Invariant measures and long time behaviour for the Benjamin-Ono equation III
aa r X i v : . [ m a t h . A P ] M a y , INVARIANT MEASURES AND LONG TIME BEHAVIOUR FORTHE BENJAMIN-ONO EQUATION III
YU DENG, NIKOLAY TZVETKOV, AND NICOLA VISCIGLIA
Abstract.
We complete the program developed in our previous works aim-ing to construct an infinite sequence of invariant measures of gaussian typeassociated with the conservation laws of the Benjamin-Ono equation. Introduction
Our goal here is to complete the program developed in our previous works [10,21, 23, 24, 25] aiming to construct an infinite sequence of invariant measures ofgaussian type associated with the conservation laws of the Benjamin-Ono equation.The Benjamin-Ono equation reads(1.1) ∂ t u + H ∂ x u + u∂ x u = 0 , where H denotes the Hilbert transform. We consider (1.1) with periodic boundaryconditions, i.e. the spatial variable x is on the one dimensional torus. It is wellknown (see e.g. [12]) that at least formally the solutions of (1.1) satisfy an infinitenumber of conservation laws of the form(1.2) E k/ ( u ) = k u k H k/ + R k/ ( u ) , k = 0 , , , · · · , where R k/ is a sum of terms homogeneous in u of order larger or equal to three(but contains ”less derivatives”). Despite of this remarkable algebraic property,which indicates that the Benjamin-Ono equation is ”integrable”, there are someimportant analytical difficulties (related to the non local nature of (1.1)) to developthe inverse scattering method for (1.1). In particular we are aware of no referenceimplementing integrability methods in the context of the periodic Benjamin-Onoequation.As already noticed in our previous works, the conservation laws (1.2) may beused to construct invariant measures supported by Sobolev spaces of increasingsmoothness. This in turn implies some insides on the long time behavior of thesolution of (1.1). Recall that the idea of using a conserved quantity to constructgaussian type invariant measures goes back to [11]. It was then further developedby many authors including [2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 16, 17, 18, 22, 23, 24, 25, 26].Let us now briefly recall the construction of gaussian type measures associatedwith (1.2). These measures are absolutely continuous with respect to the gaussianmeasure µ k/ induced by(1.3) ϕ k/ ( x, ω ) = X n ∈ Z \{ } g n ( ω ) | n | k/ e i nx , where, ( g n ( ω )) is a sequence of centered standard complex gaussian variables suchthat g n = g − n and ( g n ( ω )) n> are independent. For any N ≥ k ≥ R > we introduce the function(1.4) F k/ ,N,R ( u ) = (cid:16) k − Y j =0 χ R ( E j/ ( π N u )) (cid:17) χ R ( E ( k − / ( π N u ) − α N ) e − R k/ ( π N u ) where α N = P Nn =1 cn for a suitable constant c , π N denotes the projector on Fouriermodes n such that | n | ≤ N , χ R is a cut-off function defined as χ R ( x ) = χ ( x/R )with χ : R → R a smooth, compactly supported function such that χ ( x ) = 1 forevery | x | < k ∈ N with k ≥ µ k/ measurable function F k/ ,R ( u ) such that ( F k/ ,N,R ( u )) N ≥ converges to F k/ ,R ( u )in L q ( dµ k/ ) for every 1 ≤ q < ∞ . This in particular implies that F k/ ,R ( u ) ∈ L q ( dµ k/ ).Set dρ k/ ,R ≡ F k/ ,R ( u ) dµ k/ . Then we have that [ R> supp( ρ k/ ,R ) = supp( µ k/ )and one may conjecture that ρ k/ ,R is invariant under a well defined flow of (1.1).This conjecture was proved for k = 1 in [10] and for k ≥ Theorem 1.1.
The measures dρ ,R and dρ / ,R are invariant under the flow as-sociated with the Benjamin-Ono equation (1.1) established in [13] . There are two main sources of difficulties to proof the invariance of ρ k/ ,R .The first one, presented only for k ≥
2, is that even if E k/ is conserved quantityfor (1.1) it is no longer conserved by the approximated versions of (1.1). This dif-ficulty was resolved for k ≥ ρ k/ ,R . For k ≥ k = 4 isslightly more delicate and already appeals to a dispersive effect). For k = 2 , ρ k/ ,R is a delicate issue which is resolved in[13]. The case k = 1 is even more delicate and was resolved in [10] which in turn ledto the invariance of ρ / ,R because in the case k = 1 the first mentioned difficultyis absent.In view of the above discussion, we see that the cases k = 2 , NVARIANT MEASURES FOR BO 3 dimensional approximating problems. In many situations this approximation canbe obtained by projection on the Fourier modes with at most N frequencies, viathe sharp Dirichlet projectors π N , and letting N → ∞ . It is however not quiteclear whether in the case of the Benjamin-Ono equation at low level of regularity this family of finite dimensional problems approximate well the true solution of theBenjamin-Ono equation. To overcome this difficulty we use the idea of [7, 8, 10]and we project the equation by using a family of smoothed projectors S ǫN with ǫ > N ∈ N (see the next pargraph). However, in contrast with the casetreated in [10], where it is sufficient to work with a fixed parameter ǫ and letting N → ∞ , in the situation we face in this paper, it is important to consider both ǫ → N → ∞ , which requires a considerable care. In particular it is of crucialimportance that in Proposition 6.1 below, we have a bound proportional to t whichenables to glue local bounds on time intervals with very poor dependence on ε . Inother words, the fact that we do not sacrify any time integration and that we onlyexploit the random oscillations of the initial data in the estimates on the measureevolution is of importance for the analysis in this paper.We conclude this introduction by fixing some notations. We denote by B ( X ) theBorel sets of the topological space X and by B M ( Y ) the ball of radius M , centeredat the origin of a Banach space Y . For every fixed ǫ ∈ (0 ,
1) we denote by ψ ǫ asmooth function ψ ǫ : R → R such that ψ ǫ ( x ) = 1 for x ∈ [0 , (1 − ǫ )] , ψ ǫ ( x ) = 0 for x > , (1.5) k ψ ǫ k L ∞ = 1 and ψ ǫ ( x ) = ψ ǫ ( | x | ) . We denote by S ǫN the Fourier multiplier:(1.6) S ǫN ( X j ∈ Z a j e ijx ) = X j ∈ Z a j ψ ǫ ( jN ) e ijx . We also denote by Φ( t ) the flow associated with (1.1) (well-defined on H s , s ≥ ǫN ( t ) the flow on H s , s ≥ ∂ t u + H ∂ x u + S ǫN ( S ǫN u · S ǫN u x ) = 0 . (1.7)Since the x mean value is conserved by the flow of (1.1), we shall only considersolutions of (1.1) and of its approximated version (1.7) with vanishing zero Fouriermode (this is the case in (1.3) as well). Acknowledgement.
N.T. is supported by ERC Grant Dispeq, N.V. is sup-ported by FIRB grant Dinamiche Dispersive.2.
Deterministic Theory
In this section we prove the following deterministic result.
Proposition 2.1.
Let < ǫ < , σ > σ ′ > and M > be fixed, so that σ issmall enough. We have, for some T = T ( ǫ, σ, σ ′ , M ) > , C = C ( ǫ, σ, σ ′ , M ) > that: sup φ ∈ B M ( H / − σ ′ ) sup | t |≤ T k Φ ǫN ( t ) φ − Φ( t ) φ k H / − σ ≤ CN − θ , where θ = θ ( σ, σ ′ ) > . YU DENG, NIKOLAY TZVETKOV, AND NICOLA VISCIGLIA
The spaces.
Let the standard X s,b space be defined by k u k X s,b = X n ∈ Z Z R h n i s h τ − | n | n i b | ( F x,t u )( n, τ ) | d τ, and the Y s space be k u k Y s = X n ∈ Z h n i s (cid:18) Z R | ( F x,t u )( n, τ ) | d τ (cid:19) . We then define the space U ′ by k u k U ′ = k u k X − / , / + k u k Y / − σ ′ , and U is defined by replacing σ ′ with σ . The space ( U ′ ) T is defined by k u k ( U ′ ) T = sup {k v k U ′ : v | [ − T,T ] = u | [ − T,T ] } , while U T and X s,b,T are defined similarly.In the proof below we will denote s = 1 / − σ, s ′ = 1 / − σ ′ , r = 1 / σ = 1 − s. The norms we will control in the bootstrap estimate will be X s ′ ,r,T and X s,r,T forthe gauged function w , and ( U ′ ) T and U T for the original function u .2.2. Linear bounds.
The content of this section is well-known. We just record ithere for the reader’s convenience. In the sequel χ ∈ C ∞ ( R ) is a fixed function suchthat χ ( t ) = 1 for | t | < χ ( t ) = 0 for | t | > Proposition 2.2.
We have the following bounds :(1) the Strichartz estimates: k u k L t,x . k u k X , / , and k u k L t,x . k u k X σ,r . (2) the bound for the Duhamel evolution: kE u k X s,b . k u k X s,b − , < b < where the Duhamel operator is defined by E u ( t ) = χ ( t ) Z t χ ( s ) e ( t − s ) H∂ xx u ( s ) d s. (3) the short-time bound: for T ≤ and < b < b ′ < / we have k χ ( T − t ) u k X s,b . T b ′ − b k u k X s,b ′ . (4) fixed-time estimates: sup t k u ( t ) k H / − σ . min( k u k X / − σ , / σ , k u k Y / − σ ) . (5) the linear bounds: k χ ( t ) e ± H∂ xx φ k X s,b ∩ Y s . k φ k H s for any s and b .Proof. These are well-known properties of X s,b spaces, see [20]. (cid:3) NVARIANT MEASURES FOR BO 5
The gauge transform.
The use of gauge transforms (allowing to weak theimpact of the derivative loss) in the context of the Benjamin-Ono equation wasinitiated by Tao [19]. The computation in this section is a much simplified versionof that in [10]. We still need the notation from [10], namely that m ij representsthe sum m i + · · · + m j for i ≤ j .In this section we are fixing an ǫ and an N ; so we will denote S ǫN simply by S and ψ ǫ by ψ . Let u = Φ ǫN ( t )( φ ) be the global solution to (1.7) (with initial data φ of zero mean value). Let z be the unique mean zero antiderivative of u , andconsider the operators P : g ( Sz ) · g and Q : g ( Su ) · g . By abusing notationwe will also call them Sz and Su . The exponential M = exp (cid:18) i2 SP S (cid:19) = ∞ X λ =0 λ ! (i / λ ( SP S ) λ is defined as a power series; we will then define v = M u ; w = π > ( M u ) . The goal is to prove the following
Lemma 2.1.
