A pedestrian approach to the invariant Gibbs measures for the 2-d defocusing nonlinear Schrödinger equations
aa r X i v : . [ m a t h . A P ] J u l A PEDESTRIAN APPROACH TO THE INVARIANT GIBBSMEASURES FOR THE 2- d DEFOCUSING NONLINEARSCHR ¨ODINGER EQUATIONS
TADAHIRO OH AND LAURENT THOMANN
Abstract.
We consider the defocusing nonlinear Schr¨odinger equations on the two-dimensional compact Riemannian manifold without boundary or a bounded domain in R . Our aim is to give a pedagogic and self-contained presentation on the Wick renormal-ization in terms of the Hermite polynomials and the Laguerre polynomials and constructthe Gibbs measures corresponding to the Wick ordered Hamiltonian. Then, we constructglobal-in-time solutions with initial data distributed according to the Gibbs measure andshow that the law of the random solutions, at any time, is again given by the Gibbsmeasure. Contents
1. Introduction 21.1. Nonlinear Schr¨odinger equations 21.2. Gibbs measures 21.3. Wick renormalization 51.4. Invariant dynamics for the Wick ordered NLS 82. Construction of the Gibbs measures 122.1. Hermite polynomials, Laguerre polynomials, and Wick ordering 132.2. White noise functional 162.3. Wiener chaos estimates 192.4. Nelson’s estimate 203. On the Wick ordered nonlinearity 214. Extension to 2- d manifolds and domains in R ν N M = T and m = 3 37References 40 Mathematics Subject Classification.
Key words and phrases. nonlinear Schr¨odinger equation; Gibbs measure; Wick ordering; Hermite poly-nomial; Laguerre polynomial; white noise functional. Introduction
Nonlinear Schr¨odinger equations.
Let ( M , g ) be a two-dimensional compact Rie-mannian manifold without boundary or a bounded domain in R . We consider the defo-cusing nonlinear Schr¨odinger equation (NLS): ( i∂ t u + ∆ g u = | u | k − uu | t =0 = φ, ( t, x ) ∈ R × M , (1.1)where ∆ g stands for the Laplace-Beltrami operator on M , k = 2 m ≥ u : R × M −→ C .The aim of this article is to give a pedagogic and self-contained presentation on theconstruction of an invariant Gibbs measure for a renormalized version of (1.1). In particular,we present an elementary Fourier analytic approach to the problem in the hope that this willbe accessible to readers (in particular those in dispersive PDEs) without prior knowledge inquantum field theory and/or stochastic analysis. In order to make the presentation simpler,we first detail the case of the flat torus M = T , where T = R / (2 π Z ). Namely, we consider ( i∂ t u + ∆ u = | u | k − uu | t =0 = φ, ( t, x ) ∈ R × T . (1.2)The equation (1.2) is known to possess the following Hamiltonian structure: ∂ t u = − i ∂H∂u , (1.3)where H = H ( u ) is the Hamiltonian given by H ( u ) = 12 ˆ T |∇ u | dx + 1 k ˆ T | u | k dx. (1.4)Moreover, the mass M ( u ) = ˆ T | u | dx is also conserved under the dynamics of (1.2).1.2. Gibbs measures.
Given a Hamiltonian flow on R n : ( ˙ p j = ∂H∂q j ˙ q j = − ∂H∂p j (1.5)with Hamiltonian H ( p, q ) = H ( p , · · · , p n , q , · · · , q n ), Liouville’s theorem states that theLebesgue measure Q nj =1 dp j dq j on R n is invariant under the flow. Then, it follows fromthe conservation of the Hamiltonian H that the Gibbs measures e − βH ( p,q ) Q nj =1 dp j dq j areinvariant under the dynamics of (1.5). Here, β > with the exception of the Wiener chaos estimate (Lemma 2.6). NVARIANT GIBBS MEASURES FOR THE 2- d DEFOCUSING NLS 3 of the form: “ dP (2 m )2 = Z − exp( − βH ( u )) du ” (1.6)to be invariant under the dynamics of (1.2). As it is, (1.6) is merely a formal expressionand we need to give a precise meaning. From (1.4), we can write (1.6) as“ dP (2 m )2 = Z − e − m ´ | u | m dx e − ´ |∇ u | dx du ” . (1.7)This motivates us to define the Gibbs measure P (2 m )2 as an absolutely continuous (proba-bility) measure with respect to the following massless Gaussian free field: dµ = e Z − exp (cid:0) − ´ |∇ u | dx (cid:1) du. In order to avoid the problem at the zeroth frequency, we instead considerthe following massive Gaussian free field: dµ = e Z − e − ´ |∇ u | dx − ´ | u | dx du. (1.8)in the following. Note that this additional factor replaces − H ( u ) by − H ( u ) − M ( u ) inthe formal definition (1.6) of P (2 m )2 . In view of the conservation of mass, however, we stillexpect P (2 m )2 to be invariant if we can give a proper meaning to P (2 m )2 .It is well known that µ in (1.8) corresponds to a mean-zero Gaussian free field on T .More precisely, µ is the mean-zero Gaussian measure on H s ( T ) for any s < Q s = (Id − ∆) − s . Recall that a covariance operator Q of a mean-zeroprobability measure µ on a Hilbert space H is a trace class operator, satisfying ˆ H h f, u i H h h, u i H dµ ( u ) = h Qf, h i H (1.9)for all f, h ∈ H .We can also view the Gaussian measure µ as the induced probability measure under themap: ω ∈ Ω u ( x ) = u ( x ; ω ) = X n ∈ Z g n ( ω ) p | n | e in · x , (1.10)where { g n } n ∈ Z is a sequence of independent standard complex-valued Gaussian randomvariables on a probability space (Ω , F , P ). Namely, functions under µ are represented bythe random Fourier series given in (1.10). Note that the random function u in (1.10)is in H s ( T ) \ L ( T ) for any s <
0, almost surely. Thus, µ is a Gaussian probabilitymeasure on H s ( T ) for any s <
0. Moreover, it is easy to see that (1.9) with H = H s ( T ) In the following, Z , Z N , and etc. denote various normalizing constants so that the corresponding mea-sures are probability measures when appropriate. For simplicity, we set β = 1 in the following. See [33] for a discussion on the Gibbs measures anddifferent values of β > Strictly speaking, there is a factor of (2 π ) − in (1.10). For simplicity of the presentation, however, wedrop such harmless 2 π hereafter. Namely, g n has mean 0 and Var( g n ) = 1. T. OH AND L. THOMANN Q s = (Id − ∆) − s , s <
0, follows from (1.10). Indeed, we have ˆ H s h f, u i H s h h, u i H s dµ ( u ) = E (cid:20) X n ∈ Z b f ( n ) g n ( ω ) h n i − s X m ∈ Z b h ( m ) g m ( ω ) h m i − s (cid:21) = X n ∈ Z b f ( n ) b h ( n ) h n i − s = h Q s f, h i H s . (1.11)Here, h · i = (1 + | · | ) . Note that the second equality in (1.11) holds even for s ≥
0. For s ≥
0, however, µ is not a probability measure on H s ( T ). Indeed, we have µ ( L ( T )) = 0.The next step is to make sense of the Gibbs measure P (2 m )2 in (1.7). First, let us brieflygo over the situation when d = 1. In this case, µ defined by (1.8) is a probability measureon H s ( T ), s < . Then, it follows from Sobolev’s inequality that ´ T | u ( x ; ω ) | k dx is finitealmost surely. Hence, for any k >
2, the Gibbs measure: dP ( k )1 = Z − e − k ´ T | u | k dx dµ (1.12)is a probability measure on H s ( T ), s < , absolutely continuous with respect to µ . More-over, by constructing global-in-time dynamics in the support of P ( k )1 , Bourgain [6] provedthat the Gibbs measure P ( k )1 is invariant under the dynamics of the defocusing NLS for k >
2. Here, by invariance, we mean that P ( k )1 (cid:0) Φ( − t ) A (cid:1) = P ( k )1 ( A ) (1.13)for any measurable set A ∈ B H s ( T ) and any t ∈ R , where Φ( t ) : u ∈ H s ( T ) u ( t ) =Φ( t ) u ∈ H s ( T ) is a well-defined solution map, at least almost surely with respect to P ( k )1 .McKean [25] gave an independent proof of the invariance of the Gibbs measure when k = 4,relying on a probabilistic argument. See Remark 1.7 below for the discussion on the focusingcase. Over the recent years, there has been a significant progress in the study of invariantGibbs measures for Hamiltonian PDEs. See, for example, [24, 6, 27, 25, 7, 8, 44, 41, 42,13, 15, 31, 32, 43, 40, 30, 34, 21, 10, 19, 37, 12].The situation for d = 2 is entirely different. As discussed above, the random function u in (1.10) is not in L ( T ) almost surely. This in particular implies that ˆ T | u ( x ; ω ) | k dx = ∞ (1.14)almost surely for any k ≥
2. Therefore, we can not construct a probability measure of theform: dP ( k )2 = Z − e − k ´ T | u | k dx dµ. (1.15)Thus, we are required to perform a (Wick) renormalization on the nonlinear part | u | k ofthe Hamiltonian. This is a well studied subject in the Euclidean quantum field theory, atleast in the real-valued setting. See Simon [38] and Glimm-Jaffe [23]. Also, see Da Prato-Tubaro [18] for a concise discussion on T , where the Gibbs measures naturally appear inthe context of the stochastic quantization equation. NVARIANT GIBBS MEASURES FOR THE 2- d DEFOCUSING NLS 5
Wick renormalization.
There are different ways to introduce the Wick renormal-ization. One classical way is to use the Fock-space formalism, where the Wick ordering isgiven as the reordering of the creation and annihilation operators. See [38, 26, 20] for moredetails. It can be also defined through the multiple Wiener-Ito integrals. In the following,we directly define it as the orthogonal projection onto the Wiener homogeneous chaoses(see the Wiener-Ito decomposition (2.5) below) by using the Hermite polynomials and the(generalized) Laguerre polynomials, since this allows us to introduce only the necessary ob-jects without introducing cumbersome notations and formalism, making our presentationaccessible to readers without prior knowledge in the problem.Before we study the Wick renormalization for NLS, let us briefly discuss the Wick renor-malization on T in the real-valued setting. We refer to [18] for more details. We assumethat u is real-valued. Then, the random function u under µ in (1.8) is represented by therandom Fourier series (1.10) conditioned that g − n = g n . Given N ∈ N , let P N be theDirichlet projection onto the frequencies {| n | ≤ N } and set u N = P N u , where u is asin (1.10). Note that, for each x ∈ T , the random variable u N ( x ) is a mean-zero real-valuedGaussian with variance σ N := E [ u N ( x )] = X | n |≤ N
11 + | n | ∼ log N. (1.16)Note that σ N is independent of x ∈ T . Fix an even integer k ≥
4. We define the Wickordered monomial : u kN : by : u kN : = H k ( u N ; σ N ) , (1.17)where H k ( x ; σ ) is the Hermite polynomial of degree k defined in (2.1). Then, one can showthat the limit ˆ T : u k : dx = lim N →∞ ˆ T : u kN : dx (1.18)exists in L p ( µ ) for any finite p ≥
1. Moreover, one can construct the Gibbs measure: dP ( k )2 = Z − e − k ´ T : u k : dx dµ as the limit of dP ( k )2 ,N = Z − N e − k ´ T : u kN : dx dµ. The key ingredients of the proof of the above claims are the Wiener-Ito decomposition of L ( H s ( T ) , µ ) for s <
0, the hypercontractivity of the Ornstein-Uhlenbeck semigroup, andNelson’s estimate [28, 29].For our problem on NLS (1.2), we need to work on complex-valued functions. In thereal-valued setting, the Wick ordering was defined by the Hermite polynomials. In thecomplex-valued setting, we also define the Wick ordering by the Hermite polynomials, butthrough applying the Wick ordering the real and imaginary parts separately.Let u be as in (1.10). Given N ∈ N , we define u N by u N = P N u = X | n |≤ N b u ( n ) e in · x , T. OH AND L. THOMANN where P N is the Dirichlet projection onto the frequencies {| n | ≤ N } as above. Then, for m ∈ N , we define the Wick ordered monomial : | u N | m : by: | u N | m : = : (cid:0) (Re u N ) + (Im u N ) (cid:1) m := m X ℓ =0 (cid:18) mℓ (cid:19) : (Re u N ) ℓ : : (Im u N ) m − ℓ ) : . (1.19)It turns out, however, that it is more convenient to work with the Laguerre polynomialsin the current complex-valued setting; see Section 2. Recall that the Laguerre polynomials L m ( x ) are defined through the following generating function: G ( t, x ) := 11 − t e − tx − t = ∞ X m =0 t m L m ( x ) , (1.20)for | t | < x ∈ R . For readers’ convenience, we write out the first few Laguerrepolynomials in the following: L ( x ) = 1 , L ( x ) = − x + 1 , L ( x ) = ( x − x + 2) ,L ( x ) = ( − x + 9 x − x + 6) , L ( x ) = ( x − x + 72 x − x + 24) . (1.21)More generally, the L m are given by the formula L m ( x ) = m X ℓ =0 (cid:18) mℓ (cid:19) ( − ℓ ℓ ! x ℓ . (1.22)Given σ >
0, we set L m ( x ; σ ) := σ m L m (cid:0) xσ (cid:1) . (1.23)Note that L m ( x ; σ ) is a homogenous polynomial of degree m in x and σ . Then, given N ∈ N , we can rewrite the Wick ordered monomial : | u N | m : defined in (1.19) as: | u N | m : = ( − m m ! · L m ( | u N | ; σ N ) , (1.24)where σ N is given by σ N = E [ | u N ( x ) | ] = X | n |≤ N
11 + | n | ∼ log N, (1.25)independently of x ∈ T . See Lemma 2.1 for the equivalence of (1.19) and (1.24).For N ∈ N , let G N ( u ) = 12 m ˆ T : | P N u | m : dx. (1.26)Then, we have the following proposition. Proposition 1.1.
