows one to answer to a stronger version of an open question of Aldous [Ann. Probab., vol. 25, pp. 812--854, 1997]: we prove that not only the sequence of (rescaled) masses, seen as a process indexed by the time $lambda$, converges in distribution to the reordered sequence of lengths of the excursions above the current minimum of a Brownian motion with parabolic drift $(B_t+lambda t - t^2/2, tgeq 0)$, but we also construct a version of the standard augmented multiplicative coalescent of Bhamidi, Budhiraja and Wang [Probab. Theory Rel., to appear] using an additional Poisson point process.
We introduce and solve a new type of quadratic backward stochastic differential equation systems defined in an infinite time horizon, called emph{ergodic BSDE systems}. Such systems arise naturally as candidate solutions to characterize forward performance processes and their associated optimal trading strategies in a regime switching market. In addition, we develop a connection between the solution of the ergodic BSDE system and the long-term growth rate of classical utility maximization problems, and use the ergodic BSDE system to study the large time behavior of PDE systems with quadratic growth Hamiltonians.
We consider a one-dimensional, weakly asymmetric, boundary driven exclusion process on the interval $[0,N]cap Z$ in the super-diffusive time scale $N^2 epsilon^{-1}_N$, where $1ll epsilon^{-1}_N ll N^{1/4}$. We assume that the external field and the chemical potentials, which fix the density at the boundaries, evolve smoothly in the macroscopic time scale. We derive an equation which describes the evolution of the density up to the order $epsilon_N$.
We consider the population genetics problem: how long does it take before some member of the population has $m$ specified mutations? The case $m=2$ is relevant to onset of cancer due to the inactivation of both copies of a tumor suppressor gene. Models for larger $m$ are needed for colon cancer and other diseases where a sequence of mutations leads to cells with uncontrolled growth.
Full likelihood inference under Kingmans coalescent is a computationally challenging problem to which importance sampling (IS) and the product of approximate conditionals (PAC) method have been applied successfully. Both methods can be expressed in terms of families of intractable conditional sampling distributions (CSDs), and rely on principled approximations for accurate inference. Recently, more general $Lambda$- and $Xi$-coalescents have been observed to provide better modelling fits to some genetic data sets. We derive families of approximate CSDs for finite sites $Lambda$- and $Xi$-coalescents, and use them to obtain approximately optimal IS and PAC algorithms for $Lambda$-coalescents, yielding substantial gains in efficiency over existing methods.
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