Space-time statistical solutions for an inhomogeneous chain of harmonic oscillators
aa r X i v : . [ m a t h - ph ] F e b Space–time statistical solutionsfor an inhomogeneous chain of harmonic oscillators
T.V. Dudnikova
Keldysh Institute of Applied MathematicsRussian Academy of ScienceMoscow 125047 Russia e-mail: [email protected]
Abstract
We consider an one-dimensional inhomogeneous harmonic chain consisting of twodifferent semi-infinite chains of harmonic oscillators. We study the Cauchy problem withrandom initial data. Under some restrictions on the interaction between the oscillatorsof the chain and on the distribution of the initial data, we prove the convergence ofspace-time statistical solutions to a Gaussian measure.
Key words: inhomogeneous chain of harmonic oscillators, Cauchy problem, randominitial data, space-time statistical solutions, weak convergence of measuresAMS Subject Classification 2010: 82Cxx, 37K60, 60G60, 37A25, 60Fxx
Introduction
We consider an infinite one-dimensional harmonic chain of particles having nearest-neighborinteractions and unit mass. We assume that the particles located at points x = 1 , , . . . havethe same interaction force constants ν + > κ + ≥ x = − , − , . . . have constants ν − > κ − ≥ κ ≥ κ = κ ± , in general. Therefore, the displacementof the particle located at a point x ∈ Z from its equilibrium position obeys the followingequations: ¨ u ( x, t ) = ( ν ∆ L − κ ) u ( x, t ) , x ≥ , t > , ¨ u (0 , t ) = ν ( u (1 , t ) − u (0 , t )) + ν − ( u ( − , t ) − u (0 , t )) − κ u (0 , t ) , t > , ¨ u ( x, t ) = ( ν − ∆ L − κ − ) u ( x, t ) , x ≤ − , t > . (1.1)Here u ( x, t ) ∈ R , ∆ L denotes the second derivative on Z = { , ± , ± , . . . } :∆ L u ( x ) = u ( x + 1) − u ( x ) + u ( x − , x ∈ Z . For system (1.1), we study the Cauchy problem with the initial data u ( x,
0) = u ( x ) , ˙ u ( x,
0) = v ( x ) , x ∈ Z . (1.2)Formally, this system is Hamiltonian with the Hamiltonian functional of the formH( u, ˙ u ) = H + ( u, ˙ u ) + H − ( u, ˙ u ) + H ( u, ˙ u ) , H ± ( u, ˙ u ) := 12 X ± x ≥ (cid:16) | ˙ u ( x, t ) | + ν ± | u ( x ± , t ) − u ( x, t ) | + κ ± | u ( x, t ) | (cid:17) , H ( u, ˙ u ) := 12 (cid:16) | ˙ u (0 , t ) | + X ± ν ± | u ( ± , t ) − u (0 , t ) | + κ | u (0 , t ) | (cid:17) . We consider two cases of equations (1.1). In the first case, we assume that the harmonic chainis homogeneous and κ ± , κ > ν ± =: ν > κ ± = κ =: κ > . (1.3)In the second one, we impose condition C on the coefficients ν ± > κ , κ ± ≥ κ − ≤ κ + .Put a ± := p ν ± + κ ± and K ± ( ω ) := 12 (cid:0) κ − + κ (cid:1) + 12 q ω − κ ± q ω − a ± for ω ∈ R : | ω | ≥ a ± ; K ( ω ) := 12 (cid:0) κ − + κ (cid:1) − q κ − ω q a − ω for ω ∈ R : | ω | ≤ κ + (if κ + > . Condition C.
For different values of κ ± and ν ± , the constant κ satisfies the followingrestrictions: κ < K + ( a − ) , if a − ≥ a + ; κ < K − ( a + ) , if a + ≥ a − ; κ > K ( κ − ) , if κ − = 0; κ > K − ( κ + ) or κ < K ( a − ) , if a − ≤ κ + ; κ = 0 , if κ − = κ + = 0 . κ + = κ − , then condition C implies that κ ∈ (cid:18) κ − , κ − + 2 max { ν − , ν + } q | ν − − ν | (cid:19) and ν − = ν + . Thus, condition C excludes the case when two semi-infinite parts of the chain are identical,i.e., when κ + = κ − and ν + = ν − .We assume that the initial data Y belong to the phase space H α , α ∈ R , defined below. Definition 1.1 (i) ℓ α ≡ ℓ α ( Z ) , α ∈ R , is the Hilbert space of real-valued sequences u ( x ) , x ∈ Z , with the norm k u k α = (cid:16) X x ∈ Z h x i α u ( x ) (cid:17) / < ∞ , h x i := (1 + x ) / . Below we use also the notation ℓ ≡ ℓ .(ii) H α = ℓ α × ℓ α is the Hilbert space of pairs Y = ( u ( x ) , v ( x )) of real-valued sequences u ( x ) and v ( x ) endowed with the norm k Y k α = k u k α + k v k α < ∞ . (iii) Write C kα = C k ( R ; ℓ α ) , k = 0 , , α ∈ R . Introduce the seminorms in C kα by the rule ||| u ( · , · ) ||| α,k,T = max | t |≤ T k X r =0 k ∂ rt u ( · , t ) k α , T > . (1.4) (iv) Denote by R the operator R : H α → C α such that ( RY )( x, t ) = u ( x, t ) , (1.5) where u ( x, t ) is the solution to problem (1.1)–(1.2) with the initial data Y = ( u , v ) . Below we assume that α < − / C holds and α < − / Y is a random function. Denote by µ a Borelprobability measure on H α giving the distribution of Y . Definition 1.2
Introduce a Borel probability measure P on the space C α by the rule P ( ω ) = µ ( R − ω ) for any Borel set ω ∈ B ( C α ) . Here and below B ( X ) denotes the σ -algebra of Borel sets of a topological space X . Themeasure P is called a space-time statistical solution to problem (1.1)–(1.2) corresponding tothe initial measure µ . Denote by { P τ , τ ∈ R } the following family of measures P τ ( ω ) = P ( S − τ ω ) for any ω ∈ B ( C α ) , τ ∈ R . Here S τ denotes the shift operator in time, S τ ( u ( x, t )) = u ( x, t + τ ) , τ ∈ R . (1.6)2he main goal of the paper is to prove that the measures P τ weakly converge as τ → ∞ to a limit on the space C α , P τ ⇁ P ∞ , τ → ∞ . (1.7)This means the convergence of the integrals Z C α f ( u ) P τ ( du ) → Z C α f ( u ) P ∞ ( du ) as τ → ∞ for any bounded continuous functional f on C α . Furthermore, the limit measure P ∞ is aGaussian measure on the space C α supported by the solutions to problem (1.1). Thus, theconvergence (1.7) can be considered as an analog of the central limit theorem for a class ofsolutions to the equations (1.1). The proof of convergence (1.7) is based on the results of [6]and used the technique of [11, 16]. Also, we check that the group S τ is mixing w.r.t. themeasure P ∞ , i.e., for any f, g ∈ L ( C α , P ∞ ) ,lim τ →∞ Z f ( S τ u ) g ( u ) P ∞ ( du ) = Z f ( u ) P ∞ ( du ) Z g ( u ) P ∞ ( du ) . (1.8)For models described by partial differential equations, the long-time behavior of space-time statistical solutions was studied by Komech and Ratanov [11] for wave equations andRatanov [14] for parabolic equations. For Klein–Gordon equations, the result was obtained in[2]. The time evolution and ergodic properties of infinite harmonic crystals were studied byLanford, Lebowitz [12] and by van Hemmen [8]. For the one-dimensional chains of harmonicoscillators, the behavior of statistical solutions µ t := [ U ( t )] ∗ µ as t → ∞ , where U ( t ) standsfor the solving operator of problem (1.1)–(1.2), was investigated in [6]. In this paper, we extendthese results to the space-time statistical solutions of problem (1.1). Introduce the notation Y ( x ) = ( Y ( x ) , Y ( x )) ≡ ( u ( x ) , v ( x )) , Y ( t ) = ( Y ( t ) , Y ( t )) ≡ ( u ( · , t ) , ˙ u ( · , t )) . Theorem 2.1 (see [4, Theorem 2.1]) Let κ ± , κ ≥ , ν ± > and Y ∈ H α , α ∈ R .Then the Cauchy problem (1.1)–(1.2) has a unique solution Y ( t ) ∈ C ( R , H α ) . The operator U ( t ) : Y → Y ( t ) is continuous in H α . Furthermore, there exist constants C, B < ∞ suchthat k U ( t ) Y k α ≤ Ce B | t | k Y k α , t ∈ R . (2.1) Corollary 2.2
