Single step estimates getdoc9b2f. 332KB Jun 04 2011 12:04:44 AM

We will introduce the following operators to break up the evolution into smaller pieces: L ℓ,λ = A1, λ4n 1 2 p ℓ , ˆ L ℓ,λ = A1, λ4n 1 2 s ℓ , S ℓ,λ = L ˆ T ℓ ℓ,λ ˆ T −1 ℓ A p ℓ p ℓ+1 1 + X ℓ p ℓ −1 , 0 ˆ T ℓ , 41 ˆ S ℓ,λ = ˆ L T ℓ ℓ,λ € T ℓ −1 A1 + Y ℓ s ℓ −1 , 0 T ℓ+1 Š . Then ϕ ℓ+1 = ϕ ℓ∗ L ˆ T ℓ ℓ ˆ Q ℓ € S ℓ,0 Š ˆ Q ℓ ˆ L T ℓ ℓ Q ℓ €ˆ S ℓ,0 Š Q ℓ = ϕ ℓ∗ € S ℓ,λ Š ˆ Q ℓ €ˆ S ℓ,λ Š Q ℓ . 42 We also introduce the relative regularized phase function and target phase function: α ℓ,λ := ϕ ℓ,λ − ϕ ℓ,0 , α ⊙ ℓ,λ := ϕ ⊙ ℓ,λ − ϕ ⊙ ℓ,0 . 43 5 SDE limit for the phase function Let F ℓ denote the σ-field generated by ϕ j, λ , j ≤ ℓ − 1. Then ϕ ℓ,λ is a Markov chain in ℓ with respect to F ℓ . Indeed, the relation 42 shows that ϕ ℓ+1,λ = h ℓ,λ ϕ ℓ+1,λ , X ℓ , Y ℓ where h ℓ,λ is a deterministic function depending on ℓ and λ. Since X ℓ , Y ℓ are independent of F ℓ it follows that E ” ϕ ℓ+1,λ |F ℓ — = E ” ϕ ℓ+1,λ |ϕ ℓ,λ — . We will show that this Markov chain converges to a diffusion limit after proper normalization. In order to do this we will estimate E ” ϕ ℓ+1,λ − ϕ ℓ,λ |F ℓ — and E ” ϕ ℓ+1,λ − ϕ ℓ,λ ϕ ℓ+1,λ ′ − ϕ ℓ,λ ′ |F ℓ — using the angular shift lemma, Lemma 12. To simplify the computations we introduce ‘intermediate’ values for the process ϕ ℓ,λ by breaking the evolution operator in 42 into two parts: ϕ ℓ+ 1 2 , λ = ϕ ℓ∗ € S ℓ,λ Š ˆ Q ℓ , F ℓ+ 1 2 = σF ℓ ∪ {ϕ ℓ+ 1 2 , λ }. Note that ϕ ℓ,λ is still a Markov chain if we consider it as a process on the half integers. Remark 13. We would like to note that the ‘half-step’ evolution rules ϕ ℓ,λ → ϕ ℓ+ 1 2 , λ , ϕ ℓ+ 1 2 , λ → ϕ ℓ+1,λ are very similar to the one-step evolution of the phase function ϕ in [13]. Indeed, in [13], the evolution of ϕ is of the type ϕ ℓ+1 = ϕ ℓ∗ €˜ S ℓ,λ Š ˜ Q ℓ where ˜ S is an affine transformation and ˜ Q is a rotation similar to our S, ˆ S and Q, ˆ Q. In our case the evolution of ϕ ℓ+1,λ is the composition of two transformations with similar structure. The main difficulties in our computations are caused by the fact that Q and ˆ Q are rather different which makes the oscillating terms more complicated.

