The theory of rough paths via one-forms and the extension of an argument of Schwartz to rough differential equations
aa r X i v : . [ m a t h . C A ] J un The theory of rough paths via one-forms and the extension of anargument of Schwartz to rough differential equations
Terry J. Lyons, Danyu Yang ∗ August 11, 2018
Abstract
We give an overview of the recent approach to the integration of rough paths that reduces theproblem to classical Young integration [13]. As an application, we extend an argument of Schwartz[11] to rough differential equations, and prove the existence, uniqueness and continuity of the solution,which is applicable when the driving path takes values in nilpotent Lie group or Butcher group.
For each p ∈ [1 , ∞ ) Banach introduced a metric for measuring degrees of roughness in paths with valuesin Banach spaces known as p -variation. The paths of finite 1-variation are dense in the space of pathsof finite p -variation for each p ≥
1. Where when p = 1 the paths are weakly differentiable almost surelyand they engage with the classical Newtonian calculus for example making sense of line integrals: R t ∈ [0 ,T ] τ t ⊗ dσ t . Young [13] extended the integration so that if τ has finite q -variation and σ is continuous and has finite p -variation where p − + q − > R τ ⊗ dσ is well defined. In particular, if σ is of finite p -variation for p < R σ ⊗ dσ is meaningfully defined. Young’s original definition was directed towards definite integrals. Lyons [6]considered the case of indefinite integrals and the related context of controlled systems of differentialequations: dy t = f ( y t ) dσ t , y = a, (1)established the existence and uniqueness of the solution, and also the continuity of the solution in thedriving signal. Lyons’ integral requires the finite p -variation of σ , the finite Lip ( γ ) norm of f , and p − + γp − >
1. The methods rely strongly on Young’s approach, but a careful examination reveals thatthe arguments also rely critically on the notion of the Lipschitz function and on the division lemma forthem (Proposition 1.26 [8]).
Lemma 1 (Division Property)
For Banach spaces U and W , suppose f : U → W is Lip ( γ ) for some γ > . Then there exists h : U × U → L ( U , W ) which is Lip ( γ − such that f ( x ) − f ( y ) = h ( x, y ) ( x − y ) , ∀ x, y ∈ U , ∗ The authors would like to acknowledge the support of the Oxford-Man Institute and the support provided by ERCadvanced grant ESig (agreement no. 291244). or at least has its jumps in different times to τ nd for some constant C depending only on γ and U , k h k Lip( γ − ≤ C k f k Lip( γ ) . The bound p < p -variation paths and Lip ( γ ) functions form local algebras, and y in (1) also has finite p -variation. Fromthis it is clear that the space of integrals of σ , including all spaces of solutions to differential equationsdriven by σ , is closed under addition, and the pointwise multiplication is explicitly given byfor y t = a + Z s ∈ [0 ,t ] f ( y s ) dσ s and ˆ y t = ˆ a + Z s ∈ [0 ,t ] ˆ f (ˆ y s ) dσ s , y t ˆ y t = Z s ∈ [0 ,t ] (cid:16) f ( y s ) ˆ y s + y s ˆ f (ˆ y s ) (cid:17) dσ s + a ˆ a .This remark is implicit in establishing the existence, uniqueness and continuity theorems since it underpinsthe operations used in Picard iteration and other approximation strategies. In fact it is easy to show thatcomposition of an integral of σ with a smooth function is also an integral of σ (the chain rule).In further work [7], Lyons extended the integral of Young to the case p ≥
2, showed how the notionof bounded variation paths naturally admits a generalization to p -rough paths for any p ∈ [1 , ∞ ), andestablished an integral, existence, uniqueness and continuity theorem for differential equations controlledby weak geometric p -rough paths when f is Lip ( γ ) and γ > p . Young’s tricks, the division lemma andthe algebraic manipulations of Picard iteration were all important ingredients. The main surprise overthe case p < p -roughness. The space is quite different to that envisaged by Banach.In this short note we summarize a new approach to the case p ≥
2, which could be viewed as aproper extension of Lyons’ original approach, and is somewhere between the original arguments whichemphasized the rough paths and the perspective of Gubinelli which emphasized more the space of possibleintegrands for a given path that (in his context) are referred to as controlled rough paths. We explain howa clear perspective about a Lipschitz function f which allows one to (quite simply) reduce the problemof defining a rough line integral Z s ∈ [0 ,t ] f ( σ s ) dσ s to the integral of a slowly varying one-form t → ˆ f ( σ t ) against a rapidly varying path σ t in a way thatsatisfies Young’s conditions.The key understanding comes from repositioning the integral so that σ is a path in a nilpotent groupand h t = ˆ f ( σ t ) is a closed one-form on that group that varies more slowly with time than σ . Whenlooked at in the correct way, Young’s strategy applies and Z s ∈ [0 ,t ] h s dσ s is well defined. Apart from the clarity this understanding gives, it captures the linearity of the integralagainst a path in a convenient way, and actually leads to the introduction of the integral of any q -variationpath with values in the closed one-forms against σ . It is not surprising that the class of these integrals isagain closed under addition, pointwise multiplication and composition with smooth functions. What ismore surprising is that it is (by construction) rich enough to include the original integral Z s ∈ [0 ,t ] f ( σ s ) dσ s . As a result, differential equations against rough paths, etc. are easily deduced. It is surprising because s f ( σ s ) is certainly not in general of finite q -variation for any q satisfying1 p + 1 q > p ≥ Polynomial functions A polynomial function of degree n is a globally defined function whose( n + 1)th derivative exists and is identically zero. We intentionally avoid the definition as a power seriesaround a point, and we could choose different reference points and have different representations of the same polynomial. More specifically, for Banach spaces V and U , we say p : V → U is a polynomialfunction of degree (at most) n if D n +1 p ≡
0. For any y ∈ V , we can represent p as a power series around y : p ( x ) = n X k =0 (cid:0) D k p (cid:1) ( y ) ( x − y ) ⊗ k k ! , ∀ x ∈ V , ∀ y ∈ V ,but the value of p does not vary with y . We would like to emphasize that p is a function defined on theaffine space V , it has no natural graded algebraic structure, there is no particular choice of base pointassociated with it, and there does not exist a translation invariant norm on the space of polynomialfunctions.Just as in linear algebra, where one keeps the concept of linear map separated from the matrix onegets after fixing a particular choice of basis, it is conceptually essential that we distinguish the polynomialfunction as an object from any representation of it via its Taylor series around a chosen point.For Banach space U and integer n ≥
0, let P ( n ) ( U ) denote the space of polynomial functions of degree n taking values in U . Lipschitz functions
By using the polynomial functions (rather than power series), we can shiftthe classical viewpoint of the Lipschitz function as a function taking values in power series to a functiontaking values in polynomial functions. This modification gives rise naturally to a way to compare therepresentations of polynomial functions, and reduces a Lipschitz function to a ”slowly-varying” polynomialfunction. The first author would like to thank Youness Boutaib for sharing his understanding of Lipschitzfunctions with him.
