A non-standard evolution problem arising in population genetics
aa r X i v : . [ m a t h . A P ] M a r A NON-STANDARD EVOLUTION PROBLEM ARISING INPOPULATION GENETICS
FABIO A. C. C. CHALUB AND MAX O. SOUZA
Abstract.
We study the evolution of the probability density of an asexual, one locus pop-ulation under natural selection and random evolution. This evolution is governed by aFokker-Planck equation with degenerate coefficients on the boundaries, supplemented by apair of conservation laws. It is readily shown that no classical or standard weak solutiondefinition yields solvability of the problem. We provide an appropriate definition of weaksolution for the problem, for which we show existence and uniqueness. The solution displaysa very distinctive structure and, for large time, we show convergence to a unique stationarysolution that turns out to be a singular measure supported at the endpoints. An exponentialrate of convergence to this steady state is also proved. Introduction
A classical problem in population genetics is to study the evolution of a mutant gene. Astandard approach to this problem is to consider a finite size population and to define adiscrete dynamics for the evolution of the probability density of such a population. Usually,such models are Markov chains, in which the only absorbing states are the two pure ones.Therefore, one expects, for large time, convergence to one of these two states and, dependingon which state is achieved, one says that the mutation has been either fixed or lost. For largepopulations, it is natural to ask for a continuous model that approximates this evolution. Ina number of different ways, one arrives at a Fokker-Plank equation that describes either theevolution of the probability density (the so-called forward Kolmogorov equation), or whatis sometimes called the transient fixation probability (the backward Kolmogorov equation).
Mathematics Subject Classification.
Primary 95D15; Secondary 35K65.
Key words and phrases. gene fixation, evolutionary dynamics, degenerate parabolic equations, boundary-coupled weak solutions.The authors want to thank Peter Markowich for helpful comments, and an anonymous referee for pointingout a mistake in an earlier version of Proposition 4 and for general comments that improved the presentation.FACCC is partially supported by FCT/Portugal, grants POCI/MAT/57546/2004, PTDC/MAT/68615/2006and PTDC/MAT/66426/2006. MOS is partially supported by FAPERJ/Brazil grants 170.382/2006 and110.174/2009. MOS also thanks the support and hospitality of FCT/UNL and Complexo Interdisciplinar/UL.Part of this work has been done during the
Special Semester on Quantitative Biology Analyzed by Mathematics ,organized by RICAM, Austrian Academy of Sciences.
From a mathematical point of view, it is interesting to notice that, for the fixation proba-bilities, it is easy to specify the appropriate non-homogeneous boundary conditions which,after subtraction of an appropriate multiple of a stationary solution, are recast as Dirichletconditions. Nevertheless, this does not seem to be the case for the probability evolution.Since it must conserve mass, in many cases a condition of null probability current at the end-points is used (e.g. [22]). For a thorough introduction to the several aspects of mathematicalpopulation genetics, we refer the reader to the monographs by [3, 12]For a class of problems, however, these Fokker-Plank equations turn out to have degeneratecoefficients at the boundaries, the classical Kimura equation (cf. [17]) being the archetypalexample. For the backward one, this is not a problem since the infinitesimal generator is, verygenerally, self-adjoint. For the forward equation, however, the underlying spectral problem isof the limit-point type and, thus, no boundary conditions can be enforced. In particular, onecannot control the flux of the solutions across the boundary of the corresponding domain, andthe existence of conservation laws are not to be expected in general. This is an old issue inthe study of diffusions, and it has been tackled by [13], where the so called lateral conditionsare derived, in order to ensure that the forward and backward equations are adjoint to eachother. With these lateral conditions, however, the forward equation looses its differentialcharacter, and this led to a prevalence of the backward equation in the study of diffusions(particularly after [14]). We shall see below that is possible to ensure the duality of thebackward and forward equations, while maintaining the differential character of the forwardequation, within the framework of weak solutions.We shall study the forward Kolmogorov equation(1) ( ∂ t p ( t, x ) = ∂ x ( F ( x ) p ( t, x )) − ∂ x ( G ( x ) p ( t, x )) , x ∈ (0 , , t > p (0 , x ) = p ( x )with F positive in (0 , G vanishingat the endpoints . Typical examples are F ( x ) = G ( x ) = x (1 − x ) (forward Kimura) and F ( x ) = x (1 − x ), G ( x ) = x (1 − x )( ηx + β ) (forward Kimura with frequency selection; see [5]).Equation (1) is supplemented by the following conservation laws:dd t Z p ( t, x ) d x = 0 , (2a) dd t Z ψ ( x ) p ( t, x ) d x = 0 , (2b)where ψ satisfies(3) F ( x ) ψ ′′ + G ( x ) ψ ′ = 0 , ψ (0) = 0 , and ψ (1) = 1 . More precise statements on the hypothesis made upon F and G are deferred to section 2. NON-STANDARD EVOLUTION PROBLEM 3
Remark 1.
