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With the tools of information geometry, we can express relations between marginals of a joint distribution in geometric terms. We develop this framework in the context of population genetics and use this to interpret the famous Ohta–Kimura formula (cf. Ohta and Kimura in Genet Res 13(01):47–55, 1969) and discuss its generalizations for linkage equilibria in Wright–Fisher models with recombination with several loci. The state space associated with the Ohta–Kimura model is simply a Riemannian manifold of constant positive curvature. Furthermore, the equilibria states for recombination can be interpreted geometrically as a product of spheres. In the case of only 2 loci, we also derive the behavior of the mutual information between these two loci.

We develop a general solution for the Fokker-Planck (Kolomogorov) equation representing the diffusion limit of the Wright-Fisher model of random genetic drift for an arbitrary number of alleles at a single locus. From this solution, we can readily deduce information about the evolution of a Wright-Fisher population., 20 pages, no figure

We develop a global and hierarchical scheme for the forward Kolmogorov (Fokker-Planck) equation of the diffusion approximation of the Wright-Fisher model of population genetics. That model describes the random genetic drift of several alleles at the same locus in a population. The key of our scheme is to connect the solutions before and after the loss of an allele. Whereas in an approach via stochastic processes or partial differential equations, such a loss of an allele leads to a boundary singularity, from a biological or geometric perspective, this is a natural process that can be analyzed in detail. Our method depends on evolution equations for the moments of the process and a careful analysis of the boundary flux.

We derive and apply a partial differential equation for the moment generating function of the Wright-Fisher model of population genetics.

The diffusion approximation of the Wright-Fisher model of population genetics leads to partial differentiable equations, the so-called Kolmogorov equations, with an operator that degenerates at the boundary. Standard tools do not apply, and in fact, solutions lack regularity properties. In this paper, we develop a regularising blow-up scheme for a certain class of solutions of the backward Kolmogorov equation, the iteratively extended global solutions presented in \cite{THJ5}, and establish their uniqueness. As the model describes the random genetic drift of several alleles at the same locus from a backward perspective, the singularities result from the loss of an allele. While in an analytical approach, this causes substantial difficulties, from a biological or geometric perspective, this is a natural process that can be analyzed in detail. The presented scheme regularises the solution via a tailored successive transformation of the domain., Comment: arXiv admin note: text overlap with arXiv:1406.5146

In this paper, we develop the mathematical structure of the Wright–Fisher model for evolution of the relative frequencies of two alleles at a diploid locus under random genetic drift in a population of fixed size in its simplest form, that is, without mutation or selection. We establish a new concept of a global solution for the diffusion approximation (Fokker–Planck equation), prove its existence and uniqueness and then show how one can easily derive all the essential properties of this random genetic drift process from our solution. Thus, our solution turns out to be superior to the local solution constructed by Kimura.