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Introduction Results Recurrence of PLP Tools LCLT Standard pairs Coupling lemma Line of proofs Proof of tail lemma

Limit Theorems for locally perturbed Lorentzprocesses

Domokos Szasz(joint with Dima Dolgopyat and Tamas Varju)

Mathematics and BilliardsOrleans, March 25, 2008

Motivation

Random Walk is a stochastic model of Brownian Motion (cf.invariance principle a la Erdos-Kac, Donsker, Prohorov).Periodic Lorentz Process is a deterministic model of BrownianMotion.The problem to be treated is about the Lorentz process (planar orquasi-one-dimensional). Our aim is twofold: first we want to solvea question raised by Sinai in 1981. On the other hand, by treatingvarious problems for the Lorentz process our/my goal is to finallywork out methods that enable us to make probability theory orstatistical physics for the Lorentz process or more generaldeterministic models.

Motivation

Lorentz Process

Lorentz proces - billiard dynamics (uniform motion + specularreflection) (Ω,T , µ)

Q = Zd \ ∪∞

i=1Oi is the configuration space of the Lorentzprocess (Lorentz flow), where the closed sets Oi are pairwisedisjoint, strictly convex with C3−smooth boundaries

Ω = Q × S+ is its phase space of the billiard map (whereQ = ∂Q and S+ is the hemisphere of outgoing unit velocities)

T : Ω → Ω its discrete time billiard map (the so-calledPoincare section map)

µ the T -invariant (infinite) Liouville-measure on Ω

Lorentz Process

Q = Zd \ ∪∞

Lorentz Process

Q = Zd \ ∪∞

Lorentz Process

Q = Zd \ ∪∞

Periodic Lorentz Process → Sinai Billiard

If the scatterer configuration Oii is Zd -periodic, then the

corresponding dynamical system will be denoted by(Ωper = Qper × S+,Tper , µper ) and it makes sense to factorize itby Z

d to obtain a Sinai billiard (Ω0 = Q0 × S+,T0, µ0). Thenatural projection Ω → Q (and analogously for Ωper and for Ω0)will be denoted by πq.

Assume finite horizon

Periodic Lorentz Process → Sinai Billiard

corresponding dynamical system will be denoted by(Ωper = Qper × S+,Tper , µper ) and it makes sense to factorize itby Z

d to obtain a Sinai billiard (Ω0 = Q0 × S+,T0, µ0). Thenatural projection Ω → Q (and analogously for Ωper and for Ω0)will be denoted by πq.

Assume finite horizon

Locally Perturbed Lorentz Process

Assume d = 2 and Q = Qper outside a bounded domain. Select aninitial point x0 = (q0, v0) ∈ Ω according to a compactly supportedprobability measure µ, absolutely continuous with respect to theLiouville measure µ. Then T nx0 = (qn, vn)|n ∈ Z is the Lorentztrajectory and the resulting configuration process qn|n ≥ 0 willbe called a finite modification of the FHLP.

Definition

Assume qn ∈ Rd |n ≥ 0 is a random trajectory. Then its

diffusively scaled variant ∈ C [0, 1] (or ∈ C [0,∞]) is defined asfollows: for N ∈ Z+ denoteWN( j

qj√N

(0 ≤ j ≤ N or j ∈ Z+) and define otherwise

WN(t)(t ∈ [0, 1] or R+) as its piecewise linear, continuousextension.

Locally Perturbed Lorentz Process

Assume d = 2 and Q = Qper outside a bounded domain. Select aninitial point x0 = (q0, v0) ∈ Ω according to a compactly supportedprobability measure µ, absolutely continuous with respect to theLiouville measure µ. Then T nx0 = (qn, vn)|n ∈ Z is the Lorentztrajectory and the resulting configuration process qn|n ≥ 0 willbe called a finite modification of the FHLP.

Definition

Assume qn ∈ Rd |n ≥ 0 is a random trajectory. Then its

diffusively scaled variant ∈ C [0, 1] (or ∈ C [0,∞]) is defined asfollows: for N ∈ Z+ denoteWN( j

qj√N

(0 ≤ j ≤ N or j ∈ Z+) and define otherwise

WN(t)(t ∈ [0, 1] or R+) as its piecewise linear, continuousextension.