In population genetics, the function ψ is referred to as the fixation probability.Condition (2a) is usually stated (or assumed), in the literature of population genetics, butcondition (2b) is not. These conditions have been derived in [5] , when obtaining the forwardKimura equation, with frequency selection, as a large population limit of the so called Moranprocess (cf. [23] ). See also [28] for an alternative approach. Before we proceed, we want to clarify the nature of the conservation laws given by (2).The backward equation and (formal) adjoint of (1) is given by(4) ( ∂ t f = F ( x ) ∂ x f ( t, x ) + G ( x ) ∂ x f ( t, x ) x ∈ (0 , , t > f (0 , x ) = f ( x ) . It is readily seen that any stationary solution to (4) is a linear combination of a constantand ψ ( x ). Therefore, the conservation laws (2) are related to the kernel of the infinitesimalgenerator of (4). Finally, it should be mentioned that, if (1) is a correct approximation of thebiological process, then one expects that the probability mass accumulates at the endpoints,as t goes to infinity [17].The goal of this work is to clarify in what sense a solution to (1) that satisfies (2) exists,and how it behaves for large time. In contradistinction with [13], which uses classical functionspaces and has to modify equations (1) and (4) in order to obtain the duality relation, we shallalways work with these equations, but in more general, non-normed, distributional spaces.This also differs from recent work in degenerate equations, as for instance: the controlability ofdegenerate heat equations [21], with solutions in weighted Sobolev spaces; entropy solutionsof Fokker-Planck from multilane traffic flow [10], where the conditions that might lead toconcentration at the end points are explicitly avoided; and from the qualitative studies by[1]. See [15] for a general discussion of degenerate diffusion equations. We mention also theclassical monographs [4, 9].Equation (1), with F ( x ) = x (1 − x ) and G ( x ) = x (1 − x )( ηx + β ) has been studied inreference [5], where a proof of existence and uniqueness in the sense of definition 1 is given,under the assumption of interior regularity. An announcement that also includes other resultswas made in [6]. See also [7]. More recently, the same problem has been studied throughskillful, but formal, calculations in [22], with conditions of null probability current (formally)imposed. Thus, this work can be seen as complementary to the work by [13] by giving adifferential formulation to the forward-backward duality for degenerate diffusions. Also, itcan be regarded as an extension of [5, 6], and as a rigorous proof of the formal calculationsin [22].The main results of the paper can be outlined as follows: let BM + ([0 , , FABIO A. C. C. CHALUB AND MAX O. SOUZA
Theorem 1 (outline) . For a given p ∈ BM + ([0 , , there exists a unique solution p toEquation (1), in a sense to be made precise in definition 1, with p ∈ L ∞ (cid:0) [0 , ∞ ); BM + ([0 , (cid:1) and such that p satisfies the conservations laws (2). The solution can be written as p ( t, x ) = q ( t, x ) + a ( t ) δ + b ( t ) δ , where δ y denotes the singular measure supported at y , and q ∈ C ∞ ( R + ; C ∞ ([0 , is aclassical solution to (1). We also have that a ( t ) and b ( t ) , belong to C ([0 , ∞ )) ∩ C ∞ ( R + ) . Inparticular, we have that p ∈ C ∞ ( R + ; BM + ([0 , ∩ C ∞ ( R + ; C ∞ ((0 , . For large time, we have that lim t →∞ q ( t, x ) = 0 , uniformly, and that a ( t ) and b ( t ) are mono-tonically increasing functions such that: a ∞ := lim t →∞ a ( t ) = Z (1 − ψ ( x )) p ( x ) d x and b ∞ := lim t →∞ b ( t ) = Z ψ ( x ) p ( x ) d x, Moreover, we have that lim t →∞ p ( t, · ) = a ∞ δ + b ∞ δ , with respect to the Radon metric. Finally, the convergence rate is exponential. Remark 2.
