Strategy of the proof

for any function f ∈ C 2 K ◦ ∩ C K satisfying α ∂ ∂ x α 2 f x = 0 x ∈ K ◦ . 4.4.14 Formula 4.4.13 says that harmonic functions of a component evolve under the semigroup associated with the evolution in 4.4.3 as if the diffusion function g is zero. The assumption of local equilibrium now leads to the relation E[ f X N i β i t + s] = f ˆθ, 4.4.15 which may be described by saying that the ‘harmonic mean’ of X N i β i t + s is ˆθ. We next note that the function x → dg ∗ x + |x − ˆθ| 2 4.4.16 is harmonic. Therefore, combining 4.4.15 and 4.4.11, we find that µ i E[gX N i β i t + s] = g ∗ ˆ θ − E[g ∗ X N i β i t + s]. 4.4.17 STEP 4: We will show that µ i ∼ σ k i i → ∞. Hence 4.4.17 becomes σ k i E[gX N i β i t + s] ∼ g ∗ ˆ θ − E[g ∗ X N i β i t + s] i → ∞. 4.4.18 Since σ k i tends to infinity and the right-hand side of 4.4.18 is bounded by g ∗ ∞ , it follows that E[gX N i β i t + s] tends to zero as i → ∞. This means that, with high probability, the components X N i ξ β i t + s of the system are concentrated near the boundary of K , i.e., the system clusters. Since g ∗ is continuous on K and zero on ∂ K , it follows that also E[g ∗ X N i β i t + s] tends to zero as i → ∞. Hence, using 4.4.18 once more we see that lim i →∞ σ k i E[gX N i β i t + s] = g ∗ ˆ θ . 4.4.19 STEP 5: We now consider the k i -block {ξ ∈ N i : ξ ≤ k i }. The k i − 1-blocks that the k i -block consists of, N i in total, all reach equilibrium on the time scale T i , and they do so independently of each other. Hence we expect a law of large numbers to apply. In particular, we expect that lim i →∞ Var N −k i i ξ : ξ≤k i σ k i gX N i ξ β i t + s = 0. 4.4.20 Inspecting the definition of ˆ G i in 4.2.2, we see that 4.4.19 and 4.4.20 imply the convergence of ˆ G i to g ∗ ˆ X t , as claimed in 4.1.30. In what follows we will have to turn the heuristic reasoning in Steps 1–5 into a solid proof. The main difficulty we have to overcome is that, as i tends to infinity, not only does T i our time scale tend to infinity, so do N i and k i . We therefore cannot really say that the system X N i ξ β i t + s ξ : ξ≤k i −1, s∈[0,T i ] 4.4.21 tends to equilibrium as T i → ∞, because the space it lives on changes with i. We will have to find a way to measure how close the system is to equilibrium. We will do so by looking at the system at an exponentially distributed random time with mean T i , rather than at a fixed time T i , i.e., we effectively take a Laplace transform with respect to the time variable. For the distribution of the system at this random time we will derive an equilibrium equation with an error term. Extending a technique first used in Den Hollander Swart [21], we will reformulate the heuristic line of reasoning that led us to formula 4.4.19 in such a way that it depends on the equilibrium equation only. In this way we are able to control the errors that were made to derive 4.4.19. In order to justify also 4.4.20 in some rigorous form, we condition the system on one k i − 1-block and show that this has a negligable effect on the behavior of other k i − 1-blocks. In this way we are finally able to justify formula 4.1.30 rigorously.

4.4.2 Definitions

For each N ≥ 0 and x ∈ K N let X N be a solution of the system of stochastic differential equations in 4.1.5 with initial condition X N ξ = x ξ ξ ∈ N , 4.4.22 and for each i ∈ N pick ξ i ∈ N i such that ξ i = k i , 4.4.23 and write E ω [ · ] for the conditional expectation E ω [ · ] := E · X N i ξ i t t ≥0 = ωt t ≥0 , 4.4.24 where ω ∈ C x ξi , K [0, ∞, the space of all continuous functions ω : [0, ∞ → K satisfying ω0 = x ξ i . We choose a regular version of E ω [ · ] with the property that for every ω ∈ C x ξi , K [0, ∞ and for every f ∈ C 2 fin K Ni \{ξ i } and under the conditional law, the process Mt t ≥0 is a martingale, where Mt : = f X N i t − f X N i − t ξ =ξ i ,α ∞ k =1 c N i k −1 X N i , k,α ξ s − X N i ,α ξ s ∂ ∂ x α ξ f X N i s ds − t ξ =ξ i gX N i ξ s α ∂ ∂ x α ξ 2 f X N i s ds. 4.4.25 Here X N i , k ξ t as usual denotes the block average X N i , k ξ t : = 1 N k i η : η−ξ≤k X N i η t k ≥ 0, 4.4.26 where in the sum over η the term with η = ξ i is in this case given by X N i ξ i t = ωt t ≥ 0. 4.4.27 For each i ∈ N we introduce a stopping time τ i possibly defined on an extension of our probability space independent of the process X N i and of the Brownian motions {B N i ξ t } ξ ∈ N t ≥0 and exponentially distributed with mean λ i . Here the λ i are positive numbers that we will later choose in a suitable way. In what follows, R is a constant such that |x − y| ≤ R for all x, y ∈ K .

