Rough paths, regularity structures and renormalisation

Transcription

1 Rough paths, regularity structures and renormalisation Lorenzo Zambotti (LPMA) Santander, July 2017

2 Many thanks to The organisers Martin Hairer Massimiliano Gubinelli Kurusch Ebrahimi-Fard, Dominique Manchon, Frédéric Patras Yvain Bruned Carlo Bellingeri, Henri Elad Altman, Nikolas Tapia...

3 Integration and Multiplication Let f, g : [0, T] R two continuous functions. What does it mean to define the integral T when f, g are not differentiable? 0 f r ġ r dr Important example: g = B with (B t ) t 0 a Brownian motion. Starting point of the Rough Paths theory (Terry Lyons, Massimiliano Gubinelli). Example of a more general problem: given a distribution (ġ) and a non-smooth function ( f ), how can we define their product? Namely a distribution f ġ.

4 Local approximation If g is of class C 1, then we define I t := 0 f r ġ r dr, Then we have I 0 = 0 and for 0 s t T We write I t I s f s (g t g s ) = s t [0, T]. ( f r f s ) ġ r dr = o( t s ). I 0 = 0, I t I s = f s (g t g s ) + R st, R st = o( t s ). These properties characterise (I t ) t [0,T], since if we have I 1 and I 2 then setting I 12 := I 1 I 2 which implies I 12 constant. I 12 t I 12 s = o( t s )

5 Local approximation Let us still study the formula I 0 = 0, I t I s = f s (g t g s ) + R st, R st = o( t s ). If we compute for 0 s u t T which does not depend on I. R st R su R ut = ( f u f s )(g t g u ) Therefore the existence of I is equivalent to the existence of R such that the above formula holds.

6 A cochain complex Les us define for n 1 n := {(t 1,..., t n ) [0, T] n : t 1 t n }, C n := {f : n R continuous}, n+1 δ n : C n C n+1, (δ n f ) t1...t n+1 = ( 1) n+2 k f t1... t k Then we have δ n+1 δ n 0 (exercise!) k=1 \...t n+1. if g C n+1 and δ n+1 g = 0, then g = δ n f with f C n (exercise!). In particular we have an exact cochain complex δ R C 1 δ 1 2 δ C2 3 C3

7 Local approximation Therefore, existence of I C 1 such that I 0 = 0, (δ 1 I) st = f s (g t g s ) + o( t s ), where (δ 1 I) st = I t I s, is equivalent to the existence of R C 2 such that (δ 2 R) sut = ( f u f s )(g t g u ), where (δ 2 R) sut = R st R su R ut, R st = o( t s ). Gubinelli calls I the integral, A st := f s (g t g s ) the germ, and R st the remainder.

8 The sewing lemma For α > 0 and h C n we set h α := h(t 1,..., t n ) sup (t 1,...,t n) n t n t 1 α and we say that h C α n if h α < +. We also set C α+ n := β>α C β n. Theorem (Gubinelli) There exists a unique map Λ : C 1+ 3 δ 2 C 2 C 1+ 2 such that δ 2 Λ = id C 1+ 3 δ 2 C 2. Moreover Λ satifies for all α > 1 ΛB α K α B α, B C 1+ 3 δ 2 C 2. Proof. See the first lecture sheet of MG

9 A first application: Young integration Theorem If f C α, g C β (standard Hölder spaces) with α + β > 1 then there exists a unique pair (I, R) C β C α+β 2 such that The map I 0 = 0, I t I s = f s (g t g s ) + R st. C α C β ( f, g) I C β is the unique continuous extension of C 1 C 1 ( f, g) 0 f ġ du C 1.

10 Proof Existence. Setting A st := f s (g t g s ) C β 2, we already know that (δ 2 A) sut = ( f u f s )(g t g u ), 0 s t T, so that (δ 2 A) sut C u s α t u β C t s α+β. Setting R := Λδ 2 A C α+β 2 then A + R C β 2 and δ 2 (A + R) = δ 2 A δ 2 Λδ 2 A = 0, so that A + R = δ 1 I with I C β. Uniqueness. If I 1, I 2 then It 12 Is 12 = o( t s ). Continuity. The estimate follows from I C β f C α g C β Λδ 2 A α+β K α+β δ 2 A α+β, δ 2 A C α+β 3 δ 2 C 2. in the Sewing Lemma.

11 Dyadic approximation Let us consider for t n i := i2 n T and n 0 Then, since t n+1 2i = t n i, I n t I n+1 t = 2 n i=1 2 n i=1 It n = 1 (t n i t) A t n i 1 ti n. 2 n i=1 1 (t n i t) (δ 2 A) t n+1 2i 2 tn+1 ( ) A t n i 1 ti n A t n+1 A 2i 2 tn+1 2i 1 t n+1 2i 1 tn+1 2i 2i 1 tn+1 2i 2 n(α+β 1) which is summable. Then we obtain that I n t I t as n + (see again MG )

12 If α = β > 1/2 Theorem If f, g C α, with α > 1/2 then there exists a unique pair (I, R) C α C 2α 2 such that I 0 = 0, I t I s = f s (g t g s ) + R st. In the above situation, we write I t =: I [0,t] ( f, g) =: 0 f dg. Then uniqueness yields the Integration by parts formula since I [0,t] ( f, g) + I [0,t] (g, f ) = f t g t f 0 g 0, f t g t f s g }{{} s = f s (g t g s ) + g s ( f t f s ) + ( f }{{} t f s )(g t g s ) }{{} I t I s A st R st.

