Renormalisation in regularity structures Lorenzo Zambotti Univ. - PowerPoint PPT Presentation

Renormalisation in regularity structures Lorenzo Zambotti Univ. Paris 6 (based on work by Martin Hairer and on joint work with Yvain Bruned and M.H.) February 2016, Potsdam Lorenzo Zambotti February 2016, Potsdam

Summary ◮ Stochastic Partial Differential Equations ◮ Taylor expansions ◮ Renormalization groups ◮ Hopf algebras and co-modules ◮ Labelled trees and forests ◮ Feynman diagrams ◮ . . . Lorenzo Zambotti February 2016, Potsdam

Notations For x = ( x 1 , . . . , x d ) ∈ R d and k = ( k 1 , . . . , k d ) ∈ N d we write d x k := � x k i i ∈ R . i = 1 X = ( X 1 , . . . , X d ) denotes a variable, and X k the abstract monomial d X k := � X k i i . i = 1 A monomial is a function Π x X k : R d → R Π x X k ( y ) := ( y − x ) k . Lorenzo Zambotti February 2016, Potsdam

Taylor expansions The Taylor expansion of the function y �→ y k around the fixed base point x is k � k � y k = ( y − x + x ) k = � x k − i ( y − x ) i i i = 0 k ( y − x ) i ∂ i y k � ∂ i y k � k ! � � � ( k − i )! x k − i . = , = � � ∂ y i ∂ y i i ! � � y = x y = x i = 0 Therefore the abstract Taylor expansion of y �→ y k around x is k � k � x k − i X i = ( X + x ) k ∈ R [ X ] . � U ( x ) := i i = 0 Moreover we recover the function y �→ y k y k = [Π x U ( x )] ( y ) . Lorenzo Zambotti February 2016, Potsdam

Change of the base point If we set, for x , z ∈ R d , Γ xz : R [ X ] �→ R [ X ] k � k � Γ xz X k = ( X + x − z ) k = � ( x − z ) k − i X i , i i = 0 then it is easy to see that U ( x ) = Γ xz U ( z ) . Indeed U ( x ) = ( X + x ) k = ( X + z + x − z ) k = Γ xz U ( z ) . Lorenzo Zambotti February 2016, Potsdam

Change of the base point The operator k � k � Γ xz X k = ( X + x − z ) k = � ( x − z ) k − i X i . i i = 0 gives a rule to transform a classical Taylor expansion centered at z of a fixed polynomial into one centered at x . This definition satisfies the simple properties Π z = Π x Γ xz , Γ xx = Id , Γ xy Γ yz = Γ xz , deg (Γ xz X k − X k ) < k � Γ xz X k − X k � i ≤ C � x − z � k − i . Lorenzo Zambotti February 2016, Potsdam

Classical polynomials Given a global function y �→ y k , we can associate to each x its Taylor expansion around x U ( x ) = ( X + x ) k = Γ x 0 X k = Γ x 0 U ( 0 ) . By linearity, we obtain that U �→ R [ X ] is the Taylor expansion of a (classical) polynomial P ( · ) if and only if U ( x ) − Γ xz U ( z ) ≡ 0 and in this case deg ( P ) P ( i ) ( z ) � X i . U ( z ) = i ! i = 0 In particular for all x , y , z Π x U ( x )( y ) ≡ Π z U ( z )( y ) = P ( y ) . Lorenzo Zambotti February 2016, Potsdam

H¨ older functions A function u : R d → R is said to be of class C k + β if it is everywhere k -times differentiable with (bounded) derivatives and the k -th derivative is β -H¨ older continuous. In fact this is equivalent to requiring that for all x there exists a polynomial P x ( · ) of degree k such that | u ( y ) − P x ( y ) | ≤ C | y − x | k + β (1) and in this case necessarily � k k � u ( i ) ( x ) u ( i ) ( x ) ( y − x ) i = Π x � � X i P x ( y ) = ( y ) . i ! i ! i = 0 i = 0 Lorenzo Zambotti February 2016, Potsdam

