By M. Aizenman (Chief Editor)

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**Extra resources for Communications In Mathematical Physics - Volume 263**

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Recalling that P = P0 (dξ ) Pξ , this implies ξ EP0 EPξ (Xt · a)2 = EP (Xt · a)2 0 (20) , which gives a way to calculate the diffusion matrix D from the distribution P on . 2 of [DFGW], it is enough to verify the following hypothesis: (a) the environment process is reversible and ergodic; (b) the random variables X[s,t] , 0 ≤ s < t are in L1 ( , P); (c) the mean forward velocity exists: ϕ(ξ ) := L2 −lim t↓0 (d) the martingale Xt − t 0 1 EPξ (Xt ) . t (21) ds ϕ(ξs ) is in L2 ( , P). Let us assume ρ12 < ∞.

E. ξ , (32) where ∇x f is defined in (17), and, moreover, 1 f, (−L)f P0 = 2 P0 (dξ ) ξˆ (dx) c0,x (ξ ) (∇x f (ξ ))2 . (33) Proof. 1]. 2, Chap. 8]; (ii) L is symmetric because Tt is self-adjoint; (iii) the spectrum of L is included in (−∞, 0] by contractivity of the semigroup. Note that (iii) also implies that L is non-positive. We use the abbreviation Lp for Lp = Lp (N0 , P0 ), p = 2 or ∞. s. of (32). Due to Lemma 2, EP0 (λ20 ) < ∞ and in particular P0 (dξ ) ( f )(ξ ) thus implying that 2 ≤ 4 f 2 ∞ EP 0 λ20 < ∞ , : L∞ → L2 is a well-defined operator.

2 (note that, for a suitable positive constant c, ξˆ (dx)cˆ0,x ≤ c λ0 (ξ ) for any ξ ∈ N0 , thus allowing to exclude explosion phenomena from the results of Appendix A). Finally, given ξˆ ∈ ˆ , Xt (ξˆ ) is defined as in (19). Proof. Note that c0,x (ξ ) ≥ e−rc −4 β Ec c˜0,x (ξ ) , where c˜x,y (ξ ) := χ |Ex | ≤ Ec , |Ey | ≤ Ec , |x − y| ≤ rc ) , x, y ∈ ξˆ . Then (16) implies that (a · Da) ≥ e−rc −4 β Ec g(a), where g(a) := inf f ∈L∞ (N0 ,P0 ) P0 (dξ ) ξˆ (dx) c˜0,x (ξ ) a · x + ∇x f (ξ ) 2 ≥0. By the same arguments used in the proof of Proposition 3 one can show that there is a unique self–adjoint operator L˜ on L2 (N0 , P0 ) such that ˜ )(ξ ) := (Lf ξˆ (dx) c˜0,x (ξ ) ∇x f (ξ ) , ∀ f ∈ L∞ (N0 , P0 ).

### Communications In Mathematical Physics - Volume 263 by M. Aizenman (Chief Editor)

by John

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