Invariant Gibbs measures for the three-dimensional wave equation - PowerPoint PPT Presentation

Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity Bjoern Bringmann University of California, Los Angeles October 6th, 2020

“Baby” equation Nonlinear wave equations (main result) The Gibbs measure Local dynamics Global dynamics

“Baby” equation

Linear and nonlinear oscillator d 2 u d t 2 � u � λ ☎ u 3 ✏ 0 . (Osc) λ ✏ 0: Harmonic oscillator. λ → 0: Duffing oscillator. We define the Hamiltonian H : R 2 Ñ R by H ♣ q , p q ✏ p 2 2 � q 2 2 � λ q 4 4 . Using the variables q ✏ u and p ✏ d u d t , we obtain that ✂ q ✂ ❇ p H ✡ ✂ ✡ ✡ d p (Osc) ð ñ ✏ ✏ . ✁ q ✁ λ q 3 p ✁❇ q H d t

The Gibbs measure (1) Conservation of energy : For any initial data q ♣ 0 q , p ♣ 0 q P R , H ♣ q ♣ t q , p ♣ t qq ✏ H ♣ q ♣ 0 q , p ♣ 0 qq for all t P R . (2) Liouville’s theorem: Since ♣ q , p q ÞÑ ♣❇ p H , ✁❇ q H q is divergence- free, the solution map ♣ q ♣ 0 q , p ♣ 0 qq P R 2 Ñ ♣ q ♣ t q , p ♣ t qq P R 2 preserves the Lebesgue measure d q d p . Theorem (Invariance). The Gibbs measure d µ ✏ Z ✁ 1 exp ♣✁ H ♣ q , p qq d q d p is invariant under the Hamiltonian flow.

Example: Harmonic Oscillator Ñ Gaussian measure H ♣ q , p q ✏ p 2 2 � q 2 2 . Gibbs measure: ✁ 1 2 q 2 ✁ 1 d µ ✏ Z ✁ 1 exp ✁ 2 p 2 ✠ d q d p . Ñ Gaussian measure. Hamiltonian flow: ✂ q ♣ t q ✡ ✂ cos ♣ t q ✡ ✂ q ♣ 0 q ✡ ✁ sin ♣ t q ✏ . p ♣ t q sin ♣ t q cos ♣ t q p ♣ 0 q Ñ Rotation in phase-space. Then: Theorem (Invariance) Ñ (Standard) Gaussian measure is rotation invariant.

Example: Duffing oscillator H ♣ q , p q ✏ p 2 2 � q 2 2 � λ q 4 4 . Gibbs measure: ✁ 1 2 q 2 ✁ λ 4 q 4 ✁ 1 d µ ✏ Z ✁ 1 exp ✁ 2 p 2 ✠ d q d p ✁ λ ✁ 1 2 q 2 ✁ 1 ✏ Z ✁ 1 exp ✁ 4 q 4 ✠ ✁ 2 p 2 ✠ exp d q d p ❧♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♥ ↕ 1 is absolutely continuous with respect to the Gaussian measure. Hamiltonian ODE: d q d p d t ✏ ✁ q ✁ λ q 3 . d t ✏ p , Ñ Complicated (Jacobi elliptic functions).

Gibbs measure: Implications Most Hamiltonian ODEs cannot be solved explicitly and individual solutions are difficult to analyze. However, Theorem (Invariance) ñ Information on typical solutions. For example, one can use: ✌ Poincar´ e recurrence theorem. ✌ Furstenberg multiple recurrence theorem.

Nonlinear wave equations (main result)

Nonlinear wave equations ✁❇ 2 t u ✁ u � ∆ x u ✏ u p ♣ t , x q P R ✂ T d (NLW) ✁❇ 2 t u ✁ u � ∆ x u ✏ ♣ V ✝ u 2 q u ♣ t , x q P R ✂ T d (HNLW) In the Hartree-nonlinearity, V : T d Ñ R is an interaction potential. Hamiltonian structure: H ♣ u , ❇ t u q ✏ 1 ➺ ⑤❇ t u ⑤ 2 � ⑤ u ⑤ 2 � ⑤ ∇ u ⑤ 2 ✠ ✁ d x � V ♣ u q , 2 T d 1 V ♣ u q ✏ 1 ➺ ➺ T d u p � 1 d x T d ♣ V ✝ u 2 q u 2 d x . with V ♣ u q ✏ or p � 1 4 Then: Hamiltonian � Symplectic form Ñ (NLW) and (HNLW).

