 # Degenerate Diffusion Operators Arising in Population Biology (AM-185)

Charles L. Epstein
Rafe Mazzeo
Pages: 320
https://www.jstor.org/stable/j.ctt24hpqf

1. Front Matter
(pp. i-vi)
(pp. vii-x)
3. Preface
(pp. xi-xiv)
4. Chapter One Introduction
(pp. 1-22)

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$N\hspace{3pt}+\hspace{3pt}1$possible types, then the configuration space for the resultant continuous Markov process is typically the$N$-simplex

$\mathscr{S}_{N}\hspace{3pt}=\hspace{3pt}\left\{\left$$x_{1},..., x_{N}\right$$\hspace{3pt}:\hspace{3pt}x_{j}\hspace{3pt}\geq\hspace{3pt}0\hspace{5pt}\text{and}\hspace{5pt}x_{1}\hspace{3pt}+...+\hspace{3pt}x_{N}\hspace{3pt}\leq 1\right\}. (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...

5. I Wright-Fisher Geometry and the Maximum Principle
• Chapter Two Wright-Fisher Geometry
(pp. 25-33)

We begin by describing the class of spaces we work on and then the structure of the operators we consider.

The natural domains of definition for generalized Kimura diffusions are polyhedra in Euclidean space or, more generally, abstract manifolds with corners. In order to set notation and fix ideas, we review this class of objects here and discuss the main properties about them that will be needed below. A more complete discussion can be found in [35, 36].

The standard$N$-dimensional Euclidean space is denoted$\mathbb{R}^{N}$. For any$n\hspace{5pt}=\hspace{5pt}1,\hspace{3pt}.\hspace{3pt}.\hspace{3pt}.,\hspace{3pt}N$, let us set$m\hspace{5pt}=\hspace{5pt}N\hspace{3pt}-\hspace{3pt}n$and define the positive...

• Chapter Three Maximum Principles and Uniqueness Theorems
(pp. 34-48)

One of the most important features associated with any scalar parabolic or elliptic problem is the use of the maximum principle. Although this seems to give only qualitative properties of solutions, it can actually be used to deduce many quantitative results, including even the parabolic Schauder estimates. On a more basic level, it is the key ingredient in proving uniqueness of solutions to such an equation. We now develop the maximum principle and its main consequences, both for the model operators$\partial_{t}\hspace{3pt}-\hspace{3pt}L_{b, m}$on an open orthant, and for the general Kimura diffusion operators$\partial_{t}\hspace{3pt}-\hspace{3pt}L$on a compact manifold with corners,...

6. II Analysis of Model Problems
• Chapter Four The Model Solution Operators
(pp. 51-63)

In this chapter we introduce the model heat kernels$k_{b, m}^{t}$, i.e., the solution operators for the model problems$\partial_{t}\hspace{3pt}-\hspace{3pt}L_{b, m}$, and then prove a sequence of basic estimates for these operators which are direct generalizations of the estimates for the 1-dimensional version of this problem considered in . We recall and slightly extend the$\mathscr{C}^{0}$and$\mathscr{C}^{k}$theory in the 1-dimensional case proved in  and then derive the straightforward extensions of these results to higher dimensions. This sets the stage for the more difficult Hölder estimates for solutions, which is carried out in the next several chapters, and...

• Chapter Five Degenerate Hölder Spaces
(pp. 64-77)

The starting point to implement this perturbation theory is a description of the various function spaces we shall be using. As described above, we seek function spaces on the domain$P$for which the diffusion associated to a general Kimura diffusion operator>$L$is well posed. More pragmatically, we wish to define spaces on which one can prove analogues of the standard parabolic Schauder estimates, so that we can pass from the model to more general operators. This chapter is devoted to a description of the various spaces on which this is possible, and to an explanation of the relationships...

• Chapter Six Hölder Estimates for the 1-dimensional Model Problems
(pp. 78-106)

In this and the following three chapters we establish Hölder estimates for the solutions of the model problems, i.e., ω such that

$\left$$\partial_{t}\hspace{3pt}-\hspace{3pt}L_{b, m}\right$$\omega\hspace{3pt}=\hspace{3pt}g\hspace{5pt}\text{with}\hspace{5pt}\omega\left$$p, 0\right$$\hspace{3pt}=\hspace{3pt}f\left$$p\right$$, (6.1)$

where$f$and$g$belong to the anisotropic Hölder spaces introduced in Chapter 5. It may appear that we are taking a circuitous path, by first considering the 1-dimensional case, then pure corner models,$\mathbb{R}_{+}^{n}$, followed by Euclidean models$\left$$\mathbb{R}^{m}\right$$$before finally treating the general case,$\mathbb{R}_{+}^{n}\hspace{3pt}\times\hspace{3pt}\mathbb{R}^{m}$. In fact, all cases need to be treated, and in the end nothing is really wasted. We give a detailed treatment of the 1-dimensional case, both because it...

