The Gaussian free field, Gibbs measures and NLS on planar domains
|
|
- Paulina Byrd
- 6 years ago
- Views:
Transcription
1 The Gaussian free field, Gibbs measures and on planar domains N. Burq, joint with L. Thomann (Nantes) and N. Tzvetkov (Cergy) Université Paris Sud, Laboratoire de Mathématiques d Orsay, CNRS UMR 8628 LAGA, Paris-13, Séminaire intercontinental, feb 20th, 2014
2 Purpose Study the dynamics of solutions of non linear Schrödinger equations in a planar domain, M R 2, with Dirichlet or Neumann boundary conditions, (i t + )u u 2 u = 0, u t=0 = u 0, u M = 0, (resp. ν u M = 0), from a statistical point of view, i.e. u 0 is a random variable or (equivalently, endow the space of initial data with a probability measure, µ 0. Well posedness on the support of the measure (almost sure WP): definition of a flow Φ(t). Statistical properties of the measure propagated by the flow, µ(t) = Φ(t) (µ 0 ): continuity, recurence, growth of Sobolev norms,...
3 The Gaussian free field Consider a sequence of (complex) independent Gaussian random variables g k N (0, 1) (of law 1 π e z 2 dz )and for e k, ( + 1)e k = λ 2 k e k, e k M = 0, n N, the random variable u 0 = g k e k (x). λ n N k equivalently, consider the map ω (Ω, p) u = g k (ω) e k (x), λ n N k and endow the space of distributions D (M), with the image of p by this map.
4 The GFF continued equivalently, write u = n N u k e k (x), identify D (M) with the space of sequences U = (u k ) C N (satisfying some temperance growth conditions), and endow C N with the probability measure + λ 2 k k=1 π e λ2 k u k 2 du k. Lemma The GFF is for any ɛ > 0 supported by H ɛ (M): µ 0 (H ɛ (M) = 1, but µ 0 (L 2 (M)) = 0.
5 Wick re-ordering, Bourgain s result If u is a solution to, then v = e it( u 2 L 2 1) u is a solution to (i t + 1)v ( v 2 2 v 2 L 2)v = 0, v t=0= u 0. (R) Theorem (Bourgain 1996) There exists δ > 0 such that for µ 0 -almost every u 0, there exists a unique (global in time) solution of (R) on the torus T 2, u = Φ(t)u 0 in e it u 0 + X δ,1/2+. Furthermore there exists a function G L 1 (dµ 0 ), positive on the support of µ 0 such that the measure dν = G(u)dµ is invariant by the flow Φ(t). ν is a Gibbs measure. Recall u X s,b = D t b e it u L 2 t ;Hx s
6 Our result Let M R 2 smooth bounded domain, and : u 4 L 4 := u 2 2 u 2 L 2 2 L 2 4 u 4 L 2. Theorem (N.B. L. Thomann, N. Tzvetkov) e : u 4 L 4 is µ 0 a.s. finite, and in L 1 (dµ 0 ). Furthermore, there exists for µ 0 a.e. initial data u 0 a global in time solution to (R), with Dirichlet (resp. Neumann) bdry conditions, and the flow Φ(t) satisfies Φ(t) (e : u 4 L 4 dµ 0 ) = e : u 4 L 4 dµ 0. Rk 1: No uniqueness in our result: weak solutions. But contrarily to usual (deterministic) weak solutions some kind of uniqueness remains: for any such flow get same invariant measure Rk 2 Work in progress: hope to get strong solutions.
7 Wick re-ordering on the torus For u = e it( ) u 0 = n Z 2 u 2 u = n 1,n 2,n 3 g n n ein x n 2t, n = n 2 + 1, g n1g n2g n3 n 1 n 2 n 3 ei(n 1 n 2 +n 3 ) x ( n 1 2 n n 3 2 )t g n 2 g m = +2 eim x ( m 2 )t n 2 m n 1 n 2,n 3 n 2 n,m n g n 2 g n n 3 e in x ( n 2 )t ( u 2 2 u 2 L 2)u g n1g n2g n3 = n 1 n 2 n 3 ei(n 1 n 2 +n 3 ) x n n 1 n 2,n 3 n 2 g n 2 g n e in x n 3
8 Wick re-ordering on T 2 : analysis Last term is a.s in H 2 : E( n g n 2 g n n 3 e in x 2 H δ ) = n Z 2 E( g n 6 ) n 6 2δ < + if 6 2δ > 2 First term is a.s. in X 1/2, 1/2+. Indeed, E( 2 X s, b) n 1 n 2,n 3 n 2 ( 1 = E (1 + n 1 2 n n 3 2 k 2 ) b k = k n 1 n 2,n 3 n 2,n 1 n 2 +n 3 =k ( E n 1 n 2,n 3 n 2 g n1 g n2 g n3 2) n 1 n 2 n 3 k s m 1 m 2,m 3 m 2 g n1 g n2 g n3 g m1 g m2 g m3 )
9 Since Gaussian are independent, complex and have mean equal to 0, and since n 1 n 2, n 3 n 2, expectancy vanishes unless (n 1, n 2, n 3 ) = (m 1, m 2, m 3 ) or (n 1, n 2, n 3 ) = (m 3, m 2, m 1 ) E( n Z 2 1 n 2 n 1 n 2,n 3 n 2 2 X s, b) n 1 n 2,n 3 n 2 k=n 1 n 2 +n 3 k 2s n 1 2 n 2 2 n (1 + n 1 2 n n 3 2 k 2 ) 2b diverges logarithmically. Factor 1 (1 + n 1 2 n n 3 2 k 2 ) 2b gives additional convergence which can be exchanged to compensate the k 2s term, hence end up in X s,b, for some s > 0 a level at which Cauchy theory well posed! This (plus quite some work) proves Bourgain s Theorem.