It follows from (2.1) that for any Y ∈ H α , ||| RY ||| α, ,T ≤ C ( T ) k Y k α , ∀ T > , where the operator R is defined in (1.5). Below we assume that α < − / α < − / C holds. 3 .1 Conditions on the initial measure Definition 2.3 (i) A measure µ is called translation invariant (or space homogeneous) if µ ( S h B ) = µ ( B ) for any B ∈ B ( H α ) and h ∈ Z , where S h Y ( x ) = Y ( x + h ) , x ∈ Z .(ii) For a probability measure µ on H α , we denote by ˆ µ its characteristic functional(Fourier transform), ˆ µ (Ψ) ≡ Z exp( i h Y, Ψ i ) µ ( dY ) , Ψ ∈ S . Here
Ψ = (Ψ , Ψ ) ∈ S := S ⊕ S , S := S ( Z ) , where S ( Z ) denotes a space of real quicklydecreasing sequences, h Y, Ψ i = X i =0 , X x ∈ Z Y i ( x )Ψ i ( x ) , Y = ( Y , Y ) , Ψ = (Ψ , Ψ ) . Below we use also the notation h Y, Ψ i ± := P i =0 , P x ∈ Z ± Y i ( x )Ψ i ( x ) , Z ± := { x ∈ Z : ± x ≥ } .(iii) A measure µ is called Gaussian (of zero mean) if its characteristic functional has theform ˆ µ (Ψ) = exp {−Q (Ψ , Ψ) / } , where Q is a real-valued nonnegative quadratic form in S . We assume that the initial data Y ( x ) in (1.2) is a measurable random function withvalues in ( H α , B ( H α )) . Recall that µ is a Borel probability measure on H α which is thedistribution of Y . Let E stand for the mathematical expectation w.r.t. this measure. Denoteby Q ( x, y ) = (cid:0) Q ij ( x, y ) (cid:1) i,j =0 , the correlation matrix of the measure µ , where Q ij ( x, y ) := E (cid:0) Y i ( x ) Y j ( y ) (cid:1) ≡ Z Y i ( x ) Y j ( y ) µ ( dY ) , x, y ∈ Z , i, j = 0 , , and by Q (Ψ , Ψ) a real-valued quadratic form on S with the matrix kernel Q ( x, y ) .We impose conditions S1 – S4 on the initial measure µ . S1 µ has zero mean value, i.e., E ( Y ( x )) = 0 , x ∈ Z . S2 The correlation functions Q ij ( x, y ) satisfy the bound | Q ij ( x, y ) | ≤ h ( | x − y | ) , (2.2)where h is a nonnegative bounded function and h ( r ) ∈ L (0 , + ∞ ) . S3 The correlation matrix Q ( x, y ) satisfies the following condition Q ( x + y, y ) → (cid:26) q − ( x ) as y → −∞ q + ( x ) as y → + ∞ (cid:12)(cid:12)(cid:12)(cid:12) x ∈ Z . (2.3)Here q ± ( x ) = (cid:0) q ij ± ( x ) (cid:1) i,j =0 , stand for correlation matrices of some translation invariantmeasures µ ± with zero mean in H α . Definition 2.4
Let A be an interval in Z . Denote by σ ( A ) a σ -algebra in H α generated bythe initial data Y ( x ) with x ∈ A . Introduce the Ibragimov mixing coefficient of the measure µ by the rule ϕ ( r ) ≡ sup | µ ( A ∩ B ) − µ ( A ) µ ( B ) | µ ( B ) . ere the supremum is taken over all sets A ∈ σ ( A ) , B ∈ σ ( B ) with µ ( B ) > , and allintervals A , B ⊂ Z with distance ρ ( A , B ) ≥ r . The measure µ satisfies Ibragimov’s stronguniform mixing condition if ϕ ( r ) → as r → ∞ (cf. [9, Definition 17.2.2]). S4 µ has a finite “mean energy density”, i.e., sup x ∈ Z E | Y ( x ) | ≤ e < ∞ . Moreover, µ satisfies Ibragimov’s strong uniform mixing condition, and ϕ / ( r ) ∈ L (0 , + ∞ ) . Lemma 2.5 (i) Condition S2 implies that for any Φ , Ψ ∈ H , |Q (Φ , Ψ) | ≡ |h Q ( x, y ) , Φ( x ) ⊗ Ψ( y ) i| ≤ C k Φ k k Ψ k . (2.4) This follows from the bound (2.2) applying either the Shur test (see, e.g., [13, p.223]) orYoung’s inequality (see, e.g., [15, Theorem 0.3.1]).(ii) It follows from conditions S1 – S3 that q ij ± ∈ ℓ , i, j = 0 , . Hence, ˆ q ij ± ∈ C ( T ) .Assertions (i) and (ii) are proved in [5, Lemma 5.1].(iii) Conditions S1 and S4 imply the bound (2.2) with the function h ( r ) = Ce ϕ / ( r ) .This follows from [9, Lemma 17.2.3].(iv) The correlation functions Q ij have the property: Q ij ( x, y ) = Q ji ( y, x ) , i, j = 0 , .Then, the correlation functions q ij ± from condition S3 satisfy the relation q ii ± ( − x ) = q ii ± ( x ) , q ± ( x ) = q ± ( − x ) , x ∈ Z . (2.5) Denote by P a space of real-valued functions v ( x, t ) which are infinite differentiable in t andquickly decrease in t and x ,sup t ∈ R sup x ∈ Z h x i M h t i N | ∂ rt v ( x, t ) | ≤ C < ∞ for any M, N and r ≥ . Let [ · , · ] stand for the inner product in L ( R ; ℓ ) (or in its extensions),[ u , u ] = X x ∈ Z + ∞ Z −∞ u ( x, t ) u ( x, t ) dt. Definition 2.6
Denote by Q Pτ ( x , x , t , t ) , x , x ∈ Z , t , t ∈ R , the correlation functionsof the measures P τ , τ ∈ R , introduced in Definition 1.2, i.e., for any v , v ∈ P , Q Pτ ( v , v ) := [ Q Pτ , v ⊗ v ] = Z [ u, v ][ u, v ] P τ ( du )= X x ,x ∈ Z + ∞ Z −∞ dt ∞ Z −∞ Q Pτ ( x , x , t , t ) v ( x , t ) v ( x , t ) dt , τ ∈ R . The main result of the paper is the following theorem.5 heorem 2.7
Let α < − / and condition C hold. Then the following assertions hold.(i) Let conditions S1 – S3 be fulfilled. Then the correlation functions of P τ converge to alimit as τ → ∞ . Moreover, for any v , v ∈ P , Q Pτ ( v , v ) → Q P ∞ ( v , v ) as τ → ∞ , (2.6) where Q P ∞ ( v , v ) = Q P,ν ∞ ( T Ω ′ ~v , T Ω ′ ~v ) , (2.7) ~v i := ( v i , , the quadratic form Q P,ν ∞ is defined in (3.18) below, the operators Ω ′ and T aredefined in (3.32) and (3.39), respectively.(ii) Let conditions S1 , S3 , and S4 be fulfilled. Then the convergence (1.7) holds. Thelimit measure P ∞ is a Gaussian measure on the space C α supported by the solutions to prob-lem (1.1).(iii) The measure P ∞ is invariant w.r.t. the shifts in time, and the convergence (1.8)holds. Remark . If the initial measure µ is Gaussian, then convergence (1.7) follows from conver-gence (2.6). Furthermore, the weak convergence of the measures P τ doesn’t imply, in general,the convergence of their correlation matrices. Therefore, the last fact we prove separately. Theorem 2.7 is proved in Section 3. In Appendix, we consider the homogeneous case (1.3)and prove the similar results.
Theorem 2.8
Let α < − / and condition (1.3) hold. Then all assertions of Theorem 2.7remain true with the limiting correlation function Q P ∞ ( x , x , t , t ) of the following form Q P ∞ ( x , x , t , t ) = q P ∞ ( x − x , t − t ) . The Fourier transform of q P ∞ ( x, t ) w.r.t. variable x ( x → θ ) is of the form ˆ q P ∞ ( θ, t ) = cos( φ ( θ ) t ) ˆ q ∞ ( θ ) − sin( φ ( θ ) t ) φ − ( θ ) ˆ q ∞ ( θ ) , (2.8) where φ ( θ ) := p ν (2 − θ ) + κ , and ˆ q ij ∞ ( θ ) = ˆ q ij ∞ , + ( θ ) + ˆ q ij ∞ , − ( θ ) , i, j = 0 , , (2.9) with ˆ q ij ∞ , ± ( θ ) defined similarly to (3.11) but with φ ( θ ) instead of φ ± ( θ ) . We divide the proof of Theorem 2.7 into two steps:
Step 1 : Instead of problem (1.1)–(1.2) we first study a simpler “unperturbed” problem (3.1)with zero condition at origin and prove the results similar to Theorem 2.7, see Section 3.1.
