Über die Autorin bzw. den Autor

Charles L. Epstein is the Thomas A. Scott Professor of Mathematics at the University of Pennsylvania. Rafe Mazzeo is professor of mathematics at Stanford University.

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Degenerate Diffusion Operators Arising in Population Biology

PRINCETON UNIVERSITY PRESS

Contents

Preface..........................................................................xi1 Introduction...................................................................1I Wright-Fisher Geometry and the Maximum Principle...............................232 Wright-Fisher Geometry.........................................................253 Maximum Principles and Uniqueness Theorems.....................................34II Analysis of Model Problems....................................................494 The Model Solution Operators...................................................515 Degenerate Hölder Spaces..................................................646 Hölder Estimates for the 1-dimensional Model Problems.....................787 Hölder Estimates for Higher Dimensional Corner Models.....................1078 Hölder Estimates for Euclidean Models.....................................1379 Hölder Estimates for General Models.......................................143III Analysis of Generalized Kimura Diffusions....................................17910 Existence of Solutions........................................................18111 The Resolvent Operator........................................................21812 The Semi-group on C0(P)............................................235A Proofs of Estimates for the Degenerate 1-d Model...............................251Bibliography.....................................................................301Index............................................................................305

Chapter One

In population genetics one frequently replaces a discrete Markov chain model, which describes the random processes of genetic drift, with or without selection, and mutation with a limiting, continuous time and space, stochastic process. If there are N + 1 possible types, then the configuration space for the resultant continuous Markov process is typically the N-simplex

L_N = {(x₁, ..., x_N) : x_j ≥ 0 and x₁ + ··· + x_N ≤ 1}. (1.1)

If a different scaling is used to define the limiting process, different domains might also arise. As a geometrical object the simplex is quite complicated. Its boundary is not a smooth manifold, but has a stratified structure with strata of codimensions 1 through N. The codimension 1 strata are

[summation]_1,l = {x_l = 0} [union] L_N for l = 1, ..., N, (1.2)

along with

[summation]_1,0 = {x₁ + ··· + x_N = 1} [union] L_N. (1.3)

Components of the stratum of codimension 1 < l ≤ N arise by choosing integers 0 ≤ i₁ < ··· < i_l ≤ N and forming the intersection:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.4)

The simplex is an example of a manifold with corners. The singularity of its boundary significantly complicates the analysis of differential operators acting functions defined in L_N.

In the simplest case, without mutation or selection, the limiting operator of the Wright-Fisher process is the Kimura diffusion operator, with formal generator:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.5)

This is the "backward" Kolmogorov operator for the limiting Markov process. This operator is elliptic in the interior of L_N but the coefficient of the second order normal derivative tends to zero as one approaches a boundary. We can introduce local coordinates (x₁, y₁, ..., y_N-1) near the interior of a point on one of the faces of [summation]_1,l, so that the boundary is given locally by the equation x₁ = 0, and the operator then takes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.6)

where the matrix c_lm is positive definite. To include the effects of mutation, migration and selection, one typically adds a vector field:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.7)

where V is inward pointing along the boundary of L_N. In the classical models, if only the effect of mutation and migration are included, then the coefficients {b_i(x)} can be taken to be linear polynomials, whereas selection requires at least quadratic terms.

The most significant feature is that the coefficient of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] vanishes exactly to order 1. This places L_Kim outside the classes of degenerate elliptic operators that have already been analyzed in detail. For applications to Markov processes the difficulty that presents itself is that it is not possible to introduce a square root of the coefficient of the second order terms that is Lipschitz continuous up to the boundary. Indeed the best one can hope for is Hölder-1/2. The uniqueness of the solutions to either the forward Kolmogorov equation, or the associated stochastic differential equation, cannot then be concluded using standard methods.

Even in the presence of mutation and migration, the solutions of the heat equation for this operator in 1-dimension was studied by Kimura, using the fact that L_Kim + V preserves polynomials of degree d for each d. In higher dimensions it was done by Karlin and Shimakura by showing the existence of a complete basis of polynomial eigenfunctions for this operator. This in turn leads to a proof of the existence of a polynomial solution to the initial value problem for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with polynomial initial data. Using the maximum principle, this suffices to establish the existence of a strongly continuous semi-group acting on C⁰, and establish many of its basic properties, see [38]. This general approach has been further developed by Barbour, Etheridge, Ethier, and Griffiths, see [17, 2, 16, 25].

As noted, if selection is also included, then the coefficients of V are at least quadratic polynomials, and can be quite complicated, see [9]. So long as the second order part remains L_Kim, then a result of Ethier, using the Trotter product formula, makes it possible to again define a strongly continuous semi-group, see [18]. Various extensions of these results, using a variety of functional analytic frameworks, were made by Athreya, Barlow, Bass, Perkins, Sato, Cerrai, Clément, and others, see [1, 4, 6, 7, 8].

For example Cerrai and Clément constructed a semi-group acting on C⁰([0, 1]^N), with the coefficient a_ij of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] assumed to be of the form

a_ij (x) = m(x)A_ij (x_i, x_j). (1.8)

Here m(x) is strictly positive. In [1, 4, 3], Bass and Perkins along with several collaborators, study a class of equations, similar to that defined below. Their work has many points of contact with our own, and we discuss it in greater detail at the end of Section 1.5.

We have not yet said anything about boundary conditions, which would seem to be a serious omission for a PDE on a domain with a boundary. Indeed,...