A quick introduction to regularity structures

Transcription

1 A quick introduction to regularity structures Lorenzo Zambotti (LPMA, Univ. Paris 6) Workshop "Singular SPDEs"

2 Foreword The theory of Regularity Structures by Martin Hairer (2014) is a major advance in Stochastic Analysis. The original paper is long and difficult, and contains a number of deep new ideas. My aim is to convey a few of them, discussing an example which does not cover the whole extent of the theory but is already significant enough. Today I want to discuss the first main idea: a new point of view on Taylor expansions. The aim is to obtain a framework for the multiplication of random (Schwartz) distributions.

3 Some notations: the heat kernel For x = (x 1, x 2 ) R 2 we define the heat kernel G : R 2 R G(x) = 1 (x2 >0) Given k = (k 1, k 2 ) N 2 we define G (k) (x) = k 1 ( 1 exp x 2 ) 1. 2πx2 2x 2 x k 1 1 k 2 x k 2 2 G(x). In the following, we use x, y, z R 2. For k = (k 1, k 2 ) N 2 we define x k = x k 1 1 x k 2 2.

4 Parabolic scaling The heat kernel has a very important scaling property: G(δx 1, δ 2 x 2 ) = 1 δ G(x 1, x 2 ), δ > 0. This motivates the following definitions: x y s := x 1 y 1 + x 2 y 2 1/2, x = (x 1, x 2 ) R 2, k s := k 1 + 2k 2, k = (k 1, k 2 ) N 2. With these definitions, in what follows one can almost forget that x, y, z are not one-dimensional.

5 Abstract Monomials We define the following family T of symbols: 1, X {X 1, X 2 }, 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 ΞI k (Ξ), I((I 1 (Ξ)) 2 ) To a symbol τ we associate a real number τ called its homogeneity: Ξ = α < 3/2, X 1 = 1, X 2 = 2, 1 = 0 τ 1 τ n = τ τ n, I k (τ) = τ + 2 k s. Let T 0 = {τ : τ > Ξ, if τ = τ 1 τ 2 then τ i T 0, if τ = I k (τ 3 ) then τ 3 T 0 } and H be the space of linear combinations of elements in T 0.

6 The Π x operators We fix a bounded smooth function ξ and define recursively continuous generalized monomials Π x τ around the base point x Π x X(y) = (y x), Π x Ξ(y) = ξ(y), Π x (τ 1 τ n )(y) = n Π x τ i (y), i=1 I k (τ) Π x I k (τ)(y) = (G (k) Π x τ)(y) Then we can see that i=0 Π x τ(y) C y x τ s (y x) i (G (i+k) Π x τ)(x) i! and we can interpret analytically τ as the homogeneity of the monomial Π x τ.

7 Taylor series The Taylor series of the function y y k around the fixed base point x is y k = (y x + x) k = k i=0 ( ) k x k i (y x) i, i i.e. ( ) k y k = Π 0 X k (y) = Π x k x k i X i (y) = Π x [(X ] + x) k (y) i i=0 and more generally by translation ( ) k Π z X k = Π x k (x z) k i X i = Π x [(X ] + x z) k. i i=0

8 Taylor series The formula ( ) k Π z X k = Π x k (x z) k i X i = Π x [(X ] + x z) k. i i=0 gives a rule to transform a classical Taylor series centered at x into one centered at z, with the definition Γ xz : H H, Γ xz X k = (X + x z) k = k i=0 ( ) k (x z) k i X i. i This definition satisfies the simple properties Γ xy Γ yz = Γ xz, Π z = Π x Γ xz, Γ xz X k X k < k, Γ xz X k X k i C x z k i

9 The Γ operators In general we have the recursive definition of Γ xz : H H Γ xz X = X + (x z), Γ xz Ξ = Ξ, Γ xz τ i = i i Γ xz I k (τ) = I k (Γ xz τ) j< τ +2 k On can check again the compatibility condition Γ xz τ i (X + x z)j (Π x I k+j (Γ xz τ))(z) j! Π z = Π x Γ xz and the properties Γ xy Γ yz = Γ xz, Γ xz τ τ < τ, Γ xz τ τ l C x z τ l s, l < τ. The couple (Π, Γ) is called a model.