Step 2 : In Section 3.2, we introduce a “wave” operator Ω , which allows us to reduce the“perturbed” problem (1.1)–(1.2) to the problem (3.1).6 .1 Unperturbed problem
Consider the following problem ¨ z ( x, t ) = ( ν ± ∆ L − κ ± ) z ( x, t ) , ± x ≥ , t > ,z (0 , t ) = 0 , t ≥ ,z ( x,
0) = u ( x ) , ˙ z ( x,
0) = v ( x ) , x = 0 . (3.1) Lemma 3.1 (see [6, Lemma 2.1]) Let α ∈ R . Then for any Y ≡ ( u , v ) ∈ H α there exists aunique solution Z ( t ) ≡ ( z ( · , t ) , ˙ z ( · , t )) ∈ C ( R , H α ) to problem (3.1). Furthermore, the operator U ( t ) : Y Z ( t ) is continuous in H α , and k U ( t ) Y k α ≤ Ce B | t | k Y k α , t ∈ R . The solution to problem (3.1) consists of two solutions to the initial–boundary value prob-lems in Z + and Z − with zero boundary condition at x = 0 . Therefore, the solution to (3.1)has a form ( U ( t ) Y ) i ( x ) = P j =0 , P y ∈ Z + G ijt, + ( x, y ) Y j ( y ) for x ∈ Z + , P j =0 , P y ∈ Z − G ijt, − ( x, y ) Y j ( y ) for x ∈ Z − , (3.2)where Y ( x ) ≡ u ( x ) , Y ( x ) ≡ v ( x ) , and the Green function G t, ± ( x, y ) = ( G ijt, ± ( x, y )) i,j =0 is a matrix-valued function with the entries of the form G ijt, ± ( x, y ) := G ijt, ± ( x − y ) − G ijt, ± ( x + y ) , x, y ∈ Z ± , G ijt, ± ( x ) ≡ π Z T e − ixθ ˆ G ijt, ± ( θ ) dθ, (3.3) (cid:16) ˆ G ijt, ± ( θ ) (cid:17) i,j =0 = (cid:18) cos ( φ ± ( θ ) t ) sin ( φ ± ( θ ) t ) /φ ± ( θ ) − φ ± ( θ ) sin ( φ ± ( θ ) t ) cos ( φ ± ( θ ) t ) (cid:19) , (3.4) φ ± ( θ ) = q ν ± (2 − θ ) + κ ± . (3.5)In particular, φ ± ( θ ) = 2 ν ± | sin( θ/ | if κ ± = 0 . Note that G ijt, ± (0 , y ) ≡ G ijt, ± ( − x ) = G ijt, ± ( x ) . Definition 3.2
Introduce a measure ν = µ { Y ∈ H α : Y (0) = 0 } . Denote by ν t , t ∈ R ,a Borel probability measure on H α giving the distribution of the solution U ( t ) Y to prob-lem (3.1), i.e., ν t ( B ) = ν ( U ( − t ) B ) for any B ∈ B ( H α ) . The correlation matrix of ν t isdenoted as Q νt ( x, y ) = (cid:0) Q ν,ijt ( x, y ) (cid:1) i,j =0 , , Q ν,ijt ( x, y ) := Z Y i ( x ) Y j ( y ) ν t ( dY ) , x, y ∈ Z , t ∈ R . The correlation matrix Q νt has the following property. Lemma 3.3
Let condition S2 hold. Then sup t ∈ R | Q νt ( x, y ) | ≤ p C + C | x | p C + C | y | , x, y ∈ Z , (3.6) where the constants C and C do not depend on x, y , and C = 0 if κ − κ + = 0 . roof We check (3.6) only for x, y ∈ Z + . For another values of x, y the proof is similar.Using Definition 3.2 and representation (3.2), we obtain that for x, y ∈ Z + , t ∈ R , i, j = 0 , Q ν,ijt ( x, y ) = Z ( U ( t ) Y ) i ( x ) ( U ( t ) Y ) j ( y ) ν ( dY )= X k,l =0 , X x ′ ,y ′ ∈ Z + G ikt, + ( x, x ′ ) Q ν,kl ( x ′ , y ′ ) G jlt, + ( y, y ′ ) = h Q ν ( · , · ) , Φ ix ( · , t ) ⊗ Φ jy ( · , t ) i + , where Φ ix ( x ′ , t ) := (cid:0) G i t, + ( x, x ′ ) , G i t, + ( x, x ′ ) (cid:1) . Hence, applying (2.4), one obtains (cid:12)(cid:12) Q ν,ijt ( x, y ) (cid:12)(cid:12) ≤ C k Φ ix ( · , t ) k k Φ jy ( · , t ) k , where the constant C does not depend on x, y, t . On the other hand, the Parseval identityand (3.3) imply k Φ ix ( · , t ) k = 1 π Z T sin ( xθ ) (cid:16) | ˆ G i t, + ( θ ) | + | ˆ G i t, + ( θ ) | (cid:17) dθ, x ∈ Z + , i = 0 , . Hence, by (3.4), we have k Φ x ( · , t ) k ≤ C < ∞ and k Φ x ( · , t ) k ≤ Z T sin ( xθ ) (cid:18) C + C φ ( θ ) (cid:19) dθ ≤ C + C x, x ∈ N , (3.7)where the constants C and C do not depend on t ∈ R and x ∈ N . Moreover, C = 0 if κ + = 0 , by (3.5). If κ + = 0 , then φ ( θ ) = 4 ν sin ( θ/
2) and the bound in the r.h.s. of (3.7)follows from Fei´er’s theorem (see, e.g., [10]). (cid:3)
Corollary 3.4
Let α < − / if κ − κ + = 0 , and α < − otherwise. Then sup t ∈ R Z k Y k α ν t ( dY ) ≤ C < ∞ . (3.8)Indeed, applying the bound (3.6) gives Z k Y k α ν t ( dY ) = X x ∈ Z h x i α (cid:0) Q ν, t ( x, x ) + Q ν, t ( x, x ) (cid:1) ≤ X x ∈ Z h x i α ( C + C | x | ) ≤ C ( α ) < ∞ , by the choice of the α .Introduce the limiting matrix Q ν ∞ ( x, y ) by the rule Q ν ∞ ( x, y ) = Q ∞ , + ( x, y ) if x, y > ,Q ∞ , − ( x, y ) if x, y < , , (3.9)where Q ∞ , ± ( x, y ) := q ∞ , ± ( x − y ) − q ∞ , ± ( x + y ) − q ∞ , ± ( − x − y ) + q ∞ , ± ( − x + y ) , x, y ∈ Z ± . (3.10)8he Fourier transforms of the entries of q ∞ , ± ( x ) , x ∈ Z , have the formˆ q ∞ , ± ( θ ) = (cid:0) ˆ q ± ( θ ) + ˆ q ± ( θ ) φ − ± ( θ ) (cid:1) ± i sign( θ ) φ − ± ( θ ) (cid:0) ˆ q ± ( θ ) − ˆ q ± ( θ ) (cid:1) , ˆ q ∞ , ± ( θ ) = φ ± ( θ )ˆ q ∞ , ± ( θ ) , ˆ q ∞ , ± ( θ ) = − ˆ q ∞ , ± ( θ ) = ± i sign( θ ) φ ± ( θ )ˆ q ∞ , ± ( θ ) , (3.11)where θ ∈ T if κ ± = 0 and θ ∈ T \ { } otherwise, the functions q ij ± , i, j = 0 , q ± from condition (2.3), φ ± ( θ ) are defined in (3.5). Remark . By (2.5), ˆ q ii ± ( − θ ) = ˆ q ii ± ( θ ) and ˆ q ± ( − θ ) = ˆ q ± ( θ ) . Then, (3.11) gives ˆ q ii ∞ , ± ( − θ ) = ˆ q ii ∞ , ± ( θ ) , ˆ q ij ∞ , ± ( − θ ) = ˆ q ji ∞ , ± ( θ ) = − ˆ q ij ∞ , ± ( θ ) if i = j, i, j = 0 , . Therefore, by (3.9) and (3.10), Q ij ∞ , ± ( x, y ) = 0 and Q ν,ij ∞ ( x, y ) = 0 if i = j , Q ii ∞ , ± ( x, y ) = 2 π Z T ˆ q ii ∞ , ± ( θ ) sin( xθ ) sin( yθ ) dθ = Q ν,ii ∞ ( x, y ) , x, y ∈ Z ± . Denote by Q νt (Ψ , Ψ) , t ∈ R , a real-valued quadratic form on S = [ S ( Z )] with the matrixkernel Q νt ( x, y ) . Using (3.9), we have Q ν ∞ (Ψ , Ψ) = h Q ν ∞ ( x, y ) , Ψ( x ) ⊗ Ψ( y ) i = X ± h Q ∞ , ± ( x, y ) , Ψ( x ) ⊗ Ψ( y ) i ± . (3.12)The following theorem was proved in [6]. Theorem 3.5
Let α < − / if κ − κ + = 0 , and α < − otherwise. Then the followingassertions hold. (i) Let conditions S1 – S3 be fulfilled. Then the correlation functions of themeasures ν t converge to a limit: Q νt ( x, y ) := Z (cid:16) Y ( x ) ⊗ Y ( y ) (cid:17) ν t ( dY ) → Q ν ∞ ( x, y ) , t → ∞ , x, y ∈ Z , where the limiting correlation matrix Q ν ∞ ( x, y ) is of the form (3.9).(ii) Let conditions S1 , S3 and S4 be fulfilled. Then the measures ν t converge weakly toa limit measure as t → ∞ on the space H α . The limit measure ν ∞ is Gaussian with zeromean value and its characteristic functional is ˆ ν ∞ (Ψ) := Z e i h Y, Ψ i ν ∞ ( dY ) = exp (cid:26) − Q ν ∞ (Ψ , Ψ) (cid:27) , Ψ ∈ S , where the quadratic form Q ν ∞ is defined in (3.12). Below we will use an additional property of the quadratic form Q νt , t ∈ R . To state it wefirst introduce auxiliary spaces. Definition 3.6