• Chapter Seven Hölder Estimates for Higher Dimensional Corner Models
(pp. 107-136)

The estimates proved in the previous chapter form a solid foundation for proving analogous results in higher dimensions for model operators of the form

$L_{b, m}\hspace{3pt}=\hspace{3pt}\sum_{j = 1}^{n}[x_{j} \partial_{x_j}^{2}\hspace{3pt}+\hspace{3pt}b_{j} \partial_{x_j}]\hspace{3pt}+\hspace{3pt}\sum_{k = 1}^{m} \partial_{y_k}^{2}, (7.1)$

here$b \in \mathbb{R}_{+}^{-n}$. In this context we exploit the fact that the solution operator for$L_{b, m}$is a product of solution operators for 1-dimensional problems.

In 2-dimensions we can write

$u\left$$x_{1}, x_{2}, t\right$$\hspace{3pt}-\hspace{3pt}u\left$$y_{1}, y_{2}, t\right$$\hspace{3pt}=\hspace{3pt}[u\left$$x_{1}, x_{2}, t\right$$\hspace{3pt}-\hspace{3pt}u\left$$x_{1}, y_{2}, t\right$$]\hspace{3pt}+\hspace{3pt}[u\left$$x_{1}, y_{2}, y\right$$\hspace{3pt}-\hspace{3pt}u\left$$y_{1}, y_{2}, t\right$$], \left$$7.2\right$$$

and in$n\hspace{3pt}\gt\hspace{3pt}2$dimensions we rewrite$u\left$$x, t\right$$\hspace{3pt}-\hspace{3pt}u\left$$y, t\right$$$as

$u(x, t)\hspace{3pt}-\hspace{3pt}u(y, t)\hspace{3pt}=\hspace{3pt}\sum_{j = 0}^{n-1}[u(x^{\prime}_{j}, x_{j+1}, y^{\prime\prime}_{j}, {t})\hspace{3pt}-\hspace{3pt}u(x^{\prime}_{j}, y_{j+1}, y^{\prime\prime}_{j}, t)], (7.3)$

where:

$\left. {x^{\prime}_{j}\hspace{3pt}=\hspace{3pt}\left$$x_{1},..., x_{j}\right$$\hspace{5pt}\text{if}\hspace{5pt} 1\hspace{3pt}\leq\hspace{3pt}j\hspace{5pt}\text{and}\hspace{5pt} \emptyset\hspace{5pt}\text{if}\hspace{5pt} j\hspace{3pt}\leq\hspace{3pt}0,\atop x^{\prime\prime}_{j}\hspace{3pt}=\hspace{3pt}\left$$x_{j+2},..., x_{n}\right$$\hspace{5pt}\text{if}\hspace{5pt} j\hspace{3pt}\lt\hspace{3pt}n\hspace{3pt}-\hspace{3pt}1\hspace{5pt} \text{and}\hspace{5pt} \emptyset\hspace{5pt} \text{if}\hspace{5pt} j\hspace{3pt}\geq\hspace{3pt}n\hspace{3pt}-\hspace{3pt}1.} (7.4)$

In this way we are reduced to estimating these differences 1-variable-at-a-time, which, in light of Lemma 6.1.1 suffices.

In the proofs of the 1-dimensional estimates the only facts about the data we use...

• Chapter Eight Hölder Estimates for Euclidean Models
(pp. 137-142)

The Euclidean model problems are given by

$\partial_{t}u(y, t)\hspace{3pt}-\hspace{3pt}\sum_{j = 1}^{m} \partial_{y_j}^{2}u(y, t)\hspace{3pt}=\hspace{3pt}g(y, t)\hspace{5pt}\text{with}\hspace{5pt}u(y, 0)\hspace{3pt}=\hspace{3pt}f(y). (8.1)$

The 1-dimensional solution kernel is

$k_{t}^{e}(x, y)\hspace{3pt}=\hspace{3pt}\frac {e^{\frac{|x-y|^2}{4t}}}{\sqrt{4\pi t}}, (8.2)$

and the solution to the equation in (8.1), vanishing at$t\hspace{3pt}=\hspace{3pt}0$, is given by

$u(y, t)\hspace{3pt}=\hspace{3pt}\int_{0}^{t}\int_{-\infty}^{\infty}...\int_{-\infty}^{\infty}\prod_{j = 1}^{m}k_{t-s}^{e}(y_j, z_j)g(z, s)dzds. (8.3)$