Let m ≥ be an integer. Then, { G N ( u ) } N ∈ N is a Cauchy sequence in L p ( µ ) for any p ≥ . More precisely, there exists C m > such that k G M ( u ) − G N ( u ) k L p ( µ ) ≤ C m ( p − m N for any p ≥ and any M ≥ N ≥ . NVARIANT GIBBS MEASURES FOR THE 2- d DEFOCUSING NLS 7
Proposition 1.1 states that we can define the limit G ( u ) as G ( u ) = 12 m ˆ T : | u | m : dx = lim N →∞ G N ( u ) = 12 m lim N →∞ ˆ T : | P N u | m : dx and that G ( u ) ∈ L p ( µ ) for any finite p ≥
2. This allows us to define the Wick orderedHamiltonian: H Wick ( u ) = 12 ˆ T |∇ u | dx + 12 m ˆ T : | u | m : dx (1.27)for an integer m ≥
2. In order to discuss the invariance property of the Gibbs measures,we need to overcome the following two problems.(i) Define the Gibbs measure of the form“ dP (2 m )2 = Z − e − H Wick ( u ) − M ( u ) du ” , (1.28)corresponding to the Wick ordered Hamiltonian H Wick .(ii) Make sense of the following defocusing Wick ordered NLS on T : i∂ t u + ∆ u = : | u | m − u : , ( t, x ) ∈ R × T , (1.29)arising as a Hamiltonian PDE: ∂ t u = − i∂ u H Wick . In particular, we need to give aprecise meaning to the Wick ordered nonlinearity : | u | m − u :.Let us first discuss Part (i). For N ∈ N , let R N ( u ) = e − G N ( u ) = e − m ´ T : | u N | m : dx and define the truncated Gibbs measure P (2 m )2 ,N by dP (2 m )2 ,N := Z − N R N ( u ) dµ = Z − N e − m ´ T : | u N | m : dx dµ, (1.30)corresponding to the truncated Wick ordered Hamiltonian: H N Wick ( u ) = 12 ˆ T |∇ u | dx + 12 m ˆ T : | u N | m : dx. (1.31)Note that P (2 m )2 ,N is absolutely continuous with respect to the Gaussian free field µ .We have the following proposition on the construction of the Gibbs measure P (2 m )2 as alimit of P (2 m )2 ,N . Proposition 1.2.
Let m ≥ be an integer. Then, R N ( u ) ∈ L p ( µ ) for any p ≥ with auniform bound in N , depending on p ≥ . Moreover, for any finite p ≥ , R N ( u ) convergesto some R ( u ) in L p ( µ ) as N → ∞ . In particular, by writing the limit R ( u ) ∈ L p ( µ ) as R ( u ) = e − m ´ T : | u | m : dx , Proposition 1.2 allows us to define the Gibbs measure P (2 m )2 in (1.28) by dP (2 m )2 = Z − R ( u ) dµ = Z − e − m ´ T : | u | m : dx dµ. (1.32)Then, P (2 m )2 is a probability measure on H s ( T ), s <
0, absolutely continuous to theGaussian field µ . Moreover, P (2 m )2 ,N converges weakly to P (2 m )2 . T. OH AND L. THOMANN
Invariant dynamics for the Wick ordered NLS.
In this subsection, we study thedynamical problem (1.29). First, we consider the Hamiltonian PDE corresponding to thetruncated Wick ordered Hamiltonian H N Wick in (1.31): i∂ t u N + ∆ u N = P N (cid:0) : | P N u N | m − P N u N : (cid:1) . (1.33)The high frequency part P ⊥ N u N evolves according to the linear flow, while the low frequencypart P N u N evolves according to the finite dimensional system of ODEs viewed on theFourier side. Here, P ⊥ N is the Dirichlet projection onto the high frequencies {| n | > N } .Let µ = µ N ⊗ µ ⊥ N , where µ N and µ ⊥ N are the marginals of µ on E N = span { e in · x } | n |≤ N and E ⊥ N = span { e in · x } | n | >N , respectively. Then, we can write P (2 m )2 ,N in (1.30) as P (2 m )2 ,N = b P (2 m )2 ,N ⊗ µ ⊥ N , (1.34)where b P (2 m )2 ,N is the finite dimensional Gibbs measure defined by d b P (2 m )2 ,N = b Z − N e − m ´ T : | P N u N | m : dx dµ N . (1.35)Then, it is easy to see that P (2 m )2 ,N is invariant under the dynamics of (1.33); see Lemma 5.1below. In particular, the law of u N ( t ) is given by P (2 m )2 ,N for any t ∈ R .For N ∈ N , define F N ( u ) by F N ( u ) = P N (cid:0) : | P N u | m − P N u : (cid:1) . (1.36)Then, assuming that u is distributed according to the Gaussian free field µ in (1.8), thefollowing proposition lets us make sense of the Wick ordered nonlinearity : | u | m − u :in (1.29) as the limit of F N ( u ). Proposition 1.3.
Let m ≥ be an integer and s < . Then, { F N ( u ) } N ∈ N is a Cauchysequence in L p ( µ ; H s ( T )) for any p ≥ . More precisely, given ε > with s + ε < , thereexists C m,s,ε > such that (cid:13)(cid:13) k F M ( u ) − F N ( u ) k H s (cid:13)(cid:13) L p ( µ ) ≤ C m,s,ε ( p − m − N ε (1.37) for any p ≥ and any M ≥ N ≥ . In the real-valued setting, the nonlinearity corresponding to the Wick ordered Hamiltonianis again given by a Hermite polynomial. Indeed, from (1.17), we have k ∂ u N (cid:0) : u kN : (cid:1) = k ∂ u N H k ( u N ; σ N ) = H k − ( u N ; σ N ) , since ∂ x H k ( x ; ρ ) = kH k − ( x ; ρ ); see (2.3). The situation is slightly different in the complex-valued setting. In the proof of Proposition 1.3, the generalized Laguerre polynomials L ( α ) m ( x )with α = 1 plays an important role. See Section 3.We denote the limit by F ( u ) = : | u | m − u : and consider the Wick ordered NLS (1.29).When m = 2, Bourgain [7] constructed almost sure global-in-time strong solutions andproved the invariance of the Gibbs measure P (4)2 for the defocusing cubic Wick ordered NLS.See Remark 1.6 below. The main novelty in [7] was to construct local-in-time dynamicsin a probabilistic manner, exploiting the gain of integrability for the random rough linearsolution. By a similar approach, Burq-Tzvetkov [14, 15] constructed almost sure global-in-time strong solutions and proved the invariance of the Gibbs measure for the defocusing NVARIANT GIBBS MEASURES FOR THE 2- d DEFOCUSING NLS 9 subquintic nonlinear wave equation (NLW) posed on the three-dimensional ball in the radialsetting.On the one hand, when m = 2, there is only an ε -gap between the regularity of thesupport H s ( T ), s <
0, of the Gibbs measure P (4)2 and the scaling criticality s = 0 (and theregularity s > m ≥
3, the gap between the regularity of the Gibbs measure P (2 m )2 and the scalingcriticality is slightly more than 1 − m − ≥ . At present, it seems very difficult to closethis gap and to construct strong solutions even in a probabilistic setting.In the following, we instead follow the approach presented in the work [12] by the secondauthor with Burq and Tzvetkov. This work, in turn, was motivated by the works ofAlbeverio-Cruzeiro [1] and Da Prato-Debussche [17] in the study of fluids. The main ideais to exploit the invariance of the truncated Gibbs measures P (2 m )2 ,N for (1.33), then toconstruct global-in-time weak solutions for the Wick ordered NLS (1.29), and finally toprove the invariance of the Gibbs measure P (2 m )2 in some mild sense.Now, we are ready to state our main theorem. Theorem 1.4.
Let m ≥ be an integer. Then, there exists a set Σ of full measure withrespect to P (2 m )2 such that for every φ ∈ Σ , the Wick ordered NLS (1.29) with initialcondition u (0) = φ has a global-in-time solution u ∈ C ( R ; H s ( T )) for any s < . Moreover, for all t ∈ R , the law of the random function u ( t ) is givenby P (2 m )2 . There are two components in Theorem 1.4: existence of solutions and invariance of P (2 m )2 .A precursor to the existence part of Theorem 1.4 appears in [11]. In [11], the second authorwith Burq and Tzvetkov used the energy conservation and a regularization property underrandomization to construct global-in-time solutions to the cubic NLW on T d for d ≥
3. Themain ingredient in [11] is the compactness of the solutions to the approximating PDEs.In order to prove Theorem 1.4, we instead follow the argument in [12]. Here, the mainingredient is the tightness (= compactness) of measures on space-time functions, emanatingfrom the truncated Gibbs measure P (2 m )2 ,N and Skorokhod’s theorem (see Lemma 5.7 below).We point out that Theorem 1.4 states only the existence of a global-in-time solution u without uniqueness.Theorem 1.4 only claims that the law L ( u ( t )) of the H s -valued random variable u ( t )satisfies L ( u ( t )) = P (2 m )2 for any t ∈ R . This implies the invariance property of the Gibbs measure P (2 m )2 in somemild sense, but it is weaker than the actual invariance in the sense of (1.13).In fact, the result of Theorem 1.4 remains true in a more general setting. Let ( M , g ) bea two-dimensional compact Riemannian manifold without boundary or a bounded domainin R . We consider the equation (1.1) on M (when M is a domain in R , we impose theDirichlet or Neumann boundary condition). Assume that k = 2 m for some integer m ≥ setting, by incorporating the geometric information such as the eigenfunction estimates. Inparticular, it is worthwhile to note that the variance parameter σ N in (1.25) now depends on x ∈ M in this geometric setting and more care is needed. Once we establish the analoguesof Propositions 1.1, 1.2, and 1.3, we can proceed as in the flat torus case. Namely, thesepropositions allow us to define a renormalized Hamiltonian: H Wick ( u ) = 12 ˆ M |∇ u | dx + 12 m ˆ M : | u | m : dx, and a Gibbs measure P (2 m )2 as in (1.28). Moreover, we are able to give a sense to NLS witha Wick ordered nonlinearity: ( i∂ t u + ∆ g u = : | u | m − u : u | t =0 = φ, ( t, x ) ∈ R × M . (1.38)In this general setting, we have the following result. Theorem 1.5.
Let m ≥ be an integer. Then, there exists a set Σ of full measure withrespect to P (2 m )2 such that for every φ ∈ Σ , the Wick ordered NLS (1.38) with initialcondition u (0) = φ has a global-in-time solution u ∈ C ( R ; H s ( M )) for any s < . Moreover, for all t ∈ R , the law of the random function u ( t ) is givenby P (2 m )2 . Theorems 1.4 and 1.5 extend [12, Theorem 1.11] for the defocusing Wick ordered cubicNLS ( m = 2) to all defocusing nonlinearities (all m ≥ m . See Appendix A for anexample of an concrete combinatorial argument for m = 3 in the case M = T , followingthe methodology in [7, 12]. In order to overcome this combinatorial difficulty, we introducethe white noise functional (see Definition 2.2 below) and avoid combinatorial argumentsof increasing complexity in m , allowing us to prove Propositions 1.1 and 1.3 in a concisemanner. In order to present how we overcome the combinatorial complexity in a clearmanner, we decided to first discuss the proofs of Propositions 1.1, 1.2, and 1.3 in the caseof the flat torus T (Sections 2 and 3). This allows us to isolate the main idea. We thendiscuss the geometric component and prove the analogues of Propositions 1.1, 1.2, and 1.3in a general geometric setting (Section 4). Remark 1.6.
Let m = 2 and M = T . Then, the Wick ordered NLS (1.29) can be formallywritten as i∂ t u + ∆ u = ( | u | − σ ∞ ) u, (1.39)where σ ∞ is the (non-existent) limit of σ N ∼ log N as N → ∞ .Given u as in (1.10), define θ N = ffl T | P N u | dx − σ N , where ffl T f ( x ) dx = π ´ T f ( x ) dx .Then, it is easy to see that the limit θ ∞ := lim N →∞ θ N exists in L p ( µ ) for any p ≥
1. Thus,by setting v ( t ) = e itθ ∞ u ( t ), we can rewrite (1.39) as i∂ t v + ∆ v = ( | v | − ffl T | v | dx ) v. (1.40) NVARIANT GIBBS MEASURES FOR THE 2- d DEFOCUSING NLS 11
Note that k v k L = ∞ almost surely. Namely, (1.40) is also a formal expression for thelimiting dynamics. In [7], Bourgain studied (1.40) and proved local well-posedness below L ( T ) in a probabilistic setting.If v is a smooth solution to (1.40), then by setting w ( t ) = e − it ffl T | v | dx v ( t ), we see that w is a solution to the standard cubic NLS: i∂ t w + ∆ w = | w | w. (1.41)This shows that the Wick ordered NLS (1.39) and (1.40) are “equivalent” to the standardcubic NLS in the smooth setting. Note that this formal reduction relies on the fact that theWick ordering introduces only a linear term when m = 2. For m ≥
3, the Wick orderingintroduces higher order terms and thus there is no formal equivalence between the standardNLS (1.2) and the Wick ordered NLS (1.29).
Remark 1.7.
So far, we focused on the defocusing NLS. Let us now discuss the situationin the focusing case: i∂ t u + ∆ u = −| u | k − u with the Hamiltonian given by H ( u ) = 12 ˆ T d |∇ u | dx − k ˆ T d | u | k dx. In the focusing case, the Gibbs measure can be formally written as dP ( k ) d = Z − e − H ( u ) du = Z − e k ´ T d | u | k dx dµ. The main difficulty is that ´ T d | u | k dx is unbounded. When d = 1, Lebowitz-Rose-Speer [24]constructed the Gibbs measure P ( k )1 for 2 < k ≤
6, by adding an extra L -cutoff. Then,Bourgain [6] constructed global-in-time flow and proved the invariance of the Gibbs measurefor k ≤ . See also McKean [25].When d = 2, the situation becomes much worse. Indeed, Brydges-Slade [9] showed thatthe Gibbs measure P (4)2 for the focusing cubic NLS on T can not be realized as a probabilitymeasure even with the Wick order nonlinearity and/or with a (Wick ordered) L -cutoff.In [8], Bourgain pointed out that an ε -smoothing on the nonlinearity makes this problemwell-posed and the invariance of the Gibbs measure may be proven even in the focusingcase. Remark 1.8.