10 The problem on a manifold Wick Reordering is in the folklore of quantum field theory. Best (most efficient) approach seems to be via Fock representation Fock representation seems to be not so well suited to X s,b analysis (possible development?) Take boundary into account (should not be a serious problem in the context of Fock representation though) Keep the elementary approach and understand at that level the compensations which allow for Wick re-ordering
11 Back to Wick re-ordering: The case of S 2 Take e n Hilbert base of spherical harmonics, eigenvalues λ 2 n. u 2 u = n 1,n 2,n 3 g n1g n2g n3 n 1 n 2 n 3 e i(λ2 n1 λ2 n2 +λ2 n3 )t e n1 (x)e n2 (x)e n3 (x) g n 2 g m = + 2 n 2 m e iλ2 m t e n 2 (x)e m (x) n 1 n 2,n 3 n 2 n,m n g n 2 g n e iλ2 n t e n 3 n 2 (x)e n (x) As in the previous analysis, last term OK. Main issues: e n 2 (x) 1 e n1 e n2 e n3 is not an eigenfunction
12 The Weyl formula on S 2 Let E k = Vect {e n ; λ 2 n = k(k + 1)} an eigenspace, and e n1, e n2k+1 any orthonormal basis of E k. Proposition Let e(x, y, k) = 2k+1 p=1 e n p (x)e np (y). x S 2, e(x, x, k) = 2k + 1 Vol (S 2 ). In a mean value sense, the eigenfunctions are constant on S 2 Proof: e(x, y, k) is the kernel of the orthogonal projector on E k. Hence for any isometry T, e(x, Ty) = e(t 1 x, y). Theorem (Van der Kam, Zelditch, N.B-G. Lebeau) There exists orthonormal basis (e n,k ) having for any p < + uniformly bounded L p norms (actually most ONB are such)
13 Back to Wick re-ordering, analysis of 2nd term n,m g n 2 g m n 2 m e iλ2 m t e n 2 (x)e m (x) = ( g n 2 1)g m e iλ2 n 2 m t e n 2 (x)e m (x) m n,m + g m t n 2 m e iλ2 m e n 2 (x)e m (x) k n,e n E k,m = I + g m n 2 m e iλ2 m t e m (x) k n,e n E k,m = I + g n 2 g m t n 2 m e iλ2 m e m (x) n,m + n,m (1 g n 2 )g m e iλ2mt e n 2 m (x) = I + II + III m
14 and n Indeed, since I + III = n ( g n 2 1) n 2 II = e it u 2 L 2eit u ( g n 2 1) ( e n 2 n 2 (x) 1)e it u ( e n 2 (x) 1) is a.s finite (renormalizable). n m E ( ( g n 2 1)( g m 2 1) ) = E( g n 2 1)E( g m 2 1) = 0, ( E n ( g n 2 1) ( e n 2 n 2 (x) 1) 2 ) = ( E n n ( g n 2 1) 2 n 4 ( e n 2 (x) 1) 2 ) n 1 n 4 < +
15 Back to Wick reordering: analysis of the first term With γ(n 1, n 2, n 3, p)) = e S 2 n1 e n2 e n3 e p dx, ( E g n1 g n2 g ) n3 n1 n n1 n2 1 n 2 n 3 e i(λ2 λ2 n2 +λ2 n3 )t e n1 (x)e n2 (x)e n3 (x) 2 X s,b n 3 n 2 = E( k,p p n 1 n 2 n 3 n 2 p 2s k 2b n 1 n 2,n 3 n 2 k λ 2 n 1 +λ 2 n 2 λ 2 n 3 [0,1) g n1 g n2 g n3 n 1 n 2 n 3 γ(n 1, n 2, n 3, p) 2 ) p 2s λ 2 n 1 + λ 2 n 2 λ 2 n 3 2b n 1 2 n 2 2 n 3 2 γ(n 1, n 2, n 3, p)) 2 Difference with T 2 : additional sum (in p), but presence of γ 2, which is invariant wrt permutations and in l 1 : choice of bases, γ(n 1, n 2, n 3, p)) 2 = e n1 e n2 e n3 2 L 2 (M) C < + p
16 General manifolds Use Weyl formula (Volume of manifold normalized to 1) e(x, λ, µ) = e n 2 (x) µ λ n<λ Theorem (Hormander Th ) Let d(x) be the distance of the point x M to the boundary M. There exists C > 0 such that for any λ > 1 and x M satisfying d(x, M) λ 1/2, any d [0, 1], we have e(x, λ + dλ 1/2, λ) d 2π λ3/2 Cλ, (1) Theorem (Sogge) There exists C > 0 such that for any λ > 1 and x M e(x, λ + 1, λ) Cλ, (2)
17 The strategy of proof Regularize the system by cutting in the non linearity the frequencies higher than N (λ n > N) and get global in time solutions. Get a flow Φ N (t). Define a family ν N = e 1 2 : u N 4 L 4 : dµ 0 of probability measures invariant by the flows Φ N. Show that these measures converge (in a sense to be precised) to a limit measure ν 0 = e 1 2 : u 4 L 4 : dµ 0 Pass to the limit (in a sense to be precised) in the family of solutions Φ N (t)u 0 Φ(t)u 0 Show that the flow Φ(t) solves (R) and leaves the measure ν 0 invariant Strategy quite standard in parabolic settings (see e.g. Da Prato-Debussche),
