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Then the diffusion matrix D for the ξ process (Xt )t≥0 in Theorem 2 admits the following lower bound D ≥ ρc −rc −4 β Ec e Dc (rc , Ec ) , ρ Mott Law as Lower Bound for a Random Walk in a Random Environment 41 where (a · Dc (rc , Ec ) a) := EPˆ c (a · ψc a) 0 ∞ − 2 dt ϕc · a , et Lc ϕc · a 0 Pˆ 0c , (51) and Lc is the unique self–adjoint operator on L2 (Nˆ 0 , Pˆ 0c ) such that (Lc f )(ξˆ ) = ξˆ (dx) cˆ0,x ∇x f (ξˆ ) , ∀ f ∈ L∞ (Nˆ 0 , Pˆ 0c ) . (52) One can prove by the same arguments used in the proof of Proposition 3 that Lc is well-defined and self–adjoint.
G. [DFGW]. Proof. We give the proof for the continuous–time process, the discrete–time case being similar. We first verify the symmetric property pt (ξ |ξ ) = pt (ξ |ξ ). Actually, thanks to the construction of the dynamics given in Sect. 2, one can show that for any positive integer n and any ξ = ξ (0) , ξ (1) , . . , ξ (n−1) , ξ (n) = ξ ∈ N0 , Pξ n∗ (t) = n, ξR1 =ξ (1) , . . , ξRn =ξ (n) =Pξ n∗ (t) = n, ξR1 = ξ (n−1) , . . , ξRn =ξ (0) , where, given ξ ∈ , R1 (ξ ) < R2 (ξ ) < . . denote the jump times of the path ξ .
If ξ ∈ W, the transition probabilities are p(ξ |ξ ) := P˜ ξn+1 = ξ |ξn = ξ = λ−1 0 (ξ )c0,x (ξ ) 0 if ξ = Sx ξ , otherwise . Note that, due to (28) and the symmetry of the jump rates (2), λ0 (ξ )p(ξ |ξ ) = λ0 (ξ )p(ξ |ξ ). Mott Law as Lower Bound for a Random Walk in a Random Environment 35 Proposition 2. Let ρ2 < ∞. e. EP f (ξ0 )g(ξt ) = EP g(ξ0 )f (ξt ) ∀ f, g ∈ F(N0 ) , ∀ t > 0 , (29) and is (time) ergodic if P is ergodic. Similarly, the discrete-time Markov process (ξn )n≥0 ˜ is reversible and is (time) ergodic.