Reduction to a spectral bound

5.2.1 Reduction to a spectral bound

Let BΩ denote the Banach space of bounded measurable functions on Ω equipped with the supre- mum norm k · k ∞ . Given V ∈ BΩ, let λV = lim t →∞ 1 t log E ν ρ exp Z t V ξ s ds 5.36 denote the associated Lyapunov exponent. The limit in 5.36 exists and coincides with the upper boundary of the spectrum of the self-adjoint operator L + V on L 2 ν ρ , written λV = sup SpL + V . 5.37 Lemma 5.5. For all t 0 and all bounded and piecewise continuous V : [0, t] → BΩ, E ν ρ exp Z t V u ξ u du ≤ exp Z t λV s ds . 5.38 Proof. In the proof we will assume that s 7→ V s is continuous. The extension to piecewise continuous s 7→ V s will be straightforward. Let 0 = t t 1 · · · t r = t be a partition of the interval [0, t]. Then Z t V u ξ u du ≤ r X k=1 Z t k t k −1 V t k −1 ξ s ds + r X k=1 max s ∈[t k −1 ,t k ] kV s − V t k −1 k ∞ t k − t k −1 ≤ r X k=1 Z t k t k −1 V t k −1 ξ s ds + t max k=1, ··· ,r max s ∈[t k −1 ,t k ] kV s − V t k −1 k ∞ . 5.39 Let S V t t ≥0 denote the semigroup generated by L + V on L 2 ν ρ with inner product · , · and norm k · k. Then S V t = e t λV . 5.40 Using the Markov property, we find that E ν ρ ‚ exp – r X k=1 Z t k t k −1 V t k −1 ξ s ds ™Œ = S V t0 t 1 S V t1 t 2 −t 1 · · · S V t r−1 t r −t r −1 1 1, 1 1 ≤ S V t0 t 1 S V t1 t 2 −t 1 ··· S V t r−1 t r −t r −1 = exp – r X k=1 λ V t k −1 t k − t k −1 ™ . 5.41 Combining 5.39 and 5.41, we arrive at log E ν ρ Z t V s ξ s ds ≤ r X k=1 λ V t k −1 t k − t k −1 + t max k=1, ··· ,r max s ∈[t k −1 ,t k ] V s − V t k −1 ∞ . 5.42 Since the map V 7→ λV from BΩ to R is continuous which can be seen e.g. from 5.40 and the Feynman-Kac representation of S V t , the claim follows by letting the mesh of the partition tend to zero. 2114 Lemma 5.6. For all α, T, R, t, κ 0, E ν ρ ,0 ‚ exp – α κ ⌊tR κ ⌋ X k=1 Z kR κ k−1R κ du V k,u ξ u κ ™Œ ≤ E X ‚ exp – ⌊tR κ ⌋ X k=1 Z kR κ k−1R κ du λ k,u ™Œ 5.43 with λ k,u = λ κ,X k,u = lim t →∞ 1 t log E ν ρ exp α κ Z t ds V κ,X k,u ξ s κ , 5.44 where u ∈ [k − 1R κ , kR κ ], k = 1, 2, · · · , ⌊tR κ ⌋. Proof. Apply Lemma 5.5 to the potential V u η = ακV k,u η for u ∈ [k − 1R κ , kR κ ] with ξ u u ≥0 replaced by ξ u κ u ≥0 , and take the expectation w.r.t. E X . The spectral bound in Lemma 5.6 enables us to estimate the expression in 5.35 from above by finding upper bounds for the expectation in 5.44 with a time-independent potential V k,u . This goes as follows. Fix κ, X , k and u, and abbreviate b φ = αV κ,X k,u . 5.45 Let Q t t ≥0 be the semigroup generated by 1 κL, and define b ψ = Z M d r Q r b φ 5.46 with M = 3K1[ κ]κ 3 5.47 for a large constant K 0. Then − 1 κ L b ψ = b φ − Q M b φ 5.48 with Q r b φ η = α R κ Z k+1R κ kR κ ds X y ∈Z 3 p 6T 1[ κ]+ s −u+r κ X s , y η y − ρ = α X y ∈Z 3 Ξ r y[η y − ρ] 5.49 and Ξ r x = Ξ κ,X k,u,r x = 1 R κ Z k+1R κ kR κ ds p 6T 1[ κ]+ s −u+r κ X s , x. 5.50 As in Section 2, we introduce new probability measures P new η by an absolute continuous transforma- tion of the probability measures P η , in the same way as in 2.12–2.13 with ψ and A replaced by b ψ and 1κL, respectively. Under P new η , ξ t κ t ≥0 is a Markov process with generator 1 κ L new f = e − 1 κ b ψ 1 κ L e 1 κ b ψ f − e − 1 κ b ψ 1 κ Le 1 κ b ψ f . 5.51 2115 Since η 7→ b ψη is bounded, we have, similarly as in Proposition 2.1 with q = r = 2, λ κ,X k,u ≤ lim sup t →∞ 1 2t log E 5 k,u t + lim sup t →∞ 1 2t log E 6 k,u t 5.52 with E 5 k,u t = E 5 k,u κ, X ; t = E new ν ρ exp 2 κ Z t d r e − 1 κ b ψ Le 1 κ b ψ − L 1 κ b ψ ξ r κ 5.53 and E 6 k,u t = E 6 k,u κ, X ; t = E new ν ρ exp 2 κ Z t d r Q M b φ ξ r κ , 5.54 where E new ν ρ = R Ω ν ρ dη E new η , and we suppress the dependence on the constants T , K, R.

5.2.2 Two further lemmas