Uniqueness problems Particle models and diffusion limits

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 The domain of A is the space of all C 2 -functions on K p that depend on finitely many coordinates only, and a point x ∈ K p we denote as x = x α i α =1,...,p i ∈ . 1.1.57 Infinite systems of interacting diffusion processes of this type and their generaliza- tions are the main subject of study in this dissertation. For definiteness let us write down the operator A above in restricted coordi- nates. With ˆ K p as in 1.1.21 and ˆ K p the space of all points x = x i i ∈ with x i ∈ ˆ K p , we have for any C 2 -function ˆ f that depends on finitely many of the x i only A ˆ f x : = i αβ w p,q αβ x i ∂ 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, 1.1.58 where for any x ∈ ˆ K p and α, β = 1, . . . , p − 1 w p,q αβ x = ⊂{1,...,p} | |=q ∋α,β g xqδ αβ − 1, 1.1.59 with g x as in 1.1.25. A short look at section 1.1.3 learns us that in particular w p,2 αβ x = x α δ αβ − x β w p, p αβ x = γ x γ 1 − γ x γ pδ αβ − 1. 1.1.60

1.1.7 Other models

We briefly mention here some diffusion models that are closely related to the p- type q-tuple models. We start with two models with non-compact state space. Feller’s branching diffusion Consider an urn with balls of one color. With rate one each ball is with equal probabilities replaced by either two or zero balls ‘critical branching’. In the right scaling, this process converges to a diffusion on [0, ∞ whose generator extends the operator A f x : = x ∂ 2 ∂ x 2 f x. 1.1.61 Extensions to more colors are immediate. These models can be viewed as a limit of the p-type 2-tuple model when one of the types occupies almost all the urn.