The solution to the homogeneous initial value problem with$v\left$$y, 0\right$$\hspace{3pt}=\hspace{3pt}f\left$$y\right$$$, is given by

$v(y, t)\hspace{3pt}=\hspace{3pt}\int_{-\infty}^{\infty}...\int_{-\infty}^{\infty}\prod_{j = 1}^{m}k_{t}^{e}(y_j, z_j)f(z)dz. (8.4)$

For fixed$y, v\left$$y, t\right$$$extends analytically in$t$to define a function in$S_{0}$. The Hölder estimates for the solutions of this problem are, of course, classical. In this chapter we state the estimates and the 1-dimensional kernel estimates needed to prove them.

The solutions of the problems

$\partial_{t}v\left$$y, t\right$$\hspace{3pt}-\hspace{3pt}\sum_{j=1}^{m}\partial_{yj}^{2}v\left$$y, t\right$$\hspace{3pt}=\hspace{3pt}0\hspace{5pt}\text{with}\hspace{5pt}v\left$$y, t\right$$\hspace{3pt}=\hspace{3pt}f\left$$y\right$$ \in \mathscr{C}^{k, \gamma} \left$$\mathbb{R}^{m}\right$$ (8.5)$

and

$\partial_{t}u\left$$y, t\right$$\hspace{3pt}-\hspace{3pt}\sum_{j=1}^{m}\partial_{yj}^{2}u\left$$y, t\right$$\hspace{3pt}=\hspace{3pt}g\left$$y, t\right$$ \in \mathscr{C}^{k, \gamma} \left$$\mathbb{R}^{m} \times \mathbb{R}_{+}\right$$\hspace{5pt}\text{with}\hspace{5pt}u\left$$y, t\right$$\hspace{3pt}=\hspace{3pt}0, (8.6)$

are well known to...

• Chapter Nine Hölder Estimates for General Models
(pp. 143-178)

We now turn to the task of estimating solutions to heat equations defined by the operators of the form:

$L_{b, m}\hspace{3pt}=\hspace{3pt}\sum_{n}^{j = 1}[x_{j}\partial_{x_j}^{2}\hspace{3pt}+\hspace{3pt}b_{j}\partial_{x_j}]\hspace{3pt}+\hspace{3pt}\sum_{k = 1}^{m}\partial_{y_k}^{2}. (9.1)$

The general model operator on$\mathbb{R}_{n}^{+} \times \mathbb{R}^{m}$, denoted$L_{b, m}$, is labeled by a non-negative$n$-vector$b\hspace{3pt}=\hspace{3pt}\left$$b_{1},..., b_{n}\right$$$, and$m$, the dimension of the corner. We use$x$-variables to denote points in$\mathbb{R}_{n}^{+}$and$y$-variables to denote points in$\mathbb{R}^{m}$. If we have a function of these variables$f\left$$x, y\right$$$then, as before, we estimate differences$f\left$$x^{2}, y^{2}\right$$\hspace{3pt}-\hspace{3pt}f\left(x^{1}, y^{1}\right\)$1-variable-at-a-time. We first observe that

$f\left$$x^{2}, y^{2}\right$$\hspace{3pt}-\hspace{3pt}f\left(x^{1}, y^{1}\right\)\hspace{3pt}=\hspace{3pt}\left$f\left$$x^{2}, y^{2}\right$$\hspace{3pt}-\hspace{3pt}f\left$$x^{1}, y^{2}\right$$\right]\hspace{3pt}+\hspace{3pt}\left\[f\left$$x^{1}, y^{2}\right$$\hspace{3pt}-\hspace{3pt}f\left$$x^{1}, y^{1}\right$$\right]; (9.2) each term in brackets can then be written as a telescoping sum: f\left$$x^{2}, y^{2}\right$$\hspace{3pt}-f\hspace{3pt}\left$$x^{1}, y^{1}\right$$\hspace{3pt}=\hspace{3pt}\left\{\sum_{j=0}^{n-1}\left\[f\left$$x_{j}^{2\prime}, x_{j+1}^{2}, x_{j}^{1\prime\prime}, y^{2}\right$$\hspace{3pt}-\hspace{3pt}f\left$$x_{j}^{2\prime}, x_{j+1}^{1}, x_{j}^{1\prime\prime}, y^{2}\right$$\right$\right\}\hspace{3pt}+\hspace{3pt}\left\{\sum_{l=0}^{m-1}\left\[f\left$$x^{1}, y_{l}^{2\prime}, y_{l+1}^{2}, y_{l}^{1\prime\prime}\right$$\hspace{3pt}-\hspace{3pt}f\left$$x^{1}, y_{l}^{2\prime}, y_{l+1}^{2}, y_{l}^{1\prime\prime}\right$$\right]\right], (9.3)$...