Nonlinear wave equations Question: Since (NLW) and (HNLW) exhibit a Hamiltonian struc- ture, do they also have invariant Gibbs measures? Three parts of the question: (1) Can the Gibbs measure be constructed rigorously? (2) What are the properties of the Gibbs measure? (3) Can the invariance of the Gibbs measure be proven? Similar questions can be posed for other Hamiltonian PDEs, such as nonlinear Schr¨ odinger equations.

Overview of current results Theorem. Existence and invariance of the Gibbs measure (after a renormalization). Dim. & Nonlinearity Wave Schr¨ odinger d ✏ 1 , ⑤ u ⑤ p ✁ 1 u Friedlander ‘85, Bourgain ‘94 Zhidkov ‘94 ⑤ u ⑤ 2 u d ✏ 2, Bourgain ‘96 Oh-Thomann ‘18 d ✏ 2, ⑤ u ⑤ p ✁ 1 u Deng-Nahmod-Yue ‘19 β → 1: Oh-Okamoto- β → 2: Bourgain ‘97 d ✏ 3, ♣ V β ✝ ⑤ u ⑤ 2 q u Tolomeo ‘20 β → 1 ④ 2: Feasible. β → 0: B. ‘20 β → 0: Open. d ✏ 3, ⑤ u ⑤ 2 u Open (Extremely) open The (periodic) interaction potential V β : T 3 Ñ R behaves like ⑤ x ⑤ ✁♣ 3 ✁ β q . Smaller β Ñ higher difficulty.

Main result ✁❇ 2 t u ✁ u � ∆ x u ✏ : ♣ V ✝ u 2 q u : ♣ t , x q P R ✂ T 3 . (HNLW) ✌ V ✏ V β is a periodic version of ⑤ x ⑤ ✁♣ 3 ✁ β q . ✌ : ♣ V ✝ u 2 q u : is a renormalization of ♣ V ✝ u 2 q u . Theorem (B. ‘20). The Gibbs measure corresponding to (HNLW) exists and, for 0 ➔ β ➔ 1 ④ 2, is mutually singular with respect to the Gaussian free field. Furthermore, it is invariant under (HNLW). Remark. This is the only theorem on the invariance of a singular Gibbs measure for any dispersive equation. The singularity heavily affects the global (but not local) dynamics. B., Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity I/II (65 and 133 pages)

The Gibbs measure

Gaussian free field ✁ 1 ➺ d g ✏ Z ✁ 1 exp ✁ T d ⑤ u ⑤ 2 � ⑤ ∇ u ⑤ 2 d x ✠ d u 2 For any n P Z d , we define ① n ② ✏ ❛ 1 � ⑤ n ⑤ 2 . By using the “Fourier” transformation g n ➳ ① n ② e i ① n , x ② , ♣ g n q n P Z d ÞÑ u ✏ n P Z d we can view d g as the push-forward of the Gaussian measure ✁ ⑤ g n ⑤ 2 ✁ ✠ Z ✁ 1 â exp d g n . n 2 n P Z d Spatial regularity: g n s ➔ 1 ✁ d ① n ② e i ① n , x ② P C s ➳ x ♣ T d q a.s. ð ñ 2 . n P Z d

Gibbs measure and the Gaussian free field The Gibbs measure is (formally) given by ✁ 1 ➺ ✁ T d ⑤ u ⑤ 2 � ⑤ ∇ u ⑤ 2 d x ✠ d µ ❜ ♣ u , u t q ✏ Z ✁ 1 � ✟ exp ✁ V ♣ u q exp d u 0 2 ✁ 1 ➺ ✁ ✠ ❜ Z ✁ 1 T d ⑤❇ t u ⑤ 2 d x exp d u t . 1 2 Using our Gaussian free field, we (rigorously) define g ❜ ✏ g ❜ � ✟ ① ∇ ② # g . Idea: Show that the potential energy V is g -a.s. finite and P L 1 ♣ g q③t 0 ✉ . � ✟ exp ✁ V ♣ u q Then, we can (rigorously) define d µ ❜ ④ d g ❜ ✏ Z ✁ 1 exp � ✟ ✁ V ♣ u q .