H¨ older functions If we define k u ( i ) ( x ) X i ∈ R [ X ] , � U ( x ) = i ! i = 0 then we obtain   k − i k u ( i + j ) ( z ) X i �  u ( i ) ( x ) − � ( x − z ) j U ( x ) − Γ xz U ( z ) =  i ! j ! i = 0 j = 0 and in particular u ∈ C k + β iff for all i ≤ k � U ( x ) − Γ xz U ( z ) � i ≤ C � x − z � k + β − i . We say that U ∈ D γ if U : R d → R [ X ] takes values in the span of monomials with degree strictly less than γ and for all i < γ � U ( x ) − Γ xz U ( z ) � i ≤ C � x − z � γ − i . Lorenzo Zambotti February 2016, Potsdam

Differential equations This gives a characterization of H¨ older functions u in terms of their Taylor sum U and the operators Γ xz . In general u ( x ) = Π x U ( x )( x ) , ( reconstruction ) u ( y ) − Π x U ( x )( y ) � = 0 , Π x U ( x ) � = Π z U ( z ) . For instance, if d = 1 then the ODE with α -H¨ older coefficient b : R → R du dx = b ( u ( x )) , u ( 0 ) = u 0 ∈ R can be coded by U ∈ D 1 + α where U ( x ) = u ( x ) + b ( u ( x )) X , x ∈ R . Lorenzo Zambotti February 2016, Potsdam

Generalized Taylor expansions Regularity Structures are a far-reaching generalization of the previous construction. We want to add new monomials representing random distributions and to solve stochastic (partial) differential equations. Lorenzo Zambotti February 2016, Potsdam

Generalized Taylor expansions Regularity Structures are a far-reaching generalization of the previous construction. We want to add new monomials representing random distributions and to solve stochastic (partial) differential equations. For instance, let ξ = ξ ( x ) is a space-time white noise on R d , i.e. a centered Gaussian field such that x , y ∈ R d . E ( ξ ( x ) ξ ( y )) = δ ( x − y ) , A concrete realisation: for all ψ ∈ L 2 ( R d − 1 ) and t ∈ R � � [ 0 , t ] × R d − 1 ψ ( x ) ξ ( x ) d x := B k ( t ) � e k , ψ � , k where ( B k ) k is an IID sequence of Brownian motions and ( e k ) k is a complete orthonormal system in L 2 ( R d − 1 ) . Lorenzo Zambotti February 2016, Potsdam

The stochastic heat equation Let v : R d → R solve the heat equation with external forcing x ∈ R d , ∂ t v = ∆ v + ξ, where d � ∂ 2 ∂ t = ∂ x 1 , ∆ := x i . i = 2 The properties of this ”process” depend heavily on the dimension, since  < + ∞ , d = 2 � t C d  Var ( v ( x )) = ds d − 1 s 0 2 = + ∞ , d ≥ 3  so that for d ≥ 3 the solution is a random distribution. Lorenzo Zambotti February 2016, Potsdam

Singular stochastic PDEs If ∇ = ( ∂ x i , i = 2 , . . . , d ) then for a class of equations x ∈ R × R d − 1 ∂ t u = ∆ u + F ( u , ∇ u , ξ ) , Lorenzo Zambotti February 2016, Potsdam

Singular stochastic PDEs If ∇ = ( ∂ x i , i = 2 , . . . , d ) then for a class of equations x ∈ R × R d − 1 ∂ t u = ∆ u + F ( u , ∇ u , ξ ) , ∂ t u = ∆ u + ( ∇ u ) 2 + ξ, ( KPZ ) x ∈ R × R , ∂ t u = ∆ u + f ( u ) ( ∇ u ) 2 + g ( u ) ξ, ( gKPZ ) x ∈ R × R , x ∈ R × R 2 , ( PAM ) ∂ t u = ∆ u + u ξ, ∂ t u = ∆ u − u 3 + ξ, (Φ 4 x ∈ R × R 3 . 3 ) Lorenzo Zambotti February 2016, Potsdam