13 If α = β 1/2 However, if α = β 1/2 then neither existence nor uniqueness. This problem is revelant for stochastic integration and SDEs: X t = X 0 + with (B t ) t 0 a standard Brownian motion. 0 σ(x s ) db s In particular, we can not apply the Sewing Lemma to the germ A st := f s (g t g s ) since 2α 1 and therefore in general δ 2 A / C We need to change the germ A in such a way that δ 2 A C 1+ 3.

14 Modifying the germ Note that the result of the integration map is supposed to satisfy I t I s = f s (g t g s ) + R st, R C 2α 2. Then we could assume that also f satisfies f t f s = f s(g t g s ) + R st, R C 2α 2. If Y C 2 is such that (δ 2 Y) sut = (g u g s )(g t g u ), setting then A st := f s (g t g s ) + f sy st, (δ 2 A) sut = ( f u f s f s(g u g s ))(g }{{} t g u ) C 3α R su If 1/3 < α 1/2 we are in the setting of the Sewing Lemma. 3.

15 Rough paths For g C α, we want Y C 2 such that (δ 2 Y) sut = (g u g s )(g t g u ). In fact, for g : [0, T] R it is enough to set Y st := 1 2 (g t g s ) 2, since (a + b) 2 a 2 b 2 = 2ab. This is a natural choice, which moreover shows how much all this is related to generalised Taylor expansions. However it is not the only possible choice, nor necessarily the most desirable. As we ll see below, Itô integration is not covered by this setting. In fact, for any such Y we can set Y := Y + δ 1 h and Y still has the desired property. Note that Y st = 1 2 (g t g s ) 2 belongs to C2 2α. For reasons which will be clear later, we require this property for all Y.

16 Rough and controlled paths Let us summarise: given α ]1/3, 1/2] and g C α, we call a pair (g, Y) C α C 2α 2 a Rough Path if (δ 2 Y) sut = (g u g s )(g t g u ), 0 s u t T. A pair ( f, f ) C α C α is controlled by g if f t f s f s(g t g s ) t s 2α. We denote by D 2α g the space of paths controlled by g.

17 Integration of controlled paths In this setting, we can apply the Sewing Lemma to the germ A st := f s (g t g s ) + f sy st and define the integral I C α such that δ 1 I = A Λδ 2 A, I 0 = 0. Then the integration map acts (continuously) on controlled paths D 2α g ( f, f ) (I, f ) D 2α g.

18 Brownian motion in R Let us suppose that g B, a standard Brownian motion in R. Then for all α < 1/2, a.s. B C α. We fix α ]1/3, 1/2]. We set Y st = 1 2 (B t B s ) 2. For all α < 1/2, a.s. Y C α 2. A path controlled by B is ( f, f ) C α C α such that f t f s f s(b t B s ) t s 2α, 0 s t T. For all such ( f, f ) there exists a unique I C α such that I 0 = 0 and I t I s f s (B t B s ) f sy st t s 3α, 0 s t T. Moreover I t I s f s (B t B s ) t s 2α, 0 s t T. If the Stratonovich integral 0 f s db s is well defined, it is equal to I.

19 Brownian motion in R Let us suppose that g B, a standard Brownian motion in R. Then for all α < 1/2, a.s. B C α. We fix α ]1/3, 1/2]. We set Y st = 1 2 [(B t B s ) 2 (t s)]. For all α < 1/2, a.s. Y C α 2. A path controlled by B is ( f, f ) C α C α such that f t f s f s(b t B s ) t s 2α, 0 s t T. For all such ( f, f ) there exists a unique I C α such that I 0 = 0 and I t I s f s (B t B s ) f sy st t s 3α, 0 s t T. Moreover I t I s f s (B t B s ) t s 2α, 0 s t T. If the Itô integral 0 f s db s is well defined, it is equal to I.

20 Multi-dimensional (rough) paths It is important to extend the above setting to functions g : [0, T] R d. If α ]1/3, 1/2] and g C α, we call (g i, Y ij, 1 i, j d), with (g i, Y ij ) C α C 2α 2 a Rough Path if for all i, j (δ 2 Y ij ) sut = (g i u g i s)(g j t g j u), 0 s u t T. We say that ( f, f i ) C α (C α ) d is controlled by g if f t f s i f i s (g i t g i s) t s 2α. We denote by D 2α g the space of paths controlled by g. In this setting, we can apply the Sewing Lemma to the germ A j st := f s (g j t g j s) + i f s i Yst ij and define the integral I i C α such that δ 1 I j = A j Λδ 2 A j, I j 0 = 0.

21 Multi-dimensional (rough) paths First, this allows to cover SDEs in R d X t = X σ(x s ) db s, X, B C([0, T]; R d ), σ : R d R d R d. Furthermore, the situation is more insteresting and complicated, since there is no canonical choice for the off-diagonal terms (δ 2 Y ij ) sut = (g i u g i s)(g j t g j u), i j. It is always possible to find Y ij C 2 satisfying this, take e.g. Yst ij = g i s(g j t g j s). However in general this choice does not satisfy the analytical requirement Y ij C 2α 2. Therefore existence of Rough Paths over a path g : [0, T] R d is not obvious.

22 Brownian motion in R d Let us suppose that g i B i, with B = (B 1,..., B d ) a standard Brownian motion in R d. We fix α ]1/3, 1/2]. We set Yst ij = s (Bi u B i s) db j u. For all α < 1/2, a.s. Y C2 2α (not obvious). A path controlled by B is ( f, f ) C α (C α ) d such that f t f s i f i s (B i t B i s) t s 2α, 0 s t T. For all such ( f, f ) there exists a unique I (C α ) d such that I 0 = 0 and I j t I j s f s (B j t B j s) i f i s Y ij st t s 3α, 0 s t T. If the Stratonovich integral 0 f s db s is well defined, it is equal to I.