10 Examples For instance where G(ξ) := G ξ. Another example: Γ xz I(ΞI(Ξ)) = Γ xz I(Ξ) = I(Ξ) + (G(ξ)(x) G(ξ)(z)), = G(ξG(ξ))(x) G(ξG(ξ))(z) G(ξ)(z)(G(ξ)(x) G(ξ)(z)) + (G(ξ)(x) G(ξ)(z)) I(Ξ) + I(ΞI(Ξ)) where G(ξG(ξ)) := G (ξ (G ξ)). We should check: Γ xz τ τ < τ, Γ xz τ τ l C x z τ l s, l < τ.

11 Classical polynomials Given a classical monomial y y k, we can associate to each x its Taylor series around x f (x) = (X + x) k = Γ x0 X k = Γ x0 f (0). By linearity, we obtain that f H is the Taylor series of a (classical) polynomial P( ) if and only if and in this case f (x) Γ xz f (z) 0 f (z) = deg(p) i=0 P (i) (z) i! X i. In particular Π x f (x)(y) Π z f (z)(y) = P(y) for all x, y, z.

12 Generalized polynomials We consider now a generic f (0) H and define f (x) = Γ x0 f (0), x. This is again equivalent to f (x) Γ xz f (z) 0. Examples: f (x) = x + X, Π x f (x)(y) = x + (y x) = y f (x) = Ξ, Π x f (x)(y) = ξ(y)

13 Generalized polynomials We consider now a generic f (0) H and define f (x) = Γ x0 f (0), x. This is again equivalent to f (x) Γ xz f (z) 0. Examples: f (x) = x + X, Π x f (x)(y) = x + (y x) = y f (x) = Ξ, Π x f (x)(y) = ξ(y) f (x) = G(ξ)(x) + I(Ξ), Π x f (x)(y) = G(ξ)(x) + G(ξ)(y) G(ξ)(x) = G(ξ)(y),

14 Polynomials f (x) = G(ξG(ξ))(x) + G(ξ)(x) I(Ξ) + I(ΞI(Ξ)), Π x f (x)(y) = G(ξG(ξ))(x) + G(ξ)(x)(G(ξ)(y) G(ξ)(x)) + G(ξG(ξ))(y) G(ξG(ξ))(x) G(ξ)(x)(G(ξ)(y) G(ξ)(x)) = G(ξG(ξ))(y) where G(ξG(ξ)) := G (ξ (G ξ)).

15 Hölder functions Back to the classical situation, a function g 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-hölder continuous. In fact this is equivalent to requiring that for all x there exists a polynomial P x ( ) of degree k such that and in this case necessarily g(y) P x (y) C y x k+β P x (y) = k i=0 g (i) (x) (y x) i. i!

16 If we define then we obtain f (x) Γ xz f (z) = f (x) = k i=0 X i i! k i=0 and in particular g C k+β iff In this case g (i) (x) i! X i H, g i k i (x) j=0 g (i+j) (z) j! f (x) Γ xz f (z) i C x z k+β i. (x z) j g(x) = Π x f (x)(x), g(y) Π x f (x)(y) 0, (reconstruction) Π x f (x) Π z f (z).

17 Functional norm In the general case, for γ > 0 we say that f D γ if f takes values in the linear span of the symbols with homogeneity < γ and for all β < γ f (x) Γ xy f (y) β C f x y γ β s This innocent-looking condition can be in practice very complicated, because of the presence of the Γ operators. It is a notion of Hölder regularity in this setting of generalized monomials. If f takes values in sums of X k, then the definition is equivalent to the classical C γ -regularity (for γ / N). The coefficient multiplying X k is then the k-th derivative of f divided by k!. This definition is inspired by Gubinelli s theory of controlled rough paths.