CLT for Periodic and Perturbed Lorentz Processes

Theorem

Bunimovich-Sinai, 1981: The diffusively scaled variant WN(t) ofthe periodic Lorentz process converges weakly to a Wiener processWD2(t) with a non-degenerate covariance matrix D2.

Sinai’s conjecture, 1981: The same statement holds for thelocally perturbed periodic Lorentz process (finite horizon,d = 2)

CLT for Periodic and Perturbed Lorentz Processes

Theorem

Bunimovich-Sinai, 1981: The diffusively scaled variant WN(t) ofthe periodic Lorentz process converges weakly to a Wiener processWD2(t) with a non-degenerate covariance matrix D2.

Sinai’s conjecture, 1981: The same statement holds for thelocally perturbed periodic Lorentz process (finite horizon,d = 2)

Random Walks

Let Sn = X1 + X2 + · · · + Xn; n ≥ 0 be the simple symmetricrandom walk (SSRW) on Z

d ; d ≥ 1. According to the CLT, itsdiffusively scaled variant converges weakly to a Wiener process.

Toy-model: a locally perturbed version of the SSRW : in a finitedomain we perturb the transition probabilities P(Xn = ±ej) = 1

2dof the SSRW. For simplicity perturb at the origin, only.Elementary: the local time Nn = Cardk ≤ n|Sk = 0 is ≍ O(

if d = 1, and it is ≍ O(log n) if d = 2.

Picture, Szasz-Telcs, 1981: for d = 2 the effect of theperturbations is eaten up by the scaling, and one gets the sameCLT as for the unperturbed SSRW, whereas for d = 1 the effect isof the same order as the scaling, and one expects in the limit askew (biased) Wiener process (see Harrison-Shepp, 1981).

Random Walks

Locally Perturbed Lorentz ProcessTheorem 1

Theorem

For the locally perturbed Lorentz Process, as N → ∞,WN(t) ⇒ WD2(t) (weak convergence in C [0,∞]), where WD2(t)is the Wiener process with the non-degenerate covariance matrixD2. The limiting covariance matrix coincides with that for theunmodified periodic Lorentz process.

Reflected Periodic Lorentz Process in the Half-planeTheorem 2

Periodic Lorentz-process in the half-plane z1 ≥ 0.Nevertheless, in the horizontal direction this is a local perturbation,only!Warning: Scatterers intersecting z1 = 0 are deleted, thus, infact, the horizon is infinite.

Theorem

Consider the diffusively scaled variant WN(t) ∈ R+ × R of a FHLPqnn≥0 in a halfplane z1 ≥ 0. Then, as N → ∞, WN(t) convergesweakly to a non-degenerate Brownian motion reflected at thez2-axis.

Reflected Periodic Lorentz Process in the Half-planeTheorem 2

Periodic Lorentz-process in the half-plane z1 ≥ 0.Nevertheless, in the horizontal direction this is a local perturbation,only!Warning: Scatterers intersecting z1 = 0 are deleted, thus, infact, the horizon is infinite.

Theorem

Consider the diffusively scaled variant WN(t) ∈ R+ × R of a FHLPqnn≥0 in a halfplane z1 ≥ 0. Then, as N → ∞, WN(t) convergesweakly to a non-degenerate Brownian motion reflected at thez2-axis.

Quasi-one-dimensional modelsHalf-strip, Thm 3

In the next theorem we consider the half-strip R+ × [0, 1]. Thespecular reflection at the vertical boundary piece z1 = 0 will playthe role of the local perturbation.

Theorem

Consider a FHLP z1,nn≥0 in a halfstrip and let WN(t) ∈ R+ beits diffusively scaled variant. Then, as N → ∞, WN(t) convergesweakly to a non-degenerate Brownian motion reflected at 0.

