Random billiards with cosine reflection law Stationary random tube: quenched invariance principles Finite tube: crossing probabilities and crossing time Knudsen billiards with drift A strongly transient RWRE with unbounded jumps Knudsen billiards and random walks in random environment with unbounded jumps Random billiards and RWRE
Random billiards with cosine reflection law Stationary random tube: quenched invariance principles Finite tube: crossing probabilities and crossing time Knudsen billiards with drift A strongly transient RWRE with unbounded jumps Outline Random billiards with cosine reflection law Stationary random tube: quenched invariance principles Finite tube: crossing probabilities and crossing time Knudsen billiards with drift A strongly transient RWRE with unbounded jumps Random billiards and RWRE
Random billiards with cosine reflection law Stationary random tube: quenched invariance principles Finite tube: crossing probabilities and crossing time Knudsen billiards with drift A strongly transient RWRE with unbounded jumps (Comets, Popov, Schütz, Vachkovskaia, ARMA, 2009) The model can be informally described in the following way: ◮ A particle moves with constant speed inside some d -dimensional domain ◮ When it hits the boundary, it is reflected in some random direction, not depending on the incoming direction, and keeping the absolute value of its speed Random billiards and RWRE
Random billiards with cosine reflection law Stationary random tube: quenched invariance principles Finite tube: crossing probabilities and crossing time Knudsen billiards with drift A strongly transient RWRE with unbounded jumps ∂ D ξ 0 ξ 2 ξ 4 ( X t , V t ) D ( X 0 , V 0 ) ξ 3 ξ 1 Notations: ◮ X t ∈ D is the location of the process at time t , and V t ∈ S d − 1 is the corresponding direction; ◮ ξ n ∈ ∂ D , n = 0 , 1 , 2 , . . . are the points where the process hits the boundary. Random billiards and RWRE
Random billiards with cosine reflection law Stationary random tube: quenched invariance principles Finite tube: crossing probabilities and crossing time Knudsen billiards with drift A strongly transient RWRE with unbounded jumps Reflection: cosine reflection law: n ( x ) α x The density of the outgoing direction is proportional to cos α Random billiards and RWRE
Random billiards with cosine reflection law Stationary random tube: quenched invariance principles Finite tube: crossing probabilities and crossing time Knudsen billiards with drift A strongly transient RWRE with unbounded jumps For the case of cosine reflection law (finite domains): Theorem (assuming that the boundary is Lipschitz and a.e. continuously differentiable) ◮ The stationary measure of the random walk ξ n is uniform on ∂ D . ◮ The stationary measure of the process ( X t , V t ) is the product of uniform measures on D and S d − 1 . Proof: follows from the reversibility (the transition density is symmetric). Random billiards and RWRE
Random billiards with cosine reflection law Stationary random tube: quenched invariance principles Finite tube: crossing probabilities and crossing time Knudsen billiards with drift A strongly transient RWRE with unbounded jumps (Comets, Popov, Schütz, Vachkovskaia, AP , 2010) ω α r ω ( α, u ) n ω ( α, u ) ω ( α, u ) 0 e Λ Stationary random tube in R d Random billiards and RWRE
Random billiards with cosine reflection law Stationary random tube: quenched invariance principles Finite tube: crossing probabilities and crossing time Knudsen billiards with drift A strongly transient RWRE with unbounded jumps Let Z ( m ) be the polygonal interpolation of n / m �→ m − 1 / 2 ξ n · e · (the discrete time random walk). We denote by Q the stationary measure for the environment seen from the particle (there is an explicit formula for Q ). Theorem Assume Conditions L, P , R (“nice boundary”), and suppose that the second moment of the jump projected on the horizontal � � direction b Q is finite. Then, there exists a constant σ > 0 such that for P -almost all ω , σ − 1 Z ( m ) converges in law, under P ω , to · Brownian motion as m → ∞ . Random billiards and RWRE
Random billiards with cosine reflection law Stationary random tube: quenched invariance principles Finite tube: crossing probabilities and crossing time Knudsen billiards with drift A strongly transient RWRE with unbounded jumps The corresponding result for the continuous time Knudsen Z ( s ) stochastic billiard is available too. Define ˆ = s − 1 / 2 X st · e . t Theorem � � Assume Conditions L, P , R, and suppose that b Q < ∞ . Denote σ Γ( d 2 + 1 ) Z ˆ σ = � � , π 1 / 2 Γ( d + 1 2 ) | ω 0 | P d where σ is from the above Theorem and Z is the normalizing constant from the definition of Q . Then, for P -almost all ω , σ − 1 ˆ Z ( s ) ˆ converges in law to Brownian motion as s → ∞ . · Random billiards and RWRE