We have the evolution equation (2.1) ( ∂ t − i ∂ xx ) w = N + N , where ( N ) n = X λ ≥ C λ X n = n + n + m λ Λ · w n w n λ Y i =1 u m i m i , and ( N ) n = X λ ≥ C λ X n = n + n + n + m λ Λ · u n u n u n λ Y i =1 u m i m i . Here | C λ | . C λ /λ ! , some of the w may be replaced by ¯ w , and that | Λ | . min( | n | , | n | , | n | ) , | Λ | . , where we use the notation v n = ˆ v ( n ) .Proof. The evolution equation satisfied by w can be computed as follows:( ∂ t − i ∂ xx ) w = π > M ( ∂ t − i ∂ xx ) u + π > [ ∂ t , M ] u − i π > [ ∂ xx , M ] u = − π > ( M π < u xx ) + π > (cid:0) [ ∂ t , M ] u − i (cid:2) ∂ x , [ ∂ x , M ] (cid:3) u (cid:1) + π > ( − M S ( Su · Su x ) − ∂ x , M ] u x )= − π > ∂ x ( M π < u x )(2.2) − π > ([ ∂ x , M ] − i2 M ( SQS )) u x (2.3) + 2i π > [ ∂ x , M ] π < u x + π > (cid:0) [ ∂ t , M ] − i (cid:2) ∂ x , [ ∂ x , M ] (cid:3)(cid:1) u. (2.4)Expanding M as the power series and writing in Fourier space, we see that theterm in (2.2) has the form M , where( M ) n = X λ ≥ C λ X n + m ,λ = n ,n > >n ( n n u n ) · Φ · λ Y i =1 u m i m i , YU DENG, NIKOLAY TZVETKOV, AND NICOLA VISCIGLIA where | C λ | ≤ C λ /λ !, and Φ is a bounded factor. Notice that one of m i must be atleast | n | + | n | in size, we can rearrange the indices and rewrite this as( M ) n = X λ ≥ C λ X n + n + m ,λ = n Λ · u n u n · λ Y i =1 u m i m i , where Λ verifies the bound | Λ | ≤ min( h n i , h n i , h n i ) . Next, to analyze the term in (2.3), notice that [ ∂ x , SP S ] = SQS , we have that[ ∂ x , M ] − i2 M ( SQS ) = R, where R = X λ,µ ≥ λ + µ + 1)! (i / λ + µ +1 ( SP S ) λ [ SQS, ( SP S ) µ ] . The last commutator can be written as a power of
SP S , multiplied by[
SP S, SQS ] = SP [ S , Q ] S + SQ [ P, S ] S, multiplied by another power of SP S . We will only consider the first term, since thesecond one is similar. First we may commute ∂ x with a power of SP S and move itleft; since [ ∂ x , P ] = Q , the error term will be of form M , where( M ) n = X λ ≥ C λ X n + n + n + m ,λ = n Λ · u n u n u n · λ Y i =1 u m i m i , where | Λ | . v = ( SP S ) µ − u for some µ , we have([ S , Q ] ∂ x v ) n = i X n = n + m n (cid:0) ψ ( n /N ) − ψ ( n /N ) (cid:1) ψ ( m /N ) u m v n . Plugging in the expression of v n in terms of u , we obtain that([ S , Q ] ∂ x v ) n = i X n = n + m ,µ n (cid:0) ψ ( n /N ) − ψ ( n /N ) (cid:1) ψ ( m /N ) u m u n µ Y i =2 u m i m i , where n = n + m ,µ . Now in this sum, if h m i i & h n i for some i ≥
2, it would beof form M ; otherwise, if h n i & h n i , it would be of form M , and if h n i ≪ h n i ,then we have h n i ≪ h n i also, so by swapping n and m and using symmetry, wefind that this term will be of form M also.Next we consider the second term in (2.4). Recall from Leibniz rule that[ ∂ x , M ] = X λ,µ ≥ λ + µ + 1)! (i / λ + µ +1 ( SP S ) λ ( SQS )( SP S ) µ . If we then commute this with ∂ x again and the commutator hits one SP S factor,we will get the same cubic term as M above. Therefore, let y = ∂ t F − i ∂ x u, we only need to consider the part X λ,µ ≥ λ + µ + 1)! (i / λ + µ +1 π > ( SP S ) λ ( S ( Sy ) S )( SP S ) µ u. NVARIANT MEASURES FOR BO 7
We then have from our equation that y = − π < u x − π =0 S ( Su ) . The second term in the above equation corresponds to a term of form M ; for thefirst term above, we will combine it with the first term of line (2.4) to obtain (herewe omit the summation in λ and µ which does not affect the estimate anyway) N = π > [( SP S ) λ S ( Su ) S ( SP S ) µ ( π < u x ) − ( SP S ) λ S ( Sπ < u x ) S ( SP S ) µ ( u )] . Writing this in Fourier space, we can check that( N ) n = X n = n + n + m σ ,n > >n Λ · u n u n σ Y i =1 u m i m i , where σ = λ + µ , Λ is nonzero only if all variables are . N , and that | Λ | . N − | n | · ( | n | + | m | + · · · + | m σ | ) . Therefore, depending on whether max | m i | & min( | n | , | n | ) or not, we can alsoinclude this term in either M or M .Now we need to transform M and M further into N and N . We will leave M as it is, and further consider an M term X n + n + m ,λ = n Λ · u n u n · λ Y i =1 u m i m i . Recall that u = M − v = ∞ X λ =0 λ ! ( − i / λ ( SP S ) λ v, we have that for positive n , u n = X λ ≥ C λ X n = n + m λ Φ · v n λ Y i =1 u m i m i , where | C λ | . C λ /λ !, and | Φ | .
1. Since u = ¯ u , for negative n we have u n = X λ ≥ C λ X n = n + m λ Φ · (¯ v ) n λ Y i =1 u m i m i . Clearly we may do the same for n . If n n ≤
0, then there must be some i sothat | m i | & | n | (which cancels the Λ factor), therefore this counts as a term of N ,upon substituting v by u again. If n n >
0, then we may replace the v on theright hand side by w , since we know that w (resp. ¯ w ) is supported in the positive(resp. negative) frequencies, so we get N .In any case we have reduced each of (2.2), (2.3) and (2.4) to either N or N ,this completes the proof. (cid:3) YU DENG, NIKOLAY TZVETKOV, AND NICOLA VISCIGLIA
The bootstrap estimate.
In this section we prove the main a priori estimate,namely the following
Proposition 2.3.
Let ǫ and M be fixed. For each N , let u N be the solution to(1.7), with initial data u N (0) = φ , where k φ k H s ′ ≤ M . If N = ∞ we assume u ∞ solves (1.1). Moreover, let w N and w ∞ be the corresponding gauge transforms.Then, when T is small enough depending on ǫ and M , we can find some functions f u N , g w N and f u ∞ , g w ∞ extending u N , u ∞ and w N , w ∞ on [ − T, T ] , such that k e u N k U ′ + k e u ∞ k U ′ + N θ k f u N − f u ∞ k U . ǫ,M and k e w N k X s ′ ,r + k e w ∞ k X s ′ ,r + N θ k g w N − g w ∞ k X s,r . ǫ,M , where θ > is some constant independent of N . Notice that sup t k u ( t ) k H s . k u k U , which follows from proposition 2.2, we can see that Proposition 2.1 is a consequenceof Proposition 2.3. Proof.
We only consider the bound for u N (and denote u N by u ), since the boundfor u ∞ follows from a similar (and much easier) estimate, and the bound for thedifference u N − u ∞ follows from a standard procedure of taking differences .In order to initiate the bootstrap, the first step is to bound the norm k u N k U T and k w N k X s ′ ,r,T for very small T . By the standard arguments in X s,b theory,together with part (5) of Proposition 2.2, we know that this reduces to proving k w (0) k H s ′ . M
1. However, from the expression of the gauge transform we knowthat | w (0) n | . X µ ≥ C µ µ ! X n = n + m ,µ | u (0) n | · µ Y i =1 | u (0) m i | m i . Let the sum over m i be y n − n , then we have X l h l i / | y l | . X m , ··· ,m µ µ Y i =1 h m i i − / | u (0) m i | . k u (0) k µH / σ . M , and s ′ < /
2, so we can easily deduce that k w (0) k H s ′ . X l | y l | · (cid:18) X n h n + l i s ′ | u (0) n | (cid:19) / . M . Suppose we have constructed some e w and e u for some time T , satisfying thedesired inequalities, we now need to improve these inequalities, with the same T ,provided that T ≪ ǫ,M
1. We will first construct a new e w , and this is done simplyusing the equation (2.1). We will define w ∗ = χ ( t ) e i ∂ xx w (0) + E ( N + N ) , where E is the Duhamel operator as in Proposition 2.2, and N and N are con-structed using χ ( t ) e w and χ ( T − t ) e u respectively ; however, we will denote these two Since we are not using any energy estimate which may not be compatible with takingdifferences.