In a recent paper [36], we also studied the defocusing nonlinear wave equa-tions (NLW) in two spatial dimensions (with an even integer k = 2 m ≥ ρ ≥ ( ∂ t u − ∆ g u + ρu + u k − = 0( u, ∂ t u ) | t =0 = ( φ , φ ) , ( t, x ) ∈ R × M (1.42)and its associated Gibbs measure: dP (2 m )2 = Z − exp( − H ( u, ∂ t u )) du ⊗ d ( ∂ t u )= Z − e − m ´ u m dx e − ´ ( ρu + |∇ u | ) dx du ⊗ e − ´ ( ∂ t u ) d ( ∂ t u ) . (1.43)As in the case of NLS, the Gibbs measure in (1.43) is not well defined in the two spatialdimensions. Namely, one needs to consider the Gibbs measure P (2 m )2 associated to the Wick ordered Hamiltonian as in (1.32) and study the associated dynamical problem given bythe following defocusing Wick ordered NLW: ∂ t u − ∆ u + ρu + : u k − : = 0 . (1.44)In the case of the flat torus M = T with ρ >
0, we showed that the defocusing Wickordered NLW (1.44) is almost surely globally well-posed with respect to the Gibbs measure P (2 m )2 and that the Gibbs measure P (2 m )2 is invariant under the dynamics of (1.44). Fora general two-dimensional compact Riemannian manifold without boundary or a boundeddomain in R (with the Dirichlet or Neumann boundary condition), we showed that ananalogue of Theorem 1.5 (i.e. almost sure global existence and invariance of the Gibbsmeasure P (2 m )2 in some mild sense) holds for (1.44) when ρ >
0. In the latter case with theDirichlet boundary condition, we can also take ρ = 0.In particular, our result on T is analogous to that for the defocusing cubic NLS on T [7], where the main difficulty lies in constructing local-in-time unique solutions almostsurely with respect to the Gibbs measure. We achieved this goal for any even k ≥ T , such smoothing is not available and theconstruction of unique solutions with the Gibbs measure as initial data remains open forthe (super)quintic case. Remark 1.9.
In [6, 37], Bourgain ( k = 2 ,
3) and Richards ( k = 4) proved invariance ofthe Gibbs measures for the generalized KdV equation (gKdV) on the circle: ∂ t u + ∂ x u = ± ∂ x ( u k ) , ( t, x ) ∈ R × T . (1.45)In [35], the authors and Richards studied the problem for k ≥
5. In particular, by followingthe approach in [12] and this paper, we proved almost sure global existence and invarianceof the Gibbs measuresin some mild sense analogous to Theorem 1.4 for (i) all k ≥ k = 5 in the focusing case. Note that there is no need to apply arenormalization for constructing the Gibbs measures for this problem since the equation isposed on T . See [24, 6].This paper is organized as follows. In Sections 2 and 3, we present the details of theproofs of Propositions 1.1, 1.2, and 1.3 in the particular case when M = T . We thenindicate the changes required to treat the general case in Section 4. In Section 5, we proveTheorems 1.4 and 1.5. In Appendix A, we present an alternative proof of Proposition 1.1when m = 3 in the case M = T , performing concrete combinatorial computations.2. Construction of the Gibbs measures
In this section, we present the proofs of Propositions 1.1 and 1.2 and construct the Gibbsmeasure P (2 m )2 in (1.32). One possible approach is to use the Fock-space formalism in quan-tum field theory [38, 23, 26, 20]. As mentioned above, however, we present a pedestrianFourier analytic approach to the problem since we believe that it is more accessible to awide range of readers. The argument presented in this section and the next section (onProposition 1.3) follows the presentation in [18] with one important difference; we work in In the case of NLW, we only need to use the Hermite polynomials since we deal with real-valuedfunctions.
NVARIANT GIBBS MEASURES FOR THE 2- d DEFOCUSING NLS 13 the complex-valued setting and hence we will make use of the (generalized) Laguerre poly-nomials instead of the Hermite polynomials. Their orthogonal properties play an essentialrole. See Lemmas 2.4 and 3.2.2.1.
Hermite polynomials, Laguerre polynomials, and Wick ordering.
First, recallthe Hermite polynomials H n ( x ; σ ) defined through the generating function: F ( t, x ; σ ) := e tx − σt = ∞ X k =0 t k k ! H k ( x ; σ ) (2.1)for t, x ∈ R and σ >
0. For simplicity, we set F ( t, x ) := F ( t, x ; 1) and H k ( x ) := H k ( x ; 1) inthe following. Note that we have H k ( x, σ ) = σ k H k (cid:0) σ − x (cid:1) . (2.2)From (2.1), we directly deduce the following recursion relation ∂ x H k ( x ; σ ) = kH k − ( x ; σ ) , (2.3)for all k ≥
0. This allows to compute the H k , up to the constant term. The constant termis given by H k (0 , σ ) = ( − k (2 k − σ k and H k +1 (0 , σ ) = 0 , for all k ≥
0, where (2 k − k − k − · · · · (2 k )!2 k k ! and ( − x = 0. For readers’ convenience, we writeout the first few Hermite polynomials in the following: H ( x ; σ ) = 1 , H ( x ; σ ) = x, H ( x ; σ ) = x − σ,H ( x ; σ ) = x − σx, H ( x ; σ ) = x − σx + 3 σ . The monomial x k can be expressed in term of the Hermite polynomials: x k = [ k ] X m =0 (cid:18) k m (cid:19) (2 m − σ m H k − m ( x ; σ ) . (2.4)Fix d ∈ N , let H = R d . Then, consider the Hilbert space Γ H = L ( Q H , µ d ; C ) endowedwith the Gaussian measure dµ d = (2 π ) − d exp( −| x | / dx , x = ( x , . . . , x d ) ∈ R d . Wedefine a homogeneous Wiener chaos of order k to be an element of the form H k ( x ) = d Y j =1 H k j ( x j ) , where k = k + · · · + k d and H k j is the Hermite polynomial of degree k j defined in (2.1).Denote by Γ k ( H ) the closure of homogeneous Wiener chaoses of order k under L ( R d , µ d ). Indeed, the discussion presented here also holds for d = ∞ in the context of abstract Wiener spaces.For simplicity, however, we restrict our attention to finite values for d . Here, Q H = R d when d < ∞ . When d = ∞ , we set Q H to be an appropriate extension of H such that( H , Q H , µ ∞ ) forms an abstract Wiener space with H as the Cameron-Martin space. Then, we have the following Wiener-Ito decomposition: L ( Q H , µ d ; C ) = ∞ M k =0 Γ k ( H ) . (2.5)Given a homogeneous polynomial P k ( x ) = P k ( x , . . . , x d ) of degree k , we define theWick ordered polynomial : P k ( x ) : to be its projection onto H k . In particular, we have: x kj : = H k ( x j ) and : Q dj =1 x k j j : = Q dj =1 H k j ( x j ) with k = k + · · · + k d .Now, let g be a standard complex-valued Gaussian random variable. Then, g can bewritten as g = h √ + i h √ , where h and h are independent standard real-valued Gaussianrandom variables. We investigate the Wick ordering on | g | m for m ∈ N , that is, theprojection of | g | m onto H m . When m = 1, | g | = ( h + h ) is Wick-ordered into: | g | : = ( h −
1) + ( h −
1) = | g | − . (2.6)When m = 2, | g | = ( h + h ) = ( h + 2 h h + h ) is Wick-ordered into: | g | : = ( h − h + 3) + ( h − h −
1) + ( h − h + 3)= ( h + 2 h h + h ) − h + h ) + 2= | g | − | g | + 2 . When m = 3, a direct computation shows that | g | = 18 ( h + h ) = 18 ( h + 3 h h + 3 h h + h )is Wick-ordered into: | g | : = H ( h ) + H ( h ) H ( h ) + H ( h ) H ( h ) + H ( h )= | g | − | g | + 18 | g | − . In general, we have : | g | m : = 12 m m X ℓ =0 (cid:18) mℓ (cid:19) H ℓ ( h ) H m − ℓ ( h )= m X ℓ =0 (cid:18) mℓ (cid:19) H ℓ (Re g ; ) H m − ℓ (Im g ; ) , (2.7)where we used (2.2) in the second equality. It follows from the rotational invariance of thecomplex-valued Gaussian random variable that : | g | m : = P m ( | g | ) for some polynomial P m of degree m with the leading coefficient 1. This fact is, however, not obvious from (2.7).The following lemma shows that the Wick ordered monomials : | g | m : can be expressedin terms of the Laguerre polynomials (recall the definition (1.20)). This is (equivalent to) the Fock space in quantum field theory. See [38, Chapter I]. In particular, theFock space F ( H ) = L ∞ k =0 H ⊗ k sym C is shown to be equivalent to the Wiener-Ito decomposition (2.5). In theFock space formalism, the Wick renormalization can be stated as the reordering of the creation operatorson the left and annihilation operator on the right. We point out that while much of our discussion can berecast in the Fock space formalism, our main aim of this paper is to give a self-contained presentation (asmuch as possible) accessible to readers not familiar with the formalism in quantum field theory. Therefore,we stick to a simpler Fourier analytic and probabilistic approach. NVARIANT GIBBS MEASURES FOR THE 2- d DEFOCUSING NLS 15
Lemma 2.1.
Let m ∈ N . For a complex valued mean-zero Gaussian random variable g with Var( g ) = σ > , we have : | g | m : = m X ℓ =0 (cid:18) mℓ (cid:19) H ℓ (Re g ; σ ) H m − ℓ (Im g ; σ )= ( − m m ! · L m ( | g | ; σ ) . (2.8) As a consequence, the Wick ordered monomial : | u N | m : defined in (1.19) satisfies (1.24) for any N ∈ N .Proof. The first equality follows from (2.7) and scaling with (2.2). Moreover, by scalingwith (1.23) and (2.2), we can assume that g is a standard complex-valued Gaussian randomvariable with g = Re g and g = Im g . Define H m ( | g | ) and L m ( | g | ) by H m ( | g | ) = m X ℓ =0 (cid:18) mℓ (cid:19) H ℓ ( g ; ) H m − ℓ ( g ; ) , L m ( | g | ) = ( − m m ! · L m ( | g | ) . (2.9)Then, (2.8) follows once we prove the following three properties: H ( | g | ) = L ( | g | ) = | g | − , (2.10) ∂ ∂g∂g H m ( | g | ) = m H m − ( | g | ) , ∂ ∂g∂g L m ( | g | ) = m L m − ( | g | ) , (2.11) E [ H m ( | g | )] = E [ L m ( | g | )] = 0 , (2.12)for all m ≥
2. Noting that both H m ( | g | ) and L m ( | g | ) are polynomials in | g | , the threeproperties (2.10), (2.11), and (2.12) imply that H m ( | g | ) = L m ( | g | ) for all m ∈ N .The first property (2.10) follows from (2.6) and (1.21). Next, we prove (2.11) for H m ( | g | ).From ∂ g = ( ∂ g − i∂ g ) and ∂ g = ( ∂ g + i∂ g ), we have ∂ ∂g∂g = 14 ∆ g ,g , where ∆ g ,g denotes the usual Laplacian on R in the variables ( g , g ). Then, recallingthat ∂ x H k ( x ; σ ) = kH k − ( x ; σ ), we have ∂ ∂g∂g H m ( | g | ) = 14 ∆ g ,g H m ( | g | )= 14 m X ℓ =1 (cid:18) mℓ (cid:19) ℓ (2 ℓ − H ℓ − ( g ; ) H m − ℓ ( g ; )+ 14 m − X ℓ =0 (cid:18) mℓ (cid:19) (2 m − ℓ )(2 m − ℓ − H ℓ ( g ; ) H m − ℓ − ( g ; )= m m − X ℓ =0 (cid:18) m − ℓ (cid:19) H ℓ ( g ; ) H m − − ℓ ( g ; ) . As for the second identity in (2.11), thanks to the formula (1.22), we get ∂ ∂g∂g L m ( | g | ) = ( − m m !4 m X ℓ =0 (cid:18) mℓ (cid:19) ( − ℓ ℓ ! ∆ g ,g ( g + g ) ℓ = ( − m − m ! m X ℓ =1 (cid:18) mℓ (cid:19) ( − ℓ − ℓ ! ℓ | g | ℓ − = m L m − ( | g | ) . This proves (2.11). The property (2.12) follows from (i) independence of g and g togetherwith the orthogonality of H k ( x ) and the constant function 1 under e − x dx and (ii) theorthogonality of L m ( x ) and the constant function 1 under R + e − x dx Let u be as in (1.10). Fix x ∈ T . Letting e g n = g n e in · x , we see that { e g n } n ∈ N is a sequenceof independent standard complex-valued Gaussian random variables. Then, given N ∈ N ,Re u N ( x ) and Im u N ( x ) are mean-zero real-valued Gaussian random variables with variance σ N , while u N ( x ) is a mean-zero complex-valued Gaussian random variable with variance σ N , Then, it follows from (1.19) with (1.17) and (2.8) that: | u N ( x ) | m : = m X ℓ =0 (cid:18) mℓ (cid:19) H ℓ (Re u ( x ); σ N ) H m − ℓ (Im u ( x ); σ N )= ( − m m ! · L m ( | u N ( x ) | ; σ N ) , verifying (1.24). This proves the second claim in Lemma 2.1. (cid:3) White noise functional.