18 The approximating systems Let Π N be the orthogonal projector on the space Let Φ N (t)u 0 be the solution of Vect (e n ; λ n N). (i t + )u Π N ( ( ΠN (u) 2 2 Π N (u) 2 L 2)Π N(u) ) = 0, u t=0 = u 0, u M = 0, (resp. ν u M = 0) Hamiltonian system with Hamiltonian 1 H = 2 xu 2 dx Π N(u) 4 L Π N(u) 4 L 2 M
19 Formally, the GFF µ 0 is, in the coordinate system given by the identification u = n u ne n (x) given by dµ 0 = + λ 2 + k k=1 π e λ2 k u k 2 du k = e λ2 k u k 2 + λ 2 k k=1 π du k k=1 = e P k λ2 k u k 2 + λ 2 k k=1 π du x 2 k = e L 2 + λ 2 k k=1 π du k, and ν N = e 1 2 : u N 4 L 4 : dµ 0 = e 2H(u) + λ 2 k k=1 π du k is (at least formally) invariant (because the Hamiltonian itself is invariant and a Hamiltonian system can be seen as an ODE with a divergence free vector field hence the (infinite product of) Lebesgue measures is also invariant
20 Passing to the limit ν N ν Definition S separable complete metric space, (ρ N ) N 1 probability measures on Borel σ algebra B(S). (ρ N ) on (S, B(S)) is tight if ε > 0, K ε S compact such that ρ N (K ε ) 1 ε for all N 1. Theorem (Prokhorov) (ρ N ) N 1 is tight iff it is weakly compact, i.e. there is a subsequence (N k ) k 1 and a limit measure ρ such that for every bounded continuous function f : S R, f (x)dρ Nk (x) = f (x)dρ (x). lim k S Rk. Weak convergence implies convergence in law S
21 A tight sequence of probability measures Let T > 0 and define a probability measure on the space of (space time) functions f (t, x) p N by the image measure of ν N by the map u 0 Φ N (t)u 0. Proposition The sequence p N is for any ɛ > 0 tight on C 0 ([0, T ]; H ɛ (M)). Proof: Fix 0 < ɛ < ɛ. the measure ν 0 is supported by H ɛ which embeds compactly in H ɛ. This gains compactness in space. Then use equation to gain compactness in time.
22 Passing to the limit Φ N (t) Φ(t) Setting: we have a family of r.v. X Nk = Φ Nk (t)u 0 such that the laws of X Nk, p Nk are weakly convergent to a probability measure p. We d like to deduce that the r.v. X Nk are a.s. convergent. This is False (take X Nk = ( 1) N k X). However Theorem (Skorohod) Assume that S is a separable metric space. Let (ρ N ) N 1 and ρ be probability measures on S. Assume that ρ N ρ weakly. Then there exists a probability space on which there are S valued random variables (Y N ) N 1, Y such that L(Y N ) = ρ N for all N 1, L(Y ) = ρ and Y N Y a.s.
23 Passing to the limit in the equation I Need first to check that the r.v. Y N satisfy the same equation as X N = Φ N (t): Consider the r.v. Z N = ((i t + )Y N Π N ( ( ΠN (Y N ) 2 2 Π N (Y N ) 2 L 2)Π N(Y N ) ) All the functions appearing in the r.h.s. are continuous from C 0 ((0, T ); H ɛ ) to S = H 1 ((0, T ); H 2 ɛ ). Hence L(Z N ) ( = L ((i t + )Y N Π N ( ΠN (Y N ) 2 2 Π N (Y N ) 2L 2)Π N(Y N ) )) ( = L ((i t + )X N Π N ( ΠN (X N ) 2 2 Π N (X N ) 2L 2)Π N(X N ) )) Hence Z N = 0 a.s. = L(0) = δ 0
24 Passing to the limit in the equation II Need to pass to the limit in Π N ( ( ΠN (X N ) 2 2 Π N (X N ) 2 L 2)Π N(X N ) ). Idea is: almost sure convergence in H ɛ, and estimate in probability of a stronger term namely : u 4 L 4 :. Gives convergence in probability of the non linear term. Hence convergence a.s. for a subsequence.
