Renormalization analysis of catalytic Wright-Fisher diffusions

Recently, several authors have studied maps where a function, describing the local diffusion matrix of a diffusion process with a linear drift towards an attraction point, is mapped into the average of that function with respect to the unique invariant measure of the diffusion process, as a function of the attraction point. Such mappings arise in the analysis of infinite systems of diffusions indexed by the hierarchical group, with a linear attractive interaction between the components. In this context, the mappings are called renormalization transformations. We consider such maps for catalytic Wright-Fisher diffusions. These are diffusions on the unit square where the first component (the catalyst) performs an autonomous Wright-Fisher diffusion, while the second component (the reactant) performs a Wright-Fisher diffusion with a rate depending on the first component through a catalyzing function. We determine the limit of rescaled iterates of renormalization transformations acting on the diffusion matrices of such catalytic Wright-Fisher diffusions.


Introduction and main result
Several authors [BCGdH95,BCGdH97,dHS98,Sch98,CDG04] have studied maps where a function, describing the local diffusion matrix of a diffusion process, is mapped into the average of that function with respect to the unique invariant measure of the diffusion process itself. Such mappings arise in the analysis of infinite systems of diffusion processes indexed by the hierarchical group, with a linear attractive interaction between the components [DG93a, DG96,DGV95]. In this context, the mappings are called renormalization transformations. We follow this terminology. For more on the relation between hierarchically interacting diffusions and renormalization transformations, see Appendix A.1.
Formally, such renormalization transformations can be defined as follows.
Definition 1.1 (Renormalization class and transformation) Let D ⊂ R d be nonempty, convex, and open. Let W be a collection of continuous functions w from the closure D into the space M d + of symmetric non-negative definite d × d real matrices, such that λw ∈ W for every λ > 0, w ∈ W. We call W a prerenormalization class on D if the following three conditions are satisfied: (i) For each constant c > 0, w ∈ W, and x ∈ D, the martingale problem for the operator A c,w x is well-posed, where w ij (y) ∂ 2 ∂y i ∂y j f (y) (y ∈ D), (1.1) and the domain of A c,w x is the space of real functions on D that can be extended to a twice continuously differentiable function on R d with compact support.
(ii) For each c > 0, w ∈ W, and x ∈ D, the martingale problem for A c,w x has a unique stationary solution with invariant law denoted by ν c,w x .
If W is a prerenormalization class, then we define for each c > 0 and w ∈ W a matrix-valued function F c w on D by We say that W is a renormalization class on D if in addition: (iv) For each c > 0 and w ∈ W, the function F c w is an element of W.
If W is a renormalization class and c > 0, then the map F c : W → W defined by (1.2) is called the renormalization transformation on W with migration constant c. In (1.1), w is called the diffusion matrix and x the attraction point. ✸ Remark 1.2 (Associated SDE) It is well-known that D-valued (weak) solutions y = (y 1 , . . . , y d ) to the stochastic differential equation (SDE) In the present paper, we concern ourselves with the following renormalization class on [0, 1] 2 . and set W l,r cat := {w α,p : α > 0, p ∈ H l,r } (l, r = 0, 1). ✸ By Remark 1.2, solutions y = (y 1 , y 2 ) to the martingale problem for A c,w α,p x can be represented as solutions to the SDE (i) dy 1 (1.7) We call y 1 the Wright-Fisher catalyst with resampling rate α and y 2 the Wright-Fisher reactant with catalyzing function p.
Conditions (1.10) (i) and (ii) are satisfied, for example, for c k = (1 + γ * ) −k . Note that the functions p * 0,0,γ * and p * 1,1,γ * are independent of γ * ≥ 0. We believe that on the other hand, p * 0,1,γ * is not constant as a function of γ * , but we have not proved this. The function p * 0,1,0 is the unique nonnegative solution to the equation · ∞ . We let B(E) denote the space of all bounded Borel measurable real functions on E. We write C + (E) and C [0,1] (E) for the spaces of all f ∈ C(E) with f ≥ 0 and 0 ≤ f ≤ 1, respectively, and define B + (E) and B [0,1] (E) analogously. We let M(E) denote the space of all finite measures on E, equipped with the topology of weak convergence. The subspaces of probability measures is denoted by M 1 (E). We write N (E) for the space of finite counting measures, i.e., measures of the form ν = m i=1 δ x i with x 1 , . . . , x m ∈ E (m ≥ 0). We interpret ν as a collection of particles, situated at positions x 1 , . . . , x m . For µ ∈ M(E) and f ∈ B(E) we use the notation µ, f := E f dµ and |µ| := µ(E). By definition, D E [0, ∞) is the space of cadlag functions w : [0, ∞) → E, equipped with the Skorohod topology. We denote the law of a random variable y by L(y). If y = (y t ) t≥0 is a Markov process in E and x ∈ E, then P x denotes the law of y started in y 0 = x. If µ is a probability law on E then P µ denotes the law of y started with initial law L(y 0 ) = µ. For time-inhomogeneous processes, we use the notation P t,x or P t,µ to denote the law of the process started at time t with initial state y t = x or initial law L(y t ) = µ, respectively. We let E x , E µ , . . . etc. denote expectation with respect to P x , P µ , . . ., respectively.

Renormalization classes on compact sets 2.1 Some general facts and heuristics
In this section, we explain that our main result is a special case of a type of theorem that we believe holds for many more renormalization classes on compact sets in R d . Moreover, we describe some elementary properties that hold generally for such renormalization classes. The proofs of Lemmas 2.1-2.8 can be found in Section 5.1 below.
In particular, x → ν c,w x is a continuous probability kernel on D, and F c w ∈ C(D, M d + ) for all c > 0 and w ∈ W. Recall from Definition 1.1 that λw ∈ W for all w ∈ W and λ > 0. The reason why we have included this assumption is that it is convenient to have the next scaling lemma around, which is a consequence of time scaling. (ii) F λc (λw) = λF c w (λ, c > 0, w ∈ W, x ∈ D). (2.1) The following simple lemma will play a crucial role in what follows. the effective boundary of D (associated with w). If y is a solution to the martingale problem for the operator d i,j=1 w ij (y) ∂ 2 ∂y i ∂y j (i.e., the operator in (1.1) without the drift), then, by martingale convergence, y t converges a.s. to a limit y ∞ ; it is not hard to see that y ∞ ∈ ∂ w D a.s. The next lemma says that the effective boundary is invariant under renormalization.
Lemma 2.4 (Invariance of effective boundary) One has ∂ Fcw D = ∂ w D for all w ∈ W, c > 0.
For example, for diffusion matrices w from the renormalization class W = W cat , there occur four different effective boundaries, depending on whether w ∈ W 1,1 cat , W 0,1 cat , W 1,0 cat , or W 0,0 cat . These effective boundaries are depicted in Figure 1. The statement from Theorem 1.4 (a) that F c (W l,r cat ) ⊂ W l,r cat is just the translation of Lemma 2.4 to the special set-up there. From now on, let W be a renormalization class, i.e., W satisfies also condition (iv) from Definition 1.1. Fix a sequence of (positive) migration constants (c k ) k≥0 . By definition, the iterated probability kernels K w,(n) associated with a diffusion matrix w ∈ W (and the constants (c k ) k≥0 ) are the probability kernels on D defined inductively by The next lemma follows by iteration from Lemmas 2.1 and 2.3. It their essence, this lemma and Lemma 2.6 below go back to [BCGdH95].
Lemma 2.5 (Basic properties of iterated probability kernels) For each w ∈ W, the K w,(n) are continuous probability kernels on D. Moreover, for all x ∈ D, i, j = 1, . . . , d, and n ≥ 0, the mean and covariance matrix of K w,(n) x are given by (2.6) We equip the space C(D, M 1 (D)) of continuous probability kernels on D with the topology of uniform convergence (since M 1 (D) is compact, there is a unique uniform structure on M 1 (D) generating the topology). For 'nice' renormalization classes, it seems reasonable to conjecture that the kernels K w,(n) converge as n → ∞ to some limit K w, * in C(D, M 1 (D)). If this happens, then formula (2.6) (ii) tells us that the rescaled renormalized diffusion matrices s n F (n) w converge uniformly on D to the covariance matrix of K w, * . This gives a heuristic explanation why we need to rescale the iterates F (n) w with the scaling constants s n from (1.9) to get a nontrivial limit in (1.11). We now explain the relevance of the conditions (1.10) (i) and (ii) in the present more general context. If the iterated kernels converge to a limit K w, * , then condition (1.10) (i) guarantees that this limit is concentrated on the effective boundary: Lemma 2.6 (Concentration on the effective boundary) If s n −→ n→∞ ∞, then for any (dy)f (y) = 0. (2.7) In fact, the condition s n → ∞ guarantees that the corresponding system of hierarchically interacting diffusions with migration constants (c k ) k≥0 clusters in the local mean field limit, see [DG93a, Theorem 3] or Appendix A.1 below.
To explain also the relevance of condition (1.10) (ii), we observe that using Lemma 2.2, we can convert the rescaled iterates s n F (n) into (usual, not rescaled) iterates of another transformation. For this purpose, it will be convenient to modify the definition of our scaling constants s n a little bit. Fix some β > 0 and put (2.9) Using (2.1) (ii), one easily deduces that where γ n := 1 s n c n (n ≥ 0). (2.11) We can reformulate the conditions (1.10) (i) and (ii) in terms of the constants (γ n ) n≥0 . Indeed, it is not hard to check 1 that equivalent formulations of condition (1.10) (i) are: Since s n+1 /s n = 1 + γ n we see moreover that, for any γ * ∈ [0, ∞], equivalent formulations of condition (1.10) (ii) are: (2.13) If 0 < γ * < ∞, then, in the light of (2.10), we expect s n F (n) w to converge to a fixed point of the transformation F γ * . If γ * = 0, the situation is more complex. In this case, we expect the orbit s n F (n) w → s n+1 F (n+1) w → · · · , for large n, to approximate a continuous flow, the generator of which is (2.14) To see that the right-hand side of this equation equals the left-hand side if w is twice continuously differentiable, one needs a Taylor expansion of w together with the moment formulas (2.2) for ν . Under condition condition (2.12) (iii), we expect this continuous flow to reach equilibrium.
In the light if these considerations, we are led to at the following general conjecture.
Conjecture 2.7 (Limits of rescaled renormalized diffusion matrices) Assume that s n → ∞ and s n+1 /s n → 1 + γ * for some γ * ∈ [0, ∞]. Then, for any w ∈ W, (2.16) We call (2.16) (ii), which is in some sense the γ * → 0 limit of the fixed point equation In particular, one may hope that for a given effective boundary, the equations in (2.16) have a unique solution. Our main result (Theorem 1.4) confirms this conjecture for the renormalization class W cat and for γ * < ∞. In the next section, we discuss numerical evidence that supports Conjecture 2.7 in the case γ * = 0 for other renormalization classes on compacta as well.
In previous work on renormalization classes, fixed shapes have played an important role. By definition, for any prerenormalization class W, a fixed shape is a subclassŴ ⊂ W of the formŴ = {λw : λ > 0} with 0 = w ∈ W, such that F c (Ŵ) ⊂Ŵ for all c > 0. The next lemma describes how fixed shapes for renormalization classes on compact sets typically arise.
(c) If the w * γ * for different values of γ * are not constant multiples of each other, then W contains no fixed shapes.
Note that by Theorem 1.4, W 0,1 cat is a renormalization class satisfying the general assumptions of Lemma 2.8. The unique solution of (2.16) (i) in W 0,1 cat is of the form w * = w 1,p * where p * = p * 0,1,γ * . We conjecture that the p * 0,1,γ * for different values of γ * are not constant multiples of each other, and, as a consequence, that W 0,1 cat contains no fixed shapes. Many facts and conjectures that we have discussed can be generalized to renormalization classes on unbounded D, but in this case, the second moments of the iterated kernels K w,(n) may diverge as n → ∞. As a result, because of formula (2.6) (ii), the s n may no longer be the right scaling factors to find a nontrivial limit of the renormalized diffusion matrices; see, for example, [BCGdH97].