Definition 2 (Stein)
Let V and U be two Banach spaces. For γ > , denote n := ⌊ γ ⌋ (the largestinteger which is strictly less than γ ). For a closed set K in V , we say f is a Lipschitz function of degree γ on K , if f : K → P ( n ) ( U ) ,and for some constant M > , sup x ∈K k f ( x ) x k ∞ + sup x,y ∈K max j =0 , ,...,n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:0) D j ( f ( x ) − f ( y )) (cid:1) x k x − y k γ − j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ M. Some explanatory points are in order:1. For x ∈ K , f ( x ) is a polynomial function of degree n , and we denote by f ( x ) x the degree- n Taylorseries of f ( x ) around x . Similarly, for j = 0 , , . . . , n , (cid:0) D j ( f ( x ) − f ( y )) (cid:1) is a polynomial functionof degree n − j and (cid:0) D j ( f ( x ) − f ( y )) (cid:1) x denotes its degree-( n − j ) Taylor series around x .2. For each x ∈ K , f ( y )
7→ k f ( y ) x k ∞ is a norm on P ( n ) ( U ). These norms are equivalent, and if K iscompact then they are uniformly equivalent.3. The Lip ( γ ) norm k f k Lip( γ ) is defined to be the smallest M satisfying the inequality.3. Suppose N is a neighborhood of x and N ⊆ K . Then F : N → U defined by y ( f ( y )) ( y ) for y ∈ N is a C γ function ( n times differentiable with the n th derivative ( γ − n )-H¨older) and f ( x ) isthe polynomial function that matches F to degree n at x : (cid:0) D j ( f ( x ) − F ) (cid:1) ( x ) = 0 , j = 0 , , . . . , n .While in comparison with the notion of C γ functions, Lipschitz functions make perfect sense evenwhen K is of finite cardinality.5. The space of Lipschitz functions forms an algebra.6. Whitney’s extension theorem was extended by Stein [12] to these generalized Lipschitz functions.He proved that there is a constant C d and a linear extension operator so that any Lip ( γ ) function f on a closed set K in R d can be extended to a Lip ( γ ) function g on R d where k g k Lip( γ ) ≤ C d k f k Lip( γ ) .The crucial and somewhat counter-intuitive remark associated with Lipschitz functions is the follow-ing. Remark 3
Suppose p is a polynomial function of degree m and γ > is a real number. When γ > m , p is associated with a constant Lip ( γ ) function f : K → P ( m ) ( U ) defined by f ( x ) := p , ∀ x ∈ K .When γ ≤ m , p gives rise to a non-constant Lip ( γ ) function f ( x ) x ( z ) = ⌊ γ ⌋ X l =0 (cid:0) D l p (cid:1) ( x ) ( z − x ) ⊗ l l ! , ∀ z ∈ V , ∀ x ∈ K ,since ⌊ γ ⌋ < m . Remark 4
This transformation of polynomials into constant functions in a different function space, andmore generally, smooth functions into slowly changing functions, can be seen at the heart of the success ofthe rough path integral. Rough path integration traditionally integrates a
Lip ( p + ε − one-form againsta (weak geometric) p -rough path. Lifting of polynomial one-forms to closed one-forms
For integer n ≥
1, the step- n nilpotentLie group G n has a natural graded algebraic structure, and accommodates weak geometric p -rough pathsfor p < n + 1. G is an abelian group which is isomorphic to a Banach space, and fits naturally into thechain G = { e } π և G π և . . . π և G n π և . . . . If σ is a path of finite length taking values in G , then thereis a natural lift σ ˆ σ (the signature of σ ), which takes a path in G into a horizontal path in G n .We have defined polynomial functions and Lipschitz functions. A polynomial one-form or a Lipschitzone-form is a polynomial function or Lipschitz function taking values in one-forms.Suppose p is a polynomial one-form on G , and we would like to lift p to a one-form p ∗ on G n so that Z p ( σ ) dσ = Z p ∗ (ˆ σ ) d ˆ σ .A simple choice is to let p ∗ be the pullback of p through the projection π . Then the equality holdsbecause σ = π ˆ σ and has nothing to do with the fact that ˆ σ is the ”horizontal lift” of σ . Actually, beinga ”horizontal lift” adds an extra ingredient which we will exploit in a crucial way. If ω is any one-formon G n which has the horizontal directions in its kernel, then Z p ∗ (ˆ σ ) d ˆ σ = Z ( p ∗ + ω ) (ˆ σ ) d ˆ σ. The key point is that we can select ω such that p ∗ + ω is a closed one-form, and the selection only dependson p and not on ˆ σ . 4 heorem 5 For n ≥ and a polynomial one-form p of degree n − , there exists a unique one-form ω on G n , which is orthogonal to the horizontal directions and p ∗ + ω is a closed one-form on G n . The proof of this theorem is actually not hard: we can give one possible choice of ω , and since p ∗ + ω does not depend on ˆ σ , any two choices must coincide.While we should specify what we mean by a closed one-form on a group. Roughly speaking, closedone-forms are characterized by zero integral along closed curves, and a one-form on a connected domainis closed if it can be integrated against any continuous path on the domain, and the value of the integralonly depends on the end points of the path. A one-form is closed is equivalent to the exact equalitybetween the one-step and two-steps estimates. Integrals often correspond to closed one-forms because ofthe property R [ s.t ] = R [ s,u ] + R [ u,t ] , and this property is actually behind the fact that the lifted polynomialone-form is closed. In term of mathematical expression, we say β on group G taking values in anotheralgebra is closed (or cocyclic), if β ( a, b ) β ( ab, c ) = β ( a, bc ) , ∀ a, b, c ∈ G .By lifting a path to a horizontal path and a polynomial one-form to a closed one-form on the nilpotentLie group, we replace a general integral by the integral of a closed one-form. The integral of a closedone-form has the nice property that it does not depend on the fine structure of the path but only on itsend points. In particular, the integral makes sense for any continuous path and has no (further) regularityassumption. Integrating slowly-varying closed one-forms
