The uniqueness of hierarchically extended backward solutions of the Wright-Fisher model

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.


Introduction
The Wright-Fisher model [13,39] models the most basic component of mathematical population genetics, i.e. genetic dri . In a nite population of xed size, parents are randomly sampled and pass on the alleles they are carrying to the o spring generation. By repeating this process over many (non-overlapping) generations, the model describes the evolution of the probabilities of the di erent alleles in the population. In the basic setting, the model covers a single locus only. Extensions to several loci are possible, as is the inclusion of mutation, selection, or a spatial population structure. This has driven the research in mathematical population genetics ( [4,11]), inspired by the pioneering work of Kimura [23][24][25].
Nevertheless, the original model remains of considerable mathematical interest, in particular when we follow Kimura and consider the di usion approximation. This di usion approximation leads to a model with an in nite population size and continuous time. Its dynamics may then be described by the so-called forward and backward Kolmogorov equations. The forward equation is a partial di erential equation of parabolic type and describes the evolution of the model over time. The backward equation, which is the adjoint of the former w. r. t. a suitable product, in contrast, models a process that runs backward in time as it describes the probability of ancestral states. According to this biological interpretation, the equation is not parabolic. Mathematically, however, we can easily convert it into a parabolic equation by simply replacing −t by t. Putting it simply, a backward equation with a nal condition can be converted into a forward equation with an initial condition. More serious mathematical di culties arise from the fact that both equations become degenerate at the boundary.
This paper investigates solutions of the Kolmogorov backward equation for the relative frequencies 0 ≤ p i ≤ 1 of the alleles i = 0, . . . , n , that is ∂ ∂p i ∂p j u(p, t) =: L * n u(p, t) (1.1) The frequency p 0 does not appear in (1.1) because of the normalization n i=0 p i = 1. When one of the frequencies p i becomes 0, the corresponding coe cient also becomes 0. Thus, the di erential operator in (1.9) becomes degenerate at the boundary of our domain, the probability simplex n = {(p 1 , . . . , p n ) : p i > 0, n j=1 p j < 1}. In fact, a suitable extension of the solution of (1.1) to the boundary of n and the investigation of its properties will be our main concern and achievement.
There have been two lines of research on these Kolmogorov equations, one with tools from the theory of stochastic processes, see for instance [7,9,10,22], as well as with tools from the theory of partial di erential equations [5,6]. By its general nature, this approach is capable of certain existence, uniqueness and regularity results, but cannot come up with explicit formulas, for instance for the expected time of loss of an allele. Therefore, the second line of research uses less general tools, but makes detailed use of the speci c and explicit structure of the model. This has also included the global aspect, that is, connecting the solutions in the interior of the simplex and on its boundary faces, and a number of representation formulas has been derived. This aspect is also covered to some extent in Section 5.10 of [11] as well as in [4], but we wish to illustrate certain results in more detail and with a di erent focus.
In the literature, using an observation of [31], one usually writes the Kolmogorov backward operator in the form * using the variables (x 0 , x 1 , . . . , x n ) with n j=0 x j = 1 in place of L * n u(p, t) (cf. equation (1.1)) with (p 1 , . . . , p n ) and p 0 = 1 − n i=1 p i implicitly determined (for our notation, see Section 2.1, in particular (2.2) and (1.11). That is, one includes the variable x 0 and works on the simplex {x 0 +x 1 +. . . x n = 1, x i ≥ 0}. This formulation has the advantage of being symmetric w. r. t. all x i , but the downside is that the operator invokes more independent variables than the dimension of the space on which it is de ned. Thus, the elliptic operator becomes degenerate. Here, we have opted to work with L * n , but for the comparison with the literature, we shall utilize the version (1.2).
Much of the literature to be referenced here is based on the observation of Wright [40] that the degeneracy at the boundary may be removed if one includes mutation. More precisely, let the mutation rate m ij be the probability that when allele i is selected for o spring, the o spring carries the mutant j instead of i; furthermore, one de nes m ii = − j =i m ij . The corresponding Kolmogorov backward operator then becomes * Wright [40] discovered that calculations may be considerably simpli ed by assuming m ij = 1 2 µ j > 0 for i = j, (1.4) that is, the mutation rates depend only on the target gene (the factor 1 2 is inserted solely for purposes of normalization) and are positive. With (1.4), ( µ i x j ∂ ∂x j . (1.5) In this case, one obtains a unique stationary distribution for the Wright-Fisher di usion, given by the Dirichlet distribution with parameters µ 0 , . . . , µ n . A further simpli cation occurs when µ 0 = · · · = µ n =: µ > 0, (1.6) i.e., when all mutation rates are identical. The assumption (1.4) that the mutation rates only depend on the target gene is not very plausible from the biological perspective (the mutation rate should rather depend on the initial instead of the target gene, but (1.6) remedies that de cit in a certain sense), but in the present context the more crucial issue is the assumption of positivity. Several papers have studied this model and derived explicit formulas for the transition density of the process with generator (1.5) including [3, 8, 14-16, 28, 32-34]; these, however, were rather of a local nature, as they did not connect solutions in the interior and the boundary strata of the domain. Furthermore, Kingman's coalescent [26] has proven to be a very useful tool in this line of research, that is, the method of tracing lines of descent back into the past and analyzing their merging patterns (for a quick introduction to that theory, see also [21]). In particular, some of these formulas likewise extend to the limiting case µ = 0 in (1.6); Ethier and Gri ths [8] showed that the following formula for the transition density which had previously been derived under the assumption µ > 0, also applies to the case µ = 0. Here, Dir is the Dirichlet distribution, and d 0 M (t) is the number of equivalence classes of lines of descent of length M at time t in Kingman's coalescent for which analytical formulas have been derived in [34]. (1.7) has been studied further in many subsequent papers, for instance [16]. However, the Dirichlet distribution in (1.7) becomes singular when y approaches the boundary of K.