For any sequence ψ , we introduce odd sequences ψ − and ψ + by the rule ψ ± ( x ) = ψ ( x ) for ± x > , for x = 0 , − ψ ( − x ) for ± x < . (3.13)9 efine the Hilbert space ℓ ( κ ) := { ψ ∈ ℓ : ˆ ψ + φ − ( θ ) , ˆ ψ − φ − − ( θ ) ∈ L ( T ) } with the norm k ψ k ℓ ( κ ) := k ψ k ℓ + X ± (cid:13)(cid:13) F − θ → x [ φ − ± ( θ )] ∗ ψ ± (cid:13)(cid:13) ℓ . Introduce the space H ( κ ) := ℓ ( κ ) × ℓ with the norm k Ψ k H ( κ ) := k Ψ k ℓ ( κ ) + k Ψ k ℓ , Ψ = (Ψ , Ψ ) . In particular, if κ − κ + = 0 , then H ( κ ) = H = ℓ × ℓ . Remark . By (3.13), in the Fourier transform, | b ψ ± ( θ ) | ≤ C | sin θ | P ± x> | x || ψ ( x ) | , θ ∈ T .Hence, if P x ∈ Z | x || ψ ( x ) | < ∞ , then b ψ ± φ − ± ∈ C ( T ) . In particular, ℓ − α ⊂ ℓ ( κ ) for α < − / ,since P x ∈ Z | x || ψ ( x ) | ≤ C k ψ k − α by the Cauchy–Schwartz inequality. Lemma 3.7
The quadratic forms Q νt (Ψ , Ψ) and the characteristic functionals ˆ ν t (Ψ) , t ∈ R ,are equicontinuous in H ( κ ) . Proof
For t ∈ R , introduce a formal adjoint operator U ′ ( t ) to the solving operator U ( t ) , h U ( t ) Y, Ψ i = h Y, U ′ ( t )Ψ i , Y ∈ H α , Ψ ∈ S . Then, the action of the group U ′ ( t ) coincides with the action of U ( t ) up to the order of thecomponents. Namely, U ′ ( t )Ψ = (cid:16) ˙ ψ ( · , t ) , ψ ( · , t ) (cid:17) , where ψ ( x, t ) is a solution to problem (3.1)with the initial data ( u , v ) = (Ψ , Ψ ) . Using (2.4), we have Q νt (Ψ , Ψ) = Q ν ( U ′ ( t )Ψ , U ′ ( t )Ψ) ≤ C k U ′ ( t )Ψ k . On the other hand, by (3.2) and (3.3),( U ′ ( t )Ψ) j ( y ) = X i =0 X x ∈ Z ± G ijt, ± ( x, y )Ψ i ( x ) = X i =0 X x ∈ Z G ijt, ± ( x − y )Ψ i ± ( x ) for y ∈ Z ± . Here we use notation (3.13). Therefore, applying the Parseval identity and (3.4), we obtain k U ′ ( t )Ψ k ≤ C X ± Z T (cid:16) (1 + φ − ± ( θ )) | ˆΨ ± ( θ ) | + | ˆΨ ± ( θ ) | (cid:17) dθ ≤ C k Ψ k H ( κ ) . Hence, Q νt (Ψ , Ψ) ≤ C k Ψ k H ( κ ) uniformly in t ∈ R . (3.14)This implies the equicontinuity of the characteristic functionals ˆ ν t (Ψ) , t ∈ R . Indeed, by theCauchy–Schwartz inequality and (3.14), one obtains | ˆ ν t (Ψ ) − ˆ ν t (Ψ ) | = (cid:12)(cid:12)(cid:12) Z (cid:0) e i h Y, Ψ i − e i h Y, Ψ i (cid:1) ν t ( dY ) (cid:12)(cid:12)(cid:12) ≤ Z (cid:12)(cid:12) e i h Y, Ψ − Ψ i − (cid:12)(cid:12) ν t ( dY ) ≤ Z |h Y, Ψ − Ψ i| ν t ( dY ) ≤ sZ |h Y, Ψ − Ψ i| ν t ( dY )= p Q νt (Ψ − Ψ , Ψ − Ψ ) ≤ C k Ψ − Ψ k H ( κ ) . (cid:3) efinition 3.8 (i) Denote by R the operator R : H α → C α such that ( R Y )( x, t ) = z ( x, t ) , where z ( x, t ) is the solution to problem (3.1) with the initial data Y = ( Y , Y ) ≡ ( u , v ) .Then, by (3.2), ( R Y )( x, t ) = P j =0 , P y ∈ Z + G jt, + ( x, y ) Y j ( y ) for x ∈ Z + , P j =0 , P y ∈ Z − G jt, − ( x, y ) Y j ( y ) for x ∈ Z − . (3.15) In particular, ( R Y ) (0 , t ) = 0 for all t .(ii) Introduce a Borel probability measure P ν on the space C α as P ν ( ω ) = ν ( R − ω ) , ∀ ω ∈ B ( C α ) . This measure is called a space-time statistical solution to problem (3.1).(iii) Denote by { P ντ , τ ∈ R } the family of measures defined by the rule P ντ ( ω ) = P ν ( S − τ ω ) , ∀ ω ∈ B ( C α ) , τ ∈ R . In this section, we prove the following theorem.
Theorem 3.9
Let α < − / if κ − κ + = 0 , and α < − otherwise. Then the followingassertions hold. (i) Let conditions S1 and S2 be fulfilled. Then the bounds are true: sup τ ≥ Z ||| z ||| α, ,T P ντ ( dz ) ≤ C ( α ) < ∞ , ∀ T > , (3.16) where the constant C ( α ) does not depend on T > .(ii) Let conditions S1 – S3 be fulfilled. Then for any v , v ∈ P , Q P,ντ ( v , v ) := Z [ z, v ][ z, v ] P ντ ( dz ) → Q P,ν ∞ ( v , v ) , τ → ∞ . (3.17) Here Q P,ν ∞ ( v , v ) := [ Q P,ν ∞ , v ⊗ v ] , (3.18) where the limiting correlation matrix Q P,ν ∞ is of a form Q P,ν ∞ ( x , x , t , t ) = Q P,ν ∞ , + ( x , x , t , t ) if x , x > ,Q P,ν ∞ , − ( x , x , t , t ) if x , x < , otherwise , t , t ∈ R . (3.19) Here Q P,ν ∞ , ± ( x , x , t , t ) := 2 π Z T cos ( φ ± ( θ )( t − t )) ˆ q ∞ , ± ( θ ) sin( x θ ) sin( x θ ) dθ, (3.20) where ˆ q ∞ , ± is defined in (3.11).(iii) Let conditions S1 , S3 and S4 be fulfilled. Then the measures P ντ converge weakly toa limiting measure P ν ∞ on the space C α as τ → ∞ . The characteristic functional of P ν ∞ is ˆ P ν ∞ ( v ) ≡ Z e i [ z,v ] P ντ ( dz ) = exp (cid:26) − Q P,ν ∞ ( v, v ) (cid:27) , v ∈ P , (3.21) where the quadratic form Q P,ν ∞ is defined in (3.18)–(3.20). roof (i) At first, note that P ντ ( ω ) = ν τ ( R − ω ) for any ω ∈ B ( C α ) and τ > , (3.22)where ν τ is defined in Definition 3.2. Hence, the bound (3.16) follows from (3.8), because Z ||| z ||| α, ,T P ντ ( dz ) = Z ||| R Y ||| α, ,T ν τ ( dY ) = sup | s |≤ T Z k U ( s ) Y k α ν τ ( dY )= sup | s |≤ T Z k Y k α ν s + τ ( dY ) ≤ C ( α ) < ∞ . (ii) Let z ≡ z ( · , t ) be a solution to problem (3.1). Then, for any v ∈ P ,[ z, v ] = [ R Y , v ] = h Y , R ′ v i , (3.23)where R ′ is an adjoint operator to the operator R , R ′ v = (( R ′ v ) , ( R ′ v ) ) , and( R ′ v ) j ( y ) = P x ∈ Z + + ∞ R −∞ G jt, + ( x, y ) v ( x, t ) dt if y ∈ Z + , P x ∈ Z − + ∞ R −∞ G jt, − ( x, y ) v ( x, t ) dt if y ∈ Z − , j = 0 , , (3.24)by (3.15). In particular, using (3.3), we have ( R ′ v ) j (0) = 0 . Below we use the notation k v k L ( R ; X ) := + ∞ Z −∞ k v ( · , t ) k X dt for v ( · , t ) ∈ L ( R ; X ) with X = ℓ or X = ℓ ( κ ) . We state the additional properties of the operator R ′ in the following lemma. Lemma 3.10 (i) If κ − κ + = 0 , then R ′ v ∈ S for any v ∈ P . (ii) For any v ∈ L ( R ; ℓ ( κ )) , k R ′ v k H ( κ ) ≤ C k v k L ( R ; ℓ ( κ )) . (3.25) Proof