Since the integral of a closed one-form againstany continuous path is well-defined, we could weaken the requirement on the one-form and strengthenthe regularity assumption on the path in such a way that the integral still makes sense. For example, inthe case of classical integral, we can integrate a constant one-form against any continuous path becauseconstant one-forms are closed. Then if we weaken the requirement on the one-form and strengthen therequirement on the path in such a way that their regularities ”compensate” each other, then the integralstill makes sense as Young integral [13]. In the case of Young integral, we actually vary the constantone-form with time and get a path taking values in constant one-forms, which is more clearly seen inthe proof of the existence of the integral where we keep comparing the constant one-forms from differenttimes based on their effect on the future increment of the driving path.Constant one-form on Banach space is just a special example of closed one-forms. More generally,suppose we have a family of closed one-forms on a differential manifold or on a topological group. Fora given path taking values in the manifold or group, if the closed one-form varies with time in such away that the one-form and the path have compensated Young regularities, then the integral should stillmakes sense.As we mentioned above, a Lipschitz one-form could be viewed as a slowly-varying polynomial one-form, and that there exists a canonical lift of a polynomial one-form to a closed one-form on the nilpotentLie group. Hence we can lift a Lipschitz one-form to a slowly-varying closed one-form on the nilpotent Liegroup. More specifically, suppose α is a Lipschitz one-form on G . Then based on our argument above, α can be viewed as a slowly-varying polynomial one-form. Suppose σ is an underlying reference path.Then the evolution of σ gives a natural order (or say time), and α along σ is a ”slowly-time-varying”polynomial one-form with each α σ t a polynomial one-form. If we denote by ˆ σ t ∈ G n the horizontal lift ofthe path σ t ∈ G and denote by β ˆ σ t the closed one-form lift of the polynomial one-form α σ t , then we canrewrite the integral of a Lipschitz one-form against σ as the integral of a time-varying closed one-formagainst ˆ σ : Z α ( σ t ) dσ t = Z α σ t ( σ t ) dσ t = Z β ˆ σ t (ˆ σ t ) d ˆ σ t . When σ is of finite length, this algebraic/geometrical reformulation seems unnecessary. The point isthat for general path ˆ σ of finite p -variation taking values in G [ p ] , the integral R β ˆ σ (ˆ σ ) d ˆ σ still makes sense(the rough integral) while the classical Riemann sum integral R α ( σ ) dσ does not have a proper meaning.5 heorem 6 Suppose α is a Lip ( p + ǫ − one-form for some ǫ > . Then there exists β taking valuesin closed (or say cocyclic) one-forms on G [ p ] , such that for any σ t ∈ G of finite length with horizontallift ˆ σ t ∈ G [ p ] , we have Z α ( σ t ) dσ t = Z β ˆ σ t (ˆ σ t ) d ˆ σ t ,Moreover, the integral R β ˆ σ t (ˆ σ t ) d ˆ σ t is well-defined for any continuous path ˆ σ of finite p -variation takingvalues in G [ p ] and the integral is continuous with respect to ˆ σ in p -variation metric. Conclusion
Based on our formulation, to make sense of the rough integral, all we need is thecompensated Young regularity between two dual paths: one takes values in the group and the othertakes values in the closed (cocyclic) one-forms on the group. By viewing the Lipschitz functions asslowly-varying polynomial functions and by lifting the polynomial one-forms to closed one-forms, weencapsulate the nonlinearity of the integral to the structure of the group and to the closed one-forms onthe group so that the idea behind the generalized integral is clearer and bears a similar form to the linearYoung integral.
Suppose U , V and W are Banach spaces and p ≥ A and B are Banach algebras and G is a topological group in A . We denote by L ( A , B ) theset of continuous linear mappings from A to B , and we denote by C ( G , L ( A , B )) the set of continuousmappings from G to L ( A , B ). Definition 7 (Cocyclic One-Form)
We say β ∈ C ( G , L ( A , B )) is a cocyclic one-form, if there existsa topological group H in B such that β ( a, b ) ∈ H for all a, b ∈ G and β ( a, b ) β ( ab, c ) = β ( a, bc ) , ∀ a, b, c ∈ G .We denote the set of cocyclic one-forms by B ( G , H ) (or B ( G ) ). Since a Banach space U is canonically embedded in the Banach algebra { ( c, u ) | c ∈ R , u ∈ U} withmultiplication ( c, u ) ( r, v ) = ( cr, ru + cv ), we denote by B ( G , U ) the set of cocyclic one-forms takingvalues in U satisfying β ( a, b ) + β ( ab, c ) = β ( a, bc ) for all a, b, c in G .For p ≥
1, we denote by [ p ] the integer part of p . As in [9], we equip the tensor powers of V withadmissible norms and assume T ([ p ]) ( V ) = R ⊕ V ⊕ · · · ⊕ V ⊗ [ p ] is a graded Banach algebra equipped withthe norm k·k := P [ p ] k =0 k π k ( · ) k ( π k denotes the projection to V ⊗ k ), and the multiplication on T ([ p ]) ( V ) isinduced by a finite family of linear projective mappings denoted by P [ p ] ; G [ p ] is a closed topological groupin T ([ p ]) ( V ) whose linear span is T ([ p ]) ( V ) and whose projection to R is 1.When G [ p ] is the nilpotent Lie group over V , P [ p ] = { π k } [ p ] k =0 with π k ( ab ) = P kj =0 π j ( a ) ⊗ π k − j ( b )for k = 0 , , . . . , [ p ] and for a, b ∈ T ([ p ]) ( V ). When G [ p ] is the Butcher group over R d , P [ p ] is the setof labelled forests of degree less or equal to [ p ] and σ ( ab ) = P c P c ( σ ) ( a ) R c ( σ ) ( b ) for σ ∈ P [ p ] andfor a, b ∈ T ([ p ]) ( R d ) where the sum is over all admissible cuts of the forest σ . For more details see[10, 7, 1, 2, 4].We equip G [ p ] with the norm |·| := P [ p ] k =1 k π k ( · ) k k and define the p -variation of a continuous path g : [0 , T ] → G [ p ] by k g k p − var, [0 ,T ] := sup D,D ⊂ [0 ,T ] (cid:16)P k,t k ∈ D (cid:12)(cid:12) g − t k g t k +1 (cid:12)(cid:12) p (cid:17) p .We denote by C p − var (cid:0) [0 , T ] , G [ p ] (cid:1) the set of continuous paths of finite p -variation on [0 , T ] taking valuesin G [ p ] . (The exact form of norm on G [ p ] is not important, and the integral can be defined as long as thenorm on the group and the norm on the one-form ”compensate” each other.)6or α ∈ L (cid:0) T [ p ] ( V ) , U (cid:1) , we denote k α ( · ) k := sup v ∈ T [ p ] ( V ) , k v k =1 k α ( v ) k , k α ( · ) k k := sup v ∈V ⊗ k , k v k =1 k α ( v ) k , k = 1 , , . . . , [ p ] .We say ω : { ( s, t ) | ≤ s ≤ t ≤ T } → R + is a control, if ω is continuous, non-negative, vanishes on thediagonal and satisfies ω ( s, u ) + ω ( u, t ) ≤ ω ( s, t ) for 0 ≤ s ≤ u ≤ t ≤ T . As in [9], for g ∈ C ([0 , T ] , G [ p ] )and β : [0 , T ] → B ( G [ p ] , U ), if the limit existslim | D |→ ,D = { t k } nk =0 ⊂ [0 ,T ] β ( g , g ,t ) β t ( g t , g t ,t ) · · · β t n − (cid:0) g t n − , g t n − ,T (cid:1) with g s,t := g − s g t ,then we define the limit to be the integral R T β u ( g u ) dg u . Definition 8 (Dominated Path)
