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A Trotter-type approach to infinite rate mutually catalytic branching

Dawson and Perkins [Ann. Probab. 26 (1988) 1088--1138] constructed a stochastic model of an interacting two-type population indexed by a countable site space which locally undergoes a mutually catalytic branching mechanism. In Klenke and Mytnik [Preprint (2008), arXiv:0901.0623], it is shown that as the branching rate approaches infinity, the process converges to a process that is called the infinite rate mutually catalytic branching process (IMUB). It is most conveniently characterized as the solution of a certain martingale problem. While in the latter reference, a noise equation approach is used in order to construct a solution to this martingale problem, the aim of this paper is to provide a Trotter-type construction. The construction presented here will be used in a forthcoming paper, Klenke and Mytnik [Preprint (2009)], to investigate the long-time behavior of IMUB (coexistence versus segregation of types). This paper is partly based on the Ph.D. thesis of the second author (2008), where the Trotter approach was first introduced.

1 Introduction and main results

Background and Motivation
In [4] Dawson and Perkins studied a stochastic model of mutually catalytic (continuous state) branching. Two populations live on a countable site space S and the amount of population of type i = 1, 2 at time t at site k ∈ S is denoted by Y i,t (k) ∈ [0, ∞). The populations migrate according to a deterministic heat flow like dynamic that is characterised by the (symmetric) q-matrix A of a Markov chain on S. Locally, the populations undergo critical continuous state branching with a rate that is proportional to the size of the other type at the same place. Formally, this model can be described by a system of stochastic differential equations: Here (W i (k), k ∈ S, i = 1, 2) is an independent family of one-dimensional Brownian motions and Y 0 is chosen from a suitable subspace of ([0, ∞) 2 ) S . The parameter γ ≥ 0 can be thought of as being the branching rate for this model. Dawson and Perkins showed that there is a unique weak solution of (1.1) and studied the longtime behaviour of this model. They also constructed the analogous model in the continuous setting on R instead of S.
For the model with S = Z and A the q-matrix of symmetric nearest neighbour random walk, the model tends to a state with spatially segregated types. In an approach to describe quantitatively the cluster growth, a space and time rescaling argument suggests that it is useful to study the limit as γ → ∞ first. Studying this limit requires a formal description of the limit process X, construction of the limit process and establishing convergence of Y as γ → ∞.
This programme is carried out for a process with S a singleton in [8] and for a countable site space S in [9]. Furthermore, in [10] the longtime behaviour is studied which shows a dichotomy between coexistence and segregation of types depending on the potential properties of the matrix A.
In [9] the process X is characterised both via a martingale problem and as the solution of a system of stochastic differential equations of jump type. While the construction of X was performed via constructing approximate solutions of the stochastic differential equations, here the aim is to present a different approach via a Trotter approximation scheme.
The main idea is described via the following heuristics. Denote by a t the matrix of time t transition probabilities of the continuous time Markov chain with q-matrix A. Furthermore, let Q t (y, dy ′ ) denote the transition kernel for equation (1.1) with A = 0. It is not hard to see that Q t converges as t → ∞ to some kernel Q. In fact, if A = 0, then all colonies evolve independently, and each colony is a time transformed planar Brownian motion in (0, ∞) 2 stopped when it hits the boundary. Hence Q is the product of the harmonic measures of planar Brownian motions in the upper right quadrant. Now let ε > 0 and define X ε 0 = Y 0 and inductively let X ε (k+1)ε be distributed, given X ε kε like Q(a ε X ε kε , dy ′ ). This amounts to an interlaced dynamics where deterministic heat flow and random infinite rate branching alternate. The main result of this paper is that in fact the processes X ε converge as ε → 0 to the infinite rate mutually catalytic branching process X constructed in [9]. In Section 1.2 and 1.3 we give a formal description of this X.
The idea of using a Trotter type approach for the construction of the infinite rate mutually catalytic branching process is taken from the PhD thesis [12] and parts of the strategy of proof are based on that thesis.