The coefficients of the singular measures, a ( t ) and b ( t ) are, respectively, theextinction and the fixation probabilities. Also, notice that the decomposition of p does notfollows immediately from the linearity of (1). As a matter of fact, neither of the summandsare, per se, a solution to (1) in the sense of definition 1. Heuristically, as (1) is uniformlyparabolic in each proper compact set of the unit interval, the parabolic operator erodes theinterior density of the initial measure, which is then transferred into the boundaries andabsorbed by the singular measures there. The outline of the paper is as follows: in section 2, we present background results for theclassical (in a broad sense) solutions to (1). In section 3, we introduce an appropriate definitionof a weak solution and show that any solution of this type must satisfy the conservation laws(2). We also show that, with this formulation, (5) is indeed the adjoint of (1). In section 4,we present the proofs of existence and uniqueness. Section 5 discusses the convergence to themeasures supported at the endpoints as time goes to infinity.
NON-STANDARD EVOLUTION PROBLEM 5 Preliminaries
For the convenience of the reader, we present in this section some material that will beuseful in the sequel.Let
F, G : [0 , → R be smooth, and assume that(1) F has single zeros at x = 0 and at x = 1, and F ( x ) >
0, for x ∈ (0 , G has zeros at x = 0 and x = 1.Hadamard’s lemma (cf. [2]) then yields F ( x ) = x (1 − x )Ψ( x ) , Ψ( x ) > x ∈ [0 ,
1] and G ( x ) = x (1 − x )Π( x )Let us write, Ξ( x ) = Π( x )Ψ( x ) . Then we can rewrite (4) as(5) ( ∂ t f = x (1 − x )Ψ( x ) (cid:2) ∂ x f ( t, x ) + Ξ( x ) ∂ x f ( t, x ) (cid:3) x ∈ (0 , , t > f (0 , x ) = f ( x )The stationary solutions of (5) are linear combinations of a constant and ψ ( x ) = c − Z x e − R s Ξ( r ) d r d s, c = Z e − R s Ξ( r ) d r d s. Existence of classical solutions to (1) can be established by Fourier series, and this is easierdone by writing (1) in selfadjoint form. Let(6) e R x Ξ( s ) d s w = x (1 − x )Ψ( x ) p. Then (1) becomes(7) ∂ t w = x (1 − x )Ψ( x ) (cid:26) ∂ x w − (cid:2) ′ + Ξ (cid:3) w (cid:27) . Remark 3.
Since the standard maximum principle holds for C , solutions of (7), we findthat, if the initial condition is nonnegative, then w ( t, · ) is also nonnegative. Moreover, thisholds also for p ( t, · ) . Consider the associated spectral problem:(8) − ϕ ′′ + (cid:2) ′ + Ξ (cid:3) ϕ = λθ ( x ) ϕ,ϕ (0) = ϕ (1) = 0 , θ ( x ) = x ) x (1 − x ) . Sturm-Liouville theory for singular problems allows us to conclude that (8) is a self-adjointoperator in L ([0 , , θ ( x )d x ), with a complete set of eigenfunctions. In what follows, all L spaces will be with respect to θ ( x )d x , and we shall write ( · , · ) and k · k for the corrrespondinginner product and norm, respectively. We also recall, see [8, 25] for instance, thatlim j →∞ λ j j = K. An important property of (8), which is proved in Appendix A.1, is given by
Lemma 1.