For g ∈ C p − var (cid:0) [0 , T ] , G [ p ] (cid:1) and Banach space U , we say a continu-ous path ρ : [0 , T ] → U is dominated by g , if there exists β : [0 , T ] → B (cid:0) G [ p ] , U (cid:1) which satisfies, for some M > , control ω and θ > , k β t ( g t , · ) k ≤ M , ∀ t ∈ [0 , T ] , k ( β t − β s ) ( g t , · ) k k ≤ ω ( s, t ) θ − kp , ∀ ≤ s ≤ t ≤ T , k = 1 , , . . . , [ p ] ,such that ρ t = ρ + R t β u ( g u ) dg u for t ∈ [0 , T ] . Based on the definition of dominated paths, we introduce an operator norm on the space of one-formsto quantify the convergence of one-forms (associated with Picard iterations).For g ∈ C p − var (cid:0) [0 , T ] , G [ p ] (cid:1) and control ω , we say g is controlled by ω if k g k pp − var, [ s,t ] ≤ ω ( s, t ) forall s < t . Definition 9 (Operator Norm)
For g ∈ C p − var (cid:0) [0 , T ] , G [ p ] (cid:1) controlled by ω and β : [0 , T ] → B (cid:0) G [ p ] , U (cid:1) ,we define, for γ > , k β k γ := sup t ∈ [0 ,T ] k β t ( g t , · ) k + max k =1 ,..., ⌊ γ ⌋ sup ≤ s ≤ t ≤ T k ( β t − β s ) ( g t , · ) k k ω ( s, t ) γ − kp . Suppose k β k γ < ∞ . When γ increases, the integrability of β increases. In the extreme case that γ tends to infinity, β is compelled to be a constant cocyclic one-form, so is integrable against any continuouspath. If γ > p − σ : [0 , T ] → U such that k σ t − σ s − β s ( g s , g s,t ) k ≤ C k g k γp − var, [ s,t ] for all s < t , then σ is a weakly controlled path introduced by Gubinelli [3]. When γ > p , β is integrableagainst g and t R t β ( g u ) dg u is a dominated path. Definition 10
Suppose there exists a mapping I ′ ∈ L ( T ([ p ]) ( V ) , T ([ p ]) ( V ) ⊗ ) which satisfies I ′ (1) = I ′ ( V ) = 0 , I ′ (cid:0) V ⊗ k (cid:1) ⊆ V ⊗ ( k − ⊗ V , k = 2 , . . . , [ p ] ,and (with ′ n, denoting the projection of T ([ p ]) ( V ) ⊗ to P [ p ] − k =1 V ⊗ k ⊗ V ) I ′ ( ab ) = I ′ ( a ) + 1 ′ n, (( a ⊗ a ) I ′ ( b )) + 1 ′ n, (( a − ⊗ ( a ( b − , ∀ a, b ∈ G [ p ] . Due to the special form of the dominated paths in Picard iterations, we only need the mapping I ′ (instead of I as in [9]) for the recursive integrals to make sense. Roughly speaking, the mapping I isused to define the iterated integral of two dominated (controlled) paths, and corresponds to a universalcontinuous linear mapping which has the ”formal” expression: I ( a ) = Z T ( g ,u − ⊗ δg ,u , g ∈ C (cid:0) [0 , T ] , G [ p ] (cid:1) , a = g ,T , ∀ a ∈ G [ p ] .7he mapping I ′ encodes part of the information of I , is used to define the integral of a dominated(controlled) path against the first level of the given group-valued path, and corresponds to a universalcontinuous linear mapping with the formal expression: I ′ ( a ) = Z T ( g ,u − ⊗ δx u , x := π ( g ) , g ∈ C (cid:0) [0 , T ] , G [ p ] (cid:1) , a = g ,T , ∀ a ∈ G [ p ] .In particular, I ′ is well-defined for degree-[ p ] nilpotent Lie group and degree-[ p ] Butcher group forany p ≥ Lemma 11
Suppose g ∈ C p − var (cid:0) [0 , T ] , G [ p ] (cid:1) is controlled by ω , β : [0 , T ] → B (cid:0) G [ p ] , L ( V , W ) (cid:1) satisfies k β k γ < ∞ for some γ ∈ ( p − , [ p ]] ,and there exists ϕ : [0 , T ] → L ( V , W ) which satisfies for some M > , k ϕ t − ϕ s − β s ( g s , g s,t ) k ≤ M k β k γ ω ( s, t ) γp , ∀ ≤ s < t ≤ T . (2) If we define η : [0 , T ] → B (cid:0) G [ p ] , W (cid:1) by η t ( a, b ) := ϕ t π (cid:0) g − t a ( b − (cid:1) + β t ( g t , · ) π ( · ) I ′ (cid:0) g − t a ( b − (cid:1) , ∀ a, b ∈ G [ p ] , ∀ t ∈ [0 , T ] ,then for some structural constant c (depending on the mapping I ′ ), sup ≤ t ≤ T k η t ( g t , · ) k ≤ sup ≤ t ≤ T k ϕ t k + c k β k γ ,and there exists a constant C = C ( M, p, ω (0 , T )) such that k ( η t − η s ) ( g t , · ) k k ≤ C k β k γ ω ( s, t ) γ +1 − kp , ∀ s < t , k = 1 , , . . . , [ p ] .As a consequence, k η k γ +1 < ∞ and t R t η u ( g u ) dg u is a dominated path. Proof.
It is clear that for some constant c depending on I ′ , k η t ( g t , · ) k ≤ k ϕ t k + c k β t ( g t , · ) k ≤ sup ≤ t ≤ T k ϕ t k + c k β k γ , ∀ t ∈ [0 , T ] .For s < t and v ∈ R ⊕ V ⊕ · · · ⊕ V ⊗ [ p ] (calculation or based on the proof in [9]), we have( η t − η s ) ( g t , v ) = ( ϕ t − ϕ s − β s ( g s , g s,t )) π ( v ) + ( β t − β s ) ( g t , · ) π ( · ) I ′ ( v ) (3)+ P σ ∈P [ p ] , | σ | =[ p ] β s ( g s , σ ( g s,t )) π ( v )+ P [ p ] k =2 P σ ∈P [ p ] , | σ |≥ [ p ]+1 − k β s ( g s , σ ( g s,t ) · ) π ( · ) I ′ ( π k ( v )) .Since k β k γ < ∞ and I ′ (cid:0) V ⊗ k (cid:1) ⊆ V ⊗ ( k − ⊗ V , k = 2 , . . . , [ p ], we have, for some structural constant C depending on the norm of the mapping I ′ ,sup v ∈V ⊗ k , k v k =1 k ( β t − β s ) ( g t , · ) π ( · ) I ′ ( v ) k ≤ C k β k γ ω ( s, t ) γ +1 − kp , k = 1 , , . . . , [ p ] .Moreover, for s < t , k β s ( g s , σ ( g s,t )) k ≤ k β k γ ω ( s, t ) [ p ] p , ∀ σ ∈ P [ p ] , | σ | = [ p ] , k β s ( g s , σ ( g s,t ) · ) π ( · ) I ′ ( π k ( · )) k ≤ k β k γ (1 ∨ ω (0 , T )) ω ( s, t ) [ p ]+1 − kp , ∀ σ ∈ P [ p ] , | σ | ≥ [ p ] + 1 − k .8ence, since γ ≤ [ p ], combined with (3) and (2), for some C = C ( M, p, ω (0 , T )), we have k ( η t − η s ) ( g t , · ) k k ≤ C k β k γ ω ( s, t ) γ +1 − kp , ∀ s < t , k = 1 , , . . . , [ p ] .For γ ≥ ⌊ γ ⌋ denotes the largest integer which is strictly less than γ . For σ i ∈ P [ p ] , i = 1 , . . . , l , | σ | + · · · + | σ l | ≤ [ p ], we denote by σ ∗ · · · ∗ σ l the continuous linear mapping from V ⊗ ( | σ | + ··· + | σ l | ) to V ⊗| σ | ⊗ · · · ⊗ V ⊗| σ l | satisfying ( σ ∗ · · · ∗ σ l ) ( a ) = σ ( a ) ⊗ · · · ⊗ σ l ( a ) for all a ∈ G [ p ] (see [9] for moredetails). Definition 12 ( β ( f ( ρ )) ) Let ρ · = ρ + R · β ( g ) dg : [0 , T ] → U be a dominated path and f : U → W bea