Quasi-one-dimensional modelsHalf-strip, Thm 3

In the next theorem we consider the half-strip R+ × [0, 1]. Thespecular reflection at the vertical boundary piece z1 = 0 will playthe role of the local perturbation.

Theorem

Consider a FHLP z1,nn≥0 in a halfstrip and let WN(t) ∈ R+ beits diffusively scaled variant. Then, as N → ∞, WN(t) convergesweakly to a non-degenerate Brownian motion reflected at 0.

Quasi-one-dimensional modelsThermostatted motion with local external field, Thm 4

Next we consider a particle in a whole strip in the presence of acompactly supported thermostatted field a la Chernov-Eyink--Lebowitz-Spohn. Namely we assume that between the collisionsthe motion of the particle is determined by

v = E (q) − (E (q), v)

(v , v)v . (1)

(Easy calculation: ddt

< v , v >= 0, so |v | = const.)

Theorem

Consider a FHLP z1,nn≥0 in the strip in the presence of a smalland compactly supported external field E and let WN(t) ∈ R be itsdiffusively scaled variant. Then, as N → ∞, WN(t) convergesweakly to a skew Brownian motion.

Quasi-one-dimensional modelsThermostatted motion with local external field, Thm 4

Next we consider a particle in a whole strip in the presence of acompactly supported thermostatted field a la Chernov-Eyink--Lebowitz-Spohn. Namely we assume that between the collisionsthe motion of the particle is determined by

v = E (q) − (E (q), v)

(v , v)v . (1)

(Easy calculation: ddt

< v , v >= 0, so |v | = const.)

Theorem

Consider a FHLP z1,nn≥0 in the strip in the presence of a smalland compactly supported external field E and let WN(t) ∈ R be itsdiffusively scaled variant. Then, as N → ∞, WN(t) convergesweakly to a skew Brownian motion.

Papers

D. Dolgopyat, D. Szasz, T. Varju:

DSzV1 Recurrence properties of planar Lorentz processes, (periodiccase), Duke Math. J. next issue;

DSzV2 Limit theorems for perturbed planar Lorentz processes, pp.37, submitted,

Periodic Lorentz Process: Reminder

Lorentz proces - billiard dynamics (uniform motion + specularreflection)

Q = Zd \ ∪∞

Periodic Lorentz Process: Notations

corresponding dynamical system is(Ωper = Qper × S+,Tper , µper ).

Its factor wrt Zd is a Sinai billiard: (Ω0 = Q0 × S+,T0, µ0).

The natural projection Ω → Q (and analogously for Ωper andfor Ω0) will be denoted by πq. Identify Ω0 with the zero-thcell, and denote Qm = Q0 + m (m ∈ Z

Let κ(x) = πq(f (x)) − πq(x) : Ω(0) → R2 be the free flight

vector, and

Sn(x) =

n−1∑

κ(f k(0)(x))

vector, and

Sn(x) =

n−1∑

κ(f k(0)(x))

vector, and

Sn(x) =

n−1∑

κ(f k(0)(x))

vector, and

Sn(x) =

n−1∑

κ(f k(0)(x))

Excursions and Local Time, [DSzV1]

Let m(S) = m if S ∈ Qm, and

τ = minn > 0 : m(Sn) = 0 (i . e. τ : Ω → N)

For simplicity here only d = 2.

Theorem

There is a constant c such that µ0(τ > n) ∼ clog n

Let Nn(x) = Cardk ≤ n : m(Sk) = 0.

Theorem

Assume x is distributed according to µ0. Then cNn

log nconverges

weakly to a mean 1 exponential distribution.

τ = minn > 0 : m(Sn) = 0 (i . e. τ : Ω → N)

Theorem

log nconverges

τ = minn > 0 : m(Sn) = 0 (i . e. τ : Ω → N)

Theorem

log nconverges

τ = minn > 0 : m(Sn) = 0 (i . e. τ : Ω → N)

Theorem

log nconverges

PreliminariesHyperbolicity

A natural DT0-invariant field Cux of unstable cones (and dually

also a field Csx of stable ones) of the form c1 ≤ dφ

dq≤ c2 (or

−c2 ≤ dφdq

≤ −c1 respectively) where 0 < c1 < c2 are suitableconstants.