The operator defined by (8) is positive-definite.
We shall write ϕ j , j = 0 , , , . . . , for the eigenfunctions of (8), with corresponding eigen-value λ j , and normalization k ϕ j k = 1. Also, for the spectral problem that arises in theoriginal problem, we shall write(9) e R x Ξ( s ) d s ϕ j = x (1 − x )Ψ( x ) q j . We shall need the asymptotic behavior of the eigenfunctions for large λ j . Lemma 2.
There exists positive constants C and C , independent of j , such that (10) k ϕ j k ∞ ≤ C and k q j k ∞ ≤ C λ / j . The proof of Lemma 2 is given in Appendix A.2Finally, as in [27] for instance, we shall denote, for s >
0, the spaces D s = φ ∈ L ([0 , , θ d x ) (cid:12)(cid:12) ∞ X j =0 d φ ( j ) λ s/ j ϕ j ∈ L ([0 , , θ d x ) , d φ ( j ) = ( φ, ϕ j ) , with norm given by k φ k s = ∞ X j =0 d φ ( j ) λ sj . Since Radon measures are distributions of order less or equal to zero, we have—cf. [27] withminor modifications—that:
Proposition 1.
The initial value problem defined by Equation (7) and w (0 , x ) = w ( x ) , with w ∈ BM + ((0 , has the solution (11) w ( t, x ) = X j ≥ c w ( j ) e − tλ j ϕ j ( x ) , c w ( j ) = ( w , ϕ j ) , which is unique in the class C ∞ ( R + ; C ∞ ([0 , . Remark 4.
It can be shown that, any standard weak solution definition to (7) will lead tothe solution above—see for instance [11, 18] . Therefore, none of the conservation laws (2)can hold, and no classical-weak solution to (1–2) exists.
NON-STANDARD EVOLUTION PROBLEM 7 Weak solution and duality formulation
We now make precise what we mean by a weak solution to (1).
Definition 1.
A weak solution to (1) will be a function in L ∞ ([0 , ∞ ); BM ([0 , that sat-isfies − Z ∞ Z p ( t, x ) ∂ t φ ( t, x )d x d t = Z ∞ Z p ( t, x ) x (1 − x )Ψ( x ) (cid:2) ∂ x φ ( t, x ) + Ξ( x ) ∂ x φ ( t, x ) (cid:3) d x d t + Z p ( x ) φ (0 , x )d x, where φ ( t, x ) ∈ T = C ∞ c ([0 , ∞ ) × [0 , . Remark 5.
Notice that the test functions in definition 1 are required to be of compact supportin [0 , and not in (0 , as usual. Similar definitions have been given in other contexts; see [19, 20] , where they are termed boundary-coupled weak solutions .Definition 1 can be recast in the framework of usual distribution theory, by identifying aRadon measure with a compactly supported distribution of nonpositive order (see [26] ). Inthis case, the distribution can act in C ∞ ( R ) , but it is entirely determined by its behavior inthe support; see for instance [16] . A glance at Definition 1 shows that, on the integral on the right hand side, the test function φ is applied to the operator on the right hand side of (4). Thus, one could expect that anysolution that satisfies (1) in the sense defined above, also satisfies the conservation laws (2). Proposition 2.
Let p ∈ L ∞ ([0 , ∞ ); BM ([0 , . If χ ( x ) is a stationary solution of (4), thenthe quantity η ( t ) = Z χ ( x ) p ( t, x ) d x is constant in time.Proof. Let ζ ( t ) ∈ C ∞ c ((0 , ∞ )). Then, φ ( t, x ) = ζ ( t ) χ ( x ) is an appropriate test function. Onsubstituting φ ( t, x ) in Definition 1, we find that − Z ∞ η ( t ) ζ ′ ( t )d t = 0 . Thus η ( t ) is constant almost everywhere. (cid:3) FABIO A. C. C. CHALUB AND MAX O. SOUZA
Remark 6.