Lip ( γ ) function for some γ > p − . We define β ( f ( ρ )) : [0 , T ] → B (cid:0) G [ p ] , W (cid:1) by, for a, b ∈ G [ p ] and s ∈ [0 , T ] , β ( f ( ρ )) s ( a, b ) = ⌊ γ ⌋ X l =1 l ! (cid:0) D l f (cid:1) ( f ( ρ s )) β s ( g s , · ) ⊗ l X σ i ∈P [ p ] , | σ | + ··· + | σ l |≤ [ p ] ( σ ∗ · · · ∗ σ l ) (cid:0) g − s a ( b − (cid:1) . Definition 13 (Integral)
Suppose ρ : [0 , T ] → U is a path dominated by g ∈ C p − var (cid:0) [0 , T ] , G [ p ] (cid:1) and f : U → L ( V , W ) is a Lip ( γ ) function for some γ > p − . With β ( f ( ρ )) in Definition 12, if we define β : [0 , T ] → B (cid:0) G [ p ] , W (cid:1) by β s ( a, b ) = f ( ρ s ) π (cid:0) g − s a ( b − (cid:1) + β ( f ( ρ )) s ( g s , · ) ⊗ π ( · ) I ′ (cid:0) g − s a ( b − (cid:1) , ∀ a, b ∈ G [ p ] , ∀ s , (4) then β is integrable against g and we define the integral R f ( ρ ) dx : [0 , T ] → W by Z t f ( ρ u ) dx u := Z t β u ( g u ) dg u , ∀ t ∈ [0 , T ] . That β is integrable against g follows from Lemma 11. When G [ p ] is the nilpotent Lie group, theintegral coincides with the first level of the rough integral in [7]. When G [ p ] is the Butcher group theintegral coincides with the integral in [4]. Definition 14 (Solution)
For γ + 1 > p ≥ , suppose g ∈ C p − var (cid:0) [0 , T ] , G [ p ] (cid:1) and f : U → L ( V , U ) isa Lip ( γ ) function. We say y is a solution to the rough differential equation (with x := π ( g ) ) dy = f ( y ) dx , y = ξ ∈ U , (5) if y is a path dominated by g , and y · = ξ + R · f ( y u ) dx u with the integral defined in Definition 13. Since dominated paths are defined through integrable one-forms, instead of formulating the fixed-point problem in the space of paths as in Definition 14, we could also formulate the fixed-point problemin the space of integrable one-forms, and y is called a solution to (5) if the one-form associated with y isa fixed point of the mapping β ˆ β where ˆ β is the one-form associated with R f ( y ) dx . Schwartz gave a beautiful proof in [11] of the convergence of the series of Picard iterations for SDEs.Instead of working with contraction mapping on small intervals and pasting the local solutions together,he used the iterative expression of the differences between the n th and ( n + 1)th Picard iterations andproved that the sequence of differences decay factorially on the whole interval. Put in the simplest form,his argument can be summarized as follows. Suppose f is Lip (1) and consider the SDE: dX t = f ( X t ) dB t , X = ξ .9e define the series of Picard iterations by X n +1 t = ξ + R t f ( X nu ) dB u with X t ≡ ξ . Then by using Itˆo’sisometry and the Lipschitz property of f , we have E (cid:16)(cid:12)(cid:12) X n +1 t − X nt (cid:12)(cid:12) (cid:17) = E Z t (cid:12)(cid:12) f ( X nu ) − f (cid:0) X n − u (cid:1)(cid:12)(cid:12) du ≤ k f k Z t E (cid:16)(cid:12)(cid:12) X nu − X n − u (cid:12)(cid:12) (cid:17) du .By iterating this process, we obtain a factorial decay and the global convergence of the Picard series.We will try to extend his argument to RDEs. However, there are several points to pay attention to:generally, Lip (1) is insufficient for rough integral to be well-defined and it is illegitimate to take modulusinside the rough integral; there is no L space and no Itˆo’s isometry for general rough paths, so the factorialdecay can not be obtained in a similar way. We will rely critically on the Division Property of Lipschitzfunctions, and rely critically on the factorial decay of the Signature of a rough path [7]. In particular, weprove that the one-forms associated with the differences between the n th and ( n + 1)th Picard iterationsdecay factorially in operator norm as n tends to infinity on the whole interval. As a consequence, theone-forms associated with the Picard iterations converge in operator norm, which implies the convergenceof the Picard iterations and the convergence of their group-valued enhancements. By using the factorialdecay of the iterated integrals, we can prove the solution is unique. The continuity of the solution withrespect to the driving noise follows from the uniform convergence of the Picard iterations for the roughdifferential equations whose driving rough paths are uniformly bounded in p -variation.Let U and V be two Banach spaces. Definition 15 (Picard Iterations)
For γ + 1 > p ≥ , suppose g ∈ C p − var (cid:0) [0 , T ] , G [ p ] (cid:1) , f : U → L ( V , U ) is a Lip ( γ ) function and ξ ∈ U . We define the series of Picard iterations associated with therough differential equation dy = f ( y ) dx , y = ξ , by y nt := ξ + Z t f (cid:0) y n − u (cid:1) dx u , ∀ t ∈ [0 , T ] , with y t ≡ ξ . Definition 16
We define ζ n : [0 , T ] → B (cid:0) G [ p ] , U (cid:1) , n ≥ , by ζ ns ( a, b ) = f (cid:0) y n − s (cid:1) π (cid:0) g − s a ( b − (cid:1) + β (cid:0) f (cid:0) y n − (cid:1)(cid:1) s ( g s , · ) π ( · ) I ′ (cid:0) g − s a ( b − (cid:1) , ∀ a, b ∈ G [ p ] , ∀ s, where β (cid:0) f (cid:0) y n − (cid:1)(cid:1) is defined in term of y n − · = ξ + R · ζ n − ( g ) dg as in Definition 12 with ζ ≡ . Then based on the definition of the integral in Definition 13, y n · = ξ + R · ζ n ( g ) dg , n ≥
0, and { y n } ∞ n =0 are paths dominated by g . Lemma 17
Suppose g ∈ C p − var (cid:0) [0 , T ] , G [ p ] (cid:1) is controlled by ω , and f : U → L ( V , U ) is a Lip ( γ ) function for some γ ∈ ( p − , [ p ]] . Then there exists a constant C = C ( p, γ, k f k Lip( γ ) , ω (0 , T )) such that sup n ≥ k ζ n k γ +1 ≤ C. Proof.