23 Brownian motion in R d Let us suppose that g i B i, with B = (B 1,..., B d ) a standard Brownian motion in R d. We fix α ]1/3, 1/2]. We set Yst ij = s (Bi u B i s) db j u. For all α < 1/2, a.s. Y C2 2α (not obvious). A path controlled by B is ( f, f ) C α (C α ) d such that f t f s i f i s (B i t B i s) t s 2α, 0 s t T. For all such ( f, f ) there exists a unique I (C α ) d such that I 0 = 0 and I j t I j s f s (B j t B j s) i f i s Y ij st t s 3α, 0 s t T. If the Itô integral 0 f s db s is well defined, it is equal to I.

24 Remarks In the Young situation (α > 1/2), f and g play symmetric rôles. The integral is a bilinear functional If α 1/2, the pair (g, Y) is a non-linear object by the constraint on δ 2 Y. In particular, rough paths are non-linear objects. This is where algebra gets into the picture. On the other hand, for a fixed rough path, controlled paths form a linear space and the integral is a linear functional. The off-diagonal terms Yst ij = s (Bi u B i s) db j u, i j, are defined using Stochastic calculus. Since δ 2 Y ij C 1 3, the Sewing Lemma can not be used to define them.

25 Remarks Another fundamental remark: the analytical bound in the Sewing Lemma implies that the integral is continuous w.r.t. ( f, g, Y). This implies that solutions to a Rough Differential Equation are continuous w.r.t. the underlying rough path. This was the motivation of Terry Lyons when he introduced Rough Paths in the first place, and it is called the Continuity of the Itô-Lyons map. (Hans Föllmer wrote in the 80s a famous note conjecturing this kind of results) In the classical theory of stochastic calculus and SDEs, one has in general only measurability of the Itô map.

26 Lower regularity If we want to consider a path g : [0, T] R with even lower regularity, say g C α with α ]1/4, 1/3], then we have to modify further the germ. We assume that ( f, f, f ) (C α ) 3 satisfies f t f s = f s(g t g s ) + f s (g t g s ) 2 2 Then the germ satisfies A st := f s (g t g s ) + f s (g t g s ) R st, R C 3α 2. + f s (g t g s ) 3 3! (δ 2 A) sut = R su (g t g u ) (f t f s f s (g t g u ) 2 (g t g s )). }{{} 2 =:R st In order to apply the Sewing Lemma, we need that R C 2α 2.

27 Lower regularity If we want to consider a path g : [0, T] R with even lower regularity, say g C α with α ]1/4, 1/3], then we have to modify further the germ. We assume that ( f, f, f ) (C α ) 3 satisfies f t f s = f s(g t g s ) + f s (g t g s ) 2 2 f t f s = f s (g t g s ) + R st, Then the germ satisfies (exercise...) A st := f s (g t g s ) + f s (g t g s ) R st, R C 3α 2, R C 2α 2. + f s (g t g s ) 3 3! (δ 2 A) sut = R su (g t g u ) R (g t g u ) 2 su. 2 If 1/4 < α 1/3 we are in the setting of the Sewing Lemma.

28 Compact notations Let α ]0, 1[ and g C α. We set X n st := 1 n! (g t g s ) n, s, t [0, T], n 0. By Newton s binomial theorem n X n st = X k su X n k ut, s, u, t [0, T] k=0 (a convolution product...). Note that X n C nα 2 and n 1 (δ 2 X n ) sut = X k su X n k ut, s, u, t [0, T]. k=1 Now we define N as the largest integer such that Nα 1, i.e. N = 1/α. We say that Z : [0, T] R {0,...,N 1} is controlled by X if N 1 Zt n = Zs k X k n st + R n st, n {0,..., N 1}, R n C (N n)α 2. k=n

29 Compact notations N 1 Then the germ A st := Zs k X k+1 st k=0 k=0 N 1 [ (δ 2 A) sut = Z k s (X k+1 st X k+1 su ) Zu k Xut k+1 ] N 1 = k=0 N 1 = i=0 Z k s k+1 i=1 N 1 Xut i+1 k=i N 1 X k+1 i su X i ut Z k s X k i su N 1 = X i+1 [ ut Z i u R i su i=0 N 1 = R i su X i+1 ut i=0 k=0 N 1 i=0 N 1 ] i=0 Z k u X k+1 ut Z i u X i+1 ut Z i u X i+1 ut C (N i+i+1)α 3 C satisfies

30 Compact notations We define as above I by I 0 = 0 and δ 1 I = A Λδ 2 A, R := Λδ 2 A. If we set Z : [0, T] R {0,...,N 1} by Z 0 t = I t, Z n t := Z n 1 t, n {1,..., N 1}, then Z is a controlled path. Indeed N 1 Z t 0 k=0 Z n t = Z n 1 t = N 2 Z s k X k st = I t I s N 1 k=n 1 i=0 Z i s X i+1 st N 1 Zs k X k n+1 st +R n st = = [δ 1 I A] st +Zs N 1 X N st C2 Nα. k=n Z k s X k n st +R n st+zs N 1 X N n st.