NVARIANT MEASURES FOR BO 9 functions simply by w and u below. Using Proposition 2.2 again, we now only needto bound kN k X s ′ ,r − + kN k X s ′ ,r − . To bound N , we use duality to reduce the bounding the following expression J = X µ C µ X n = n + m µ Z ξ = ξ + η µ Λ · h n i s ′ ( F x,t v )( n , ξ ) ×× Y j =1 ( F x,t u )( n j , ξ j ) · µ Y i =1 ( F x,t u )( m i , η i ) m i , note the abuse of notation by replacing χ ( T − t ) e u with u . Here we assume that v ∈ X , / − σ ⊂ L t,x (even after taking absolute value in Fourier space), and wemay assume without loss of generality that | n | . | n | (the case | n | . | m i | ismuch easier). Now use that h ∂ x i − s ′ u ∈ L t,x , and that u ∈ Y s ′ ⊂ L t,x when σ issmall enough, and that ∂ − x u ∈ L ∞ t,x (all hold after taking absolute value in Fourierspace), we could simply take absolute value of every term in J , then switch to ( t, x )space, then use H¨older to bound J . The gain T θ will come from Proposition 2.2and the time cutoff χ ( T − t ) (the same happens below).Now let us consider the harder part N . We may omit the summation in µ , andwe only need to consider a sum of type J = X n = n + n + m µ h n i s ′ h n i − s ′ h n i − s ′ min ≤ j ≤ h n j i ×× Z ξ = ξ + ξ + η µ +Ξ F ( n , ξ ) G ( n , ξ ) G ( n , ξ ) µ Y i =1 H ( m i , η i ) m i . Here Ξ = | n | n − | n | n − | n | n and F is defined by F ( n, ξ ) = ( F x,t v )( n, ξ + | n | n )with v as above, G and H are defined in the same way, corresponding to functions h ∂ x i s ′ w and u respectively. Moreover, we may assume in the summation thatmin h n j i ≫ max h m i i , since otherwise we can bound this term in the same way as N . In this situation we can check algebraically that | Ξ | ∼ max j h n j i · min h n j i . Let max h n j i = A and min h n j i = B , then the weight h n i s ′ h n i − s ′ h n i − s ′ min ≤ j ≤ h n j i . B / ;moreover, one of ξ j or η i must be & AB by our bound on Ξ.Let | ξ j | & AB for some j , say j = 1 (the other cases being similar). Notice that v ∈ X , / − σ ⊂ L t,x , and also h ∂ x i s ′ w ∈ L t,x (the function that determines G ), andthat we can cancel the weight B / by a power h ξ i / , so we still have h ∈ X , / ⊂ L t,x by interpolation, where( F t,x h )( n, ξ + | n | n ) = h ξ i / G ( n, ξ ) . Now we simply use the above arguments to cancel the weight, then switch to the( x, t ) space and ue H¨older, bounding the F factor in L t,x , one G factor in L t,x andthe other in L t,x , and all H factors in appropriate spaces.If | η i | & AB for some i (say i = 1), then we will use the X − / , / bound for u , which implies h ∈ X / , / ⊂ L t,x by H¨older, where ( F x,t h )( n, ξ + | n | n ) = | n | − h ξ i / H ( n, ξ ) . Moreover the h ξ i / factor cancels the weight, so we simply bound F and both G factors in L t,x (using part (1) of Proposition 2.2 and interpolation), bound the H factor corresponding to m in L t,x , then bound the other factors in appropriatenorms.Finally we should improve the bound on u . We must be careful here, since wewill not use the evolution equation of u ; however, let us postpone this issue to theend, and first see how we can bound the ( U ′ ) T norm of u .Bounding the Y s ′ norm is easy; since u = M − v we can write u as a linearcombination of spacetime shifts ( n, β ) of v with coefficients that are summableeven after multiplying by h n i / (this can be proved in the same way as in theanalysis of w (0) before), and we know that a spacetime shift ( n, β ) increases the Y s norm by a factor . h n i s ′ .Now we need to bound the X − / , / norm of u . Clearly we may restrict to π > u , so by the formula u = M − v and duality we only need to bound J = X n = n + m µ Z β = β + η µ +Ξ h n i − / h β i / × (2.5) × h n i − s ′ h β i − r F ( n , β ) G ( n , β ) µ Y i =1 H ( m i , η i ) m i , provided h m i i ≪ h n i for each i , and H is as above, F and G are bounded in L t,x .If instead h m i i & max( h n i , h n i ) for some i , then we should have J = X n = n + m µ Z β = β + η µ +Ξ h n i − / h β i / × (2.6) × h n i / h β i − / F ( n , β ) G ( n , β ) µ Y i =1 H ( m i , η i ) m i . In both cases we haveΞ = | n | n − | n | n − | m | m − · · · − | m µ | m µ . From the equation we know that either h β i & h β i , or h η i i & h β i for some i , or | β | . | Ξ | .In case (2.5), if h β i . h β i , then we can cancel the two powers, then bound F and G in L t,x , the other factors in L ∞ t,x ; if h β i . h η i i for some i , then we simplyinvoke the X − / , / bound for u and make similar arguments; if | β | . | Ξ | , noticethat | Ξ | . | n | · max i | m i | , NVARIANT MEASURES FOR BO 11 we can use this to cancel the weight, then bound F in L t,x , G in L t,x , the otherfactors in appropriate spaces.In case (2.6), we must have some i , so that | m i | ∼ A is larger than any otherparameter. We may assume in the worst case that | n | ∼ A (since when | n | ∼ A or | m j | ∼ A we will gain more due to the powers we have), and the maximum ofall other parameters is B . Then again we have either h β i . h β i or h β i . h η i i or | β | . | Ξ | . AB . In the first case we cancel the weight, then bound F and G in L t,x , in the second case, we make similar arguments as before, using the X − / , / norm of u ; in the third case we can cancel the weight and gain at least A / , sothe proof still goes through.Finally let us discuss how to obtain an improved estimate without using theevolution equation for u . We argue as in [10], first choose some large K dependingon the bound M ′ appearing in the bootstrap assumption, but still smaller than T − ; then by decomposing π > u into π >K u and π [0 ,K ] u , we can bound the slightlyweaker norm k ∂ − σ/ x u ∗ k U ′ of some other extension u ∗ of u by O M (1) (in fact, thebound for π >K part is trivial since we can gain a power of K , and for the π [0 ,K ] partwe will use the evolution equation for u ). Then we use the formula u = M − v andwrite v = w + π ≤ v . We will use u ∗ to realize the operator M , use some extension w ∗ of w that is bounded by O M (1) as we just proved. Then we should be able tobound the output function by O M (1) as above, except for the part where we have π ≤ v (which is bounded only by O M ′ (1) instead of O M (1)). But in this case wemust have | m i | ≥ K for some i , so we gain a small power of K which cancels the O M ′ (1) loss.In this way we can complete the proof of the proposition. (cid:3) Some useful orthogonality relations
In this section we recall for the sake of completeness some useful results from[25] on the orthogonality of multilinear products of Gaussian variables g k ( ω ) thatappear in (1.3). Introduce the sets : A ( n ) = { ( j , ..., j n ) ∈ Z n | j k = 0 , k = 1 , ..., n, n X k =1 j k = 0 } , ˜ A ( n ) = { ( j , ..., j n ) ∈ A ( n ) | j k = − j l , ∀ k, l } , ˜ A c ( n ) = A ( n ) \ ˜ A ( n ) and˜ A c,j ( n ) = { ( j , ..., j n ) ∈ ˜ A c ( n ) | j = j l = − j m for some 1 ≤ l = m ≤ n } . Proposition 3.1.
Assume that ( j , ..., j n ) , ( i , ..., i n ) ∈ ˜ A ( n ) , { j , ..., j n } 6 = { i , ..., i n } , then R g j ...g j n g i ...g i n dp = 0 . Proposition 3.2.
Let i, j > be fixed and assume ( j , j , j , j , j ) ∈ ˜ A c,j (5) , ( i , i , i , i , i ) ∈ ˜ A c,i (5) , (cid:8) { j , j , j , j , j } \ { j, − j } (cid:9) = (cid:8) { i , i , i , i , i } \ { i, − i } (cid:9) , then R g j g j g j g j g j g i g i g i g i g i dp = 0 . The proof of the propositions above are based on the following lemma.
Lemma 3.1.
Let (3.1) ( j , ..., j n ) , ( i , ..., i n ) ∈ A ( n ) , { j , ..., j n } 6 = { i , ..., i n } be such that: (3.2) Z g j ...g j n g i ...g i n dp = 0 . Then there exist ≤ l, m ≤ n , with l = m and such that at least one of the followingoccurs: either i l = − i m or j l = − j m .Proof. By (3.1) we get the existence of l ∈ { , ...n } such that:(3.3) |{ k = 1 , ..., n | i l = i k }| 6 = |{ k = 1 , ..., n | j k = i l }| where | . | denotes the cardinality. Next we introduce N l = { k = 1 , ..., n | i k = − i l } , M l = { k = 1 , ..., n | i k = i l } , P l = { k = 1 , ..., n | j k = i l } , L l = { k = 1 , ..., n | j k = − i l } . Notice that M l = ∅ since it contains at least the element l , and also by (3.3) |M l | 6 = |P l | . We can assume |M l | > |P l | (the case |M l | < |P l | is similar). Ouraim is to prove that N l = ∅ . Next assume by the absurd that N l = ∅ , then byindependence we get Z g j ...g j n g i ...g i n dp (3.4) = Z | g i l | |P l | ¯ g |M l | + |L l |−|P l | i l dp Z (cid:0) Π k / ∈M l g i k (cid:1)(cid:0) Π h/ ∈L l ∪P l g j h (cid:1) dp = 0where at the last step we used |M l | + |L l | − |P l | >
0. Hence we get an absurd by(3.2). (cid:3) On the approximation of the measures dρ ,R and dρ / ,R We first introduce the modified energies: E ǫN ( u ) = k u k H − k S ǫN u k H + E ( S ǫN u ) , (4.1) G ǫN ( u ) = k u k H / − k S ǫN u k H / + E / ( S ǫN u ) , (4.2)and the approximating modified densities: F ǫN,R = χ R ( k π N u k L ) × χ R ( k π N u k H / − α N + 1 / Z ( S ǫN u ) dx )(4.3) × exp( k S ǫN u k H − E ( S ǫN u )) ,H ǫN,R = χ R ( k π N u k L ) × χ R ( k π N u k H / + 1 / Z ( S ǫN u ) dx )(4.4) × χ R ( E ǫN ( π N u ) − α N ) × exp( k S ǫN u k H / − E / ( S ǫN u )) . We recall the explicit expressions of E and E / : E ( u ) = k u k H + 34 Z u H ∂ x u + 18 Z u and E / ( u ) = k u k H / − ( Z uu x + 12 u ( H u x ) ) − Z ( 13 u H u x + 14 u H ( uu x )) − Z u . Next we prove that as N → ∞ the measures F ǫN,R dµ (for ǫ > dρ ,R and H ǫN,R dµ / converge to dρ / ,R (in a strong sense). Proposition 4.1.
Let
R, σ > and ǫ > be fixed, then: (4.5) lim N →∞ sup A ∈B ( H / − σ ) | Z A F ǫ N,R dµ − Z A dρ ,R | = 0 , (4.6) lim N →∞ sup A ∈B ( H − σ ) | Z A H ǫ N,R dµ − Z A dρ / ,R | = 0 . The next lemma will be of importance in the sequel.
Lemma 4.1.
For every fixed
R > , ǫ > , p ∈ [1 , ∞ ) we have sup N { (cid:13)(cid:13) F ǫ N,R (cid:13)(cid:13) L p ( dµ ) , (cid:13)(cid:13) H ǫ N,R (cid:13)(cid:13) L p ( dµ / ) } < ∞ . The proof follows modulo minor changes in the argument presented in the anal-ysis in [23]. The only difference is that in this paper we use smoothed projectors S ǫ N in the definition of the approximating measures, while in [23] we use the sharpprojectors π N . This difference however does not affect the argument presented in[23]. Lemma 4.2.