25 Toward a generalization of Bourgain s result on any smooth bounded domain? Need a nice local Cauchy theory at the level of regularity where the measure is supported (rest = more or less automatic) Standard (X s,b ) WP thresholds for cubic : T 2 rational s > 0 (Bourgain) S 2 s > 1 4 (N. B, Gérard, Tzvetkov) T 2 irrational s > 1 3 (Catoire Wang) Compact surfaces (without bdry) s > 1 2 (N. B-Gérard-Tzvetkov) Compact surfaces (with bdry) s > 2 3 (Anton, Blair-Smith-Sogge) Control 1st iteration in X s,1/2+0, s > s c Perform X s,b analysis in the Fock representation
Almost sure global existence and scattering for the one-dimensional NLS
Almost sure global existence and scattering for the one-dimensional NLS Nicolas Burq 1 Université Paris-Sud, Université Paris-Saclay, Laboratoire de Mathématiques d Orsay, UMR 8628 du CNRS EPFL oct 20th
More informationGlobal solutions for the cubic non linear wave equation
Global solutions for the cubic non linear wave equation Nicolas Burq Université Paris-Sud, Laboratoire de Mathématiques d Orsay, CNRS, UMR 8628, FRANCE and Ecole Normale Supérieure Oxford sept 12th, 2012
More informationProbabilistic Sobolev embeddings, applications to eigenfunctions estimates
Contemporary Mathematics Probabilistic Sobolev embeddings, applications to eigenfunctions estimates Nicolas Bur and Gilles Lebeau Abstract. The purpose of this article is to present some probabilistic
More informationEigenvalues and eigenfunctions of the Laplacian. Andrew Hassell
Eigenvalues and eigenfunctions of the Laplacian Andrew Hassell 1 2 The setting In this talk I will consider the Laplace operator,, on various geometric spaces M. Here, M will be either a bounded Euclidean
More informationAlmost Sure Well-Posedness of Cubic NLS on the Torus Below L 2
Almost Sure Well-Posedness of Cubic NLS on the Torus Below L 2 J. Colliander University of Toronto Sapporo, 23 November 2009 joint work with Tadahiro Oh (U. Toronto) 1 Introduction: Background, Motivation,
More informationFrom the N-body problem to the cubic NLS equation
From the N-body problem to the cubic NLS equation François Golse Université Paris 7 & Laboratoire J.-L. Lions golse@math.jussieu.fr Los Alamos CNLS, January 26th, 2005 Formal derivation by N.N. Bogolyubov
More informationFocal points and sup-norms of eigenfunctions
Focal points and sup-norms of eigenfunctions Chris Sogge (Johns Hopkins University) Joint work with Steve Zelditch (Northwestern University)) Chris Sogge Focal points and sup-norms of eigenfunctions 1
More informationNotions such as convergent sequence and Cauchy sequence make sense for any metric space. Convergent Sequences are Cauchy
Banach Spaces These notes provide an introduction to Banach spaces, which are complete normed vector spaces. For the purposes of these notes, all vector spaces are assumed to be over the real numbers.
More informationStrichartz estimates for the Schrödinger equation on polygonal domains
estimates for the Schrödinger equation on Joint work with Matt Blair (UNM), G. Austin Ford (Northwestern U) and Sebastian Herr (U Bonn and U Düsseldorf)... With a discussion of previous work with Andrew
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informationStrichartz Estimates for the Schrödinger Equation in Exterior Domains
Strichartz Estimates for the Schrödinger Equation in University of New Mexico May 14, 2010 Joint work with: Hart Smith (University of Washington) Christopher Sogge (Johns Hopkins University) The Schrödinger
More information1 Math 241A-B Homework Problem List for F2015 and W2016
1 Math 241A-B Homework Problem List for F2015 W2016 1.1 Homework 1. Due Wednesday, October 7, 2015 Notation 1.1 Let U be any set, g be a positive function on U, Y be a normed space. For any f : U Y let
More informationA class of domains with fractal boundaries: Functions spaces and numerical methods
A class of domains with fractal boundaries: Functions spaces and numerical methods Yves Achdou joint work with T. Deheuvels and N. Tchou Laboratoire J-L Lions, Université Paris Diderot École Centrale -
More informationb i (µ, x, s) ei ϕ(x) µ s (dx) ds (2) i=1
NONLINEAR EVOLTION EQATIONS FOR MEASRES ON INFINITE DIMENSIONAL SPACES V.I. Bogachev 1, G. Da Prato 2, M. Röckner 3, S.V. Shaposhnikov 1 The goal of this work is to prove the existence of a solution to
More informationQuantum ergodicity. Nalini Anantharaman. 22 août Université de Strasbourg
Quantum ergodicity Nalini Anantharaman Université de Strasbourg 22 août 2016 I. Quantum ergodicity on manifolds. II. QE on (discrete) graphs : regular graphs. III. QE on graphs : other models, perspectives.
More informationMeasurable functions are approximately nice, even if look terrible.
Tel Aviv University, 2015 Functions of real variables 74 7 Approximation 7a A terrible integrable function........... 74 7b Approximation of sets................ 76 7c Approximation of functions............
More informationNormal form for the non linear Schrödinger equation
Normal form for the non linear Schrödinger equation joint work with Claudio Procesi and Nguyen Bich Van Universita di Roma La Sapienza S. Etienne de Tinee 4-9 Feb. 2013 Nonlinear Schrödinger equation Consider
More informationInégalités de dispersion via le semi-groupe de la chaleur
Inégalités de dispersion via le semi-groupe de la chaleur Valentin Samoyeau, Advisor: Frédéric Bernicot. Laboratoire de Mathématiques Jean Leray, Université de Nantes January 28, 2016 1 Introduction Schrödinger
More information9 Radon-Nikodym theorem and conditioning
Tel Aviv University, 2015 Functions of real variables 93 9 Radon-Nikodym theorem and conditioning 9a Borel-Kolmogorov paradox............. 93 9b Radon-Nikodym theorem.............. 94 9c Conditioning.....................
