Definitions Convergence of the diffusion rate

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 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 .