We observe that standard spectral theory shows that both the infinitesimal gen-erators of (1) and (4) can be appropriately defined in a domain dense in L ((0 , such thatthey are adjoints of each other. However, in this case, equation (1) will not be the forwardKolmogorov equation associated to (4) . On the other hand, in the sense of the pairing used indefinition 1, (4) with f ( t, · ) ∈ C ∞ c ([0 , is the adjoint of (1) with p ( t, · ) ∈ BM ([0 , . Thus,we recover the usual interpretation of the conservation laws given by the kernel of the adjoint. Existence and uniqueness
In what follows, it will be convenient to decompose a compact distribution, or a Radonmeasure, as the sum of a distribution without singular support at the endpoints, and twodistributions singularly supported at the endpoints. We shall write E ′ to denote the space ofcompactly supported distributions in R . Lemma 3.
Let ν ∈ E ′ , with supp( ν ) = [0 , . Then, the setwise decomposition [0 ,
1] = { } ∪ (0 , ∪ { } , yields a decomposition of ν , namely ν = ν + µ + ν , where ν i is a compact distribution supported at x = i , and sing supp ( µ ) ⊂ (0 , . Moreover,if ν is a Radon measure, then µ ∈ BM ((0 , , and ν i = c i δ i , with c i ∈ R , are singularmeasures with support at x = i .Proof. Let ζ ǫi , i = 0 , , , { [0 , ǫ ) , (1 − ǫ, , ( ǫ, − ǫ ) } . Let φ ∈ C ∞ c ([0 , ν i , i = 0 , µ by Z ν i φ ( x ) d x := lim ǫ → Z ζ ǫi νφ ( x ) d x, i = 0 , , Z µφ ( x ) d x := lim ǫ → Z ζ ǫ νφ ( x ) d x. Then clearly ν = ν + µ + ν . Also, it is readily seen that sing supp( µ ) ⊂ (0 , ζ ǫ ( x ) = 1and ζ ǫ n ) ( x ) = 0, n ≥
1, for x ∈ [0 , ǫ ), we find that ν is supported at x = 0, with a similarargument holding for ν . Moreover, since a Radon measure is inner regular, the restrictionof ν to (0 ,
1) yields a Radon measure in (0 , (cid:3) For the initial condition, we shall write p = a δ + q + b δ , NON-STANDARD EVOLUTION PROBLEM 9 to denote the corresponding decomposition. Also, in order to show the existence of a solutionto (1) in the sense of definition 1, we shall temporarily consider p ∈ L ∞ ([0 , ∞ ); E ′ ), withsupport in [0 , p = p + q + p , for the decomposition of p .We now show that q must be, as a matter of fact, much more regular. Proposition 3 (Interior regularity) . Assume that q ∈ BM + ((0 , . If a solution to (1)exists, then q ( t, x ) must be the unique classical solution in the sense of section 2, with q (0 , x ) = q ( x ) , i.e., (12) q ( t, x ) = ∞ X j =0 b q ( j ) q j e − λ j t , with q j given by (9) , and b q ( j ) is j -th Fourier coefficient of q . In particular, q ∈ C ∞ (cid:0) R + ; C ∞ ([0 , (cid:1) . Proof.