31 Iterated integrals In four celebrated papers (1954, 1957, 1958, 1971) Kuo-Tsai Chen discovered that the family of iterated integrals of a smooth path in R d has a number of algebraic properties. Let s t and X : [s, t] R d a smooth path. Set X st () := 1, X st (i 1... i n ) = = s s Ẋ in r n dr n rn with n N, i k {1,..., d}. s X sr (i 1... i n 1 ) Ẋr in dr Ẋ i n 1 r n 1 dr n 1 r2 s Ẋ i 1 r1 dr 1, Then X st is in the dual V of the vector space V spanned by all finite words {(a 1... a n )} n 0 with letters in {1,..., d} (tensor algebra). Example: X st ( } i. {{.. } i ) = 1 n! (Xi t Xs) i n. n

32 Bialgebra On V we have a bialgebra structure (defined by Frédéric on Monday) the shuffle product : V V V iσ jτ = i(σ jτ) + j(iσ τ). the deconcatenation coproduct : V V V (i 1... i n ) := n (i 1... i k ) (i k+1... i n ) k=0 associativity (id ) = ( id) coassociativity (id ) = ( id) unit 1 : R V, (id 1)(v, r) = (1 id)(r, v) = rv counit 1 : V R, (id 1 ) = (1 id) = id compatibility (a b) = ( a) ( b) grading V = n 0 V n where V n is the span of the words with n letters.

33 Convolution product Note the recursive formulae () = () (), (τi) = (id i) τ + τi (). If V has a coproduct, then on V we can define the convolution product : V V V which is associative with unit 1. E.g. (A B)(τ) := (A B) τ X su X ut, τ = X su X ut, τ, τ V. Important remark: is commutative if and only if is cocommutative. (Deconcatenation is not cocommutative)

34 Hopf Algebra If V is a bialgebra and we have a linear map A : V V (antipode) such that for all τ V (A id) τ = (id A) τ = 1 1 (τ) then V is called a Hopf Algebra. In our case: A(i 1... i n ) = ( 1) n (i n... i 1 ). Let G V the space of characters (multiplicative functionals): g V, g(a b) = g(a) g(b), a, b V. If V is a Hopf Algebra then G is a group for the convolution product (g 1 g 2 )(τ) := (g 1 g 2 ) τ with inverse g 1 = g A and identity 1.

35 Concatenation If u [s, t] then X [s,t] := (X r, r [s, t]) is the concatenation of X [s,u] and X [u,t]. We write X [s,t] = X [s,u] X [u,t]. Setting r n+1 := t, r 0 := s, we have X st (i 1... i n ) = n = = k=0 s Ẋ in r n dr n rn s n X su (i 1... i k ) X ut (i k+1... i n ). k=0 Namely X st = X su X ut, Ẋ i r2 n 1 r n 1 dr n 1 Ẋ i 1 r1 dr 1 1 (rk u<r k+1 ) s X st, τ = X su X ut, τ = X su X ut, τ, τ V.

36 Shuffle Note now that 1 (s<r1 < <r n<t) 1 (s<rn+1 < <r n+m <t) = σ Sh(n,m) where Sh(n, m) is the set of all σ S n+m such that σ 1 (1) < σ 1 (2) <... < σ 1 (n), 1 (s<rσ(1) < <r σ(n+m) <t) σ 1 (n + 1) < σ 1 (n + 2)... < σ 1 (n + m). This yields the multiplicativity w.r.t. the shuffle product X st, τ 1 X st, τ 2 = X st, τ 1 τ 2 (i 1... i n ) (i n+1... i n+m ) = (i σ(1)... i σ(n+m) ). σ Sh(n,m)

37 Geometric rough paths Chen proved that X is a V -valued function with the following properties for all s u t: X st (τ) = (X su X ut ) τ, τ V, i.e. X su X ut = X st. X st (τ 1 τ 2 ) = X st (τ 1 )X st (τ 2 ). (Notations from [Hairer-Kelly 2013]). Therefore X is a flow of characters. Terry Lyons defined [1998] a (weak) geometric rough path of regularity α > 0 as a V -valued function X satisfying the above properties plus some control on the modulus of continuity sup s t [ X st (i 1... i n ) / t s nα ] < +, for all (i 1... i n ) V. Remarks: Smooth paths are dense. X st (i) = X i t X i s for some X i C α, since i is primitive in V.

38 Rough integration and differential equations Terry Lyons proved that this setting allows to give a deterministic theory of integration w.r.t. dx and to solve differential equations dy = α(y) dx, obtaining continuity of the Itô-Lyons map X Y and even X Y, although the map X Y is in general only measurable. This result includes Brownian integration, both in the sense of Itô and Stratonovich (although the Itô rough path is not geometric), but not more general rough paths. Note that setting X t := X 0t, we have X st = X 1 s where A is the antipode. X t = (X s A) X t

39 The Stratonovich Rough Path is geometric Let (B i t) i 1,t 0 be independent Brownian Motions. We set X st () := 1 and for n 1 X st (i 1... i n ) := s X sr (i 1... i n 1 ) db in r. We claim that this defines a.s. a geometric rough path.

40 The Chen relation A recurrence proof: let us set τ = (i 1,..., i n 1 ) and τ i = (i 1,..., i n 1, i). Then X st, τ i = = s u s (X sr τ) db i r (X sr τ) db i r + = X su, τ i + = X su, τ i + X su u u (X sr τ) db i r X su X ur, τ db i r u X ur db i r, τ = X su X ut, τ i 1 + (id i) τ = X su X ut, τ i.

