Modification of branched Rough Paths Nikolas Tapia, joint work w. - PowerPoint PPT Presentation

Modification of branched Rough Paths Nikolas Tapia, joint work w. Lorenzo Zambotti (Paris) 23 Jan. 2019 N. Tapia Modification of branched Rough Paths 23 Jan. 2019 1 / 25

Introduction Introduction N. Tapia Modification of branched Rough Paths 23 Jan. 2019 2 / 25

Introduction Rough paths were introduced by Terry Lyons near the end of the 90’s to deal with stochastic integration (and SDEs) in a path-wise sense. Some years later Massimiliano Gubinelli introduced controlled rough paths, and brached Rough Paths a decade after Lyons’ work. In 2014, Martin Hairer introduced Regularity Structures which generalize branched Rough Paths. All of these objects consist of a mixture of algebraic and analytic properties. N. Tapia Modification of branched Rough Paths 23 Jan. 2019 3 / 25

Introduction A crucial tool in Regularity Structures is the renormalization step. This step relies on knowledge of the group of automorphisms of the space of models. In this setting, an answer has been given by Bruned, Hairer and Zambotti (2016) for stationary models. Now we will discuss the same problem for branched Rough Paths. Some work on this has already been carried by Bruned, Chevyrev, Friz and Preiß (2017). N. Tapia Modification of branched Rough Paths 23 Jan. 2019 4 / 25

Geometric rough paths Geometric rough paths N. Tapia Modification of branched Rough Paths 23 Jan. 2019 5 / 25

Geometric rough paths Geometric rough paths (signatures) have recently found a number of applications in Data Analysis and Statistical Learning. For a “smooth” path x : [ 0 , 1 ] → � d , one defines its signature S ( x ) : [ 0 , 1 ] 2 → T ( � d ) ∗ as ∫ t ∫ t n − 1 ∫ t 1 d x i 1 u 1 d x i 2 u 2 · · · d x i n � S ( x ) s , t , e i 1 ··· i n � = · · · u n s s s i.e. S ( x ) is the collection of all iterated integrals of the components of x . Here, e i 1 ··· i n ≔ e i 1 ⊗ · · · ⊗ e i n is a basis element of T ( � d ) = � ⊕ � d ⊕ ( � d ⊗ � d ) ⊕ · · · For example: � S ( x ) s , t , e i � = x i t − x i s ∫ t � S ( x ) s , t , e ii � = ( x i t − x i s ) 2 s ) d x j ( x i u − x i � S ( x ) s , t , e ij � = u , 2 s N. Tapia Modification of branched Rough Paths 23 Jan. 2019 6 / 25

Geometric rough paths The family of iterated integrals satisfies the so-called shuffle relation , implied by the integration-by-parts formula: � S ( x ) s , t , e i 1 ··· i n ✁ e i i +1 ··· i n + m � = � S ( x ) s , t , e i 1 ··· i n �� S ( x ) s , t , e i n +1 ··· i n + m � . For example, for n = 1 , m = 1 we recover integration by parts: ∫ t ∫ u ∫ t ∫ u ∫ t ∫ t u 1 d x j d x j d x j d x i u 1 d x i d x i u 2 + u 2 = u . u s s s s s s It also satisfies the following identity, called Chen’s rule , a generalization of ∫ u ∫ t ∫ t s : s + u = � S ( x ) s , t , e i 1 ··· i n � = � S ( x ) s , u , e i 1 ··· i n � + � S ( x ) u , t , e i 1 ··· i n � n − 1 � � S ( x ) s , u , e i 1 ··· i j �� S ( x ) u , t , e i j +1 ··· i n � . + j =1 N. Tapia Modification of branched Rough Paths 23 Jan. 2019 7 / 25

Geometric rough paths The vector space T ( � d ) = � ⊕ � d ⊕ ( � d ⊗ � d ) ⊕ · · · can be made into an algebra in two ways: the tensor (or concatenation) product, and the shuffle product . It also carries two coproducts: the deconcatenation coproduct ∆ and the deshuffling coproduct ∆ ✁ . In fact, ( T ( � d ) , ⊗ , ∆ ✁ ) and ( T ( � d ) , ✁ , ∆ ) are Hopf algebras, dual to one another. The signature S ( x ) of a smooth path is a family of linear maps on T ( � d ) , i.e. an element of T ( � d ) ∗ ≔ T ( ( � d ) ) . The above properties can be summarized by saying that, for each s < t the element S ( x ) is an algebra morphism (shuffle relation) satisfying S ( x ) s , u ⊗ S ( x ) u , t for all s < u < t . N. Tapia Modification of branched Rough Paths 23 Jan. 2019 8 / 25

Geometric rough paths A classical theorem by Young tells us that the integration operator ∫ 1 I ( f , g ) ≔ f s d g s 0 can be extended continuously from C 0 × C 1 → C 1 to C α × C β → C β if and only if α + β > 1 . Thus, finding the signature S ( x ) as above is only possible for paths in C α for α > 1 2 . Theorem (Lyons–Victoir (2007)) 2 with α − 1 � � and x ∈ C α , there exists a map X : [ 0 , 1 ] 2 → T ( Given α < 1 ( � d ) ) such that X s , t is multiplicative, X s , u ⊗ X u , t = X s , t and |� X s , t , e i 1 ··· i k �| � | t − s | kγ . N. Tapia Modification of branched Rough Paths 23 Jan. 2019 9 / 25

Branched rough paths Branched rough paths N. Tapia Modification of branched Rough Paths 23 Jan. 2019 10 / 25