4.4.3 Block immobility

Lemma 4.4.1 For i ∈ N let X N i be a solution of 4.1.5 with initial condition 4.4.22. There exists a constant M such that for all i ∈ N , k ≥ 1, x ∈ K Ni , ω ∈ C ξ i , K [0, ∞ and t ≥ 0 E ω X N i , k t − x k 2 ≤ M t N k i . 4.4.28 Proof of Lemma 4.4.1: We first treat the case that k ≥ k i . In this case X N i , k t = 1 N k i ω t + ξ : ξ≤k ξ =ξ i X N i ξ t , 4.4.29 and X N i , k t − x k 2 = 1 N 2k i ξ : ξ≤k X N i ξ t − x ξ 2 ≤ 1 N 2k i ξ : ξ≤k ξ =ξ i X N i ξ t − x ξ 2 + 2N k i − 1 N 2k i R 2 . 4.4.30 Let f ∈ C 2 fin K Ni \{ξ i } be given by f y : = 1 N 2k i ξ : ξ≤k ξ =ξ i y ξ − x ξ 2 . 4.4.31 Then the fact that the process M in 4.4.25 is a martingale implies that E ω 1 N 2k i ξ : ξ≤k ξ =ξ i X N i ξ t − x ξ 2 = 1 N 2k i E ω t ξ =ξ i ,α ∞ l =1 c N i l −1 X N i , l,α ξ s − X N i ,α ξ s 1 {ξ≤k} 2 η : η≤k η =η i X N i ,α η s − x ξ ds + 1 N 2k i E ω t ξ =ξ i gX N i ξ s1 {ξ≤k} 2d ds. 4.4.32 Here 1 N 2k i E ω t ξ =ξ i ,α ∞ l =1 c N i l −1 X N i , l,α ξ s − X N i ,α ξ s 1 {ξ≤k} 2 η : η≤k η =η i X N i ,α η s − x ξ ds ≤ E ω t α ∞ l =1 c N i l −1 1 N k i ξ : ξ≤k X N i , l,α ξ s − X N i ,α ξ s 2 1 N k i η : η≤k X N i ,α η s − x ξ ds + E ω t ∞ l =1 c N i l −1 2N k i − 1 N 2k i R 2 ds ≤ E ω t α ∞ l =1 c N i l −1 X N i , l ∨k,α ξ s − X N i , k,α ξ s 2 X N i , k,α η s − x ξ ds + t2 2N k i − 1 N 2k i R 2 ≤ 4t c N i k R 2 + 4t 1 N k i R 2 ≤ 8R 2 t N k i , 4.4.33 where 1 N 2k i E ω t ξ =ξ i gX N i ξ s1 {ξ≤k} 2d ds ≤ 2dg ∞ t N k i . 4.4.34 This completes the proof for k ≥ k i . By inspection we see that our bounds are also valid for k k i . Corollary 4.4.2 For i ∈ N let X N i be a solution of 4.1.5 with initial condition 4.4.22 and let τ i be as in Section 4.4.2. There exists a constant M such that for all i ∈ N , k ≥ 1, x ∈ K Ni and ω ∈ C ξ i , K [0, ∞ E ω X N i , k τ i − x k 2 ≤ M λ i N k i . 4.4.35 Proof of Corollary 4.4.2: Condition on τ i and use Lemma 4.4.1 to get E ω X N i , k τ i − x k 2 = ∞ E ω X N i , k t − x k o 2 λ −1 i e −tλ i dt ≤ ∞ M t N k i λ −1 i e −tλ i dt = M λ i N k i . 4.4.36

4.4.4 An approximate equilibrium equation

Write i : = {ξ ∈ N i : ξ ≤ k i − 1} i ∈ N 4.4.37 for the k i − 1-block around the origin in the hierarchical group N i . For ˆ θ ∈ K and i ∈ N , we introduce an operator A i ˆθ with domain D A i ˆθ : = C 2 K i and A i ˆθ f x : = ξ ∈ i k i −1 k =1 c N i k −1 α [x k,α ξ − x α ξ ] ∂ ∂ x α ξ f x + ξ ∈ i gx ξ α ∂ ∂ x α ξ 2 f x + ξ ∈ i c N i k i −1 α [ ˆ θ α − x α ξ ] ∂ ∂ x α ξ f x = ξ ∈ i η ∈ i a i η − ξ α [x α η − x α ξ ] ∂ ∂ x α ξ f x + ξ ∈ i gx ξ α ∂ ∂ x α ξ 2 f x + ξ ∈ i c N i k i −1 α [ ˆ θ α − x α ξ ] ∂ ∂ x α ξ f x, 4.4.38