Let φ ∈ C ∞ c ([0 , ∞ ) × (0 , − Z ∞ Z q ( t, x ) ∂ t φ ( t, x ) d x d t = Z ∞ Z q ( t, x ) x (1 − x )Ψ( x ) (cid:2) ∂ x φ ( t, x ) + Ξ( x ) ∂ x φ ( t, x ) (cid:3) d x d t + Z ∞ q ( x ) φ (0 , x ) d x. The result now follows by taking testing functions of the form φ ( t, x ) = ζ ( t )e − R x Ξ( s ) d s ˜ φ ( x ) , ζ ∈ C c ([0 , ∞ )) and ˜ φ ∈ C c ((0 , , and then we use a standard Galerkin approximation procedure. (cid:3) Before we proceed, we observe that, since p and p are distributions supported on asingleton, we must have, for some integers M and M ′ , that(13) p ( t, x ) = M X k =0 a k ( t ) δ ( k )0 + M ′ X k =0 b k ( t ) δ ( k )1 + q ( t, x ) , where δ ( k ) x denotes the k -th distributional derivative of the singleton supported measure. Theorem 2 (Existence and uniqueness) . The unique solution of (1) in the sense of defini-tion 1, with initial condition p ∈ BM + ([0 , is given by p ( t, x ) = q ( t, x ) + a ( t ) δ + b ( t ) δ , with q ( t, x ) given by (12). Moreover, we have a ( t ) = Ψ(0) Z t q ( s, s + a and b ( t ) = Ψ(1) Z t q ( s, s + b . Proof.
First, we define e T = { φ ∈ C ∞ c ((0 , ∞ ) × [0 , } . For l >
0, we also define, e T l, = { φ ∈ C c ([0 , ∞ ) × [0 , | ∂ nx φ ( t,
0) = 0 , ≤ n < l } , with a similar definition for e T l, . Notice that, for r > s , e T r, ⊂ e T s, .On substituting (13) in definition 1, with φ ∈ e T , using that q is smooth for t > − Z ∞ " M X k =0 a k ( t ) ∂ t ∂ kx φ ( t,
0) + M ′ X k =0 b k ( t ) ∂ t ∂ kx φ ( t, d t = Z ∞ [ q ( t, φ ( t,
0) + q ( t, φ ( t, t ++ Z ∞ M X k =0 a k ( t ) ∂ kx ( x (1 − x )Ψ( x ) ∂ x φ ( t, x )) (cid:12)(cid:12) x =0 d t + Z ∞ M X k =0 a k ( t ) ∂ kx ( x (1 − x )Π( x ) ∂ x φ ( t, x )) | x =0 d t ++ Z ∞ M ′ X k =0 b k ( t ) ∂ kx ( x (1 − x )Ψ( x ) ∂ x φ ( t, x )) (cid:12)(cid:12) x =1 d t + Z ∞ M ′ X k =0 b k ( t ) ∂ kx ( x (1 − x )Π( x ) ∂ x φ ( t, x )) | x =1 d t. Restricting somewhat further, for φ ∈ e T M +1 , , we find that0 = Z ∞ a M ( t ) ∂ x [ x (1 − x )Ψ( x )] x =0 ∂ M +1 x φ ( t,
0) d t. Thus a M ( t ) = 0, and the sum can be only up to M −
1. Repeating the argument inductivelyyields M = 0. An analogous argument yields M ′ = 0. Thus only a ( t ) and b ( t ) can benonzero. We now drop the subscripts and determine their values. Applying definition 1 to φ ∈ e T , such that φ ( t,
1) = 0, we find that − Z ∞ a ( t ) ∂ t φ ( t,
0) d t = Z ∞ q ( t, φ ( t,
0) d t. Integrating by parts the corresponding relation for a ( t ), we obtain Z ∞ a ( t ) ∂ t φ ( t,
0) d t = Z ∞ (cid:18) Ψ(0) Z t q ( s,
0) d s + a (cid:19) ∂ t φ ( t,
0) d t. Hence a ( t ) − Ψ(0) Z t q ( s,
0) d s − a = const , NON-STANDARD EVOLUTION PROBLEM 11 everywhere, in as much as the integral is continuous. Since a (0) = a , the identity follows. Asimilar calculation also shows that b ( t ) = Ψ(1) Z t q ( s,
1) d s + b . Uniqueness follows from proposition 3 and from the expressions for a ( t ) and b ( t ). Finally,notice that, since q ( t, x ) ≥
