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We study the time evolution of a chain of nonlinear oscillators. We focus on the fractal features of the spectral entropy and analyze its characteristic intermediate timescales as a function of the nonlinear coupling. A Brownian motion is…
In this work we describe a new model for the evolution of a diploid structured population backwards in time that allows for large migrations and uneven offspring distributions. The model generalizes both the mean-field model of Birkner et…
Co-localization of networks of genes in the nucleus is thought to play an important role in determining gene expression patterns. Based upon experimental data, we built a dynamical model to test whether pure diffusion could account for the…
We review recent progress in the understanding of the role of multiple- and simultaneous multiple merger coalescents as models for the genealogy in idealised and real populations with exceptional reproductive behaviour. In particular, we…
The goal of this paper is to prove rigorous results for the behavior of genealogies in a one-dimensional long range biased voter model introduced by Hallatschek and Nelson [25]. The first step, which is easily accomplished using results of…
We study the evolution of the population genealogy in the classic neutral Moran Model of finite size and in discrete time. The stochastic transformations that shape a Moran population can be realized directly on its genealogy and give rise…
We consider an extension of the noisy $N$-Branching Random Walk that models the evolution of a population subject to natural selection. We show the existence of a critical value for the noise which separates the limiting genealogical…
Nonlinear stochastic differential equations generating signals with 1/f spectrum have been used so far to describe socio-economical systems. In this paper we consider the motion of a Brownian particle in an inhomogeneous environment such…
The expansion of a population into new habitat is a transient process that leaves its footprints in the genetic composition of the expanding population. How the structure of the environment shapes the population front and the evolutionary…
This paper is concerned mainly with the macroscopic fractal behavior of various random sets that arise in modern and classical probability theory. Among other things, it is shown here that the macroscopic behavior of Boolean coverage…
Standard neutral population genetics theory with a strictly fixed population size has important limitations. An alternative model that allows independently fluctuating population sizes and reproduces the standard neutral evolution is…
The symbiotic branching model is a spatial population model describing the dynamics of two interacting types that can only branch if both types are present. A classical result for the underlying stochastic partial differential equation…
This paper is the second of a series devoted to the study of the dynamics of the spectrum of large random matrices. We study general extensions of the partial differential equation arising to characterize the limit spectral measure of the…
Distribution of a Brownian motion conditioned to start from the boundary of an open set $G$ and to stay in $G$ for a finite period of time is studied. Characterizations of such distributions in terms of certain singular stochastic…
Statistical properties of the site frequency spectrum associated with Lambda-coalescents are our objects of study. In particular, we derive recursions for the expected value, variance, and covariance of the spectrum, extending earlier…
We consider the spatial Lambda-Fleming-Viot process model for frequencies of genetic types in a population living in R^d, in the special case in which there are just two types of individual, labelled 0 and 1. At time zero, everyone in the…
The site frequency spectrum describes variation among a set of n DNA sequences. Its i'th entry (i=1,2,...,n-1) is the number of nucleotide sites at which the mutant allele is present in i copies. Under selective neutrality, random mating,…
We study a model for the entanglement of a two-dimensional reflecting Brownian motion in a bounded region divided into two halves by a wall with three or more small windows. We map the Brownian motion into a Markov Chain on the fundamental…
The Brownian web is a random variable consisting of a Brownian motion starting from each space-time point on the plane. These are independent until they hit each other, at which point they coalesce. Tsirelson mentions this model in his…
The phylogenetic effective sample size is a parameter that has as its goal the quantification of the amount of independent signal in a phylogenetically correlated sample. It was studied for Brownian motion and Ornstein-Uhlenbeck models of…