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

By N. Chernov, D. Dolgopyat

A classical version of Brownian movement involves a heavy molecule submerged right into a fuel of sunshine atoms in a closed box. during this paintings the authors examine a second model of this version, the place the molecule is a heavy disk of mass M 1 and the gasoline is represented via only one element 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 via convex (scattering) partitions of the box. The authors turn out that the location and pace 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 approach depend upon the preliminary stipulations. The proofs are according to robust hyperbolicity of the underlying dynamics, quickly decay of correlations in platforms with elastic collisions (billiards), and strategies of averaging concept

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Similar quantities at the point xi . For any curve W we denote by W (x, x ) the segment of W between the points x, x ∈ W and by (x, x )W the angle between the tangent vectors to the curve W at x and x . 6 (Distortion bounds). Under the above assumptions, if the following bound holds for i = 0 with some C0 = c > 0, then it holds for all i = 1, . . 7 (Curvature bounds). Under the above assumptions, if the following bound holds for i = 0 with some C0 = c > 0, then it holds for all i = 1, . . , n with some Ci = C > c (independent of i and n) |Wi (xi , xi )| (xi , xi )Wi ≤ Ci |Wi |2/3 The proofs of these two propositions are quite lengthy.

Let π0 denote the natural projection of Ω onto Ω0 . Note that, for each Q, V the projection π0 : ΩQ,V → Ω0 is one-to-one. 7)). We turn back to u-curves W ⊂ M. For any such curve, W = π0 (˜ π − (W)) is a smooth or piecewise smooth curve in Ω0 , whose components we also call u-curves. Any such curve is described by a smooth function ϕ = ϕ(r). Let dx = (dQ, dV, dq, dv) be the postcollisional tangent vector to W at a moment of collision. Its projection under the derivative D(π0 ◦ π ˜ − ) is a tangent vector to W , which we denote by (dr, dϕ).

7) dx(t + 0) ≤ ϑ dx(t + 0) The notation t + 0, t + 0 refer to the postcollisional vectors. The proof easily follows from the previous equations. 3). 4. 5), instead of v, and respectively dw instead of dv. Then the vector w changes by the same rule w+ = Rn (w− ) for both types of collisions (at ∂D and ∂P(Q)). At collisions with ∂D, the vector dw = dv will change by the rule dw+ = Rn dw− + Θ+ (dq + ) while at collisions with the heavy disk, the vector dw = dv − dV will change by a similar rule dw+ = Rn dw− + Θ+ (dq + − dQ+ ) Furthermore, the expressions for Θ+ and Θ− will be identical for both types of collisions.

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