Next, we define the white noise functional. Let w ( x ; ω ) bethe mean-zero complex-valued Gaussian white noise on T defined by w ( x ; ω ) = X n ∈ Z g n ( ω ) e in · x . Definition 2.2.
The white noise functional W ( · ) : L ( T ) → L (Ω) is defined by W f ( ω ) = h f, w ( ω ) i L x = X n ∈ Z b f ( n ) g n ( ω ) . (2.13)for a function f ∈ L ( T ).Note that this is basically the periodic and higher dimensional version of the classicalWiener integral ´ ba f dB . It can also be viewed as the Gaussian process indexed by f ∈ L ( T ). See [38, Model 1 on p. 19 and Model 3 on p. 21]. For each f ∈ L ( T ), W f is acomplex-valued Gaussian random variable with mean 0 and variance k f k L . Moreover, wehave E (cid:2) W f W h ] = h f, h i L x for f, h ∈ L ( T ). In particular, the white noise functional W ( · ) : L ( T ) → L (Ω) is anisometry. Lemma 2.3.
Given f ∈ L ( T ) , we have ˆ Ω e Re W f ( ω ) dP ( ω ) = e k f k L . (2.14) NVARIANT GIBBS MEASURES FOR THE 2- d DEFOCUSING NLS 17
Proof.
Noting that Re g n and Im g n are mean-zero real-valued Gaussian random variableswith variance , it follows from (2.13) that ˆ Ω e Re W f ( ω ) dP ( ω ) = Y n ∈ Z π ˆ R e Re b f ( n ) Re g n − (Re g n ) d Re g n × ˆ R e Im b f ( n ) Im g n − (Im g n ) d Im g n = e k f k L . (cid:3) The following lemma on the white noise functional and the Laguerre polynomials playsan important role in our analysis. In the following, we present an elementary proof, usingthe generating function G in (1.20). See also Folland [22]. Lemma 2.4.
Let f, h ∈ L ( T ) such that k f k L = k h k L = 1 . Then, for k, m ∈ Z ≥ , wehave E (cid:2) L k ( | W f | ) L m ( | W h | ) (cid:3) = δ km |h f, h i| k . (2.15) Here, δ km denotes the Kronecker delta function. First, recall the following identity: e u = 1 √ π ˆ R e xu − x dx. (2.16)Indeed, we used a rescaled version of (2.16) in the proof of Lemma 2.3. Proof of Lemma 2.4.
Let G be as in (1.20). Then, for any − < t, s <
0, from (2.16) andLemma 2.3, we have ˆ Ω G ( t, | W f ( ω ) | ) G ( s, | W h ( ω ) | ) dP ( ω ) = 11 − t − s ˆ Ω e − t − t | W f | − s − s | W h | dP ( ω )= 11 − t − s π ˆ R e − x x y y × ˆ Ω exp (cid:16) Re W q − t − t ( x − ix ) f + q − s − s ( y − iy ) h (cid:17) dP dx dx dy dy = 11 − t − s π ˆ R e − x x − t ) − y y − s ) × e Re (cid:0) q − t − t q − s − s ( x − ix )( y + iy ) h f,h i (cid:1) dx dx dy dy
28 T. OH AND L. THOMANN
By a change of variables and applying (2.16), we have= 14 π ˆ R e − y y ˆ R e √ ts ( y Re h f,h i− y Im h f,h i ) x − x dx × ˆ R e √ ts ( y Re h f,h i + y Im h f,h i ) x − x dx dy dy = 12 π ˆ R e − y y e ts |h f,h i| ( y + y ) dy dy = 11 − ts |h f, h i| = ∞ X k =0 t k s k |h f, h i| k . (2.17)In the second to the last equality, we used the fact that ts |h f, h i| < . Hence, it followsfrom (1.20) and (2.17) that ∞ X k =0 t k s k |h f, h i| k = ∞ X k,m =0 t k s m ˆ Ω L k ( | W f ( ω ) | ) L m ( | W h ( ω ) | ) dP ( ω ) . By comparing the coefficients of t k s m , we obtain (2.15). (cid:3) Now, we are ready to make sense of the nonlinear part of the Wick ordered Hamiltonian H Wick in (1.27). We first present the proof of Proposition 1.1 for p = 2. Recall that G N ( u ) = 12 m ˆ T : | P N u | m : dx. Then, we have the following convergence property of G N ( u ) in L ( µ ). Lemma 2.5.
Let m ≥ be an integer. Then, { G N ( u ) } N ∈ N is a Cauchy sequence in L ( H s ( T ) , µ ) . More precisely, there exists C m > such that k G M ( u ) − G N ( u ) k L ( µ ) ≤ C m N (2.18) for any M ≥ N ≥ . Given N ∈ N , let σ N be as in (1.25). For fixed x ∈ T and N ∈ N , we define η N ( x )( · ) := 1 σ N X | n |≤ N e n ( x ) p | n | e n ( · ) , (2.19) γ N ( · ) := X | n |≤ N
11 + | n | e n ( · ) , (2.20)where e n ( y ) = e in · y . Note that k η N ( x ) k L ( T ) = 1 (2.21)for all (fixed) x ∈ T and all N ∈ N . Moreover, we have h η M ( x ) , η N ( y ) i L ( T ) = 1 σ M σ N γ N ( y − x ) = 1 σ M σ N γ N ( x − y ) , (2.22)for fixed x, y ∈ T and N, M ∈ N with M ≥ N . NVARIANT GIBBS MEASURES FOR THE 2- d DEFOCUSING NLS 19
Proof of Lemma 2.5.
Let m ≥ N ∈ N and x ∈ T , it followsfrom (1.10), (2.13), and (2.19) that u N ( x ) = σ N u N ( x ) σ N = σ N W η N ( x ) . (2.23)Then, from (1.24) and (2.23), we have: | u N | m : = ( − m m ! σ mN L m (cid:18) | u N | σ N (cid:19) = ( − m m ! σ mN L m (cid:0)(cid:12)(cid:12) W η N ( x ) (cid:12)(cid:12) (cid:1) . (2.24)From (2.24), Lemma 2.4, and (2.22), we have(2 m ) k G M ( u ) − G N ( u ) k L ( µ ) = ( m !) ˆ T x × T y ˆ Ω h σ mM L m (cid:0)(cid:12)(cid:12) W η M ( x ) (cid:12)(cid:12) (cid:1) L m (cid:0)(cid:12)(cid:12) W η M ( y ) (cid:12)(cid:12) (cid:1) − σ mM σ mN L m (cid:0)(cid:12)(cid:12) W η M ( x ) (cid:12)(cid:12) (cid:1) L m (cid:0)(cid:12)(cid:12) W η N ( y ) (cid:12)(cid:12) (cid:1) − σ mM σ mN L m (cid:0)(cid:12)(cid:12) W η N ( x ) (cid:12)(cid:12) (cid:1) L m (cid:0)(cid:12)(cid:12) W η M ( y ) (cid:12)(cid:12) (cid:1) + σ mN L m (cid:0)(cid:12)(cid:12) W η N ( x ) (cid:12)(cid:12) (cid:1) L m (cid:0)(cid:12)(cid:12) W η N ( y ) (cid:12)(cid:12) (cid:1)i dP dxdy = ( m !) ˆ T x × T y (cid:2) ( γ M ( x − y )) m − ( γ N ( x − y )) m (cid:3) dxdy = ( m !) ˆ T (cid:2) ( γ M ( x )) m − ( γ N ( x )) m (cid:3) dx ≤ C m ˆ T (cid:12)(cid:12) γ M ( x ) − γ N ( x ) (cid:12)(cid:12) · (cid:2) | γ M ( x ) | m − + | γ N ( x ) | m − (cid:3) dx. (2.25)In the second equality, we used the fact that γ N is a real-valued function.From (2.20), we have (cid:13)(cid:13) γ M − γ N (cid:13)(cid:13) L = (cid:18) X N< | n |≤ M | n | ) (cid:19) . N . (2.26)By Hausdorff-Young’s inequality, we have (cid:13)(cid:13) | γ N | m − (cid:13)(cid:13) L = k γ N k m − L m − ≤ X | n |≤ N | n | ) m − m − ! m − ≤ C m < ∞ (2.27)uniformly in N ∈ N . Then, (2.18) follows from (2.25), (2.26), and (2.27). (cid:3) Wiener chaos estimates.
In this subsection, we complete the proof of Proposi-tion 1.1. Namely, we upgrade (2.18) in Lemma 2.5 to any finite p ≥
2. Our main tool isthe following Wiener chaos estimate (see [38, Theorem I.22]).
Lemma 2.6.
Let { g n } n ∈ N be a sequence of independent standard real-valued Gaussianrandom variables. Given k ∈ N , let { P j } j ∈ N be a sequence of polynomials in ¯ g = { g n } n ∈ N of degree at most k . Then, for p ≥ , we have (cid:13)(cid:13)(cid:13)(cid:13) X j ∈ N P j (¯ g ) (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) ≤ ( p − k (cid:13)(cid:13)(cid:13)(cid:13) X j ∈ N P j (¯ g ) (cid:13)(cid:13)(cid:13)(cid:13) L (Ω) . (2.28) Observe that the estimate (2.28) is independent of d ∈ N . By noting that P j (¯ g ) ∈ L kℓ =0 Γ ℓ ( H ), this lemma follows as a direct corollary to the hypercontractivity of theOrnstein-Uhlenbeck semigroup due to Nelson [28].We are now ready to present the proof of Proposition 1.1. Proof of Proposition 1.1.
Let m ≥ ≤ p ≤
2, Proposition 1.1 followsfrom Lemma 2.5. In the following, we consider the case p >
2. From (1.22), (1.24),and (1.26), we have G M ( u ) − G N ( u ) = ( − m m !2 m m X ℓ =1 (cid:18) mℓ (cid:19) ( − ℓ ℓ ! Σ ℓ . Here, Σ ℓ is given byΣ ℓ = σ mM σ ℓM X Γ ℓ (0) | n j |≤ M ℓ Y j =1 g ∗ n j p | n j | − σ mN σ ℓN X Γ ℓ (0) | n j |≤ N ℓ Y j =1 g ∗ n j p | n j | , where Γ k and g ∗ n j are defined byΓ k ( n ) = { ( n , . . . , n k ) ∈ Z k : n − n + · · · + ( − k n k = n } , (2.29) g ∗ n j = ( g n j if j is odd, g n j if j is even. (2.30)Noting that Σ ℓ is a sum of polynomials of degree 2 ℓ in { g n } n ∈ Z , Proposition 1.1 followsfrom Lemmas 2.5 and 2.6. (cid:3) Nelson’s estimate.
In this subsection, we prove Proposition 1.2. Our main toolis the so-called Nelson’s estimate, i.e. in establishing an tail estimate of size λ >
0, wedivide the argument into low and high frequencies, depending on the size of λ . See (2.32)and (2.34). What plays a crucial role here is the defocusing property of the Hamiltonianand the logarithmic upper bound on − G N ( u ), which we discuss below.For each m ∈ N , there exists finite a m > − m L m ( x ) ≥ − a m for all x ∈ R .Then, it follows from (1.23), (1.24), (1.25), and (1.26) that there exists some finite b m > − G N ( u ) = − m ˆ T : | P N u | m : dx ≤ b m (log N ) m (2.31)for all N ≥
1. Namely, while G N ( u ) is not sign definite, − G N ( u ) is bounded from aboveby a power of log N . This is where the defocusing property of the equation (1.33) plays anessential role. Proof of Proposition 1.2.
Let m ≥ c m,p , C m > µ (cid:0) p | G M ( u ) − G N ( u ) | > λ (cid:1) ≤ C m e − c m,p N m λ m (2.32)for all M ≥ N ≥ p ≥
1, and all λ >
0. See, for example, [43, Lemma 4.5].
NVARIANT GIBBS MEASURES FOR THE 2- d DEFOCUSING NLS 21
We first show that R N ( u ) = e − G N ( u ) is in L p ( µ ) with a uniform bound in N . We have k R N ( u ) k pL p ( µ ) = ˆ H s e − pG N ( u ) dµ ( u )= ˆ ∞ µ ( e − pG N ( u ) > α ) dα ≤ ˆ ∞ µ ( − pG N ( u ) > log α ) dα. Hence, it suffices to show that there exist
C, δ > µ ( − pG N ( u ) > log α ) ≤ Cα − (1+ δ ) (2.33)for all α > N ∈ N . Given λ = log α >
0, choose N ∈ R such that λ = 2 pb m (log N ) m .Then, it follows from (2.31) that µ (cid:0) − pG N ( u ) > λ (cid:1) = 0 (2.34)for all N < N . For N ≥ N , it follows from (2.31) and (2.32) that there exist δ m,p > C m,p > µ (cid:0) − pG N ( u ) > λ (cid:1) ≤ µ (cid:0) − pG N ( u ) + pG N ( u ) > λ − pb m (log N ) m (cid:1) ≤ µ (cid:0) − pG N ( u ) + pG N ( u ) > λ (cid:1) ≤ C m e − c ′ m,p N m λ m = C m e − c ′ m,p λ m e e cmλ m ≪ C m,p e − (1+ δ m,p ) λ (2.35)for all N ≥ N . This shows that (2.33) is satisfied in this case as well. Hence, we have R N ( u ) ∈ L p ( µ ) with a uniform bound in N , depending on p ≥ G N ( u ) converges to G ( u ) in measure with respect to µ . Then, as a compositionof G N ( u ) with a continuous function, R N ( u ) = e − G N ( u ) converges to R ( u ) := e − G ( u ) inmeasure with respect to µ . In other words, given ε >
0, defining A N,ε by A N,ε = (cid:8) | R N ( u ) − R ( u ) | ≤ ε (cid:9) , we have µ ( A cN,ε ) →
0, as N → ∞ . Hence, by Cauchy-Schwarz inequality and the fact that k R k L p , k R N k L p ≤ C p uniformly in N ∈ N , we obtain k R − R N k L p ( µ ) ≤ k ( R − R N ) A N,ε k L p ( µ ) + k ( R − R N ) A cN,ε k L p ( µ ) ≤ ε (cid:0) µ ( A N,ε ) (cid:1) p + k R − R N k L p ( µ ) (cid:0) µ ( A cN,ε ) (cid:1) p ≤ Cε, for all sufficiently large N . This completes the proof of Proposition 1.2. (cid:3) On the Wick ordered nonlinearity
In this section, we present the proof of Proposition 1.3. The main idea is similar to thatin Section 2 but, this time, we will make use of the generalized Laguerre functions L ( α ) m ( x ).The generalized Laguerre polynomials L ( α ) m ( x ) are defined through the following generatingfunction: G α ( t, x ) := 1(1 − t ) α +1 e − tx − t = ∞ X m =0 t m L ( α ) m ( x ) , (3.1) for | t | < x ∈ R . From (3.1), we obtain the following differentiation rule; for ℓ ∈ N , d ℓ dx ℓ L ( α ) m ( x ) = ( − ℓ L ( α + ℓ ) m − ℓ ( x ) . (3.2)Given N ∈ N , let u N = P N , where u is as in (1.10). Let m ≥ F N ( u ) = P N (cid:0) : | P N u | m − P N u : (cid:1) = ( − m m ! σ mN · m P N ∂ u N n L m (cid:16) | u N | σ N (cid:17)o = ( − m +1 ( m − σ m − N · P N n L (1) m − (cid:16) | u N | σ N (cid:17) u N o . (3.3) Remark 3.1.