PreliminariesSingularities

Primary singularities:tangencies ∪ (their images underT n

0 ) : n ∈ Z.

Secondary singularities: further cutting by boundaries ofhomogeneity strips: For k ≥ k0 let

Hk = (r , φ) :π

2− k−2 < φ <

2− (k + 1)−2,

H−k = (r , φ) :π

2− k−2 < −φ <

2− (k + 1)−2,

H0 = (r , φ) : −(π

2− k−2

0 ) < φ <π

2− k−2

PreliminariesSingularities

Primary singularities:tangencies ∪ (their images underT n

0 ) : n ∈ Z.

Secondary singularities: further cutting by boundaries ofhomogeneity strips: For k ≥ k0 let

Hk = (r , φ) :π

2− k−2 < φ <

2− (k + 1)−2,

H−k = (r , φ) :π

2− k−2 < −φ <

2− (k + 1)−2,

H0 = (r , φ) : −(π

2− k−2

0 ) < φ <π

2− k−2

PreliminariesStandard pairs

Definition

A standard pair is a pair ℓ = (γ, ρ) where γ is a homogeneousunstable curve and ρ is a homogeneous density on γ.

An unstable curve is homogeneous if

it does not intersect any singularity (i. e. neither primary norsecondary ones);it satisfies a (distorsion) bound.

A probability density ρ on a homogeneous unstable curve γ iscalled a homogeneous density if its logarithm satisfies aHolder-type density bound

| log ρ(x) − log ρ(y)| ≤ L2θs+(x ,y)

where θ < 1 and s+(x , y) is the first time T s0 (x) and T s

0 (y)are separated by a singularity.

Definition

Growth lemma-1

Let γ be a homogeneous curve and for n ≥ 1 and x ∈ γ let rn(x)denote the distance of the point T n

0 (x) from the nearest boundarypoint of the H-component γn(x) containing T n

0 (x).

Growth lemma-1

(Growth lemma, Chernov, 2002, 2006). If k0 is sufficiently large,then

(a) there are constants β1 ∈ (0, 1) and β2 > 0 such that for anyε > 0 and any n ≥ 1

mesℓ(x : rn(x) < ε) ≤ (β1Λ)nmes(x : r0(x) < ε/Λn) + β2ε

(b) there are constants β3, β4 > 0, such that ifn ≥ β3| log length(γ)|, then for any ε > 0 and any n ≥ 1 onehas

mesℓ(x : rn(x) < ε) ≤ β4ε

Growth lemma-2’Markovian’ form!

(Growth lemma-cont’d, Chernov-Dolgopyat, 2005)

(c) If ℓ = (γ, ρ) is a standard pair, then

Eℓ(A T n0 ) =

cαnEℓαn(A)

where cαn > 0,∑

α cαn = 1 and ℓαn = (γαn, ραn) are standardpairs where γαn = γn(xα) for some xα ∈ γ and ραn is thepushforward of ρ up to a multiplicative factor.

(d) If n ≥ β3| log length(ℓ)|, then

length(ℓαn)<ε

cαn ≤ β4ε.

Local limit theoremCLT

The CLT for the Lorentz process (BS81, BCS91) states that thereis a positive definite matrix D2 such that Sn/(

√detD2

converges to a 2-dimensional standard Gaussian distribution. Infact, by using the shorthand κn = κ(f n(x)), we have then

D2 = µ0(κ0 ⊗ κ0) + 2

∞∑

µ0(κ0 ⊗ κn). (2)

Local limit theoremProperties

Theorem

(Szasz-Varju, 2004) Let x be distributed on Ω0 according to µ0.Let the distribution of m(Sn(x)) be denoted by Υn. There is aconstant c such that

limn→∞

nΥn → c−1l

where l is the counting measure on the integer lattice Z2 and →

stands for vague convergence.

Remark In fact, c−1 = 1

2π√

detD2.

Local limit theoremJoint distributions

The following result is a slight extension of of the LCLT and canbe proven similarly.