Numerical solutions to the asymptotic fixed point equation
Let t → w(t, · ) be a solution to the continuous flow with the generator in (2.14), i.e., w is an M d + -valued solution to the nonlinear partial differential equation Solutions to (2.18) are quite easy to simulate on a computer. We have simulated solutions for all kind of diffusion matrices (including nondiagonal ones) on the unit square [0, 1] 2 , with the effective boundaries 1-6 depicted in Figure 2. For all initial diffusion matrices w(0, · ) we tried, the solution converged as t → ∞ to a fixed point w * . In all cases except case 6, the fixed point was unique. The fixed points are listed in Figure 2. The functions p * 0,1,0 and q * from Figure 2 are plotted in Figure 3. Here p * 0,1,0 is the function from Theorem 1.4 (c). The fixed points for the effective boundaries in cases 1,2, and 4 are the unique solutions of equation (1.12) (ii) from Theorem 1.4 in the classes W 1,1 cat , W 0,1 cat , and W 0,0 cat , respectively. The simulations suggest that the domain of attraction of these fixed points (within the class of "all" diffusion matrices on [0, 1] 2 ) is actually a lot larger than the classes W 1,1 cat , W 0,1 cat , and W 0,0 cat . case effective boundary fixed points w * of (2.18) The function q * from case 3 satisfies q * (x 1 , 1) = x 1 (1 − x 1 ) and is zero on the other parts of the boundary. In contrast to what one might perhaps guess in view of case 2, q * is not of the form q * (x 1 , x 2 ) = f (x 2 )x 1 (1 − x 1 ) for some function f .
Case 5 is somewhat degenerate since in this case the fixed point is not continuous. The only case where the fixed point is not unique is case 6. Here, m can be any positive definite matrix, while g * , depending on m, is the unique solution on (0, 1) 2 of the equation 1 + 1 2 2 i,j=1 m ij ∂ 2 ∂x i ∂x i g * (x) = 0, with zero boundary conditions.

Previous rigorous results
In this section we discuss some results that have been derived previously for renormalization classes on compact sets.
In order for the iterates in (2.19) to be well-defined, Theorem 2.9 assumes that a renormalization class W of diffusion matrices w on D with effective boundary {e 0 , . . . , e d } is given. The problem of finding a nontrivial example of such a renormalization class is open in dimensions greater than one. In the one-dimensional case, however, the following result is known.
Lemma 2.10 [DG93b] (Renormalization class on the unit interval) The set is a renormalization class on [0, 1].
About renormalization of isotropic diffusions, the following result is known. Below, ∂D := D\D denotes the topological boundary of D.
Theorem 2.11 [dHS98] (Universality class of isotropic models) Let D ⊂ R d be open, bounded, and convex and let m ∈ M d + be fixed and (strictly) positive definite. Set w * ij (x) := m ij g * (x), where g * is the unique solution of 1 + 1 2 ij m ij ∂ 2 ∂x i ∂x j g * (x) = 0 for x ∈ D and g * (x) = 0 for x ∈ ∂D. Assume that W is a renormalization class on D such that w * ∈ W and such that each w ∈ W is of the form for some g ∈ C(D) satisfying g > 0 on D and g = 0 on ∂D. Let (c k ) k≥0 be migration constants such that s n → ∞ as n → ∞. Then, for all w ∈ W, uniformly on D, The proof of Theorem 2.11 follows the same lines as the proof of Theorem 2.9, with the difference that in this case one needs to generalize the first moment formula (2.6) (i) in the now converges to the m-harmonic measure on ∂D with mean x, and this implies (2.22).
Again, in dimensions d ≥ 2, the problem of finding a 'reasonable' class W satisfying the assumptions of Theorem 2.11 is so far unresolved. The problem with verifying conditions (i)-(iv) from Definition 1.1 in an explicit set-up is that (i) and (ii) usually require some smoothness of w, while (iv) requires that one can prove the same smoothness for F c w, which is difficult.
The proofs of Theorems 2.9 and 2.11 are based on the same principle. For any diffusion matrix w, let H w denote the class of w-harmonic functions, i.e., functions h ∈ C(D) satisfying If w belongs to one of the renormalization classes in Theorems 2.9 and 2.11, then H w has the property that T c e., the operator in (1.1) without the diffusion part. In this case we say that w has invariant harmonics; see [Swa00]. As a consequence, one can prove that the iterated kernels satisfy D K w,(n) x (dy)h(y) = h(x) for all h ∈ H w and x ∈ D. If s n → ∞, then this implies that K w,(n) x converges to the unique H w -harmonic measure on ∂ w D with mean x. Diffusion matrices from W cat do not in general have invariant harmonics. Therefore, to prove Theorem 1.4, we need new techniques.
Note that in the renormalization classes from Theorems 2.9 and 2.11, the unique attraction point w * does not depend on γ * . Therefore, by Lemma 2.8, these renormalization classes contain a unique fixed shape, which is given by {λw * : λ > 0}.

Connection with branching theory
From now on, we focuss on the renormalization class W cat . We will show that for this renormalization class, the rescaled renormalization transformations F γ from (2.9) can be expressed in terms of the log-Laplace operators of a discrete time branching process on [0, 1]. This will allow us to use techniques from the theory of spatial branching processes to verify Conjecture 2.7 for the renormalization class W cat in the case γ * < ∞.

Poisson-cluster branching processes
We first need some concepts and facts from branching theory. Finite measure-valued branching processes (on R) in discrete time have been introduced by Jiřina [Jir64]. We need to consider only a special class. Let E be a separable, locally compact, and metrizable space. We call a continuous map Q from E into M 1 (M(E)) a continuous cluster mechanism. By definition, an M(E)-valued random variable X is a Poisson cluster measure on E with locally finite intensity measure µ and continuous cluster mechanism Q, if its log-Laplace transform satisfies For given µ and Q, such a Poisson cluster measure exists, and is unique in distribution, provided that the right-hand side of (3.1) is finite for f = 1. It may be constructed as X = i χ x i , where i δ x i is a (possibly infinite) Poisson point measure with intensity µ, and given x 1 , x 2 , . . ., the χ x 1 , χ x 2 , . . . are independent random variables with laws Q(x 1 , · ), Q(x 2 , · ), . . ., respectively. Now fix a finite sequence of functions q k ∈ C + (E) and continuous cluster mechanisms Q k (k = 1, . . . , n), define and assume that sup x∈E U k 1(x) < ∞ (k = 1, . . . , n). Then U k maps B + (E) into B + (E) for each k, and for each M(E)-valued initial state X 0 , there exists a (time-inhomogeneous) Markov chain (X 0 , . . . , X n ) in M(E), such that X k , given X k−1 , is a Poisson cluster measure with intensity q k X k−1 and cluster mechanism Q k . It is not hard to see that We call X = (X 0 , . . . , X n ) the Poisson-cluster branching process on E with weight functions q 1 , . . . , q n and cluster mechanisms Q 1 , . . . , Q n . The operator U k is called the log-Laplace operator of the transition law from X k−1 to X k . Note that we can write (3.4) in the suggestive form Here, if µ is an M(E)-valued random variable, then Pois(µ) denotes an N (E)-valued random variable such that conditioned on µ, Pois(µ) is a Poisson point measure with intensity µ.