Here, ∂ u denotes the usual differentiation in u viewing u and u as independentvariables. This is not to be confused with ∂H∂u in (1.3). Note that ∂H∂u in (1.3) comes from thesymplectic structure of NLS and the Gˆateaux derivative of H . More precisely, we can viewthe dynamics of NLS (1.2) as a Hamiltonian dynamics with the symplectic space L ( T )and the symplectic form ω ( f, g ) = Im ˆ f ( x ) g ( x ) dx . Then, we define ∂H∂u by dH | u ( φ ) = ω (cid:16) φ, − i ∂H∂u (cid:17) , where dH | u ( φ ) is the the Gˆateaux derivative given by dH | u ( φ ) = ddε H ( u + εφ ) (cid:12)(cid:12) ε =0 .The following lemma is an analogue of Lemma 2.4 for the generalized Laguerre polyno-mials L (1) m ( x ) and plays an important role in the proof of Proposition 1.3. Lemma 3.2.
Let f, h ∈ L ( T ) such that k f k L = k h k L = 1 . Then, for k, m ∈ Z ≥ , wehave E h L (1) k ( | W f | ) W f L (1) m ( | W h | ) W h i = δ km ( k + 1) |h f, h i| k h f, h i . (3.4) Here, δ km denotes the Kronecker delta function. Besides (2.16), we will use the following identity: ue u = 1 √ π ˆ R xe xu − x dx. (3.5)This follows from differentiating (2.16) in u . Proof of Lemma 3.2.
Let G be as in (3.1) with α = 1. Let − < t <
0. From (2.16)and (3.5), we have G ( t, | W f | ) W f = 1(1 − t ) Re W f e − t − t (cid:0) (Re W f ) +(Im W f ) (cid:1) + i (1 − t ) Im W f e − t − t (cid:0) (Re W f ) +(Im W f ) (cid:1) = 1 √− t (1 − t ) π ˆ R ( x + ix ) e − x x e q − t − t ( x Re W f + x Im W f ) dx dx . NVARIANT GIBBS MEASURES FOR THE 2- d DEFOCUSING NLS 23
Given x , x , y , y ∈ R , let x = x + ix and y = y + iy . Then, for any − < t, s < ˆ Ω G ( t, W f ( ω )) W f ( ω ) G ( s, W h ( ω )) W h ( ω ) dP ( ω )= 1 √− t (1 − t ) √− s (1 − s ) π ˆ R xye − | x | | y | × ˆ Ω exp (cid:16) Re W q − t − t xf + q − s − s yh (cid:17) dP dx dx dy dy = 1 √− t (1 − t ) √− s (1 − s ) π ˆ R xye − | x | − t ) − | y | − s ) × e Re (cid:0) q − t − t q − s − s xy h f,h i (cid:1) dx dx dy dy By a change of variables and applying (2.16) and (3.5), we have= 12 √ ts π ˆ R xye − | x | − | y | e √ ts Re( xy h f,h i ) dx dx dy dy = h f, h i π ˆ R | y | e − (1 − ts |h f,h i| ) | y | dy dy By integration by parts, we have= h f, h i − ts |h f, h i| π ˆ R e − (1 − ts |h f,h i| ) | y | dy dy = h f, h i (1 − ts |h f, h i| ) = ∞ X k =0 ( k + 1) t k s k |h f, h i| k h f, h i . (3.6)Hence, it follows from (3.1) and (3.6) that ∞ X k =0 ( k + 1) t k s k |h f, h i| k h f, h i = ∞ X k,m =0 t k s m ˆ Ω L (1) k ( | W f ( ω ) | ) W f L (1) m ( | W h ( ω ) | ) W h dP ( ω ) . By comparing the coefficients of t k s m , we obtain (3.4). (cid:3) As a preliminary step to the proof of Proposition 1.3, we first estimate the size of theFourier coefficient of F N ( u ). Lemma 3.3.
Let m ≥ be an integer. Then, for any θ > , there exists C m,θ > suchthat kh F N ( u ) , e n i L x k L ( µ ) ≤ C m,θ | n | ) (1 − θ ) (3.7) for any n ∈ Z and any N ∈ N . Moreover, given positive ε < and any < θ ≤ − ε ,there exists C m,θ,ε > such that kh F M ( u ) − F N ( u ) , e n i L x k L ( µ ) ≤ C m,θ,ε N ε (1 + | n | ) (1 − θ − ε ) (3.8) for any n ∈ Z and any M ≥ N ≥ . Proof.
We first prove (3.7). Let m ≥ N ∈ N . From (3.3) with (2.23),we have F N ( u ) = ( − m +1 ( m − σ m − N · P N n L (1) m − (cid:0)(cid:12)(cid:12) W η N ( x ) (cid:12)(cid:12) (cid:1) W η N ( x ) o . (3.9)Clearly, h F N ( u ) , e n i L x = 0 when | n | > N . Thus, we only need to consider the case | n | ≤ N .From Lemma 3.2 with (3.9), (2.21) and (2.22), we have kh F N ( u ) , e n i L x k L ( µ ) = (cid:2) ( m − (cid:3) σ m − N ˆ T x × T y e n ( x ) e n ( y ) × ˆ Ω L (1) m − (cid:0)(cid:12)(cid:12) W η N ( x ) (cid:12)(cid:12) (cid:1) W η N ( x ) L (1) m − (cid:0)(cid:12)(cid:12) W η N ( y ) (cid:12)(cid:12) (cid:1) W η N ( y ) dP dxdy = m !( m − ˆ T x × T y | γ N ( x − y ) | m − γ N ( x − y ) e n ( x − y ) dxdy = C m F (cid:2) | γ N | m − γ N (cid:3) ( n ) . (3.10)Let Γ m − ( n ) be as in (2.29). For ( n , . . . , n m − ) ∈ Γ m − ( n ), we have max j | n j | & | n | .Thus, we have F (cid:2) | γ N | m − γ N (cid:3) ( n ) = X Γ m − ( n ) | n j |≤ N m − Y j =1
11 + | n j | ≤ d m,θ | n | ) − θ . (3.11)Hence, (3.7) follows from (3.10) and (3.11).Next, we prove (3.8). Let M ≥ N ≥
1. Proceeding as before with (3.9), Lemma 3.2,and (2.22), we have kh F M ( u ) − F N ( u ) , e n i L x k L ( µ ) = C m n [0 ,M ] ( | n | ) F (cid:2) | γ M | m − γ M (cid:3) ( n ) − [0 ,N ] ( | n | ) F (cid:2) | γ N | m − γ N (cid:3) ( n ) o = C m [0 ,N ] ( | n | ) n F (cid:2) | γ M | m − γ M (cid:3) ( n ) − F (cid:2) | γ N | m − γ N (cid:3) ( n ) o + C m ( N,M ] ( | n | ) F (cid:2) | γ M | m − γ M (cid:3) ( n ) . (3.12)On the one hand, noting that | n | > N , we can use (3.11) to estimate the second term onthe right-hand side of (3.12), yielding (3.8). On the other hand, noting that (cid:12)(cid:12)(cid:12) F (cid:2) | γ M | m − γ M (cid:3) ( n ) − F (cid:2) | γ N | m − γ N (cid:3) ( n ) (cid:12)(cid:12)(cid:12) ≤ X Γ m − ( n ) | n j |≤ M max j | n j |≥ N m − Y j =1
11 + | n j | ≤ d m,θ N , | n | ) − θ , we can estimate the first term on the right-hand side of (3.12) by (3.8). (cid:3) Next, we use the Wiener chaos estimate (Lemma 2.6) to extend Lemma 3.3 for any finite p ≥ NVARIANT GIBBS MEASURES FOR THE 2- d DEFOCUSING NLS 25
Corollary 3.4.
Let m ≥ be an integer. Then, for any θ > , there exists C m,θ > suchthat kh F N ( u ) , e n i L x k L p ( µ ) ≤ C m,θ ( p − m − | n | ) (1 − θ ) (3.13) for any n ∈ Z and any N ∈ N . Moreover, given positive ε < and any < θ ≤ − ε ,there exists C m,θ,ε > such that kh F M ( u ) − F N ( u ) , e n i L x k L p ( µ ) ≤ C m,θ,ε ( p − m − N ε (1 + | n | ) (1 − θ − ε ) (3.14) for any n ∈ Z and any M ≥ N ≥ .Proof. Let m ≥ p >
2. From (3.3) with (1.22), we have F N ( u ) = | u | m − u + m − X ℓ =0 a m,ℓ,N | u | ℓ − u. Recalling (2.29) and (2.30), we have h F N ( u ) , e n i L x = m X ℓ =0 a m,ℓ,N X Γ ℓ − ( n ) | n j |≤ N ℓ − Y j =1 g ∗ n j p | n j | . (3.15)Noting that the right-hand side of (3.15) is a sum of polynomials of degree (at most)2 m − { g n } n ∈ Z , the bound (3.13) follows from Lemma 3.3 and 2.6. The proof of (3.14)is analogous and we omit the details. (cid:3) Finally, we present the proof of Proposition 1.3.
Proof of Proposition 1.3.
Let s <
0. Choose sufficiently small θ > s + θ < p ≥
2. Then, it follows from Minkowski’s integral inequality and (3.13) that (cid:13)(cid:13) k F N ( u ) k H s (cid:13)(cid:13) L p ( µ ) ≤ (cid:18) X n ∈ Z h n i s kh F N ( u ) , e n i L x k L p ( µ ) (cid:19) . ( p − m − (cid:18) X n ∈ Z h n i − θ +2 s (cid:19) ≤ C m,p < ∞ since s + θ <
0. Similarly, given ε > s + ε <
0, choose sufficiently small θ > s + θ + ε <
0. Then, from (3.14), we have (cid:13)(cid:13) k F M ( u ) − F N ( u ) k H s (cid:13)(cid:13) L p ( µ ) ≤ (cid:18) X n ∈ Z h n i s kh F M ( u ) − F N ( u ) , e n i L x k L p ( µ ) (cid:19) . ( p − m − N ε (cid:18) X n ∈ Z h n i − θ +2 ε +2 s (cid:19) . ( p − m − N ε since s + θ + ε <
0. This proves (1.37). (cid:3) Extension to 2- d manifolds and domains in R Let ( M , g ) be a two-dimensional compact Riemannian manifold without boundary or abounded domain in R . In this section, we discuss the extensions of Propositions 1.1, 1.2,and 1.3 to M .Let { ϕ n } n ∈ N be an orthonormal basis of L ( M ) consisting of eigenfunctions of − ∆ g (with the Dirichlet or Neumann boundary condition when M is a domain in R ) withthe corresponding eigenvalues { λ n } n ∈ N , which we assume to be arranged in the increasingorder. Then, by Weyl’s asymptotics, we have λ n ≈ n . (4.1)See, for example, [45, Chapter 14].Let { g n ( ω ) } n ∈ N be a sequence of independent standard complex-valued Gaussian randomvariables on a probability space (Ω , F , P ). We define the Gaussian measure µ as the inducedprobability measure under the map: ω ∈ Ω u ( x ) = u ( x ; ω ) = X n ∈ N g n ( ω )(1 + λ n ) ϕ n ( x ) . (4.2)Note that all the results in Sections 2 and 3 still hold true in this general context withexactly the same proofs, except for Lemma 2.5 and Lemma 3.3, where we used standardFourier analysis on T . In the following, we will instead use classical properties of thespectral functions of the Laplace-Beltrami operator.Let us now define the Wick renormalization in this context. Let u be as in (4.2). Given N ∈ N , we define the projector P N by u N = P N u = X λ n ≤ N b u ( n ) ϕ n . We also define σ N by σ N ( x ) = E [ | u N ( x ) | ] = X λ n ≤ N | ϕ n ( x ) | λ n . log N, (4.3)where the last inequality follows from [12, Proposition 8.1] and Weyl’s law (4.1). Unlike σ N defined in (1.25) for the flat torus T , the function σ N defined above depends on x ∈ M .Note that σ N ( x ) > x ∈ M . The Wick ordered monomial : | u N | m : is then definedby : | u N | m : = ( − m m ! · L m ( | u N | ; σ N ) . (4.4)By analogy with (2.19) and (2.20) we define η N ( x )( · ) := 1 σ N ( x ) X λ n ≤ N ϕ n ( x ) p λ n ϕ n ( · ) , (4.5) γ N ( x, y ) := X λ n ≤ N ϕ n ( x ) ϕ n ( y )1 + λ n , (4.6)for x, y ∈ M . We simply set γ = γ ∞ when N = ∞ . NVARIANT GIBBS MEASURES FOR THE 2- d DEFOCUSING NLS 27
From the definition (4.3) of σ N , we have k η N ( x ) k L ( M ) = 1 for all x ∈ M . Moreover, wehave h η M ( x ) , η N ( y ) i L ( M ) = 1 σ M ( x ) σ N ( y ) γ N ( x, y ) (4.7)for all x, y ∈ M and M ≥ N .We now introduce the spectral function of the Laplace-Beltrami operator on M as π j ( x, y ) = X λ n ∈ ( j − ,j ] ϕ n ( x ) ϕ n ( y ) , for x, y ∈ M and j ∈ Z ≥ . From [39, (1.3) and (1.5) with q = ∞ ], we have the bound π j ( x, x ) ≤ C ( j + 1), uniformly in x ∈ M . Therefore, by Cauchy-Schwarz inequality, weobtain | π j ( x, y ) | ≤ X λ n ∈ ( j − ,j ] | ϕ n ( x ) || ϕ n ( y ) | ≤ C ( j + 1) , (4.8)uniformly in x, y ∈ M .Let σ be a weighted counting measure on Z ≥ defined by σ = P ∞ j =0 ( j + 1) δ j , where δ j is the Dirac delta measure at j ∈ Z ≥ . We define the operator L by L : c = { c j } ∞ j =0 ∞ X j =0 c j π j . Then, we have the following boundedness of the operator L . Lemma 4.1.