The first assertion follows from (3.24) and formulas (3.3)–(3.5). To prove assertion (ii),we apply (3.24), notation (3.13) for v ( x, t ) , and equations (3.3) and obtain( R ′ v ) j ( y ) = X x ∈ Z + ∞ Z −∞ G jt, ± ( x − y ) v ± ( x, t ) dt for y ∈ Z ± . Hence, by the Parseval identity and (3.4), we have k ( R ′ v ) j k ℓ ≤ X ± + ∞ Z −∞ k ˆ G jt, ± ( θ )ˆ v ± ( θ, t ) k L ( T ) dt ≤ X ± + ∞ Z −∞ k φ − j ± ( θ )ˆ v ± ( θ, t ) k L ( T ) dt. k ( R ′ v ) k ℓ ≤ C k v k L ( R ; ℓ ) . If κ − κ + = 0 , then the same bound is valid for ( R ′ v ) .If κ − κ + = 0 , then k ( R ′ v ) k ℓ ≤ C k v k L ( R ; ℓ ( κ )) . Furthermore, k ( R ′ v ) k ℓ ( κ ) = k ( R ′ v ) k ℓ + X ± (cid:13)(cid:13)(cid:13) F − [ φ − ± ] ∗ (cid:0) ( R ′ v ) (cid:1) ± (cid:13)(cid:13)(cid:13) ℓ ≤ + ∞ Z −∞ k v ( · , t ) k ℓ dt + X ± + ∞ Z −∞ k φ − ± ( θ )ˆ v ± ( θ, t ) k L ( T ) dt. This implies the bound (3.25). In particular, R ′ v ∈ H ( κ ) for any v ∈ P . Lemma 3.10 isproved. (cid:3) We return to the proof of assertion (ii) of Theorem 3.9. For any v , v ∈ P , Q P,ντ ( v , v ) = Z [ R Y, v ][ R Y, v ] ν τ ( dY ) = Z h Y, R ′ v ih Y, R ′ v i ν τ ( dY )= h Q ντ ( x, y ) , ( R ′ v )( x ) ⊗ ( R ′ v )( y ) i ≡ Q ντ ( R ′ v , R ′ v ) . If κ − κ + = 0 , then R ′ v i ∈ S and Theorem 3.5 (i) implies Q P,ντ ( v , v ) = Q ντ ( R ′ v , R ′ v ) → Q ν ∞ ( R ′ v , R ′ v ) , τ → ∞ . (3.26)If κ ± = 0 , then convergence (3.26) follows from Theorem 3.5 (i) and Lemmas 3.7 and 3.10,because the space S is dense in H ( κ ) .It remains to check formula (3.20). Using (3.26), (3.12), (3.9) and (3.24), we have Q P,ν ∞ ( v , v ) = Q ν ∞ ( R ′ v , R ′ v ) = h Q ν ∞ ( y , y ) , R ′ v ( y ) ⊗ R ′ v ( y ) i = X ± X x ,x ∈ Z ± + ∞ Z −∞ dt ∞ Z −∞ Q P,ν ∞ , ± ( x , x , t , t ) v ( x , t ) v ( x , t ) dt , where, by definition, Q P,ν ∞ , ± ( x , x , t , t ) := X i,j =0 , X y ,y ∈ Z ± Q ij ∞ , ± ( y , y ) G it , ± ( x , y ) G jt , ± ( x , y )for x , x ∈ Z ± , t , t ∈ R . Hence, (3.19) holds. Using formulas (3.3) and (3.10) and theParseval identity, we obtain Q P,ν ∞ , ± ( x , x , t , t ) = X i,j =0 , X y ,y ∈ Z q ij ∞ , ± ( y − y ) G it , ± ( x , y ) G jt , ± ( x , y )= 42 π X i,j =0 , Z T ˆ q ij ∞ , ± ( θ ) ˆ G it , ± ( θ ) ˆ G jt , ± ( θ ) sin( x θ ) sin( x θ ) dθ (3.27)for ± x , ± x > t , t ∈ R . Applying (3.11) and (3.4), we obtain Q P,ν ∞ , ± ( x , x , t , t ) = 2 π Z T (cid:26) cos ( φ ± ( θ )( t − t )) ˆ q ∞ , ± ( θ ) − sin ( φ ± ( θ )( t − t )) φ ± ( θ ) ˆ q ∞ , ± ( θ ) (cid:27) × sin( x θ ) sin( x θ ) dθ. q ∞ , ± ( θ ) are odd and φ ± ( θ ) are even.(iii) According to the methods of [16], to establish the weak convergence of the measures P ντ on the space C α it is enough to prove the following two assertions:(A1) The family of measures { P ντ , τ ∈ R } is weakly compact in C α ;(A2) The characteristic functionals of P ντ converge to a limit as τ → ∞ .The first (second) assertion provides the existence (resp., uniqueness) of the limit measures P ν ∞ . Proof of assertion (A1) : To prove the weak compactness of the family { P ντ , τ ∈ R } , weverify that this family satisfies the following conditions (a) and (b) of the Prokhorov Theorem(see, e.g., [7]):(a) sup { P ντ , τ ∈ R } < ∞ ,(b) for any ε > K ε in C β such that sup τ P ντ ( C β \ K ε ) < ε .Condition (a) holds since P ντ are probability measures. To check condition (b), we apply thetechnique of [16, Theorem XII.5.2]. For k = 0 , T > C kα,T the space of thefunctions t → u ( · , t ) ∈ ℓ α , t ∈ [0 , T ] , for which the norm (1.4) is finite. For any T >
M > K ( T, M ) := { u ∈ C α,T : ||| u ||| α, ,T ≤ M } . Below we choose M ≡ M ( T ) by a special way. The sets K ( T, M ) are uniformly bounded anduniformly equicontinuous. Since the embedding of the spaces ℓ α in ℓ β is compact if α > β ,then the sets K ( T, M ) are precompact in C β,T by the Dubinskii embedding theorems (see,e.g., [1] or [16, Theorem IV.4.1]) using the Arzel`a–Ascoli theorem (see, e.g., [17, Ch.3, § T > J T : C β → C β,T of the restriction of the functions u ( x, t ) ∈ C β from Z × R into Z × [ − T, T ] . Applying the Chebyshev inequality and the bound (3.16), weobtain P ντ { C β \ J − T K ( T, M ) } ≤ Z ||| u ||| α, ,T P ντ ( du ) /M ≤ C ( α ) /M , (3.28)where by K we denote the closure of K in the topology of the metrizable space C β . For any ε > M = M ε ( T ) such that C ( α ) ∞ X T =1 M ε ( T ) < ε. Set K ε := ∞ T T =1 J − T K ( T, M ε ( T )) . Then the bound (3.28) implies the condition (b). Proof of assertion (A2) : Applying (3.22), (3.23), Theorem 3.5 (ii) and Lemmas 3.7 and3.10, we obtain that for every v ∈ P ,ˆ P ντ ( v ) = Z e i [ z,v ] P ντ ( dz ) = Z e i h Y ,R ′ v i ν τ ( dY ) → exp (cid:26) − Q ν ∞ ( R ′ v, R ′ v ) (cid:27) , τ → ∞ . The assertion (iii) of Theorem 3.9 is proved. (cid:3) .2 Perturbed problem The key role in the proof of convergence (1.7) for problem (1.1) plays the following lemma.
Lemma 3.11 (see [6, Lemma 4.3]) Let Y ∈ H α , α < − / , and conditions C , S1 , and S2 hold. Then there exists a linear bounded operator Ω : H → H α such that the followingrepresentation holds U ( t ) Y ( x ) = Ω( U ( t ) Y )( x ) + δ ( x, t ) , where E k δ ( · , t ) k α ≤ C h t i − . (3.29) Here U ( t ) Y ≡ ( u ( · , t ) , ˙ u ( · , t )) is a solution to problem (1.1)–(1.2), the operator Ω is of theform Ω Y = Y + Γ Y, (Γ Y )( x ) := (cid:0) h Y, ¯ Γ ( x, · ) i , h Y, ¯ Γ ( x, · ) i (cid:1) , x ∈ Z , (3.30) where ¯ Γ j ( x, y ) , j = 0 , , is a vector-valued function of the form ¯ Γ j ( x, y ) = + ∞ R Γ ± x ( s ) (cid:16) U ′ ( − s ) G j (cid:17) ( y ) ds if ± x > , G j ( y ) if x = 0 , y ∈ Z . Here G j ( y ) := ν ± G j ± ( y ) for ± y ≥ , G j ± ( y ) := + ∞ Z N ( j ) ( s ) g ± ( y, − s ) ds, y ∈ Z , g ± ( y, t ) := (cid:16) G t, ± ( ± , y ) , G t, ± ( ± , y ) (cid:17) ,N (0) ( s ) ≡ N ( s ) , N (1) ( s ) ≡ ˙ N ( s ) , the functions N ( s ) and Γ ± x ( s ) are constructed in [6]. Theysatisfy the following bounds: | N ( s ) | ≤ C h s i − / , X x ∈ Z ± \{ } h x i α | Γ ± x ( s ) | ≤ C h s i − , s ∈ R , α < − / . Corollary 3.12
Let α < − / . Then there is a bounded linear operator Ω ′ : H − α → H such that for any Ψ ∈ S we have h U ( t ) Y , Ψ i = h U ( t ) Y , Ω ′ Ψ i + δ ( t ) , where E | δ ( t ) | ≤ C h t i − k Ψ k − α . (3.31) The operator Ω ′ is of a form Ω ′ Ψ = Ψ + Γ ′ Ψ , (Γ ′ Ψ)( y ) := X j =0 h ¯ Γ j ( · , y ) , Ψ j ( · ) i , Ψ = (Ψ , Ψ ) . (3.32) Remark . As shown in [6], k ¯ Γ j ( x, · ) k ∈ H α ∀ α < − / , where k · k ≡ k · k H . Furthermore,using the similar reasonings as in [6] one can check that k ¯ Γ j ( x, · ) k H ( κ ) ∈ H α for any α < − / , j = 0 , . Therefore, k Γ ′ Ψ k H ( κ ) ≤ C k Ψ k − α and k Ω ′ Ψ k H ( κ ) ≤ C k Ψ k − α ∀ Ψ ∈ H − α . (3.33)Before to prove Theorem 2.7 we state the results concerning the statistical solutions µ t toproblem (1.1). 15 efinition 3.13 µ t is a Borel probability measure in H α which gives the distribution of Y ( t ) , µ t ( B ) = µ ( U ( − t ) B ) for any B ∈ B ( H α ) , t ∈ R . The correlation functions of the measure µ t are defined as Q ijt ( x, y ) = E (cid:0) Y i ( x, t ) Y j ( y, t ) (cid:1) , i, j = 0 , , x, y ∈ Z , t ∈ R . Here Y i ( x, t ) are the components of the solution Y ( t ) = ( Y ( · , t ) , Y ( · , t )) = ( u ( · , t ) , ˙ u ( · , t )) .Denote by Q t the quadratic form with the matrix kernel ( Q ijt ( x, y )) i,j =0 , , Q t (Ψ , Ψ) = Z |h Y, Ψ i| µ t ( dY ) = X i,j =0 , (cid:10) Q ijt ( x, y ) , Ψ i ( x )Ψ j ( y ) (cid:11) , t ∈ R , Ψ = (Ψ , Ψ ) ∈ S . Lemma 3.14