Let ǫ > be fixed and σ > be small. For every sequence N k in N ,there exists a subsequence N k h such that: (4.7) F ǫ N kh ,R ( u ) − F N kh ,R ( u ) → , a.e. (w.r.t. dµ ) u ∈ H / − σ , (4.8) H ǫ N kh ,R ( u ) − H N kh ,R ( u ) → , a.e. (w.r.t. dµ / ) u ∈ H − σ , where F N,R = χ R ( k π N u k L ) χ R (cid:16) k π N u k H / − α N + 1 / Z ( π N u ) dx (cid:17) e − R ( π N u ) and H N,R = χ R ( k π N u k L ) χ R ( E / ( π N u )) χ R ( E ( π N u ) − α N ) e − R / ( π N u ) are two of the functions introduced in (1.4) .Proof. First we focus on the proof of (4.7). Notice that if we prove(4.9) k Z ( S ǫ N u ) H ∂ x ( S ǫ N u ) − Z ( π N u ) H ∂ x ( π N u ) k L ( dµ ) → N → ∞ , then up to subsequence we get | Z ( S ǫ N u ) H ∂ x ( S ǫ N u ) − Z ( π N u ) H ∂ x ( π N u ) | → , a.e. (w.r.t. dµ ) u ∈ H / − σ . On the other hand(4.10) Z ( π N u ) − Z ( S ǫ N u ) → , ∀ u ∈ H / − σ , provided that σ > H / − σ ⊂ L . Hencesummarizing we get(4.11) | R ǫ N kh ( u ) − R N kh ( u ) | → h → ∞ a.e. (w.r.t. dµ ) u ∈ H / − σ , where: R N ( u ) = 3 / Z ( π N u ) H ∂ x ( π N u ) + 1 / Z ( π N u ) and R ǫ N = 3 / Z ( S ǫ N u ) H ∂ x ( S ǫ N u ) + 1 / Z ( S ǫ N u ) . Recall also that following [23] one can show that there exists L such that Z ( π N u ) H ∂ x ( π N u ) → L in L ( dµ ), in particular we have up to subsequence convergence a.e. w.r.t. dµ and hence we can assume that up to subsequence R N ( u ) is bounded a.e. w.r.t. dµ .By combining this fact with (4.11) we deduce:(4.12) lim k →∞ exp( − R ǫ N kh ( u )) − exp( − R N kh ( u )) = 0 , a.e. (w.r.t. dµ ) u ∈ H / − σ . On the other hand we have Z ( π N u ) − Z ( S ǫ N u ) → , ∀ u ∈ H / − σ , and hence(4.13) lim N →∞ h χ R ( k π N u k H / − α N + 1 / Z ( S ǫ N u ) ) − χ R ( k π N u k H / − α N + 1 / Z ( π N u ) ) i = 0 a.e. (w.r.t. dµ ) u ∈ H / − σ . We conclude by combining (4.12) and (4.13).Next we focus on (4.9), whose proof follows by (cid:13)(cid:13) X ( j,k,l ) ∈ Z < | i || j | , | k |≤ Nj + k + l =0 | j || k | (1 − ψ ǫ ( j/N ) ψ ǫ ( k/N ) ψ ǫ ( l/N )) g j g k g l (cid:13)(cid:13) L ω → N → ∞ , that in turn, by an orthogonality argument (as in [25]), is equivalent to: X ( j,k,l ) ∈ Z < | j | , | k | , | l |≤ Nj + k + l =0 | j | | k | | − ψ ǫ ( j/N ) ψ ǫ ( k/N ) ψ ǫ ( l/N ) | → N → ∞ . Notice that due to the cut–off ψ ǫ we can restrict the sum on the set { ( j, k, l ) ∈ Z | j + k + l = 0 , < | j | , | k | , | l | ≤ N, max {| j | /N, | k | /N, | l | /N } ≥ (1 − ǫ ) } and hence we can control the sum above by X | j | , | k | > (1 − ǫ ) N/ | j | | k | → N → ∞ . The proof of (4.8) is similar to the proof of (4.7), provided that we show:(4.14) k Z ( S ǫ N u )( S ǫ N u x ) − ( π N u )( π N u x ) k L ( dµ ) → N → ∞ , NVARIANT MEASURES FOR BO 15 (4.15) k Z ( S ǫ N u )( H S ǫ N u x ) − ( π N u )( H π N u x ) k L ( dµ ) → N → ∞ , and | Z ( S ǫ N u ) H ( S ǫ N u ) x − Z ( π N u ) H ( π N u ) x | → , (4.16) | Z ( S ǫ N u ) H ( S ǫ N uS ǫ N u x ) − Z ( π N u ) H ( π N uπ N u x ) | → , (4.17) | Z ( S ǫ N u ) − Z ( π N u ) | → , (4.18) a.e. (w.r.t. dµ ) u ∈ H − σ The proof of (4.18) follows by the Sobolev embedding H − σ ⊂ L . To prove (4.16)(and by a similar argument (4.17)) we use the following inequality (that follows byfractional integration by parts, see page 283 in [23]): | Z v v v H ∂ x v dx | ≤ C ( k v k L ∞ k v k L ∞ k v k H / k v k H / + k v k L ∞ k v k L ∞ k v k H / k v k H / + k v k L ∞ k v k L ∞ k v k H / k v k H / )and hence (4.16) follows provided that k S ǫ N u − π N u k L ∞ → k S ǫ N u − π N u k H / → dµ / ) u ∈ H − σ . The second estimate is trivial and the first onefollows since we can select ρ, p > W ρ,p ⊂ L ∞ and also k v k W ρ,p < ∞ a.e. (w.r.t. dµ ) u ∈ H − σ (see Proposition 4.2 in [23] ). The proofof (4.14) and (4.15) follows the same orthogonality argument as the proof of (4.9).More precisely we get X ( j,k,l ) ∈ Z ,j + k + l =00 < | j | , | k | , | l |≤ N max {| j | /N, | k | /N, | l | /N }≥ (1 − ǫ ) | j || k || l | . X < | j | , | k |≤ N, | l | >N (1 − ǫ ) | j || k || l | + X j + k + l =00 < | j | , | k |≤ N, | j | >N (1 − ǫ ) | j || k || l | = O ( 1 N α )for some α >
0. In the last estimate we used [21] (end of page 500). (cid:3)
Proof of Proposition 4.1.
The proof of (4.5) and (4.6) since now on are the same,hence we focus on the first one. It is sufficient to prove that given any sequence N k in N there exists a subsequence N k h such that (4.5) occurs. Recall thatlim N →∞ sup A ∈B ( H / − σ ) | Z A F N,R dµ − Z A dρ ,R | = 0 . By combining Lemma 4.2 with the Egoroff theorem we get that, up to subsequence,for every ǫ > ǫ ⊂ H / − σ , with σ >
0, such that µ (Ω ǫ ) < − ǫ and F ǫ N,R ( u ) − F N,R ( u ) → L ∞ (Ω ǫ ) . As a consequence we get(4.19) | Z A ∩ Ω ǫ F ǫ N,R ( u ) dµ − Z A ∩ Ω ǫ F N,R ( u ) dµ | < ǫ for N > N ( ǫ ) . On the other hand by the H¨older inequality | Z A ∩ Ω cǫ F ǫ N,R ( u ) dµ − Z A ∩ Ω cǫ F N,R ( u ) dµ | (4.20) . sup N (cid:16) k F ǫ N,R k L ( dµ ) + k F N,R k L ( dµ ) (cid:17) × | µ (Ω cǫ ) | / . ǫ / where we used Lemma 4.1 and [23, Proposition 6.5]. The proof follows by combining(4.19) with (4.20). (cid:3) A-priori Gaussian bounds w.r.t. dµ Recall that for every ǫ > ψ ǫ any function that satisfies (1.5) andby S ǫN the associated multiplier defined by (1.6). The main aim of this section isthe proof of the following result. Proposition 5.1.
Let us denote by S the family of operators S ǫN , for every N ∈ N , ǫ > . Then we have: k Z Sϕ H ( Sϕ x ) SϕSϕ x − Sϕ H ( Sϕ x ) S ( SϕSϕ x ) k L ( dµ ( ϕ )) = O ( √ ǫ );(5.1) k Z ( Sϕ ) ( SϕSϕ x ) − ( Sϕ ) S ( SϕSϕ x ) k L ( dµ ( ϕ )) = O ( ǫ ) + O ( ln N √ N ) . (5.2)The estimate (5.1) is equivalent to:(5.3) k X ( a,b,c,d ) ∈A N (4) Λ ǫN ( a, b, c, d ) sign ( d ) | a || b | g a g b g c g d k L ω = O ( √ ǫ )where g e are the Gaussian independent variables in (1.3),(5.4) Λ ǫN ( a, b, c, d ) = ψ ǫ ( aN ) ψ ǫ ( bN ) ψ ǫ ( cN ) ψ ǫ ( dN )[ ψ ǫ ( a + cN ) − A N (4) = { ( a, b, c, d ) ∈ Z | < | a | , | b | , | c | , | d | ≤ N, a + b + c + d = 0 } . The proof of (5.3) (and hence (5.1)) is splitted in several lemmas.
Lemma 5.1.
We have P ( a,b,c,d ) ∈B ǫN Λ ǫN ( a, b, c, d ) sign ( d ) | a || b | g a g b g c g d = 0 , where B ǫN = { ( a, b, c, d ) ∈ A N (4) | < | a | , | b | , | c | , | d | ≤ N (1 − ǫ ) } for every N ∈ N , ǫ > .Proof. Let us fix ( a, b, c, d ) ∈ B ǫN . First we assume c, d > c, d < a + b + c + d = 0 we deduce that min { a, b } <
0. Sincewe are assuming 0 < | a | , | b | , | c | , | d | ≤ N (1 − ǫ ), we get | a + c | = | b + d | < N (1 − ǫ ).Hence we obtain by the cut–off properties of ψ ǫ that Λ ǫN ( a, b, c, d ) = 0.Hence we have to consider the case c · d <
0. Under the extra assumption a = b wededuce that the vectors ( a, b, c, d ) , ( a, b, d, c ) , ( b, a, c, d ) , ( b, a, d, c ) are distinct andbelong to B ǫN . Moreover g a g b g c g d = g a g b g d g c = g b g a g c g d = g b g a g d g c and by simplealgebra (recall a + b + c + d = 0) we get:1 | a || b | [Λ ǫN ( a, b, c, d ) sign ( d ) + Λ ǫN ( a, b, d, c ) sign ( c ) NVARIANT MEASURES FOR BO 17 +Λ ǫN ( b, a, c, d ) sign ( d ) + Λ ǫN ( b, a, d, c ) sign ( c )] = 0 . The same argument works for a = b (in fact in this case ( a, a, c, d ) , ( a, a, d, c ) ∈ B ǫN are distinct since c and d have opposite sign, g a g a g c g d = g a g a g d g c and we havethe identity a Λ ǫN ( a, a, c, d ) sign ( d ) + a Λ ǫN ( a, a, d, c ) sign ( c ) = 0). The proof isconcluded. (cid:3) Lemma 5.2.
