A PEDESTRIAN APPROACH TO THE INVARIANT GIBBS MEASURES FOR THE 2-d DEFOCUSING NONLINEAR SCHRÖDINGER EQUATIONS

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1 A PEDESTRIAN APPROACH TO THE INVARIANT GIBBS MEASURES FOR THE -d DEFOCUSING NONLINEAR SCHRÖDINGER EQUATIONS TADAHIRO OH AND LAURENT THOMANN Abstract. We consider the defocusing nonlinear Schrödinger equations on the twodimensional 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 renormalization in terms of the Hermite polynomials and the Laguerre polynomials and construct the Gibbs measures corresponding to the Wick ordered Hamiltonian. Then, we construct global-in-time solutions with initial data distributed according to the Gibbs measure and show that the law of the random solutions, at any time, is again given by the Gibbs measure. Contents. Introduction.. Nonlinear Schrödinger equations.. Gibbs measures.3. Wick renormalization 5.4. Invariant dynamics for the Wick ordered NLS 8. Construction of the Gibbs measures.. Hermite polynomials, Laguerre polynomials, and Wick ordering 3.. White noise functional 6.3. Wiener chaos estimates 9.4. Nelson s estimate 0 3. On the Wick ordered nonlinearity 4. Extension to -d manifolds and domains in R 6 5. Proof of Theorem.4 and Theorem Extending the truncated Gibbs measures onto space-time functions Tightness of the measures ν N Proof of Theorem.5 35 Appendix A. Example of a concrete combinatorial argument: the case M = T and m = 3 37 References Mathematics Subject Classification. 35Q55. Key words and phrases. nonlinear Schrödinger equation; Gibbs measure; Wick ordering; Hermite polynomial; Laguerre polynomial; white noise functional.

2 T. OH AND L. THOMANN. Introduction.. Nonlinear Schrödinger equations. Let M, g be a two-dimensional compact Riemannian manifold without boundary or a bounded domain in R. We consider the defocusing nonlinear Schrödinger equation NLS: { i t u + g u = u k u t, x R M,. u t=0 = φ, where g stands for the Laplace-Beltrami operator on M, k = m 4 is an even integer, and the unknown is the function u : R M C. The aim of this article is to give a pedagogic and self-contained presentation on the construction of an invariant Gibbs measure for a renormalized version of.. In particular, we present an elementary Fourier analytic approach to the problem in the hope that this will be accessible to readers in particular those in dispersive PDEs without prior knowledge in quantum 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/πZ. Namely, we consider { i t u + u = u k u t, x R T.. u t=0 = φ, The equation. is known to possess the following Hamiltonian structure: t u = i H u,.3 where H = Hu is the Hamiltonian given by Hu = u dx + u k dx..4 T k T Moreover, the mass Mu = u dx T is also conserved under the dynamics of.... Gibbs measures. Given a Hamiltonian flow on R n : {ṗj = H q j q j = H p j.5 with Hamiltonian Hp, q = Hp,, p n, q,, q n, Liouville s theorem states that the Lebesgue measure n j= dp jdq j on R n is invariant under the flow. Then, it follows from the conservation of the Hamiltonian H that the Gibbs measures e βhp,q n j= dp jdq j are invariant under the dynamics of.5. Here, β > 0 denotes the reciprocal temperature. NLS. is a Hamiltonian PDE, where the Hamiltonian is conserved under its dynamics. Thus by drawing an analogy to the finite dimensional setting, one expects the Gibbs measure with the exception of the Wiener chaos estimate Lemma.6.

3 INVARIANT GIBBS MEASURES FOR THE -d DEFOCUSING NLS 3 of the form: dp m = Z exp βhudu.6 to be invariant under the dynamics of.. 3 As it is,.6 is merely a formal expression and we need to give a precise meaning. From.4, we can write.6 as dp m = Z e m u mdx e u dx du..7 This motivates us to define the Gibbs measure P m as an absolutely continuous probability measure with respect to the following massless Gaussian free field: dµ = Z exp u dx du. In order to avoid the problem at the zeroth frequency, we instead consider the following massive Gaussian free field: dµ = Z e u dx u dx du..8 in the following. Note that this additional factor replaces Hu by Hu Mu in the formal definition.6 of P m. In view of the conservation of mass, however, we still expect P m to be invariant if we can give a proper meaning to P m. It is well known that µ in.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 < 0 with the covariance operator Q s = Id +s. Recall that a covariance operator Q of a mean-zero probability measure µ on a Hilbert space H is a trace class operator, satisfying f, u H h, u H dµu = Qf, h H.9 H for all f, h H. We can also view the Gaussian measure µ as the induced probability measure under the map: 4 ω Ω ux = ux; ω = n Z g n ω + n ein x,.0 where {g n } n Z is a sequence of independent standard 5 complex-valued Gaussian random variables on a probability space Ω, F, P. Namely, functions under µ are represented by the random Fourier series given in.0. Note that the random function u in.0 is in H s T \ L T for any s < 0, almost surely. Thus, µ is a Gaussian probability measure on H s T for any s < 0. Moreover, it is easy to see that.9 with H = H s T In the following, Z, ZN, and etc. denote various normalizing constants so that the corresponding measures are probability measures when appropriate. 3 For simplicity, we set β = in the following. See [33] for a discussion on the Gibbs measures and different values of β > 0. 4 Strictly speaking, there is a factor of π in.0. For simplicity of the presentation, however, we drop such harmless π hereafter. 5 Namely, gn has mean 0 and Varg n =.