Let α < − / and conditions C , S1 , and S2 be fulfilled. Then the followingbound holds sup t ∈ R Z k Y k α µ t ( dY ) = sup t ∈ R E k U ( t ) Y k α ≤ C < ∞ . (3.34) Proof . As shown in [4], for any α < − / k U ′ ( t ) ¯ Γ j ( x, · ) k ∈ H α uniformly in t ∈ R , i.e.,sup t ∈ R (cid:13)(cid:13)(cid:0) k U ′ ( t ) ¯ Γ j ( x, · ) k (cid:1)(cid:13)(cid:13) α ≡ sup t ∈ R X x ∈ Z h x i α (cid:13)(cid:13) U ′ ( t ) ¯ Γ j ( x, · ) (cid:13)(cid:13) < ∞ . (3.35)We check that sup t ∈ R E k Ω U ( t ) Y k α ≤ C < ∞ . (3.36)Indeed, applying (3.30), (3.8), (2.4) and (3.35) gives E k Ω U ( t ) Y k α ≤ E k U ( t ) Y k α + X j =0 E k|h Y ( · ) , U ′ ( t ) ¯ Γ j ( x, · ) ik α ≤ C + X j =0 X x ∈ Z h x i α Q (cid:0) U ′ ( t ) ¯ Γ j ( x, · ) , U ′ ( t ) ¯ Γ j ( x, · ) (cid:1) ≤ C + C X j =0 X x ∈ Z h x i α k U ′ ( t ) ¯ Γ j ( x, · ) k ≤ C < ∞ . Therefore, (3.29) and (3.36) imply the bound (3.34). (cid:3)
Theorem 3.15 (see [6, Theorems 2.3, 2.4]) Let α < − / and condition C hold. Then thefollowing assertions are fulfilled.(i) Let conditions S1 – S3 hold. Then for all Ψ ∈ S , lim t →∞ E |h Y ( t ) , Ψ i| = Q ∞ (Ψ , Ψ) = Q ν ∞ (Ω ′ Ψ , Ω ′ Ψ) , (3.37) where the quadratic form Q ν ∞ is introduced in (3.12).(ii) Let conditions S1 , S3 and S4 hold. Then the measures µ t weakly converge to aGaussian measure µ ∞ as t → ∞ on H α . The characteristic functional of µ ∞ is of a form ˆ µ ∞ (Ψ) = exp {−Q ∞ (Ψ , Ψ) / } , Ψ ∈ S . emark . It follows from the bounds (3.14) and (3.33) that sup t ∈ R Q νt (Ω ′ Ψ , Ω ′ Ψ) ≤ C k Ω ′ Ψ k H ( κ ) ≤ C k Ψ k − α ∀ Ψ ∈ H − α . In particular, the r.h.s. of (3.37) is defined for any Ψ ∈ S . Proof of Theorem 2.7:
At first, using Lemma 3.11, we estimate [ S τ u, v ] , where v ∈ P , u ≡ u ( x, t ) is a solution to problem (1.1), S τ is defined in (1.6). Set ~v := ( v,
0) . For any v ∈ P , we have [ S τ u, v ] = [ S τ z, T Ω ′ ~v ] + δ τ , where E ( δ τ ) = o (1) , τ → ∞ , (3.38) z ≡ z ( x, t ) is a solution to problem (3.1), and the operator T is defined by the rule T Φ := Φ − ˙Φ for Φ ≡ Φ( t ) = (Φ ( t ) , Φ ( t )) . (3.39)To prove (3.38), we first write [ S τ u, v ] in a form[ S τ u, v ] = + ∞ Z −∞ h U ( t + τ ) Y , ~v ( · , t ) i dt = + ∞ Z −∞ h U ( t + τ ) Y , Ω ′ ~v ( · , t ) i dt + δ τ , (3.40)where E ( δ τ ) = o (1) , τ → ∞ . The bound (3.40) follows from Corollary 3.12 because E ( δ τ ) ≤ (cid:16) + ∞ Z −∞ q E |h U ( t + τ ) Y , ~v ( · , t ) i − h U ( t + τ ) Y , Ω ′ ~v ( · , t ) i| dt (cid:17) ≤ C (cid:16) + ∞ Z −∞ h t + τ i − / k v ( · , t ) k − α dt (cid:17) = o (1) as τ → ∞ . (3.41)Secondly, we rewrite the integral in the r.h.s. of (3.40) using notation (3.39): + ∞ Z −∞ h U ( t + τ ) Y , Ω ′ ~v ( · , t ) i dt = + ∞ Z −∞ h z ( · , t + τ ) , T Ω ′ ~v ( · , t ) i dt = [ S τ z, T Ω ′ ~v ] . This implies representation (3.38). Further, using (3.38), we obtain that for v , v ∈ P , Q Pτ ( v , v ) = Z [ u, v ][ u, v ] P τ ( du ) = Q P,ντ ( T Ω ′ ~v , T Ω ′ ~v ) + δ ′ τ , (3.42)where the quadratic form Q P,ντ is introduced in (3.17), δ ′ τ = o (1) as τ → ∞ . Note that T Ω ′ ~v i
6∈ P , in general, and we can not apply convergence (3.17) immediately.At first, using the equality Q P,ντ ( w , w ) = Q ντ ( R ′ w , R ′ w ) , we obtain Q Pτ ( v , v ) = Q ντ ( R ′ T Ω ′ ~v , R ′ T Ω ′ ~v ) + o (1) , τ → ∞ . Then, the convergence of Q Pτ ( v , v ) to a limit as τ → ∞ follows from the following facts:17i) the quadratic form Q ντ (Ψ , Ψ) converges to a limit for any Ψ ∈ S (Theorem 3.5 (i));(ii) S is dense in H ( κ ) ;(iii) the quadratic forms Q ντ (Ψ , Ψ) , τ ∈ R , are equicontinuous in H ( κ ) (Lemma 3.7);(iv) R ′ T Ω ′ ~v ∈ H ( κ ) for any v ∈ P .Hence, it remains to check the last fact. By (3.32) and (3.39), R ′ T Ω ′ ~v = R ′ (cid:0) v + (Γ ′ ~v ) − ∂ t (Γ ′ ~v ) (cid:1) . Due to (3.33), we have k Γ ′ ~v ( · , t ) k H ( κ ) ≡ k (Γ ′ ~v ) ( · , t ) k ℓ ( κ ) + k (Γ ′ ~v ) ( · , t ) k ℓ ≤ C k v ( · , t ) k − α . (3.43)Since v + (Γ ′ ~v ) ∈ L ( R ; ℓ ( κ )) , then the bound (3.25) gives k R ′ (cid:0) v + (Γ ′ ~v ) (cid:1) k H ( κ ) ≤ C k v + (Γ ′ ~v ) k L ( R ; ℓ ( κ )) ≤ C k v k L ( R ; ℓ − α ) . However, we can not apply bound (3.25) with (Γ ′ ~v ) instead of v , because (Γ ′ ~v ) ( · , t ) ∈ ℓ for any t by (3.43), but (Γ ′ ~v ) ( · , t ) ℓ ( κ ) , in general. Now we study R ′ ˙ w with w := (Γ ′ ~v ) .Note first that R ′ v ( y ) = R ′± v ( y ) for y ∈ Z ± with( R ′± v ) j ( y ) := X x ∈ Z ± + ∞ Z −∞ G jt, ± ( x, y ) v ( x, t ) dt = X x ∈ Z + ∞ Z −∞ G jt, ± ( x − y ) v ± ( x, t ) dt, where we use notation (3.13). Then, in the Fourier transform,( [ R ′± ˙ w ) j ( θ ) = + ∞ Z −∞ b G jt, ± ( θ ) ∂ t b w ± ( θ, t ) dt = − + ∞ Z −∞ b G jt, ± ( θ ) b w ± ( θ, t ) dt, θ ∈ T , by (3.4). Hence, using the Parseval equality, we have k R ′± ˙ w k H ( κ ) ≤ C X j =0 , k ( [ R ′± ˙ w ) j k L ( T ) + C k ( [ R ′± ˙ w ) φ − ± k L ( T ) ≤ C + ∞ Z −∞ k b w ± ( · , t ) k L ( T ) dt ≤ C k w k L ( R ; ℓ ) . Therefore, k R ′ ∂ t (Γ ′ ~v ) k H ( κ ) ≤ X ± k R ′± ∂ t (Γ ′ ~v ) k H ( κ ) ≤ C k (Γ ′ ~v ) k L ( R ; ℓ ) ≤ C k v k L ( R ; ℓ − α ) by (3.43). Hence, R ′ T Ω ′ ~v ∈ H ( κ ) . This completes the proof of assertion (i) of Theorem 2.7.Assertion (ii) of Theorem 2.7 follows from the following lemma.18 emma 3.16 (1) Let conditions C , S1 , and S2 hold. Then the family of the measures { P τ , τ ∈ R } is weakly compact in the space C β , with any β < α < − / , and the boundholds: sup τ ≥ Z ||| u ||| α, ,T P τ ( du ) ≤ C ( α ) < ∞ , (3.44) where the constant C ( α ) does not depend on T > .(2) Let conditions C , S1 , S3 , and S4 hold. Then for every v ∈ P , the characteristicfunctionals of P τ converge to a limit as τ → ∞ , ˆ P τ ( v ) ≡ Z e i [ u,v ] P τ ( du ) → ˆ P ∞ ( v ) , τ → ∞ . (3.45) Here ˆ P ∞ ( v ) = ˆ P ν ∞ ( T Ω ′ ~v ) , where ˆ P ν ∞ is defined in (3.21). Proof