Shimakura in [33] came up with the somewhat less explicit formula Here, the λ m are the eigenvalues of the elliptic operator, and E m is the projection onto the corresponding eigenspace, and the index K enumerates the faces of the simplex. This solution is de ned on the entire simplex, and it matches all (independent) data on di erent boundary strata as t tends to 0. However, the transitions from a face into one of its boundary faces are not accounted for in this scheme, which considers the solutions on the individual faces separately. Thus, (1.8) is simply a decomposition into the various modes of the solutions of a linear PDE, summed over all faces of the simplex. In particular, Shimakura's solution satis es corresponding regularity properties. Altogether, this illustrates the rather local character of the solution scheme.
In the present paper, we want to get a more detailed analytical picture of the behavior at the boundary and investigate global solutions, speci cally their uniqueness, on the entire state space including its strati ed boundary. In an important recent work, Epstein and Mazzeo [5,6] have developed PDE techniques to tackle the issue of solving PDEs on a manifold with corners that degenerate at the boundary with the same leading terms as the Kolmogorov backward equation for the Wright-Fisher model (1.1) in the closure of the probability simplex in ( n ) −∞ = n × (−∞, 0). A crucial ingredient of their analysis is the construction of appropriate function spaces. In our context, their spaces C k,γ WF ( n ) would consist of k times continuously di erentiable functions whose kth derivatives are Hölder continuous with exponent γ w. r. t. the Fisher metric. In terms of the Euclidean metric on the simplex, this means that a weaker Hölder exponent (essentially γ 2 ) is required in the normal than in the tangential directions at the boundary. Using this framework, they then show that if the initial values are of class C k,γ WF ( n ), then there exists a unique solution in that class. This result is very satisfactory from the perspective of PDE theory (see e.g. [20]). In the situation that we are facing in this paper, however, the data and the solutions are not even continuous, let alone of some class C 0,γ ( n ), as we want to study the boundary transitions. Likewise, the (stationary) uniqueness assertion does not apply, which Epstein and Mazzeo have established for largely regular (in particular, globally continuous) solutions by a modi ed version the Hopf boundary point Lemma and some maximum principle (yielding a similar, but more general result as Proposition 10.2 in [19]).
The same also holds for other works which treat uniqueness issues in the context of degenerate PDEs, but are not adapted to the very speci c class of solutions at hand. This includes the extensive work by Feehan [12] where -amongst other issues -the uniqueness of solutions of elliptic PDE whose di erential operator degenerates along a certain portion of the boundary ∂ 0 of the domain is established: for a problem with a partial Dirichlet boundary condition, where the boundary data are only given on ∂ \∂ 0 , a so-called secondorder boundary condition is applied for the degenerate boundary area; this condition says that a solution needs to be such that the leading terms of the di erential operator continuously vanishes towards ∂ 0 , while the solution itself is also of class C 1 up to ∂ 0 . Within this framework, Feehan than shows that -under certain technical assumptions -degenerate operators satisfy a corresponding maximum principle for the partial boundary condition, which assures the uniqueness of a solution. Although this in principle may also apply to solutions of Wright-Fisher di usion equations, this does not entirely cover the situation at hand, since, if n ≥ 2, L * only partially degenerates towards the boundary (instances of codimension 1). More precisely, its degeneracy behaviour is stepwise, corresponding to the strati ed boundary structure of the domain n , and hence does not satisfy the requirements for Feehan's scenario. Furthermore, in the language of [12], the intersection of the regular and the degenerate boundary part ∂∂ 0 , would encompass a hierarchically iterated boundarydegeneracy structure, which is beyond the scope of that work. Therefore, in this paper, we continue the detailed investigation of the boundary behavior of solutions of the (extended) Kolmogorov backward equation (1.1) started in [19] (the concept of solution developed there expands a notion of solution which is well-known in the literature (cf. [27,29])). In analytical terms, the issue is the regularity of solutions at singularities of the boundary, that is, where two or more faces of the simplex n meet. When considering particular extension paths from the boundary into the interior of the simplex (they have nothing to do, however, with Kingman's coalescent lines of descent as utilized in some of the literature discussed above), these may result in boundary singularities at certain strata of the boundary of the domain, and we are interested in the directions in which the singularities of the boundary of the simplex are approached from the interior, because we want to resolve these boundary singularities.
