The p-type q-tuple model with migration

isomorfic to K p −1 , it is then possible to prove strong uniqueness of solutions to 1.1.48 by induction. For this technique, see Example 3.1.8 below. For c sufficiently large, it is possible to show that solutions to the martingale problem for A p,q + B θ, c never hit the boundary ∂ K p , so that solutions to 1.1.48 are strongly unique for all time. For this technique, see Theorem 2.2.9 below. However, for c positive but not too large the process X with probability one reaches the boundary ∂ K p in a finite time. In fact, it hits ∂ K infinitely often in a finite time, each time bouncing back from it. About processes with such behavior very little is known. Certainly the standard strong uniqueness results do not apply, except in the one-dimensional i.e. 2-type case. For the p-type 2-tuple model, there are several ways to circumvent this prob- lem. For example, it is possible to find a root σ that is lower-triangular. This corresponds the fact that each subselection of the colors is itself a Markov process following a Wright-Fisher diffusion. Thus, the idea is that one can first prove strong uniqueness for one color, using one-dimensional techniques, then prove strong uniqueness for the second color conditional on the first one, and so on. In another approach, one can prove weak uniqueness for the p-type 2-tuple model with migration by more or less explicitly calculating all moments of X t. Here one uses the fact that A p,2 + B θ, c maps a polynomial of degree n into a polynomial of degree at most n. Thus, the time evolution of all moments up to n-th order is described by a closed system of equations, that is easily seen to have a unique solution. 3 This technique has the advantage that it also proves that the closure of A p,2 + B θ, c generates a Feller semigroup. For general p-type q-tuple models, one can see that A p,q maps a polynomial of degree n into a polynomial of degree n − 2 + q, and hence for q ≥ 3 the time evolution of moments up to n-th order cannot be expressed in a closed system. It seems that duality techniques involving moments that are known to work in certain other models also fail here, and uniqueness of solutions to the martingale problem for A p,q + B θ, c with p, q ≥ 3, for general θ ∈ K p and c 0, is still an open problem.

1.1.6 Interacting p-type q-tuple models

We now consider a collection of urns, indexed by a finite or countable Abelian group , with group operation i + j inverse −i unit element 0. 1.1.49 3 In fact, this solution can be represented in terms of a dual process, which is sometimes handy in calculations. But here this duality is not essential. For example, may be the n-dimensional integer lattice Z n , or a finite part of Z n with periodic boundary conditions. We will also frequently consider the case that = N , the N -dimensional hierarhical group see below. We fill the urns according to a Poisson process, where balls of color α occur with intensity ρθ α , with θ α α =1,...,p = θ ∈ K p and ρ 0. Thus the total number of balls in each urn is Poisson distributed with mean ρ, and a given ball is with proba- bility θ α of color α. We introduce the following migration mechanism between our urns. We assume that balls independently of each other perform continuous-time random walks on , where a ball in urn j ∈ jumps to urn i ∈ with rate a j − i. 1.1.50 Here the migration kernel a : → [0, ∞ is a function satisfying i ai ∞. 1.1.51 We further assume that every q-tuple of balls present at a certain moment in an urn is subject to the resampling mechanism descibed in section 1.1.1 with rate ρ 2 −q . Let us write Y ρ ,α i t for the number of balls of color α in urn i at time t, and let us consider the process X ρ = X ρ t t ≥0 = X ρ ,α i t α =1,...,p i ∈, t≥0 1.1.52 given by X ρ ,α i t : = 1 ρ Y α i t. 1.1.53 Then we expect X ρ to converge, as ρ → ∞, to a diffusion process X = X t t ≥0 = X α i t α =1,...,p i ∈, t≥0 , 1.1.54 with initial condition X i = θ i ∈ , 1.1.55 that solves the martingale problem for the operator A, given by A f x : = i ⊂{1,...,p} | |=q γ ∈ x γ α,β ∈ qδ αβ − 1 ∂ 2 ∂ x α i ∂ x β i f x + i j,α a j − ix α j − x α i ∂ ∂ x α i f x. 1.1.56