Similarly to (3.22), we have P τ ( ω ) = µ τ ( R − ω ) for any ω ∈ B ( C α ) and τ > , (3.46)where µ τ is defined in Definition 3.13. To prove the bound (3.44), we apply (3.46) and obtain Z ||| u ||| α, ,T P τ ( du ) = Z ||| RY ||| α, ,T µ τ ( dY ) = sup | s |≤ T Z k U ( s ) Y k α µ τ ( dY )= sup | s |≤ T Z k Y k α µ s + τ ( dY ) ≤ C ( α ) < ∞ by the bound (3.34). The bound (3.44) and the Prokhorov theorem imply the weak compact-ness of the measures family { P τ , τ ∈ R } in the space C β , β < α . This can be proved by asimilar method as in the proof of Theorem 3.9 (iii).To prove (3.45), we use the inequality (cid:12)(cid:12) e iξ − (cid:12)(cid:12) ≤ | ξ | for ξ ∈ R and bounds (3.38), (3.41)and obtain that (cid:12)(cid:12)(cid:12) ˆ P τ ( v ) − ˆ P ντ ( T Ω ′ ~v ) (cid:12)(cid:12)(cid:12) ≤ E | δ τ | ≤ p E ( δ τ ) ≤ C + ∞ Z −∞ h t + τ i − / k v ( · , t ) k − α dt → , τ → ∞ . It remains to apply Theorem 3.9 (iii) and obtain thatˆ P ντ ( T Ω ′ ~v ) → ˆ P ν ∞ ( T Ω ′ ~v ) , τ → ∞ . (3.47)However, T Ω ′ ~v
6∈ P , in general. More precisely, convergence (3.47) follows from the followingfacts:(i) the equality ˆ P ντ ( T Ω ′ ~v ) = ˆ ν τ ( R ′ T Ω ′ ~v ) holds by (3.22) and (3.23);(ii) ˆ ν τ (Ψ) converges to a limit as τ → ∞ for any Ψ ∈ S (Theorem 3.5 (ii));(iii) S is dense in H ( κ ) (evidently);(iv) the characteristic functionals ˆ ν τ (Ψ) , τ ∈ R , are equicontinuous in H ( κ ) (Lemma 3.7);(v) R ′ T Ω ′ ~v ∈ H ( κ ) for any v ∈ P (this was proved above).Lemma 3.16 is proved. (cid:3) This completes the proof of assertion (ii) of Theorem 2.7. Assertion (iii) of Theorem 2.7 isproved in next section. 19 .3 Mixing property of the limit measure P ∞ We first prove the convergence (1.8) for the measure P ν ∞ . The invariance of P ν ∞ w.r.t. thegroup S τ , τ ∈ R , follows from Theorem 3.9 (iii). Lemma 3.17
The group S τ is mixing w.r.t. the measure P ν ∞ , i.e., for any f, g ∈ L ( C α , P ν ∞ ) , lim τ →∞ Z f ( S τ z ) g ( z ) P ν ∞ ( dz ) = Z f ( z ) P ν ∞ ( dz ) Z g ( z ) P ν ∞ ( dz ) . (3.48) In particular, the group S τ is ergodic w.r.t. the measure P ν ∞ , i.e., lim T →∞ T T Z f ( S τ z ) dτ = Z f ( z ) P ν ∞ ( dz ) (mod P ν ∞ ) . Proof
Since P ν ∞ is a Gaussian measure with zero mean value, it is enough to prove (see [7])that for any v , v ∈ P , I τ := Z [ S τ z, v ][ z, v ] P ν ∞ ( dz ) → τ → ∞ . (3.49)Using (3.23), (3.24), and (3.9), we obtain I τ = Z (cid:10)
Y, R ′ S − τ v (cid:11) h Y, R ′ v i ν ∞ ( dY ) = (cid:10) Q ν ∞ ( y , y ) , ( R ′ S − τ v )( y ) ⊗ ( R ′ v )( y ) (cid:11) = X ± X x ,x ∈ Z ± + ∞ Z −∞ dt ∞ Z −∞ A τ, ± ( x , x , t , t ) v ( x , t ) v ( x , t ) dt , (3.50)where, by definition, A τ, ± ( x , x , t , t ) := X i,j =0 , X y ,y ∈ Z ± Q ij ∞ , ± ( y , y ) G it + τ, ± ( x , y ) G jt , ± ( x , y ) . Similarly to (3.27), we have A τ, ± ( x , x , t , t ) := X i,j =0 , X y ,y ∈ Z q ij ∞ , ± ( y − y ) G it + τ, ± ( x , y ) G jt , ± ( x , y )= 2 π X i,j =0 , Z T ˆ q ij ∞ , ± ( θ ) ˆ G it + τ, ± ( θ ) ˆ G jt , ± ( θ ) sin( x θ ) sin( x θ ) dθ = 2 π Z T cos ( φ ± ( θ )( t + τ − t )) ˆ q ∞ , ± ( θ ) sin( x θ ) sin( x θ ) dθ. (3.51)Hence, applying Lemma 2.5 (ii), formulas (3.11), and Fei´er’s theorem (if κ ± = 0 ), we obtain | A τ, ± ( x , x , t , t ) | ≤ C Z T (cid:12)(cid:12) ˆ q ∞ , ± ( θ ) sin( x θ ) sin( x θ ) (cid:12)(cid:12) dθ ≤ C + C ( | x | + | x | ) , (3.52)20here the constants C and C do not depend on x , x ∈ Z ± and C = 0 if κ ± = 0 .Since v , v ∈ P , it follows from (3.50) and (3.52) that to prove (3.49) it suffices to check theconvergence A τ, ± ( x , x , t , t ) → τ → ∞ , (3.53)for fixed values of x , x ∈ Z ± \ { } and t , t ∈ R . We denote by R τ, ± the integrand in ther.h.s. of (3.51) and rewrite it in the form R τ, ± ( θ ) ≡ R τ, ± ( θ ; x , x , t , t ) = cos ( φ ± ( θ )( t + τ − t )) ˆ q ∞ , ± ( θ ) sin( x θ ) sin( x θ )= cos ( φ ± ( θ ) τ ) a ± ( θ ) + sin ( φ ± ( θ ) τ ) b ± ( θ ) , where a ± ( θ ) ≡ a ± ( θ ; x , x , t , t ) = cos ( φ ± ( θ )( t − t )) ˆ q ∞ , ± ( θ ) sin( x θ ) sin( x θ ) ,b ± ( θ ) ≡ b ± ( θ ; x , x , t , t ) = − sin ( φ ± ( θ )( t − t )) ˆ q ∞ , ± ( θ ) sin( x θ ) sin( x θ ) , and a ± , b ± ∈ L ( T ) . Choose a δ > f ( θ ) + g ( θ ) = 1 ,where f and g are nonnegative functions in C ∞ ( T ) , supp f ⊂ O δ (0) , supp g ∩ O δ/ (0) = ∅ .We split A τ, ± into the sum of two integrals, A τ, ± = 2 π Z T f ( θ ) R τ, ± ( θ ) dθ + 2 π Z T g ( θ ) R τ, ± ( θ ) dθ ≡ A fτ, ± + A gτ, ± . On the one hand, ∀ ε > ∃ δ > | A fτ, ± | ≤ Cε uniformly in τ . On the otherhand, the phase functions φ ± ( θ ) are smooth and φ ′± ( θ ) = 0 on the support of the g . Hence,the oscillatory integrals in A gτ, ± vanish by the Lebesgue–Riemann Theorem. Therefore, theconvergence (3.53) holds and Lemma 3.17 is proved.Now we check assertion (1.8) for the limit measure P ∞ . We prove that for any v , v ∈ P I ′ τ := Z [ S τ u, v ][ u, v ] P ∞ ( du ) → τ → ∞ . (3.54)Applying (2.7) gives I ′ τ = Z [ z, T Ω ′ ~v ( · , t − τ )] [ z, T Ω ′ ~v ] P ν ∞ ( dz ) = Z [ S τ z, T Ω ′ ~v ] [ z, T Ω ′ ~v ] P ν ∞ ( dz ) . Put f i ( z ) := [ z, T Ω ′ ~v i ] , i = 1 , f i ∈ L ( C α , P ν ∞ ) , since Z | f i ( z ) | P ν ∞ ( dz ) = Z [ u, v i ] P ∞ ( du ) < ∞ by (2.7). Therefore, we apply (3.48) and obtain I ′ τ = Z f ( S τ z ) f ( z ) P ν ∞ ( dz ) → Z f ( z ) P ν ∞ ( dz ) Z f ( z ) P ν ∞ ( dz ) as τ → ∞ . Finally, Z f i ( z ) P ν ∞ ( dz ) = Z [ u, v i ] P ∞ ( du ) = 0 , because P ∞ has zero mean value. Hence, (3.54) holds. This completes the proof of Theo-rem 2.7. (cid:3) ppendix: Homogeneous harmonic chain Let condition (1.3) hold. Then the problem (1.1)–(1.2) becomes (cid:26) ¨ u ( x, t ) = ( ν ∆ L − κ ) u ( x, t ) , x ∈ Z , t > ,u ( x,
0) = u ( x ) , ˙ u ( x,