Gibbs measure and the Gaussian free field Regularity: s ➔ 1 ✁ d ④ 2. Difficulty: If d ➙ 2, then s ➔ 0. Ñ V ♣ u q ✏ ✽ g -a.s. The previous idea can (only) be implemented if (i): d ✏ 1, (ii): d ✏ 2, (iii): d ✏ 3 and β → 1 ④ 2. In (ii) and (iii), the potential energy V needs to be renormalized. This is indicated by writing : V : instead of V . Remark. If µ ❜ ✦ g ❜ , we can use Gaussian initial data in the local theory for (NLW) and (HNLW).

Stochastic quantization Stochastic quantization: The Gibbs measure is formally invariant under the stochastic heat equation ❄ ❇ t u � u ✁ ∆ u ✏ : u 3 : � (Heat) 2 η, u ♣ 0 q ✏ φ. where η is space-time white noise. Stoch. Quant. (Heat) (NLW) Gibbs measure Physics: Nelson ‘66, Parisi-Wu ‘81. Mathematics: Da Prato-Debussche ‘03, Hairer-Matetski ‘15, Mourrat-Weber ‘17, Gubinelli-Hofmanov´ a ‘18, . . . Variational approach: Barashkov-Gubinelli ‘18, ‘20. Similar spirit but relies on stochastic control theory. Ñ Used here.

Gibbs measure for the 3d Hartree nonlinearity V ♣ u q ✏ 1 ➺ V ♣ x q ✓ ⑤ x ⑤ ✁♣ 3 ✁ β q . T d ♣ V ✝ u 2 q u 2 d x and 4 The Gibbs measure µ ❜ corre- Theorem (B. ‘20, Measures). sponding to a (renormalization of) V exists and, for 0 ➔ β ➔ 1 ④ 2, is mutually singular with respect to the Gaussian free field. Furthermore, there exists a reference measure ν ❜ , an ambient prob- : Ω Ñ C ✁ 1 ④ 2 ✁ ǫ ♣ T 3 q ability space ♣ Ω , F , P q , and random functions x : Ω Ñ C 1 ④ 2 � β ✁ ǫ ♣ T 3 q such that and x µ ❜ ✦ ν ❜ , ν ❜ ✏ Law P g ❜ ✏ Law P ♣ q . � ✟ � and , Remark. Oh-Okamoto-Tolomeo ‘20 independently obtained a sim- ilar result. Why the “dots”? Ñ Stay tuned!

Local dynamics

Local dynamics Theorem (B. ‘20, LWP). On any small interval, (HNLW) is well-posed with high probability under the Gibbs measure. Replace Gibbs measure Ñ Reference measure ν ❜ ✏ Law P ♣ � q . We write u r t s ✏ ♣ u ♣ t q , ❇ t u ♣ t qq . Recall that (HNLW) is given by ★ ✁❇ 2 t u ✁ u � ∆ x u ✏ : ♣ V ✝ u 2 q u : ♣ t , x q P R ✂ T 3 , u r 0 s ✏ � . Deterministic critical regularity: s c ✏ 1 ④ 2 ✁ β . Regularity of : s ➔ ✁ 1 ④ 2. Ñ Almost a full derivative below the deterministic theory.

Ansatz � ✝ Ansatz: u ✏ � X � Y , where: ✌ is the linear evolution of . ✝ ✌ is the “first” Picard iterate, which solves ✟ ✝ � ✟ 2 ✟ ✁ ❇ 2 � � t ✁ 1 � ∆ ✏ : V ✝ : ✌ X has regularity 1 ④ 2 ✁ but exhibits a para-controlled structure , as introduced by Gubinelli, Imkeller, and Perkowski. ✌ Y is a smooth remainder at regularity 1 ④ 2 � which contains . The threshold s ✏ 1 ④ 2 determines whether the multiplication by is well-defined. Mourrat, Weber, Xu, Construction of Φ 4 3 diagrams for pedestrians.