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4. Mean forward velocity and infinitesimal square displacement. Proposition 4. Let P satisfy ρ12 < ∞ and let ϕ be the Rd -valued function on N0 and ψ be the function on N0 with values in the real symmetric d × d matrices, respectively defined by ϕ(ξ ) = ξˆ (dx) c0,x (ξ ) x , (a · ψ(ξ )a) = ξˆ (dx) c0,x (ξ ) (a · x)2 . (39) (i) ϕ(ξ ) is in L2 (N0 , P0 ) and is equal to the mean forward velocity given by the convergent L2 -strong limit (21). (ii) (a · ψ(ξ )a) is in L2 (N0 , P0 ) and is equal to the infinitesimal mean square displacement given by the convergent L2 -strong limit (22).

12]) and can be described roughly as follows: After arriving at site y ∈ ξˆ , the particle waits an exponential time with parameter λy (ξ ) and then jumps to another site z ∈ ξˆ with probability pξ (z|y) := cy,z (ξ ) . λy (ξ ) (26) More precisely, consider ξ ∈ N0 such that 0 < λz (ξ ) < ∞ for any z ∈ ξˆ and set ξ ˜ ξ := supp(ξˆ ) N . A generic path in ˜ ξ is denoted by X˜ nξ . Given x ∈ ξˆ , let P˜ x n≥0 be the distribution on ˜ ξ of a discrete–time random walk on supp(ξˆ ) starting in x and having transition probabilities p ξ (z|y).

Of (72) is equal to (1) (1) ϕc , et Lc ϕc Pˆ c . 0 ξˆ Lemma 5. Let τN and τN,x be defined as in (68) and (71), and let M = 2N − 2[N α ]. Then ξˆ ξˆ ξˆ lim EPˆ c mN χ (x ∈ CM ) PN,x τN ≤ t = 0, (73) ξˆ lim EPˆ c mN χ (x ∈ CM ) Pˆ c ˆ τN,x ≤ t Sx ξ N↑∞ = 0. (74) N↑∞ Mott Law as Lower Bound for a Random Walk in a Random Environment 49 Proof. One can check by a coupling argument that the two expectations in (73) and (74) coincide: for each N ∈ N+ , ξˆ ∈ Nˆ and x ∈ CM ∩ supp(ξˆ ), one can define a probability ξˆ × ˆ such that measure µ on N ξˆ µ(A × ˆ ) = PN,x (A), ξˆ N µ( × B) = Pˆ c ˆ (B), ∀A ∈ B( Sx ξ ξˆ N ), ∀B ∈ B( ˆ ), ξˆ ξˆ and such that, µ almost surely, τN (ω) = τN,x (ξˆ ) and ωs = x+Xs (ξˆ ) for any 0 ≤ s < τN .