Download Brownian Brownian motion. I by N. Chernov, D. Dolgopyat PDF

By N. Chernov, D. Dolgopyat

A classical version of Brownian movement comprises a heavy molecule submerged right into a fuel of sunshine atoms in a closed box. during this paintings the authors research a 2nd model of this version, the place the molecule is a heavy disk of mass M 1 and the fuel is represented by means of only one aspect particle of mass m = 1, which interacts with the disk and the partitions of the box through elastic collisions. Chaotic habit of the debris is ensured by means of convex (scattering) partitions of the box. The authors end up that the placement and speed of the disk, in a suitable time scale, converge, as M, to a Brownian movement (possibly, inhomogeneous); the scaling regime and the constitution of the restrict method depend upon the preliminary stipulations. The proofs are in response to robust hyperbolicity of the underlying dynamics, quick decay of correlations in structures with elastic collisions (billiards), and techniques of averaging idea

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Due ˆ , call them W ˜ and W ˜ , such that dist(FQ (W W to our choice of ε4 and the expansion of u-curves by a factor ≥ ϑ−1 , the ˜ ) into two u-curves of length > c4 ε + 5c2 ε point FQ (x) divides FQ (W m each. Since FQ is discontinuous at FQ (x), our inductive assumption ˜ ), a contradiction. 3) and x ∈ Ω. 31) εn (x, Q) = max Q − Q(F i x) + 1/M 0≤i≤n where Q(y) denotes the Q-coordinate of a point y ∈ Ω. For a u-curve W ⊂ Ω we put εn (W, Q) = sup εn (x, Q) x∈W Recall that the Q coordinate varies by < Const/M on u-curves, so that the map F n acts on a u-curve W ⊂ Ω similarly to the action of FQn on its projection πQ (W ), if εn (W, Q) is small.

Wk with some k ≤ KnA . For each Wj we put Wj∗ = Wj ∩ FQd (W∗ ) and estimate mesi (Wj ) |A(z ) − A(z )| d mesi ≤ C ϑ εγ |Wj | Wj ∗ |Wj | d ≤ C ϑd εγ mesi (Wj ) |Wj |βA 0 t−βA dt 52 4. STANDARD PAIRS where C , C > 0 are some constants. 48) W |A(z ) − A(z )| d mesi ≤ Const KnβAA ϑd εγ mesi (W ) , |W |βA where we first used the homogeneity of the measure mesi to estimate mesi (Wj ) ≤ Const |Wj | mesi (W ) |W | and then by Jensen’s inequality obtain |Wj |1−βA ≤ KnβAA |W |1−βA . 50) W ⊂Fi (γ) mesi (W ) ≤ Const |W |βA [rn−k (x)]−βA dρ(x) ≤ Const γ (we remind the reader that βA < 1).

For every α and x ∈ γα and n ≥ 0 denote by rn (x) the distance from the point F n (x) to the nearest endpoint of the H-component of F n (γα ) to which the point F n (x) belongs. 18) then imply ˜ mesγ x : rn (x) < ε ≤ eβ mesγ (γ ) −1 mesγ x : rn (x) < ε 42 4. 3). 5. Perturbative analysis. Recall that the billiard-type map FQ,V : ΩQ,V → ΩQ,V is essentially independent of V and can be identified with FQ : ΩQ → ΩQ via πQ,0 ◦ FQ,V = FQ ◦ πQ,0 . Furthermore, the spaces ΩQ are identified with the r, ϕ coordinate space Ω0 by the projection π0 .

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