N. Chernov, D. Dolgopyat's Brownian Brownian motion. I PDF
By N. Chernov, D. Dolgopyat
A classical version of Brownian movement contains a heavy molecule submerged right into a fuel of sunshine atoms in a closed box. during this paintings the authors learn a 2nd model of this version, the place the molecule is a heavy disk of mass M 1 and the fuel is represented through 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 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 strategy rely on the preliminary stipulations. The proofs are in response to powerful hyperbolicity of the underlying dynamics, quickly decay of correlations in structures with elastic collisions (billiards), and techniques of averaging concept
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Additional info for Brownian Brownian motion. I
For any (Q, V ) and (Q , V ) we have ¯ ¯ ,V ) ≤ C A(Q, V ) − A(Q + nA Q−Q + V −V V + V + n2A /M for some C = C(A) > 0. Proof. 9) on the entire space Ω, the estimate would be trivial. 1. As a result, two error terms appear, denoted by E1 + E2 , where E1 comes from the fact that the functions AQ,V and AQ ,V have diﬀerent singularity sets SQ,V and SQ ,V , and E2 comes from the growing local Lipschitz constant near these singularity sets. The error term E1 is bounded by 2 A ∞ Area(G), where G is the region swept by the singularity set SQ,V as it transforms to SQ ,V when (Q, V ) continuously change to (Q , V ).
23), we observe that a small H-curve γ may be cut into several pieces by the singularities of F, which are made by grazing (tangential) collisions with the scatterers and the disk P(Q). At each of them, γ is sliced into two parts – one hits the scatterer (or the disk) and 38 4. STANDARD PAIRS gets reﬂected (almost tangentially) and the other misses the collision (passes by). The reﬂecting part is further subdivided into countably many H-components by the boundaries of the homogeneity sections Hk .
18) e−β ≤ ≤ eβ . 19) ˜ (x, x )W ≤ eβ ∀x, x ∈ W. Now consider an H-curve W0 such that Wi = F i (W0 ) is an H-curve for every i = 1, . . , n. Let mes0 be an absolutely continuous measure on W0 with some density ρ0 with respect to the measure induced by the |·|-norm. Then mesi = F i (mes0 ) is a measure on the curve Wi with 36 4. STANDARD PAIRS some density ρi for each i = 1, . . , n. 8 (Density bounds). Under the above assumptions, if the following bound holds for i = 0 with some C0 = c > 0, it holds for all i = 1, .
Brownian Brownian motion. I by N. Chernov, D. Dolgopyat