Degenerate Diffusion Operators Arising in Population Biology: 185 (Annals of Mathematics Studies, 185) - Hardcover

Epstein, Charles L.; Mazzeo, Rafe

Hardcover

ISBN 10: 069115712X ISBN 13: 9780691157122

Verlag: Princeton University Press, 2013

Inhaltsangabe

This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the Martingale problem and therefore the existence of the associated Markov process. Charles Epstein and Rafe Mazzeo use an "integral kernel method" to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high codimensional strata of the boundary. Epstein and Mazzeo establish the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Holder spaces. They show that the semigroups defined by these operators have holomorphic extensions to the right half-plane. Epstein and Mazzeo also demonstrate precise asymptotic results for the long-time behavior of solutions to both the forward and backward Kolmogorov equations.

Über diesen Titel

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

By Charles L. Epstein Rafe Mazzeo

PRINCETON UNIVERSITY PRESS

Copyright © 2013 Princeton University Press
All right reserved.
ISBN: 978-0-691-15712-2

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

Introduction

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, one would expect that there would be an infinite dimensional space of solutions to the homogeneous equation. It is possible to formulate local boundary conditions that assure uniqueness, but, in some sense, this is not necessary. As a result of the degeneracy of the principal part, uniqueness for these types of equations can also be obtained as a consequence of regularity alone! We illustrate this in the simplest 1-dimensional case, which is the equation, with b(0) ≥ 0, b(1) ≤ 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.9)

If we assume that [partial derivative]_xv(x, t) extends continuously to [0, 1] × (0, ∞) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.10)

then a simple maximum principle argument shows that the solution is unique. In our approach, such regularity conditions naturally lead to uniqueness, and little effort is expended in the consideration of boundary conditions. In Chapter 3 we prove a generalization of the Hopf boundary point maximum principle that demonstrates, in the general case, how regularity implies uniqueness.

1.1 GENERALIZED KIMURA DIFFUSIONS

In his seminal work, Feller analyzed the most general closed extensions of operators, like those in (1.9), which generate Feller semi-groups in 1-dimension, see [20]. He analyzes the resolvent kernel, using methods largely restricted to ordinary differential equations. In [10], Chen and Stroock use probabilistic methods to prove estimates on the fundamental solution of the parabolic equation, [partial derivative]_tu = x(1 - x) [partial derivative]²_xu.

Up to now very little is known, in higher dimensions, about the analytic properties of the solution to the initial value problem for the heat equation

[partial derivative]t v - (L_Kim + V )v = 0 in (0, ∞) × L_N with v(0, x) = f (x). (1.11)

Indeed, if we replace L_Kim with a qualitatively similar second order part, which does not take one of the forms described above, then even the existence of a solution is not known. In this monographwe introduce a very flexible analytic framework for studying a large class of equations, which includes all standard models, of this type appearing in population genetics, as well as the SIR model for epidemics, see [19, 38], and models that arise inMathematical Finance, see [21]. Our approach is to introduce non-isotropic Hölder spaces with respect to which we establish sharp existence and regularity results for the solutions to heat equations of this type, as well as the corresponding elliptic problems. Using the Lumer-Phillips theorem we conclude that the C⁰-graph closure of this operator generates a strongly continuous semi-group.

The approach here is an extension of our work on the 1-d case in [15], which allows us to prove existence, uniqueness and regularity results for a class of higher dimensional, degenerate diffusion operators. While our methods also lead to a precise description of the heat kernel in the 1-dimensional case, this has proved considerably more challenging in higher dimensions. It is hoped that a combination of the analytic techniques used here, and the probabilistic techniques from [10] will lead to good descriptions of the heat kernels in higher dimensions.

Our analysis applies to a class of operators that we call generalized Kimura diffusions, which act on functions defined on manifolds with corners. Such spaces generalize the notion of a regular convex polyhedron in R^N, e.g., the simplex. Working in this more general context allows for a great deal of flexibility, which proves indispensable in the proof of our basic existence result.

Locally a manifold with corners, P, can be described as a subset of R^N defined by inequalities. Let {p_k(x) : k = 1, ..., K} be smooth functions in the unit ball B₁(0) [subset] R^N, vanishing at 0, with {dp_k(0) : k = 1, ..., K} linearly independent; clearly K ≤ N. Locally P is diffeomorphic to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.12)

We let [summation]_k = P [union] {x : p_k (x) = 0}; suppose that [summation]_k contains a non-empty, open (N - 1)-dimensional hypersurface and that dp_k is non-vanishing in a neighborhood of [summation]_k. The boundary of P is a stratified space, where the strata of codimension n locally consists of points where the boundary is defined by the vanishing of n functions with independent gradients. The components of the codimension 1 part of the bP are called faces. As in (1.4), the codimension-n stratum of bP is formed from intersections of n faces.