Theorem

For each fixed k the following holds:If n1, n2 . . . nk → ∞, then

µ0(m(Sn1) = m(Sn1+n2) = · · · = m(Sn1+n2+···+nk) = 0) ∼

Standard pairsProperties

Notation: instead of T(0) use f(0). Let ℓ be a standard pair,

A ∈ H (Holder) and take n such that | log length(ℓ)| < n1/2−δ.(a) (Asymptotic equidistribution, Chernov, 2006) There is aconstant

Eℓ(A f n0 ) −

≤ Constθn| log length(ℓ)|

(b) (Correlation decay, Chernov, 2005) Let A,B ∈ H have zeromean. Then

Eℓ(AnBn) = nDA,B + O(| log2 length(ℓ)|)

DA,B =

∞∑

j=−∞

A(x)B(f j0 x)dµ0(x).

Eℓ(A f n0 ) −

DA,B =

∞∑

j=−∞

Eℓ(A f n0 ) −

DA,B =

∞∑

j=−∞

(c) (Weak invariance principle, Chernov, 2006) Let x be distributedaccording to ℓ and wn(t) be defined by

=Si√n

with linear interpolation in between. Then, as n → ∞, wn

converges weakly (in C ([0, 1] → R2) to the 2-dimensional

Brownian Motion with zero mean and covariance matrix D2 givenby (2). Moreover, for the Prohorov metric, known to be equivalentto the weak convergence, the preivous convergence is unform forstandard pairs satisfying the condition | log length(ℓ)| < n1/2−δ.Finally, a similar result holds for the LLP, too.

Standard pairsProbabilistic laws on standard pairs

(d) (Large deviation bound, Chernov-Dolgopyat, 2005) If1 < R < n1/6−δ then

Pℓ(|An − n

Adµ0| ≥ R√

n) ≤ c1e−c2R

(e) (Local limit, [DSzV1]) If A is a Holder continuous function onΩ supported on Ω0, then

nEℓ(A f n) → c−1

Adµ(0)

where c is the constant from subsection 10. For the LLP we have

√nEℓ(A f n) → c−1

Adµ(0)

Standard pairsProbabilistic laws on standard pairs

(d) (Large deviation bound, Chernov-Dolgopyat, 2005) If1 < R < n1/6−δ then

Pℓ(|An − n

Adµ0| ≥ R√

n) ≤ c1e−c2R

(e) (Local limit, [DSzV1]) If A is a Holder continuous function onΩ supported on Ω0, then

nEℓ(A f n) → c−1

Adµ(0)

where c is the constant from subsection 10. For the LLP we have

√nEℓ(A f n) → c−1

Adµ(0)

Coupling lemma 1Lorentz process

Assume A ∈ H. In general, for a standard pair ℓ = (γ, ρ), denoteby [ℓ] the value of m ∈ Z

2 for which γ ∈ Ωm.

Given δ0 > 0 there exist constants q > 0, n0 ≥ 1 C > 0, θ < 1 andκ > 0 such that for any m ∈ Z

2 and arbitrary pair of standardpairs ℓ1 = (γ1, ρ1), ℓ2 = (γ2, ρ2) satisfying for some m ∈ Z

[ℓ1] = [ℓ2] = m (3)

and length(ℓj) ≥ δ0, there exist probability measures ν1, ν2

supported on f n0γ1 and f n0γ2 respectively, a constant c , families ofstandard pairs ℓβj = (γβj , ρβj)β and positive constantscβjβ : j = 1, 2, satisfying

Coupling lemma 1Lorentz process, continued

(i) Eℓj(Af n0) = cνj(A)+

cβjEℓβj(A) j = 1, 2

with c ≥ q;(ii) There exist a measure preserving mapπ : (γ1, f

−n0∗ ν1) → (γ2, f

−n0∗ ν2) such that for every n ≥ n0

d(f nx , f nπx) ≤ Cθn (4)

(iii) For every ρ > 0

β: length(ℓβj)<ρ

cβj ≤ Const(δ0)ρκ j = 1, 2.

