Definitions Introduction and main result

By g ∗ we denote the unique solution continuous on K and twice continuously differentiable on K ◦ of the equation − 1 2 g ∗ = 1 on K ◦ g ∗ = 0 on ∂ K , 4.1.17 with = α ∂ ∂ x α 2 the Laplacian. By D K [0, ∞ we denote the space of cadlag functions from [0, ∞ to K , equipped with the Skorohod topology. We use the symbol ⇒ to denote weak convergence of processes in path space, i.e., the weak convergence of their laws as probability measures on D K [0, ∞. d = 1: For one-dimensional K , exact results have been derived about the asymp- totic behavior of ˆ X N,k . Let K = [0, 1]. Let H be the class of Lipschitz continuous functions g : [0, 1] → [0, ∞ satisfying gx = 0 ⇔ x ∈ {0, 1}. Then 4.1.16 has a unique strong solution for every g ∈ H , x ∈ K and c ∈ [0, ∞. We cite the following result from Dawson and Greven [10]. Proposition 4.1.1 If g ∈ H , then for every k ≥ 0 ˆX N,k ⇒ Z g k , c k θ as N → ∞, 4.1.18 where c k : = σ k c k , and g k ∈ H is the function g k : = σ k F c k −1 ◦ · · · F c 1 ◦ F c g. 4.1.19 Here F c : H → H is a renormalization transformation given by F c gx : = [0,1] gyν g,c x d y x ∈ [0, 1], 4.1.20 where ν g,c x is the unique equilibrium distribution of 4.1.16. In Baillon et al. [1] the behavior of the function g k was studied in the limit as k → ∞. The main result of that paper is the following: Proposition 4.1.2 For any g ∈ H lim k →∞ sup x ∈[0,1] |g k x − g ∗ x | = 0, 4.1.21 where the function g ∗ is given by g ∗ x : = x1 − x x ∈ [0, 1]. 4.1.22 Since c k → c ∗ = c 1 − c as k → ∞, 4.1.23 we may combine Propositions 4.1.1 and 4.1.2 to obtain see [21] ˆX N,k t ⇒ Z g ∗ , c ∗ θ N → ∞, k → ∞, 4.1.24 where the limits need to be taken in the order indicated. The fact that large blocks are always governed by the same diffusion function g ∗ , regardless of the diffusion function g for single components, is described by saying that systems of the form 4.1.5 exhibit universal behavior on large space-time scales. d ≥ 2: In den Hollander and Swart [21] we conjectured that the result in 4.1.24 generalizes to higher-dimensional K , where the function g ∗ is given by 4.1.17. However, we were only able to prove some partial results in this direction. The main difficulty we encountered was that it is very hard to prove for a function g that the transformations F c k are well-defined on g and on all its iterates F g, F c ◦ F g and so on. This requires that uniqueness holds for solutions of 4.1.16 for all x ∈ K , and not only for g itself, but also for its iterates. This is often already a problem for g = g ∗ In the present paper, we do not attempt to prove seperate limit theorems for N → ∞ and k → ∞ such as Proposition 4.1.1 and Proposition 4.1.2, but instead let N and k tend to infinity together. In the case of one-dimensional K , the Propo- sitions 4.1.1 and 4.1.2 already imply, through formula 4.1.24, that it is possible to choose N i , k i , tending to infinity as i → ∞, such that ˆX N i , k i t ⇒ Z g ∗ , c ∗ θ as i → ∞. 4.1.25 In this paper, we try to generalize this result to higher-dimensional K . Moreover, we investigate how N i and k i must be chosen for the convergence in 4.1.24 to hold. In what follows, we fix integers N i ≥ 2, k i ≥ 1 i ∈ N tending to infinity as i → ∞. We write ˆX i : = ˆX N i , k i β i : = β N i , k i ˆ F i t : = F i β i t , 4.1.26 where F i t t ≥0 is the filtration generated by the process X N i , k i . Unfortunately, we do not know yet how to prove 4.1.25 in the case of higher-dimensional K . How- ever, we can show that the drift and the diffusion rate of the process ˆ X N i , k i converge to the functions x → c ∗ θ −x and x → g ∗ x, respectively. The following scaling theorem is our main result. Theorem 4.1.3 Let N i ≥ 2, k i ≥ 1 i ∈ N be integers tending to infinity as i → ∞ such that lim i →∞ k i log N i = 0, 4.1.27 and assume that g ∗ ∈ C 1 K . Then there exist ˆ F i t t ≥0 -adapted processes ˆ B i = ˆ B i t t ≥0 and ˆ G i = ˆ G i t t ≥0 with sample paths in D R d [0, ∞, D R [0, ∞, re- spectively, such that for each f ∈ C 2 K the process M i t t ≥0 , given by M i t : = f ˆX i t − t α ˆB i,α s ∂ ∂ x α f ˆX i s + ˆ G i s α ∂ ∂ x α 2 f ˆX i s ds 4.1.28 is an ˆ F i t t ≥0 -martingale. Moreover, for each T 0 lim i →∞ E T ˆB i t − c ∗ θ − ˆX i t 2 dt = 0. 4.1.29 and lim i →∞ E T ˆG i t − g ∗ ˆ X i t dt 2 = 0. 4.1.30 Formula 4.1.28 identifies ˆ B i and ˆ G i as the drift and the diffusion rate of the process ˆ X i . Thus, formulas 4.1.29 and 4.1.30 show that these local character- istics of the process ˆ X i converge, as i → ∞, and that their limits are universal in the diffusion function g for single components. The fact that this happens for all N i , k i satisfying condition 4.1.27 is new even in the case of one-dimensional K . Theorem 4.1.3 is a universality result of the type we were originally after in den Hollander and Swart [21]. The author believes that also the convergence in 4.1.25 holds under condition 4.1.27, but there are at present two technical obstacles to proving a result of that form. The first difficulty comes from the fact that uniqueness of solutions to 4.1.16 for arbitrary g = g ∗ , c = c ∗ and x = θ remains an open problem. The following partial results are known. 1. Strong uniqueness is known to hold for θ ∈ K ◦ , c ∗ sufficiently large, and K satisfying mild regularity conditions Theorem 1.9 in Den Hollander Swart [21].