More informationDecouplings and applications
April 27, 2018 Let Ξ be a collection of frequency points ξ on some curved, compact manifold S of diameter 1 in R n (e.g. the unit sphere S n 1 ) Let B R = B(c, R) be a ball with radius R 1. Let also a
More informationGlobal Harmonic Analysis and the Concentration of Eigenfunctions, Part III:
Global Harmonic Analysis and the Concentration of Eigenfunctions, Part III: Improved eigenfunction estimates for the critical L p -space on manifolds of nonpositive curvature Christopher D. Sogge (Johns
More informationON STRICHARTZ ESTIMATES FOR SCHRÖDINGER OPERATORS IN COMPACT MANIFOLDS WITH BOUNDARY. 1. Introduction
ON STRICHARTZ ESTIMATES FOR SCHRÖDINGER OPERATORS IN COMPACT MANIFOLDS WITH BOUNDARY MATTHEW D. BLAIR, HART F. SMITH, AND CHRISTOPHER D. SOGGE 1. Introduction Let (M, g) be a Riemannian manifold of dimension
More informationFocal points and sup-norms of eigenfunctions
Focal points and sup-norms of eigenfunctions Chris Sogge (Johns Hopkins University) Joint work with Steve Zelditch (Northwestern University)) Chris Sogge Focal points and sup-norms of eigenfunctions 1
More informationON THE PROBABILISTIC CAUCHY THEORY FOR NONLINEAR DISPERSIVE PDES
ON THE PROBABILISTIC CAUCHY THEORY FOR NONLINEAR DISPERSIVE PDES ÁRPÁD BÉNYI, TADAHIRO OH, AND OANA POCOVNICU Abstract. In this note, we review some of the recent developments in the well-posedness theory
More informationA PEDESTRIAN APPROACH TO THE INVARIANT GIBBS MEASURES FOR THE 2-d DEFOCUSING NONLINEAR SCHRÖDINGER EQUATIONS
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
More informationFunctional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability...
Functional Analysis Franck Sueur 2018-2019 Contents 1 Metric spaces 1 1.1 Definitions........................................ 1 1.2 Completeness...................................... 3 1.3 Compactness......................................
More informationMath The Laplacian. 1 Green s Identities, Fundamental Solution
Math. 209 The Laplacian Green s Identities, Fundamental Solution Let be a bounded open set in R n, n 2, with smooth boundary. The fact that the boundary is smooth means that at each point x the external
More informationZdzislaw Brzeźniak. Department of Mathematics University of York. joint works with Mauro Mariani (Roma 1) and Gaurav Dhariwal (York)
Navier-Stokes equations with constrained L 2 energy of the solution Zdzislaw Brzeźniak Department of Mathematics University of York joint works with Mauro Mariani (Roma 1) and Gaurav Dhariwal (York) LMS
More informationSolutions: Problem Set 4 Math 201B, Winter 2007
Solutions: Problem Set 4 Math 2B, Winter 27 Problem. (a Define f : by { x /2 if < x
More informationRecall that any inner product space V has an associated norm defined by
Hilbert Spaces Recall that any inner product space V has an associated norm defined by v = v v. Thus an inner product space can be viewed as a special kind of normed vector space. In particular every inner
More informationStrauss conjecture for nontrapping obstacles
Chengbo Wang Joint work with: Hart Smith, Christopher Sogge Department of Mathematics Johns Hopkins University Baltimore, Maryland 21218 wangcbo@jhu.edu November 3, 2010 1 Problem and Background Problem
More informationGlobal Harmonic Analysis and the Concentration of Eigenfunctions, Part II:
Global Harmonic Analysis and the Concentration of Eigenfunctions, Part II: Toponogov s theorem and improved Kakeya-Nikodym estimates for eigenfunctions on manifolds of nonpositive curvature Christopher
More informationON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS
Bendikov, A. and Saloff-Coste, L. Osaka J. Math. 4 (5), 677 7 ON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS ALEXANDER BENDIKOV and LAURENT SALOFF-COSTE (Received March 4, 4)
More informationControl from an Interior Hypersurface
Control from an Interior Hypersurface Matthieu Léautaud École Polytechnique Joint with Jeffrey Galkowski Murramarang, microlocal analysis on the beach March, 23. 2018 Outline General questions Eigenfunctions
More informationOverview of normed linear spaces
20 Chapter 2 Overview of normed linear spaces Starting from this chapter, we begin examining linear spaces with at least one extra structure (topology or geometry). We assume linearity; this is a natural
More informationl(y j ) = 0 for all y j (1)
Problem 1. The closed linear span of a subset {y j } of a normed vector space is defined as the intersection of all closed subspaces containing all y j and thus the smallest such subspace. 1 Show that
More informationRecent developments in mathematical Quantum Chaos, I
Recent developments in mathematical Quantum Chaos, I Steve Zelditch Johns Hopkins and Northwestern Harvard, November 21, 2009 Quantum chaos of eigenfunction Let {ϕ j } be an orthonormal basis of eigenfunctions
More informationShort note on compact operators - Monday 24 th March, Sylvester Eriksson-Bique
Short note on compact operators - Monday 24 th March, 2014 Sylvester Eriksson-Bique 1 Introduction In this note I will give a short outline about the structure theory of compact operators. I restrict attention
More informationErgodic Theorems. Samy Tindel. Purdue University. Probability Theory 2 - MA 539. Taken from Probability: Theory and examples by R.
Ergodic Theorems Samy Tindel Purdue University Probability Theory 2 - MA 539 Taken from Probability: Theory and examples by R. Durrett Samy T. Ergodic theorems Probability Theory 1 / 92 Outline 1 Definitions
More informationMATHS 730 FC Lecture Notes March 5, Introduction
1 INTRODUCTION MATHS 730 FC Lecture Notes March 5, 2014 1 Introduction Definition. If A, B are sets and there exists a bijection A B, they have the same cardinality, which we write as A, #A. If there exists
More informationVariations on Quantum Ergodic Theorems. Michael Taylor
Notes available on my website, under Downloadable Lecture Notes 8. Seminar talks and AMS talks See also 4. Spectral theory 7. Quantum mechanics connections Basic quantization: a function on phase space
More information3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due ). Show that the open disk x 2 + y 2 < 1 is a countable union of planar elementary sets. Show that the closed disk x 2 + y 2 1 is a countable
More informationZdzislaw Brzeźniak. Department of Mathematics University of York. joint works with Mauro Mariani (Roma 1) and Gaurav Dhariwal (York)
Navier-Stokes equations with constrained L 2 energy of the solution Zdzislaw Brzeźniak Department of Mathematics University of York joint works with Mauro Mariani (Roma 1) and Gaurav Dhariwal (York) Stochastic
More informationKAM for NLS with harmonic potential
Université de Nantes 3rd Meeting of the GDR Quantum Dynamics MAPMO, Orléans, 2-4 February 2011. (Joint work with Benoît Grébert) Introduction The equation : We consider the nonlinear Schrödinger equation
More informationReal Analysis Problems
Real Analysis Problems Cristian E. Gutiérrez September 14, 29 1 1 CONTINUITY 1 Continuity Problem 1.1 Let r n be the sequence of rational numbers and Prove that f(x) = 1. f is continuous on the irrationals.