We have sup N k X ( a,b,c,d ) ∈C ǫN Λ ǫN ( a, b, c, d ) sign ( d ) | a || b | g a g b g c g d k L ω = O ( √ ǫ ) , where C ǫN = { ( a, b, c, d ) ∈ A N (4) | max {| a | , | b |} > N (1 − ǫ ) } and N ∈ N , ǫ > .Proof. Assume for simplicity that | a | > N (1 − ǫ ) (the case | b | > N (1 − ǫ ) is similar).Next we split C ǫN = ˜ C ǫN ∪ ˜ C ǫ,cN where˜ C ǫN = { ( a, b, c, d ) ∈ C ǫN | a = − b, a = − c, a = − d, b = − c, b = − d, c = − d } and ˜ C ǫ,cN = C ǫN \ ˜ C ǫN . Then by orthogonality and Proposition 3.1 we can estimate k X ( a,b,c,d ) ∈ ˜ C ǫN Λ ǫN ( a, b, c, d ) ( sign ( d )) | a || b | g a g b g c g d k L ω . ( X N (1 − ǫ ) < | a | 1) = O ( ǫ ) . Concerning the sum on the set ˜ C ǫ,cN notice that:˜ C ǫ,cN = (cid:0) { ( a, − a, a, − a ) || a | 6 = 0 } ∪ { ( a, a, − a, − a ) || a | 6 = 0 } ∪ { ( a, − a, b, − b ) , | a | 6 = | b |}∪{ ( a, b, − a, − b ) , | a | 6 = | b |} ∪ { ( a, b, − b, − a ) , | a | 6 = | b |} (cid:1) \ C ǫN . As a consequence we get ( a, b, c, d ) ∈ ˜ C ǫ,cN implies ( − a, − b, − c, − d ) ∈ ˜ C ǫ,cN and more-over g a g b g c g d = g − a g − b g − c g − d . Since we have the identity sign ( d )Λ ǫN ( a, b, c, d ) + sign ( − d )Λ ǫN ( − a, − b, − c, − d ) = 0 , it is easy to deduce that X ( a,b,c,d ) ∈ ˜ C ǫ,cN Λ ǫN ( a, b, c, d ) sign ( d ) | a || b | g a g b g c g d = 0 . (cid:3) Lemma 5.3. We have P ( a,b,c,d ) ∈D ǫN Λ ǫN ( a, b, c, d ) sign ( d ) | a || b | g a g b g c g d = 0 , where D ǫN = { ( a, b, c, d ) ∈ A N (4) | < | a | , | b | ≤ N (1 − ǫ ) , | c | , | d | > N (1 − ǫ ) } for every N ∈ N , ǫ > .Proof. We notice that if ( a, b, c, d ) ∈ D ǫN then c · d < 0. In fact assume by theabsurd that c, d > c, d < | c + d | > N (1 − ǫ ) and it implies | a + b | > N (1 − ǫ ). This is in contradiction with | a + b | ≤ | a | + | b | ≤ N (1 − ǫ ).The proof can be concluded arguing as in Lemma 5.1 in the case c · d < (cid:3) Lemma 5.4. We have sup N k X ( a,b,c,d ) ∈E ǫN Λ ǫN ( a, b, c, d ) sign ( d ) | a || b | g a g b g c g d k L ω = O ( √ ǫ ) , where E ǫN = { ( a, b, c, d ) ∈ A N (4) | < | a | , | b | , | c | ≤ N (1 − ǫ ) , | d | > N (1 − ǫ ) } [ { ( a, b, c, d ) ∈ A N (4) | < | a | , | b | , | d | ≤ N (1 − ǫ ) , | c | > N (1 − ǫ ) } , for every N ∈ N and ǫ > .Proof. Arguing as in Lemma 5.1 in the case c · d < 0, we can restrict to the case( a, b, c, d ) ∈ E ǫN with c · d > 0. Next we split the sum on two constraints (seeSection 3 for the definition of ˜ A (4) and ˜ A c (4)): E ǫN ∩ ˜ A (4) ∩ { ( a, b, c, d ) | c · d > } and E ǫN ∩ ˜ A c (4) ∩ { ( a, b, c, d ) | c · d > } . By combining an orthogonality argument with Proposition 3.1 we can estimate thesum on the first constraint by (cid:0) X | a + b | >N (1 − ǫ ) a b (cid:1) · (cid:0) X N (1 − ǫ ) < | d |≤ N (cid:1) = O ( ǫ ) , where we used that c · d > | a + b | = | c + d | > N (1 − ǫ ). Concerning thesum on the second constraint we have E ǫN ∩ ˜ A c (4) ∩ { ( a, b, c, d ) | c · d > }⊂ E ǫN \ (cid:0) { ( a, b, − a, − b ) } ∪ { ( a, b, − b, − a ) ∪ { ( a, − a, b, − b ) } (cid:1) = ∅ . (cid:3) Proof of Proposition 5.1. The proof of (5.3) (and hence (5.1)) follows by combiningLemma 5.1, 5.2, 5.3, 5.4.Next we focus on the proof of (5.2), that can be written as follows:(5.6) k X ( a,b,c,d,e ) ∈A N (5) Γ ǫN ( a, b, c, d, e ) sign ( e ) | a || b || c || d | g a g b g c g d g e k L ω = O ( ǫ ) + O ( ln N √ N ) , where:(5.7) Γ ǫN ( a, b, c, d, e ) = ψ ǫ ( aN ) ψ ǫ ( bN ) ψ ǫ ( cN ) ψ ǫ ( dN ) ψ ǫ ( eN )[1 − ψ ǫ ( d + eN )] , and(5.8) A N (5) = { ( a, b, c, d, e ) ∈ Z | < | a | , | b | , | c | , | d | , | e | ≤ N, a + b + c + d + e = 0 } . Due to the cut-off properties of ψ ǫ the sum in (5.6) can be replaced by the sum onthe set A N (5) ∩ { ( a, b, c, d, e ) || d + e | > N (1 − ǫ ) } = A N (5) ∩ { ( a, b, c, d, e ) || a + b + c | > N (1 − ǫ ) } . Next we split(5.9) A N (5) = ˜ A N (5) ∪ ˜ A cN (5) , NVARIANT MEASURES FOR BO 19 where: ˜ A N (5) = { ( a, b, c, d, e ) ∈ A N (5) | a / ∈ {− b, − c, − d, − e } , b / ∈ {− c, − d, − e } , c / ∈ {− d, − e } , d = − e } and ˜ A cN (5) = A N (5) \ ˜ A N (5). By orthogonality and Proposition 3.1 we get: k X ( a,b,c,d,e ) ∈ ˜ A N (5) | a + b + c | >N (1 − ǫ ) Γ ǫN ( a, b, c, d, e ) sign ( e ) | a || b || c || d | g a g b g c g d g e k L ω . X < | a | , | b | , | c | , | d |≤ N, | a + b + c | >N (1 − ǫ ) | a | | b | | c | | d | = O ( 1 N ) . Concerning the sum on the set ˜ A cN (5) ∩ { ( a, b, c, d, e ) || a + b + c | > N (1 − ǫ ) } we firstconsider the splitting:˜ A cN (5) = ˜ A cN,a = − b ∪ ˜ A cN,a = − c ∪ ˜ A cN,a = − d ∪ ˜ A cN,a = − e ∪ ˜ A cN,b = − c ∪ ˜ A cN,b = − d ∪ ˜ A cN,b = − e ∪ ˜ A cN,c = − d ∪ ˜ A cN,c = − e ∪ ˜ A cN,d = − e , where(5.10) ˜ A cN,a = − b = { ( a, b, c, d, e ) ∈ A N (5) | a = − b } and analogous definition for the other sets. First notice that˜ A cN,d = − e ∩ { ( a, b, c, d, e ) || d + e | > N (1 − ǫ ) } = ∅ . Moreover by orthogonality and Proposition 3.2 we can estimate the sum on the set˜ A cN,a = − b ∩ { ( a, b, c, d, e ) || a + b + c | > N (1 − ǫ ) } (notice that since a = − b we get | c | > N (1 − ǫ )) by: X < | a | 0. In the first case the sum on the r.h.s. of (5.11) can be estimated by O ( ln N √ N ) and in the second case it can be controlled by: X NVARIANT MEASURES FOR BO 21 Almost Invariance of F ǫN,R dµ The main result of this section is the following proposition, where F ǫN,R is definedin (4.3). Proposition 6.1. Let σ, R > be fixed. Then for every δ > there exists N = N ( δ ) > and ǫ = ǫ ( δ ) > such that | Z A F ǫN,R dµ − Z Φ ǫN ( t ) A F ǫN,R dµ | ≤ δt for every A ∈ B ( H / − σ ) and for every t .Remark . Notice that the proposition follows provided that we showsup t,A | ddt Z Φ ǫN ( t ) A F ǫN,R dµ | → ǫ → , N → ∞ . Using that the modified energies associated with E and E / are true conserva-tion lows for the approximated flows, we obtain that the proof of Proposition 6.1can be completed by combining Remark 6.1 with Proposition 5.4 in [24], once weproof the following statement. Proposition 6.2. We have the flowing estimate: lim ǫ → (cid:16) lim sup N →∞ (cid:13)(cid:13) ddt E ǫN ( π N Φ ǫN ( t ) ϕ ) t =0 (cid:13)(cid:13) L ( dµ ( ϕ )) (cid:17) = 0 , where the energies E ǫN ( u ) are defined by (4.1) .Proof. In the sequel we make computations with N ∈ N , ǫ > S ǫN = S , E ǫN = ˜ E . Notice that if wedenote by u ( t ) = π N Φ ǫN ( t ) ϕ the solution to (1.7), then Su ( t ) solves(6.1) Su t + H Su xx + ( SuSu x ) + ( S − Id )( SuSu x ) = 0 . Next we recall that(6.2) ˜ E ( u ) = E ( Su ) + ( k u k H − k Su k H ) , where E ( u ) = k u k H + 3 / Z u H ∂ x u + 1 / Z u . Arguing as in [23] and recalling that Su solves (6.1), then we get: ddt E ( Su ) t =0 =(6.3) 2 Z Sϕ x ( Id − S )( SϕSϕ x ) x + 3 / Z ( Sϕ ) H ( Sϕ x )( Id − S )( SϕSϕ x )+3 / Z ( Sϕ ) H (( Id − S )( SϕSϕ x ) x ) + 1 / Z ( Sϕ ) ( Id − S )( SϕSϕ x ) . Moreover we have (use that u ( t ) solves (1.7)): ddt ( k u k H − k Su k H ) t =0 =(6.4) 2 Z ( −H ϕ xx − S ( SϕSϕ x )) x ϕ x − Z ( −H Sϕ xx − S ( SϕSϕ x )) x Sϕ x = 2 Z SϕSϕ x Sϕ xx − Z SϕSϕ x S ϕ xx (here we used integration by parts and R v H v = 0). By combining (6.2), (6.3),(6.4) we get: ddt ˜ E ( π N Φ ǫN ( t ) ϕ ) t =0 = 3 / Z ( Sϕ ) H ( Sϕ x )( Id − S )( SϕSϕ x )+ 3 / Z ( Sϕ ) H (( Id − S )( SϕSϕ x ) x ) + 1 / Z ( Sϕ ) ( Id − S )( SϕSϕ x ) , and by integration by parts ... = 3 / Z Sϕ H ( Sϕ x )( SϕSϕ x ) − / Z Sϕ H ( Sϕ x ) S ( SϕSϕ x ) − / Z SϕSϕ x H ( SϕSϕ x ) + 3 / Z SϕSϕ x H S ( SϕSϕ x )+ 1 / Z ( Sϕ ) ( SϕSϕ x ) − / Z ( Sϕ ) S ( SϕSϕ x ) . By using the property R v H v = 0 we deduce ... = 3 / Z Sϕ H ( Sϕ x )( Id − S )( SϕSϕ x ) + 1 / Z ( Sϕ ) ( Id − S )( SϕSϕ x ) . We conclude by Proposition 5.1. (cid:3) Invariance of dρ ,R The proof of the invariance of dρ ,R follows via standard arguments (see e.g.