0, we have that both a and b are increasing. (cid:3) Large time behavior
We now present some results for the behavior of the solution in the large time limit.Let us define b ∞ := Z ψ ( x ) p ( x ) d x = b + Z ψ ( x ) q ( x ) d x,a ∞ := Z p ( x ) d x − b ∞ = a + Z (1 − ψ ( x )) q ( x ) d x. Using the conservation laws (2), we also have b ∞ := Z ψ ( x ) p ( t, x ) d x = b ( t ) + Z ψ ( x ) q ( t, x ) d x,a ∞ := Z p ( t, x ) d x − b ∞ = a ( t ) + Z (1 − ψ ( x )) q ( t, x ) d x. Since 0 ≤ ψ ≤ q ( t, · ) ≥
0, we have that both a ∞ − a ( t ) and b ∞ − b ( t ) are nonneg-ative. From the representation given by (11), we have that lim t →∞ k q ( t, x ) k ∞ = 0. Hence,lim t →∞ a ( t ) = a ∞ and lim t →∞ b ( t ) = b ∞ .Moreover, since q ( t, x ) ≥
0, we have a ∞ − a ( t ) + b ∞ − b ( t ) = Z q ( t, x ) d x = k q ( t, · ) k , which also yields the inequalities a ∞ − a ( t ) ≤ k q ( t, · ) k and b ∞ − b ( t ) ≤ k q ( t, · ) k . The behavior of the L norm of q is given by the following result: Proposition 4.
Let p be the solution to (1) with an initial condition with q ∈ BM + ((0 , and let λ be the smallest eigenvalue of (8). Then we have that lim t →∞ e λ t k q ( t, · ) k = C ∞ . In addition, if we assume that w = x (1 − x )Ψ( x )e − R x Ξ( s ) d s q ∈ BM + ((0 , ∩ D s , s > , then there exists C ,s > such that || q ( t, · ) || ≤ C ,s k w k s e − λ t . In particular, the same limit property and bounds apply to a ∞ − a ( t ) and b ∞ − b ( t ) .Proof. For the first part, recall that q ( t, x ) = ∞ X j =0 c w ( j )e − λ j t q j ( x ) . Let us write Q j = Z q j ( x ) d x. We observe that Q j = (e − R x Ξ( s ) d s , ϕ j ) . However, since e R x Ξ( s ) d s L ([0 , , θ d x ), we do not have an immediate bound for | Q j | .On the other hand, we observe that q j satisfies − λ j q j ( x ) = ∂ x [ x (1 − x )Ψ( x ) q j ( x )] − ∂ x [ x (1 − x )Π( x ) q j ( x )] , which integrated yields Q j = Ψ(0) q j (0) + Ψ(1) q j (1) λ j . For large j , (10) guarantees that we must then have | Q j | ≤ C λ − / j . Thus, for t >
0, we havee λ t k q ( t, · ) k = Q c w (0) + ∞ X j =1 Q j c w ( j )e − ( λ j − λ ) t , and the result follows with C ∞ = Q c w (0).For the second part, let us define the auxiliary functions: α s ( x ) = ∞ X j =0 \ w ( j ) λ s/ j ϕ j ( x ) and β s ( t, x ) = ∞ X j =0 Q j λ − s/ j ϕ j ( x )e − λ j t . Then, we have that k q ( t, · ) k = ( α s , β s ( t, · )) ≤ k α s k k β s ( t, · ) k == k w k s e − λ t k e λ t β s ( t, · ) k ≤ C ,s k w k s e − λ t , with C ,s = k β s (0 , · ) k . (cid:3) NON-STANDARD EVOLUTION PROBLEM 13
Theorem 3 (Exponential convergence) . Let ρ denote the Radon metric, and let p ∞ = a ∞ δ + b ∞ δ . Under the same hypothesis of proposition 4, we have that (14) lim t →∞ e λ t ρ ( p, p ∞ ) ≤ C ∞ . With the additional hypothesis, we have that (15) ρ ( p, p ∞ ) ≤ C ,s k w k s e − λ t . In particular, (14) implies convergence in the Wasserstein metric.Proof.