Let ≤ q ≤ . Then, the operator L defined above is continuous from ℓ q ( Z ≥ , σ ) into L q ′ ( M ) . Here, q ′ denotes the H¨older conjugate of q .Proof. By interpolation, it is enough to consider the endpoint cases q = 1 and q = 2. • Case 1: q = 1. Assume that c ∈ ℓ ( Z ≥ , σ ). Then, from (4.8), we get | L ( c )( x, y ) | ≤ ∞ X j =0 | c j || π j ( x, y ) | ≤ C ∞ X j =0 ( j + 1) | c j | = k c k ℓ ( σ ) . for all x, y ∈ M . This implies the result for q = 1. • Case 2: q = 2. Assume that c ∈ ℓ ( Z ≥ , σ ). By the orthogonality of the eigenfunctions ϕ n , we have ˆ M | L ( c )( x, y ) | dx = ∞ X j =0 | c j | π j ( y, y ) . (4.9)From (4.8) and (4.9), we deduce that ˆ M | L ( c )( x, y ) | dxdy ≤ C ∞ X j =0 ( j + 1) | c j | = k c k ℓ ( σ ) . This implies the result for q = 2. (cid:3) Next, we extend the definition of γ N to general values of s : γ s,N ( x, y ) := X λ n ≤ N ϕ n ( x ) ϕ n ( y )(1 + λ n ) s for x, y ∈ M . When N = ∞ , we simply set γ s = γ s, ∞ as before. Note that when s = 2, γ ,N and γ correspond to γ N and γ defined in (4.6). Lemma 4.2.
Let s > . Then, the sequence { γ s,N } N ∈ N converges to γ s in L p ( M ) for all ≤ p < − s when s ≤ and ≤ p ≤ ∞ when s ≥ . Moreover, for the same range of p ,there exist C > and κ > such that k γ s,M − γ s,N k L p ( M ) ≤ CN κ , (4.10) for all M ≥ N ≥ .Proof. Given M ≥ N ≥
1, define α N,M ( x, y ) and β N,M ( x, y ) by α N,M ( x, y ) : = γ s,M ( x, y ) − γ s,N ( x, y )= X N<λ n ≤ M ϕ n ( x ) ϕ n ( y )(1 + λ n ) s = M X j = N +1 X λ n ∈ ( j − ,j ] ϕ n ( x ) ϕ n ( y )(1 + λ n ) s (4.11)and β N,M ( x, y ) : = M X j = N +1 j ) s X λ n ∈ ( j − ,j ] ϕ n ( x ) ϕ n ( y ) = M X j = N +1 π j ( x, y )(1 + j ) s . Let us first estimate the difference α N,M − β N,M : | α N,M ( x, y ) − β N,M ( x, y ) | ≤ M X j = N +1 X λ n ∈ ( j − ,j ] (cid:12)(cid:12)(cid:12)(cid:12) λ n ) s − j ) s (cid:12)(cid:12)(cid:12)(cid:12) | ϕ n ( x ) || ϕ n ( y ) |≤ C M X j = N +1 j s +1 X λ n ∈ ( j − ,j ] | ϕ n ( x ) || ϕ n ( y ) | . Then, by (4.8), we obtain | α N,M ( x, y ) − β N,M ( x, y ) | ≤ CN s − . (4.12)Next, we estimate β N,M . Define a sequence c = { c j } ∞ j =0 by setting c j = ( j ) s , if N + 1 ≤ j ≤ M, , otherwise.Note that c ∈ ℓ q ( N , σ ) for s < q ≤
2. Hence, it follows from Lemma 4.1 that, given any2 ≤ p < − s , there exist C > κ > k β N,M k L p ( M ) = (cid:13)(cid:13)(cid:13)(cid:13) M X j = N +1 π j (1 + j ) s (cid:13)(cid:13)(cid:13)(cid:13) L p ( M ) ≤ C (cid:18) M X j = N +1 j + 1(1 + j ) s p ′ (cid:19) p ′ ≤ CN κ . (4.13)The desired estimate (4.10) follows from (4.11), (4.12), and (4.13). (cid:3) As in the case of the flat torus, define G N , N ∈ N , by G N ( u ) = 12 m ˆ M : | P N u | m : dx. Then, we have the following extension of Proposition 1.1
NVARIANT GIBBS MEASURES FOR THE 2- d DEFOCUSING NLS 29
Proposition 4.3.
Let m ≥ be an integer. Then, { G N ( u ) } N ∈ N is a Cauchy sequence in L p ( µ ) for any p ≥ . More precisely, there exists C m > such that k G M ( u ) − G N ( u ) k L p ( µ ) ≤ C m ( p − m N (4.14) for any p ≥ and any M ≥ N ≥ . As in Section 2, we make use of the white noise functional on L ( M ). Let w ( x ; ω ) bethe mean-zero complex-valued Gaussian white noise on M defined by w ( x ; ω ) = X n ∈ N g n ( ω ) ϕ ( x ) . We then define the white noise functional W ( · ) : L ( M ) → L (Ω) by W f = h f, w ( ω ) i L ( M ) = X n ∈ N b f ( n ) g n ( ω ) . (4.15)Note that Lemma 2.3 and hence Lemma 2.4 also hold on M . Proof.
Thanks to the Wiener chaos estimate (Lemma 2.6), we are reduced to the case p = 2.Given N ∈ N and x ∈ T , it follows from (4.3), (4.5), and (4.15) that u N ( x ) = σ N ( x ) u N ( x ) σ N ( x ) = σ N ( x ) W η N ( x ) . (4.16)Then, from (4.4) and (4.16), we have: | u N | m : = ( − m m ! σ mN L m (cid:18) | u N | σ N (cid:19) = ( − m m ! σ mN L m (cid:0)(cid:12)(cid:12) W η N ( x ) (cid:12)(cid:12) (cid:1) . (4.17)Hence, from (4.17), Lemma 2.4, and (4.7), we have(2 m ) k G M ( u ) − G N ( u ) k L ( µ ) = ( m !) ˆ M x ×M y ˆ Ω h σ mM ( x ) σ mM ( y ) L m (cid:0)(cid:12)(cid:12) W η M ( x ) (cid:12)(cid:12) (cid:1) L m (cid:0)(cid:12)(cid:12) W η M ( y ) (cid:12)(cid:12) (cid:1) − σ mM ( x ) σ mN ( y ) L m (cid:0)(cid:12)(cid:12) W η M ( x ) (cid:12)(cid:12) (cid:1) L m (cid:0)(cid:12)(cid:12) W η N ( y ) (cid:12)(cid:12) (cid:1) − σ mN ( x ) σ mM ( y ) L m (cid:0)(cid:12)(cid:12) W η N ( x ) (cid:12)(cid:12) (cid:1) L m (cid:0)(cid:12)(cid:12) W η M ( y ) (cid:12)(cid:12) (cid:1) + σ mN ( x ) σ mN ( y ) L m (cid:0)(cid:12)(cid:12) W η N ( x ) (cid:12)(cid:12) (cid:1) L m (cid:0)(cid:12)(cid:12) W η N ( y ) (cid:12)(cid:12) (cid:1)i dP dxdy = ( m !) ˆ M x ×M y (cid:2) | γ M ( x, y ) | m − | γ N ( x, y ) | m (cid:3) dxdy. The desired estimate (4.14) for p = 2 follows from H¨older’s inequality and Lemma 4.2. (cid:3) Remark 4.4.
Observe that the renormalization procedure (4.4) uses less spectral informa-tion than the one used in [12, Section 8] for the case m = 2. Namely, the approach in [12]needed an explicit expansion of the spectral function (see [12, Proposition 8.7]), but theinequality (4.8) is enough in the argument above.The function γ defined in (4.6) is the Green function of the operator 1 − ∆. It is well-known (see for example Aubin [2, Theorem 4.17]) that it enjoys the bound | γ ( x, y ) | ≤ C (cid:12)(cid:12) log( d ( x, y )) (cid:12)(cid:12) , (4.18) where d ( x, y ) is the distance on M between the points x, y ∈ M . The bound (4.18) impliesthat γ ∈ L p ( M ) for all 1 ≤ p < ∞ . However, we do not know whether γ N (which is theGreen function of a spectral truncation of 1 − ∆) satisfies a similar bound, uniformly in N .This could have given an alternative proof. We refer to [12, Remark 8.4] for a discussionon these topics.All the definitions and notations from (1.28) to (1.36) have obvious analogues in thegeneral case of the manifold M , and thus we do not redefine them here.For N ∈ N , let R N ( u ) = e − G N ( u ) = e − m ´ M : | u N | m : dx . In view of (4.3) and (4.17), the logarithmic upper bound (2.31) on − G N ( u ) also holds onthe manifold M . Hence, by proceeding as in the case of the flat torus, we have the followinganalogue of Proposition 1.2. Proposition 4.5.
Let m ≥ be an integer. Then, R N ( u ) ∈ L p ( µ ) for any p ≥ with auniform bound in N , depending on p ≥ . Moreover, for any finite p ≥ , R N ( u ) convergesto some R ( u ) in L p ( µ ) as N → ∞ . We conclude this section by the following analogue of Proposition 1.3, which enables usto define the Wick ordered nonlinearity : | u | m − u : on the manifold M . Proposition 4.6.
Let m ≥ be an integer and s < . Then, { F N ( u ) } N ∈ N defined in (1.36) and (3.3) is a Cauchy sequence in L p ( µ ; H s ( M )) for any p ≥ . More precisely, there exist κ > and C m,s,κ > such that (cid:13)(cid:13) k F M ( u ) − F N ( u ) k H s (cid:13)(cid:13) L p ( µ ) ≤ C m,s,κ ( p − m − N κ (4.19) for any p ≥ and any M ≥ N ≥ .Proof. Given
N, n ∈ N , define J N,n by J N,n = m !( m − ˆ M x ×M y | γ ,N ( x, y ) | m − γ ,N ( x, y ) ϕ n ( x ) ϕ n ( y ) dxdy. Then, proceeding as in (3.10) and (3.12) with (3.9), Lemma 3.2, and (4.7), we obtain kh F M ( u ) − F N ( u ) , ϕ n i L x k L ( µ ) = [0 ,N ] ( λ n ) (cid:0) J M,n − J N,n (cid:1) + ( N,M ] ( λ n ) J M,n for M ≥ N ≥
1. With ε = − s >
0, we then obtain (cid:13)(cid:13) k F M ( u ) − F N ( u ) k H − ε (cid:13)(cid:13) L ( µ ) = X n ≥ λ n ) ε kh F M ( u ) − F N ( u ) , ϕ n i L x k L ( µ ) = X λ n ≤ N λ n ) ε ( J M,n − J N,n ) + X N<λ n ≤ M λ n ) ε J M,n = C m ˆ M x ×M y (cid:0) | γ ,M | m − γ ,M − | γ ,N | m − γ ,N (cid:1) γ ε,N ( x, y ) dxdy + C m ˆ M x ×M y | γ ,M | m − γ ,M (cid:0) γ ε,M − γ ε,N (cid:1) ( x, y ) dxdy =: A N,M + B N,M . NVARIANT GIBBS MEASURES FOR THE 2- d DEFOCUSING NLS 31
In the following, We only bound the term B N,M , since the first term A N,M can behandled similarly. Set h∇ x i = (1 − ∆ x ) . Then, noting that h∇ x i − ε γ ε = γ ε and that h∇ x i − ε γ = γ ε , it follows from Cauchy-Schwarz inequality and the fractional Leibnizrule that B N,M = C m ˆ M x ×M y h∇ x i − ε (cid:0) | γ ,M | m − γ ,M ( x, y ) (cid:1) h∇ x i − ε ( γ ε,M − γ ε,N )( x, y ) dxdy = C m ˆ M x ×M y h∇ x i − ε (cid:0) | γ ,M | m − γ ,M ( x, y ) (cid:1) ( γ ε,M − γ ε,N )( x, y ) dxdy. ≤ C m (cid:13)(cid:13) h∇ x i − ε (cid:0) | γ ,M | m − γ ,M (cid:1)(cid:13)(cid:13) L ( M ) k γ ε,M − γ ε,N k L ( M ) . (cid:13)(cid:13) γ ε,M (cid:13)(cid:13) L pε ( M ) (cid:13)(cid:13) γ ,M (cid:13)(cid:13) m − L qε ( M ) k γ ε,M − γ ε,N k L ( M ) with p ε = − ε/ and q ε = 8( m − /ε . Hence, from Lemma 4.2 we conclude that B N,M ≤ C m,ε N κ . By estimating A N,M in an analogous manner, we obtain (cid:13)(cid:13) k F M ( u ) − F N ( u ) k H − ε (cid:13)(cid:13) L ( µ ) ≤ C m,ε N κ . (4.20)The bound (4.19) for general p ≥ (cid:3) Proof of Theorem 1.4 and Theorem 1.5
In this section, we present the proof of Theorem 1.5 on a manifold M (which containsa particular case of the flat torus stated in Theorem 1.4). Fix an integer m ≥ s < P (2 m )2 ,N are invariant underits dynamics. Then, we construct a sequence { ν N } N ∈ N of probability measures on space-time functions such that their marginal distributions at time t are precisely given by thetruncated Gibbs measures P (2 m )2 ,N . In Subsection 5.2, we prove a compactness propertyof { ν N } N ∈ N so that ν N converges weakly up to a subsequence. In Subsection 5.3, bySkorokhod’s theorem (Lemma 5.7), we upgrade this weak convergence of ν N to almost sureconvergence of new C ( R ; H s )-valued random variables, whose laws are given by ν N , andcomplete the proof of Theorem 1.5.5.1. Extending the truncated Gibbs measures onto space-time functions.