[24]) by the following proposition. Proposition 7.1. Let σ > small enough and ¯ t > be fixed. Then for everycompact set A ⊂ H / − σ we have: (7.1) Z A dρ ,R ≤ Z Φ(¯ t ) A dρ ,R . Proof. We fix M > A ⊂ B M ( H / − σ ) and we choose L > t )( B M ( H / − σ ) ⊂ B L ( H / − σ )for every t ∈ [0 , ¯ t ] (the existence of L follows by [13]).Next we fix k > N k ∈ N and ǫ k > | Z A F ǫ k N,R dµ − Z Φ ǫkN ( t ) A F ǫ k N,R dµ | ≤ t/k, ∀ N > N k , ∀ t. On the other hand we have by Proposition 2.1 the existence of t = t ( L, k ) > C = C ( L, k ) > u ∈ B L ( H / − σ ) t ∈ [0 ,t ] k Φ ǫ k N ( t ) u − Φ( t ) u k H / − σ ′ ≤ CN − θ and hence(7.4) Z Φ ǫkN ( t ) A dρ ,R ≤ Z Φ( t ) A + B CN − θ ( H / − σ ′ ) dρ ,R , ∀ t ∈ [0 , t ] . NVARIANT MEASURES FOR BO 23 In turn by combining (7.3) with Proposition 4.1 we get the existence of ˜ N k ∈ N such that(7.5) | Z A dρ ,R − Z Φ ǫkN ( t ) A dρ ,R | ≤ t /k ∀ N > ˜ N k , ∀ t. By combining (7.4) with (7.5) we get(7.6) Z A dρ ,R ≤ Z Φ( t ) A + B CN − θ ( H / − σ ′ ) dρ ,R + 3 t /k, ∀ t ∈ [0 , t ]that by taking the limit as N → ∞ gives: Z A dρ ,R ≤ Z Φ( t ) A dρ ,R + 3 t /k, ∀ t ∈ [0 , t ] . It is sufficient to iterate the bound at most [¯ t/t ] + 1 times and to take the limit as k → ∞ in order to get (7.1) (notice that we can iterate thanks to (7.2)). (cid:3) The measures H ǫN,R dµ / and the invariance of dρ / ,R The proof of the invariance of dρ / ,R is similar to the proof of the invariance of dρ ,R , once we establish the following analogue of Proposition 6.1. Proposition 8.1. Let σ, R > be fixed. Then for every δ > there exists N = N ( δ ) > and ǫ = ǫ ( δ ) > such that | Z A H ǫN,R dµ / − Z Φ ǫN ( t ) A H ǫN,R dµ / | ≤ δt for every A ∈ B ( H − σ ) and for every t . Its proof follows by the following analogue of Proposition 6.2. Proposition 8.2. We have the following estimate: lim ǫ → (cid:16) lim sup N →∞ (cid:13)(cid:13) ddt G ǫN ( π N Φ ǫN ( t ) ϕ ) t =0 (cid:13)(cid:13) L ( dµ / ( ϕ )) (cid:17) = 0 , where G ǫN ( u ) are defined in (4.2) . In turn we split the proof of this proposition in several steps. Proposition 8.3. We have the following estimates: k Z Sϕ ( H Sϕ x ) H ( Id − S )( SϕSϕ x ) x k L ( dµ / ( ϕ )) = O ( r ln NN ) + O ( ǫ ) , (8.1) k Z ( Sϕ x ) ( Id − S )( SϕSϕ x ) k L ( dµ / ( ϕ )) = O ( s ln NN ) + O ( √ ǫ ) , (8.2) k Z ( H Sϕ x ) ( Id − S )( SϕSϕ x ) k L ( dµ / ( ϕ )) = O ( s ln NN ) + O ( √ ǫ ) , (8.3) where S = S ǫN . We split the proof of (8.1) in several lemmas. Notice that we have k Z Sϕ ( H Sϕ x ) H ( Id − S )( SϕSϕ x ) x k L ( dµ / ) = k X ( a,b,c,d ) ∈A N (4) ∆ ǫN ( a, b, c, d ) | c + d | sign ( d ) | a | / | b | / | c | / | d | / g a g b g c g d k L ω , where g e are the Gaussian independent variables in (1.3),(8.4) ∆ ǫN ( a, b, c, d ) = ψ ǫ ( aN ) ψ ǫ ( bN ) ψ ǫ ( cN ) ψ ǫ ( dN )[1 − ψ ǫ ( c + dN )]and(8.5) A N (4) = { ( a, b, c, d ) ∈ Z | < | a | , | b | , | c | , | d | ≤ N, a + b + c + d = 0 } . Lemma 8.1. We have P ( a,b,c,d ) ∈B ǫN | c + d | sign ( d ) | a | / | b | / | c | / | d | / ∆ ǫN ( a, b, c, d ) g a g b g c g d = 0 where: B ǫN = { ( a, b, c, d ) ∈ A N (4) | < | a | , | b | , | c | , | d | ≤ N (1 − ǫ ) } for any N ∈ N , ǫ > .Proof. Notice that in the case sign ( b ) · sign ( d ) < | c + d | sign ( d ) | a | / | b | / | c | / | d | / ∆ ǫN ( a, b, c, d ) + | a + b | sign ( b ) | a | / | b | / | c | / | d | / ∆ ǫN ( c, d, a, b ) = 0 . Hence we can assume that sign ( b ) = sign ( d ). By the condition a + b + c + d = 0 weget that at least one of the following occurs: either sign ( b ) = sign ( a ) or sign ( c ) = sign ( d ). In any case we have | a + b | = | c + d | < N (1 − ǫ ) and hence we concludeby the cut–off properties of ψ ǫ that ∆ ǫN ( a, b, c, d ) = 0. (cid:3) Lemma 8.2. We have k X ( a,b,c,d ) ∈C ǫN | c + d | sign ( d ) | a | / | b | / | c | / | d | / ∆ ǫN ( a, b, c, d ) g a g b g c g d k L ω = O ( r ln NN ) + O ( ǫ ) , where: C ǫN = { ( a, b, c, d ) ∈ A N (4) | max {| a | , | c |} > N (1 − ǫ ) } for every N ∈ N , ǫ > .Proof. We will only treat the case | a | > N (1 − ǫ ) (for | c | > N (1 − ǫ ) the sameargument works). Notice that by the cut–off property of ψ ǫ we can work on the set C ǫN ∩ {| a | > N (1 − ǫ ) } ∩ {| a + b | = | c + d | > N (1 − ǫ ) } .We argue as in Lemma 5.2 and we split C ǫN = ˜ C ǫN ∪ ˜ C ǫ,cN . When we sum on the set˜ C ǫN then we can combine an orthogonality argument with Proposition 3.1 and withthe identity d = − a − b − c , and we are reduced to: X | a | >N (1 − ǫ )0 < | b | , | c |≤ N | a | | b || c | + X | a | >N (1 − ǫ )0 < | b | , | c |≤ N | a + b + c || a | | b || c | ≤ X | a | >N (1 − ǫ )0 < | b | , | c |≤ N | a | | b || c | + X | a | >N (1 − ǫ )0 < | b | , | c |≤ N | a | | b || c | + X | a | >N (1 − ǫ )0 < | b | , | c |≤ N | a | | c | + X | a | >N (1 − ǫ )0 < | b | , | c |≤ N | a | | b || c | = O ( ln NN ) . Concerning the sum on C ǫ,cN we work on the set: { ( a, a, − a, − a ) || a | 6 = 0 } ∪ { ( a, b, − a, − b ) , | a | 6 = | b |} ∪ { ( a, b, − b, − a ) , | a | 6 = | b |} (cid:1) . NVARIANT MEASURES FOR BO 25 Notice that | a + b | sign ( − b ) | a | / | b | / | a | / | b | / ∆ ǫN ( a, b, − a, − b ) + | a + b | sign ( b ) | a | / | b | / | c | / | d | / ∆ ǫN ( − a, − b, a, b ) = 0and hence we have to consider the sum on the set: { ( a, a, − a, − a ) || a | 6 = 0 } ∪ { ( a, b, − b, − a ) , | a | 6 = | b |} . We can conclude by combining the Minkowski inequality with the following esti-mates: X | a | >N (1 − ǫ ) | a | + X < | a | , | b |≤ N | a | >N (1 − ǫ ) | a + b | >N (1 − ǫ ) | a + b || a | | b | ≤ O ( 1 N ) + X < | b |≤ NN (1 − ǫ ) < | a |≤ N | a || b | + X < | b |≤ NN (1 − ǫ ) < | a |≤ N | a | | b | = O ( ln NN ) + O ( ǫ ) . (cid:3) Lemma 8.3. We have X ( a,b,c,d ) ∈D ǫN | c + d | sign ( d ) | a | / | b | / | c | / | d | / ∆ ǫN ( a, b, c, d ) g a g b g c g d = 0 , where: D ǫN = { ( a, b, c, d ) ∈ A N (4) | < | a | , | c | ≤ N (1 − ǫ ) , max {| b | , | d |} > N (1 − ǫ ) } for every N ∈ N , ǫ > .Proof. Arguing as in Lemma 8.1 we can assume that sign ( b ) = sign ( d ), and alsoby the cut-off property of ψ ǫ we can restrict to the set | a + b | = | c + d | > N (1 − ǫ ).We claim that D ǫN ∩ { sign ( b ) = sign ( d ) } ∩ {| a + b | > N (1 − ǫ ) } = ∅ , and it willconclude the proof. We can assume b, d > b, d < < min { b, d } ≤ N (1 − ǫ ). In fact in case it isnot true then we get b + d > N (1 − ǫ ) which implies | a + c | = b + d > N (1 − ǫ ),and it is in contradiction with 0 < | a | , | c | ≤ N (1 − ǫ ). We assume for simplicity0 < b ≤ N (1 − ǫ ). Moreover by the condition | a + b | > N (1 − ǫ ) we get (since | a | < N (1 − ǫ ) and 0 < b ≤ N (1 − ǫ )) a > 0. By combining a + b + c + d = 0and a, b, d > 0, we get c < 0. Hence we have | c | = a + b + d and it is absurd since0 < | c | ≤ N (1 − ǫ ) and a + b + d > d ≥ N (1 − ǫ ). (cid:3) Proof of Proposition 8.3. The proof of (8.1) follows by combining Lemma 8.1, 8.2and 8.3. Concerning the proof of (8.2) first notice that k Z ( Sϕ x ) ( Id − S )( SϕSϕ x ) k L ( dµ / ) (8.6) = k X ( a,b,c,d ) ∈A N (4) ∆ ǫN ( a, b, c, d ) sign ( a ) sign ( b ) sign ( d ) | a | / | b | / | c | / | d | / g a g b g c g d k L ω , where ∆ ǫN ( a, b, c, d ) and A N (4) are defined in (8.4) and (8.5). Next (followingSection 3) we split A N (4) = ˜ A N (4) ∪ ˜ A cN (4), where:˜ A N (4) = { ( a, b, c, d ) ∈ A N (4) | a = − b, a = − c, a = − d, b = − c, b = − d, c = − d } and˜ A cN (4) = { ( a, a, − a, − a ) || a | 6 = 0 }∪ { ( a, b, − a, − b ) , | a | 6 = | b |}∪ { ( a, b, − b, − a ) , | a | 6 = | b |} (cid:1) , (notice that we have excluded the vectors ( a, − a, b, − b ) that in principle belongto A N (4), but due to the cut-off property of ψ ǫ they give a trivial contributionin the sum in (8.6)). When we consider the sum on ˜ A N (4), and we recall that∆ ǫN ( a, b, c, d ) = 0 when | c + d | ≤ N (1 − ǫ ), then by orthogonality and Proposition 3.1we are reduced to: X < | a | , | b | , | c | , | d |≤ N | a + b | = | c + d | >N (1 − ǫ ) | a || b || c | | d | ≤ O (ln N ) X < | c | , | d |≤ N (1 − ǫ ) | c + d | >N (1 − ǫ ) | c | | d | + O (ln N ) X < | c | , | d |≤ N | c | >N (1 − ǫ ) | c + d | >N (1 − ǫ ) | c | | d | + X < | c | , | d |≤ N | d | >N (1 − ǫ ) | a + b | = | c + d | >N (1 − ǫ ) | a || b || c | | d | . The first and second terms on the r.h.s. are O ( ln NN ) (in particular the first term isestimated by Lemma 3.3 in [24]). Concerning the third sum we can estimate it by X n>N (1 − ǫ ) (cid:0) X < | a | , | b | We shall also need the following estimates. Proposition 8.4. We have the following estimates: k Z ( Sϕ ) ( H Sϕ x )[( Id − S )( SϕSϕ x )] k L ( dµ / ) = O ( r ln NN ) + O ( √ ǫ ) , k Z ( Sϕ ) H [( Id − S )( SϕSϕ x ) x ] k L ( dµ / ) = O ( r ln NN ) + O ( √ ǫ ) , k Z ( Sϕ ) H ( SϕSϕ x )[( Id − S )( SϕSϕ x )] k L ( dµ / ) = O ( r ln NN ) + O ( √ ǫ ) , k Z ( Sϕ ) H [ Sϕ x ( Id − S )( SϕSϕ x )] k L ( dµ / ) = O ( r ln NN ) + O ( √ ǫ ) k Z ( Sϕ ) H [ Sϕ ( Id − S )( SϕSϕ x ) x ] k L ( dµ / ) = O ( r ln NN ) + O ( √ ǫ ) . Proof. We focus on the first estimate (the others can be treated by the same argu-ments as below). We have to prove k X ( a,b,c,d,e ) ∈A N (5) sign ( e )Γ ǫN ( a, b, c, d, e ) | a | / | b | / | c | / | d | / | e | / g a g b g c g d g e k L ω (8.8) = O ( r ln NN ) + O ( √ ǫ ) , where Γ ǫN ( a, b, c, d, e ) and A N (5) are defined in (5.7) and (5.8). Next we split A N (5) = ˜ A N (5) ∪ ˜ A cN (5) (see (5.9)) and we get by Proposition 3.1 k X ( a,b,c,d,e ) ∈ ˜ A N (5) sign ( e )Γ ǫN ( a, b, c, d, e ) | a | / | b | / | c | / | d | / | e | / g a g b g c g d g e k L ω (8.9) ≤ X < | a | , | b | , | c | , | d | , | e |≤ N | d + e | >N (1 − ǫ ) | a | | b | | c || d | | e | . (cid:0) X | a | , | b | , | c |≤ N | a + b + c | >N (1 − ǫ ) | a | | b | | c | (cid:1) · (cid:0) X | d | , | e |≤ N | d + e | >N (1 − ǫ ) | d | | e | (cid:1) (here the constraint | d + e | > N (1 − ǫ ) comes from the cut–off ψ ǫ ). Next noticethat X | d | , | e |≤ N | d + e | >N (1 − ǫ ) | d | | e | (8.10) ≤ X < | d | , | e |≤ N (1 − ǫ ) | d + e | >N (1 − ǫ ) | d | | e | + X | d |≥ N (1 − ǫ ) | d + e | >N (1 − ǫ ) | d | | e | + X N (1 − ǫ ) ≤| e |≤ N | d + e | >N (1 − ǫ ) | d | | e | = O ( ln NN ) + O ( ǫ )where we have used Lemma 3.3 of [24] to estimate the first term on the r.h.s. Bya similar argument X | a | , | b | , | c |≤ N | a + b + c | >N (1 − ǫ ) | a | | b | | c | = O ( ln NN ) + O ( ǫ ) . Hence the l.h.s. in (8.9) can be estimated by O ( ln NN ) + O ( ǫ ). Next we considerthe set(8.11) ˜ A cN,a = − b = { ( a, b, c, d, e ) ∈ A N (5) | a = − b } . By Proposition 3.2 we can estimate the sum on the set ˜ A cN,a = − b ∩ { ( a, b, c, d, e ) || a + b + c | > N (1 − ǫ ) } by X < | a |≤ N | a | (cid:0) X | d + e | >N (1 − ǫ ) , < | d | , | e |≤ N,N (1 − ǫ ) < | c |≤ N | c || d | | e | (cid:1) . √ ǫ (cid:0) X | d + e | >N (1 − ǫ ) , < | d | , | e |≤ N | d | | e | (cid:1) (the condition | c | > N (1 − ǫ ) comes from a + b + c = c on the set a = − b ), and by(8.10) we can continue the estimate ... ≤ O ( √ ǫ )[ O ( r ln NN ) + O ( √ ǫ )] . We can treat the sum on ˜ A cN,a = − c by the following estimates: X < | a |≤ N | a | (cid:0) X | d + e | >N (1 − ǫ ) , < | d | , | e |≤ N | b | | d | | e | (cid:1) . (cid:0) X | d + e | >N (1 − ǫ ) , < | d | , | e |≤ N | d | | e | (cid:1) = O ( r ln NN ) + O ( √ ǫ ) , NVARIANT MEASURES FOR BO 29 where we used again (8.10). Concerning the sum on ˜ A cN,a = − d , we are reduced tothe estimate: X < | a |≤ N | a | (cid:0) X | a + b + c | >N (1 − ǫ )0 < | d |≤ N | b | | c || e | (cid:1) (8.12) ≤ X < | a |≤ N | a | [ (cid:0) X | a + b + c | >N (1 − ǫ ) | e + d | >N (1 − ǫ )0 < | e | , | d |≤ N (1 − ǫ ) | b | | c || d | | e | (cid:1) + (cid:0) X | a + b + c | >N (1 − ǫ ) | e + d | >N (1 − ǫ ) | d |≥ N (1 − ǫ ) | b | | c || d | | e | (cid:1) / ]+ X < | a |≤ N | a | (cid:0) X | a + b + c | >N (1 − ǫ ) | e + d | >N (1 − ǫ ) N (1 − ǫ ) ≤| e |≤ N | b | | c || d | | e | (cid:1) / . Notice that the first two terms on the r.h.s. give a contribution O ( ln N √ N ) (in par-ticular to estimate the first term we used Lemma 3.3 in [24]). Concerning the lastterm, due to the fact a = − d and | − d + b + c | > N (1 − ǫ ), it can be controlled by X < | a |≤ N | a | (cid:0) X | a + b + c | >N (1 − ǫ ) | e + d | >N (1 − ǫ ) N (1 − ǫ ) ≤ | e | ≤ N | b | | c || d | | e | (cid:1) / ≤ X < | a |≤ N | a | √ ǫ (cid:0) X |− d + b + c | >N (1 − ǫ ) | b | | c || d | (cid:1) / = O ( √ ǫ )[ O ( r ln NN ) + O ( √ ǫ )] . At the last step we have used the estimate X |− d + b + c | >N (1 − ǫ ) | b | | c || d | = O ( ǫ ) + O ( ln NN ) , that in turn follows by splitting the constraint in four subdomains given by: {|− d + b + c | > N (1 − ǫ ) , | d | , | b | , | c | ≤ N (1 − ǫ ) } , {|− d + b + c | > N (1 − ǫ ) , | d | > N (1 − ǫ ) } , {| − d + b + c | > N (1 − ǫ ) , | b | > N (1 − ǫ ) } , {| − d + b + c | > N (1 − ǫ ) , | c | > N (1 − ǫ ) } . The sum on the first constraint can be estimated by Lemma 3.2 in [25], the estimateon the second and third one are trivial, and the last one gives a contribution O ( ǫ ).Concerning the sum on ˜ A cN,a = − e , notice that the constraint | d + e | > N (1 − ǫ ) isequivalent to | d − a | > N (1 − ǫ ) and hence we are reduced to treat: X < | a |≤ N | a | (cid:0) X | d − a | >N (1 − ǫ )0 < | b | , | c | , | d |≤ N | b | | c || d | (cid:1) (8.13) ≤ √ ln N (cid:0) X | a |≥ N (1 − ǫ ) / | a | + X | a |≤ N (1 − ǫ ) / | a | ( X | d |≥ N (1 − ǫ ) / | d | ) (cid:1) = O ( √ ln NN ) . On the set ˜ A cN,b = − c the constraint | a + b + c | > N (1 − ǫ ) becomes | a | > N (1 − ǫ )and hence the estimate follows by X < | b |≤ N | b | (cid:0) X | a | >N (1 − ǫ )0 < | d | , | e |≤ N | a | | d | | e | (cid:1) = O ( √ ln NN ) . Notice that the sum on the sets ˜ A cN,b = − d and ˜ A cN,b = − e are similar to the sum on˜ A cN,a = − d and ˜ A cN,a = − e , that have been already treated. Next we focus on the sumon the set ˜ A cN,c = − d . In this case the constraint | a + b + c | > N (1 − ǫ ) becomes | a + b − d | > N (1 − ǫ ) and we are reduced to estimate X < | d |≤ N | d | (cid:0) X | a + b − d | >N (1 − ǫ )0 < | d | , | e |≤ N | a | | b | | e | (cid:1) ≤ √ ln N (cid:0) X | d |≥ N (1 − ǫ ) / | d | + X | a + b |≥ N (1 − ǫ ) / | a | | b | (cid:1) = O ( √ ln NN ) . The last sum to be considered is on the set ˜ A cN,c = − e , where the constraint | d + e | >N (1 − ǫ ) can be rewritten as | d − c | > N (1 − ǫ ), hence we conclude by the followingestimates: X < | c |≤ N | c | (cid:0) X < | a | , | b | , | c | , | d |≤ N | d − c | >N (1 − ǫ ) | a | | b | | d | (cid:1) ≤ X < | c |≤ N (1 − ǫ ) (cid:0) X < | d |≤ N | d − c | >N (1 − ǫ ) | c | | d | (cid:1) + X N (1 − ǫ ) ≤| c |≤ N | c | ≤ X < | c |≤ N (1 − ǫ ) (cid:0) X < | d |≤ N (1 − ǫ ) | d − c | >N (1 − ǫ ) | c | | d | (cid:1) + X < | c |≤ N (1 − ǫ ) (cid:0) X | d |≥ N (1 − ǫ ) | d − c | >N (1 − ǫ ) | c | | d | (cid:1) + O ( ǫ ) = O ( 1 √ N ) + O ( ǫ ) , where we used Lemma 3.3 in [25] to estimate the first sum on the r.h.s. (cid:3) Proposition 8.5. We have the following estimates: k Z ( Sϕ ) ( Id − S )( SϕSϕ x ) k L ( dµ / ) = O ( 1 √ N ) . Proof. We have to show k X ( a,b,c,d,e,f ) ∈A N (6) sign ( f )Λ ǫN ( a, b, c, d, e, f ) g a g b g c g d g e g f | a | / | b | / | c | / | d | / | e | / | f | / k L ω = O ( 1 √ N ) , (8.14)where:Λ ǫN ( a, b, c, d, e, f ) = ψ ( aN ) ǫ ψ ǫ ( bN ) ψ ǫ ( cN ) ψ ǫ ( dN ) ψ ǫ ( eN ) ψ ǫ ( fN )[1 − ψ ǫ ( | e + f | N )]and A N (6) = { ( a, b, c, d, e, f ) ∈ Z | < | a | , | b | , | c | , | d | , | e | , | f | ≤ N, a + b + c + d + e + f = 0 } . NVARIANT MEASURES FOR BO 31 By the Minkowski inequality and by the cut–off property of ψ ǫ we can estimate thel.h.s. in (8.14) by X ( a,b,c,d,e,f ) ∈A N (6) | a + b + c + d | >N (1 − ǫ ) | a | / | b | / | c | / | d | / | e | / | f | / (8.15) ≤ X < | a | , | b | , | c | , | d | , | e |≤ N | a + b + c + d | >N (1 − ǫ ) | a | / | b | / | c | / | d | / | e | / = O ( 1 √ N )where we used | a + b + c + d | > N (1 − ǫ ) implies max {| a | , | b | , | c | , | d |} ≥ N/ − ǫ ). (cid:3) Proof of Proposition 8.2. For simplicity we denote ˜ G = G ǫN and S = S ǫN , then weget for a general function v ˜ G ( v ) = E / ( Sv ) + ( k v k H / − k Sv k H / ) , (8.16)where E / ( u ) = k u k H / − ( Z uu x + 12 u ( Hu x ) )(8.17) − Z ( 13 u Hu x + 14 u H ( uu x )) − Z u . Arguing as in Proposition 6.2 and by using the notation u ( t ) = π N Φ ǫN ( t ) ϕ we get: ddt E / ( Su ) t =0 = [ Z Su t H Sϕ xxx + Z Sϕ H Su txxx (8.18) − ( Z Su t ( Sϕ x ) + 12 Su t ( H Sϕ x ) ) − ( Z SϕSϕ x Su tx + Sϕ ( H Sϕ x ) H Su tx ) − Z ( Sϕ ) Su t H Sϕ x − Z 13 ( Sϕ ) H Su tx − Z ( Sϕ ) Su t H ( SϕSϕ x ) − Z ( Sϕ ) H ( Su t Sϕ x ) − Z ( Sϕ ) H ( SϕSu tx ) − Z ( Sϕ ) Su t ] Su t =( Id − S ) SϕSϕ x . Moreover by using the equation solved by u ( t, x ) we get: ddt ( k u k H / − k Su k H / ) t =0 = Z ( −H ϕ xx − S ( SϕSϕ x )) H ϕ xxx + Z ϕ H ( −H ϕ xx − S ( SϕSϕ x )) xxx + Z ( H Sϕ xx + S ( SϕSϕ x )) H Sϕ xxx + Z Sϕ H ( H Sϕ xx + S ( SϕSϕ x )) xxx . By properties of H and integration by parts we get: ddt ( k u k H / − k Su k H / ) t =0 (8.19) = Z ( − S ( SϕSϕ x )) H ϕ xxx + Z ϕ H ( − S ( SϕSϕ x )) xxx + Z ( S ( SϕSϕ x )) H Sϕ xxx + Z Sϕ H ( S ( SϕSϕ x )) xxx = 2 Z ϕ H ( − S ( SϕSϕ x )) xxx + 2 Z ( S ( SϕSϕ x )) H Sϕ xxx . By combining (8.18), (8.19) with (8.16) we deduce that the terms of order threein the expression ddt ( ˜ G ( π N Φ ǫN ( t ) ϕ ) t =0 give a trivial contribution, as the followingcomputation shows:cubic terms in (8.18)+ (8.19)= 2 Z ϕ H ( − S ( SϕSϕ x )) xxx + 2 Z ( S ( SϕSϕ x )) H Sϕ xxx + Z ( Id − S )( SϕSϕ x ) H Sϕ xxx + Z Sϕ H ( Id − S )( SϕSϕ x ) xxx = − Z ϕ H ( S ( SϕSϕ x )) xxx + 2 Z ( S ( SϕSϕ x )) H Sϕ xxx + Z ( SϕSϕ x ) H Sϕ xxx + Z Sϕ H ( SϕSϕ x ) xxx − Z S ( SϕSϕ x ) H Sϕ xxx − Z Sϕ H S ( SϕSϕ x ) xxx = 0 . Next by combining again (8.18), (8.19) with (8.16) we compute the contribution to ddt ( ˜ G ( π N Φ ǫN ( t ) ϕ ) t =0 given by terms of order four:quartic terms in (8.18)+ (8.19)= − Z 32 ( Id − S )( SϕSϕ x )( Sϕ x ) − SϕSϕ x ( Id − S )( SϕSϕ x ) x − Z 12 ( Id − S )( SϕSϕ x )( H Sϕ x ) − Sϕ ( H Sϕ x ) H ( Id − S )( SϕSϕ x ) x = − Z 32 ( Id − S )( SϕSϕ x )( Sϕ x ) − Z 12 ( Id − S )( SϕSϕ x )( H Sϕ x ) − Sϕ ( H Sϕ x ) H ( Id − S )( SϕSϕ x ) x , where we used2 Z SϕSϕ x ( Id − S )( SϕSϕ x ) x = Z ∂ x (( Id − S ) / ( SϕSϕ x )) = 0 . NVARIANT MEASURES FOR BO 33 By Proposition 8.3 we can estimate the terms above. Next we focus on the quinticterms: quintic terms in (8.18)+ (8.19)= − Z ( Sϕ ) ( H Sϕ x )[( Id − S )( SϕSϕ x )] − Z ( Sϕ ) H [( Id − S )( SϕSϕ x ) x ] − Z ( Sϕ ) H ( SϕSϕ x )[( Id − S )( SϕSϕ x )] − Z ( Sϕ ) H [ Sϕ x ( Id − S )( SϕSϕ x )] − Z ( Sϕ ) H [ Sϕ ( Id − S )( SϕSϕ x ) x ]Notice that those terms can be controlled by Proposition 8.4. Next we notice thatsixth order terms in (8.18)+ (8.19)= − Z ( Sϕ ) ( Id − S )( SϕSϕ x )and they can be controlled thanks to Proposition 8.5. (cid:3) References [1] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applica-tions to nonlinear evolution equations. II. The KdV-equation , Geom. Funct. Anal., 3, (1993),209-262.[2] J. Bourgain , Periodic nonlinear Schr¨odinger equation and invariant measures , Comm. Math.Phys., 166, (1994) 1-26.[3] J. Bourgain, Invariant measures for the 2d-defocusing nonlinear Schr¨odinger equation ,Comm. Math. Phys., 176 (1996) 421-445.[4] J. Bourgain and A. Bulut. Invariant Gibbs measure evolution for the radial nonlinear waveequation on the 3D ball , to appear in Journal of Functional Analysis.[5] J. Bourgain and A. Bulut. Almost sure global well posedness for the radial nonlinearSchrodinger equation on the unit ball I: the 2D case , to appear in Annales IHP.[6] J. Bourgain and A. Bulut. Almost sure global well posedness for the radial nonlinearSchrodinger equation on the unit ball II: the 3D case , to appear in J. Eur. Math. Soc.[7] N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations II.A global existence result , Invent. Math., 173 (2008) 477-496.[8] N. Burq, L. Thomann and N. Tzvetkov, Long time dynamics for the one dimensional nonlinear Schr¨odinger equation , to appear in Ann. Institut Fourier.[9] Y. Deng, Two dimensional NLS equation with random radial data , to appear in Anal. PDE.[10] Y. Deng, Invariance of the Gibbs measure for the Benjamin-Ono equation , to appear in J.Eur. Math. Soc.[11] L. Lebowitz, R. Rose, E. Speer, Statistical dynamics of the Nonlinear Schr¨odinger equation ,J. Stat. Phys. 50 (1988) 657-687.[12] Y. Matsuno, Bilinear transformation method , Academic Press, 1984.[13] L. Molinet, Global well-posendess in L for the periodic Benjamin-Ono equation , Amer. J.Math. 130 (2008) 635-685.[14] L. Molinet and D. Pilod, The Cauchy problem of the Benjamin-Ono equation in L revisited ,Anal. PDE, 5 (2012), 365-395.[15] A. Nahmod, T.Oh, L. Rey-Bellet, G. Staffilani, Invariant weighted Wiener measures andalmost sure global well-posedness for the periodic derivative NLS , J. Eur. Math. Soc. 14(2012), 1275-1330.[16] T. Oh, Invariance of the Gibbs measure for the Schr¨odinger-Benjamin-Ono system , SIAMJ. Math. Anal., 41 (2009/10) 2207-2225.[17] G. Richards, Invariance of the Gibbs measure for the periodic quartic gKdV , Preprint 2013.[18] A-S. de Suzzoni, Invariant mesure for the cubic non linear wave equation on the unit ball of R , Dynamics of PDE, 8, (2011) 127-147. [19] T. Tao, Global well-posedness of the Benjamin-Ono equation in H , J. Hyperbolic Diff.Equations, 1 (2004) 27-49.[20] T. Tao, Nonlinear dispersive equations: local and global analysis , CBMS Regional ConferenceSeries in Mathematics , vol. 106, 2006.[21] N. Tzvetkov, Construction of a Gibbs measure associated to the periodic Benjamin-Onoequation , Probab. Theory Relat. Fields 146 (2010) 481-514.[22] N. Tzvetkov, Invariant measures for the defocusing NLS , Ann. Inst. Fourier 58 (2008) 2543-2604.[23] N. Tzvetkov, N. Visciglia Gaussian measures associated to the higher order conservation lawsof the Benjamin-Ono equation , Ann. Scient. Ec. Norm. Sup. 46 (2013) 249-299.[24] N. Tzvetkov, N. Visciglia Invariant measures and long time behaviour for the Benjamin-Onoequation , International Mathematics Research Notices 2013; doi: 10.1093/imrn/rnt094.[25] N. Tzvetkov, N. Visciglia Invariant measures and long time behaviour for the Benjamin-Onoequation II , J. Math. Pures Appl. 2014; doi:10.1016/j.matpur.2014.03.009[26] P. Zhidkov, KdV and Nonlinear Schr¨odinger equations : qualitative theory , Lecture notes inMathematics 1756, Springer, 2001. Department of Mathematics, Princeton University, Princeton, NJ 08544 E-mail address : [email protected] D´epartement de Math´ematiques, Universit´e de Cergy-Pontoise, 2, avenue AdolpheChauvin, 95302 Cergy-Pontoise Cedex, France and Institut Universitaire de France E-mail address : [email protected] Universit`a Degli Studi di Pisa, Dipartimento di Matematica, Largo B. Pontecorvo 5I - 56127 Pisa. Italy E-mail address ::