Branched rough paths Let ( H , · , ∆ ) be the Butcher–Connes–Kreimer Hopf algebra. As an algebra, H is the commutative polynomial algebra over the set of non-planar trees decorated by some alphabet A . The product is simply the disjoint union of forests, e.g. d c · d c g = g f f b b e e a a The empty forest 1 acts as the unit. The coproduct ∆ is described in terms of admissible cuts. For example d + d ⊗ d c = c + c d ⊗ b c + b d ⊗ d ⊗ ∆ ′ b + c b c ⊗ a b b a a a a a N. Tapia Modification of branched Rough Paths 23 Jan. 2019 11 / 25

Branched rough paths Definition (Gubinelli (2010)) A branched Rough Path is a map X : [ 0 , 1 ] 2 → H ∗ such that each X s , t is an algebra morphism and |� X st , τ �| � | t − s | γ | τ | . X su ⋆ X ut = X st , Example: let ( B t ) t ≥ 0 be a Brownian motion, set � X st , � ≔ B t − B s and ∫ t � X su , τ 1 � · · · � X su , τ k � d B u . � X st , [ τ 1 · · · τ k ]� = s N.B.: This definition can be uniquely extended such that X s , t is an algebra morphism. N. Tapia Modification of branched Rough Paths 23 Jan. 2019 12 / 25

Branched rough paths Delta maps Let C k be the continuous functions in k variables vanishing when consecutive variables coincide. Gubinelli (2003) defines an exact cochain complex δ 1 δ 2 δ 3 0 → � → C − − → C − − → C − − → · · · 1 2 3 that is δ k +1 ◦ δ k = 0 and im δ k = ker δ k +1 . Remark If F ∈ ker δ 2 then there exists f ∈ C 1 such that F st = f t − f s . If C ∈ ker δ 3 then there exists F ∈ C 2 such that C sut = F st − F su − F ut . In general, none of these operators are injective: if F = G + δ k − 1 H then δ k F = δ k G . N. Tapia Modification of branched Rough Paths 23 Jan. 2019 13 / 25

Branched rough paths The Sewing Lemma Can do more if we restrict to smaller spaces: let C µ 2 be the F ∈ C 2 such that | F st | � F � µ ≔ sup | t − s | µ < ∞ . s < t Similarly, C µ 3 are the C ∈ C 3 such that � C � µ < ∞ for some suitable norm. Theorem (Gubinelli (2004)) There is a unique linear map Λ : C 1+ ∩ ker δ 3 → C 1+ such that δ 2 Λ = id . In each of 3 2 C µ 3 for µ > 1 it satisfies 1 � Λ C � µ ≤ 2 µ − 2 � C � µ . N. Tapia Modification of branched Rough Paths 23 Jan. 2019 14 / 25

Branched rough paths The Sewing Lemma Chen’s rule reads � X st , τ � = � X su , τ � + � X ut , τ � + � X su ⊗ X ut , ∆ ′ τ � . or δ 2 F τ sut = � X su ⊗ X ut , ∆ ′ τ � where F τ st ≔ � X st , τ � . 3 is such that the bound for X implies δ 2 F τ ∈ C γ | τ | The norm on C . 3 The integer N ≔ ⌊ γ − 1 ⌋ is special. Let G N denote the multiplicative maps on the truncated space N ≔ � { τ : | τ | ≤ N } H N. Tapia Modification of branched Rough Paths 23 Jan. 2019 15 / 25

Branched rough paths The Sewing Lemma Theorem (Gubinelli (2010)) Suppose X : [ 0 , 1 ] 2 → G N satisfies |� X st , τ �| � | t − s | γ | τ | . Then there exists a unique X : [ 0 , 1 ] 2 → H ∗ such that ˆ multiplicative extension ˆ � X N = X . � H Proof. Suppose | τ | = N + 1 is a tree and set C τ sut = � X su ⊗ X ut , ∆ ′ τ � . First one shows that C τ ∈ ker δ 3 by using the coassociativity of ∆ ′ . The bound above implies that C τ ∈ C γ | τ | . 3 Therefore C τ lies in the domain of Λ and we can set � X st , τ � ≔ ( Λ C τ ) st . Continue inductively. � N. Tapia Modification of branched Rough Paths 23 Jan. 2019 16 / 25

Results Results N. Tapia Modification of branched Rough Paths 23 Jan. 2019 17 / 25

Results Action The previous argument works only because γ | τ | > 1 i.e. | τ | > N . If γ | τ | ≤ 1 , for any g τ ∈ C γ | τ | (Hölder space) the function G τ st ≔ F τ st + δ 1 g τ st sut = � X su ⊗ X ut , ∆ ′ τ � . also satisfies δ 2 G τ Let X and X ′ be two BRPs coinciding on � { 1 , . . . , d } . st ≔ � X ′ Fix a tree τ with | τ | = 2 and let F τ st ≔ � X st , τ � , G τ st , τ � . Then δ 2 F τ = δ 2 G τ so there is g τ ∈ C 1 such that F τ st = G τ st + δ 1 g τ st . Moreover g τ ∈ C 2 γ . N. Tapia Modification of branched Rough Paths 23 Jan. 2019 18 / 25

Results Action This suggests that there might be an action of D γ ≔ {( g τ ) | τ |≤ N : g τ ∈ C γ | τ | , g τ 0 = 0 } on the space BRP γ of branched Rough Paths. Theorem (T.-Zambotti (2018)) Let γ ∈ ( 0 , 1 ) such that γ − 1 � � . There is a regular action of D γ on BRP γ . N. Tapia Modification of branched Rough Paths 23 Jan. 2019 19 / 25