The formal generator is a degenerate elliptic operator of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.13)

Here A_ij (x) is a smooth, symmetric matrix valued function in P. The second order term is positive definite in the interior of P and degenerates along the hypersurface boundary components in a specific way. For each k

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.14)

On the other hand,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.15)

The first order part of L is an inward pointing vector field

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.16)

We call a second order partial differential operator defined on P, which is non-degenerate elliptic in int P, with this local description near any boundary point a generalized Kimura diffusion.

If p is a point on the stratum of bP of codimension n, then locally there are coordinates (x₁, ..., x_n; y₁, ..., y_m) so that p corresponds to (0; 0), and a neighborhood, U, of p is given by

U = {(x; y) [member of] [0, 1)ⁿ × (-1, 1)^m}. (1.17)

In these local coordinates a generalized Kimura diffusion, L, takes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (1.18)

(a_ij) and (c_kl) are symmetric matrices, the matrices (a_ii) and (c_kl) are strictly positive definite. The coefficients {[??](x; y)} are non-negative along bP [union] U, so that first order part is inward pointing.

Let P be a compact manifold with corners and L a generalized Kimura diffusion defined on P. Broadly speaking, our goal is to prove the existence, uniqueness and regularity of solutions to the equation

([partial derivative]_t - L)u = g in P × (0, ∞) with u(p, 0) = f (p), (1.19)

with certain boundary behavior along bP × [0, ∞), for data g and f satisfying appropriate regularity conditions. These results in turn can be used to prove the existence of a strongly continuous semi-group acting on C⁰(P), with formal generator L. This is the "backward Kolmogorov equation." The solution to the "forward Kolmogorov equation," ([partial derivative]_t - L*)m = v, is then given by the adjoint semi-group, canonically defined on a (non-dense) domain in [C⁰(P)]' = M(P), the space of finite Borel measures on P.

1.2 MODEL PROBLEMS

The problem of proving the existence of solutions to a class of PDEs is essentially a matter of finding a good class of model problems, for which existence and regularity can be established, more or less directly, and then finding a functional analytic setting in which to do a perturbative analysis of the equations of interest. The model operators for Kimura diffusions are the differential operators, defined on Rⁿ₊ × R^m, by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.20)

Here b = (b₁, ..., b_n) is a non-negative vector.

We have not been too explicit about the boundary conditions that we impose along bRⁿ₊ × R^m. This condition can be defined by a local Robin-type formula involving the value of the solution and its normal derivative along each hypersurface boundary component of bP. For b > 0, the 1-dimensional model operator, L_b = x[partial derivative]²_x + b[partial derivative]_x, has two indicial roots

β₀ = 0, β₁ = 1 - b, (1.21)

that is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.22)

If b [not equal to] 1, then the boundary condition,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.23)

excludes the appearance of terms like x^1-b in the asymptotic expansion of solutions along x = 0. In fact, this condition insures that u is as smooth as possible along the boundary: if g = 0 and f has m derivatives then the solution to (1.19), satisfying (1.23), does as well. This boundary condition can be encoded as a regularity condition, that is u(·, t) [member of] C¹([0, ∞)) [union] C²((0, ∞)), with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.24)

for t > 0. We call the unique solution to a generalized Kimura diffusion, satisfying this condition, or its higher dimensional analogue, the regular solution. The majority of this monograph is devoted to the study of regular solutions.

In applications to probability one often seeks solutions to equations of the form Lw = g, where w satisfies a Dirichlet boundary condition: w [??] p = h. Our uniqueness results often imply that these equations cannot have a regular solution, for example, when g ≥ 0. In the classical case the solutions to these problems can sometimes be written down explicitly, and are seen to involve the non-zero indicial roots. Usually these satisfy the other natural boundary condition, a la [20]. In 1-dimension, when b [not equal to] 0, 1 it is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.25)

and allows for solutions that are O(x^1-b) as x -> 0⁺. These are not smooth up to the boundary, even if the data is. The adjoint of L is naturally defined as an operator on M(P), the space finite Borel measures on P. It is more common to study this operator using techniques from probability theory, see [40].

(Continues...)

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Details zur Ausgabe

• Verlag: Princeton University Press
• Erscheinungsdatum: 2013
• ISBN 10: 069115712X
• ISBN 13: 9780691157122
• Einband: Tapa dura
• Sprache: Englisch
• Anzahl der Seiten: 320
• Kontakt zum Hersteller: Nicht verfügbar