In contrast to the approaches discussed above that invoke strong tools from the theory of stochastic processes, our approach is not stochastic, but analytic and geometric in nature, which means that the spirit of our approach is rather related to that of [5,6]. In contrast to that approach, however, we develop geometric constructions, within the framework of information geometry, that is, the geometry of probability distributions, see [1,2], in order to have an approach that on one hand is naturally capable of studying such generalizations as indicated above, but on the other hand can still derive explicit formulas. This is part of a general research program, see [17][18][19][35][36][37][38]. The biological interpretation provides a key to some of our technical arguments. Alleles can disappear from the population in di erent order. We work on the n-dimensional probability simplex, which represents the relative frequencies of (n+1) alleles in a population. Its k-dimensional boundary faces F 1 , F 2 represent population states with only (k+1) alleles. When two such faces meet in a (k−1)-dimensional subface F 0 , we can approach each point in F 0 from either F 1 or F 2 , but this then corresponds to di erent orders of the loss events. Therefore, the resulting limits might a priori be di erent. Therefore, there arises the issue of continuity of the solution of our process on faces of codimension 2 or higher in the boundary of the probability simplex. And for such discontinuous solutions, the maximum principle does not apply to show uniqueness. For this reasons, we need to combine biologically correct continuity and extension assumptions with a careful blow-up process that resolves the remaining ambiguities. This constitutes the main technical achievement of this paper.
Let us now describe in more speci c terms what we achieve in this paper. Based on the previous work [19], we continue the analysis of solutions of the (extended) Kolmogorov backward equation for the di usion approximation of the Wright-Fisher model for U( · , t) ∈ C 2 p n for each xed t ∈ (−∞, 0) and U(p, · ) ∈ C 1 ((−∞, 0)) for each xed p ∈ n resp. the stationary (extended) Kolmogorov backward equation is the corresponding backward operator.
Emerging solutions of the backward Kolmogorov equation may be interpreted as probability distributions over ancestral states yielding some given current state of allele frequencies with time running backward as indicated by the name. Such an ancestral state could have possessed more alleles than the current state, as on the path towards that latter state, some alleles that had been originally present in the population could have been lost. In analytical terms, one could assume that such a loss of allele event is continuous, in the sense that the relative frequency of the corresponding allele simply goes to 0. Geometrically, however, this means that the process from the interior of a probability simplex enters into some boundary stratum and henceforth stays there. Also, when two or more alleles got lost, they could have disappeared in di erent orders from the population. A corresponding global and hierarchical solution for the Kolmogorov backward equation that persists and stays regular across di erent such loss of allele events in the past was constructed in the preceding paper [19] (Propositions 8.1 f.), which was technically rather involved: Proposition (pathwise extension of solutions, informal version of Proposition 3.2). Let k, n ∈ N with 0 ≤ k < n, and let u I k be a proper solution of the Kolmogorov backward equation (1.9) restricted to and a global extension U i k ,...,i n I k in k≤d≤n being an analogous extension of the nal condition f = f I k in (I k ) k ; in particular, we have U i k ,...,i n I k k .
This result allows us to extend any solution of the Kolmogorov backward equation that lives on a certain stratum ("proper solution") to all corresponding strata of higher dimension of some larger domain. The obtained extension then solves the analogous problem in the entire larger domain, for a nal condition which is likewise an extension of the original nal condition. In terms of the Wright-Fisher model, where a proper solution models ancestral states of a certain set of alleles over time, this scheme yields corresponding ancestral states on all those sets from which the original set can be reached by a (multiple) loss of alleles. These states are again modelled over time; the nal condition which is met by the solution corresponds to an analogous extension of the original target set which spans all relevant higher-dimensional strata. The scheme is versatile and can be applied for all potential losses of alleles.
The result improves results in the literature (cf. [27,29], and is indispensable for a complete understanding and a rigorous solution of the Kolmogorov backward equation. The present paper completes this approach by establishing the uniqueness for this class of solution in the stationary case. The key is the degeneracy at the boundary of the Kolmogorov equations. While from an analytical perspective, this presents a profound di culty for obtaining boundary regularity of the solutions of the equations, from a biological or geometric perspective, this is very natural because it corresponds to the loss of some alleles from the population in nite time by random dri . And from a stochastic perspective, this has to happen almost surely. For this reason, the above equations are not accessible by standard theory (cf. e.g. [30]), because the square root of the coe cients of the second order terms of L * is not Lipschitz continuous up to the boundary. As a consequence, in particular the uniqueness of solutions to the above Kolmogorov backward equations may not be derived from standard results. Instead, such degenerate equations arising from population biology have been analyzed by Epstein and Mazzeo (cf. [5,6]) only recently. While their aim was to develop a general and widely applicable theory, we rather focus on the speci c properties of the Wright-Fisher model to obtain results that do not readily follow from the general theory. We shall derive the regularity and uniqueness of a certain class of solutions that are the hierarchically extended solutions of the Kolmogorov backward equation developed in [19].
Our aim is the global regularity in the closure of the domain, and this will be achieved by resolving any incompatibilities between di erent boundary strata. For that purpose, we shall construct an appropriate transformation of the relevant part of the domain (i.e. the simplex n , cf. below). As a result, we can work on a domain that is a product of a simplex and a cube. Thereby, the iteratively extended solutions are turned into corresponding solutions of the transformed equation, which are then of su cient global regularity; in particular, they are globally continuous. For generic iteratively extended solutions this does not yet yield a corresponding regularity. However, the transformation scheme is still applicable, and the transformation image may be extended that way as well (see 7.3 (iii)). In any case, before such a transformation, a solution may be highly irregular, and certainly not globally continuous on the closed simplex.