The renormalization branching process
We will now construct a Poisson-cluster branching process on [0, 1] of a special kind, and show that the rescaled renormalization transformations on W cat can be expressed in terms of the log-Laplace operators of this branching process. By Lemma 5.4 below, for each γ > 0 and x ∈ [0, 1], the SDE has a unique (in law) stationary solution. We denote this solution by (y γ x (t)) t∈R . Let τ γ be an independent exponentially distributed random variable with mean γ, and set Define constants q γ and continuous (by Corollary 5.10 below) cluster mechanisms Q γ by and let U γ denote the log-Laplace operator with (constant) weight function q γ and cluster mechanism Q γ , i.e., We now establish the connection between renormalization transformations on W cat and log-Laplace operators.
Proposition 3.1 (Identification of the renormalization transformation) Let F γ be the rescaled renormalization transformation on W cat defined in (2.9). Then Fix a diffusion matrix w α,p ∈ W cat and migration constants (c k ) k≥0 . Define constants s n and γ n as in (2.8) and (2.11), respectively, where β := 1/α. Then Proposition 3.1 and formula (2.10) show that Here U γ n−1 , . . . , U γ 0 are the log-Laplace operators of the Poisson-cluster branching process X = (X −n , . . . , X 0 ) with weight functions q γ n−1 , . . . , q γ 0 and cluster mechanisms Q γ n−1 , . . . , Q γ 0 . We call X (started at some time −n in an initial law L(X −n )) the renormalization branching process. By formulas (3.4) and (3.11), the study of the limiting behavior of rescaled iterated renormalization transformations on W cat reduces to the study of the renormalization branching process X in the limit n → ∞.

Convergence to a time-homogeneous process
Let X = (X −n , . . . , X 0 ) be the renormalization branching process introduced in the last section.
If the constants (γ k ) k≥0 satisfy n γ n = ∞ and γ n → γ * for some γ * ∈ [0, ∞), then X is almost time-homogeneous for large n. More precisely, we will prove the following convergence result.
where ⇒ denotes weak convergence of laws on path space, k n (t) := min{k : 0 ≤ k ≤ n, n−1 l=k γ l ≤ t}, and Y 0 is the super-Wright-Fisher diffusion with activity and growth parameter both identically 1 and initial law L(Y 0 0 ) = µ.
The super-Wright-Fisher diffusion was studied in [FS03]. By definition, Y 0 is the time-homogeneous Markov process in M[0, 1] with continuous sample paths, whose Laplace functionals are given by (3.14) Here U 0 t f = u t is the unique mild solution of the semilinear Cauchy equation For a further study of the renormalization branching process X and its limiting processes Y γ * (γ * ≥ 0) we will use the technique of embedded particle systems, which we explain in the next section.

Weighted and Poissonized branching processes
In this section, we explain how from a Poisson-cluster branching process it is possible to construct other branching processes by weighting and Poissonization. We first need to introduce spatial branching particle systems in some generality. Let E again be separable, locally compact, and metrizable. For ν ∈ N (E) and f ∈ B [0,1] (E), we adopt the notation f 0 := 1 and f ν : (3.16) We call a continuous map x → Q(x, · ) from E into M 1 (N (E)) a continuous offspring mechanism.
Fix continuous offspring mechanisms Q k (1 ≤ k ≤ n), and let (X 0 , . . . , X n ) be a Markov chain in N (E) such that, given that X k−1 = m i=1 δ x i , the next step of the chain X k is a sum of independent random variables with laws Q k (x i , · ) (i = 1, . . . , m). Then We call U k the generating operator of the transition law from X k−1 to X k , and we call X = (X 0 , . . . , X n ) the branching particle system on E with generating operators U 1 , . . . , U n . It is often useful to write (3.17) in the suggestive form Here, if ν is an N (E)-valued random variable and f ∈ B [0,1] (E), then Thin f (ν) denotes an N (E)-valued random variable such that conditioned on ν, Thin f (ν) is obtained from ν by independently throwing away particles from ν, where a particle at x is kept with probability f (x). One has the elementary relations where D = denotes equality in distribution. We are now ready to describe weighted and Poissonized branching processes. Let X = (X 0 , . . . , X n ) be a Poisson-cluster branching process on E, with continuous weight functions q 1 , . . . , q n , continuous cluster mechanisms Q 1 , . . . , Q n , and log-Laplace operators U 1 , . . . , U n given by (3.2) and satisfying (3.3). Let Z k x denote an M(E)-valued random variable with law Q k (x, · ). Let h ∈ C + (E) be bounded, h = 0, and put Proposition 3.3 (Weighting of Poisson-cluster branching processes) Assume that there exists a constant K < ∞ such that U k h ≤ Kh for all k = 1, . . . , n. Then there exists a Poisson-cluster branching process X h = (X h 0 , . . . , X h n ) on E h with weight functions (q h 1 , . . . , q h n ) given by q h k := q k /h, continuous cluster mechanisms Q h 1 , . . . , Q h n given by

21)
and log-Laplace operators U h 1 , . . . , U h n satisfying The processes X and X h are related by Proposition 3.4 (Poissonization of Poisson-cluster branching processes) Assume that U k h ≤ h for all k = 1, . . . , n. Then there exists a branching particle system and generating operators U h 1 , . . . , U h n satisfying The processes X and X h are related by Here, the right-hand side of (3.24) is always a probability measure, despite that it may happen that q k (x)/h(x) > 1. The (straightforward) proofs of Propositions 3.3 and 3.4 can be found in Section 7.1 below. If (3.23) holds then we say that X h is obtained from X by weighting with density h. If (3.26) holds then we say that X h is obtained from X by Poissonization with density h. Proposition 3.4 says that a Poisson-cluster branching process X contains, in a way, certain 'embedded' branching particle systems X h . Poissonization relations for superprocesses and embedded particle systems have enjoyed considerable attention, see [FS04] and references therein.

Extinction versus unbounded growth for embedded particle systems
In this section we explain how embedded particle systems can be used to prove Theorem 1.4. Throughout this section (γ k ) k≥0 are positive constants such that n γ n = ∞ and γ n → γ * for some γ * ∈ [0, ∞), and X = (X −n , . . . , X 0 ) is the renormalization branching process on [0, 1] defined in Section 3.2. We write (3.27) In view of formula (3.11), in order to prove Theorem 1.4, we need the following result.
In our proof of Proposition 3.5, we will use embedded particle systems X h = (X h −n , . . . , X h 0 ) obtained from X by Poissonization with certain h taken from the classes H 1,1 , H 0,0 , and H 0,1 .
Lemma 3.6 (Embedded particle system with h 1,1 ) The constant function h 1,1 (x) := 1 is U γ -harmonic for each γ > 0. The corresponding embedded particle system In (3.29) and similar formulas below, ⇒ denotes weak convergence of probability measures on [0, ∞]. Thus, (3.29) says that for processes started with one particle on the position x at times −n, the number of particles at time zero converges to infinity as n → ∞.
Here, a branching particle system X is called critical if each particle produces on average one offspring (in each time step and independent of its position). Formula (3.30) says that the embedded particle system X h 0,0 gets extinct during the time interval {−n, . . . , 0} with probability tending to one as n → ∞. We can summarize Lemmas 3.6 and 3.7 by saying that the embedded particle system associated with h 1,1 grows unboundedly while the embedded particle system associated with h 0,0 becomes extinct as n → ∞. We will also consider an embedded particle systems X h 0,1 for a certain h 0,1 taken from H 0,1 . It turns out that this system either gets extinct or grows unboundedly, each with a positive probability. In order to determine these probabilities, we need to consider embedded particle systems for the time-homogeneous processes Y γ * (γ * ∈ [0, ∞)) from (3.12) and (3.13). If h ∈ H 0,1 is U γ * -superharmonic for some γ * > 0, then Poissonizing the process Y γ * with h yields a branching particle system on (0, 1] which we denote by then Poissonizing the super-Wright-Fisher diffusion Y 0 with h yields a continuous-time branching particle system on (0, 1], which we denote by Lemma 3.8 (Embedded particle system with h 0,1 ) The function h 0,1 (x) := 1 − (1 − x) 7 is U γ -superharmonic for each γ > 0. The corresponding embedded particle system X h 0,1 on (0, 1] satisfies is compact in the topology of weak convergence, there is a unique uniform structure compatible with the topology, and therefore it makes sense to talk about uniform convergence of M1[0, ∞]valued functions (in this case, x → P −n,δx |X locally uniformly for all x ∈ (0, 1], where (3.33) We now explain how Lemmas 3.6-3.8 imply Proposition 3.5. In doing so, it will be more convenient to work with weighted branching processes than with Poissonized branching processes. A little argument (which can be found in Lemma 7.12 below) shows that Lemmas 3.6-3.8 are equivalent to the next proposition.
4 Discussion, open problems where α > 0 is a constant, p is a nonnegative function on [0, 1] satisfying p(0) = 0 and p(1) > 0, and (B i ξ ) i=1,2 ξ∈Z 2 is a collection of independent Brownian motions. We call x a system of linearly interacting catalytic Wright-Fisher diffusions with catalyzation function p. It is expected that x clusters, i.e., x(t) converges in distribution as t → ∞ to a limit (x ξ (∞)) ξ∈Z 2 such that x ξ (∞) = x 0 (∞) for all ξ ∈ Z 2 and x 0 (∞) takes values in the effective boundary associated with the diffusion matrix w α,p (see (2.3)). Heuristic arguments, based on renormalization, yield a formula for the clustering distribution L(x 0 (∞)) in terms of the diffusion matrix w * which is the unique solution of the asymptotic fixed point equation (2.16) (ii) in the renormalization class W 0,1 cat ; see Conjecture A.3 in Appendix A.2 below. The present paper is inspired by the work of Greven, Klenke and Wakolbinger [GKW01]. They study a model that is closely related to (4.1), but where x 1 is replaced by a voter model. They show that their model clusters and determine its clustering distribution L(x 0 (∞)), which turns out to coincide with the mentioned prediction for (4.1) based on renormalization theory. In fact, they believe their results to hold for the model in (4.1) too, but they could not prove this due to certain technical difficulties that a [0, 1]-valued catalyst would create, compared to the simpler {0, 1}-valued voter model.
The work in [GKW01] not only provides the main motivation for the present paper, but also inspired some of our techniques for proving Theorem 1.4. This concerns in particular the proof of Proposition 3.1, which makes the connection between renormalization transformations and a branching process. We hope that conversely, our techniques may shed some light on the problems left open by [GKW01], in particular, the question whether their results stay true if the voter model catalyst is replaced by a Wright-Fisher catalyst. It seems plausible that their results may not hold for the model in (4.1) if the catalyzing function p grows too fast at 0. On the other hand, our proofs suggest that p with a finite slope at 0 should be OK. (In particular, while deriving formula (3.40), we use that p can be bounded from above by r + h 0,1 for some r + > 0, which requires that p has a finite slope at 0.) Our results are also interesting in the wider program of studying renormalization classes in the sense of Definition 1.1. We conjecture that the class W 0,1 cat , unlike all renormalization classes studied previously, contains no fixed shapes (see the discussion following Lemma 2.8).
In fact, we expect this to be the usual situation. In this sense, the renormalization classes studied so far were all of a special type.