5.1 Single step estimates

Throughout the rest of the proof we will use the notation k = n − ℓ. We will need to rescale the discrete time by n in order to get a limit, we will use t = ℓn and also introduce ˆ st = p 1 − t. 328 We start with the identity p ℓ ℑ ˆ ρ ℓ = s ℓ ℑρ ℓ = È s 2 ℓ − µ 2 n − m + n 2 4 µ 2 n = È n − µ 2 n − m + n 2 4 µ 2 n − ℓ − 1 2 = p n − ℓ = p k = p n ˆ st. Note that this means that ρ ℓ = ± r n − n − 12 n − ℓ − 12 + i r n − ℓ n − ℓ − 12 = ± r n 1 n 1 + k + i È k n 1 + k , 44 ˆ ρ ℓ = r m − n − 12 m − ℓ − 12 + i r n − ℓ m − ℓ − 12 = r m 1 m 1 + k + i È k m 1 + k 45 where the sign in ℜρ ℓ is positive if µ n p m − n and negative otherwise. For the angular shift estimates we need to consider Z ℓ,λ := i. S −1 ℓ,λ − i = ˆ ρ ℓ X ℓ p n ˆ st · p ℓ+1 p ℓ + − λ 4n ˆ st + ˆ ρ ℓ p ℓ+1 − p ℓ p ℓ ℑ ˆ ρ ℓ =: V ℓ + v ℓ , ˆ Z ℓ,λ := i.ˆ S −1 ℓ,λ − i = ρ ℓ Y ℓ p n ˆ st + − λ 4n ˆ st + ρ ℓ+1 − ρ ℓ ℑρ ℓ =: ˆ V ℓ + ˆ v ℓ . 46 Here V ℓ , ˆ V ℓ are the random and v ℓ , ˆ v ℓ are the deterministic parts of the appropriate expressions. We have the following estimates for the deterministic parts by Taylor expansion: v ℓ,λ = v λ t n + Ok −2 , v λ t = − λ 4ˆ st − ˆ ρt 2ptˆ st , |v ℓ,λ | ≤ c k , ˆ v ℓ,λ = ˆ v λ t n + Ok −2 , ˆ v λ t = − λ 4ˆ st + d d t ρt ℑρt , |ˆv ℓ,λ | ≤ c k , where pt = p n t = p m n − t and ρt = ρ n t, ˆ ρt = ˆ ρ n t are defined by equations 44 and 45 with ℓ = n t. For the random terms from 30 we get EV ℓ = E ˆ V ℓ = O k −12 n − ℓ −32 , EV 2 ℓ = 1 n q 1 t + O k −1 n − ℓ −1 , E ˆ V 2 ℓ = 1 n q 2 t + O k −1 n − ℓ −1 , E|V 2 ℓ | = E| ˆ V 2 ℓ | = 1 n q 3 t + O k −1 n − ℓ −1 , E|V d ℓ |, E| ˆ V 2 ℓ | = O k −d2 , d = 3, 4, where the constants in the error term only depend on β and q 1 t = 2 ˆ ρt 2 βˆ st 2 , q 2 t = 2 ρt 2 βˆ st 2 , q 3 t = 2 βˆ st 2 . 47 We introduce the notations ∆ 1 2 f x, λ = f x+ 1 2 , λ − f x, λ , ∆ f x, λ = f x+1, λ − f x, λ and we also set for ℓ ∈ Z + η ℓ = ρ 2 ˆ ρ 2 ρ 2 1 ˆ ρ 2 1 . . . ρ 2 ℓ ˆ ρ 2 ℓ . The following proposition is the analogue of Proposition 22 in [13]. 329 Proposition 14. For ℓ ≤ n we have E ” ∆ 1 2 ϕ ℓ,λ ϕ ℓ,λ = x — = 1 n b 1 λ t + 1 n osc 1 + O k −32 = O k −1 E ” ∆ 1 2 ϕ ℓ,λ ∆ 1 2 ϕ ℓ,λ ′ ϕ ℓ,λ = x, ϕ ℓ,λ ′ = y — = 1 n a 1 t, x, y + 1 n osc 2 + O k −32 E ” ∆ 1 2 ϕ ℓ+ 1 2 , λ ϕ ℓ+ 1 2 , λ = x — = 1 n b 2 λ t + 1 n osc 3 + O k −32 = O k −1 E ” ∆ 1 2 ϕ ℓ+ 1 2 , λ ∆ 1 2 ϕ ℓ+ 1 2 , λ ′ ϕ ℓ+ 1 2 , λ = x, ϕ ℓ+ 1 2 , λ ′ = y — = 1 n a 2 t, x, y + 1 n osc 4 + O k −32 , E ” |∆ 1 2 ϕ ℓ,λ | d ϕ ℓ,λ = x — , E ” |∆ 1 2 ϕ ℓ+ 1 2 , λ | d ϕ ℓ,λ = x — = O k −d2 , d = 2, 3 where t = ℓn , b 1 λ = λ 4ˆ s + ℜ ˆ ρ 2pˆ s + ℑ ˆ ρ 2 2 βˆ s 2 , b 2 λ = λ 4ˆ s − ℜ d d t ρ ℑρ + ℑρ 2 2 βˆ s 2 , a 1 = 1 βˆ s 2 ℜ ” e ix − y — + 1+ ℜ ˆ ρ 2 βˆ s 2 , a 2 = 1 βˆ s 2 ℜ ” e ix − y — + 1 + ℜρ 2 βˆ s 2 . The oscillatory terms are osc 1 = ℜ−v ℓ − iq 1 2e −i x ˆ ρ −2 ℓ η ℓ + ℜie −2i x ˆ ρ −4 ℓ η 2 ℓ q 1 4, osc 2 = q 3 ℜe −i x ˆ ρ −2 ℓ η ℓ + e −i y ˆ ρ −2 ℓ η ℓ 2 + ℜq 1 e −i x ˆ ρ −2 ℓ η ℓ + e −i y ˆ ρ −2 ℓ η ℓ + e −ix+ y ˆ ρ −4 ℓ η 2 ℓ 2, osc 3 = ℜ−ˆv ℓ − iq 2 2e −i x η ℓ + ℜie −2i x η 2 ℓ q 2 4, osc 4 = q 3 ℜe −i x η ℓ + e −i y η ℓ 2 + ℜq 2 e −i x η ℓ + e −i y η ℓ + e −ix+ y η 2 ℓ 2. Proof. We start with the identity ϕ ℓ+ 1 2 , λ − ϕ ℓ,λ = ϕ ℓ+1,λ∗ ˆ Q −1 ℓ − ϕ ℓ,λ∗ ˆ Q −1 ℓ = ϕ ℓ,λ∗ ˆ Q −1 ℓ S ℓ,λ − ϕ ℓ,λ∗ ˆ Q −1 ℓ = ashS ℓ,λ , e i ϕ ℓ,λ ¯ η ℓ ˆ ρ −2 ℓ , −1. Here we used the definition of the angular shift with the fact that S ℓ,λ and any affine transfor- mation will preserve ∞ ∈ H which corresponds to −1 in U. A similar identity can be proved for ∆ 1 2 ϕ ℓ+ 1 2 , λ . The proof now follows exactly the same as in [13], it is a straightforward application of Lemma 12 using the estimates on v ℓ,λ , ˆ v ℓ,λ , V ℓ , ˆ V ℓ .

5.2 The continuum limit