3 on large scales Jean-Christophe Mourrat Hendrik Weber - PowerPoint PPT Presentation

Dynamical ϕ 4 3 on large scales Jean-Christophe Mourrat Hendrik Weber Mathematics Institute University of Warwick Paths to, from and in renormalization Potsdam, 11 Feb. 2016

Stochastic quantisation equation ∂ t ϕ = △ ϕ − ϕ 3 − A ϕ + ξ p.2

Stochastic quantisation equation ∂ t ϕ = △ ϕ − ϕ 3 − A ϕ + ξ ξ space-time white noise, i.e. centred Gaussian E ξ ( t , x ) ξ ( t ′ , x ′ ) = δ ( t − t ′ ) δ ( x − x ′ ). Spatial dimension d = 2 or d = 3. A ∈ R real parameter. p.2

Stochastic quantisation equation ∂ t ϕ = △ ϕ − ϕ 3 − A ϕ + ξ ξ space-time white noise, i.e. centred Gaussian E ξ ( t , x ) ξ ( t ′ , x ′ ) = δ ( t − t ′ ) δ ( x − x ′ ). Spatial dimension d = 2 or d = 3. A ∈ R real parameter. Invariant measure, ϕ 4 model, formally given by − 1 � ϕ 4 + 2 A ϕ 2 dx � � µ ∝ exp ν ( d ϕ ) 4 ν distribution of Gaussian free field. p.2

Aim of this talk ∂ t ϕ = △ ϕ − ϕ 3 − A ϕ + ξ Problem: ξ very irregular ⇒ ϕ distribution valued. Renormalisation procedure (= removing infinite constants) necessary when dealing with nonlinearity. p.3

Aim of this talk ∂ t ϕ = △ ϕ − ϕ 3 − A ϕ + ξ Problem: ξ very irregular ⇒ ϕ distribution valued. Renormalisation procedure (= removing infinite constants) necessary when dealing with nonlinearity. Local theory available: d = 2 da Prato-Debussche ’03. d = 3 Hairer ’14, Catellier-Chouk ’14. p.3

Aim of this talk ∂ t ϕ = △ ϕ − ϕ 3 − A ϕ + ξ Problem: ξ very irregular ⇒ ϕ distribution valued. Renormalisation procedure (= removing infinite constants) necessary when dealing with nonlinearity. Local theory available: d = 2 da Prato-Debussche ’03. d = 3 Hairer ’14, Catellier-Chouk ’14. Main result of this talk: Global theory d = 2 existence and uniqueness on [0 , ∞ ) × R 2 . d = 3 existence and uniqueness on [0 , ∞ ) × T 3 . p.3

Why is this interesting? ∂ t ϕ = △ ϕ − ϕ 3 − A ϕ + ξ Relation to QFT. p.4

Why is this interesting? ∂ t ϕ = △ ϕ − ϕ 3 − A ϕ + ξ Relation to QFT. Interesting dynamics: Arise as scaling limits (Presutti et al. 90s , Mourrat-W. ’14). Similar properties to Ising model (has phase transition, ergodicity properties...). p.4

Why is this interesting? ∂ t ϕ = △ ϕ − ϕ 3 − A ϕ + ξ Relation to QFT. Interesting dynamics: Arise as scaling limits (Presutti et al. 90s , Mourrat-W. ’14). Similar properties to Ising model (has phase transition, ergodicity properties...). Method in a nutshell: Only non-linear term has right sign – strong non-linear damping term. p.4

Why is this interesting? ∂ t ϕ = △ ϕ − ϕ 3 − A ϕ + ξ Relation to QFT. Interesting dynamics: Arise as scaling limits (Presutti et al. 90s , Mourrat-W. ’14). Similar properties to Ising model (has phase transition, ergodicity properties...). Method in a nutshell: Only non-linear term has right sign – strong non-linear damping term. Difficulty: How to extract this in presence of random distributions, infinite constants, etc. p.4

Why is this interesting? ∂ t ϕ = △ ϕ − ϕ 3 − A ϕ + ξ Relation to QFT. Interesting dynamics: Arise as scaling limits (Presutti et al. 90s , Mourrat-W. ’14). Similar properties to Ising model (has phase transition, ergodicity properties...). Method in a nutshell: Only non-linear term has right sign – strong non-linear damping term. Difficulty: How to extract this in presence of random distributions, infinite constants, etc. This is a PDE talk. p.4

Two-dimensional case: Da Prato-Debussche 2003 Stochastic step: solution of stochastic heat equation: ∂ t = △ + ξ. 2 ❀ 3 ❀ distributions in C 0 − . Can construct and . All , , p.5

Two-dimensional case: Da Prato-Debussche 2003 Stochastic step: solution of stochastic heat equation: ∂ t = △ + ξ. 2 ❀ 3 ❀ distributions in C 0 − . Can construct and . All , , Deterministic step: u = ϕ − . ∂ t u = △ u − ( + u ) 3 � u 3 + 3 u 2 + 3 u + � . = △ u − p.5

Two-dimensional case: Da Prato-Debussche 2003 Stochastic step: solution of stochastic heat equation: ∂ t = △ + ξ. 2 ❀ 3 ❀ distributions in C 0 − . Can construct and . All , , Deterministic step: u = ϕ − . ∂ t u = △ u − ( + u ) 3 � u 3 + 3 u 2 + 3 u + � . = △ u − Multiplicative inequality: If α < 0 < β with α + β > 0 � � � � � � � τ u C α � � τ � u C β . � � � C α p.5