Open problems
The general program of studying renormalization classes in the sense of Definition 1.1 contains a wealth of open problems. In our proofs, we make heavy use of the single-way nature of the catalyzation in (1.7), in particular, the fact that y 1 is an autonomous process which allows one to condition on y 1 and consider y 2 as a process in a random environment created by y 1 . As soon as one leaves the single-way catalytic regime one runs into several difficulties, both technically (it is hard to prove that a given class of matrices is a renormalization class in the sense of Definition 1.1) and conceptually (it is not clear when solutions to the asymptotic fixed shape equation (2.16) (ii) are unique). Therefore, it seems at present hard to verify the complete picture for renormalization classes on the unit square that arises from the numerical simulations described in Section 2.2 and Figures 2 and 3, unless one or more essential new ideas are added.
In this context, the study of the nonlinear partial differential equation (2.18) and its fixed points seems to be a challenging problem. This may be a hard problem from an analytic point of view, since the equation is degenerate and not in divergence form. For the renormalization class W cat , the quasilinear equation (2.18) reduces to the semilinear equation (3.15), which is analytically easier to treat and moreover has a probabilistic interpretation in terms of a superprocess. For a study of the semilinear equation (3.15) we refer to [FS03]. We do not know whether solutions to equation (2.18) can in general be represented in terms of a stochastic process of some sort.
Even for the renormalization class W cat , several interesting problems are left open. One of the most urgent ones is to prove that the functions p * 0,1,γ * are not constant in γ * , and therefore, by Lemma 2.8 (c), W 0,1 cat contains no fixed shapes. Moreover, we have not investigated the iterated renormalization transformations in the regime γ * = ∞. Also, we believe that the convergence in (3.28) (ii) does not hold if the condition that p is Lipschitz is dropped, in particular, if p has an infinite slope at 0 or an infinite negative slope at 1. For p ∈ H 0,0 , it seems plausible that a properly rescaled version of the iterates U (n) p converges to a universal limit, but we have not investigated this either. Finally, we have not investigated the convergence of the iterated kernels K w,(n) from (2.4) (in particular, we have not verified Conjecture A.2) for the renormalization class W cat .
Our methods, combined with those in [BCGdH95], can probably be extended to study the action of iterated renormalization transformations on diffusion matrices of the following more general form (compared to (1.4)): where g : [0, 1] → R is Lipschitz, g(0) = g(1) = 0, g > 0 on (0, 1), and p ∈ H as before. This would, however, require a lot of extra technical work and probably not generate much new insight. The numerical simulations mentioned in Section 2.2 suggest that many diffusion matrices of an even more general form than (4.2) also converge under renormalization to the limit points w * from Theorem 1.4, but we don't know how to prove this.

Part II
Outline of Part II In Section 5, we verify that W cat is a renormalization class, we prove Proposition 3.1, which connects the renormalization transformations F c to the log-Laplace operators U γ , and we collect a number of technical properties of the operators U γ that will be needed later on. In Section 6 we prove Theorem 3.2 about the convergence of the renormalization branching process to a time-homogeneous limit. In Section 7, we prove the statements from Section 3.5 about extinction versus unbounded growth of embedded particle systems, with the exception of Lemma 3.7, which is proved in Section 8. In Section 9, finally, we combine the results derived by that point to prove our main theorem.

The renormalization class W cat
In this section we prove Theorem 1.4 (a) and Proposition 3.1, as well as Lemmas 2.1-2.8 from Section 2. The section is organized according to the techniques used. Section 5.1 collects some facts that hold for general renormalization classes on compact sets. In Section 5.2 we use the SDE (1.7) to couple catalytic Wright-Fisher diffusions. In Section 5.3 we apply the moment duality for the Wright-Fisher diffusion to the catalyst and to the reactant conditioned on the catalyst. In Section 5.4 we prove that monotone concave catalyzing functions form a preserved class under renormalization.

Renormalization classes on compact sets
In this section, we prove the lemmas stated in Section 2. Recall that D ⊂ R d is open, bounded, and convex, and that W is a prerenormalization class on D, equipped with the topology of uniform convergence.
Proof of Lemma 2.1 To see that (x, c, w) → ν c,w x is continuous, let (x n , c n , w n ) be a sequence converging in D × (0, ∞) × W to a limit (x, c, w). By the compactness of D, the sequence (ν cn,wn xn ) n≥0 is tight, and each limit point ν * satisfies Therefore, by [EK86,Theorem 4.9.17], ν * is an invariant law for the martingale problem associated with A c,w x . Since we are assuming uniqueness of the invariant law, ν * = ν c,w x and therefore ν cn,wn xn ⇒ ν c,w x . The continuity of F c w(x) is a simple consequence of the continuity of ν c,w x . Proof of Lemma 2.2 Formula (2.1) (i) follows from the fact that rescaling the time in solutions (y t ) t≥0 to the martingale problem for A c,w x by a factor λ has no influence on the invariant law. Formula (2.1) (ii) is a direct consequence of formula (2.1) (i). Proof of Lemma 2.4 If x ∈ ∂ w D, then y t := x (t ≥ 0) is a stationary solution to the martingale problem for A c,w x , and therefore ν c,w is not a stationary solution to the martingale problem for A c,w x and therefore D ν c,w x (dy)|y − x| 2 > 0. Let tr(w(y)) := i w ii (y) denote the trace of w(y).
From now on assume that W is a renormalization class. Note that where we denote the composition of two probability kernels K, L on D by Proof of Lemma 2.5 This is a direct consequence of Lemmas 2.1 and 2.3. In particular, the relations (2.6) follow by iterating the relations (2.2).
Proof of Lemma 2.6 Recall that tr(w(y)) denotes the trace of w(y). Formulas (2.5) and Since D is compact, the left-hand side of this equation is bounded uniformly in x ∈ D and n ≥ 1, and therefore, since we are assuming s n → ∞, Since w is symmetric and nonnegative definite, tr(w(y)) is nonnegative, and zero if and only if y ∈ ∂ w D. If f ∈ C(D) satisfies f = 0 on ∂ w D, then, for every ε > 0, the sets C m := {x ∈ D : |f (x)| ≥ ε + m tr(w(x))} are compact with C m ↓ ∅ as m ↑ ∞, so there exists an m (depending on ε) such that |f | < ε + m tr(w). Therefore, x (dy)tr(w(y)) = ε. (5.6) Since ε > 0 is arbitrary, (2.7) follows.

Coupling of catalytic Wright-Fisher diffusions
In this section we verify condition (i) of Definition 1.1 for the class W cat , and we prepare for the verification of conditions (ii)-(iv) in Section 5.3. In fact, we will show that the larger class W cat := {w α,p : α > 0, p ∈ C + [0, 1]} is also a renormalization class, and the equivalents of Theorem 1.4 (a) and Proposition 3.1 remain true for this larger class. (We do not know, however, if the convergence statements in Theorem 1.4 (b) also hold in this larger class; see the discussion in Section 4.2.) For each c ≥ 0, w ∈ W cat and x ∈ [0, 1] 2 , the operator A c,w x is a densely defined linear operator on C([0, 1] 2 ) that maps the identity function into zero and, as one easily verifies, satisfies the positive maximum principle. Since [0, 1] 2 is compact, the existence of a solution to the martingale problem for A c,w x , for each [0, 1] 2 -valued initial condition, now follows from general theory (see [RW87], Theorem 5.23.5, or [EK86, Theorem 4.5.4 and Remark 4.5.5]).
We are therefore left with the task of verifying uniqueness of solutions to the martingale problem for A c,w x . By [EK86, Problem 4.19, Corollary 5.3.4, and Theorem 5.3.6], it suffices to show that solutions to (1.7) are pathwise unique.
where in both equations B is the same Brownian motion. If y 0 ≤ỹ 0 a.s., then Proof This is an easy adaptation of a technique due to Yamada and Watanabe [YW71]. Since One easily verifies that φ n (x), xφ ′ n (x), and xφ ′′ n (x) are nonnegative and converge, as n → ∞, to x ∨ 0, x ∨ 0, and 0, respectively. By Itô's formula: (5.11) Here the terms in (ii) are nonpositive, and hence, letting n → ∞ and using the elementary the properties of φ n , and the fact that the process P is uniformly bounded, we find that by our assumption that y 0 ≤ỹ 0 . This shows that y t ≤ỹ t a.s. for each fixed t ≥ 0, and by the continuity of sample paths the statement holds for all t ≥ 0 almost surely. Proof Let (y 1 , y 2 ) and (ỹ 1 ,ỹ 2 ) be solutions to (1.7) relative to the same pair (B 1 , B 2 ) of Brownian motions, with (y 1 0 , y 2 0 ) = (ỹ 1 0 ,ỹ 2 0 ). Applying Lemma 5.1, with inequality in both directions, we see that y 1 =ỹ 1 a.s. Applying Lemma 5.1 two more times, this time using that y 1 =ỹ 1 a.s., we see that also y 2 =ỹ 2 a.s.
Corollary 5.4 (Ergodicity) The Markov process defined by the SDE (3.6) has a unique invariant law Γ γ x and is ergodic, i.e, solutions to (3.6) started in an arbitrary initial law L(y 0 ) satisfy L(y t ) =⇒ t→∞ Γ γ x .
Proof Since our process is a Feller diffusion on a compactum, the existence of an invariant law follows from a simple time averaging argument. Now start one solutionỹ of (3.6) in this invariant law and let y be any other solution, relative to the same Brownian motion. Corollary 5.3 then gives ergodicity and, in particular, uniqueness of the invariant law.