4 4 T. OH AND L. THOMANN Q s = Id +s, s < 0, follows from.0. Indeed, we have [ f, u H s h, u H sdµu = E H s n Z fng n ω n s m Z ] ĥmg m ω m s = n Z fn ĥn n 4s = Q s f, h H s.. Here, = +. Note that the second equality in. holds even for s 0. For s 0, however, µ is not a probability measure on H s T. Indeed, we have µl T = 0. The next step is to make sense of the Gibbs measure P m in.7. First, let us briefly go over the situation when d =. In this case, µ defined by.8 is a probability measure on H s T, s <. Then, it follows from Sobolev s inequality that T ux; ω k dx is finite almost surely. Hence, for any k >, the Gibbs measure: dp k = Z e k T u kdx dµ. is a probability measure on H s T, s <, absolutely continuous with respect to µ. Moreover, by constructing global-in-time dynamics in the support of P k, Bourgain [6] proved that the Gibbs measure P k is invariant under the dynamics of the defocusing NLS for k >. Here, by invariance, we mean that P k k Φ ta = P A.3 for any measurable set A B H s T and any t R, where Φt : u 0 H s T ut = Φtu 0 H s T is a well-defined solution map, at least almost surely with respect to P k. McKean [5] gave an independent proof of the invariance of the Gibbs measure when k = 4, relying on a probabilistic argument. See Remark.7 below for the discussion on the focusing case. Over the recent years, there has been a significant progress in the study of invariant Gibbs measures for Hamiltonian PDEs. See, for example, [4, 6, 7, 5, 7, 8, 44, 4, 4, 3, 5, 3, 3, 43, 40, 30, 34,, 0, 9, 37, ]. The situation for d = is entirely different. As discussed above, the random function u in.0 is not in L T almost surely. This in particular implies that ux; ω k dx =.4 T almost surely for any k. Therefore, we can not construct a probability measure of the form: dp k = Z e k T u kdx dµ..5 Thus, we are required to perform a Wick renormalization on the nonlinear part u k of the Hamiltonian. This is a well studied subject in the Euclidean quantum field theory, at least in the real-valued setting. See Simon [38] and Glimm-Jaffe [3]. Also, see Da Prato- Tubaro [8] for a concise discussion on T, where the Gibbs measures naturally appear in the context of the stochastic quantization equation.

5 INVARIANT GIBBS MEASURES FOR THE -d DEFOCUSING NLS 5.3. Wick renormalization. There are different ways to introduce the Wick renormalization. One classical way is to use the Fock-space formalism, where the Wick ordering is given as the reordering of the creation and annihilation operators. See [38, 6, 0] for more details. 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.5 below by using the Hermite polynomials and the generalized Laguerre polynomials, since this allows us to introduce only the necessary objects without introducing cumbersome notations and formalism, making our presentation accessible to readers without prior knowledge in the problem. Before we study the Wick renormalization for NLS, let us briefly discuss the Wick renormalization on T in the real-valued setting. We refer to [8] for more details. We assume that u is real-valued. Then, the random function u under µ in.8 is represented by the random Fourier series.0 conditioned that g n = g n. Given N N, let P N be the Dirichlet projection onto the frequencies { n N} and set u N = P N u, where u is as in.0. Note that, for each x T, the random variable u N x is a mean-zero real-valued Gaussian with variance σ N := E[u Nx] = log N..6 + n n N Note that σ N is independent of x T. Fix an even integer k 4. We define the Wick ordered monomial :u k N : by :u k N : = H k u N ; σ N,.7 where H k x; σ is the Hermite polynomial of degree k defined in.. Then, one can show that the limit :u k : dx = lim :u k N : dx.8 T N T exists in L p µ for any finite p. Moreover, one can construct the Gibbs measure: as the limit of dp k = Z e k T :u k : dx dµ dp k,n = Z N e k T :u k N : 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, and Nelson s estimate [8, 9]. For our problem on NLS., we need to work on complex-valued functions. In the real-valued setting, the Wick ordering was defined by the Hermite polynomials. In the complex-valued setting, we also define the Wick ordering by the Hermite polynomials, but through applying the Wick ordering the real and imaginary parts separately. Let u be as in.0. Given N N, we define u N by u N = P N u = ûne in x, n N

6 6 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 : = : Re u N + Im u N m : m m = :Re u l N l : :Im u N m l :..9 l=0 It turns out, however, that it is more convenient to work with the Laguerre polynomials in the current complex-valued setting; see Section. Recall that the Laguerre polynomials L m x are defined through the following generating function: Gt, x := tx e t t = t m L m x,.0 for t < and x R. For readers convenience, we write out the first few Laguerre polynomials in the following: m=0 L 0 x =, L x = x +, L x = x 4x +, L 3 x = 3! x3 + 9x 8x + 6, L 4 x = 4! x4 6x 3 + 7x 96x More generally, the L m are given by the formula m m l L m x = x l l.. l! Given σ > 0, we set l=0 L m x; σ := σ m L m x σ..3 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.9 as where σ N is given by : u N m : = m m! L m u N ; σ N,.4 σ N = E[ u N x ] = n N log N,.5 + n independently of x T. See Lemma. for the equivalence of.9 and.4. For N N, let G N u = : P N u m : dx..6 m T Then, we have the following proposition. Proposition.. 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 > 0 such that for any p and any M N. G M u G N u L p µ C m p m N

7 INVARIANT GIBBS MEASURES FOR THE -d DEFOCUSING NLS 7 Proposition. states that we can define the limit Gu as Gu = : u m : dx = lim m G Nu = T N m lim : P N u m : dx N T and that Gu L p µ for any finite p. This allows us to define the Wick ordered Hamiltonian: H Wick u = u dx + : u m : dx.7 T m T for an integer m. 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 m = Z e H Wicku Mu du,.8 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,.9 arising as a Hamiltonian PDE: t u = i u H Wick. In particular, we need to give a precise 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 and define the truncated Gibbs measure P m,n by dp m,n := Z N R Nudµ = Z N e m T : u N m : dx T : u N m : dx dµ,.30 corresponding to the truncated Wick ordered Hamiltonian: HWicku N = u dx + : u N m : dx..3 T m T Note that P m,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 m as a limit of P m,n. Proposition.. Let m be an integer. Then, R N u L p µ for any p with a uniform bound in N, depending on p. Moreover, for any finite p, R N u converges to some Ru in L p µ as N. In particular, by writing the limit Ru L p µ as Ru = e m T : u m : dx, Proposition. allows us to define the Gibbs measure P m in.8 by dp m = Z Rudµ = Z e m T : u m : dx dµ..3 Then, P m is a probability measure on H s T, s < 0, absolutely continuous to the Gaussian field µ. Moreover, P m m,n converges weakly to P.