Recall that ρ ( ν, µ ) = sup (cid:26) Z f ( x )d( ν − µ ) (cid:12)(cid:12)(cid:12)(cid:12) f ∈ C ([0 , − , (cid:27) . But, for such f we have that, when t > (cid:12)(cid:12)(cid:12)(cid:12)Z f ( x )d( p ∞ − p ( t, x )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z | d( p ∞ − p ( t, x )) |≤ Z ( a ∞ − a ( t )) δ d x + Z ( b ∞ − b ( t )) δ d x + Z | q ( t, x ) | d x = a ∞ − a ( t ) + b ∞ − b ( t ) + k q ( t, · ) k = 2 k q ( t, · ) k . Now, both (14) and (15) follows from Proposition 4. (cid:3)
Remark 7.
In many applications, the slowest decaying mode ϕ is taken to be a quasi-stationary distribution for the diffusion process. The constant C ∞ is then the total probabilitymass of such a distribution. Appendix A. Postponed proofs
A.1.
Proof of positive-definiteness of (8) . Proof of Lemma 1.
Let(16) v = e − R x Ξ( s ) d s ϕ. Then (8) becomes(17) − v ′′ − Ξ v ′ = λθ ( x ) v, v (0) = v (1) = 0 . When λ = 0, then (17) becomes the stationary version of (5), with Dirichlet boundarycondition. Its general solution, ¯ v , is given by ¯ v = c + c ψ , which does not satisfy therequired boundary conditions. Thus, zero cannot be an eigenvalue of (17). Moreover, since the transformation (16) preserves the oscillation properties of the eigenfunctions, we havethat the eigenfunction v , corresponding to the smallest eigenvalue λ , will not have anyzeros inside (0 , v > , x = x ∗ ∈ (0 , v ′ ( x ∗ ) = 0. Hence we must have − λ v ( x ∗ ) x ∗ (1 − x ∗ )Ψ( x ∗ ) = v ′′ ( x ∗ ) . Note that v ′′ ( x ∗ ) = 0, otherwise we would have λ = 0. Since it is a maximum, we must have v ′′ ( x ∗ ) <
0. Since, v ( x ∗ ) >
0, we have λ > (cid:3) A.2.
Proof of the asymptotic estimates.
Proof of Lemma 2.
For the proof, we drawn on results by [24, chapter 12] that are summarizedas follows
Theorem 4.
Let ζ (cid:18) d ζ d x (cid:19) = − θ ( x ) , ζ (0) = 0 . Also let ˆ ϕ j ( ζ ) = A ,j (cid:18) d ζ d x (cid:19) − / | ζ | / J (cid:16) λ / j | ζ | / (cid:17) , where J is the standard Bessel function of order one, and A ,j is choosen such that k ˆ ϕ j k = 1 .For large j , we have that k ϕ j − ˆ ϕ j k ∞ ≤ K F ( λ / j | ζ | / ) exp K λ / j F ( ζ ) ! F ( ζ ) λ / j , where K , K are positive constants and F , F are positive and bounded continuous functions. With this result, we can now prove the asymptotic behavior for ϕ j and q j Let A − ,j = Z (cid:18) d ζ d x (cid:19) − | ζ | J (cid:16) λ / j | ζ | / (cid:17) d x and J is the standard Bessel function of order one. Let z = λ / j | ζ | / . Then we find that Z (cid:18) d ζ d x (cid:19) − | ζ | J (cid:16) λ / j | ζ | / (cid:17) θ d x = 1 λ j Z z zJ ( z ) d z, where z = λ / j | ζ (1) | . For large j , we have from the asymptotic behavior of J at infinitythat A ,j = Cλ / j + O (1) . Also, since uJ ( u ) ≤ C p ( u ), for large u , we have (cid:13)(cid:13)(cid:13) | ζ | / J (cid:16) λ / j | ζ | / (cid:17)(cid:13)(cid:13)(cid:13) ∞ = 1 λ / j k zJ ( z ) k ∞ ≤ Cλ − / j . NON-STANDARD EVOLUTION PROBLEM 15
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