✸
We conclude this section with a lemma that prepares for the verification of condition (iv) in Definition 1.1 for the class W cat .
Lemma 5.6 (Monotone coupling of stationary Wright-Fisher diffusions) Assume that c > 0, α > 0 and 0 ≤ x ≤x ≤ 1. Then the pair of equations has a unique stationary solution (y t ,ỹ t ) t∈R . This stationary solution satisfies Proof Let (y t ,ỹ t ) t≥0 be a solution of (5.18) and let (y ′ t ,ỹ ′ t ) t≥0 be another one, relative to the same Brownian motion B. Then, by Lemma 5.3, E[|y t − y ′ t |] → 0 and also E[|ỹ t −ỹ ′ t |] → 0 as t → ∞. Hence we may argue as in the proof of Corollary 5.4 that (5.18) has a unique invariant law and is ergodic. Now start a solution of (5.18) in an initial condition such that y 0 ≤ỹ 0 . By ergodicity, the law of this solution converges as t → ∞ to the invariant law of (5.18) and using Lemma 5.1 we see that this invariant law is concentrated on {(y,ỹ) ∈ [0, 1] 2 : y ≤ỹ}. Now consider, on the whole real time axis, the stationary solution to (5.18) with this invariant law. Applying Lemma 5.1 once more, we see that (5.19) holds.

Duality for catalytic Wright-Fisher diffusions
In this section we prove Theorem 1.4 (a) and Proposition 3.1. Moreover, we will show that their statements remain true if the renormalization class W cat is replaced by the larger class W cat := {w α,p : α > 0, p ∈ C + [0, 1]}. We begin by recalling the usual moment duality for Wright-Fisher diffusions.
For γ > 0 and x ∈ [0, 1], let y be a solution to the SDE dy(t) = 1 γ (x − y(t))dt + 2y(t)(1 − y(t))dB(t), (5.20) i.e., y is a Wright-Fisher diffusion with a linear drift towards x. It is well-known that y has a moment dual. To be precise, let (φ, ψ) be a Markov process in N 2 = {0, 1, . . .} 2 that jumps as: (5.21) Then one has the following duality relation (see for example Lemma 2.3 in [Shi80] or Proposition 1.5 in [GKW01]) where 0 0 := 1. The duality in (5.22) has the following heuristic explanation. Consider a population containing a fixed, large number of organisms, that come in two genetic types, say I and II. Each pair of organisms in the population is resampled with rate 2. This means that one organism of the pair (chosen at random) dies, while the other organism produces one child of its own genetic type. Moreover, each organism is replaced with rate 1 γ by an organism chosen from an infinite reservoir where the frequency of type I has the fixed value x. In the limit that the number of organisms in the population is large, the relative frequency y t of type I organisms follows the SDE (5.20). Now E[y n t ] is the probability that n organisms sampled from the population at time t are all of type I. In order to find this probability, we follow the ancestors of these organisms back in time. Viewed backwards in time, these ancestors live for a while in the population, until, with rate 1 γ , they jump to the infinite reservoir. Moreover, due to resampling, each pair of ancestors coalesces with rate 2 to one common ancestor. Denoting the number of ancestors that lived at time t − s in the population and in the reservoir by φ s and ψ s , respectively, we see that the probability that all ancestors are of type I is E y [y n t ] = E (n,0) [y φt x ψt ]. This gives a heuristic explanation of (5.22). Since eventually all ancestors of the process (φ, ψ) end up in the reservoir, we have (φ t , ψ t ) → (0, ψ ∞ ) as t → ∞ a.s. for some N-valued random variable ψ ∞ . Taking the limit t → ∞ in (5.22), we see that the moments of the invariant law Γ γ x from Corollary 5.4 are given by: It is not hard to obtain an inductive formula for the moments of Γ γ x , which can then be solved to yield the formula Γ γ x (dy)y n = n−1 k=0 x + kγ 1 + kγ (n ≥ 1). (5.24) In particular, it follows that This is the important fixed shape property of the Wright-Fisher diffusion (see formula (2.17)). We now consider catalytic Wright-Fisher diffusions (y 1 , y 2 ) as in (1.7) with p ∈ C + [0, 1] and apply duality to the catalyst y 2 conditioned on the reactant y 1 . Let (y 1 t , y 2 t ) t∈R be a stationary solution to the SDE (1.7) with c = 1/γ. Let (φ,ψ) be a N 2 -valued process, defined on the same probability space as (y 1 , y 2 ), such that conditioned on the past path (y 1 −t ) t≤0 , the process (φ,ψ) is a (time-inhomogeneous) Markov process that jumps as: with rate 1 γφ t . (5.26) Then, in analogy with (5.22), We may interpret (5.26) by saying that pairs of ancestors in a finite population coalesce with time-dependent rate 2p(y 1 −t ) and ancestors jump to an infinite reservoir with constant rate 1 γ . Again, eventualy all ancestors end up in the reservoir, and therefore (φ t ,ψ t ) → (0,ψ ∞ ) as t → ∞ a.s. for some N-valued random variableψ ∞ . Taking the limit t → ∞ in (5.27) we find that Proof Our process being a Feller diffusion on a compactum, the existence of an invariant law follows from time averaging. We need to show uniqueness. If (y 1 , y 2 ) = y 1 t , y 2 t ) t∈R is a stationary solution, then y 1 is an autonomous process, and L(y 1 0 ) = Γ 1/c x , the unique invariant law from Corollary 5.4. Therefore, L((y 1 t ) t∈R ) is determined uniquely by the requirement that (y 1 , y 2 ) be stationary. By (5.28), the conditional distribution of y 2 0 given (y 1 t ) t≤0 is determined uniquely, and therefore the joint distribution of y 2 0 and (y 1 t ) t≤0 is determined uniquely. In particular, L(y 1 0 , y 2 0 ) = ν c,w x is determined uniquely.
Remark 5.8 (Reversibility) It seems that the invariant law ν c,w x from Lemma 5.7 is reversible. In many cases (densities of) reversible invariant measures can be obtained in closed form by solving the equations of detailed balance. This is the case, for example, for the onedimensional Wright-Fisher diffusion. We have not attempted this for the catalytic Wright-Fisher diffusion. ✸ The next proposition implies Proposition 3.1 and prepares for the proof of Theorem 1.4 (a).
Proposition 5.9 (Extended renormalization class) The set W cat is a renormalization class on [0, 1] 2 , and Proof To see that W cat is a renormalization class we need to check conditions (i)-(iv) from Definition 1.1. By Lemma 5.2, the martingale problem for A c,w x is well-posed for all c ≥ 0, w ∈ W cat and x ∈ [0, 1] 2 . By Lemma 5.7, the corresponding Feller process on [0, 1] 2 has a unique invariant law ν c,w x . This shows that conditions (i) and (ii) from Definition 1.1 are satisfied. Note that by the compactness of [0, 1] 2 , any continuous function on [0, 1] 2 is bounded, so condition (iii) is automatically satisfied. Hence W is a prerenormalization class. As a consequence, for any p ∈ C + [0, 1], F γ w 1,p is well-defined by (1.2) and (2.9). We will now first prove (5.29) and then show that W cat is a renormalization class.
Fix γ > 0, p ∈ C + [0, 1], and x ∈ [0, 1] 2 . Let (y 1 t , y 2 t ) t∈R be a stationary solution to the SDE (1.7) with α = 1 and c = 1/γ. Then We are left with the task of showing that Here, by (2.2) (ii), (5.32) By (5.28), using the fact that E[y 2 0 ] = x 2 (which follows from (5.27) or more elementary from (2.6) (i)), we find that Note that P (2,0) [ψ ∞ = 1] is the probability that the two ancestors coalesce before one of them leaves the population. The probability of noncoalescence is given by where τ γ is an exponentially distributed random variable with mean γ. Combining this with (5.32) and (5.33) we find that where we have used the definition of U γ . We still have to show that W cat satisfies condition (iv) from Definition 1.1. For any α > 0 and p ∈ C + [0, 1], by scaling (Lemma 2.2) and (5.29), By Lemma 2.1, this diffusion matrix is continuous, which implies that U c α ( p α ) is continuous. Our proof of Propostion 5.9 has a corollary.
Proof of Theorem 1.4 (a) We need to show that W cat is a renormalization class and that F c maps the subclasses W l,r cat into themselves. It has already been explained in Section 2 that the latter fact is a consequence of Lemma 2.4. Since in Proposition 5.9 it has been shown that W cat is a renormalization class, we are left with the task to show that F c maps W cat into itself. By (5.29) and scaling, it suffices to show that U γ maps H into itself.
where L is the Lipschitz constant of p and we have used the same exponentially distributed τ γ for y γ x and y γ x .