8 8 T. OH AND L. THOMANN.4. Invariant dynamics for the Wick ordered NLS. In this subsection, we study the dynamical problem.9. First, we consider the Hamiltonian PDE corresponding to the truncated Wick ordered Hamiltonian H N Wick in.3: i t u N + u N = P N : PN u N m P N u N :..33 The high frequency part P N un evolves according to the linear flow, while the low frequency part P N u N evolves according to the finite dimensional system of ODEs viewed on the Fourier 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 EN = span{ein x } n >N, respectively. Then, we can write P m,n in.30 as where P m,n P m,n = P m,n µ N,.34 is the finite dimensional Gibbs measure defined by m d P,N = Ẑ N e m T : P N u N m : dx dµ N..35 Then, it is easy to see that P m,n is invariant under the dynamics of.33; see Lemma 5. below. In particular, the law of u N t is given by P m,n for any t R. For N N, define F N u by F N u = P N : PN u m P N u:..36 Then, assuming that u is distributed according to the Gaussian free field µ in.8, the following proposition lets us make sense of the Wick ordered nonlinearity : u m u : in.9 as the limit of F N u. Proposition.3. Let m be an integer and s < 0. Then, {F N u} N N is a Cauchy sequence in L p µ; H s T for any p. More precisely, given ε > 0 with s + ε < 0, there exists C m,s,ε > 0 such that F M u F N u H s L p µ C m,s,εp m N ε.37 for any p and any M N. In the real-valued setting, the nonlinearity corresponding to the Wick ordered Hamiltonian is again given by a Hermite polynomial. Indeed, from.7, we have k u N :u k N : = 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.3. The situation is slightly different in the complexvalued setting. In the proof of Proposition.3, the generalized Laguerre polynomials L α m x with α = plays an important role. See Section 3. We denote the limit by F u = : u m u : and consider the Wick ordered NLS.9. When m =, Bourgain [7] constructed almost sure global-in-time strong solutions and proved the invariance of the Gibbs measure P 4 for the defocusing cubic Wick ordered NLS. See Remark.6 below. The main novelty in [7] was to construct local-in-time dynamics in a probabilistic manner, exploiting the gain of integrability for the random rough linear solution. By a similar approach, Burq-Tzvetkov [4, 5] constructed almost sure global-intime strong solutions and proved the invariance of the Gibbs measure for the defocusing

9 INVARIANT GIBBS MEASURES FOR THE -d DEFOCUSING NLS 9 subquintic nonlinear wave equation NLW posed on the three-dimensional ball in the radial setting. On the one hand, when m =, there is only an ε-gap between the regularity of the support H s T, s < 0, of the Gibbs measure P 4 and the scaling criticality s = 0 and the regularity s > 0 of the known deterministic local well-posedness [5]. On the other hand, when m 3, the gap between the regularity of the Gibbs measure P m and the scaling criticality is slightly more than m. At present, it seems very difficult to close this gap and to construct strong solutions even in a probabilistic setting. In the following, we instead follow the approach presented in the work [] by the second author with Burq and Tzvetkov. This work, in turn, was motivated by the works of Albeverio-Cruzeiro [] and Da Prato-Debussche [7] in the study of fluids. The main idea is to exploit the invariance of the truncated Gibbs measures P m,n for.33, then to construct global-in-time weak solutions for the Wick ordered NLS.9, and finally to prove the invariance of the Gibbs measure P m in some mild sense. Now, we are ready to state our main theorem. Theorem.4. Let m be an integer. Then, there exists a set Σ of full measure with respect to P m such that for every φ Σ, the Wick ordered NLS.9 with initial condition u0 = φ has a global-in-time solution for any s < 0. by P m. u CR; H s T Moreover, for all t R, the law of the random function ut is given There are two components in Theorem.4: existence of solutions and invariance of P m. A precursor to the existence part of Theorem.4 appears in []. In [], the second author with Burq and Tzvetkov used the energy conservation and a regularization property under randomization to construct global-in-time solutions to the cubic NLW on T d for d 3. The main ingredient in [] is the compactness of the solutions to the approximating PDEs. In order to prove Theorem.4, we instead follow the argument in []. Here, the main ingredient is the tightness = compactness of measures on space-time functions, emanating from the truncated Gibbs measure P m,n and Skorokhod s theorem see Lemma 5.7 below. We point out that Theorem.4 states only the existence of a global-in-time solution u without uniqueness. Theorem.4 only claims that the law Lut of the H s -valued random variable ut satisfies Lut = P m for any t R. This implies the invariance property of the Gibbs measure P m in some mild sense, but it is weaker than the actual invariance in the sense of.3. In fact, the result of Theorem.4 remains true in a more general setting. Let M, g be a two-dimensional compact Riemannian manifold without boundary or a bounded domain in R. We consider the equation. on M when M is a domain in R, we impose the Dirichlet or Neumann boundary condition. Assume that k = m for some integer m. In Section 4, we prove the analogues of Propositions.,., and.3 in this geometric

10 0 T. OH AND L. THOMANN setting, by incorporating the geometric information such as the eigenfunction estimates. In particular, it is worthwhile to note that the variance parameter σ N in.5 now depends on x M in this geometric setting and more care is needed. Once we establish the analogues of Propositions.,., and.3, we can proceed as in the flat torus case. Namely, these propositions allow us to define a renormalized Hamiltonian: H Wick u = u dx + : u m : dx, m M and a Gibbs measure P m as in.8. Moreover, we are able to give a sense to NLS with a Wick ordered nonlinearity: { i t u + g u = : u m u: t, x R M..38 u t=0 = φ, In this general setting, we have the following result. Theorem.5. Let m be an integer. Then, there exists a set Σ of full measure with respect to P m such that for every φ Σ, the Wick ordered NLS.38 with initial condition u0 = φ has a global-in-time solution for any s < 0. by P m. u CR; H s M Moreover, for all t R, the law of the random function ut is given Theorems.4 and.5 extend [, Theorem.] for the defocusing Wick ordered cubic NLS m = to all defocusing nonlinearities all m. While the main structure of the argument follows that in [], the main source of challenge for our problem is the more and more complicated combinatorics for higher values of m. See Appendix A for an example of an concrete combinatorial argument for m = 3 in the case M = T, following the methodology in [7, ]. In order to overcome this combinatorial difficulty, we introduce the white noise functional see Definition. below and avoid combinatorial arguments of increasing complexity in m, allowing us to prove Propositions. and.3 in a concise manner. In order to present how we overcome the combinatorial complexity in a clear manner, we decided to first discuss the proofs of Propositions.,., and.3 in the case of the flat torus T Sections and 3. This allows us to isolate the main idea. We then discuss the geometric component and prove the analogues of Propositions.,., and.3 in a general geometric setting Section 4. Remark.6. Let m = and M = T. Then, the Wick ordered NLS.9 can be formally written as M i t u + u = u σ u,.39 where σ is the non-existent limit of σ N log N as N. Given u as in.0, define θ N = ffl T P N u dx σ N, where ffl T fxdx = 4π T fxdx. Then, it is easy to see that the limit θ := lim N θ N exists in L p µ for any p. Thus, by setting vt = e itθ ut, we can rewrite.39 as i t v + v = v ffl T v dxv..40