18 General models For relevant applications, the assumptions that all Π x τ are continuous functions is far too restrictive. For instance, Π x Ξ(y) = ξ(y) is supposed to model a white noise. For ϕ C c (R 2 ) and δ > 0 we set ( ) ϕ x,δ (z) := δ 3 ϕ δ 1 (z 1 x 1 ), δ 2 (z 2 x 2 ).

19 General models A model is given by a couple (Π, Γ) where 1. for all x, Π x : H S (R 2 ) and for all ϕ C c (R 2 ) Π x τ(ϕ x,δ ) Cδ τ. 2. for all x, y, z, Γ xz : H H is such that Γ xx = Id, Γ xy Γ yz = Γ xz and Γ zx τ τ < τ, Γ zx τ τ l C z x s τ l, l < τ. 3. for all x, z: Π z = Π x Γ xz. Note that Π x is not necessarily multiplicative in general: this is in fact the main point since we deal with distributions.

20 The reconstruction theorem Our starting problem was to associate to a function u a Taylor expansion U(x) around each point x. What about the inverse problem? Given a Taylor expansion U(x) H around each point x, can we find a function u which has this expansion up to a remainder?

21 The reconstruction theorem Our starting problem was to associate to a function u a Taylor expansion U(x) around each point x. What about the inverse problem? Given a Taylor expansion U(x) H around each point x, can we find a function u which has this expansion up to a remainder? This is the content of the cornerstone of the theory, the Reconstruction Theorem: For all γ > 0 there exists a unique operator R : D γ S (R 2 ) such that Rf (y) Π x f (x)(y) C f x y γ s or, more precisely, such that Rf (ϕ x,δ ) Π x f (x)(ϕ x,δ ) Cδ γ.

22 Classical multiplication Consider g i C γ i, i = 1, 2, with γ i = k i + β i > 0 and k i g i (y) = a j i (x) (y x)j + R i (x, y), R i (x, y) C x y k i +β i. j=0 Pointwise multiplication gives, setting γ = γ 1 γ 2, g 1 g 2 (y) = b j (x) (y x) j + R(x, y), j γ R(x, y) C x y γ where b j (x) = j m=0 a m 1 (x)aj m 2 (x). After all, if g 1 and g 2 are in C 1 then g 1 g 2 is not necessarily in C 2!

23 Multiplication of modelled distributions Consider f i D γ i, γ i > 0, i = 1, 2: and α i := min j τ j i. N i f i (x) = a j i (x) τ j i, j=1 By the reconstruction theorem, setting g i := Rf i : N i g i (y) = a j i (x) Π xτ j i (y) + R i(x, y) j=1 where R i (x, y) C x y γ i s is a remainder. Can we define the product of g 1 and g 2?

24 Multiplication Suppose now that ξ is continuous ( Ξ < 0) and Π x is multiplicative. Then pointwise multiplication gives 2 N i g 1 (y)g 2 (y) = a j i (x) Π xτ j i (y) + R i(x, y) i=1 j=1 = a j 1 1 (x)a j 2 2 (x) Π x τ j 1 1 τ j 2 2 (y) + R(x, y) j 1 +j 2 γ where γ = (γ 1 + α 2 ) (γ 2 + α 1 ), α i is the minimal homogeneity of (τ j i ) j and R(x, y) C x y γ s. Of course, in general α 1 α 2 can be negative and then γ < γ 1 γ 2.

25 Abstract multiplication Then this suggests the following definition in the general case: for f i D γ i, j = 1, 2 N i f i (x) = a j i (x) τ j i, i=1 setting γ = (γ 1 + α 2 ) (γ 2 + α 1 ) and (f 1 f 2 )(x) = ai 1 (x)a2 i (x) τ i 1 τi 2. τ 1 i + τ 2 i <γ Then by a general result f 1 f 2 D γ, and if γ > 0 the reconstruction theorem gives a well-defined distribution R(f 1 f 2 ) such that R(f 1 f 2 )(y) = τ 1 i + τ 2 i <γ with R(x, y) C x y γ s. a 1 i (x)a2 i (x) Π x(τ 1 i τ 2 i )(y) + R(x, y)

26 SPDEs Let us now consider a SPDE driven by space-time white noise ξ, for instance KPZ t u = xx u + ( x u) 2 + ξ, t > 0, x R.