Monotone and concave catalyzing functions
In this section we prove that the log-Laplace operators U γ from (3.9) map monotone functions into monotone functions, and monotone concave functions into monotone concave functions. We do not know if in general U γ maps concave functions into concave functions.
Proof Our proof of Proposition 5.11 is in part based on ideas from [BCGdH97, Appendix A]. The proof is quite long and will depend on several lemmas. We remark that part (a) can be proved in a more elementary way using Lemma 5.6. We recall some facts from Hille-Yosida theory. A linear operator A on a Banach space V is closable and its closure A generates a strongly continuous contraction semigroup (S t ) t≥0 if and only if is dense for some, and hence for all α > 0. If A generates a Feller semigroup and g ∈ C(E), then the operator A + g (with domain D(A + g) := D(A)) generates a strongly continuous semigroup (S g t ) t≥0 on C(E). If g ≤ 0 then (S g t ) t≥0 is contractive. If (ξ t ) t≥0 is the Feller process with generator A, then one has the Feynman-Kac representation x ∈ E, g, u ∈ C(E)). Write (5.44) Let S denote the closure of a set S ⊂ C([0, 1] 2 ). We need the following lemma.
(5.47) It follows from Lemma 5.13 that for each fixed r, t, and z, the function x → S 0 r S g t 1(x, z) is nondecreasing if f is nonincreasing, and nondecreasing and convex if f is nonincreasing and concave. Therefore, taking the expectation over the randomness of τ γ , the claims follow from (5.46) and (5.47).
We still need to prove Lemmas 5.12 and 5.13.
Proof of Lemma 5.12 It is easy to see that the operator B from (5.43) is densely defined, satisfies the positive maximum principle, and maps the constant function 1 into 0. Therefore, by Hille-Yosida (5.41), we must show that the range R(1 − αB) is dense in C([0, 1] 2 ) for some, and hence for all α > 0. Let P n denote the space of polynomials on [0, 1] 2 of n-th and lower order, i.e., the space of functions f : [0, 1] 2 → R of the form a kl x k y l with a k,l = 0 for k + l > n.
(5.48) Set P ∞ := n P n . It is easy to see that B maps the space P n into itself, for each n ≥ 0. Since each P n is finite-dimensional, a simple argument (see [EK86, Proposition 1.3.5]) shows that the image of P ∞ under 1 − αB is dense in C([0, 1] 2 ) for all but countably many, and hence for all α > 0.
In order to prove Lemma 5.13, based on Lemma 5.14, we will show that the Laplace equation (5.49) has smooth solutions u for sufficiently many functions v. Here 'suffiently many' will mean dense in the topology of uniform convergence of functions and their derivatives up to second order. To this aim, we make C (2) ([0, 1] 2 ) into a Banach space by equipping it with the norm u (2) := u + u y + u x + u yy + 2 u xy + u xx .
(5.63) Finally e −λt S g t u = Sg t u = lim ε→0 e G ε t u (t ≥ 0, u ∈ C([0, 1] 2 )), (5.64) so (5.63) implies that for each t ≥ 0: (5.65) Using the continuity of S g t in g (which follows from Feynman-Kac (5.42)) we arrive at the statements in Lemma 5.13. 6 Convergence to a time-homogeneous process 6.1 Convergence of certain Markov chains Section 6 is devoted to the proof of Theorem 3.2. In the present subsection, we start by formulating a theorem about the convergence of certain Markov chains to continuous-time processes. In Section 6.2 we specialize to Poisson-cluster branching processes and superprocesses. In Section 6.3, finally, we carry out the necessary calculations for the specific processes from Theorem 3.2.
Let E be a compact metrizable space. Let A : D(A) → C(E) be an operator defined on a domain D(A) ⊂ C(E). We say that a process y = (y t ) t≥0 solves the martingale problem for A if y has sample paths in D E [0, ∞) and for each f ∈ D(A), the process (M f t ) t≥0 given by is a martingale with respect to the filtration generated by y. We say that existence (uniqueness) holds for the martingale problem for A if for each probability measure µ on E there is at least one (at most one (in law)) solution y to the martingale problem for A with initial law L(y 0 ) = µ. If both existence and uniqueness hold we say that the martingale problem is well-posed. For each n ≥ 0, let X (n) = (X (n) 0 , . . . , X (n) m(n) ) (with 1 ≤ m(n) < ∞) be a (time-inhomogeneous) Markov process in E with k-th step transition probabilities We assume that the P k are continuous probability kernels on E. Let (ε By definition, a space A of real functions is called an algebra if A is a linear space and f, g ∈ A implies f g ∈ A. Theorem 6.1 (Convergence of Markov chains) Assume that L(X (n) 0 ) ⇒ µ as n → ∞ for some probability law µ on E. Suppose that there exists at most one (in law) solution to the martingale problem for A with initial law µ. Assume that the linear span of D(A) contains an algebra that separates points. Assume that and lim n→∞ sup k: t for each T > 0. Then there exists a unique solution y to the martingale problem for A with initial law µ and moreover L(y (n) ) ⇒ L(y), where ⇒ denotes weak convergence of probability measures on D E [0, ∞).
Proof We apply [EK86,Corollary 4.8.15]. Fix f ∈ D(A). We start by observing that is a martingale with respect to the filtration generated by X (n) and therefore, is a martingale with respect to the filtration generated by y (n) . Put Then we can rewrite the martingale in (6.10) as (6.14) By [EK86,Corollary 4.8.15] and the compactness of the state space, it suffices to check the following conditions on φ (n) and ξ (n) : (6.15) for some N ≥ 0 and for each T > 0, r ≥ 1, 0 ≤ s 1 < · · · < s r ≤ T , and h 1 , . . . , h r ∈ H ⊂ C(E).

Convergence of certain branching processes
In this section we apply Theorem 6.1 to certain branching processes and superprocesses. Throughout this section, E is a compact metrizable space and A : D(A) → C(E) is a linear operator on C(E) such that the closure A of A generates a Feller process ξ = (ξ t ) t≥0 in E with Feller semigroup (P t ) t≥0 given by P t f (x) := E x [f (ξ t )] (t ≥ 0, f ∈ C(E)).
Let α ∈ C + (E) and β, f ∈ C(E). By definition, a function t → u t from [0, ∞) into C(E) is a classical solution to the semilinear Cauchy problem if t → u t is continuously differentiable (in C(E)), u t ∈ D(A) for all t ≥ 0, and (6.20) holds. We say that u is a mild solution to (6.20) if t → u t is continuous and Proof Results of this type are well-known, see for example [EK86,Theorem 9.4.3], [Fit88], and [ER91, Théorème 7]. Since, however, it is not completely straightforward to derive the proposition above from these references, we give a concise autonomous proof of most of our statements. Only for the continuity of sample paths we refer the reader to [Fit88, Corollary (4.7)] or [ER91, Corollaire 9]. We are going to extend G to an operatorĜ that is linear and satisfies the conditions of the Hille-Yosida Theorem (5.41). For any γ ∈ C + (E) and µ ∈ M(E), let Clust γ (µ) denote a random measure such that on {γ = 0}, Clust γ (µ) is equal to µ, and on {γ > 0}, Clust γ (µ) is a Poisson cluster measure with intensity 1 γ µ and cluster mechanism Q(x, ·) = L(τ γ(x) δ x ), where τ γ(x) is exponentially distributed with mean γ(x). It is not hard to see that Note that since V γ 1 is bounded, the previously mentioned Poisson cluster measure mentioned above is well-defined. By definition, we put Clust γ (∞) := ∞.
Define a linear operator G α on C(M(E)) ∞ ) by with as domain D(G α ) the space of all F ∈ C(M(E) ∞ ) for which the limit exists. Define a linear operator G β by with domain D(G β ) := C(M(E)) ∞ ). Define P * t : M(E) ∞ → M(E) ∞ by P * t µ, f := µ, P t f (t ≥ 0, f ∈ C(E), µ ∈ M(E)) and P * t ∞ := ∞ (t ≥ 0). Finally, let G A be the linear operator on C(M(E)) ∞ ) defined by with as domain D(G A ) the space of all F for which the limit exists. Define an operatorĜ bŷ Now let (q ε ) ε>0 be continuous weight functions and let (Q ε ) ε>0 be continuous cluster mechanisms on E. Assume that and define probability kernels K ε on E by For each n ≥ 0, let (ε . Define t (n) k and k (n) (t) as in (6.4)-(6.5). Define Theorem 6.4 (Convergence of Poisson-cluster branching processes) Assume that L(X (n) 0 ) ⇒ ρ as n → ∞ for some probability law ρ on M(E). Suppose that the constants ε (n) k fulfill (6.7). Assume that for each δ > 0, and Here ⇒ denotes weak convergence of probability measures on D M(E) [0, ∞).
Proof We apply Theorem 6.1 to the operator G, where we use the fact that if we view ) (note the compactification), equipped with the topology of weak convergence, then the induced topology on M 1 (D M(E) [0, ∞)) is again the topology of weak convergence. By Proposition 6.3, solutions to the martingale problem for G are unique. Since F f F g = F f +g and D(A) is a linear space, the linear span of the domain of G is an algebra. Using the fact that D(A) is dense in C(E) we see that this algebra separates points. Therefore, we are left with the task to check (6.8).
We observe that where Γ γ x is the equilibrium law of the process y γ x from Corollary 5.4. It follows from (5.24) that uniformly in x as γ → 0. Therefore, for any δ > 0, uniformly in x as γ → 0. Consequently, a Taylor expansion of f around x yields which, using the fact that q γ = ( 1 γ + 1), gives (6.64) This shows that (6.36) is fulfilled. In particular, for all δ > 0.

Embedded particle systems
In this section we use embedded particle systems to prove Proposition 3.5. An essential ingredient in the proofs is Proposition 7.15 (a), which will be proved in the Section 8.

Weighting and Poissonization
Proof of Proposition 3.3 Obviously q h k ∈ C + (E h ) for each k = 1, . . . , n. Since h ∈ C + (E) and h is bounded, it is easy to see that the map µ → hµ from M(E) into M(E h ) is continuous, and therefore the cluster mechanisms defined in (3.21) are continuous. Since h(x) ≤ K < ∞, the log-Laplace operators U h k satisfy (3.3). If X is started in an initial state X 0 , then the Poisson-cluster branching process X h with log-Laplace operators U h 1 , . . . , U h n started in which proves (3.23).