11 INVARIANT GIBBS MEASURES FOR THE -d DEFOCUSING NLS Note that v L = almost surely. Namely,.40 is also a formal expression for the limiting dynamics. In [7], Bourgain studied.40 and proved local well-posedness below L T in a probabilistic setting. If v is a smooth solution to.40, then by setting wt = e it ffl T v dx vt, we see that w is a solution to the standard cubic NLS: i t w + w = w w..4 This shows that the Wick ordered NLS.39 and.40 are equivalent to the standard cubic NLS in the smooth setting. Note that this formal reduction relies on the fact that the Wick ordering introduces only a linear term when m =. For m 3, the Wick ordering introduces higher order terms and thus there is no formal equivalence between the standard NLS. and the Wick ordered NLS.9. Remark.7. So far, we focused on the defocusing NLS. Let us now discuss the situation in the focusing case: i t u + u = u k u with the Hamiltonian given by Hu = u dx u k dx. T d k T d In the focusing case, the Gibbs measure can be formally written as dp k d = Z e Hu du = Z e k T d u k dx dµ. The main difficulty is that T u k dx is unbounded. When d =, Lebowitz-Rose-Speer [4] d constructed the Gibbs measure P k for < k 6, by adding an extra L -cutoff. Then, Bourgain [6] constructed global-in-time flow and proved the invariance of the Gibbs measure for k 6. See also McKean [5]. When d =, the situation becomes much worse. Indeed, Brydges-Slade [9] showed that the Gibbs measure P 4 for the focusing cubic NLS on T can not be realized as a probability measure 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 problem well-posed and the invariance of the Gibbs measure may be proven even in the focusing case. Remark.8. In a recent paper [36], we also studied the defocusing nonlinear wave equations NLW in two spatial dimensions with an even integer k = m 4 and ρ 0: { t u g u + ρu + u k = 0 t, x R M.4 u, t u t=0 = φ 0, φ, and its associated Gibbs measure: dp m = Z exp Hu, t udu d t u = Z e um m dx e ρu + u dx du e tu d t u..43 As in the case of NLS, the Gibbs measure in.43 is not well defined in the two spatial dimensions. Namely, one needs to consider the Gibbs measure P m associated to the Wick

12 T. OH AND L. THOMANN ordered Hamiltonian 6 as in.3 and study the associated dynamical problem given by the following defocusing Wick ordered NLW: t u u + ρu + :u k : = In the case of the flat torus M = T with ρ > 0, we showed that the defocusing Wick ordered NLW.44 is almost surely globally well-posed with respect to the Gibbs measure P m and that the Gibbs measure P m is invariant under the dynamics of.44. For a general two-dimensional compact Riemannian manifold without boundary or a bounded domain in R with the Dirichlet or Neumann boundary condition, we showed that an analogue of Theorem.5 i.e. almost sure global existence and invariance of the Gibbs measure P m in some mild sense holds for.44 when ρ > 0. In the latter case with the Dirichlet 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 almost surely with respect to the Gibbs measure. We achieved this goal for any even k 4 by exploiting one degree of smoothing in the Duhamel formulation of the Wick ordered NLW.44. As for the Wick ordered NLS.9 on T, such smoothing is not available and the construction of unique solutions with the Gibbs measure as initial data remains open for the superquintic case. Remark.9. In [6, 37], Bourgain k =, 3 and Richards k = 4 proved invariance of the Gibbs measures for the generalized KdV equation gkdv on the circle: t u + 3 xu = ± x u k, t, x R T..45 In [35], the authors and Richards studied the problem for k 5. In particular, by following the approach in [] and this paper, we proved almost sure global existence and invariance of the Gibbs measuresin some mild sense analogous to Theorem.4 for i all k 5 in the defocusing case and ii k = 5 in the focusing case. Note that there is no need to apply a renormalization for constructing the Gibbs measures for this problem since the equation is posed on T. See [4, 6]. This paper is organized as follows. In Sections and 3, we present the details of the proofs of Propositions.,., and.3 in the particular case when M = T. We then indicate the changes required to treat the general case in Section 4. In Section 5, we prove Theorems.4 and.5. In Appendix A, we present an alternative proof of Proposition. when m = 3 in the case M = T, performing concrete combinatorial computations.. Construction of the Gibbs measures In this section, we present the proofs of Propositions. and. and construct the Gibbs measure P m in.3. One possible approach is to use the Fock-space formalism in quantum field theory [38, 3, 6, 0]. As mentioned above, however, we present a pedestrian Fourier analytic approach to the problem since we believe that it is more accessible to a wide range of readers. The argument presented in this section and the next section on Proposition.3 follows the presentation in [8] with one important difference; we work in 6 In the case of NLW, we only need to use the Hermite polynomials since we deal with real-valued functions.

13 INVARIANT GIBBS MEASURES FOR THE -d DEFOCUSING NLS 3 the complex-valued setting and hence we will make use of the generalized Laguerre polynomials instead of the Hermite polynomials. Their orthogonal properties play an essential role. See Lemmas.4 and Hermite polynomials, Laguerre polynomials, and Wick ordering. First, recall the Hermite polynomials H n x; σ defined through the generating function: F t, x; σ := e tx σt = k=0 t k k! H kx; σ. for t, x R and σ > 0. For simplicity, we set F t, x := F t, x; and H k x := H k x; in the following. Note that we have H k x, σ = σ k Hk σ x.. From., we directly deduce the following recursion relation x H k x; σ = kh k x; σ,.3 for all k 0. This allows to compute the H k, up to the constant term. The constant term is given by H k 0, σ = k k!! σ k and H k+ 0, σ = 0, for all k 0, where k!! = k k 3 3 = k! k k! and!! = by convention. This can be easily deduced from. by taking x = 0. For readers convenience, we write out the first few Hermite polynomials in the following: H 0 x; σ =, H x; σ = x, H x; σ = x σ, H 3 x; σ = x 3 3σx, H 4 x; σ = x 4 6σx + 3σ. The monomial x k can be expressed in term of the Hermite polynomials: x k = [ k ] m=0 k m!! σ m m H k m x; σ..4 Fix d N, 7 let H = R d. Then, consider the Hilbert space 8 Γ H = L Q H, µ d ; C endowed with the Gaussian measure dµ d = π d exp x /dx, x = x,..., x d R d. We define a homogeneous Wiener chaos of order k to be an element of the form H k x = d H kj x j, j= where k = k + + k d and H kj is the Hermite polynomial of degree k j defined in.. Denote by Γ k H the closure of homogeneous Wiener chaoses of order k under L R d, µ d. 7 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. 8 Here, QH = 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.