41 Multiplicativity w.r.t. the shuffle Recall that for M, N two continuous semimartingales, the Stratonovich integral has the property M t N t M s N s = (integration by parts formula). This implies X st (i) X st ( j) = s s M r dn r + X sr (i) db j r + = X st (ij) + X st ( ji) = X st (i j). s s N r dm r X sr ( j) db i r

42 Multiplicativity w.r.t. the shuffle Let M t := X st (τi), N t := X st (σ j), t s. Then X st (τi) X st (σ j) = M t N t = = = s s X sr τ X sr (σ j) db i r + X sr (τ σj) db i r + s s s M r dn r + X sr σ X sr (τi) db j r X sr (τi σ) db j r = X st ((τ σj)i + (τi σ) j) = X st (τi σ j). s N r dm r =

43 The extension Theorem Theorem (T. Lyons) Given a (geometric) rough path X of regularity α > 0, the values (Xτ, τ V m, m > N) are uniquely determined by the values of (Xτ, τ V m, m N), where N := 1/α. Proof. We have for all τ V m (δ 2 Xτ) sut = (X su X ut ) τ where τ := τ () τ τ () is the reduced coproduct. We conclude by recurrence on the number of letters and by the Sewing Lemma since (δ 2 Xτ) C mα 3.

44 Controlled Paths Given a geometric rough path X of regularity α > 0, we say that Z : [0, T] V N 1, with N := 1/α, is a controlled path if for all words τ, σ Z τ t = σ N 1 Z σ s (X st τ ) σ + R τ st, R τ C (N τ )α 2, where τ : V R is the linear functional such that τ (σ) = 1 (τ=σ) and σ is the number of letters in σ. When the alphabet has a single letter, the condition is: We say that Z : [0, T] R {0,...,N 1} is controlled by X if N 1 Zt n = Zs k X k n st + R n st, n {0,..., N 1}, R n C (N n)α 2. k=n

45 Rough Integration Theorem If Z is a controlled path then for each letter i the germ A i st := σ N 1 Z σ s X σi st satisfies δ 2 A C (N+1)α 3. Then by the Sewing Lemma the rough integral 0 Z dx i is well defined where X i t X i s = X st (i).

46 The Itô Rough Path is not geometric Let (B i t) i 1,t 0 be independent Brownian Motions. We set X st () := 1 and E.g. i i = 2ii, X st (i i) = 2 X st (i 1... i n ) := s s X sr (i 1... i n 1 ) db in r. (B i r B i s) db i r = (B i t B i s) 2 (t s) X st (i) = B i t B i s = X st (i i) X st (i)x st (i).

47 The Itô Rough Path However we do have X st = X su X ut : setting r 0 := t, r n+1 := s X st (i 1... i n ) = n rn = db in r n = k=0 s s db i r2 n 1 r n 1 s n X su (i 1... i k ) X ut (i k+1... i n ). k=0 How can we describe the Itô Rough Path? db i 1 r1 1 (rk u<r k+1 )

48 Decorated Trees/Forests Two equivalent settings m n j j k q i i j k i i p m n j j k q i i j k i i p

49 The Connes-Kreimer Hopf algebra We consider the space H of rooted trees, with edges decorated by letters of the alphabet {1,..., d}. The identity is, the product is the identification of the roots, and the coproduct is τ = σ τ(τ/σ) σ where σ varies among all subtrees of τ with the same root as τ. This is a bialgebra and a Hopf algebra. The previous bialgebra V is canonically embedded in H: a word (i 1 i n ) is interpreted as a linear tree with n edges, the first (at the root) decorated with i n, the next with i n 1 and so on. The coproduct of H extends that of V, the product does not. This Hopf algebra was already famous in numerical analysis (!): Butcher (1972) and Hairer-Wanner (1974).

50 An example j k j k i = i + j k i j k j k + k i + j i + i (different but isomorphic representation w.r.t. that common in algebra, see Kurusch lectures).

51 A recursive formula H has a recursive structure: all elements of H are obtained from with a finite number of products and of applications of the operators τ [τ] i where we add to the root of τ a new edge with decoration i and we move the root to the new node. The coproduct has the recursive construction =, (τ 1 τ n ) = ( τ 1 ) ( τ n ) [τ] i = [τ] i + (id [ ] i ) τ. (A non-cocommutative coproduct) H is graded by the number of edges.

52 Chen and Kreimer In 1998, Dirk Kreimer gives an extension of Chen s result. He extends the iterated integrals to functionals of decorated trees in H: X st, = 1 X st, τ 1 τ n = X st, τ 1 X st, τ n X st, [τ] i = s (X su τ) Ẋ i u du and shows that X is a H -valued function with the following properties for all s u t: X st (τ) = (X su X ut ) τ, τ H, i.e. X su X ut = X st X st (τ 1 τ 2 ) = X st (τ 1 )X st (τ 2 ).

53 Examples j j k τ = i i j k i i X st (τ) = 1 X st (τ) = Xt i Xs i = s Ẋ i r dr X st (τ) = (X i t X i s)(x j t X j s)(x k t X k s ) X st (τ) = X st (τ) = s s (X j r X j s) Ẋ i r dr (X j r X j s)(x k r X k s ) Ẋ i r dr

54 A recursive proof of Chen s relation X st, [τ] i = = s u s (X sr τ) Ẋ i r dr (X sr τ) Ẋ i r dr + = X su, [τ] i + = X su, [τ] i + X su u u (X sr τ) Ẋ i r dr X su X ur, τ Ẋ i r dr u X ur Ẋ i r dr, τ = X su, [τ] i + X su X ut [ ] i, τ = X su X ut, [τ] i 1 + (id [ ] i ) τ = X su X ut, [τ] i.