More informationBrownian Motion and Conditional Probability
Math 561: Theory of Probability (Spring 2018) Week 10 Brownian Motion and Conditional Probability 10.1 Standard Brownian Motion (SBM) Brownian motion is a stochastic process with both practical and theoretical
More informationarxiv: v1 [math-ph] 19 Oct 2018
A geometrization of quantum mutual information Davide Pastorello Dept. of Mathematics, University of Trento Trento Institute for Fundamental Physics and Applications (TIFPA-INFN) via Sommarive 14, Povo
More informationStrichartz Estimates in Domains
Department of Mathematics Johns Hopkins University April 15, 2010 Wave equation on Riemannian manifold (M, g) Cauchy problem: 2 t u(t, x) gu(t, x) =0 u(0, x) =f (x), t u(0, x) =g(x) Strichartz estimates:
More informationON THE ROBIN EIGENVALUES OF THE LAPLACIAN IN THE EXTERIOR OF A CONVEX POLYGON
NANOSYSTES: PHYSICS, CHEISTRY, ATHEATICS, 15, 6 (1, P. 46 56 ON THE ROBIN EIGENVALUES OF THE LAPLACIAN IN THE EXTERIOR OF A CONVEX POLYGON Konstantin Pankrashkin Laboratoire de mathématiques, Université
More informationResearch Statement. Xiangjin Xu. 1. My thesis work
Research Statement Xiangjin Xu My main research interest is twofold. First I am interested in Harmonic Analysis on manifolds. More precisely, in my thesis, I studied the L estimates and gradient estimates
More informationChapter 7: Bounded Operators in Hilbert Spaces
Chapter 7: Bounded Operators in Hilbert Spaces I-Liang Chern Department of Applied Mathematics National Chiao Tung University and Department of Mathematics National Taiwan University Fall, 2013 1 / 84
More informationPenalized Barycenters in the Wasserstein space
Penalized Barycenters in the Wasserstein space Elsa Cazelles, joint work with Jérémie Bigot & Nicolas Papadakis Université de Bordeaux & CNRS Journées IOP - Du 5 au 8 Juillet 2017 Bordeaux Elsa Cazelles
More informationGrowth of Sobolev norms for the cubic NLS
Growth of Sobolev norms for the cubic NLS Benoit Pausader N. Tzvetkov. Brown U. Spectral theory and mathematical physics, Cergy, June 2016 Introduction We consider the cubic nonlinear Schrödinger equation
More informationEstimates on Neumann eigenfunctions at the boundary, and the Method of Particular Solutions" for computing them
Estimates on Neumann eigenfunctions at the boundary, and the Method of Particular Solutions" for computing them Department of Mathematics Australian National University Dartmouth, July 2010 Outline Introduction
More information4 Countability axioms
4 COUNTABILITY AXIOMS 4 Countability axioms Definition 4.1. Let X be a topological space X is said to be first countable if for any x X, there is a countable basis for the neighborhoods of x. X is said
More informationOn the definitions of Sobolev and BV spaces into singular spaces and the trace problem
On the definitions of Sobolev and BV spaces into singular spaces and the trace problem David CHIRON Laboratoire J.A. DIEUDONNE, Université de Nice - Sophia Antipolis, Parc Valrose, 68 Nice Cedex 2, France
More informationLecture No 1 Introduction to Diffusion equations The heat equat
Lecture No 1 Introduction to Diffusion equations The heat equation Columbia University IAS summer program June, 2009 Outline of the lectures We will discuss some basic models of diffusion equations and
More informationSpectral theory for compact operators on Banach spaces
68 Chapter 9 Spectral theory for compact operators on Banach spaces Recall that a subset S of a metric space X is precompact if its closure is compact, or equivalently every sequence contains a Cauchy
More informationGradient estimates for eigenfunctions on compact Riemannian manifolds with boundary
Gradient estimates for eigenfunctions on compact Riemannian manifolds with boundary Xiangjin Xu Department of athematics Johns Hopkins University Baltimore, D 21218 Abstract The purpose of this paper is
More informationExtrema of discrete 2D Gaussian Free Field and Liouville quantum gravity
Extrema of discrete 2D Gaussian Free Field and Liouville quantum gravity Marek Biskup (UCLA) Joint work with Oren Louidor (Technion, Haifa) Discrete Gaussian Free Field (DGFF) D R d (or C in d = 2) bounded,
More informationPartial Differential Equations
Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,
More information3 Orthogonality and Fourier series
3 Orthogonality and Fourier series We now turn to the concept of orthogonality which is a key concept in inner product spaces and Hilbert spaces. We start with some basic definitions. Definition 3.1. Let
More informationHILBERT SPACES AND THE RADON-NIKODYM THEOREM. where the bar in the first equation denotes complex conjugation. In either case, for any x V define
HILBERT SPACES AND THE RADON-NIKODYM THEOREM STEVEN P. LALLEY 1. DEFINITIONS Definition 1. A real inner product space is a real vector space V together with a symmetric, bilinear, positive-definite mapping,
More informationEigenfunction Estimates on Compact Manifolds with Boundary and Hörmander Multiplier Theorem
Eigenfunction Estimates on Compact anifolds with Boundary and Hörmander ultiplier Theorem by Xiangjin Xu A dissertation submitted to the Johns Hopkins University in conformity with the requirements for
More informationEigenvalues and Eigenfunctions of the Laplacian
The Waterloo Mathematics Review 23 Eigenvalues and Eigenfunctions of the Laplacian Mihai Nica University of Waterloo mcnica@uwaterloo.ca Abstract: The problem of determining the eigenvalues and eigenvectors