14 4 T. OH AND L. THOMANN Then, we have the following Wiener-Ito decomposition: 9 L Q H, µ d ; C = Γ k H..5 Given a homogeneous polynomial P k x = P k x,..., x d of degree k, we define the Wick ordered polynomial : P k x : to be its projection onto H k. In particular, we have : x k j : = H kx j and : d j= xk j j : = d j= H k j x j with k = k + + k d. Now, let g be a standard complex-valued Gaussian random variable. Then, g can be written as g = h + i h, where h and h are independent standard real-valued Gaussian random variables. We investigate the Wick ordering on g m for m N, that is, the projection of g m onto H m. When m =, g = h + h is Wick-ordered into k=0 : g : = h + h = g..6 When m =, g 4 = 4 h + h = 4 h4 + h h + h4 is Wick-ordered into : g 4 : = 4 h4 6h h h + 4 h4 6h + 3 = 4 h4 + h h + h 4 h + h + = g 4 4 g +. When m = 3, a direct computation shows that is Wick-ordered into In general, we have g 6 = 8 h + h 3 = 8 h6 + 3h 4 h + 3h h 4 + h 6 : g 6 : = 8 H 6h H 4h H h H h H 4 h + 8 H 6h = g 6 9 g g 6. : g m : = m = m l=0 m l=0 m l H l h H m l h m H l l Re g; H m lim g;,.7 where we used. in the second equality. It follows from the rotational invariance of the complex-valued Gaussian random variable that : g m : = P m g for some polynomial P m of degree m with the leading coefficient. This fact is, however, not obvious from.7. The following lemma shows that the Wick ordered monomials : g m : can be expressed in terms of the Laguerre polynomials recall the definition.0. 9 This is equivalent to the Fock space in quantum field theory. See [38, Chapter I]. In particular, the Fock space FH = k=0 H k sym C is shown to be equivalent to the Wiener-Ito decomposition.5. In the Fock space formalism, the Wick renormalization can be stated as the reordering of the creation operators on the left and annihilation operator on the right. We point out that while much of our discussion can be recast in the Fock space formalism, our main aim of this paper is to give a self-contained presentation as much 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.

15 INVARIANT GIBBS MEASURES FOR THE -d DEFOCUSING NLS 5 Lemma.. Let m N. For a complex valued mean-zero Gaussian random variable g with Varg = σ > 0, we have m : g m m : = H l l Re g; σ H m lim g; σ l=0 = m m! L m g ; σ..8 As a consequence, the Wick ordered monomial : u N m : defined in.9 satisfies.4 for any N N. Proof. The first equality follows from.7 and scaling with.. Moreover, by scaling with.3 and., we can assume that g is a standard complex-valued Gaussian random variable with g = Re g and g = Im g. Define H m g and L m g by m H m g m = H l l g ; H m lg ;, l=0 L m g = m m! L m g..9 Then,.8 follows once we prove the following three properties: H g = L g = g,.0 g g H m g = m H m g, g g L. m g = m L m g, E[H m g ] = E[L m g ] = 0,. for all m. Noting that both H m g and L m g are polynomials in g, the three properties.0,., and. imply that H m g = L m g for all m N. The first property.0 follows from.6 and.. Next, we prove. for H m g. From g = g i g and g = g + i g, we have g g = 4 g,g, where g,g denotes the usual Laplacian on R in the variables g, g. Then, recalling that x H k x; σ = kh k x; σ, we have g g H m g = 4 g,g H m g = m m ll H 4 l l g ; H m lg ; + 4 l= m l=0 m = m l=0 m l m lm l H l g ; H m l g ; m H l l g ; H m lg ;.

16 6 T. OH AND L. THOMANN As for the second identity in., thanks to the formula., we get g g L m g = m m! m m l 4 l g,g l! g + g l l=0 m = m m l m! l l g l = m L m g. l! l= This proves.. The property. follows from i independence of g and g together with the orthogonality of H k x and the constant function under e x dx and ii the orthogonality of L m x and the constant function under R+ e x dx Let u be as in.0. Fix x T. Letting g n = g n e in x, we see that { g n } n N is a sequence of 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.9 with.7 and.8 that m : u N x m m : = H l l Re ux; σ N H m l Im ux; σ N l=0 = m m! L m u N x ; σ N, verifying.4. This proves the second claim in Lemma.... White noise functional. Next, we define the white noise functional. Let wx; ω be the mean-zero complex-valued Gaussian white noise on T defined by wx; ω = n Z g n ωe in x. Definition.. The white noise functional W : L T L Ω is defined by W f ω = f, wω L x = n Z fngn ω..3 for a function f L T. Note that this is basically the periodic and higher dimensional version of the classical Wiener integral b a fdb. It can also be viewed as the Gaussian process indexed by f L T. See [38, Model on p. 9 and Model 3 on p. ]. For each f L T, W f is a complex-valued Gaussian random variable with mean 0 and variance f L. Moreover, we have E [ W f W h ] = f, h L x for f, h L T. In particular, the white noise functional W : L T L Ω is an isometry. Lemma.3. Given f L T, we have e Re W f ω dp ω = e 4 f L..4 Ω

17 INVARIANT GIBBS MEASURES FOR THE -d DEFOCUSING NLS 7 Proof. Noting that Re g n and Im g n are mean-zero real-valued Gaussian random variables with variance, it follows from.3 that Ω e Re W f ω dp ω = n Z e Re fn Re g n Re g n d Re g n π R e Im fn Im g n Im g n d Im g n R = e 4 f L. The following lemma on the white noise functional and the Laguerre polynomials plays an important role in our analysis. In the following, we present an elementary proof, using the generating function G in.0. See also Folland []. Lemma.4. Let f, h L T such that f L = h L =. Then, for k, m Z 0, we have E [ L k W f L m W h ] = δ km f, h k..5 Here, δ km denotes the Kronecker delta function. First, recall the following identity: e u = π R x xu e dx..6 Indeed, we used a rescaled version of.6 in the proof of Lemma.3. Proof of Lemma.4. Let G be as in.0. Then, for any < t, s < 0, from.6 and Lemma.3, we have Ω Gt, W f ω Gs, W h ω dp ω = t s = t s 4π e x +x +y +y R 4 = t s 4π Ω R 4 e Ω e t t W f s s W h dp ω exp Re W t t x ix f+ s dp dx dx dy dy s y iy h x +x t y +y s e Re t t s s x ix y +iy f,h dx dx dy dy