Random billiards with cosine reflection law Stationary random tube: quenched invariance principles Finite tube: crossing probabilities and crossing time Knudsen billiards with drift A strongly transient RWRE with unbounded jumps (Comets, Popov, Schütz, Vachkovskaia, JSP , 2010) Let � D ω H be the part of the random tube ω which lies between 0 and H : D ω � H = { z ∈ ω : z · e ∈ [ 0 , H ] } . ˆ D ℓ = { 0 } × ω 0 , ˆ D r = { H } × ω H , ω 0 is the set of points of ω 0 , from where the particle can ˜ reach ˆ D r by a path which stays within � H and ˜ D ω D ℓ := { 0 } × ˜ ω 0 C H : the event that the particle crosses the tube without going back to ˜ D ℓ Random billiards and RWRE
Random billiards with cosine reflection law Stationary random tube: quenched invariance principles Finite tube: crossing probabilities and crossing time Knudsen billiards with drift A strongly transient RWRE with unbounded jumps ˆ D r � D ω H ˜ D ℓ ˆ D ℓ � D ω H On the definition of ˜ D ℓ , � D ω H , and the event C H (a trajectory crossing the tube is shown) Random billiards and RWRE
Random billiards with cosine reflection law Stationary random tube: quenched invariance principles Finite tube: crossing probabilities and crossing time Knudsen billiards with drift A strongly transient RWRE with unbounded jumps Results ( T H is the total lifetime of the particle): � � γ d | S d − 1 | | ω 0 | ◮ E ω T H ∼ · H P 2 | ˜ ω 0 | σ 2 � � γ d | S d − 1 | ˆ | ω 0 | · 1 ◮ P ω [ C H ] ∼ P 2 | ˜ ω 0 | H 1 σ 2 · H 2 ◮ E ω ( T H | C H ) ∼ 3 ˆ As a consequence, � � � ω 0 | − 1 � H } ) ∼ H E ω ( T H I { C c 3 γ d | S d − 1 | | ω 0 | | ˜ P ∼ 2 E ω ( T H I { C H } ) P Random billiards and RWRE
Random billiards with cosine reflection law Stationary random tube: quenched invariance principles Finite tube: crossing probabilities and crossing time Knudsen billiards with drift A strongly transient RWRE with unbounded jumps (Comets, Popov, arXiv:1009.0048 ; to appear in AIHP-PS) Informal definition: ◮ the process lives in the infinite random tube ◮ the jumps in the positive direction are always accepted ◮ the jumps in the negative direction are accepted with probability e − λ u , where u is the horizontal size of the attempted jump. Random billiards and RWRE
Random billiards with cosine reflection law Stationary random tube: quenched invariance principles Finite tube: crossing probabilities and crossing time Knudsen billiards with drift A strongly transient RWRE with unbounded jumps ξ 1 ω ξ 0 Random billiards and RWRE
Random billiards with cosine reflection law Stationary random tube: quenched invariance principles Finite tube: crossing probabilities and crossing time Knudsen billiards with drift A strongly transient RWRE with unbounded jumps ξ 1 = ξ 2 ω ξ 0 Random billiards and RWRE
Random billiards with cosine reflection law Stationary random tube: quenched invariance principles Finite tube: crossing probabilities and crossing time Knudsen billiards with drift A strongly transient RWRE with unbounded jumps ξ 1 = ξ 2 ω ξ 3 ξ 0 Random billiards and RWRE
Random billiards with cosine reflection law Stationary random tube: quenched invariance principles Finite tube: crossing probabilities and crossing time Knudsen billiards with drift A strongly transient RWRE with unbounded jumps ξ 1 = ξ 2 ξ 4 ω ξ 3 ξ 0 Random billiards and RWRE
Random billiards with cosine reflection law Stationary random tube: quenched invariance principles Finite tube: crossing probabilities and crossing time Knudsen billiards with drift A strongly transient RWRE with unbounded jumps ξ 1 = ξ 2 ξ 4 ω ξ 3 ξ 0 ξ 5 Random billiards and RWRE
Random billiards with cosine reflection law Stationary random tube: quenched invariance principles Finite tube: crossing probabilities and crossing time Knudsen billiards with drift A strongly transient RWRE with unbounded jumps ξ 1 = ξ 2 ξ 4 ω ξ 3 ξ 0 ξ 5 = ξ 6 Random billiards and RWRE
Random billiards with cosine reflection law Stationary random tube: quenched invariance principles Finite tube: crossing probabilities and crossing time Knudsen billiards with drift A strongly transient RWRE with unbounded jumps ξ 1 = ξ 2 ξ 7 ξ 4 ω ξ 3 ξ 0 ξ 5 = ξ 6 Random billiards and RWRE
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