The infinite rate branching process
We start with a definition of the state spaces of our processes. Define E : We can weaken the requirement that A be a q-matrix: Let A = (A(k, l)) k,l∈S be a matrix indexed by the countable set S satisfying A(k, l) ≥ 0 for k = l We fix this β for the rest of this paper.
Define the spaces L f,2 = y ∈ ([0, ∞) 2 ) S : y(k) = 0 for only finitely many k ∈ S , as well as Finally, define the spaces Let Af (k) = l∈S A(k, l)f (l) if the sum is well-defined. Let A n denote the nth matrix power of A (note that this is well-defined and finite by (1.3)) and define Let S denote the (not necessarily Markov) semigroup generated by A, that is We will use the notation Af , S t f and so on also for [0, ∞) 2 valued functions f with the obvious coordinate-wise meaning.
For u ∈ R S define u β = k∈S |u(k)|β(k). (1.6) Note that for f ∈ L β , the expressions Af and S t f are well-defined and that (recall M from (1.4)) That is, the spaces L β and L β,2 are preserved under the dynamics of (S t ).
We will employ a martingale problem in order to characterise the infinite rate mutually catalytic branching process X ∈ D([0, ∞); L β,E ). In order to formulate this martingale problem for X conveniently, for x = (x 1 , x 2 ) ∈ R 2 and y = (y 1 , y 2 ) ∈ R 2 we introduce the lozenge product Note that x ⋄ y = y ⋄ x, hence F is symmetric. For x, y ∈ (R 2 ) S we write whenever the infinite sum is well-defined and let H(x, y) = exp( x, y ). (1.11) Note that the function H(x, y) is well-defined if either x ∈ (R 2 ) S and y ∈ L f,E or x ∈ L β,E and y ∈ L β,E ∞ . It is shown in [8,Corollary 2.4] that the vector space of finite linear combinations n i=1 α i F ( · , y i ), n ∈ N, α i ∈ C, y i ∈ E, is dense in the space C l (E; C) of bounded continuous complex valued functions on E with a limit at infinity. Hence the family H( · , y), y ∈ L f,E , is measure determining for probability measures on L β,E (but not on L β,2 ).
In [9] the following theorem was established: Note that for the uniqueness it is crucial that the single coordinates take values in E. If we would require only values in [0, ∞) 2 , then also the finite rate mutually catalytic branching process Y is a solution of the martingale problem for any γ ≥ 0. In Proposition 1.1, we will see that also our approximate process X ε is a solution to (MP) with the larger state space L β,2 .
In [10,Theorem 1.3] it was shown that the processes Y defined in (1.1) converge to X as γ → ∞ in the Meyer-Zheng topology. Hence the name infinite rate mutually catalytic branching process for X is justified.

