Block immobility Convergence of the diffusion rate

where a i ξ : = k i −1 k =ξ 1 N k i c N i k −1 ξ ∈ i . 4.4.39 Lemma 4.4.3 below gives us an estimate of how close the expectation of functions f i of large k i −1-blocks is to their equilibrium value with respect to the dynamics described by the operator A i x ki . Lemma 4.4.3 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. For i ∈ N , let f i ∈ C 2 K i , let ∇ ξ f i : = ∂ ∂ x α ξ f i α =1,...,d , 4.4.40 and let ∇ ξ f i ∞ be the supremum norm of |∇ ξ f i | the Euclidean norm of ∇ ξ f i . Then there exist constants M 1 , M 2 such that for all i ∈ N , x ∈ K Ni and ω ∈ C ξ i , K [0, ∞ E ω A i x ki f i X N i τ i ≤ 2λ −1 i f i ∞ + M 1 c k i −1 N 1 − 3 2 k i i λ 1 2 i + M 2 c k i N −k i i ξ ∈ i ∇ ξ f i ∞ . 4.4.41 Here, in the left-hand side of 4.4.41, we lift the function A i x k f i to the space K Ni in the obvious way. Proof of Lemma 4.4.3: Use the fact that the process M in 4.4.25 is a martingale and apply optional stopping to see that E ω [ f i X N i τ i ] − E ω [ f i X N i 0] = E ω τ i A f i X N i t dt , 4.4.42 where for x ∈ K Ni A f i x = ξ ∈ i ,α ∞ k =1 c N k −1 [x k,α ξ − x α ξ ] ∂ ∂ x α ξ f i x + ξ ∈ i ,α gx ξ ∂ ∂ x α ξ 2 f i x, 4.4.43 and we have lifted the function f i to K Ni in the obvious way. Note that ξ i ∈ i , so that in our summations we do not have to write ξ = ξ i . Formula 4.4.42 can be rewritten as ∞ dt E ω f i X N i t λ −1 i e −tλ i − E ω f i X N i β i t = ∞ dt E ω t du A f i X N i u λ −1 i e −tλ i = ∞ dt t du E ω A f i X N i u λ −1 i e −tλ i = −e −tλ i t du E ω A f i X N i u ∞ t =0 − ∞ dt −e −tλ i E ω A f i X N i t = ∞ dt E ω A f i X N i t e −tλ i , 4.4.44 leading to the equation E ω A f i X N i τ i = λ −1 i E ω f i X N i τ i − E ω f i X N i . 4.4.45 Here A f i X N i τ i = ξ ∈ i ∞ k =1 c N i k −1 α [X N i , k,α ξ τ i − X N i ,α ξ τ i ] ∂ ∂ x α ξ f i X N i τ i + ξ ∈ i gX N i ξ τ i α ∂ ∂ x α ξ 2 f i X N i τ i . 4.4.46 In view of our heuristic reasoning in Section 4.4.1, we write this as A f i X N i τ i = ξ ∈ i k i −1 k =1 c N i k −1 α [X N i , k,α ξ τ i − X N i ,α ξ τ i ] ∂ ∂ x α ξ f i X N i τ i + ξ ∈ i gX N i ξ τ i α ∂ ∂ x α ξ 2 f i X N i τ i + ξ ∈ i c N i k i −1 α [x k i ,α − X N i ,α ξ τ i ] ∂ ∂ x α ξ f i X N i τ i + ξ ∈ i c N i k i −1 α [X N i , k i ,α τ i − x k i ,α ] ∂ ∂ x α ξ f i X N i τ i + ξ ∈ i ∞ k =k i +1 c N i k −1 α [X N i , k,α τ i − X N i ,α ξ τ i ] ∂ ∂ x α ξ f i X N i τ i . 4.4.47 Here the first two terms represent the ‘internal’ evolution of the k i − 1-block around the origin. The third term comes from their interaction with the k i -block average, which for an appropriate choice of λ i is essentially fixed to its value at time 0. The fourth term is an error term compensating for the fact that the k i -block is not completely fixed at its value at time 0. The fifth term describes the small interaction with the k-blocks for k k i . Combining 4.4.45 and 4.4.47, we arrive at E ω A i x ki f i X N i τ i = λ −1 i E ω f i X N i τ i − E ω f i X N i β i t − c N i k i −1 ξ ∈ i E ω α [X N i , k i ,α τ i − x k i ,α ] ∂ ∂ x α ξ f i X N i τ i − ∞ k =k i +1 c N i k −1 ξ ∈ i E ω α [X N i , k,α τ i − X N i ,α ξ τ i ] ∂ ∂ x α ξ f i X N i τ i . 4.4.48 From this it follows that E ω A i x ki f i X N i τ i ≤ 2λ −1 i f i ∞ + c N i k i −1 E ω X N i , k i τ i − x k i ξ ∈ i ∇ ξ f i ∞ +2R c N i k i ξ ∈ i ∇ ξ f i ∞ , 4.4.49 where we use that N i ≥ 2 and c 1. We now note that by Corollary 4.4.2 E ω X N i , k i τ i − x k i 2 ≤ E ω X N,k τ i − x k i 2 ≤ Mλ i N −k i i . 4.4.50 Inserting this into 4.4.49, we arrive at 4.4.41.

4.4.5 Equilibrium calculations

In this section, we construct ‘test functions’ f i that we will insert into the almost- equilibrium equation in Lemma 4.4.3 to draw certain conclusions about the behav- ior of the k i − 1-blocks. We split the operator A i ˆθ in 4.4.38 as follows: A i ˆθ = B i ˆθ + C i , 4.4.51