The geometry of the space of branched Rough Paths Nikolas Tapia 1 , - PowerPoint PPT Presentation

The geometry of the space of branched Rough Paths Nikolas Tapia 1 , joint work w. Lorenzo Zambotti 2 1 NTNU Trondheim 2 Sorbonne-Unversité 14 Nov. 2018, Clermont-Ferrand N. Tapia (NTNU) The geometry of the space of branched Rough Paths 14 Nov. 2018, Clermont-Ferrand 1 / 26

Introduction Introduction N. Tapia (NTNU) The geometry of the space of branched Rough Paths 14 Nov. 2018, Clermont-Ferrand 2 / 26

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 (NTNU) The geometry of the space of branched Rough Paths 14 Nov. 2018, Clermont-Ferrand 3 / 26

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 (NTNU) The geometry of the space of branched Rough Paths 14 Nov. 2018, Clermont-Ferrand 4 / 26

Branched rough paths Branched rough paths N. Tapia (NTNU) The geometry of the space of branched Rough Paths 14 Nov. 2018, Clermont-Ferrand 5 / 26

Branched rough paths Let ( H , · , ∆ ) be the Butcher–Connes–Kreimer Hopf algebra. As an algebra, H is the commutative polynomial algebra over the set T 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 (NTNU) The geometry of the space of branched Rough Paths 14 Nov. 2018, Clermont-Ferrand 6 / 26

Branched rough paths The Hopf algebra H is graded by the number of vertices in a forest. It is also connected. Let G be the characters on H . Definition (Gubinelli (2010)) A branched Rough Path is a map X : [ 0 , 1 ] 2 → G such that X tt = ε 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. Tapia (NTNU) The geometry of the space of branched Rough Paths 14 Nov. 2018, Clermont-Ferrand 7 / 26

Branched rough paths A cochain complex 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 (NTNU) The geometry of the space of branched Rough Paths 14 Nov. 2018, Clermont-Ferrand 8 / 26

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 (NTNU) The geometry of the space of branched Rough Paths 14 Nov. 2018, Clermont-Ferrand 9 / 26

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 subcoalgebra N � H H N ≔ ( n ) . n =0 N. Tapia (NTNU) The geometry of the space of branched Rough Paths 14 Nov. 2018, Clermont-Ferrand 10 / 26

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 → G on H such that ˆ � map 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 (NTNU) The geometry of the space of branched Rough Paths 14 Nov. 2018, Clermont-Ferrand 11 / 26

Results Results N. Tapia (NTNU) The geometry of the space of branched Rough Paths 14 Nov. 2018, Clermont-Ferrand 12 / 26

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 H ( 1 ) . st ≔ � X ′ Fix 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 (NTNU) The geometry of the space of branched Rough Paths 14 Nov. 2018, Clermont-Ferrand 13 / 26

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 (NTNU) The geometry of the space of branched Rough Paths 14 Nov. 2018, Clermont-Ferrand 14 / 26

Results Action This means we have a mapping D γ × BRP γ ∋ ( g , X ) → gX ∈ BRP γ such that g ′ ( gX ) = ( g ′ + g ) X for all g , g ′ ∈ D γ and, for every pair X , X ′ ∈ BRP γ there exists a unique g ∈ D γ such that X ′ = gX . BRP γ is a principal homogeneous space for D γ . N. Tapia (NTNU) The geometry of the space of branched Rough Paths 14 Nov. 2018, Clermont-Ferrand 15 / 26

Very rough sketch of proof Very rough sketch of proof N. Tapia (NTNU) The geometry of the space of branched Rough Paths 14 Nov. 2018, Clermont-Ferrand 16 / 26

Very rough sketch of proof If γ > 1 2 the result is easy: just set � gX st , i � = � X st , i � + δg i st and � gX , τ � for | τ | ≥ 2 is given by the Sewing Lemma. If 1 3 < γ < 1 2 the action is the same in degree 1 . In degree 2 we must have su + δg j j � sut = ( δx j ut + δg i su )( δx i δ 2 � gX , i ut ) . The canonical choice (Young integral) ∫ t su + δg j ( δx j su ) d ( x i u + g i u ) s is not well defined since 2 γ < 1 . N. Tapia (NTNU) The geometry of the space of branched Rough Paths 14 Nov. 2018, Clermont-Ferrand 17 / 26

Very rough sketch of proof In higher degrees the expressions are more complicated. We handle this by constructing an anisotropic geometric Rough Path ¯ X such that � X st , τ � = � ¯ X st , ψ ( τ )� n ) , ✁ , ¯ where ψ : ( H , · , ∆ ) → ( T ( T ∆ ) is the Hairer–Kelly map. Anisotropic means that letters (trees) are allowed to have different weights. In addition to the standard grading by the number of letters we have a weight function, e.g. � c � a ⊗ ω = 3 γ . b N. Tapia (NTNU) The geometry of the space of branched Rough Paths 14 Nov. 2018, Clermont-Ferrand 18 / 26

Very rough sketch of proof More concretely, ¯ X is a character over the shuffle algebra on the alphabet T N . Single trees become letters in T ( T N ) , hence they are in degree one! Set � g ¯ X , τ � ≔ � ¯ X , τ � + δg τ . Then define � gX , τ � = � g ¯ X , ψ ( τ )� . N. Tapia (NTNU) The geometry of the space of branched Rough Paths 14 Nov. 2018, Clermont-Ferrand 19 / 26

Very rough sketch of proof Coments 1 Lifting of Chen’s rule to the Lie algebra g . If X st = exp ⋆ ( α st ) then α st = BCH ( α su , α ut ) = α su + α ut + BCH ′ ( α su , α ut ) . 2 We use an explicit BCH formula due to Reutenauer. 3 We use the Lyons–Victoir (2007) method but in a constructive way, without invoking the axiom of choice. 4 However, the action is not unique nor canonical. The construction depends on a finite number of arbitrary choices. 5 We are able to construct γ -regular H -rough paths over any x ∈ C γ ( � d ) . N. Tapia (NTNU) The geometry of the space of branched Rough Paths 14 Nov. 2018, Clermont-Ferrand 20 / 26