In the stationary case, such transformed solutions are uniquely de ned by their values on the vertices of the domain (analogously to a globally continuous solution of the original problem in n , cf. Section 6). It just needs to be shown that a complete set of boundary data on the simplex generates su cient boundary data on the larger domain which is produced by the blow-up.
Theorem (informal version of Theorem 7.3 on p. 481). Let n ∈ N + , i 0 ∈ {0, . . . , n} and a solution u {i 0 } : −→ R be given. Then an extension U of u {i 0 } to the entire simplex (cf. Proposition 3.2) is unique within the class of all extensions U which satisfy certain ' extension constraints' and additional boundary regularity of the blow-up image if n ≥ 2.

The simplex
We want to consider relative frequencies (of alleles) in a population, and therefore, we shall use the probability simplex as the corresponding state space. In this subsection, we shall introduce a suitable simplex notation as well as the appropriate function spaces (see also [18]).
Let p 0 , p 1 , . . . , p n denote the relative frequencies of alleles 0, 1, . . . , n. As we have n which is the (open) n-dimensional standard orthogonal simplex The closure of this simplex is In order to include the time parameter t ∈ (−∞, 0], we also write The boundary of the simplex ∂ n = n \ n consists of boundary strata, the faces, which are (sub-)simplices themselves, from the (n − 1)-dimensional facets down to the 0-dimensional vertices. Each subsimplex of dimension k ≤ n − 1 is isomorphic to the k-dimensional standard orthogonal simplex k . To denote a particular subsimplex, we introduce index sets 1 The index set I n may be omitted, thus n = (I n ) n . The index 0 corresponding to p 0 plays an important role: On Each of the ( n+1 k+1 ) subsets I k of I n corresponds to a boundary face For notational consistency, we also put ∂ n n = n . This boundary concept can iteratively be applied to simplices in the boundary of some Regarding the Wright-Fisher model, the simplex corresponds to the state where precisely the k + 1 alleles i 0 , . . . , i k are present in the population. The boundary ∂ k n , i.e. the union of all corresponding subsimplices, represents the state with any k + 1 alleles. In the set of alleles i 0 , . . . , i k corresponding to , the elimination of one of the alleles corresponds to a transition to ∂ k−1 , and the particular component in that boundary then indicates which of the alleles got eliminated.
We also introduce spaces of square integrable functions for our subsequent integral products on n and its faces (which will mainly be used implicitly, for details cf. [37]) 2 , (2.8) In order to de ne an extended solution on n and its faces (indicated by a capitalized U), we shall in addition need appropriate spaces of pathwise regular functions. Such a solution needs to be at least of class C 2 in every boundary stratum (actually, a solution typically always is of class C ∞ , which likewise applies to each boundary stratum). Moreover, it should stay regular at boundary transitions that reduce the dimension by one, i.e. for k . Thus, derivatives in the interior which are tangential towards the face under consideration have to smoothly connect with derivatives in that stratum, whereas for the normal derivatives, there is a smooth extension on the stratum. 2 Here, λ k stands for the k-dimensional Lebesgue measure, but when integrating over some k with 0 / ∈ I k , the measure needs to be replaced with the one induced on k by the Lebesgue measure of the containing R k+1this measure, however, will still be denoted by λ k as it is clear from the domain of integration k with either 0 ∈ I k or 0 / ∈ I k which version is actually used.
Turning from some speci c boundary transition to the entire simplex, we may require that a corresponding property applies to all possible boundary transitions within n that reduce the dimension by one, thus formally for d and its boundary of codimension 1 ∂ d−1 d ; still, a solution needs to be well-de ned everywhere. However, the class of solutions, which comply with such pathwise regularity even on the (entire) strati cation of the domain, is certainly larger than that of plain globally smooth solutions.
Correspondingly, we de ne for l ∈ N ∪ {∞} with respect to the spatial variables. Likewise, for ascending chains of (sub-)simplices with a more speci c boundary condition, we put for index sets I k ⊂ · · · ⊂ I n and again for l ∈ with respect to the spatial variables; χ denotes the characteristic function of a set, i.e. = 1 there and 0 elsewhere.

The cube
We next introduce some notation for cubes and their boundary instances and de ne for n ∈ N an n-dimensional cube n as n := (p 1 , . . . , p n ) p i ∈ (0, 1) for i = 1, . . . , n . (2.11) Analogous to n , if we wish to denote the corresponding coordinate indices explicitly, this may be done by providing the coordinate index set I ′ n := {i 1 , . . . , i n } ⊂ {1, . . . , n}, i j = i l for j = l as upper index of n , thus This is particularly useful for boundary instances of the cube (cf. below) or if for other purposes a certain ordering (i j ) j=0,...,n of the coordinate indices is needed. For n itself and if no ordering is needed, the index set may be omitted (in such a case it may be assumed I ′ n ≡ {1, . . . , n} as in equation (2.11)). Please note that a primed index set is always assumed to not contain the index 0 (resp. i 0 = 0, which we usually stipulate in case of orderings) as in (2.11), we do not have the coordinate index 0.