18 8 T. OH AND L. THOMANN By a change of variables and applying.6, we have = y +y 4π e tsy Re f,h y Im f,h x x dx R e R R = e y +y e ts f,h y +y dy dy π R = ts f, h = k=0 e tsy Re f,h +y Im f,h x x dx dy dy t k s k f, h k..7 In the second to the last equality, we used the fact that ts f, h <. Hence, it follows from.0 and.7 that t k s k f, h k = t k s m L k W f ω L m W h ω dp ω. k=0 k,m=0 By comparing the coefficients of t k s m, we obtain.5. Ω Now, we are ready to make sense of the nonlinear part of the Wick ordered Hamiltonian H Wick in.7. We first present the proof of Proposition. for p =. Recall that G N u = : P N u m : dx. m T Then, we have the following convergence property of G N u in L µ. Lemma.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 > 0 such that for any M N. G M u G N u L µ C m N.8 Given N N, let σ N be as in.5. For fixed x T and N N, we define where e n y = e in y. Note that η N x := γ N := σ N n N n N for all fixed x T and all N N. Moreover, we have η M x, η N y L T = for fixed x, y T and N, M N with M N. e n x + n e n,.9 + n e n,.0 η N x L T =. σ M σ N γ N y x = σ M σ N γ N x y,.

19 INVARIANT GIBBS MEASURES FOR THE -d DEFOCUSING NLS 9 Proof of Lemma.5. Let m be an integer. from.0,.3, and.9 that u N x = σ N u N x σ N Then, from.4 and.3, we have : u N m : = m m!σn m un L m σ N From.4, Lemma.4, and., we have m G M u G N u L µ = m! = m! T x T y T x T y Given N N and x T, it follows = σn W η N x..3 = m m!σn m L m W ηn x..4 Ω [ σm m L m W ηm x Lm W ηm y σmσ m N m L WηM m x Lm WηN y σmσ m N m L WηN m x Lm WηM y + σn m L m W ηn x Lm W ηn ] y dp dxdy [ γm x y m γ N x y m] dxdy = m! T [ γm x m γ N x m] dx C m T γ M x γ N x [ γ M x m + γ N x m ] dx..5 In the second equality, we used the fact that γ N is a real-valued function. From.0, we have γ M γ L N = + n N..6 By Hausdorff-Young s inequality, we have γn m L = γ N m L 4m N< n M n N + n 4m 4m 3 4m 3 uniformly in N N. Then,.8 follows from.5,.6, and.7. C m <.7.3. Wiener chaos estimates. In this subsection, we complete the proof of Proposition.. Namely, we upgrade.8 in Lemma.5 to any finite p. Our main tool is the following Wiener chaos estimate see [38, Theorem I.]. Lemma.6. Let {g n } n N be a sequence of independent standard real-valued Gaussian random variables. Given k N, let {P j } j N be a sequence of polynomials in ḡ = {g n } n N of degree at most k. Then, for p, we have P j ḡ p k P j ḡ.8 L Ω. j N L p Ω j N

20 0 T. OH AND L. THOMANN Observe that the estimate.8 is independent of d N. By noting that P j ḡ k l=0 Γ lh, this lemma follows as a direct corollary to the hypercontractivity of the Ornstein-Uhlenbeck semigroup due to Nelson [8]. We are now ready to present the proof of Proposition.. Proof of Proposition.. Let m be an integer. For p, Proposition. follows from Lemma.5. In the following, we consider the case p >. From.,.4, and.6, we have Here, Σ l is given by where Γ k and g n j Σ l = σm M σ l M G M u G N u = m m! m Γ l 0 n j M are defined by l j= m m l Σ l l. l! l= g n j + nj σm N σ l N Γ l 0 n j N l j= g n j + nj, Γ k n = {n,..., n k Z k : n n + + k n k = n}, {.9 gn g nj if j is odd, j = if j is even..30 g nj Noting that Σ l is a sum of polynomials of degree l in {g n } n Z, Proposition. follows from Lemmas.5 and Nelson s estimate. In this subsection, we prove Proposition.. Our main tool is the so-called Nelson s estimate, i.e. in establishing an tail estimate of size λ > 0, we divide the argument into low and high frequencies, depending on the size of λ. See.3 and.34. What plays a crucial role here is the defocusing property of the Hamiltonian and the logarithmic upper bound on G N u, which we discuss below. For each m N, there exists finite a m > 0 such that m L m x a m for all x R. Then, it follows from.3,.4,.5, and.6 that there exists some finite b m > 0 such that G N u = : P N u m : dx b m log N m.3 m T for all N. Namely, while G N u is not sign definite, G N u is bounded from above by a power of log N. This is where the defocusing property of the equation.33 plays an essential role. Proof of Proposition.. Let m be an integer. It follows from Proposition. that the following tail estimate holds: there exist c m,p, C m > 0 such that µ p G M u G N u > λ C m e cm,pn m λ m.3 for all M N, p, and all λ > 0. See, for example, [43, Lemma 4.5].