Recallthat P N is the spectral projector onto the frequencies (cid:8) n ∈ N : λ n ≤ N (cid:9) . Consider thetruncated Wick ordered NLS: i∂ t u N + ∆ u N = P N (cid:0) : | P N u N | m − P N u N : (cid:1) . (5.1)We first prove global well-posedness of (5.1) and invariance of the truncated Gibbs mea-sure P (2 m )2 ,N defined in (1.30): dP (2 m )2 ,N = Z − N R N ( u ) dµ = Z − N e − m ´ M : | u N | m : dx dµ. Lemma 5.1.
Let N ∈ N . Then, the truncated Wick ordered NLS (5.1) is globally well-posedin H s ( M ) . Moreover, the truncated Gibbs measure P (2 m )2 ,N is invariant under the dynamicsof (5.1) .Proof. We first prove global well-posedness of the truncated Wick ordered NLS (5.1). Given N ∈ N , let v N = P N u N . Then, (5.1) can be decomposed into the nonlinear evolutionequation for v N on the low frequency part { λ n ≤ N } : i∂ t v N + ∆ v N = P N (cid:0) : | v N | m − v N : (cid:1) (5.2)and a linear ODE for each high frequency λ n > N : i∂ t c u N ( n ) = λ n c u N ( n ) . (5.3)As a linear equation, any solution c u N ( n ) to (5.3) exists globally in time. By viewing (5.2)on the Fourier side, we see that (5.2) is a finite dimensional system of ODEs of dimension d N = (cid:8) n : λ n ≤ N (cid:9) , where the vector field depends smoothly on (cid:8) c u N ( n ) (cid:9) λ n ≤ N . Hence,by the Cauchy-Lipschitz theorem, we obtain local well-posedness of (5.2).With (3.3) we have ddt ˆ M | v N | dx = 2 Re ˆ M ∂ t v N v N dx = − (cid:18) i ˆ M |∇ v N | dx (cid:19) − − m +1 ( m − σ m − N Re (cid:18) i ˆ M L (1) m − (cid:16) | v N | σ N (cid:17) | v N | dx (cid:19) = 0 . In particular, this shows that the Euclidean norm (cid:13)(cid:13) { c v N ( n ) } λ n ≤ N (cid:13)(cid:13) C dN = (cid:18) X λ n ≤ N | c v N ( n ) | (cid:19) = (cid:18) ˆ M | v N | dx (cid:19) is conserved under (5.2). This proves global existence for (5.2) and hence for the truncatedWick ordered NLS (5.1).As in (1.34), write P (2 m )2 ,N = b P (2 m )2 ,N ⊗ µ ⊥ N . On the one hand, the Gaussian measure µ ⊥ N onthe high frequencies { λ n > N } is clearly invariant under the linear flow (5.3). On the otherhand, noting that (5.2) is the finite dimensional Hamiltonian dynamics corresponding to H N Wick ( v N ) with H N Wick ( v N ) = 12 ˆ M |∇ v N | dx + 12 m ˆ M : | v N | m : dx, we see that b P (2 m )2 ,N is invariant under (5.2). Therefore, the truncated Gibbs measure P (2 m )2 ,N is invariant under the dynamics of (5.1). (cid:3) Let Φ N : H s ( M ) → C ( R ; H s ( M )) be the solution map to (5.1) constructed inLemma 5.1. For t ∈ R , we use Φ N ( t ) : H s ( M ) → H s ( M ) to denote the map defined NVARIANT GIBBS MEASURES FOR THE 2- d DEFOCUSING NLS 33 by Φ N ( t )( φ ) = (cid:0) Φ N ( φ ) (cid:1) ( t ). We endow C ( R ; H s ( M )) with the compact-open topology.Namely, we can view C ( R ; H s ( M )) as a Fr´echet space endowed with the following metric: d ( u, v ) = ∞ X j =1 j k u − v k C ([ − j,j ]; H s ) k u − v k C ([ − j,j ]; H s ) . Under this topology, a sequence { u n } n ∈ N ⊂ C ( R ; H s ( M )) converges if and only if it con-verges uniformly on any compact time interval. Then, it follows from the local Lipschitzcontinuity of Φ N ( · ) that Φ N is continuous from H s ( M ) into C ( R ; H s ( M )). We now extend P (2 m )2 ,N on H s to a probability measure ν N on C ( R ; H s ( M )) by setting ν N = P (2 m )2 ,N ◦ Φ − N . Namely, ν N is the induced probability measure of P (2 m )2 ,N under the map Φ N . In particular,we have ˆ C ( R ; H s ) F ( u ) dν N ( u ) = ˆ H s F (Φ N ( φ )) dP (2 m )2 ,N ( φ ) (5.4)for any measurable function F : C ( R ; H s ( M )) → R .5.2. Tightness of the measures ν N . In the following, we prove that the sequence { ν N } N ∈ N of probability measures on C ( R ; H s ( M )) is precompact. Recall the followingdefinition of tightness of a sequence of probability measures. Definition 5.2.
A sequence { ρ n } n ∈ N of probability measures on a metric space S is tight if, for every ε >
0, there exists a compact set K ε such that ρ n ( K cε ) ≤ ε for all n ∈ N .It is well known that tightness of a sequence of probability measures is equivalent to pre-compactness of the sequence. See [3]. Lemma 5.3 (Prokhorov’s theorem) . If a sequence of probability measures on a metric space S is tight, then there is a subsequence that converges weakly to a probability measure on S . The following proposition shows that the family { ν N } N ∈ N is tight and hence, up to asubsequence, it converges weakly to some probability measure ν on C ( R ; H s ). Proposition 5.4.
Let s < . Then, the family { ν N } N ∈ N of the probability measures on C ( R ; H s ( M )) is tight. The proof of Proposition 5.4 is similar to that of [12, Proposition 4.11]. While [12, Proposi-tion 4.11] proves the tightness of { ν N } N ∈ N restricted to [ − T, T ] for each
T >
0, we directlyprove the tightness of { ν N } N ∈ N on the whole time interval.In the following, we first state several lemmas. We present the proof of Proposition 5.4at the end of this subsection. For simplicity of presentation, we use the following notations.Given T >
0, we write L pT H s for L p ([ − T, T ]; H s ). We use a similar abbreviation for otherfunction spaces in time. Let ρ be a probability measure on H s . With a slight abuse ofnotation, we use L p ( ρ ) H s to denote k φ k L p ( ρ ) H s = (cid:13)(cid:13) k φ k H s (cid:13)(cid:13) L p ( ρ ) . The first lemma provides a control on the size of random space-time functions. Theinvariance of P (2 m )2 ,N under the dynamics of (5.1) plays an important role. Lemma 5.5.
Let s < and p ≥ . Then, there exists C p > such that (cid:13)(cid:13) k u k L pT H s (cid:13)(cid:13) L p ( ν N ) ≤ C p T p , (5.5) (cid:13)(cid:13) k u k W ,pT H s − (cid:13)(cid:13) L p ( ν N ) ≤ C p T p , (5.6) uniformly in N ∈ N .Proof. By Fubini’s theorem, the definition (5.4), the invariance of P (2 m )2 ,N (Lemma 5.1), andH¨older’s inequality, we have (cid:13)(cid:13) k u k L pT H s (cid:13)(cid:13) L p ( ν N ) = (cid:13)(cid:13) k Φ N ( t )( φ ) k L pT H s (cid:13)(cid:13) L p ( P (2 m )2 ,N ) = (cid:13)(cid:13) k Φ N ( t )( φ ) k L p ( P (2 m )2 ,N ) H s (cid:13)(cid:13) L pT = (2 T ) p k φ k L p ( P (2 m )2 ,N ) H s ≤ (2 T ) p k R N k L p ( µ ) k φ k L p ( µ ) H s . (5.7)Then, (5.5) follows from (5.7) with Proposition 4.5, (4.2), and Lemma 2.6.From (5.1) and the definition of F N , we have (cid:13)(cid:13) k ∂ t u k L pT H s − (cid:13)(cid:13) L p ( ν N ) ≤ (cid:13)(cid:13) k u k L pT H s (cid:13)(cid:13) L p ( ν N ) + (cid:13)(cid:13) k F N ( u ) k L pT H s − (cid:13)(cid:13) L p ( ν N ) . (5.8)The first term is estimated by (5.5). Proceeding as in (5.7) with Propositions 4.5 and 4.6,we have (cid:13)(cid:13) k F N ( u ) k L pT H s − (cid:13)(cid:13) L p ( ν N ) ≤ (2 T ) p k R N k L p ( µ ) k F N ( φ ) k L p ( µ ) H s − ≤ C p T p . This proves (5.6). (cid:3)
Recall the following lemma on deterministic functions from [12].
Lemma 5.6 ([12, Lemma 3.3]) . Let
T > and ≤ p ≤ ∞ . Suppose that u ∈ L pT H s and ∂ t u ∈ L pT H s for some s ≤ s . Then, for δ > p − ( s − s ) , we have k u k L ∞ T H s − δ . k u k − p L pT H s k u k p W ,pT H s . Moreover, there exist α > and θ ∈ [0 , such that for all t , t ∈ [ − T, T ] , we have k u ( t ) − u ( t ) k H s − δ . | t − t | α k u k − θL pT H s k u k θW ,pT H s . We are now ready to present the proof of Proposition 5.4.
Proof of Proposition 5.4.
Let s < s < s <
0. For α ∈ (0 , C αT H s = C α ([ − T, T ]; H s ( M )) defined by the norm k u k C αT H s = sup t ,t ∈ [ − T,T ] t = t k u ( t ) − u ( t ) k H s | t − t | α + k u k L ∞ T H s . It follows from the Arzel`a-Ascoli theorem that the embedding C αT H s ⊂ C T H s is compactfor each T > p ≫ k u k C αT H s . k u k − θL pT H s k u k θW ,pT H s − . k u k L pT H s + k u k W ,pT H s − (5.9)for some α ∈ (0 , (cid:13)(cid:13) k u k C αT H s (cid:13)(cid:13) L p ( ν N ) ≤ C p T p . (5.10) NVARIANT GIBBS MEASURES FOR THE 2- d DEFOCUSING NLS 35
For j ∈ N , let T j = 2 j . Given ε >
0, define K ε by K ε = (cid:8) u ∈ C ( R ; H s ) : k u k C αTj H s ≤ c ε − T p j for all j ∈ N (cid:9) . Then, by Markov’s inequality with (5.10) and choosing c > ν N ( K cε ) ≤ c − C εT − − p j (cid:13)(cid:13) k u k C αTj H s (cid:13)(cid:13) L p ( ν N ) ≤ c − C p ε ∞ X j =1 T − j = c − C p ε < ε. Hence, it remains to prove that K ε is compact in C ( R ; H s ) endowed with the compact-open topology. Let { u n } n ∈ N ⊂ K ε . By the definition of K ε , { u n } n ∈ N is bounded in C αT j H s for each j ∈ N . Then, by a diagonal argument, we can extract a subsequence { u n ℓ } ℓ ∈ N convergent in C αT j H s for each j ∈ N . In particular, { u n ℓ } ℓ ∈ N converges uniformly in H s on any compact time interval. Hence, { u n ℓ } ℓ ∈ N converges in C ( R ; H s ) endowed with thecompact-open topology. This proves that K ε is compact in C ( R ; H s ). (cid:3) Proof of Theorem 1.5.
It follows from Proposition 5.4 and Lemma 5.3 that, passingto a subsequence, ν N j converges weakly to some probability measure ν on C ( R ; H s ( M ))for any s <
0. The following Skorokhod’s theorem tells us that, by introducing a newprobability space ( e Ω , F , e P ) and a sequence of new random variables f u N with the samedistribution ν N , we can upgrade this weak convergence to almost sure convergence of f u N .See [3]. Lemma 5.7 (Skorokhod’s theorem) . Let S be a complete separable metric space. Supposethat ρ n are probability measures on S converging weakly to a probability measure ρ . Then,there exist random variables X n : e Ω → S with laws ρ n and a random variable X : e Ω → S with law ρ such that X n → X almost surely. By Lemma 5.7, there exist another probability space ( e Ω , e F , e P ), a sequence (cid:8)g u N j (cid:9) j ∈ N of C ( R ; H s )-valued random variables, and a C ( R ; H s )-valued random variable u such that L (cid:0)g u N j (cid:1) = L ( u N j ) = ν N , L ( u ) = ν, (5.11)and g u N j converges to u in C ( R ; H s ) almost surely with respect to e P .Next, we determine the distributions of these random variables at a given time t . ByLemma 5.1, we have L ( u N j ( t )) = P (2 m )2 ,N j (5.12)for each t ∈ R . Lemma 5.8.