Two-dimensional case: Da Prato-Debussche 2003 Stochastic step: solution of stochastic heat equation: ∂ t = △ + ξ. 2 ❀ 3 ❀ distributions in C 0 − . Can construct and . All , , Deterministic step: u = ϕ − . ∂ t u = △ u − ( + u ) 3 � u 3 + 3 u 2 + 3 u + � . = △ u − Multiplicative inequality: If α < 0 < β with α + β > 0 � � � � � � � τ u C α � � τ � u C β . � � � C α Short time existence, uniqueness via Picard iteration. p.5

Non-explosion on the torus I Testing against u p − 1 � t 1 � |∇ u s | 2 � � � u t � p L p − � u 0 � p � � � u p − 2 L 1 + � u p +2 � + ( p − 1) � L 1 ds � � p s s p � 0 � t � � B ( u s , τ s ) , u p − 1 = ds . s 0 Use the sign of − u 3 to get additional “good term”. p.6

Non-explosion on the torus I Testing against u p − 1 � t 1 � |∇ u s | 2 � � � u t � p L p − � u 0 � p � � � u p − 2 L 1 + � u p +2 � + ( p − 1) � L 1 ds � � p s s p � 0 � t � � B ( u s , τ s ) , u p − 1 = ds . s 0 Use the sign of − u 3 to get additional “good term”. Bad terms: − 3 u 2 − 3 u � B , u p − 1 � � , u p − 1 � = − . p.6

Non-explosion on the torus II � u 2 , u p − 1 � � � u p +1 , Control bad term: = . 1 Duality: � � �� u p +1 , � � � u p +1 � B α 1 , 1 � � B − α ∞ , ∞ . � � � p.7

Non-explosion on the torus II � u 2 , u p − 1 � � � u p +1 , Control bad term: = . 1 Duality: � � �� u p +1 , � � � u p +1 � B α 1 , 1 � � B − α ∞ , ∞ . � � � 2 Interpolation: � u p +1 � B α 1 , 1 � � u p +1 � 1 − α �∇ ( u p +1 ) � α L 1 + � u p +1 � L 1 . L 1 p.7

Non-explosion on the torus II � u 2 , u p − 1 � � � u p +1 , Control bad term: = . 1 Duality: � � �� u p +1 , � � � u p +1 � B α 1 , 1 � � B − α ∞ , ∞ . � � � 2 Interpolation: � u p +1 � B α 1 , 1 � � u p +1 � 1 − α �∇ ( u p +1 ) � α L 1 + � u p +1 � L 1 . L 1 ∞ , ∞ finite by construction. The terms � u p +1 � 1 − α sup 0 ≤ t ≤ T � � B − α L 1 and �∇ ( u p +1 ) � α L 1 are controlled by good terms. Yields a priori bound on � u � L p , enough for non-explosion. p.7

Discussion d = 2 Solution theory on full space R 2 via approximation on large tori. Hardest part uniqueness. p.8

Discussion d = 2 Solution theory on full space R 2 via approximation on large tori. Hardest part uniqueness. We expect to be able to show tightness of orbits in Krylov Bogoliubov scheme ⇒ alternative construction of invariant measure. p.8

Discussion d = 2 Solution theory on full space R 2 via approximation on large tori. Hardest part uniqueness. We expect to be able to show tightness of orbits in Krylov Bogoliubov scheme ⇒ alternative construction of invariant measure. Cubic − ϕ : 3: could be replaced by any Wick polynomial with odd degree. p.8

Discussion d = 2 Solution theory on full space R 2 via approximation on large tori. Hardest part uniqueness. We expect to be able to show tightness of orbits in Krylov Bogoliubov scheme ⇒ alternative construction of invariant measure. Cubic − ϕ : 3: could be replaced by any Wick polynomial with odd degree. Related (but different) construction for PAM on R × R 3 by Hairer, Labbé ’15. p.8

The three dimensional case Simple da Prato-Debussche trick does not work: p.9

The three dimensional case Simple da Prato-Debussche trick does not work: , , can still be constructed but lower regularity: ∈ C − 1 ∈ C − 3 2 − , ∈ C − 1 − , 2 − . Equation for u = ϕ − � u 3 + 3 u 2 + 3 u + ∂ t u = △ u − � cannot be solved by Picard iteration. p.9

The three dimensional case Simple da Prato-Debussche trick does not work: , , can still be constructed but lower regularity: ∈ C − 1 ∈ C − 3 2 − , ∈ C − 1 − , 2 − . Equation for u = ϕ − � u 3 + 3 u 2 + 3 u + ∂ t u = △ u − � cannot be solved by Picard iteration. Next order expansion u = ϕ − + gives � u 3 + 3 u 2 + 3 u − 3 � . ∂ t u = △ u − + . . . p.9

The three dimensional case Simple da Prato-Debussche trick does not work: , , can still be constructed but lower regularity: ∈ C − 1 ∈ C − 3 2 − , ∈ C − 1 − , 2 − . Equation for u = ϕ − � u 3 + 3 u 2 + 3 u + ∂ t u = △ u − � cannot be solved by Picard iteration. Next order expansion u = ϕ − + gives � u 3 + 3 u 2 + 3 u − 3 � . ∂ t u = △ u − + . . . Still cannot be solved, because of u . Expanding further does not solve the problem. p.9

System of equations with paraproducts Catellier-Chouk: Split up remainder equation: u = v + w p.10