21 INVARIANT GIBBS MEASURES FOR THE -d DEFOCUSING NLS We first show that R N u = e G N u is in L p µ with a uniform bound in N. We have R N u p L p µ = H s e pg N u dµu = 0 + µe pg N u > αdα µ pg N u > log αdα. Hence, it suffices to show that there exist C, δ > 0 such that µ pg N u > log α Cα +δ.33 for all α > and N N. Given λ = log α > 0, choose N 0 R such that λ = pb m log N 0 m. Then, it follows from.3 that µ pg N u > λ = 0.34 for all N < N 0. For N N 0, it follows from.3 and.3 that there exist δ m,p > 0 and C m,p > 0 such that µ pg N u > λ µ pg N u + pg N0 u > λ pb m log N 0 m µ pg N u + pg N0 u > λ C m e c m,p N m 0 λ m = C m e c m,p λ m e cmλ m C m,p e +δm,pλ.35 for all N N 0. This shows that.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. By.3, G N u converges to Gu in measure with respect to µ. Then, as a composition of G N u with a continuous function, R N u = e G N u converges to Ru := e Gu in measure with respect to µ. In other words, given ε > 0, defining A N,ε by A N,ε = { R N u Ru ε }, we have µa c N,ε 0, as N. Hence, by Cauchy-Schwarz inequality and the fact that R L p, R N L p C p uniformly in N N, we obtain R R N L p µ R R N AN,ε L p µ + R R N A c N,ε L p µ ε µa N,ε p + R R N L p µ µa c N,ε p Cε, for all sufficiently large N. This completes the proof of Proposition.. 3. On the Wick ordered nonlinearity In this section, we present the proof of Proposition.3. The main idea is similar to that in Section 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 generating function: G α t, x := tx e t t α+ = m=0 t m L α m x, 3.

22 T. OH AND L. THOMANN for t < and x R. From 3., we obtain the following differentiation rule; for l N, d l dx l Lα m x = l L α+l m l x. 3. Given N N, let u N = P N, where u is as in.0. Let m be an integer. Then, from.36,.4,.3, and 3., we have F N u = P N : PN u m P N u: { } = m m!σn m m P N un L un m σ N { = m+ m!σ m N P N L un m u N }. 3.3 σ N Remark 3.. Here, u denotes the usual differentiation in u viewing u and u as independent variables. This is not to be confused with H H u in.3. Note that u in.3 comes from the symplectic structure of NLS and the Gâteaux derivative of H. More precisely, we can view the dynamics of NLS. as a Hamiltonian dynamics with the symplectic space L T and the symplectic form ωf, g = Im fxgxdx. Then, we define H u by dh u φ = ω φ, i H u where dh u φ is the the Gâteaux derivative given by dh u φ = d dε Hu + εφ ε=0. The following lemma is an analogue of Lemma.4 for the generalized Laguerre polynomials L m x and plays an important role in the proof of Proposition.3. Lemma 3.. Let f, h L T such that f L = h L =. Then, for k, m Z 0, we have [ ] E L k W f W f L m W h W h = δ km k + f, h k f, h. 3.4 Here, δ km denotes the Kronecker delta function. Besides.6, we will use the following identity: ue u x = xu xe dx. 3.5 π This follows from differentiating.6 in u. Proof of Lemma 3.. Let G be as in 3. with α =. Let < t < 0. From.6 and 3.5, we have G t, W f Re W f = t Re W f e t W t f +Im W f = + i t Im W f e t t t t 3 R, Re W f +Im W f x + ix e x +x t t e x Re W f +x Im W f dx dx. π R

23 INVARIANT GIBBS MEASURES FOR THE -d DEFOCUSING NLS 3 Given x, x, y, y R, let x = x + ix and y = y + iy. Then, for any < t, s < 0, from Lemma.3, we have G t, W f ωw f ωg s, W h ωw h ωdp ω Ω = = t t 3 t t 3 s s 3 4π xye R 4 exp Re W t Ω t xf+ s 3 s s 4π xye R 4 x + y s yh dp dx dx dy dy x t y s e Re t s t s xy f,h dx dx dy dy By a change of variables and applying.6 and 3.5, we have = ts 4π xye x y e ts Rexy f,h dx dx dy dy R 4 = f, h y e ts f,h y dy dy 4π R By integration by parts, we have f, h = ts f, h e ts f,h y dy dy π R f, h = ts f, h = k + t k s k f, h k f, h. 3.6 k=0 Hence, it follows from 3. and 3.6 that k + t k s k f, h k f, h = t k s m k=0 k,m=0 By comparing the coefficients of t k s m, we obtain 3.4. Ω L k W f ω W f L m W h ω W h dp ω. As a preliminary step to the proof of Proposition.3, we first estimate the size of the Fourier coefficient of F N u. Lemma 3.3. Let m be an integer. Then, for any θ > 0, there exists C m,θ > 0 such that F N u, e n L x L µ C m,θ + n θ 3.7 for any n Z and any N N. Moreover, given positive ε < and any 0 < θ ε, there exists C m,θ,ε > 0 such that F M u F N u, e n L x L µ C m,θ,ε N ε + n θ ε 3.8 for any n Z and any M N.

24 4 T. OH AND L. THOMANN Proof. We first prove 3.7. Let m be an integer and N N. From 3.3 with.3, we have { F N u = m+ m!σ m N P N L WηN m } x WηN x. 3.9 Clearly, F N u, e n L x = 0 when n > N. Thus, we only need to consider the case n N. From Lemma 3. with 3.9,. and., we have F N u, e n L x L µ = [ m! ] σ m N e n xe n y T x T y L m W ηn x WηN xl m W ηn y WηN ydp dxdy Ω = m!m! γ N x y m γ N x ye n x ydxdy T x T y = C m F [ γ N m γ N ] n. 3.0 Let Γ m n be as in.9. For n,..., n m Γ m n, we have max j n j n. Thus, we have F [ γ N m γ N ] n = Γ m n n j N m j= + n j d m,θ + n. 3. θ Hence, 3.7 follows from 3.0 and 3.. Next, we prove 3.8. Let M N. Proceeding as before with 3.9, Lemma 3., and., we have F M u F N u, e n L x L µ { = C m [0,M] n F [ γ M m ] γ M n [0,N] n F [ γ N m ] } γ N n = C m [0,N] n {F [ γ M m ] [ γ M n F γn m ] } γ N n + C m N,M] n F [ γ M m γ M ] n. 3. On the one hand, noting that n > N, we can use 3. to estimate the second term on the right-hand side of 3., yielding 3.8. On the other hand, noting that F [ γ M m ] [ γ M n F γn m ] γ N n Γ m n n j M max j n j N m j= + n j d m,θ maxn, + n θ, we can estimate the first term on the right-hand side of 3. by 3.8. Next, we use the Wiener chaos estimate Lemma.6 to extend Lemma 3.3 for any finite p.