We first suppose ω (0 , T ) ≤
1, and prove that there exists λ p,γ > p and γ such that when k f k Lip( γ ) ≤ λ p,γ we have sup n ≥ k ζ n k γ +1 ≤ λ p,γ . We prove it by using mathematicalinduction. Suppose for some constant λ n ∈ (0 , k ζ n k γ +1 ≤ λ n ,which holds when n = 0 since ζ ≡
0. We want to prove that there exists a constant C p,γ ≥ (cid:13)(cid:13) ζ n +1 (cid:13)(cid:13) γ +1 ≤ k f k Lip( γ ) (1 + C p,γ λ n ) := λ (1 + C p,γ λ n ) .Then when λ ∈ (0 , (2 C p,γ ) − ), if λ n ≤ λ/ (1 − C p,γ λ ) then λ (1 + C p,γ λ n ) ≤ λ/ (1 − C p,γ λ ). Since λ = 0 ≤ λ/ (1 − C p,γ λ ), we have λ n ≤ λ/ (1 − C p,γ λ ) ≤ λ for all n ≥
0. It can be checked that ζ n +1 islinear with respect to scalar multiplication of f , so we assume k f k Lip( γ ) = 1, and want to prove (cid:13)(cid:13) ζ n +1 (cid:13)(cid:13) γ +1 ≤ C p,γ λ n when k ζ n k λ +1 ≤ λ n . (6)10y following similar proof as in [9] of the stability of dominated paths under composition with regularfunctions and by using k f k Lip( γ ) = 1, ω (0 , T ) ≤ k ζ n k γ +1 ≤ λ n ∈ (0 , C p,γ > s < t , k ( β ( f ( y n )) t − β ( f ( y n )) s ) ( g t , · ) k k ≤ C p,γ λ n ω ( s, t ) γ − kp , k = 1 , , . . . , [ p ] − k f ( y nt ) − f ( y ns ) − β ( f ( y n )) s ( g s , g s,t ) k ≤ C p,γ λ n ω ( s, t ) γp .Since y n +1 · = ξ + R · f ( y n ) dx , by using Lemma 11, we have (cid:13)(cid:13)(cid:0) ζ n +1 t − ζ n +1 s (cid:1) ( g t , · ) (cid:13)(cid:13) k ≤ C p,γ λ n ω ( s, t ) γ +1 − kp , ∀ s < t , k = 1 , , . . . , [ p ] , (cid:13)(cid:13) ζ n +1 t ( g t , · ) (cid:13)(cid:13) ≤ C p,γ λ n , ∀ t ,which implies (6).For the general case, we rescale the differential equation and consider dy = ˆ f ( y ) d ˆ x , y = ξ , with c := λ − p,γ || f || Lip( γ ) , ˆ f := c − f and ˆ g := P [ p ] k =0 c k π k ( g ) with ˆ x := π (ˆ g ). Then the solution path staysunchanged, and we have || ˆ f || Lip( γ ) ≤ λ p,γ . If we denote by { β n } n the one-forms (as in Definition 16)associated with the Picard iterations of dy = ˆ f ( y ) d ˆ x , y = ξ , then it can be proved inductively that, ζ ns ( g t , v ) = β ns (ˆ g t , ˆ v ) , ∀ v ∈ R ⊕ V ⊕ · · · ⊕ V ⊗ [ p ] with ˆ v := P [ p ] k =0 c k π k ( v ) , ∀ s < t , ∀ n ≥ n ≥ k β n k γ +1 < ∞ then sup n ≥ k ζ n k γ +1 < ∞ . Denote ˆ ω ( s, t ) := c p ω ( s, t ) for s < t . We divide the interval [0 , T ] into the union of finitely many overlapping subintervals ∪ [ s i , t i ] suchthat ˆ ω ( s i , t i ) ≤ i . Because these subintervals overlap, we can paste their estimates together.Indeed, by using the cocyclic property, for s < u < t ,( β nt − β ns ) ( g t , v ) = ( β nt − β nu ) ( g t , v ) + ( β nu − β ns ) ( g u , g u,t v ) , ∀ v ∈ V ⊕ · · · ⊕ V ⊗ [ p ] ,which implies k ( β nt − β ns ) (ˆ g t , · ) k k ≤ k ( β nt − β nu ) (ˆ g t , · ) k k + P [ p ] j = k k ( β nu − β ns ) (ˆ g u , · ) k j ≤ c ˆ ω ( u, t ) γ +1 − kp + c P [ p ] j = k ˆ ω ( s, u ) γ +1 − jp ≤ c ˆ ω ( s, t ) γ +1 − kp ,where c i may depend on ˆ ω (0 , T ). Definition 18
With the Picard iterations { y n } n in Definition 15, we define z n : [0 , T ] → U , n ≥ , by z nt = y nt − y n − t , t ∈ [0 , T ] . Since { y n } n are Picard iterations which satisfy y n +1 · = ξ + R · f ( y nu ) dx u with y · ≡ ξ , by using thedivision property of f (i.e. f ( x ) − f ( y ) = h ( x, y ) ( x − y ) for all x, y ∈ U and k h k Lip( γ − ≤ C k f k Lip( γ ) ),we have the recursive expression of { z n } n : z n +1 t = R t h (cid:0) y nu , y n − u (cid:1) z nu dx u , with z t = f ( ξ ) ( x t − x ) , ∀ t ∈ [0 , T ] .By iteration, we have z n +1 t = R ··· R p ≥ , suppose g ∈ C p − var (cid:0) [0 , T ] , G [ p ] (cid:1) with x := π ( g ) and f : U → L ( V , U ) is a Lip ( γ ) function. Let h be the function obtained in the division property of f as in Lemma 1. Forintegers n ≥ l ≥ and ≤ s ≤ t ≤ T , we define η l,ns,t ∈ L ( U , U ) , l ≥ , and η ,ns,t ∈ U , recursively by η l,n +1 s,t := Z ts h (cid:0) y n +1 u , y nu (cid:1) η l,ns,u dx u ,with η l,ls,t := Z ts h (cid:0) y lu , y l − u (cid:1) dx u , l ≥ , and η , s,t := f ( ξ ) ( x t − x s ) . The integrals are well-defined based on Lemma 11 and inductive arguments. In particular, we have z n +1 t = η ,n ,t , ∀ t ∈ [0 , T ] .We define η l,ns,t for general l and s to make the induction work.Then we define the integrable one-form β l,ns, · associated with the dominated path η l,ns, · and prove that β l,ns, · decay factorially in operator norm as ( n − l ) tends to infinity.For σ , σ ∈ P [ p ] , | σ | + | σ | ≤ [ p ], we denote by σ ∗ σ the continuous linear mapping from V ⊗ ( | σ | + | σ | ) to V ⊗| σ | ⊗ V ⊗| σ | satisfying ( σ ∗ σ ) ( a ) = σ ( a ) ⊗ σ ( a ) for all a ∈ G [ p ] (see [9] for more details). Definition 20