(7.4)
Here and in similar formulas below, if in a conditional probability the symbol Pois( · ) occurs twice with the same argument, then it always refers to the same random variable (and not to independent Poisson point measures with the same intensity, for example). Using moreover (7.3) we can rewrite (3.24) as In particular, since we are assuming that h is U k -subharmonic, this shows that Q h k (x, · ) is a probability measure. Let X h be the branching particle system with offspring mechanisms Q h 1 , . . . , Q h k . Let Z h,k x be random variables such that L(Z h,k x ) = Q h k (x, · ). Then, by (3.18), (3.24), (3.20), and (7.3), To see that also (3.26) holds, just note that by (3.19), (3.25), and (3.5), Since this formula holds for all f ∈ B [0,1] (E h ), formula (3.26) follows.
Remark 7.1 (Boundedness of h) Propositions 3.3 and 3.4 generalize to the case that h is unbounded, except that in this case the cluster mechanism in (3.21) and the offspring mechanism in (3.24) need in general not be continuous. Here, in order for (3.22) and (3.25) to be well-defined, one needs to extend the definition of U k f to unbounded functions f , but this can always be done unambiguously [FS03,Lemma 9]. ✸

Sub-and superharmonic functions
This section contains a number of pivotal calculations involving the log-Laplace operators U γ from (3.9). In particular, we will prove that the functions h 1,1 , h 0,0 , and h 0,1 from Lemmas 3.6, 3.7, and 3.8, respectively, are U γ -superharmonic. We start with an observation that holds for general log-Laplace operators.
Lemma 7.2 (Constant multiples) Let U be a log-Laplace operator of the form (3.2) satisfying (3.3) and let f ∈ B + (E). Then U (rf ) ≤ rU f for all r ≥ 1, and U (rf ) ≥ rU f for all 0 ≤ r ≤ 1. In particular, if f is U -superharmonic then rf is U -superharmonic for each r ≥ 1, and if f is U -subharmonic then rf is U -superharmonic for each 0 ≤ r ≤ 1.
Proof If X is a branching process and U is the log-Laplace operator of the transition law from X 0 to X 1 then, using Jensen's inequality, for all r ≥ 1, (7.8) Since this holds for all µ ∈ M(E), it follows that U (rf ) ≤ rU f . The proof of the statements for 0 ≤ r ≤ 1 is the same but with the inequality signs reversed.
We next turn our attention to the functions h 1,1 and h 0,0 .
Lemma 7.6 (Sampling at multiple times) Fix 0 ≤ t 1 < · · · < t n = t and nonnegative integers m 1 , . . . , m n . Let y be the diffusion in (5.20). Then and between these deterministic times jumps with rates as in (5.21).
For any m ≥ 1, we put The next lemma shows that we have particular good control on the action of U γ on the functions h m .
The next result is a simple application of Lemma 7.7.
We now set out to prove that h 7 , which is the function h 0,1 from Lemma 3.8, is U γ -superharmonic. In order to do so, we will derive upper bounds on the expectation of ψ ′ ∞ . We derive two estimates: one that is good for small γ and one that is good for large γ.
In order to avoid tedious formal arguments, it will be convenient to recall the interpretation of the process (φ ′ , ψ ′ ) and Lemma 7.6. Recall from the discussion following (5.22) that (y γ x (t)) t∈R describes the equilibrium frequency of genetic type I as a function of time in a population that is in genetic exchange with an infinite reservoir. From this population we sample at times −σ k (k ≥ 0, σ k < τ γ/2 ) each time m individuals, and ask for the probability that they are not all of the genetic type II. In order to find this probability, we follow the ancestors of the sampled individuals back in time. Then φ ′ t and ψ ′ t are the number of ancestors that lived at time −t in the population and the reservoir, respectively, and E[1 − (1 − x) ψ ′ ∞ ] is the probability that at least one ancestor is of type I. Lemma 7.9 (Bound for small γ) For each γ ∈ (0, ∞) and m ≥ 1, The function χ m is concave and satisfies χ m (0) = 1 for each m ≥ 1.
Lemma 7.10 (Bound for large γ) For each γ ∈ (0, ∞) and m ≥ 1, , and therefore Unlike in the proof of the last lemma, this time we cannot fully ignore the coalescence of ancestors sampled at different times. In order to deal with this we use a trick: at time zero we introduce an extra ancestor that can only jump to the reservoir when t ≥ τ γ and there are no other ancestors left in the population. We further assume that all other ancestors do not jump to the reservoir on their own. Let ξ t be one as long as this extra ancestor is in the population and zero otherwise, and let φ ′′ t be the number of other ancestors in the population according to these new rules. Then we have at a Markov process (ξ, φ ′′ ) started in (ξ 0 , φ ′′ 0 ) = (1, m) that jumps as: (ξ t , φ ′′ (n + 1)n. Then it is not hard to see that, for an appropriate coupling, φ ′′ Proof Recall that h 0,1 (x) = h 7 (x) = 1 − (1 − x) 7 (x ∈ [0, 1]). We will show that for each γ ∈ (0, ∞). The function χ m (γ) from Lemma 7.9 satisfies Since χ m (γ) is concave in γ and satisfies χ m (0) = 1, it follows that χ m (γ) < 1 for all 0 < γ ≤ 1 and m ≥ 5. By Lemma 7.10, for all γ ≥ 1, This shows that h m is U γ -superharmonic for each γ > 0. By Lemma 7.2, for each r > 1, (7.47) By Lemma 7.3 and the monotonicity of U γ , Since the right-hand side of (7.47) is smaller than 1 for x ∈ (0, 1) and tends to m ′ /m < 1 as x → 0, since the right-hand side of (7.48) is smaller than 1 for x in an open neighborhood of 1, and since both bounds are continuous, (7.42) follows.