7. III Analysis of Generalized Kimura Diffusions
• Chapter Ten Existence of Solutions
(pp. 181-217)

We now return to the principal goal of this monograph, the analysis of a generalized Kimura diffusion operator,$L$, defined on a manifold with corners,$P$. The estimates proved in the previous chapters for the solutions to model problems, along with the adapted local coordinates introduced in Chapter 2.2, allow the use of the Schauder method to prove existence of solutions to the inhomogeneous problem

$\left$$\partial_{t}\hspace{3pt}-\hspace{3pt}L\right$$\omega\hspace{3pt}=\hspace{3pt}g\hspace{5pt}\text{in}\hspace{5pt}P\hspace{3pt}\times\hspace{3pt}\left$$0, T\right$$\hspace{5pt}\text{with}\hspace{5pt}\omega\left(x, 0\right\)\hspace{3pt}=\hspace{3pt}f. (10.1)$

Ultimately we will show that if

$f\hspace{3pt}\in\hspace{3pt}\mathscr{C}_{WF}^{k, {2+\gamma}}\hspace{3pt}(P)\hspace{5pt}\text{and}\hspace{5pt}g\hspace{3pt}\in\hspace{3pt}\mathscr{C}_{WF}^{k, \gamma} (P\hspace{3pt}\times\hspace{3pt}[0, T]), (10.2)$

then the unique solution$\omega\hspace{3pt}\in\hspace{3pt}\mathscr{C}_{W F}^{k, 2+\gamma}\hspace{3pt}(P\hspace{3pt}\times\hspace{3pt}[0, T])$. In this chapter we prove the basic existence result:

Theorem 10.0.2.For$k\hspace{3pt}\in\hspace{3pt}\mathbb{N}_{0}$and$0\hspace{3pt}\lt\hspace{3pt}\gamma\hspace{3pt}\lt\hspace{3pt}1$,if the data$f,g$satisfy...

• Chapter Eleven The Resolvent Operator
(pp. 218-234)

We have shown that$e^{tL}$, the formal solution operator for the Cauchy problem

$\left$$\partial_{t}\hspace{3pt}-\hspace{3pt}L\right$$v\hspace{3pt}=\hspace{3pt}0,\hspace{5pt}v\left$$p, 0\right$$\hspace{3pt}=\hspace{3pt}f\left$$p\right$$,$

makes sense for initial data$f\hspace{3pt}\in\hspace{3pt}\mathscr{C}_{W F}^{0, 2+\gamma}\hspace{3pt}(P)$, and that the solution belongs to$\mathscr{C}_{W F}^{0, 2+\gamma}\hspace{3pt}(P\hspace{3pt}\times\hspace{3pt}[0, \infty))$. Of course, much more is true, but the extension to merely continuous data seems to entail rather different techniques from those employed herein.

The Laplace transform of$e^{tL}$is formally the resolvent operator:

$(\mu\hspace{3pt}-\hspace{3pt}L)^{-1}\hspace{3pt}=\hspace{3pt}\int_{0}^{\infty} e^{-\mu t} e^{t L} f dt. (11.1)$

Using the Laplace transform of a parametrix for the heat kernel and a perturbative argument,we construct belowan operator$R\left$$\mu\right$$$, which depends analytically on$\mu$lying in the complement of a set$E\hspace{3pt}\sub\hspace{3pt}\mathbb{C}$which lies in...

• Chapter Twelve The Semi-group on $\mathscr{C}^{0}\left$$P\right$$$
(pp. 235-250)

In the previous chapters we have dealt almost exclusively with solutions to (10.1) with inhomogeneous terms$f$and$g$in the WF-Hölder spaces. As explained early in this monograph, the reason for working in Hölder spaces in the first place is to handle the perturbation theory in passing from the model operator to the actual one. The original problem, suggested by applications to population genetics, is to study (10.1) with$g\hspace{5pt}=\hspace{5pt}0$and$f\hspace{3pt}\in\hspace{3pt}\mathscr{C}^{0}\left$$P\right$$$. As noted earlier, the existence theory we have developed suffices to prove that the$\mathscr{C}^{0}\left$$P\right$$$-graph closure of$L$with domain$\mathscr{C}_{W F}^{0, 2+\gamma}\hspace{3pt}(P),$, for any$0\hspace{3pt}\lt\hspace{3pt}\gamma$...

8. Appendix A Proofs of Estimates for the Degenerate 1-d Model
(pp. 251-300)
9. Bibliography
(pp. 301-304)
10. Index
(pp. 305-306)