The main result
We now define the approximating process X ε in detail. In order to do so we introduce the harmonic measure (1.12) Now for fixed ε > 0, consider the stochastic process X ε with values in L β,2 with the following dynamics: Clearly, the explicit solution is (ii) At time nε, X ε has a discontinuity. Independently, each coordinate X ε nε− (k) = S ε X ε (n−1)ε (k) is replaced by a random element of E drawn according to the distribution Q X ε nε− (k) .
If, for x ∈ E S , we denote by Q(x, · ) = k∈S Q x(k) the Markov kernel of independent displacements, then (X ε nε ) n∈N0 is a Markov chain on L β,E with transition kernel Q ε (x, · ) := Q(S ε x, · ). Note that X ε is a càdlàg process with values in L β,2 (but not in L β,E !) and that, for any y ∈ L f,E , is a martingale. Furthermore, as we will show in Lemma 2.2, we have H(x ′ , y) Q(x, dx ′ ) = H(x, y) for all y ∈ L f,E and x ∈ L β,2 . As an immediate consequence, we get the following proposition.
For all x ∈ L β,E and y ∈ L f,E , and for X ε defined as above with X 0 = x, we have that We will show that X ε converges to a process that takes values in L β,E while preserving this martingale property.
The main theorem of this paper is the following.
With a little bit of good will, this construction can be interpreted as a Trotter product approach. Recall that (under suitable assumptions on the spaces and cores of the involved operators) the Trotter product formula states the following (see, e.g., [6, Corollary 6.7]): If (S t ) t≥0 , (T t ) t≥0 and (U t ) t≥0 are strongly continuous contraction semigroups with generators A, B and C = A + B, respectively, then In our setting T t = Q for all t > 0 and T 0 = id, hence (T t ) is by no means strongly continuous. Nevertheless, Theorem 1 shows that the limit exists.
A nice spin off from this construction is the following statement about the distribution of X t for fixed t.
Theorem 2 For all t ≥ 0, x ∈ L β,E and y ∈ L β ∞ , we have E x [Q Xt,y ] = Q Stx,y . In particular, for all k ∈ S, we have P x [X t (k) ∈ · ] = Q Stx(k) .
As an application of Theorem 2, we consider the interface problem in dimension d = 1. Assume that S = Z and that Af (k) = 1 2 f (k + 1) is the q-matrix of symmetric simple random walk on Z. Hence a t is the time t transition kernel of continuous time rate 1 symmetric simple random walk. Let u, v > 0 and assume that x(k) = (u, 0) for k < 0 and x(k) = (0, v) for k ≥ 0. Let X be the infinite rate mutually catalytic branching process on Z with X 0 = x. Define b t,1 := sup{k ∈ Z : X 1,t (k − 1) > 0} and b t,2 := inf{k ∈ Z : X 2,t (k) > 0}.
We conjecture that b t,1 = b t,2 almost surely. In this case, the position b t := b t,1 could be considered as the interface between the type 1 population (left) and the type 2 population (right). It is a challenging task to find out what the dynamics of (b t ) t≥0 is. By work on the finite branching rate process of [2] and [3], we should have lim sup t→∞ b t = ∞ and lim inf t→∞ b t = −∞. That is, the type at any given site changes again and again infinitely late.
Theorem 2 gives an indication what the distribution of b t is for fixed t. ({0} × (0, ∞)). By an explicit calculation using the density of Q (see Lemma 2.1), we get (1.14). The other two statements follow from the central limit theorem for a t . 2

Outline
The rest of the paper is organised as follows. In Section 2, we collect some basic facts about the harmonic measure Q and prove Proposition 1.1. In Section 3, we derive a submartingale related with X ε and show that the two types of X ε are non-positively correlated. In Section 4, we show relative compactness of the family (X ε , ε > 0). Finally, in Section 5, we finish the proofs of Theorem 1 and 2.
2 The Harmonic Measure Q

Harmonic Measure and Duality
Recall that Q x is the harmonic measure for planar Brownian motion in the upper right quadrant started at x ∈ [0, ∞) 2 and stopped upon leaving (0, ∞) 2 . If x = (u, v) ∈ (0, ∞) 2 , then the harmonic measure Q x has a one-dimensional Lebesgue density on E that can be computed explicitly Proof. This follows from explicitly computing the integral ∞ c Q (u,v) (d(0,v)) in (2.1). 2 Recall F from (1.9). Explicitly computing the Laplacian with respect to the first coordinate gives Hence for y ∈ E, the function F ( · , y) is harmonic for planar Brownian motion B and hence (F (B t , y)) t≥0 is a bounded martingale. If τ denotes the first exit time of B from (0, ∞) 2 , we infer Similarly, since linear functions are harmonic for Brownian motion and using that pth moments of (B t ) t≤τ are bounded for p < 2 (see Lemma 2.4), we can derive Note that (2.5) could also be computed explicitly using Lemma 2.3.

Recall that
3) we get immediately the following lemma.