Let e u N j and u be as above. Then, we have L (cid:0)g u N j ( t ) (cid:1) = P (2 m )2 ,N j and L ( u ( t )) = P (2 m )2 for any t ∈ R .Proof. Fix t ∈ R . Let R t : C ( R ; H s ) → H s be the evaluation map defined by R t ( v ) = v ( t ).Note that R t is continuous. From (5.12), we have P (2 m )2 ,N j = ν N j ◦ R − t . (5.13) Denoting by ν tN j the distribution of g u N j ( t ), it follows from (5.11) and (5.13) that ν tN j = ν N j ◦ R − t = P (2 m )2 ,N j . (5.14)Since g u N j converges to u in C ( R ; H s ) almost surely with respect to e P , g u N j ( t ) convergesto u ( t ) in H s almost surely. Then, denoting by ν t the distribution of u ( t ), it follows fromthe dominated convergence theorem with (5.14) that ν t ( A ) = ˆ { u ( t )( ω ) ∈ A } d e P = lim j →∞ ˆ (cid:8) g u Nj ( t )( ω ) ∈ A (cid:9) d e P = lim j →∞ P (2 m )2 ,N j ( A ) . (5.15)Therefore, from (5.15) and Proposition 4.5, we conclude that L ( u ( t )) = P (2 m )2 . (cid:3) Finally, we show that the random variable u is indeed a global-in-time distributionalsolution to the Wick ordered NLS i∂ t u + ∆ u = : | u | m − u : , ( t, x ) ∈ R × M . (5.16)Then, Theorem 1.5 follows from Lemmas 5.8 and 5.9. Lemma 5.9.
Let g u N j and u be as above. Then, g u N j and u are global-in-time distributionalsolutions to the truncated Wick ordered NLS (5.1) for each j ∈ N and to the Wick orderedNLS (5.16) , respectively.Proof. For j ∈ N , define the D ′ t,x -valued random variable X j by X j = i∂ t u N j + ∆ u N j − P N j (cid:0) : | P N j u N j | m − P N j u N j : (cid:1) . Here, D ′ t,x = D ′ ( R × M ) denotes the space of space-time distributions on R × M . Wedefine e X j for g u N j in an analogous manner. Since u N j is a solution to (5.1), we see that L D ′ t,x ( X j ) = δ , where δ denotes the Dirac delta measure. By (5.11), we also have L D ′ t,x ( e X j ) = δ , for each j ∈ N . In particular, g u N j is a global-in-time distributional solution to the truncatedWick ordered NLS (5.1) for each j ∈ N , i.e. i∂ t g u N j + ∆ g u N j = P N j (cid:0) : | P N j g u N j | m − P N j g u N j : (cid:1) in the distributional sense, almost surely with respect to e P .In view of the almost sure convergence of g u N j to u in C ( R ; H s ), we have i∂ t g u N j + ∆ g u N j −→ i∂ t u + ∆ u in D ′ ( R × M ) as j → ∞ , almost surely with respect to e P . Next, we show the almost sureconvergence of F N j (cid:0)g u N j (cid:1) to F ( u ) = : | u | m − u :. For simplicity of notation, let F j = F N j and u j = g u N j . Given M ∈ N , write F j ( u j ) − F ( u ) = (cid:0) F j ( u j ) − F ( u j ) (cid:1) + (cid:0) F ( u j ) − F M ( u j ) (cid:1) + (cid:0) F M ( u j ) − F M ( u ) (cid:1) + (cid:0) F M ( u ) − F ( u ) (cid:1) . (5.17) NVARIANT GIBBS MEASURES FOR THE 2- d DEFOCUSING NLS 37
Then, for each fixed M ≥
1, it follows from the almost sure convergence of g u N j to u in C ( R ; H s ) and the continuity of F M that the third term on the right-hand side of (5.17)converges to 0 in C ( R ; H s ) as j → ∞ , almost surely with respect to e P .Fix T > s < −
1. Arguing as in (5.7) with Proposition 4.6, we have (cid:13)(cid:13) k F ( u j ) − F M ( u j ) k L T H s (cid:13)(cid:13) L ( ν Nj ) = (cid:13)(cid:13) k F (Φ N j φ ) − F M (Φ N j φ ) k L ( P (2 m )2 ,Nj ) H s (cid:13)(cid:13) L T = (2 T ) k F ( φ ) − F M ( φ ) k L ( P (2 m )2 ,Nj ) H s . T k R N j k L ( µ ) k F ( φ ) − F M ( φ ) k L ( µ ) H s ≤ CT M − ε , (5.18)for some small ε >
0, uniformly in j ∈ N . In the third step, we used the fact that Z N & Z N = k R N ( u ) k L ( ρ ) → k R ( u ) k L ( ρ ) > N → ∞ . The fourthterm on the right-hand side of (5.17) can be treated in an analogous manner. Proceedingas in (5.18), we obtain (cid:13)(cid:13) k F j ( u j ) − F ( u j ) k L T H s (cid:13)(cid:13) L ( ν Nj ) ≤ (2 T ) k R N j k L ( µ ) k F j ( φ ) − F ( φ ) k L ( µ ) H s ≤ CT N − εj . Putting everything together, we conclude that, after passing to a subsequence, F j ( u j )converges to F ( u ) in L ([ − T, T ]; H s ) almost surely with respect to e P . Since the choiceof T > T ℓ = 2 ℓ , ℓ ∈ N . Thus, for each ℓ ≥
2, we obtain a set Ω ℓ ⊂ Ω ℓ − of full measure such that asubsequence F j ( ℓ ) ( u j ( ℓ ) )( ω ) of F j ( ℓ − ( u j ( ℓ − ) from the previous step converges to F ( u )( ω )in L ([ − T ℓ , T ℓ ]; H s ) for all ω ∈ Ω ℓ . Then, by a diagonal argument, passing to a subsequence, F j ( u j ) converges to F ( u ) in L H s almost surely with respect to e P . In particular, up toa subsequence, F j ( u j ) converges to F ( u ) in D ′ ( R × M ) almost surely with respect to e P .Therefore, u is a global-in-time distributional solution to (5.16). (cid:3) Appendix A. Example of a concrete combinatorial argument: the case M = T and m = 3In this appendix, we present a concrete combinatorial computation on the Fourier sidefor the proof of Proposition 1.1 when m = 3. The aim of this appendix is to convincereaders of increasing combinatorial complexity in m . Compare the m = 3 case presentedhere with the m = 2 case in [7]. This shows that the use of the white noise functional isessential in establishing our result for general m ≥ G N ( u ) be as in (1.26). For simplicity, we show that G N ( u ) is uniformly boundedin L ( µ ). Namely, we prove k G N ( u ) k L ( µ ) ≤ C < ∞ (A.1)independently of N ∈ N . Then, a small modification yields Proposition 1.1 for p = 2. Thegeneral case follows from the p = 2 case and the Wiener chaos estimate (Lemma 2.6). From (1.21), (1.23), (1.24), and (1.26) with (1.10), we have6 G N ( u ) = ˆ T : | u N | : dx = ˆ T | u N | − σ N | u N | + 18 σ N | u N | − σ N dx = X Γ (0) | n j |≤ N Y j =1 g ∗ n j p | n j | − (cid:18) X | n |≤ N
11 + | n | (cid:19)(cid:18) X Γ (0) | n j |≤ N Y j =1 g ∗ n j p | n j | (cid:19) + 18 (cid:18) X | n |≤ N
11 + | n | (cid:19) (cid:18) X | n |≤ N | g n | | n | (cid:19) − (cid:18) X | n |≤ N
11 + | n | (cid:19) =: I + II + III + IV , (A.2)where σ N is as in (1.25) and Γ k (0) and g ∗ n are as in (2.29) and (2.30), respectively.The basic idea is to regroup the terms in (A.2) by introducing some factorizations,and separately estimate each contribution. Given ℓ ∈ N , we say that we have a pair in n = ( n , . . . , n ℓ ) ∈ Γ ℓ (0) if n j = n j ′ for some odd j and even j ′ .Let us first consider I . Given n ∈ Γ (0), there are three cases: (i) no pair, (ii) 1 pair,and (iii) 3 pairs. Thus, write I as I = I + I + I , corresponding to the three cases: (i) no pair, (ii) 1 pair, and (iii) 3 pairs, respectively. Forsimplicity of notation, we may drop the frequency restriction | n | ≤ N in the following butit is understood that all the summations are over {| n | ≤ N } . • Case 1:
No pair. In this case, we can easily estimate the contribution from I by k I k L ( µ ) . (cid:18) X Γ (0) 6 Y j =1
11 + | n j | (cid:19) ≤ C < ∞ . (A.3) • Case 2: { n , n , n } and { n , n , n } . Thus, we haveI = 9 (cid:18) X | g n | | n | (cid:19)(cid:18) X Γ (0) n = n ,n Y j =1 g ∗ n j p | n j | (cid:19) . Combining this with II, we haveI + II = 9 (cid:18) X | g n | −
11 + | n | (cid:19)(cid:18) X Γ (0) n = n ,n Y j =1 g ∗ n j p | n j | (cid:19) − (cid:18) X
11 + | n | (cid:19)(cid:18) X | g n | | n | (cid:19) + 9 (cid:18) X
11 + | n | (cid:19)(cid:18) X | g n | (1 + | n | ) (cid:19) =: II + II + II . (A.4) NVARIANT GIBBS MEASURES FOR THE 2- d DEFOCUSING NLS 39
Note that E [ | g n | −
1] = 0. Then, by Lemma 2.6, we have k II k L ( µ ) . (cid:13)(cid:13)(cid:13)(cid:13) X | g n | −
11 + | n | (cid:13)(cid:13)(cid:13)(cid:13) L ( µ ) (cid:13)(cid:13)(cid:13)(cid:13) X Γ (0) n = n ,n Y j =1 g ∗ n j p | n j | (cid:13)(cid:13)(cid:13)(cid:13) L ( µ ) . (cid:18) X | n | ) (cid:19) (cid:18) X Γ (0) 4 Y j =1
11 + | n j | (cid:19) ≤ C < ∞ . (A.5)The terms II and II are treated with other terms in the following. • Case 3: n , n , and n : (i) n = n = n , (ii) n = n = n up to permutations, (iii) all distinct. WriteI = I + I + I , corresponding to these three cases. ◦ Subcase 3 (i): n = n = n . In this case, the contribution can be estimated by k I k L ( µ ) ≤ (cid:13)(cid:13)(cid:13)(cid:13) X | g n | (1 + | n | ) (cid:13)(cid:13)(cid:13)(cid:13) L ( µ ) . (cid:18) X | n | ) (cid:19) ≤ C < ∞ . (A.6) ◦ Subcase 3 (ii): n = n = n up to permutations. In this case, we haveI = (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) X | g n | (1 + | n | ) (cid:19)(cid:18) X m = n | g m | | m | (cid:19) = 9 (cid:18) X | g n | (1 + | n | ) (cid:19)(cid:18) X | g m | | m | (cid:19) − (cid:18) X | g n | (1 + | n | ) (cid:19) =: I + O L ( µ ) (1) . (A.7)Here, we estimated the second term as in (A.6). ◦ Subcase 3 (iii): all distinct. In this case, we haveI = 6 (cid:18) X | g n | | n | (cid:19)(cid:18) X n = n | g n | | n | (cid:19)(cid:18) X n = n ,n | g n | | n | (cid:19) = 6 (cid:18) X | g n | | n | (cid:19)(cid:18) X n = n | g n | | n | (cid:19)(cid:18) X n = n | g n | | n | (cid:19) − (cid:18) X | g n | | n | (cid:19)(cid:18) X n = n | g n | (1 + | n | ) (cid:19) = 6 (cid:18) X | g n | | n | (cid:19)(cid:18) X n = n | g n | | n | (cid:19)(cid:18) X | g n | | n | (cid:19) − (cid:18) X | g n | | n | (cid:19)(cid:18) X n = n | g n | (1 + | n | ) (cid:19) − (cid:18) X | g n | (1 + | n | ) (cid:19)(cid:18) X n = n | g n | | n | (cid:19) = 6 (cid:18) X | g n | | n | (cid:19) − (cid:18) X | g n | | n | (cid:19)(cid:18) X | g n | (1 + | n | ) (cid:19) + 12 (cid:18) X | g n | (1 + | n | ) (cid:19) =: I + I + O L ( µ ) (1) . (A.8)From (A.4), (A.7), and (A.8), we haveII + I + I = 9 (cid:18) X − | g n | | n | (cid:19)(cid:18) X | g n | (1 + | n | ) (cid:19) . Proceeding as in (A.5), we obtain k II + I + I k L ( µ ) ≤ C < ∞ . (A.9)From (A.2), (A.4), and (A.8), we haveIII + IV + II + I = 6 (cid:18) X | g n | −
11 + | n | (cid:19) . Proceeding as in (A.5), we obtain k III + IV + II + I k L ( µ ) . (cid:13)(cid:13)(cid:13)(cid:13) X | g n | −
11 + | n | (cid:13)(cid:13)(cid:13)(cid:13) L ( µ ) ≤ C < ∞ . (A.10)Finally, putting (A.2)-(A.10) together, we obtain (A.1). Remark A.1.
The above computation merely handles the nonlinear part G N ( u ) in thetruncated Wick ordered Hamiltonian. In order to prove Theorem 1.4, one still needs toestimate F N ( u ) in (1.36), which has a different combinatorial structure. For our problem,it is much more efficient to work on the physical side, using the white noise functional andthe (generalized) Laguerre polynomials. Acknowledgements.
T.O. was supported by the European Research Council (grantno. 637995 “ProbDynDispEq”). L.T. was supported by the grant “ANA´E” ANR-13-BS01-0010-03. The authors would like to thank Martin Hairer for helpful discussions. They arealso grateful to the anonymous referees for their comments.
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Tadahiro Oh, School of Mathematics, The University of Edinburgh, and The Maxwell In-stitute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings,Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom
E-mail address : [email protected] Laurent Thomann, Institut ´Elie Cartan, Universit´e de Lorraine, B.P. 70239, F-54506Vandoeuvre-l`es-Nancy Cedex, France
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