55 Branched rough paths In 2006 Massimiliano defines a branched rough path of regularity α > 0 as a function X : [0, T] 2 H s.t. X st, τ = X su X ut, τ, τ H. X st, τ 1 τ 2 = X st, τ 1 X st, τ 2. sup s t [ X st, τ / t s α τ ] < +, for all τ H, where τ is the number of edges of τ. Notations and presentation follow [Hairer-Kelly 2013]. Massimiliano also extends the analytical theory of rough SDEs to the branched case, in particular the notion of controlled paths. Since [ ] i is primitive, we have X st ([ ] i ) = X i t X i s with X i C α.

56 Itô as a Branched Rough Path X st, [τ] i = = s u s (X sr τ) db i r (X sr τ) db i r + = X su, [τ] i + u = X su, [τ] i + X su u (X sr τ) db i r X su X ur, τ db i r u X ur db i r, τ = X su, [τ] i + X su X ut [ ] i, τ = X su X ut, [τ] i 1 + (id [ ] i ) τ = X su X ut, [τ] i.

57 Itô as a Branched Rough Path Let us recall that the Itô Branched Rough Path is not geometric, since X st (i i) = (B i t B i s) 2 (t s) (B i t B i s) 2 = X st (i)x st (i). Note that now i i = 2τ with τ equal to i i which is not a product in H. On the other hand, σ = i = σσ = i i. Note that setting X t := X 0t, we have X st = X 1 s where A is the antipode in H. X t = (X s A) X t

58 The antipode A i = i. j j A i = i + i j j k j k j k A i = i i j k + i k + i j

59 Exercise Let X be the Itô Brownian rough path, with 1/3 γ < 1/2. Then for j τ = i X st τ = = s 0 (B j r B j s) db i r B j r db i r s = (X s A) X t τ. 0 B j r db i r + B i sb j s B i tb j s

60 Question Let X be the Itô Brownian rough path, with 1/3 γ < 1/2. For s t, what is X ts? For instance, if τ := What is X ts τ? i i, Xst τ = s (B i r B i s) db i r = (Bi t B i s) 2 (t s). 2

61 Question Let X be the Itô Brownian rough path, with 1/3 γ < 1/2. For s t, what is X ts? For instance, if τ := i i, Xst τ = s (B i r B i s) db i r = (Bi t B i s) 2 (t s). 2 What is X ts τ? Answer: by the Chen relation, X ts = X st A. Example: X ts τ = X st τ + X st i i = (Bi t B i s) 2 (t s) 2 + (B i t B i s) 2 = (Bi t B i s) 2 + (t s). 2

62 The extension Theorem Theorem Given a branched rough path X of regularity α > 0, the values (Xτ, τ H m, m > N) are uniquely determined by the values of (Xτ, τ H m, m N), where N := 1/α. Proof. We have for all τ H m (δ 2 Xτ) sut = (X su X ut ) τ where τ := τ τ τ is the reduced coproduct. We conclude by recurrence on the number of edges and by the Sewing Lemma since (δ 2 Xτ) C mα 3.

63 Controlled Paths Given a branched rough path X of regularity α > 0, we say that Z : [0, T] H N 1, with N := 1/α, is a controlled path if for all trees τ, σ Z τ t = σ N 1 Z σ s (X st τ ) σ + R τ st, R τ C (N τ )α 2, where τ : H R is the linear functional such that τ (σ) = 1 (τ=σ) and σ is the number of edges in σ. When the alphabet has a single letter, the condition is: We say that Z : [0, T] R {0,...,N 1} is controlled by X if N 1 Zt n = Zs k X k n st + R n st, n {0,..., N 1}, R n C (N n)α 2. k=n

64 Rough Integration Theorem If Z is a controlled path then for each letter i the germ A i st := σ N 1 Z σ s X [σ] i st satisfies δ 2 A C (N+1)α 3. Then by the Sewing Lemma the rough integral 0 Z dx i is well defined where X i t X i s = X st ([ ] i ).

65 Rough Integration Theorem If Z is a controlled path then for each letter i the germ A i st := σ N 1 Z σ s X [σ] i st satisfies δ 2 A C (N+1)α 3. Then by the Sewing Lemma the rough integral 0 Z dx i is well defined where X i t X i s = X st ([ ] i ). For further readings on Rough Paths, see the books by Peter Friz. If you want one paper to read on what I discussed until now, then I recommend [Hairer-Kelly, AIHP15].

66 Three papers Martin Hairer (2014), A theory of regularity structures, Inventiones. Yvain Bruned, M.H., L.Z. (2016), Algebraic renormalisation of regularity structures, arxiv. Ajay Chandra, M.H. (2016), An analytic BPHZ theorem for regulariy structures, arxiv. This trio of papers "gives a completely automatic black box for local existence and uniqueness theorems for a wide class of SPDEs".

67 Singular stochastic PDEs Let ξ be a space time white noise (KPZ) t u = u + ( x u) 2 + ξ, x R, (PAM) t u = u + u ξ, x R 2, (Φ 4 3) t u = u u 3 + ξ, x R 3. Even for polynomial non-linearities, we do not know how to properly define products of (random) distributions. Note that if T S (R d ) and ψ S(R d ), then we can define canonically the product ψt = Tψ S (R d ) by ψt(ϕ) = Tψ(ϕ) := T(ψϕ), ϕ S(R d ). Similar problem with stochastic integrals, as we have seen.

68 Wong-Zakai Let us consider the ODE in R d x ε = b(x ε ) + f (x ε ) Ḃ ε (1) where B ε is a smooth approximation of a BM B. Then it is well known that x ε x solution to the Stratonovich SDE dx = b(x) dt + f (x) db. In order to obtain the Itô SDE in the limit, one has to define rather d dtˆx ε = b(ˆx ε ) 1 2 Df (ˆx ε) f (ˆx ε ) + f (ˆx ε ) Ḃ ε (2) and in this case ˆx ε ˆx solution to dˆx = b(ˆx) dt + f (ˆx) db. Now, (2) is a renormalisation of (1).