27 SPDEs Let us now consider a SPDE driven by space-time white noise ξ, for instance KPZ t u = xx u + ( x u) 2 + ξ, t > 0, x R. The theory of regularity structures allows to "solve" this equation as follows: 1. ξ is replaced by a regularized version ξ ε = ρ ε ξ

28 SPDEs Let us now consider a SPDE driven by space-time white noise ξ, for instance KPZ t u = xx u + ( x u) 2 + ξ, t > 0, x R. The theory of regularity structures allows to "solve" this equation as follows: 1. ξ is replaced by a regularized version ξ ε = ρ ε ξ 2. one constructs a model (Π ε, Γ ε ) as described above

29 SPDEs Let us now consider a SPDE driven by space-time white noise ξ, for instance KPZ t u = xx u + ( x u) 2 + ξ, t > 0, x R. The theory of regularity structures allows to "solve" this equation as follows: 1. ξ is replaced by a regularized version ξ ε = ρ ε ξ 2. one constructs a model (Π ε, Γ ε ) as described above 3. the solution u ε has a generalized Taylor series U ε in D 3/2+κ,ε, κ > 0

30 SPDEs Let us now consider a SPDE driven by space-time white noise ξ, for instance KPZ t u = xx u + ( x u) 2 + ξ, t > 0, x R. The theory of regularity structures allows to "solve" this equation as follows: 1. ξ is replaced by a regularized version ξ ε = ρ ε ξ 2. one constructs a model (Π ε, Γ ε ) as described above 3. the solution u ε has a generalized Taylor series U ε in D 3/2+κ,ε, κ > 0 4. the regularized SPDE is written as a fixed point in U ε in D 3/2+κ,ε

31 SPDEs Let us now consider a SPDE driven by space-time white noise ξ, for instance KPZ t u = xx u + ( x u) 2 + ξ, t > 0, x R. The theory of regularity structures allows to "solve" this equation as follows: 1. ξ is replaced by a regularized version ξ ε = ρ ε ξ 2. one constructs a model (Π ε, Γ ε ) as described above 3. the solution u ε has a generalized Taylor series U ε in D 3/2+κ,ε, κ > 0 4. the regularized SPDE is written as a fixed point in U ε in D 3/2+κ,ε 5. a renormalization procedure is performed on the model, yielding (ˆΠ ε, ˆΓ ε )

32 SPDEs Let us now consider a SPDE driven by space-time white noise ξ, for instance KPZ t u = xx u + ( x u) 2 + ξ, t > 0, x R. The theory of regularity structures allows to "solve" this equation as follows: 1. ξ is replaced by a regularized version ξ ε = ρ ε ξ 2. one constructs a model (Π ε, Γ ε ) as described above 3. the solution u ε has a generalized Taylor series U ε in D 3/2+κ,ε, κ > 0 4. the regularized SPDE is written as a fixed point in U ε in D 3/2+κ,ε 5. a renormalization procedure is performed on the model, yielding (ˆΠ ε, ˆΓ ε ) and Ûε, û ε

33 SPDEs Let us now consider a SPDE driven by space-time white noise ξ, for instance KPZ t u = xx u + ( x u) 2 + ξ, t > 0, x R. The theory of regularity structures allows to "solve" this equation as follows: 1. ξ is replaced by a regularized version ξ ε = ρ ε ξ 2. one constructs a model (Π ε, Γ ε ) as described above 3. the solution u ε has a generalized Taylor series U ε in D 3/2+κ,ε, κ > 0 4. the regularized SPDE is written as a fixed point in U ε in D 3/2+κ,ε 5. a renormalization procedure is performed on the model, yielding (ˆΠ ε, ˆΓ ε ) and Ûε, û ε 6. one proves convergence of (ˆΠ ε, ˆΓ ε ) to (ˆΠ, ˆΓ) and of û ε to û, independent of the particular regularization