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 Feb. 6, 2019 @ MPI MiS Leipzig N. Tapia (NTNU) The geometry of the space of branched Rough Paths Feb. 6, 2019 @ MPI MiS Leipzig 1 / 31

Introduction Introduction N. Tapia (NTNU) The geometry of the space of branched Rough Paths Feb. 6, 2019 @ MPI MiS Leipzig 2 / 31

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 Feb. 6, 2019 @ MPI MiS Leipzig 3 / 31

Introduction Given x ∈ C 1 and V ∈ C ∞ , consider y t = V ( y t ) � � x t . How can we get a local description of y ? Note that, setting δψ st ≔ ψ t − ψ s , ∫ t R 1 st ≔ δy st − V ( y s ) δx st = ( V ( y u ) − V ( y s )) � x u d u = o (| t − s |) . s N. Tapia (NTNU) The geometry of the space of branched Rough Paths Feb. 6, 2019 @ MPI MiS Leipzig 4 / 31

Introduction st ≔ δy st − V ( y s ) δx st − V ′ ( y s ) V ( y s ) ( δx st ) 2 We can be more precise. Set R 2 . 2 ∫ t ∫ t ∫ u R 2 x u d u − V ′ ( y s ) V ( y s ) ( V ( y u ) − V ( y s )) � x r d r � � x u d u st = s s s ∫ t ∫ t ∫ u = V ′ ( y s ) x u d u − V ′ ( y s ) V ( y s ) x u d u + o (| t − s | 2 ) δy su � x r d r � � s s s ∫ t ∫ u ∫ t ∫ u = V ′ ( y s ) x u d u − V ′ ( y s ) V ( y s ) x u d u + o (| t − s | 2 ) x r d r � x r d r � V ( y r ) � � s s s s ∫ t ∫ u = V ′ ( y s ) x u d u + o (| t − s | 2 ) ( V ( y r ) − V ( y s )) � x r d r � s s = o (| t − s | 2 ) N. Tapia (NTNU) The geometry of the space of branched Rough Paths Feb. 6, 2019 @ MPI MiS Leipzig 5 / 31

Geometric rough paths Geometric rough paths N. Tapia (NTNU) The geometry of the space of branched Rough Paths Feb. 6, 2019 @ MPI MiS Leipzig 6 / 31

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 , 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 (NTNU) The geometry of the space of branched Rough Paths Feb. 6, 2019 @ MPI MiS Leipzig 7 / 31

Geometric rough paths The vector space T ( � d ) can be made into an algebra in two ways: the tensor (or concatenation) product, and the shuffle product . Example: e i ✁ e j = e ij + e ji , e ij ✁ e pq = e ij pq + e ipj q + e pij q + e ipqj + e piqj + e pqij . 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. N. Tapia (NTNU) The geometry of the space of branched Rough Paths Feb. 6, 2019 @ MPI MiS Leipzig 8 / 31

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 ˆ ⊗ S ( x ) u , t , ∆ e i 1 ··· i n � N. Tapia (NTNU) The geometry of the space of branched Rough Paths Feb. 6, 2019 @ MPI MiS Leipzig 9 / 31

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 . It also satisfies � X s , t , e i � = δx i st . N. Tapia (NTNU) The geometry of the space of branched Rough Paths Feb. 6, 2019 @ MPI MiS Leipzig 10 / 31

Branched rough paths Branched rough paths N. Tapia (NTNU) The geometry of the space of branched Rough Paths Feb. 6, 2019 @ MPI MiS Leipzig 11 / 31

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 Feb. 6, 2019 @ MPI MiS Leipzig 12 / 31

Branched rough paths B-Series Consider again, for smooth x and V , y t = V ( y t ) � � x t . Theorem (B-Series expansion (Gubinelli, 2010)) We have the expansion 1 � σ ( τ ) V τ ( y s )� X st , τ � δy st = τ ∈ T Here V τ is the elementary differential V [ τ 1 ··· τ k ] ( y ) = V ( k ) ( y ) V τ 1 ( y ) · · · V τ k ( y ) . Example V ( y ) = V ′ ( y ) V ( y ) , ( y ) = V ′′ ( y ) 2 V ( y ) 3 . V N. Tapia (NTNU) The geometry of the space of branched Rough Paths Feb. 6, 2019 @ MPI MiS Leipzig 13 / 31

Branched rough paths B-Series The factor � X st , τ � is defined recursively: ∫ t x u d u � X st , [ τ 1 · · · τ k ]� = � X su , τ 1 � · · · � X su , τ � � s Example: � X st , � = 1 � = 1 2 ( x t − x s ) 2 , 12 ( x t − x s ) 5 � X st , N. Tapia (NTNU) The geometry of the space of branched Rough Paths Feb. 6, 2019 @ MPI MiS Leipzig 14 / 31

Branched rough paths B-Series Let G be the multiplicative functionals (characters) on H . Definition (Gubinelli (2010)) A branched Rough Path is a map X : [ 0 , 1 ] 2 → G such that |� 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 That is: ∫ t �∫ u � �∫ u ∫ t � ( B u − B s ) 2 d B u . � X st , � = d B r d B r d B u = s s s s N. Tapia (NTNU) The geometry of the space of branched Rough Paths Feb. 6, 2019 @ MPI MiS Leipzig 15 / 31

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 Feb. 6, 2019 @ MPI MiS Leipzig 16 / 31

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 Feb. 6, 2019 @ MPI MiS Leipzig 17 / 31

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 Feb. 6, 2019 @ MPI MiS Leipzig 18 / 31