69 Regularisation Let ξ ε = ρ ε ξ a regularisation of ξ and let u ε solve t u ε = u ε + F(u ε, u ε, ξ ε ). What happens as ε 0? We need a topology such that the map ξ ε u ε is continuous ξ ε ξ as ε 0. For classical negative Sobolev spaces the first point fails. For classical positive Sobolev spaces the second point fails. The theory of regularity structures (RS) gives a framework to solve this problem.

70 The Solution Map on models Martin s theory gives a space of Models (M, d) (analog of the space of Rough Paths) a canonical lift of every smooth ξ ε to a model X ε M a continuous function Φ : M S (R d ) such that u ε = Φ(X ε ) solves the regularised equation t u ε = u ε + F(u ε, u ε, ξ ε ). The model X ε M contains a finite number of relevant explicit products (analogous to the necessary finitely many iterated integrals) e.g. ξ ε (G ξ ε ) (with G the heat kernel). These products can be ill-defined in the limit ε 0: E[ξ ε (G ξ ε )] = ρ ε G ρ ε (0) G(0) = +. Therefore in general X ε does not converge in (M, d) as ε 0.

71 Renormalised products The theory identifies a class of equations, called subcritical, for which it is enough to modify a finite number of products in order to obtain a convergent lift ˆX ε M of ξ ε. For instance ξ ε (G ξ ε ) ξ ε (G ξ ε ) E[ξ ε (G ξ ε )]. The model ˆX ε M contains all these modified (renormalised) products. Convergence in (M, d) means (simplifying a lot) convergence of all these objects as distributions. Then we define the renormalised solution by û ε := Φ(ˆX ε ).

72 An image M ˆX ˆXε X ε Φ S (R d ) ξ ξ ε û û ε u ε S (R d )

73 The general procedure One can summarize the procedure into three steps: Analytic step Construction of the space of models (M, d) and continuity of the solution map Φ : M S (R d ), [MH14] Algebraic step Renormalisation of the canonical model X ε ˆX ε M, [BHZ16] Probabilistic step Convergence in probability of the renormalised model ˆX ε to ˆX in (M, d), [CH16]. We obtain a renormalised solution û := Φ(ˆX), also the unique solution of a fixed point problem. This works for very general noises, far beyond the Gaussian case.

74 Wong-Zakai for SPDEs The analogous result for the SPDE is much more subtle: if t u ε = 2 x u ε + H(u ε ) + F(u ε ) ξ ε, x R, then u ε = Φ(X ε ) does not converge in general; necessary to renormalise the equation and study û ε := Φ(ˆX ε ): t û ε = 2 x ûε + H(û ε ) C ε F (û ε ) F(û ε ) + F(û ε ) ξ ε with C ε = E[ξ ε (G ξ ε )] ε 1. The limit û := Φ(ˆX) solves dû = ( 2 x û + H(û)) dt + F(û) dw t in the Itô sense (true for very general ξ ε, see [Chandra-Shen]). Although there is nothing singular in this SPDE, the result is far from simple and requires the full power of the theory [Hairer-Pardoux15].

75 Important messages We want to renormalise the (unknown) solution u ε = Φ(X ε ). We renormalise the (finitely many, explicit) ill-defined products and construct the renormalised model ˆX ε [BHZ16]. We prove that the renormalised model ˆX ε converges to ˆX in (M, d) [CH16]. Continuity of the solution map Φ : M S (R d ) yields convergence of the renormalised solution û ε = Φ(ˆX ε ) to û = Φ(ˆX) [MH14]. Very important: (M, d), X ε, ˆX ε and X ε ˆX ε are all non-linear. The group describing the transformation X ε ˆX ε is in general non-commutative. Renormalisation does not mean modifying the equation but choosing the correct equation.

76 Another example: KPZ The regularised version is t u ε = x 2 u ε + ( x u ε ) 2 + ξ ε which has to be renormalised to t û ε = x 2 ûε + ( x û ε ) 2 C ε + ξ ε and [ C ε = E ( x G ξ ε ) 2] 1 ε. In this case, one of the ill-defined products to be renormalised is ( x G ξ ε ) 2 ( x G ξ ε ) 2 E[( x G ξ ε ) 2 ].

77 Singular stochastic PDEs Around 2010, Martin and Massimiliano, among others, try to generalise Rough Paths to stochastic PDEs like KPZ, PAM and Φ 4. (KPZ) t u = u + ( u) 2 + ξ, (t, x) R R, (PAM) t u = u + u ξ, (t, x) R R 2, (Φ 4 3) t u = u u 3 + ξ, (t, x) R R 3. This needs two generalisations: The rough path must be parametrized by R d with d 2 X st (τ) can become a distribution, say, in t for fixed s, i.e. we want to allow that sup s t [ X st (τ) / t s ατ ] < + with α τ R. Two new theories are born: regularity structures and paraproducts.

78 Rough Paths? Consider e.g. t u ε = u ε + σ(u ε ) ξ ε, (t, x) R R. What is the associated "Rough Path" (model)? If we had before X st, [τ] i = then now it looks reasonable to replace Ẋ i u ξ ε (u, y), s s du (X su τ) Ẋ i u du 0 R G t u (x y) du dy.