In the standard topology on R n , n is open (which we always assume when writing n ), and its closure n is given by (again using the index set notation) As in the case of the simplex, the boundary ∂ n of n consists of various subcubes (faces) of descending dimensions, starting from the (n − 1)-dimensional facets down to the vertices (which represent 0-dimensional cubes). All appearing subcubes of dimension 0 ≤ k ≤ n − 1 are isomorphic to the k-dimensional standard cube k and hence will be denoted by k if it is irrelevant or clear from the context which particular subcube we consider. We also put down until (∅) 0 := (0, . . . , 0) for k = 0. If necessary, we may also identify a certain boundary face k of ∂ n for 0 ≤ k ≤ n − 1 by only giving the values of the n − k xed coordinates, i.e. with indices in I ′ n \I ′ k , which may be either 0 or 1, hence with j 1 , . . . , j n−k ∈ I ′ n , i r = i s for r = s and b 1 , . . . , b n−k ∈ {0, 1} chosen accordingly. If we wish to indicate the total k-dimensional boundary of n , i.e. the union of all k-dimensional faces belonging to n , we may write ∂ k n for k = 0, . . . , n with ∂ n n := n .
Finally, when writing products of simplices and cubes which do not span all considered dimensions, we indicate the value of the missing coordinates by curly brackets marked with the corresponding coordinate index, i.e. for I n = {i 0 , i 1 , . . . , i n } and I k ⊂ I n with i k+1 / ∈ I k we have e.g.
If coordinates are xed at 0, the corresponding entry may be omitted, e.g. we may just write Furthermore, we also introduce a (closed) cube For functions de ned on the cube, the pathwise smoothness required for an application of the corresponding Kolmogorov backward operator (cf. p. 463) may be de ned as with the simplex in equality (2.9) in [19]; hence, we put with respect to the spatial variables, implying that the operator is continuous at all boundary transitions within n . This concept likewise applies to subsets of n where needed.

Hierarchical extended solutions of the Kolmogorov backward equation
In this section, we recall the main results from [19]; for details please see also there. We de ne a class of extensions by imposing extension constraints (de nition 6.1 in [18]); more precisely, an extension is required to be smooth and constrained to vanish towards certain boundary strata:

De nition 3.1 (extension constraints). Let I d be an index set with |I
a functionū : d is said to be an extension of u satisfying the extension constraints if (i) for t < 0ū( · , t) is continuously extendable to the boundary ∂ d−1 resp. vanishes on the remainder of ∂ d−1 d and is of class C ∞ with respect to the spatial variables in with p 0 = 1 − i∈I d \{0} p i and π i k ,...,i d (p) = (p 1 , . . . ,p n ) such thatp i k = p i k + . . . + p i d , p i k+1 = . . . =p i d = 0 andp j = p j for j ∈ I d \{i k , . . . , i d }.
We combine all extensions into a function U i k ,...,i n I k in k≤d≤n with respect to the spatial variables for t < 0 as well as in C ∞ ((−∞, 0)) with respect to t, and we have being an analogous extension of the nal condition f = f I k in (I k ) k ; in particular, we have U i k ,...,i n I k k .

Motivation
To motivate the regularization scheme, we use the example of U d−2 for d = n, . . . , k + 2. This implies that the desired transformation needs to a ect all relevant dimensions in an iterative manner. In each step, one dimension from the simplex is removed and converted into a dimension of the corresponding cube component, i.e. the corresponding coordinate is released from the simplex property i p i ≤ 1. In doing so, with each iteration, the relevant component of the solution gains the required regularity at the corresponding level while all other components remain una ected; eventually, the entire solution resp. all its components are transformed such that they extend smoothly to the boundary.
Altogether, a er n − k − 1 of these steps, the relevant component of (I n ) n is converted into a cube of dimension n − k − 1, and the transformed solution is su ciently regularized; in particular, it will then smoothly extend to the full boundary.

Analysis of a simple example
A crucial aspect of our procedure is the resolution of the singularities that appear with those iteratively extended solutions. This will be done by suitable blow-up transformations. To give an example, we use a solution for n = 2 (cf. Proposition 3.2): we then have e.g.
and of course only the top-dimensional component resp. its continuous extension yields incompatibilities. Hence, we may transform 3 it viap 1 := p 1 + p 2 andp 2 := , which is produced during the transformation (cf. below). As pointed out above, for greater n, we have to recursively apply such transformations in order to resolve all appearing singularities.

The blow-up transformation and its iteration
We shall now present the details of the blow-up transformation and derive all necessary results. We start with the basic transformation (cf. also Figure 1) and proceed to the results for a suitably iterated application of this blow-up transformation, by which we can resolve all singularities of our solution. with p 0 := 1 − i∈I d \{0} p i C ∞ -di eomorphically onto withp 0 := 1 − i∈I d \{0,s}p i and altogether including their boundaries are mapped as follows: are interchanged. Thus, unless stated otherwise, in the following we shall always assume that thep s -coordinate is chosen with an orientation as given in Lemma 5.1.
That we putp s := 0 where p r + p s = 0 in Lemma 5.1 (cf. second line of equation 5.5) may appear somewhat surprising as the set p r + p s = 0 li s under the blow-up to the new boundary face, and the range ofp s on that locus is what gives this subset a larger dimension. However, the main purpose is to be able to locate the set a er the blow-up (which makes the blow-up well-de ned). In the remainder, it will be crucial to identify all strata of the domain and the data/solution given on them a er the blow-up.