With η l,ns,t in Definition 19, for integers n ≥ l ≥ and s ∈ [0 , T ) , we define the integrableone-form β l,ns, · : [ s, T ] → B (cid:0) G [ p ] , L ( U , U ) (cid:1) , l ≥ , and β ,ns, · : [ s, T ] → B (cid:0) G [ p ] , U (cid:1) (associated with η l,ns, · and η ,ns, · respectively) recursively by, for t ∈ ( s, T ] and a, b ∈ G [ p ] , β l,n +1 s,t ( a, b ) = β n +1 ,n +1 t,t ( a, b ) η l,ns,t + h (cid:0) y n +1 t , y nt (cid:1) β l,ns,t ( g t , · ) π ( · ) I ′ (cid:0) g − t a ( b − (cid:1) + β (cid:0) h (cid:0) y n +1 , y n (cid:1)(cid:1) t ( g t , · ) β l,ns,t ( g t , · ) P σ i ∈P [ p ] , | σ | + | σ |≤ [ p ] ( σ ∗ σ ) ( · ) π ( · ) I ′ (cid:0) g − t a ( b − (cid:1) , β l,ls,t ( a, b ) = h (cid:0) y lt , y l − t (cid:1) π (cid:0) g − t a ( b − (cid:1) + β (cid:0) h (cid:0) y l , y l − (cid:1)(cid:1) t ( g t , · ) π ( · ) I ′ (cid:0) g − t a ( b − (cid:1) , l ≥ , β , s,t ( a, b ) = f ( ξ ) π (cid:0) g − t a ( b − (cid:1) ,where β (cid:0) h (cid:0) y n +1 , y n (cid:1)(cid:1) is defined from (cid:0) y n +1 , y n (cid:1) t = ( ξ, ξ ) + R t (cid:0) ζ n +1 u , ζ nu (cid:1) ( g u ) dg u as in Definition 12. The notation in the definition of β l,n +1 may need some explanations. For k = 1 , . . . , [ p ] − v ∈ V ⊗ ( k +1) , we have I ′ ( v ) ∈ V ⊗ k ⊗ V . Since σ ∗ σ : V ⊗ ( | σ | + | σ | ) → V ⊗| σ | ⊗ V ⊗| σ | and π : V → V ,we have ( σ ∗ σ ) ( · ) π ( · ) I ′ ( v ) ∈ V ⊗| σ | ⊗ V ⊗| σ | ⊗ V for any v ∈ V ⊗ ( | σ | + | σ | +1) . Then in the expression β (cid:0) h (cid:0) y n +1 , y n (cid:1)(cid:1) t ( g t , · ) β l,ns,t ( g t , · ) ( σ ∗ σ ) ( · ) π ( · ) I ′ ( v ) for v ∈ V ⊗ ( | σ | + | σ | +1) ,we treat β (cid:0) h (cid:0) y n +1 , y n (cid:1)(cid:1) t ( g t , · ) as a continuous linear mapping on V ⊗| σ | and treat β l,ns,t ( g t , · ) as a con-tinuous linear mapping on V ⊗| σ | .Based on the definition of integral in Definition 13, we have η l,ns,t = R ts β l,ns,u ( g u ) dg u . In particular, z n +1 t = Z t β ,n ,u ( g u ) dg u , ∀ t ∈ [0 , T ] . Lemma 21
Suppose g ∈ C p − var (cid:0) [0 , T ] , G [ p ] (cid:1) is controlled by ω , and f : U → L ( V , U ) is a Lip ( γ ) function for some γ ∈ ( p, [ p ] + 1] . Then there exist a constant C = C ( p, γ, k f k Lip( γ ) , ω (0 , T )) such that (cid:13)(cid:13)(cid:13) β l,ns, · (cid:13)(cid:13)(cid:13) γ ≤ C n − [ p ] − l (cid:16) n − [ p ] − lp (cid:17) ! , ∀ n ≥ l + [ p ] + 1 , ∀ l ≥ , ∀ s ∈ [0 , T ) , (7) where β l,ns, · denotes t β l,ns,t introduced in Definition 20 for t ∈ [ s, T ] . roof. The constants in this proof may depend on p , γ , k f k Lip( γ ) and ω (0 , T ).Firstly, we prove that, for integers n ≥ l ≥ ≤ s ≤ u ≤ t ≤ T , β l,ns,t ( g t , v ) = β l,nu,t ( g t , v ) + P nj = l +1 β j,nu,t ( g t , v ) η l,j − s,u , ∀ v ∈ V ⊕ · · · ⊕ V ⊗ [ p ] . (8)The equality holds when n = l based on the definition of β l,ls,t . Suppose it holds when n − l ≤ s . Thenby combining the definition of β l,n +1 in Definition 20 with the inductive hypothesis (8) and by using η l,ns,t = P nj = l +1 η j,nu,t η l,j − s,u + η l,ns,u + η l,nu,t , it can be proved that (8) holds when n − l = s + 1.Without loss of generality we assume γ ∈ ( p, [ p ] + 1]. Based on Lemma 17, sup n ≥ k ζ n k [ p ]+1 < ∞ .Then since k h k Lip( γ − ≤ C k f k Lip( γ ) , by using Lemma 11, it can be proved inductively that, for some K ≥
1, sup l ≥ max n = l,...,l +[ p ] (cid:13)(cid:13)(cid:13) β l,ns, · (cid:13)(cid:13)(cid:13) γ ≤ K . (9)Then combined with η l,ns,t = R ts β l,ns,u ( g u ) dg u , we have, for some constant M > (cid:13)(cid:13)(cid:13) η l,ns,t (cid:13)(cid:13)(cid:13) ≤ M ω ( s, t ) n − l +1 p , ∀ s < t , n − l + 1 = 1 , , . . . , [ p ] , ∀ l ≥ η l,ns,t = P nj = l +1 η j,nu,t η l,j − s,u + η l,ns,u + η l,nu,t , ∀ ≤ s < u < t ≤ T , ∀ n ≥ l ≥ β = 3 p and some constant M ≥
1, (we choose β = 3 p to make the induction work) (cid:13)(cid:13)(cid:13) η l,ns,t (cid:13)(cid:13)(cid:13) ≤ M n − l +1 p ω ( s, t ) n − l +1 p β (cid:16) n − l +1 p (cid:17) ! , ∀ s < t , ∀ n ≥ l ≥
0. (10)Then we prove by induction on n − l that, for some constants K ≥ C ≥ (cid:13)(cid:13)(cid:13)(cid:16) β l,ns,t − β l,ns,u (cid:17) ( g t , · ) (cid:13)(cid:13)(cid:13) k ≤ K C n − [ p ] − lp ω ( s, t ) n − [ p ] − lp β (cid:16) n − [ p ] − lp (cid:17) ! ω ( u, t ) γ − kp , ∀ s < u < t , ∀ n ≥ l + [ p ] , ∀ l ≥
