Spatially ergodic measures Proofs of Theorem 3.1.3 and Lemma 3.1.4

Here E j P t j ˜ X j 2 = j k P t j P t kE[ ˜ X j 0 ˜ X k 0] = j k P t j P t kC k − j = i j P t j P t i + jC i = i j P t j P t i − j C i = i P 2t i C i , 3.3.39 where all infinite sums are absolutely convergent and we have used that, by the symmetry of a S , P t i = P t −i. Formula 3.3.38 and 3.3.39 show that 3.3.37 holds for i = 0. Using Lemma 3.3.3 we can easily generalize this to arbitrary i ∈ . By Lemma 3.1.4 and Lemma 3.3.1 we have the representation C t i = j P t j − iC j + 2 t P s − iE[trwX t − s]ds. 3.3.40 Taking the limit t → ∞ we get with the help of 3.3.37 that C ∞ i = lim t →∞ 2 t P s − iE[trwX t − s]ds = 2E[trwX ∞] ∞ P t − idt, 3.3.41 where we use the notation in 3.3.33. Let us assume for the moment that E[tr wX ∞] = 0. Then P[X ∞ ∈ ∂ w K ] = 1. On the other hand, 3.3.41 gives C ∞ = 0 and hence P[X ∞ = θ] = 1. This contradicts our assumption that θ ∈ ∂ w K and we conclude that E[tr wX ∞] 0. Therefore P[X ∞ ∈ ∂ w K ] 1 and the claim follows from shift-invariance. Transient a S , P[X i ∞ = X j ∞] 1 ∀i = j ∈ : Let I t t ≥0 be the random walk with kernel a S . Let τ i be the stopping time τ i : = inf{t ≥ 0 : I t = i} i ∈ . 3.3.42 It is easy to see that for all i ∈ ∞ P t − idt = P i [τ ∞] ∞ P t 0dt. 3.3.43 Let us assume that for some i = 0 we have P i [τ ∞] = 1. Then by the symmetry of the random walk, also P [τ i ∞] = 1. But this implies that the random walk starting in 0 visits 0 infinitely often, which contradicts our assumption that it is transient. It follows that P i [τ j ∞] 1 for all i = j. Combining 3.3.43 and 3.3.41 we can conclude that C ∞ i C ∞ ∀i = 0. 3.3.44 Now Cauchy-Schwarz in 3.3.35 implies that P[X i ∞ = X j ∞] 1 for all i = j.

3.4 Proofs of Theorem 3.1.5, Lemma 3.1.6

and Corollary 3.1.7

3.4.1 Potential theory

In this section we collect some elementary facts about w-harmonic functions from potential theory. We assume that for each x ∈ K , the non-interacting equation 3.1.19 has a unique weak solution X x with initial condition X x = x. We denote its last element by X x ∞. We denote the semigroup on BK associated with 3.1.19 by S t t ≥0 and we add a last element S ∞ as in 3.1.41. Restricted to the smaller domain C K ⊂ BK , the S t t ≥0 form a Feller semigroup, whose generator we denote by G. Note that D G ⊂ C K . Lemma 3.4.1 For each solution X to 3.1.19 P[X ∞ ∈ ∂ w K ] = 1. 3.4.1 Proof of Lemma 3.4.1: Since X is a bounded martingale, it converges. Now the lemma is just a special case of Theorem 3.1.3.