25 INVARIANT GIBBS MEASURES FOR THE -d DEFOCUSING NLS 5 Corollary 3.4. Let m be an integer. Then, for any θ > 0, there exists C m,θ > 0 such that F N u, e n L x L p µ C m,θ p m n θ for any n Z and any N N. Moreover, given positive ε < and any 0 < θ ε, there exists C m,θ,ε > 0 such that F M u F N u, e n L x L p µ C m,θ,ε p m for any n Z and any M N. N ε + n θ ε 3.4 Proof. Let m be an even integer. In view of Lemma 3.3, we only consider the case p >. From 3.3 with., we have Recalling.9 and.30, we have m F N u = u m u + a m,l,n u l u. F N u, e n L x = m l=0 a m,l,n l=0 Γ l n n j N l j= g n j + nj. 3.5 Noting that the right-hand side of 3.5 is a sum of polynomials of degree at most m in {g n } n Z, the bound 3.3 follows from Lemma 3.3 and.6. The proof of 3.4 is analogous and we omit the details. Finally, we present the proof of Proposition.3. Proof of Proposition.3. Let s < 0. Choose sufficiently small θ > 0 such that s + θ < 0. Let p. Then, it follows from Minkowski s integral inequality and 3.3 that FN u H s L p µ n s F N u, e n L x L p µ n Z p m n Z n +θ+s Cm,p < since s + θ < 0. Similarly, given ε > 0 such that s + ε < 0, choose sufficiently small θ > 0 such that s + θ + ε < 0. Then, from 3.4, we have FM u F N u H s L p µ n s F M u F N u, e n L x L p µ n Z p m since s + θ + ε < 0. This proves.37. N ε n +θ+ε+s p m N ε n Z

26 6 T. OH AND L. THOMANN 4. Extension to -d manifolds and domains in R Let M, g be a two-dimensional compact Riemannian manifold without boundary or a bounded domain in R. In this section, we discuss the extensions of Propositions.,., and.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 with the corresponding eigenvalues {λ n} n N, which we assume to be arranged in the increasing order. Then, by Weyl s asymptotics, we have λ n n. 4. See, for example, [45, Chapter 4]. Let {g n ω} n N be a sequence of independent standard complex-valued Gaussian random variables on a probability space Ω, F, P. We define the Gaussian measure µ as the induced probability measure under the map: ω Ω ux = ux; ω = g n ω ϕ n N + λ n n x. 4. Note that all the results in Sections and 3 still hold true in this general context with exactly the same proofs, except for Lemma.5 and Lemma 3.3, where we used standard Fourier analysis on T. In the following, we will instead use classical properties of the spectral functions of the Laplace-Beltrami operator. Let us now define the Wick renormalization in this context. Let u be as in 4.. Given N N, we define the projector P N by u N = P N u = ûnϕ n. We also define σ N by λ n N σ N x = E[ u N x ] = λ n N ϕ n x + λ n log N, 4.3 where the last inequality follows from [, Proposition 8.] and Weyl s law 4.. Unlike σ N defined in.5 for the flat torus T, the function σ N defined above depends on x M. Note that σ N x > 0 for all x M. The Wick ordered monomial : u N m : is then defined by By analogy with.9 and.0 we define : u N m : = m m! L m u N ; σ N. 4.4 η N x := σ N x γ N x, y := λ n N λ n N for x, y M. We simply set γ = γ when N =. ϕ n x + λ n ϕ n, 4.5 ϕ n xϕ n y + λ, 4.6 n

27 INVARIANT GIBBS MEASURES FOR THE -d DEFOCUSING NLS 7 From the definition 4.3 of σ N, we have η N x L M = for all x M. Moreover, we have η M x, η N y L M = γ σ M xσ N x, y 4.7 N y 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 = ϕ n xϕ n y, λ n j,j] for x, y M and j Z 0. From [39,.3 and.5 with q = ], we have the bound π j x, x Cj +, uniformly in x M. Therefore, by Cauchy-Schwarz inequality, we obtain π j x, y ϕ n x ϕ n y Cj +, 4.8 λ n j,j] uniformly in x, y M. Let σ be a weighted counting measure on Z 0 defined by σ = j=0 j + δ j, where δ j is the Dirac delta measure at j Z 0. We define the operator L by L : c = {c j } j=0 c j π j. j=0 Then, we have the following boundedness of the operator L. Lemma 4.. Let q. Then, the operator L defined above is continuous from l q Z 0, σ into L q M. Here, q denotes the Hölder conjugate of q. Proof. By interpolation, it is enough to consider the endpoint cases q = and q =. Case : q =. Assume that c l Z 0, σ. Then, from 4.8, we get Lcx, y c j π j x, y C j + c j = c l σ. j=0 for all x, y M. This implies the result for q =. Case : q =. ϕ n, we have j=0 Assume that c l Z 0, σ. By the orthogonality of the eigenfunctions M From 4.8 and 4.9, we deduce that This implies the result for q =. Lcx, y dx = M Lcx, y dxdy C c j π j y, y. 4.9 j=0 j + c j = c l σ. j=0 Next, we extend the definition of γ N to general values of s: γ s,n x, y := ϕ n xϕ n y + λ n s λ n N

28 8 T. OH AND L. THOMANN for x, y M. When N =, we simply set γ s = γ s, as before. Note that when s =, γ,n and γ correspond to γ N and γ defined in 4.6. Lemma 4.. 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 > 0 and κ > 0 such that for all M N. γ s,m γ s,n L p M C N κ, 4.0 Proof. Given M N, define α N,M x, y and β N,M x, y by and α N,M x, y := γ s,m x, y γ s,n x, y β N,M x, y := = N<λ n M M j=n+ ϕ n xϕ n y = + λ n s + j s λ n j,j] j=n+ λ n j,j] M j=n+ λ n j,j] ϕ n xϕ n y = ϕ n xϕ n y + λ n s M j=n+ π j x, y. + j s Let us first estimate the difference α N,M β N,M : M α N,M x, y β N,M x, y + λ n s + j s ϕ nx ϕ n y Then, by 4.8, we obtain C M j=n+ j s+ λ n j,j] ϕ n x ϕ n y. 4. α N,M x, y β N,M x, y C. 4. N s Next, we estimate β N,M. Define a sequence c = {c j } j=0 by setting c j = { +j s, if N + j M, 0, otherwise. Note that c l q N, σ for s < q. Hence, it follows from Lemma 4. that, given any p < s, there exist C > 0 and κ > 0 such that M β N,M L p M = π j M j + p + j s C C L p M + j s p N κ. 4.3 j=n+ j=n+ The desired estimate 4.0 follows from 4., 4., and 4.3. As in the case of the flat torus, define G N, N N, by G N u = : P N u m : dx. m M Then, we have the following extension of Proposition.