Proof of Lemma 5.1. The transformation corresponds geometrically to a scaling of the domain into thep s -direction with scaling factor 1 p r . The assertion about the transformation domains is straightforward since we have 0 ≤ p s p r +p s ≤ 1 on Likewise, the C ∞ -di eomorphism property follows since r s is smoothly di erentiable as long as p r = p r + p s > 0 and the inverse transformation ( r s ) −1 is likewise smooth. The latter is given by By this, it also becomes obvious that ( r s ) −1 maps Consequently, we obtain for an iterated application of the blow-up transformation: and altogether The n − k − 1 additional (n − 1)-dimensional faces N k+1 , . . . , N n−1 of n−k−1 are given by Explicitly, r n−k−1 s n−k−1 • . . . • r 1 s 1 is given bỹ p i 2 + · · · + p i n p i 1 + p i 2 + · · · + p i n for p i 1 + · · · + p i n > 0 0 for p i 1 + · · · + p i n = 0, (5.20) . . .
p i j + · · · + p i n p i j−1 + p i j + · · · + p i n for p i j−1 + · · · + p i n > 0 0 for p i j−1 + · · · + p i n = 0, . . . Proof. The Proposition will be proven in parallel with Propositions 5.7 and 5.8, cf. below. The next Lemma is concerned with the transformation behaviour of the operator L * n , at rst for a single blow-up step. All considerations apply to L * n in its domain n as well as, taking the restriction property of L * n (cf. [19]) into account, in the closure n resp. to the transformed operator L * n in the subsequent transformed images of the domain (the domain in question may not be stated explicitly -this will be done in Proposition 5.7): Lemma 5.5. Let I ′ n := {1, . . . , n} be an index set with r, s ∈ I ′ n and let {i 1 , . . . , i n } be an ordering of I ′ n such that r, s ∈ {i 1 , . . . , i m } for some m ≤ n. When changing coordinates (p i ) i∈I ′ n → (p i ) i∈I ′ n by r s , the operator with a ij (p) = p i (δ i j − p j ) for i, j ∈ {i 1 , . . . , i m }, a ij = 0 else for i = j is transformed into For the proof, we need the following lemma: transforms under a change of the spatial coordinates −→ , p −→p into n k,l=1ã Proof of Lemma 5.5. When changing coordinates (p i ) → (p i ), the coe cients of the 2nd order derivatives a ij transform asã while we may get additional rst order terms with coe cients i,j a ij ∂ 2pk ∂p i ∂p j (cf. Lemma 5.6). For the transformation at hand, we have (cf. equations (5.6) and (5.5)) and (cf. equation (5.7)) Therefore, (5.32) yieldsã for k, l = s, that is,ã kl (p) = a kl (p) + a kt (p)δ l r + a sl (p)δ k r + a ss (p)δ k r δ l r = p k (δ k l − p l ) − p k p s δ l r − p s p l δ k r + p s (1 − p s )δ k r δ l r =p k (δ k l −p l )  For an iterated application of the blow-up transformation, we obtain thus: Proposition 5.7. In the setting of a full blow-up transformation as in Proposition 5.4, the If in any step the coordinatep s j is chosen with alternative orientation (cf. remark 5.3),p s j , whenever it appears in the above formulae, is replaced by (1 −p s j ).
Thus, the iterated blow-up translates the (extended) Kolmogorov backward equation in n into a corresponding di erential equation in (1 −p i j ) for d = k + 2, . . . , n (5.47) 5 Please note that on boundary instances of (I n \I k+1 ) n−k−1 , i.e.p i l = 0 for some l ∈ I n \I k+1 , the corresponding summands are assumed not to appear in the right sum in equation (5.44), which may be interpreted as a result of a successive restriction. The given domain is the maximal domain for the operator as it is not de ned on the exception set n−1 j=k+1 N j (however, cf. also Lemma 7.1 for the stationary case). withπ i k+1 (p i j ) :=p i j for i j ∈ I k+1 ,π i k−1 (p i j ) := 0 else. The transformed functions u i k ,i k+1 ;i k+2 ,...,i d I k smoothly extend to respectively; consequently also U i k ,i k+1 ;i k+2 ,...,i n I k smoothly extends to If in any step the coordinatep s j is chosen with alternative orientation (cf. remark 5.3),p s j in the above formulae needs to be replaced by (1 −p s j ).