More informationAnalysis Qualifying Exam
Analysis Qualifying Exam Spring 2017 Problem 1: Let f be differentiable on R. Suppose that there exists M > 0 such that f(k) M for each integer k, and f (x) M for all x R. Show that f is bounded, i.e.,
More information3 Compact Operators, Generalized Inverse, Best- Approximate Solution
3 Compact Operators, Generalized Inverse, Best- Approximate Solution As we have already heard in the lecture a mathematical problem is well - posed in the sense of Hadamard if the following properties
More informationMET Workshop: Exercises
MET Workshop: Exercises Alex Blumenthal and Anthony Quas May 7, 206 Notation. R d is endowed with the standard inner product (, ) and Euclidean norm. M d d (R) denotes the space of n n real matrices. When
More informationOn semilinear elliptic equations with measure data
On semilinear elliptic equations with measure data Andrzej Rozkosz (joint work with T. Klimsiak) Nicolaus Copernicus University (Toruń, Poland) Controlled Deterministic and Stochastic Systems Iasi, July
More informationTopics in Harmonic Analysis Lecture 1: The Fourier transform
Topics in Harmonic Analysis Lecture 1: The Fourier transform Po-Lam Yung The Chinese University of Hong Kong Outline Fourier series on T: L 2 theory Convolutions The Dirichlet and Fejer kernels Pointwise
More informationMATH 263: PROBLEM SET 1: BUNDLES, SHEAVES AND HODGE THEORY
MATH 263: PROBLEM SET 1: BUNDLES, SHEAVES AND HODGE THEORY 0.1. Vector Bundles and Connection 1-forms. Let E X be a complex vector bundle of rank r over a smooth manifold. Recall the following abstract
More informationSEPARABILITY AND COMPLETENESS FOR THE WASSERSTEIN DISTANCE
SEPARABILITY AND COMPLETENESS FOR THE WASSERSTEIN DISTANCE FRANÇOIS BOLLEY Abstract. In this note we prove in an elementary way that the Wasserstein distances, which play a basic role in optimal transportation
More informationn [ F (b j ) F (a j ) ], n j=1(a j, b j ] E (4.1)
1.4. CONSTRUCTION OF LEBESGUE-STIELTJES MEASURES In this section we shall put to use the Carathéodory-Hahn theory, in order to construct measures with certain desirable properties first on the real line
More informationInégalités spectrales pour le contrôle des EDP linéaires : groupe de Schrödinger contre semigroupe de la chaleur.
Inégalités spectrales pour le contrôle des EDP linéaires : groupe de Schrödinger contre semigroupe de la chaleur. Luc Miller Université Paris Ouest Nanterre La Défense, France Pde s, Dispersion, Scattering
More informationGibbs measure for the periodic derivative nonlinear Schrödinger equation
Gibbs measure for the periodic derivative nonlinear Schrödinger equation Laurent homann, Nikolay zvetkov o cite this version: Laurent homann, Nikolay zvetkov. Gibbs measure for the periodic derivative
More informationOn the spectrum of the Laplacian
On the spectrum of the Laplacian S. Kesavan Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai - 600113. kesh@imsc.res.in July 1, 2013 S. Kesavan (IMSc) Spectrum of the Laplacian July 1,
More informationL p Functions. Given a measure space (X, µ) and a real number p [1, ), recall that the L p -norm of a measurable function f : X R is defined by
L p Functions Given a measure space (, µ) and a real number p [, ), recall that the L p -norm of a measurable function f : R is defined by f p = ( ) /p f p dµ Note that the L p -norm of a function f may
More informationTHE QUINTIC NONLINEAR SCHRÖDINGER EQUATION ON THREE-DIMENSIONAL ZOLL MANIFOLDS
THE QUINTIC NONLINEAR SCHRÖDINGER EQUATION ON THREE-DIMENSIONAL ZOLL MANIFOLDS SEBASTIAN HERR Abstract. Let (M, g) be a three-dimensional smooth compact Riemannian manifold such that all geodesics are
More informationSPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS
SPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS TSOGTGEREL GANTUMUR Abstract. After establishing discrete spectra for a large class of elliptic operators, we present some fundamental spectral properties
More informationA Concise Course on Stochastic Partial Differential Equations
A Concise Course on Stochastic Partial Differential Equations Michael Röckner Reference: C. Prevot, M. Röckner: Springer LN in Math. 1905, Berlin (2007) And see the references therein for the original
More information7 Complete metric spaces and function spaces
7 Complete metric spaces and function spaces 7.1 Completeness Let (X, d) be a metric space. Definition 7.1. A sequence (x n ) n N in X is a Cauchy sequence if for any ɛ > 0, there is N N such that n, m
More informationChaos, Quantum Mechanics and Number Theory
Chaos, Quantum Mechanics and Number Theory Peter Sarnak Mahler Lectures 2011 Hamiltonian Mechanics (x, ξ) generalized coordinates: x space coordinate, ξ phase coordinate. H(x, ξ), Hamiltonian Hamilton
More informationSPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS
SPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS G. RAMESH Contents Introduction 1 1. Bounded Operators 1 1.3. Examples 3 2. Compact Operators 5 2.1. Properties 6 3. The Spectral Theorem 9 3.3. Self-adjoint
More informationMeasure Theory on Topological Spaces. Course: Prof. Tony Dorlas 2010 Typset: Cathal Ormond
Measure Theory on Topological Spaces Course: Prof. Tony Dorlas 2010 Typset: Cathal Ormond May 22, 2011 Contents 1 Introduction 2 1.1 The Riemann Integral........................................ 2 1.2 Measurable..............................................