Moments of the Harmonic Measure
Since the harmonic measure Q does not possess a second moment, our proofs will rely on pth moment estimates for p ∈ (1, 2). Here we collect some of these estimates. Define arctan † as the inverse of the tangent function tan : [0, π] →R. That is, arctan † (x) = arctan(x) + π½ {x<0} .
Proof. This follows from explicitly computing the integral using (2.1).
Then for any p ∈ (0, 2), we have Furthermore, for any p ∈ (1, 2), we have Proof. By the reflection principle and independence of B 1 and B 2 , we get where N 0,t (a, b) = (2πt) −1/2 b a e −r 2 /2t dr is the centred normal distribution with variance t. Hence for any p ∈ (0, 2), (2.9) We can continue this inequality by This gives (2.7). For p ∈ (1, 2), we can continue (2.9) by Interchanging the roles of u and v in (2.10) gives (2.8). 2 Lemma 2.5 For p ∈ (1, 2), there exists a constant C p < ∞ such that for every x ∈ E and i = 1, 2, we have Proof. By the Burkholder-Davis-Gundy inequality (see, e.g., [5, Theorem VII.92]) and Lemma 2.4, (B i,t ) t≤τ is a uniformly integrable martingale. Hence, by Jensen's inequality, The claim (2.12) could be checked either by a direct computation using Lemma 2.3 or by proceeding as follows: Let B and τ be as in Lemma 2.4. Using the Burkholder-Davis-Gundy inequality and then Lemma 2.4, we get The Approximating Process X ε 3.1 Martingale property of X ε Proposition 3.1 Let x ∈ L β,E and k ∈ S. Define the process N ε,x for i = 1, 2, k ∈ S and t ≥ 0 by (i) For each i = 1, 2 and k ∈ S, the process (N ε,x i,t (k)) t≥0 is a martingale with respect to the natural filtration. In particular, A(k, k)) and note that |λ| < ∞ by assumption (1.3). Define Then Z ε i (k) andZ ε i are nonnegative submartingales.
Proof. (i) This is an immediate consequence of the definition of X ε and (2.5).
(ii) Since A(k, l) ≥ 0 for all k = l, we have Together with (2.5) this shows that Z ε i is a submartingale. As a sum of submartingales, alsoZ ε i is a submartingale. 2 Corollary 3.2 For every K, T > 0 and any set G ⊂ S, we have

(3.2)
In particular, Proof. This is an immediate consequence of Proposition 3.1 and Doob's inequality. 2
For the induction base m = n, this is true by definition of X ε . Now assume that we have shown the statement for some m ≥ 1. Using the induction hypothesis in the first line and Lemma 3.3 in the second line, we get The proof is similar to the proof of Corollary 3.4. (Note that X ε t , y ∈ [0, ∞) 2 almost surely, since X ε t ∈ L β,E almost surely.) 2

Tightness
The goal of this section is to show the following proposition. By Prohorov's theorem, in order to show relative compactness of (X ε ), it is enough to show tightness of (X ε ).
The strategy of proof is to check the compact containment condition for X ε (Lemma 4.4) and then use Aldous's tightness criterion for functions h(X ε t ), where h : L β,2 → R is Lipschitz continuous and depends on only finitely many coordinates.
We start by collecting some basic facts about compact sets and separating function spaces. The proofs of the following statements are standard and are therefore omitted here.
Now for any n ∈ N, choose a finite S n ⊂ S such that n e λT S T (x 1 + x 2 ) ½ S\Sn β < 2 −n−1 η and define B n := y ∈ L β,2 : (y 1 + y 2 )½ S\Sn β < 1/n . According to Corollary 3.2, we have P x X ε t ∈ B n for all t ∈ [0, T ] ≥ 1 − 2 −n−1 η. Now let Γ by the closure of A K ∩ ∞ n=1 B n . Then P x X ε t ∈ Γ for all t ∈ [0, T ] ≥ 1 − η and by Lemma 4.2, Γ is compact. Proof. The idea is to use Aldous's criterion for tightness in D([0, ∞); R). As h is bounded, (Y ε t ) ε>0 is tight for each t ≥ 0. Hence by Aldous's criterion (see, e.g., [1,Eq. (13)] or [7, Section VI.4a]), we need to show the following: For any η > 0 and T > 0, there exist δ > 0 and ε 0 > 0 such that for any stopping time τ ≤ T , we have sup Since h is Lipschitz continuous and depends on only finitely many coordinates, it is enough to consider the case, where h(x) = x i (k) for some k ∈ S and i = 1, 2. Using Markov's inequality, it is enough to show that for any η > 0 and T > 0 there exist δ > 0 and ε 0 > 0 such that for any stopping time τ ≤ T , we have Then