Singular stochastic PDEs If ∇ = ( ∂ x i , i = 2 , . . . , d ) then for a class of equations x ∈ R × R d − 1 ∂ t u = ∆ u + F ( u , ∇ u , ξ ) , ∂ t u = ∆ u + ( ∇ u ) 2 + ξ, ( KPZ ) x ∈ R × R , ∂ t u = ∆ u + f ( u ) ( ∇ u ) 2 + g ( u ) ξ, ( gKPZ ) x ∈ R × R , x ∈ R × R 2 , ( PAM ) ∂ t u = ∆ u + u ξ, ∂ t u = ∆ u − u 3 + ξ, (Φ 4 x ∈ R × R 3 . 3 ) Even for polynomial non-linearities, we do not know how to properly define products of (random) distributions. This is where infinities arise (see below). Lorenzo Zambotti February 2016, Potsdam

Some notations: the heat kernel Let d ≥ 2. For x = ( x 1 , . . . , x d ) ∈ R d we define the heat kernel G : R d �→ R − x 2 2 + · · · + x 2 � � 1 d G ( x ) = ✶ ( x 1 > 0 ) √ 2 π x 1 . exp 2 x 1 Given k = ( k 1 , . . . , k d ) ∈ N d we define G ( k ) ( x ) = ∂ k 1 · · · ∂ k d G ( x ) . ∂ x k 1 ∂ x k d 1 d Lorenzo Zambotti February 2016, Potsdam

Parabolic scaling The heat kernel has a very important scaling property: G ( δ 2 x 1 , δ x 2 , . . . , δ x d ) = 1 δ G ( x ) , δ > 0 . This motivates the following definitions: � x − y � s := | x 1 − y 1 | 1 / 2 + | x 2 − y 2 | + · · · + | x d − y d | , x ∈ R d , k ∈ N 2 . | k | s := 2 k 1 + k 2 + · · · + k d , Lorenzo Zambotti February 2016, Potsdam

Generalized Monomials We want to introduce new monomials which allow to approximate u locally. We need a monomial for the noise : we introduce Ξ , Π x Ξ( y ) := ξ ( y ) . Remember that Π x X k ( y ) = ( y − x ) k and | Π x X k ( y ) | ≤ � x − y � | k | s . s Then we see that the scaled degree | k | s of X k has both an algebraic and an analytic interpretation. We need a similar concept for all (abstract) monomials. Lorenzo Zambotti February 2016, Potsdam

Abstract Monomials We define the following family T of symbols (trees): ◮ 1 , X ∈ { X 1 , . . . , X d } , Ξ ∈ T ◮ if τ 1 , . . . , τ n ∈ T then τ 1 · · · τ n ∈ T (commutative and associative product) ◮ if τ ∈ T then I ( τ ) ∈ T and I k ( τ ) ∈ T (formal convolution with the heat kernel differentiated k times) Examples: I (Ξ) , X n Ξ I k (Ξ) , I (( I 1 (Ξ)) 2 ) To a symbol τ we associate a real number | τ | called its homogeneity: | Ξ | = α < − ( d + 1 ) / 2, | X 1 | = 2, | X 2 | = 1, | 1 | = 0 | τ 1 · · · τ n | = | τ 1 | + · · · + | τ n | , |I k ( τ ) | = | τ | + 2 − | k | s . Let H be the space of linear combinations of elements in T . α < 0 is chosen so that ξ is a.s. a distribution of order at least α . Lorenzo Zambotti February 2016, Potsdam

The Π operators We fix a bounded smooth function ξ and define recursively functions of y ∈ R d Π 1 ( y ) = 1 , Π X ( y ) = y , Π Ξ( y ) = ξ ( y ) , n � Π( τ 1 · · · τ n )( y ) = Π τ i ( y ) , i = 1 Π I k ( τ )( y ) = ( G ( k ) ∗ Π τ )( y ) . These are global functions which include y �→ y k . Lorenzo Zambotti February 2016, Potsdam