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One-dimensional Fisher-Wright diffusion process on the interval $(0,1)$ with mutations is considered. This is a widely known model in population genetics. The goal of the paper is an exponential recurrence of the process, which also implies…
The transition distribution of a sample taken from a Wright-Fisher diffusion with general small mutation rates is found using a coalescent approach. The approximation is equivalent to having at most one mutation in the coalescent tree of…
Diffusion theory is a central tool of modern population genetics, yielding simple expressions for fixation probabilities and other quantities that are not easily derived from the underlying Wright-Fisher model. Unfortunately, the textbook…
In a (two-type) Wright-Fisher diffusion with directional selection and two-way mutation, let $x$ denote today's frequency of the beneficial type, and given $x$, let $h(x)$ be the probability that, among all individuals of today's…
The Wright--Fisher diffusion is important in population genetics in modelling the evolution of allele frequencies over time subject to the influence of biological phenomena such as selection, mutation, and genetic drift. Simulating paths of…
In the present review we survey the properties of a transcendental function of the Wright type, nowadays known as M-Wright function, entering as a probability density in a relevant class of self-similar stochastic processes that we…
The Moran discrete process and the Wright-Fisher modelare the most popular models in population genetics. It is common tounderstand the dynamics of these models to use an approximating diffusionprocess, called Wright-Fisher diffusion. Here,…
A number of discrete time, finite population size models in genetics describing the dynamics of allele frequencies are known to converge (subject to suitable scaling) to a diffusion process in the infinite population limit, termed the…
Duality plays an important role in population genetics. It can relate results from forwards-in-time models of allele frequency evolution with those of backwards-in-time genealogical models; a well known example is the duality between the…
The Wright-Fisher diffusion is a fundamentally important model of evolution encompassing genetic drift, mutation, and natural selection. Suppose you want to infer the parameters associated with these processes from an observed sample path.…
We analyze the diffusion processes associated to equations of Wright-Fisher type in one spatial dimension. These are defined by a degenerate second order operator on the interval [0, 1], where the coefficient of the second order term…
We study a family of n-dimensional diffusions, taking values in the unit simplex of vectors with nonnegative coordinates that add up to one. These processes satisfy stochastic differential equations which are similar to the ones for the…
The stationary distribution of a sample taken from a Wright-Fisher diffusion with general small mutation rates is found using a coalescent approach. The approximation is equivalent to having at most one mutation in the coalescent tree to…
The entropy rates of the Wright-Fisher process, the Moran process, and generalizations are computed and used to compare these processes and their dependence on standard evolutionary parameters. Entropy rates are measures of the variation…
We introduce generalized notions of a divergence function and a Fisher information matrix. We propose to generalize the notion of an exponential family of models by reformulating it in terms of the Fisher information matrix. Our methods are…
Recently, several authors have studied maps where a function, describing the local diffusion matrix of a diffusion process with a linear drift towards an attraction point, is mapped into the average of that function with respect to the…
Wright-Fisher diffusions and their dual ancestral graphs occupy a central role in the study of allele frequency change and genealogical structure, and they provide expressions, explicit in some special cases but generally implicit, for the…
We study two types of stochastic processes, a mean-field spatial system of interacting Fisher-Wright diffusions with an inferior and an advantageous type with rare mutation (inferior to advantageous) and a (mean-field) spatial system of…
Generalized inheritance is used with the almost-conformal Killing equation. Examples are the flat FRW, Kasner, and deSitter metrics. The volume changes in FRW while transitioning to a stiff fluid are discussed. An inheritance current is…
Random Fisher matrices arise naturally in multivariate statistical analysis and understanding the properties of its eigenvalues is of primary importance for many hypothesis testing problems like testing the equality between two multivariate…