For the stationary components, we have in particular: while in accordance with Proposition 5.4 the domain is mapped and altogether The n − 1 additional (n − 1)-dimensional faces N 1 , . . . , N n−1 of ∂ (I ′ n ) n are given by Proof of Propositions 5.4-5.8. We prove the assertions of the three Propositions in parallel: Our aim is to transform U i k ,...,i n I k into a function that does not feature any incompatibilities and hence is su ciently regular with respect to the entire closure of the (transformed) domain. For that purpose, we shall show that the full blow-up via a repeated application of the coordinate transformation r s of Lemma 5.1 with the indices r and s to be picked as shown in each step yields the desired result for U i k ,i k+1 ;i k+2 ,...,i n I k , while the transformation behaviour of the domain and the operator is as stated in Proposition 5.4. For notational simplicity, we will usually suppress the t-component in the notation for our domains throughout this proof; for instance, we shall write (I n ) n instead of (I n ) n −∞ . Starting with the top-dimensional component of U i k ,...,i n I k , which is u i k ,...,i n I k (p, t) =ū i k ,...,i n−1 I k (π i n−1 ,i n (p), t) · p i n−1 p i n−1 + p i n = u I k (π i k ,...,i n−1 (π i n−1 ,i n (p)), t) \N n−1 since we haveã kl (p) = p k (δ k l −p l ) =p k (δ k l −p l ) for k, l = i n−1 , i n . Ifp i n is chosen with alternative orientation (cf. remark 5.3), thenp i n needs to be replaced by (1 −p i n ) everywhere.
As already indicated, the transformed solution is still not smoothly extendable to the full boundary of the transformed domain: its (n − 2)-dimensional incompatibility is resolved, but its lower-dimensional incompatibilities persist. Thus, the highest-dimensional incompatibility now is of dimension n − 3, and hence the situation is ready for another application of the blow-up transformation.
Thus, we need an iterative procedure to resolve all incompatibilities. For this purpose, we assume that a er the m-th step (m = 1, . . . , n − k − 2) an already transformed function U i k ,...,i n−m ;i n−m+1 ,...,i n I k with (note that we again associate coordinates p resp.p etc. to the domain before/a er the (m + 1)-th transition; furthermore, we will use the conventionū i k I k ≡ u I k to simplify the notation) Furthermore, we assume that the operator L * has the corresponding form To analyze the transformation behaviour of the operator, we rst note that the requirements of Lemma 5.5 on a ij are met as for i, j ∈ {i 1 , . . . , i n−m } we have a ij (p) = p i (δ i j − p j ) by equation (5.69), while all other non-diagonal coe cients vanish. Hence, by the Lemma, we have for i, j ∈ {i 1 , . . . , i n−m }ã while forã i j i j with j = n − m + 1, . . . , n we obtaiñ Likewise,ã i n−m i n−m takes the form whereas all other coe cients vanish. Altogether, this yields Thus, a er the (m+1)-st blow-up step, the structure of domain, solution and operator is the same as before, just with the index m replaced by m + 1. Eventually, a er n − k − 1 blow-up steps domain, solution and operator have attained the asserted form of the corresponding statements. In particular, the remaining u I k as a proper solution smoothly extends to the entire boundary of (I k ) k , and hence so doesū k+1 , implying that eachũ i k ,i k+1 ;i k+2 ,...,i d I k smoothly extends to Proof of Corollary 5.9. In the given setting, we haveū i 0 ,i 1 n ) which is of class C ∞ p and vanishes on the mentioned faces.
For the proof, we trace the extendability of U towards the additional faces back to that of U in (I n ) n for approaching the incompatibilities -which will be accomplished by the next lemma. Note that in the following we will use a disjoint formulation of the additional faces by putting n−j .
n corresponds to d−1 , in particular its interior corresponds to Proof. To take account of the 'additional' faces N m of (I ′ n ) n produced during the blow-up transformations, we carry out the inverse of the full blow-up transformation of Proposition 5.4 (cf. equations (5.19)-(5.22)), yielding p i 1 := p i 1 + · · · + p i n , (5.87) p i 2 + · · · + p i n p i 1 + p i 2 + · · · + p i n for p i 1 + · · · + p i n > 0 0 for p i 1 + · · · + p i n = 0, . . .

The stationary Kolmogorov backward equation and uniqueness
When we ask for the long-term behaviour of the process, i.e. which alleles are eventually lost and in which order, we are lead to the stationary Kolmogorov backward equation. Solutions of this equation have already appeared implicitly in the preceding section as extensions of solutions in ∂ 0 n since the corresponding operator L * 0 has 0 as its only eigenvalue. Although we have already developed the extended setting presented in Section 3, we start by considering some interior simplex n , (resp. the corresponding restriction of an extended solution). Then, for a solution in n , we may argue again that all eigenmodes of the solution corresponding to a positive eigenvalue vanish for t → −∞, while those corresponding to the eigenvalue zero are preserved. This implies that a solution of the Kolmogorov backward equation (1.9) in n converges uniformly to a solution of the corresponding homogeneous or stationary Kolmogorov backward equation for u ∈ C 2 ( n ) and with boundary condition f (which needs to be attained smoothly in a suitable sense). At rst sight, this looks like a boundary value problem (for some suitably chosen boundary function f , assuring the uniqueness of a solution). However, as may be expected from the previous considerations, the role of the boundary here is di erent from usual boundary value problems and again requires some extra care: On the one hand, a proper solution in n always converges to the trivial stationary solution (≡ 0), whose (continuous) extension to the boundary also vanishes at all negative times. On the other hand, any solution which extends to ∂ n is already strongly constrained by the degeneracy behaviour of the di erential operator if suitable regularity assumptions on the solution in n (cf. also (2.9)) apply: Proof. We shall proceed iteratively: Assuming that L * k U = 0 for all (I k ) k ⊂ ∂ k n , we show that this property extends to each k , and hence we obtain L * k−1 U = 0 on ∂ k−1 n . A repeated application then yields (6.2). W. l. o. g. let with L * k−1 being the restriction of L * k to k−1 . We take some p ∈ (I k−1 ) k−1 and choose a sequence (p l ) l∈N in (I k ) k with p l → p and apply this operator to U at p l ∈ (I k ) k . The resulting expression inside the bracket is (cf. equation (8.8) in [19]). Considering an arbitrary correspondingly transformed stationary Kolmogorov backward equation on the cube (which is basically analogous to the simplex, cf. the preceding considerations). The eventual result will be obtained by applying the uniqueness result for the cube to the transformed iteratively extended solutions (assuming su cient regularity if necessary). Again, this is limited to the stationary components.