79 Rough Paths? In Rough Paths X st (τ) is always an increment X st, [τ] i = The analytic property = s a (X su τ) Ẋ i u du (X su τ) Ẋ i u du s sup[ X st, τ / t s γ τ ] < + s t a (X su τ) Ẋ i u du. is recursive, since if s, t are close to each other then u [s, t] is close to s as well.

80 Rough Paths? Let us use a now notation for the addition of a new trunk: [τ] i I(τ). For SPDEs, we imagine a recursive object Π x τ(y) replacing X st (τ), such that Π x I(τ)(y) = G (Π x τ)(y) G (Π x τ)(x). (From now on, x, y are space-time variables.) What would be a reasonable analytic requirement here? If Π x τ(y) C y x τ s with τ s > 0 then we would like to have, by analogy with RPs, Π x I(τ)(y) C y x τ s+2 but this requires further assumptions on y G (Π x τ)(y).

81 Taylor sums and remainders In fact we have to modify the definition of Π x τ(y). We recall X st, [τ] i = = s a (X su τ) Ẋ i u du (X su τ) Ẋ i u du s a (X su τ) Ẋ i u du. This increment is a Taylor remainder at order 0. This suggests to go to a higher order by setting Π x I(τ)(y) = G (Π x τ)(y) (y x)k k G (Π x τ)(x). k! k I(τ) s But then we have to modify the coproduct if we want Chen s relation. It still involves extraction of a subtree at the root and contraction, but there are additional decorations that take into account the terms of the Taylor series.

82 Tree representation Recall that we are interested in a finite number of polynomial functions of ξ ε, P 1 (ξ ε ),..., P N (ξ ε ). More precisely, for a fixed ϕ Cc we consider the random variables Z i := ϕ(z) P i (ξ ε (z)) dz, i = 1,..., N. R d To each such random variable we associate a rooted tree T i. Every integration variable in Z i is a vertex in T i. Every integral kernel in Z i is an edge in T i.

83 Examples Ξ ϕ(z) ξ ε (z) dz = ϕ(z) ρ ε (z x) ξ(dx) dz x z Remark: the previous tree is absent in Rough Paths. x y I(Ξ) ϕ(z) G ξ ε (z) dz z ΞI(Ξ) ϕ(z) ξ ε (z) G ξ ε (z) dz z x y 2 y 1

84 Examples ΞI(ΞI(Ξ)) ΞI(Ξ) 2

85 Further decorations on trees We have additional decorations on trees, needed to code Π x I(τ)(y) = G (Π x τ)(y) (y x)k k G (Π x τ)(x). k! k I(τ) s n on nodes, representing powers of (y x) e on edges, representing derivatives k G of the heat kernel + Te n = ( ) 1 n (T/S) n n S e+e e S T n S,e S! n S S n S+πe S e S S

86 Distributions We have a linear space H of decorated trees, representing distributions on R d which are relevant to the given equation. Since we do not expect to multiply all distributions, H is not assumed to be an algebra. We do not expect H to have a coproduct either, so it is not clear how to define the Chen relation X xz X zy = X xy. The solution is to split X xy into two components, containing respectively functions and distributions. Remember: in Rough Paths we have X st = X 1 s X t. Then we want to differentiate the two factors, and have X 1 s behaving as a true function of s, while X t can behave as a distribution in t.

87 Comodules We consider two spaces of decorated trees, H and H + such that H + is a Hopf algebra and codes classical functions H is a linear space coding relevant explicit distributions we have a left coaction + : H H + H compatible with the coproduct of H +. Then H is a comodule over H +. For g x G + and Π : H S (R d ), Π x τ(y) := g x Π, + τ (y) is a good candidate for X xy = X 1 x X y.

88 Remarks Π x τ(y) = g x Π, + τ (y) g x G + is a character and therefore multiplicative in general Π : H S (R d ) is not multiplicative, even if it takes values in smooth functions this "freedom" of Π to be non-multiplicative is crucial in the renormalisation procedure Π is always assumed to satisfy ΠΞ = ξ ε, Π I(τ) = G Πτ the canonical choice of Π, for a regularised version ξ ε of the noise, satisfies moreover multiplicativity Π(τ 1 τ n ) = Π(τ 1 ) Π(τ n ).

89 Renormalisation We consider a third space of decorated forests, H H is a Hopf algebra and codes renormalisation of diverging subtrees we have right coactions : H H H, : H + H + H compatible with the coproduct of H, so that H and H + are comodules over H. Te n = ( ) 1 n (T/S) n n S e+e e S n S,e S! n S S n S+πe S e S S

90 Positive and negative renormalisations If we set A + (T) := {S T : S subtree with the same root as T} A (T) := {S T : S subforest of T} then + T n e = T n e = S A + (T) S A (T) ( ) 1 n (T/S) n n S e+e e n S,e S! n S S n S+πe S e S S ( ) 1 n (T/S) n n S e+e e n S,e S! n S S n S+πe S e S S

91 The renormalised model We define for l G H maps M l : H H and M l : H + H + M l (τ) := (id l) τ. We can define for l G H Π l xτ(y) := g x M l ΠM l, + τ (y) = (g x l Π l)( ) + τ(y). A compatibility condition between these coactions implies that this works well... G + is the structure group, G the renormalisation group.

92 Coactions and coproducts We have defined 2 coproducts and 3 coactions, which are all variants of just 2 operators +, : a contraction/extraction of subtrees at the root (as in Rough Paths) a contraction/extraction of subforests. We also have a non-trivial action on decorations, related to the Taylor sums, which is the same for all operators. For the Analytical theory: there is an analog of controlled paths. Several theorems replace the Sewing Lemma.