0) = v ( x ) , x ∈ Z . At first, we state results concerning the statistical solutions µ t , t ∈ R . Lemma A.1 (see [3, Theorem A]) Let α < − / and condition (1.3) hold. Then all asser-tions of Theorem 3.15 hold, where the quadratic form Q ∞ has the matrix kernel Q ∞ ( x, y ) = q ∞ ( x − y ) , (A.1) q ∞ ( x ) = F − θ → x [ˆ q ∞ ( θ )] and ˆ q ∞ ( θ ) = (ˆ q ij ∞ ( θ )) is defined in (2.9). Proof of Theorem 2.8 . Introduce the adjoint operator R ′ to the operator R defined in(1.5), [ RY, v ] = h Y, R ′ v i for Y ∈ H α and v ∈ P . (A.2)Then for v ∈ P ,( R ′ v )( x ) = (cid:16) X y ∈ Z + ∞ Z −∞ G t ( y − x ) v ( y, t ) dt, X y ∈ Z + ∞ Z −∞ G t ( y − x ) v ( y, t ) dt (cid:17) , (A.3)where G ijt is defined as G ijt, + in (3.3) and (3.4) but with φ ( θ ) instead of φ + ( θ ) . It followsfrom (A.2) and (3.46) that for v , v ∈ P , Q Pτ ( v , v ) := Z [ u, v ][ u, v ] P τ ( du ) = Z [ RY, v ][ RY, v ] µ τ ( dY )= Z h Y, R ′ v ih Y, R ′ v i µ τ ( dY ) = h Q τ ( x, y ) , R ′ v ( x ) ⊗ R ′ v ( y ) i . Since κ = 0 , then R ′ v ∈ S for any v ∈ P . Hence, we can apply Lemma A.1 and obtain Q Pτ ( v , v ) → h Q ∞ ( x, y ) , R ′ v ( x ) ⊗ R ′ v ( y ) i as τ → ∞ . Hence, Q P ∞ ( v , v ) = h Q ∞ ( x, y ) , R ′ v ( x ) ⊗ R ′ v ( y ) i . Now we check formula (2.8). Using (A.1)and (A.3), we have Q P ∞ ( x , x , t , t ) = Q P ∗ ( x − x , t , t ) , where in the Fourier transform x → θ ˆ Q P ∗ ( θ, t , t ) = X x ∈ Z e iθx Q P ∗ ( x, t , t ) = X i,j =0 ˆ G it ( θ )ˆ q ij ∞ ( θ ) ˆ G jt ( θ ) , θ ∈ T , t , t ∈ R . Using formulas ˆ q ∞ ( θ ) = φ ( θ )ˆ q ∞ ( θ ) , ˆ q ∞ ( θ ) = − ˆ q ∞ ( θ ) , and (3.4) with φ + ≡ φ , we haveˆ Q P ∗ ( θ, t , t ) = cos ( φt ) ˆ q ∞ ( θ ) cos ( φt ) − sin ( φt ) φ − ˆ q ∞ ( θ ) cos ( φt )+ cos ( φt ) ˆ q ∞ ( θ ) φ − sin ( φt ) + sin ( φt ) ˆ q ∞ ( θ ) sin ( φt )= cos ( φ ( t − t )) ˆ q ∞ ( θ ) − sin ( φ ( t − t )) φ − ˆ q ∞ ( θ )22ith φ ≡ φ ( θ ) . This implies (2.8).Now we prove the convergence (1.7) by a similar way as in Theorem 2.7. Assertion (1.7)follows from the bound (3.44) and convergence (3.45). The bound (3.44) can be proved in thesame way as in Theorem 2.7. Lemma A.1 implies that for any Ψ ∈ S ,ˆ µ t (Ψ) → ˆ µ ∞ (Ψ) as t → ∞ . Using (A.2) and taking Ψ := R ′ v , we obtainˆ P τ ( v ) = Z e i [ u,v ] P τ ( du ) ≡ Z e i h Y,R ′ v i µ τ ( dY ) → exp (cid:26) − Q ∞ ( R ′ v, R ′ v ) (cid:27) , τ → ∞ , where the quadratic form Q ∞ is introduced in Lemma A.1. Theorem 2.8 is proved. (cid:3) Now we verify the mixing property (1.8) for the limit measure P ∞ . The invariance of themeasure P ∞ w.r.t. the shifts in time and in space follows from convergence (1.7) and (A.1).Since the measure P ∞ is Gaussian with zero mean value, it is enough to prove that for any v , v ∈ P , I τ := E ∞ ([ S τ u, v ][ u, v ]) → τ → ∞ , (A.4)where E ∞ denotes the integral w.r.t. the limit measure P ∞ .Indeed, using (A.2) and (A.3), we obtain I τ = Z (cid:10)
Y, R ′ S − τ v (cid:11) h Y, R ′ v i µ ∞ ( dY ) = 12 π Z T (cid:16) ˆ q ∞ ( θ ) , \ ( R ′ S − τ v )( θ ) ⊗ \ ( R ′ v )( θ ) (cid:17) dθ = 12 π Z T dθ + ∞ Z −∞ dt ∞ Z −∞ B τ ( t , t , θ )ˆ v ( θ, t )ˆ v ( θ, t ) dt , where the function B τ ( t , t , θ ) is of the form B τ ( t , t , θ ) := X i,j =0 ˆ G it + τ ( θ )ˆ q ij ∞ ( θ ) ˆ G jt ( θ )= cos ( φ ( θ )( t + τ − t )) ˆ q ∞ ( θ ) − sin ( φ ( θ )( t + τ − t )) φ − ( θ )ˆ q ∞ ( θ ) . We represent I τ as follows: I τ = X ± π Z T e ± iφ ( θ ) τ c ± ( θ ) dθ, (A.5)where c ± ( θ ) := 12 + ∞ Z −∞ dt ∞ Z −∞ e ± iφ ( θ )( t − t ) (cid:0) ˆ q ∞ ( θ ) ± iφ − ( θ )ˆ q ∞ ( θ ) (cid:1) ˆ v ( θ, t )ˆ v ( θ, t ) dt . Note that c ± ( θ ) ∈ L ( T ) by Lemma 2.5 (ii) and formulas (3.11) with φ ± ≡ φ . Hence,the oscillatory integrals in (A.5) vanish by the Lebesgue–Riemann Theorem. Therefore, theassertions (A.4) and (1.8) hold.Similarly to (1.8), we can check the following assertion.23 emma A.2 Let S h , h ∈ Z , denote the shifts in space, S h u ( x, t ) = u ( x + h, t ) . Then, forany f, g ∈ L ( C α , P ∞ ) , lim h →∞ E ∞ f ( S h u ) g ( u ) = E ∞ f E ∞ g. Proof
Indeed, it suffices to check that I h := E ∞ ([ S h u, v ][ u, v ]) → h → ∞ . Using(A.2) and (A.3), we obtain I h = Z h Y, R ′ S − h v ih Y, R ′ v i µ ∞ ( dY ) = 12 π Z T e ihθ D ( θ ) dθ, where D ( θ ) := + ∞ Z −∞ dt ∞ Z −∞ (cid:0) cos ( φ ( t − t )) ˆ q ∞ ( θ ) − sin ( φ ( t − t )) φ − ˆ q ∞ ( θ ) (cid:1) ˆ v ( θ, t )ˆ v ( θ, t ) dt . Therefore, I h vanishes as h → ∞ by the Lebesgue–Riemann Theorem, because D ( θ ) ∈ L ( T ) . (cid:3) The following lemma generalizes the convergence (1.8).
Lemma A.3
The group S τ is mixing of order r ≥ w.r.t. the measure P ∞ , i.e., for any f , . . . , f r ∈ L r +1 ( C α , P ∞ ) , lim τ ,...,τ r →∞ Z f ( u ) f ( S τ u ) · . . . · f r ( S τ + ... + τ r u ) P ∞ ( du ) = r Y i =0 Z f i ( u ) P ∞ ( du ) . Proof
Since the measure P ∞ is Gaussian with zero mean value, it is enough to prove that forany v , . . . , v r ∈ P , I τ ,...,τ r := E ∞ ([ u, v ][ S τ u, v ] · . . . · [ S τ + ... + τ r u, v r ]) → τ → ∞ . (A.6)At first, note that (see [7, Ch.III, § E ∞ ([ u, v ] · . . . · [ u, v n ]) = (cid:26) , if n is odd, P Q E ∞ ([ u, v i ][ u, v j ]) , if n is even.Here the sum is taken over all partitions of { v , . . . , v n } into pairs, the product is taken overall pairs of the partition (the pairs that differ by the permutation of elements are consideredas one). For example, if n = 4 , there are three partitions of { v , v , v , v } into pairs and E ∞ ([ u, v ] · . . . · [ u, v ]) = b b + b b + b b , where b ij := E ∞ ([ u, v i ][ u, v j ]) . Hence, I τ ,...,τ r = 0 if r is even. If r is odd, then I τ ,...,τ r = E ∞ (cid:0) [ u, v ][ u, S − τ v ] · . . . · [ u, S − τ + ... + τ r v r ] (cid:1) = r X k =1 E ∞ (cid:0) [ u, v ][ u, S − τ + ... + τ k v k ] (cid:1) · (cid:16)X Y B ij (cid:17) , (A.7)24here the inner sum is taken over all partitions of { v , . . . , v k − , v k +1 , . . . , v r } into pairs, theproduct is taken over all pairs of the partition, and B ij := E ∞ (cid:16)(cid:2) u, S − τ + ... + τ i v i (cid:3)(cid:2) u, S − τ + ... + τ j v j (cid:3)(cid:17) , i, j = 1 , . . . , k − , k + 1 , . . . , r. Since the measure P ∞ is invariant w.r.t. S τ , B ij ≤ p E ∞ ([ u, v i ] ) q E ∞ ([ u, v j ] ) ≤ C < ∞ . Furthermore, (A.4) implies that for any k ≥ E ∞ (cid:0) [ u, v ][ u, S − τ + ... + τ k v k ] (cid:1) → τ , . . . , τ k → + ∞ . (A.8)Formulas (A.7)–(A.8) imply the convergence (A.6). (cid:3) Acknowledgment . This work was supported by the Russian Science Foundation (Grantno. 19-71-30004).
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