The uniqueness of solutions of the stationary Kolmogorov backward equation
The main application of the blow-up scheme is the uniqueness proof for the iteratively extended solutions of the Kolmogorov backward equation that satisfy the extension constraints 3.1. In the present paper, this is limited to the stationary components. First, we will discuss the uniqueness of solutions of the correspondingly transformed stationary Kolmogorov backward equation on the cube (which is basically analogous to the simplex, cf. section 10 in [19]). A er that, the main result will be stated by applying the uniqueness result for the cube to the transformed iteratively extended solutions (assuming su cient regularity if necessary).
Regarding the uniqueness of stationary solutions on the cube with the transformed Kolmogorov backward operator given by equation (5.54) and with extension U ∈ C ∞ p ( n ), we have L * U = 0 in n , (7.2) i.e. Proof. The statement is proven iteratively: Assuming that equation (7.3) holds in some (arbitrary) domain d+1 ⊂ ∂ d+1 n , we show that a corresponding formula also holds for any d ⊂ ∂ d d+1 ⊂ ∂ d n . A repeated application of the argument then yields the assertion. Let d+1 = p i 1 = b 1 , . . . ,p i n−d−1 = b n−d−1 and d = p i 1 = b 1 , . . . ,p i n−d = b n−d with i n−d = i 1 , . . . , i n−d−1 and b n−d ∈ {0, 1}. If we have i n−d <î(d + 1), then as p i n−d → 0 resp.p i n−d → 1, the value of the operator in equation (7.3) applied to U -with the occurring derivatives and the coe cients being continuous -depends continuously onp up to the boundary. Thus equation (7.3), which already has the corresponding form for d (notê ı(d) ≡î(d + 1)), also holds on d .
If we rather have i n−d >î(d + 1) and b n−d = 1, then, when choosing somep ∈ d and a sequence (p l ) l∈N in d+1 withp l →p, the expression 1 2p with L * as in equation (7.3) is uniquely determined by its values on ∂ 0 n .
Proof. The uniqueness may be shown by a successive application of the maximum principle for strata of increasing dimension, starting from ∂ 0 n . We shall rst show that in every instance of the domain d ⊂ ∂ d n for all 1 ≤ d ≤ n, the solution U| d is uniquely de ned by its values on ∂ d : If equation (7.3) on d comprises d derivative terms, it is straightforward to see that the operator is locally uniformly elliptic on d , hence satis es the assumptions of Hopf 's maximum principle. Since the solution is of class C 0 on d , this yields the desired uniqueness. If, in contrast, d is such that the operator on d comprises only d ′ < d derivative terms, we rst show the uniqueness on each d ′ -dimensional bre of d (with corresponding boundary part), which follows from an analogous consideration: Clearly, on each bre the operator is locally uniformly elliptic and the solution is continuous up to the respective boundary, thus the uniqueness likewise follows from Hopf 's maximum principle. By assembling, the desired uniqueness is then obtain for the entire d . Applying these considerations successively for ∂ 0 n , . . . , ∂ n n = n yields the desired global uniqueness.
With the blow-up scheme at hand, the preceding uniqueness result may also be conveyed to the simplex n , assuming some additional regularity for the transformation image (which still does not imply an analogous regularity for the pre-image). We nally arrive at: Consequently, also the global extension U {i 0 } as in Proposition 8.4 in [19] or in Theorem 3.3 is unique.
Proof. The assertion for the trivial case n = 1 directly follows, as U i 0 ,i 1 {i 0 } is already su ciently regular in (I 1 ) for an application of the maximum principle: In particular, it is globally continuous, and along with the locally uniform ellipticity of the Kolmogorov backward operator, the uniqueness follows. For n ≥ 2, any function U which is a solution of the n with L * as in equation (7.3). Hence, the uniqueness result of Proposition 7.2 applies and proves the uniqueness of the transformed function (and, in view of the injectivity of the blow-up, also the uniqueness of U) -for speci ed boundary data on the entire ∂ 0 n . Thus, we only need to show that these boundary data are uniquely determined by the assumptions made. This is straightforward: In accordance with Lemma 5.10, U or its corresponding continuous extension vanishes on {p i j = 1} ⊂ ∂ (I ′ n ) n , j = 1, . . . , n. As by assumption (iii) the continuous extendability applies to the entire (I ′ n ) n , U resp. its extension even vanishes on p i 1 = 1 , . . . , p i n = 1 . n ) ∩ C 0 (I ′ n ) n (this may be seen directly from equation (5.49)), it also is the unique extension.