Assume that |m1|, |m2| → ∞ and if ℓ1, ℓ2 are standard pairssatisfying

[ℓj ] = mj , length(ℓj) > |mj |−100, j = 1, 2 (5)

|m1||m2|

< 2. (6)

Given ζ > 0 and ε > 0 there exists R such that for any twostandard pairs ℓ1 = (γ1, ρ1), ℓ2 = (γ2, ρ2) satisfying the previousassumptions and |mj | > R the following holds.

Let n = |m1|2(1+ζ). There exist positive constants c and cβj ,probability measures ν1 and ν2 supported on f nγ1 and f nγ2

respectively, and families of standard pairs ℓβjβ; j = 1, 2satisfying

Eℓj(A f n) = c νj(A) +

cβjEℓβj(A) j = 1, 2 (7)

with c ≥ 1 − ε.

Theorem

Moreover there exists a measure preserving map

π : (γ1 × [0, 1], f −nν1 × λ) → (γ2 × [0, 1], f −nν2 × λ)

where λ is the Lebesgue measure on [0, 1] such that ifπ(x1, s1) = (x2, s2) then for any n ≥ n

d(f nx1, fnx2) ≤ Cθn−n,

where C , θ are the constants from our preliminary lemma.

Martingale approacha la Stroock-Varadhan

All our limiting processes behave like the Brownian Motion with aspecified boundary condition. Therefore these limiting processesare characterized by the fact that

φ(W (t)) − 1

ab=1,2

σabDabφ(W (s))ds (8)

is a martingale for a set of the functions dense in the domain ofthe generator of the corresponding process.

Therefore, for showing the convergence of a sequence of stochasticprocesses to such a Brownian Motion, by general theory it sufficesto show that the limiting process W (t) of any convergentsubsequence of the processes in question the process

φ(W (t)) − 1

ab=1,2

σabDabφ(W (s))ds (9)

is a martingale for the suitable class of functions.

In fact, these classes of functions are the following:

BM in R2 : C 2 functions of compact support (Theorem 1);

BM in a halfline: C 2 functions of compact support satisfying∂φφx

(0) = 0 (Theorem 2);

skew BM: continuous functions of compact support whichadmit C 2 extensions to (−∞, 0] and [0,∞) such that

φ′+(0) = aφ′

−(0)

where a is the skewness parameter (Theorem 3);

reflected BM in a halfplane x1 ≥ 0 : C 2 functions of compactsupport satisfying ∂φ

∂x1(0, x2) = 0 (Theorem 4).

Tail of return timesLocally perturbed planar periodic Lorentz process

Notation: for a standard pair ℓ = (γ, ρ) let [ℓ] denote an indexm ∈ Z

d such that πqγ intersects the m-th cell of the configurationspace. Fix a small δ0 > 0.

Consider planar FHLP. Fix a scatterer S and let Γ be a finite set ofscatterers. Then there exist constantsC = C (Card(Γ)) > 0, k0 = k0(Card(Γ)), ξ > 0 such that for anystandard pair ℓ such that πqγ ∩ S 6= ∅, length(ℓ) ≥ δ0 we have

qj 6∈ (S⋃

Γ) for j = k0 . . . n)

logξ n.

Weak bound on ”local time”Locally perturbed planar periodic Lorentz process

There is a constant K such that for all S

Eℓ(Card(j : qj ∈ S)) ≤ K log1+ξ N

where ξ is the constant from the previous Lemma on tail of returntimes.

Proof of Tail of return times

Without loss of generality we can assume that S is in the 0-th cell.Take a standard pair ℓ with length(ℓ) ≥ δ0. It suffices to show thatif R is sufficiently large and d([ℓ], (S

Γ)) ≥ R, then

qj 6∈ (S⋃

Γ) for j = 1 . . . n)

≥ Const

logα n. (10)

We establish (10) in case Card(Γ) = 1, the general case is similar.For fixing our ideas we also assume that d([ℓ],S) ≪ d([ℓ],Γ), theother cases are easier.