Extinction versus unbounded growth
In this section we show that Lemmas 3.6-3.8 are equivalent to Proposition 3.9. (This follows from the equivalence of conditions (i) and (ii) in Lemma 7.12 below.) We moreover prove Lemmas 3.6 and 3.8 and prepare for the proof of Lemma 3.7. We start with some general facts about log-Laplace operators and branching processes. For the next lemma, let E be a separable, locally compact, metrizable space. For n ≥ 0, let q n ∈ C + (E) be continuous weight functions, let Q n be continuous cluster mechanisms on E, and assume that the associated log-Laplace operators U n defined in (3.2) satisfy (3.3). Assume that 0 = h ∈ C + (E) is bounded and U n -superharmonic for all n, let E h := {x ∈ E : h(x) > 0}, and define generating operators ) be the one-step branching particle system with generating operator U h n . (In a typical application of this lemma, the operators U n will be iterates of other log-Laplace operators, and X (n) 0 , X (n) 1 will be the initial and final state, respectively, of a Poisson cluster branching process with many time steps.) Lemma 7.12 (Extinction versus unbounded growth) Assume that ρ ∈ C [0,1] (E h ) and put Then the following statements are equivalent: locally uniformly for x ∈ E ∀λ > 0, locally uniformly for x ∈ E (i = 1, 2).
Proof of Lemma 7.12 It is not hard to see that (i) is equivalent to locally uniformly for x ∈ E ∀0 < λ ≤ 1.
By (3.4), condition (ii) implies that locally uniformly for x ∈ E for all λ > 0, and therefore (ii) implies (iii). Obviously (iii)⇒ (i) ′ ⇒(iv) so we are done if we show that (iv)⇒(ii). Indeed, (iv) implies that locally uniformly for x ∈ E, which shows that for all 0 < c < C < ∞. Using (iv) once more we arive at (ii).
Our next lemma gives sufficient conditions for the n-th iterates of a single log-Laplace operator U to satisfy the equivalent conditions of Lemma 7.12. Let E (again) be separable, locally compact, and metrizable. Let q ∈ C + (E) be a weight function, Q a continuous cluster mechanism on E, and assume that the associated log-Laplace operator U defined in (3.2) satisfies (3.3). Let X = (X 0 , X 1 , . . .) be the Poisson-cluster branching process with log-Laplace operator U in each step, let 0 = h ∈ C + (E) be bounded and U -superharmonic, and let X h = (X h 0 , X h 1 , . . .) denote the branching particle system on E h obtained from X by Poissonization with a Usuperharmonic function h, in the sense of Proposition 3.4. Lemma 7.13 (Sufficient condition for extinction versus unbounded growth) Assume that Then the process X h started in any initial law L( , which is uniformly bounded away from one by (7.54). Therefore, P [X h k+1 = 0|X h k ] → 0 a.s. on A is only possible if the number of particles tends to infinity.
The continuity of ρ can be proved by a straightforward adaptation of the proof of [FS04, Proposition 5 (d)] to the present setting with discrete time and noncompact space E. An essential ingredient in the proof, apart from (7.54), is the fact that the map ν → P ν [X h n ∈ · ] from N (E) to M 1 (N (E)) is continuous, which follows from the continuity of Q h .
We now turn our attention more specifically to the renormalization branching process X . In the remainder of this section, (γ k ) k≥0 is a sequence of positive constants such that n γ n = ∞ and γ n → γ * for some γ * ∈ [0, ∞), and X = (X −n , . . . , X 0 ) is the Poisson cluster branching process on [0, 1] defined in Section 3.2. We put U (n) := U γ n−1 • · · · • U γ 0 . If 0 = h ∈ C[0, 1] is U γ k -superharmonic for all k ≥ 0, then X h and X h denote the branching process and the branching particle system on {x ∈ [0, 1] : h(x) > 0} obtained from X by weighting and Poissonizing with h in the sense of Propositions 3.3 and 3.4, respectively.
Proof of Lemma 3.6 By induction, it follows from Lemma 7.3 that It is not hard to see (compare the footnote at (2.12)) that Therefore, since we are assuming that n γ n = ∞, Remark 7.14 (Conditions on (γ n ) n≥0 ) Our proof of Lemma 3.6 does not use that γ n → γ * for some γ * ∈ [0, ∞). On the other hand, the proof shows that n γ n = ∞ is a necessary condition for (3.29). ✸ We do not know if the assumption that γ n → γ * for some γ * ∈ [0, ∞) is needed in Lemma 3.7. We guess that it can be dropped, but it will greatly simplify proofs to have it around. We will show that in order to prove Lemmas 3.7 and 3.8, it suffices to prove their analogues for embedded particle systems in the time-homogeneous processes Y γ * (γ * ∈ [0, ∞)). More precisely, we will derive Lemmas 3.7 and 3.8 from the following two results. Below, (U 0 t ) t≥0 is the log-Laplace semigroup of the super-Wright-Fisher diffusion Y 0 , defined in (3.15). The functions p * 0,1,γ * (γ * ∈ [0, ∞)) are defined in (3.34).
r ≥ 1, and that each function f ∈ B + [0, 1] with f (0) = 0 and f (1) > 0 can be bounded as x is the invariant law of y γ x from Corollary 5.4. In particular, setting f = 1 gives To prove (3.30), by Lemma 7.12 it suffices to show that uniformly on [0, 1] for all 0 < λ ≤ 1. We first treat the case γ * > 0. Then, by Theorem 3.2 (a), for each fixed l ≥ 1 and f ∈ C + [0, 1], uniformly on [0, 1]. Therefore, by a diagonal argument, we can find l(n) → ∞ such that Using the fact that the function h 0,0 is U γ -superharmonic for each γ > 0 and the monotonicity 8.2 A representation for the Campbell law (Local) extinction properties of critical branching processes are usually studied using Palm laws. Our proof of formula (8.1) is no exception, except that we will use the closely related Campbell laws. Loosely speaking, Palm laws describe a population that is size-biased at a given position, plus 'typical' particle sampled from that position, while Campbell laws describe a population that is size-biased as a whole, plus a 'typical' particle sampled from a random position. Let P be a probability law on N (0, 1) with N (0,1) P(dν)|ν| = 1. Then the size-biased law P size associated with P is the probability law on N (0, 1) defined by (8.5) The Campbell law associated with P is the probability law on (0, 1) × N (0, 1) defined by for all Borel-measurable A ⊂ (0, 1) and B ⊂ N (0, 1). If (v, V ) is a (0, 1) × N (0, 1)-valued random variable with law P Camp , then L(V ) = P size , and v is the position of a 'typical' particle chosen from V . Let denote the law of Y h at time n, started at time 0 with one particle at position x ∈ (0, 1). Note that by criticality, N (0,1) P x,n (dν)|ν| = 1. Using again criticality, it is easy to see that in order to prove the extinction formula (8.1), it suffices to show that lim n→∞ P x,n size {1, . . . , N } = 0 (x ∈ (0, 1), N ≥ 1). (8.8) In order to prove (8.8), we will write down an expression for P x,n Camp . Let Q h denote the offspring mechanism of Y h , and, for fixed x ∈ (0, 1), let Q h Camp (x, · ) denote the Campbell law associated with Q h (x, · ). The next proposition is a time-inhomogeneous version of Kallenberg's famous backward tree technique; see [Lie81, Satz 8.2].
Proposition 8.2 (Representation of Campbell law) Let (v k , V k ) k≥0 be the Markov process in (0, 1) × N (0, 1) with transition laws Camp (x, · ) ((x, ν) ∈ (0, 1) × N (0, 1)), (8.9) started in (v 0 , V 0 ) = (δ x , 0). Let (Y h,(k) ) k≥1 be branching particle systems with offspring mechanism Q h , conditionally independent given Formula (8.10) says that the Campbell law at time n arises in such a way, that an 'immortal' particle at positions v 0 , . . . , v n sheds off offspring V 1 − δ v 1 , . . . , V n − δ vn , distributed according to the size-biased law with one 'typical' particle taken out, and this offspring then evolve under the usual forward dynamics till time n. Note that the position of the immortal particle (v k ) k≥0 is an autonomous Markov chain. We need a bit of explicit control on Q h Camp . where the random measures Z γ * x are defined in (3.7).
Proof By the definition of the Campbell law (8.6), and (3.24), where (y γ * x (t)) t∈R is a stationary solution to the SDE (3.6) with γ = γ * . By Lemma 8.3, the transition law of the Markov chain (v k ) k≥0 from Proposition 8.2 is given by where Γ γ * x is the invariant law of y γ * x from Corollary 5.4. In the next section we will prove the following lemma.

The immortal particle
Proof of Lemma 8.4 Let K(x, dy) denote the transition kernel (on (0, 1)) of the Markov chain (v k ) k≥0 , i.e., by (8.14), K(x, dy) = (1 + γ * ) y(1 − y) x(1 − x) Γ γ * x (dy). Here the (c k ) k≥0 are positive constants satisfying k c k /N k < ∞, and x k ξ (t) denotes the k-block average around ξ: (see [DG93a,Kle96]; a similar problem is treated in [DE68]). Assuming that the law of x(t) converges weakly as t → ∞ to the law of some D Ω N -valued random variable x(∞), one expects that in the recurrent case x(∞) must have the following properties: (i) x ξ (∞) = x η (∞) a.s. ∀ξ, η ∈ Ω N , (ii) x ξ (∞) ∈ ∂ w D a.s. ∀ξ ∈ Ω N . (A.7) Here ∂ w D is the effective boundary of D, defined in (2.3). If x(t) converges in law to a limit x(∞) satisfying (A.7), then we say that x clusters. In the transient case, it is believed that solutions of (A.2) do not cluster. (For compact D these facts were proved in [Swa00].) An important tool in the study of solutions to (A.2) is the so-called interaction chain. This is the chain (x 0 0 (t), x 1 0 (t), . . .) of block-averages around the origin. Heuristic arguments suggest that in the local mean field limit N → ∞, the interaction chain converges to a certain well-defined Markov chain.
Conjecture A.1 Fix w ∈ W, θ ∈ D, and positive numbers (c k ) k≥0 such that for N large enough, k c k /N k < ∞. For all N large enough, let x N be a solution to (A.2)-(A.3), and assume that t N are constants such that, for some n ≥ 1, lim N →∞ N −n t N = T ∈ [0, ∞). Then and initial state I w −n = y T , where dy t = c n (θ − y t )dt + √ 2σ (n) (y t )dB t , y 0 = θ, (A.10) and σ (n) is a root of the diffusion matrix F (n) w.
Rigorous versions of conjecture A.1 have been proved for renormalization classes on D = [0, 1] and D = [0, ∞) in [DG93a,DG93b]. Note that the iterated kernels K w,(n) defined in (2.4) are the transition probabilities from time −n to time 0 of the interaction chain in the mean-field limit: Lemma 2.6 expresses the fact that the system x N clusters in the local mean-field limit N → ∞.
The condition s n → ∞ in Lemma 2.6 means that k≥0 1 c k = ∞, which, in a sense, is the N → ∞ limit of condition (A.6).

A.2 The clustering distribution of linearly interacting diffusions
Let D ⊂ R d be open, bounded, and convex, and let W be a renormalization class on D. Fix migration constants (c k ) k≥0 and assume that s n → ∞ and s n+1 /s n → 1 + γ * for some γ * ∈ [0, ∞]. Recall the definition of the iterated probability kernels K w,(n) in (2.4). Recall Conjecture 2.7. Assuming that the rescaled renormalized diffusion matrices s n F (n) w converge to a limit w * , we can make a guess about the limit of the iterated probability kernels K w,(n) .
(ii) If γ * = 0, then K * x = lim t→∞ P x [I 0 t ∈ · ], (A.14) where (I 0 s ) s≥0 is the diffusion process with generator d i,j=1 w * ij (y) ∂ 2 ∂y i ∂y j . 3 For γ * > 0, the situation is more complex. In this case at the right-hand side of (A.16) we expect the law D P θ [yT ∈ dx]K * x , where y solves the SDE dyt = 1 γ * (θ − yt)dt + √ 2σ * (yt)dBt and σ * is a root of the diffusion matrix w * . Note that in this case the right-hand side of (A.16) depends on T .
where K * is the kernel in (A.14).
In particular, consider the case that the migration constants (c k ) k≥0 are of the form c k = r k for some r > 0. In this case, s n+1 /s n → 1 r ∨ 1, and s n → ∞ if and only if r ≤ 1. One can check (see (A.6)) that for fixed N ≥ 2, the random walk with the kernel a in (A.5) is recurrent if and only if r ≤ 1. The critical case r = 1 corresponds to a critically recurrent random walk. For a precise definition of critical recurrence, see [Kle96, formula (1.15)]. For r = 1, we expect that the double limit in (A.16) can be replaced by a single limit. More precisely, for each fixed N ≥ 2, we expect that lim t→∞ L(x N 0 (t)) = K * θ . (A.17) In this case, we call K * θ the clustering distribution of x N . The clustering distribution of linearly interacting isotropic diffusions was studied in [Swa00]. We expect (A.17) to hold, even more generally, for all systems of linearly interacting diffusions with a critically recurrent migration mechanism. In particular, we expect (A.17) to hold for symmetric nearest-neighbor interaction on Z d in the critical dimension d = 2. If one is ready to make this enormous leap of faith, then combining Conjectures 2.7 and A.2, one arrives at the following conjecture. x η (t) − x ξ (t) dt + σ(x ξ (t))dB ξ (t), (A.18) with initial condition x ξ (0) = θ ∈ D (ξ ∈ Z 2 ). Then where (I θ s ) s≥0 is the diffusion with generator i,j w * ij (y) ∂ 2 ∂y i ∂y j and initial condition I θ 0 = θ.