More informationProbability and Measure
Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 84 Paper 4, Section II 26J Let (X, A) be a measurable space. Let T : X X be a measurable map, and µ a probability
More informationPotential Theory on Wiener space revisited
Potential Theory on Wiener space revisited Michael Röckner (University of Bielefeld) Joint work with Aurel Cornea 1 and Lucian Beznea (Rumanian Academy, Bukarest) CRC 701 and BiBoS-Preprint 1 Aurel tragically
More informationMyopic Models of Population Dynamics on Infinite Networks
Myopic Models of Population Dynamics on Infinite Networks Robert Carlson Department of Mathematics University of Colorado at Colorado Springs rcarlson@uccs.edu June 30, 2014 Outline Reaction-diffusion
More informationInverse Scattering with Partial data on Asymptotically Hyperbolic Manifolds
Inverse Scattering with Partial data on Asymptotically Hyperbolic Manifolds Raphael Hora UFSC rhora@mtm.ufsc.br 29/04/2014 Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/2014
More informationThe dynamical Sine-Gordon equation
The dynamical Sine-Gordon equation Hao Shen (University of Warwick) Joint work with Martin Hairer October 9, 2014 Dynamical Sine-Gordon Equation Space dimension d = 2. Equation depends on parameter >0.
More informationABSOLUTELY CONTINUOUS SPECTRUM OF A TYPICAL SCHRÖDINGER OPERATOR WITH A SLOWLY DECAYING POTENTIAL
ABSOLUTELY CONTINUOUS SPECTRUM OF A TYPICAL SCHRÖDINGER OPERATOR WITH A SLOWLY DECAYING POTENTIAL OLEG SAFRONOV 1. Main results We study the absolutely continuous spectrum of a Schrödinger operator 1 H
More informationRandom Process Lecture 1. Fundamentals of Probability
Random Process Lecture 1. Fundamentals of Probability Husheng Li Min Kao Department of Electrical Engineering and Computer Science University of Tennessee, Knoxville Spring, 2016 1/43 Outline 2/43 1 Syllabus
More information2 (Bonus). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due 9/5). Prove that every countable set A is measurable and µ(a) = 0. 2 (Bonus). Let A consist of points (x, y) such that either x or y is
More information16 1 Basic Facts from Functional Analysis and Banach Lattices
16 1 Basic Facts from Functional Analysis and Banach Lattices 1.2.3 Banach Steinhaus Theorem Another fundamental theorem of functional analysis is the Banach Steinhaus theorem, or the Uniform Boundedness
More informationSPECTRAL THEOREM FOR SYMMETRIC OPERATORS WITH COMPACT RESOLVENT
SPECTRAL THEOREM FOR SYMMETRIC OPERATORS WITH COMPACT RESOLVENT Abstract. These are the letcure notes prepared for the workshop on Functional Analysis and Operator Algebras to be held at NIT-Karnataka,
More informationSpectral Zeta Functions and Gauss-Bonnet Theorems in Noncommutative Geometry
Spectral Zeta Functions and Gauss-Bonnet Theorems in Noncommutative Geometry Masoud Khalkhali (joint work with Farzad Fathizadeh) Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions
More informationBrownian motion. Samy Tindel. Purdue University. Probability Theory 2 - MA 539
Brownian motion Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Brownian motion Probability Theory
More informationOn the convergence of sequences of random variables: A primer
BCAM May 2012 1 On the convergence of sequences of random variables: A primer Armand M. Makowski ECE & ISR/HyNet University of Maryland at College Park armand@isr.umd.edu BCAM May 2012 2 A sequence a :
More informationThe Schrödinger equation with spatial white noise potential
The Schrödinger equation with spatial white noise potential Arnaud Debussche IRMAR, ENS Rennes, UBL, CNRS Hendrik Weber University of Warwick Abstract We consider the linear and nonlinear Schrödinger equation
More informationGaussian Fields and Percolation
Gaussian Fields and Percolation Dmitry Beliaev Mathematical Institute University of Oxford RANDOM WAVES IN OXFORD 18 June 2018 Berry s conjecture In 1977 M. Berry conjectured that high energy eigenfunctions
More informationOPERATOR THEORY ON HILBERT SPACE. Class notes. John Petrovic
OPERATOR THEORY ON HILBERT SPACE Class notes John Petrovic Contents Chapter 1. Hilbert space 1 1.1. Definition and Properties 1 1.2. Orthogonality 3 1.3. Subspaces 7 1.4. Weak topology 9 Chapter 2. Operators
More information