Take a sufficiently small ε0 > 0. Let τ1 be the first time τ suchthat either |qτ | ≥ R1+ε0 or |qτ | ≤ Rε0. It is proven in Sections 6and 7 of [DSzV1] that for any standard pair ℓ satisfyinglength(ℓ) ≥ δ0 and [ℓ] = R we have

|qτ1 | ≥ R1+ε0 and rτ1(x) ≥ R−100)

≥ ζ (11)

where 1 − ζ ≍ ε0, and thus ζ can be made as close to 1 as neededby choosing ε0 small.

Define τk as a first time τ after τk−1 when either

|qτ | ≥ R(1+ε0)k or |qτ | ≤ Rε0(1+ε0)k−1.

Iterating (11) we get

|qτk| ≥ R(1+ε0)

and rτk(x) ≥ R−100(1+ε0)

k−1)

≥ ζk . (12)

Let k be the largest number such that

R(1+ε0)k <d(Γ, 0)

Applying (12) with k = k we see that the probability that the

particle moves (d(Γ,0)2 )1/(1+ε0) away from the origin without visiting

S is at least c1/ log(d(Γ, 0)).For crossing the region where the particle can hit Γ we need a moredelicate argument. To do so we define τ1 as a first time τ after τk

such that

|qτ | ≥ d1+ε0(Γ, 0) or |qτ | ≤ d1/(1+ε0)3(Γ,0).

Then by the argument of Lemma 6.1/(a) of Section 6 of [DSzV/1]there exists a constant c2 > 0 such that for any standard pair ℓsatisfying

|[l ]| ≥(

d(Γ, 0)

)1/(1+ε0)

and length(ℓ) ≥ d−100(Γ, 0) (13)

we have

|qτ1 | ≥ d1+ε0(Γ, 0), rτ1(x) ≥ δ0 and τ1 − τk ≤ d3(1+ε0)(Γ, 0))

≥ c2.

On the other hand, by [DSzV1], for any standard pair satisfying(13)

qj visits Γ before time d3(1+ε0)(Γ, 0))

→ 0 as ε0 → 0, d(Γ, 0) → ∞.

Hence if ε0 is sufficiently small, then we can arrange that for asuitable c3 > 0

|qτ1 | ≥ d1+ε0(Γ, 0), rτ1(x) ≥ δ0 and qjnot visits Γ before τ1

≥ c3

Next let τk be the first time τ after τk−1 such that either

|qτ | ≥ d (1+ε0)k (Γ, 0) or |qτ | ≤ d(Γ, 0).

The argument used to prove (12) shows that for any ℓ such that

|[l ]| ≥ d (1+ε0)(Γ, 0), length(ℓ) > δ0

we havePℓ(|qτk

| ≥ R(1+ε0)k

) ≥ ζk .

Taking k such that

d (1+ε0)k (Γ, 0) = n

we get part (a).

The obvious, but important extension?Multidimensional case

Kramli-Szasz, 1985: Transience of periodic Lorentz process is easyfor d ≥ 3 once a Markov-partition with good properties has beenconstructed.Peter Balint-Peter Toth, 2006:

Young-tower, i. e. a Markov-partition with good properties isconstructed under an extra condition on the scattererconfiguration;

the extra condition is plausible for generic systems, itschecking is in progress. (Role in local ergodicity fornon-algebraic billiards!)

The obvious, but important extension?Multidimensional case

Kramli-Szasz, 1985: Transience of periodic Lorentz process is easyfor d ≥ 3 once a Markov-partition with good properties has beenconstructed.Peter Balint-Peter Toth, 2006:

Young-tower, i. e. a Markov-partition with good properties isconstructed under an extra condition on the scattererconfiguration;

the extra condition is plausible for generic systems, itschecking is in progress. (Role in local ergodicity fornon-algebraic billiards!)

Extension:

Standard pair techniques for multidimensional Sinai-billiards?Question: Two small Lorentz discs among a periodic configurationof scatterers.

limit theorems for locally perturbed lorentz processes · limit theorems for locally perturbed...

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