Proof of Lemma 3.1.6 Proofs of Theorem 3.1.5, Lemma 3.1.6

We apply Lemma 3.4.3 to see that the function x → v ∗ x + |x − θ| 2 3.4.50 is w-harmonic. Lemma 3.1.6 therefore tells us that for all t ≥ 0 E[v ∗ X t] + VarX t = E v ∗ j P t j X j + Var j P t j X j . 3.4.51 By Lemma 3.3.3, Lemma 3.3.2 and the spatial ergodicity of L X 0 this implies that lim t →∞ E[v ∗ X t] + C t = v ∗ θ . 3.4.52 Combining this with 3.4.49 we see there exists a T such that for all t ≥ T ∂ ∂ t C t i ≥ j a S j − iC t j − C t i + 2λδ i 0 v ∗ θ − C t − 2ε. 3.4.53 Random walk representation: Let us define D t i : = v ∗ θ − C t i − 2ε i ∈ , t ≥ 0. 3.4.54 Then 3.4.53 can be rewritten as ∂ ∂ t D t i ≤ j a S j − iD t j − D t i − 2λδ i 0 D t t ≥ T . 3.4.55 We note that since t → C t is continuously differentiable in B, so is t → D t . Arguing as in the proof of Lemma 3.3.1, we can represent solutions of the differen- tial inequality 3.4.55 in terms of a contracting semigroup P λ t t ≥0 on B, with generator G f i : = j a S j − i f j − f i − 2λδ i 0 f 0 f ∈ B. 3.4.56 This semigroup is related to a random walk on that jumps from i to j with rate a S j − i and that is killed at the origin with rate 2λ. When P λ t j, i denotes the probability that this random walk, starting from a point i , is in j at time t, then P λ t f i = j P λ t j, i f j f ∈ B, 3.4.57 and for solutions of 3.4.55 we have the representation D T +t i ≤ j P λ t j, i D T j t ≥ 0. 3.4.58 Convergence of the covariance function: If a S is recurrent, then the random walk is killed with probability one. This means that for each i ∈ lim t →∞ j P λ t j, i = 0. 3.4.59 Combining this with 3.4.58 and using the boundedness of K we see that for each i ∈ there exists a T ′ such that for all t ≥ T ′ C t i ≥ v ∗ θ − 3ε. 3.4.60 We have thus shown that for every i ∈ lim inf t →∞ C t i ≥ v ∗ θ . 3.4.61 On the other hand, with the help of formula 3.4.52 it is easy to see that lim sup t →∞ C t ≤ v ∗ θ . 3.4.62 By Cauchy-Schwarz we have C t i ≤ C t 0 for all i ∈ compare 3.3.35, and hence lim t →∞ C t i = v ∗ θ ∀i ∈ . 3.4.63 Convergence of X t: Let S t t ≥0 be the semigroup in 3.1.40. Pick any function φ ∈ C K . By Lemma 3.4.2 φ x − S ∞ φ x = 0 x ∈ ∂ w K . 3.4.64 Formulas 3.4.52 and 3.4.63 imply that lim t →∞ E[v ∗ X t] = 0. 3.4.65 Since v ∗ is continuous, non-negative, and zero only at ∂ w K see Lemma 3.4.3 formulas 3.4.64 and 3.4.65 imply that lim t →∞ E[φX t] − E[S ∞ φ X t] = 0. 3.4.66 Now S ∞ φ ∈ H Lemma 3.4.2 and therefore Lemmas 3.1.6, 3.3.2 and 3.3.3 imply that lim t →∞ E[S ∞ φ X t] = S ∞ φθ = K Ŵ θ d xφx. 3.4.67 Thus we see that X t ⇒ X ∞ as t → ∞, 3.4.68 where the law of X ∞ is given by L X ∞ = Ŵ θ . 3.4.69 Convergence of X t: Formula 3.4.61 and Cauchy-Schwarz imply that for all i, j ∈ lim t →∞ E |X i t − X j t | 2 = 0. 3.4.70 Combining this with 3.4.68 we easily see that for each finite ⊂ the collection X i t i ∈ converges weakly to a limit X i ∞ i ∈ . By the fact that continuous functions depending on finitely many coordinates only are dense in C K see the proof of Lemma 3.2.1 this implies weak convergence of X t.

3.4.5 Proof of Corollary 3.1.7

Under the condition w = λw ∗ , most inequalities in the proof of Theorem 3.1.5 can be replaced by equalities. In fact, under the weaker condition recall that v ∗ = tr w ∗ tr w = λv ∗ , 3.4.71 we have equality in 3.4.48 with ε = 0. Since we are working with the initial condition X i = θ for all i ∈ , formula 3.4.52 strengthens to E[v ∗ X t] + C t = v ∗ θ t ≥ 0. 3.4.72 Formula 3.4.58 with T = 0 = ε and an equality sign reads v ∗ θ − C t i = j P λ t j, i v ∗ θ − C j , 3.4.73 where C j = 0 for all j. In this way we find that C t i = v ∗ θ 1 − j P λ t j, i . 3.4.74 Here 1 − j P λ t j, i is the same as the probability K λ t i appearing in 3.1.54. One can derive formula 3.1.54 in a similar way as formula 3.4.74. For that, one needs to replace the covariance function C t i by a covariance matrix function C t j − i αβ : = E[X α i t − θ α X β j t − θ β ]. 3.4.75 Generalizing Lemma 3.4.3 one then finds that, for each α, β, the function x → w ∗ αβ x 3.4.76