0, (11)which holds when n − l = [ p ] with K = K β based on (9). Suppose (11) holds when n − l = [ p ] , . . . , s forsome s ≥ [ p ]. Then when n − l = s + 1 (so n − l ≥ [ p ] + 1), for s < u < t , based on (8), we have, for any v ∈ V ⊕ · · · ⊕ V ⊗ [ p ] , (cid:16) β l,ns,t − β l,ns,u (cid:17) ( g t , v ) (12)= (cid:16) β l,nu,t − β l,nu,u (cid:17) ( g t , v ) + P nj = n − [ p ] (cid:16) β j,nu,t − β j,nu,u (cid:17) ( g t , v ) η l,j − s,u + P n − [ p ] − j = l +1 (cid:16) β j,nu,t − β j,nu,u (cid:17) ( g t , v ) η l,j − s,u = : I ( v ) + II ( v ) + III ( v ) .For I ( v ), by using (8), we have (cid:16) β l,nu,t − β l,nu,u (cid:17) ( g t , v ) = (cid:16) β l,nt,t − β l,nu,u (cid:17) ( g t , v ) + P nj = l +1 β j,nt,t ( g t , v ) η l,j − u,t = P nj = n − [ p ]+1 β j,nt,t ( g t , v ) η l,j − u,t ,where we used β j,nt,t ≡ n ≥ [ p ] + j which can be proved inductively based on the definition of β l,ns,t in Definition 20. Hence, for k = 1 , . . . , [ p ], by using that (cid:13)(cid:13)(cid:13) β j,nt,t ( g t , · ) (cid:13)(cid:13)(cid:13) k = 0, j ≤ n − k , and the factorialdecay of η l,ns,t in (10), we have, for some C ≥
1, (since γ ≤ [ p ] + 1) k I ( · ) k k = (cid:13)(cid:13)(cid:13)P nj = n − [ p ]+1 β j,nt,t ( g t , · ) η l,j − u,t (cid:13)(cid:13)(cid:13) k (13) ≤ n X j = n − k +1 (cid:13)(cid:13)(cid:13) β j,nt,t ( g t , · ) (cid:13)(cid:13)(cid:13) k M j − lp ω ( u, t ) j − lp β (cid:16) j − lp (cid:17) ! ≤ K C M n − [ p ] − lp ω ( u, t ) n − [ p ] − lp β (cid:16) n − [ p ] − lp (cid:17) ! ω ( u, t ) γ − kp .13or II ( v ), by using (9) and (10), we have k II ( · ) k k = (cid:13)(cid:13)(cid:13)P nj = n − [ p ] (cid:16) β j,nu,t − β j,nu,u (cid:17) ( g t , · ) η l,j − s,u (cid:13)(cid:13)(cid:13) k (14) ≤ n X j = n − [ p ] (cid:13)(cid:13)(cid:13)(cid:16) β j,nu,t − β j,nu,u (cid:17) ( g t , · ) (cid:13)(cid:13)(cid:13) k M j − lp ω ( s, u ) j − lp β (cid:16) j − lp (cid:17) ! ≤ K C M n − [ p ] − lp ω ( s, u ) n − [ p ] − lp β (cid:16) n − [ p ] − lp (cid:17) ! ω ( u, t ) γ − kp .For III ( v ), since [ p ] < n − j ≤ n − l − s when j = l + 1 , . . . , n − [ p ] −
1, by using the inductivehypothesis (11) and neo-classical inequality [8, 5], we have k III ( · ) k k = (cid:13)(cid:13)(cid:13)P n − [ p ] − j = l +1 (cid:16) β j,nu,t − β j,nu,u (cid:17) ( g t , · ) η l,j − s,u (cid:13)(cid:13)(cid:13) k (15) ≤ n − [ p ] − X j = l +1 K C n − [ p ] − jp ω ( u, t ) n − [ p ] − jp β (cid:16) n − [ p ] − jp (cid:17) ! M j − lp ω ( s, u ) j − lp β (cid:16) j − lp (cid:17) ! ω ( u, t ) γ − kp ≤ K pβ ( C ∨ M ) n − [ p ] − lp ω ( s, t ) n − [ p ] − lp β (cid:16) n − [ p ] − lp (cid:17) ! ω ( u, t ) γ − kp .Hence, based on (12), (13), (14) and (15), since n − [ p ] − l ≥ β = 3 p , by choosing K = K ( C ∨ β )( β to take into account of n = l + [ p ]) and C = 3 M , we have (11) holds when n − l = s + 1, and theinduction is complete.On the other hand, when n − l ≥ [ p ], based on (8) and by using β j,nt,t ≡ n ≥ [ p ] + j , we have β l,ns,t ( g t , · ) = β l,nt,t ( g t , · ) + P nj = l +1 β j,nt,t ( g t , · ) η l,j − s,t = P nj = n − [ p ]+1 β j,nt,t ( g t , · ) η l,j − s,t .Hence, by using the factorial decay of η l,ns,t in (10), we have (cid:13)(cid:13)(cid:13) β l,ns,t ( g t , · ) (cid:13)(cid:13)(cid:13) ≤ n X j = n − [ p ]+1 (cid:13)(cid:13)(cid:13) β j,nt,t ( g t , · ) (cid:13)(cid:13)(cid:13) k (cid:13)(cid:13)(cid:13) η l,j − s,t (cid:13)(cid:13)(cid:13) ≤ K C M n − [ p ] − lp ω ( s, t ) n − [ p ] − lp (cid:16) n − [ p ] − lp (cid:17) ! . (16)Then for s ∈ [0 , T ) since (cid:13)(cid:13)(cid:13) β l,ns, · (cid:13)(cid:13)(cid:13) γ := sup s ≤ t ≤ T (cid:13)(cid:13)(cid:13) β l,ns,t ( g t , · ) (cid:13)(cid:13)(cid:13) + max k =1 ,..., [ p ] sup s ≤ u ≤ t ≤ T ω ( u, t ) − ( γ − kp ) (cid:13)(cid:13)(cid:13)(cid:16) β l,ns,t − β l,ns,u (cid:17) ( g t , · ) (cid:13)(cid:13)(cid:13) k ,we have the lemma holds based on (11) and (16). Theorem 22 (Existence, Uniqueness and Continuity of the Solution)
For [ p ] + 1 ≥ γ > p ≥ ,suppose g ∈ C p − var (cid:0) [0 , T ] , G [ p ] (cid:1) is controlled by ω , f : U → L ( V , U ) is a Lip ( γ ) function and ξ ∈ U .Then the Picard iterations { y n } ∞ n =0 in Definition 15 converge uniformly on [0 , T ] to the unique solutionto the rough differential equation dy = f ( y ) dx , y = ξ ,and the solution is continuous with respect to g in p -variation norm. Moreover, there exist integrableone-forms β n : [0 , T ] → B (cid:0) G [ p ] , U (cid:1) , n ≥ , and a constant C = C ( p, γ, k f k Lip( γ ) , ω (0 , T )) > such that y nt = ξ + Z t β nu ( g u ) dg u , ∀ t ∈ [0 , T ] ,and (cid:13)(cid:13) β n +1 − β n (cid:13)(cid:13) γ ≤ C n − [ p ] (cid:16) n − [ p ] p (cid:17) ! , ∀ n ≥ [ p ] + 1 . (17)There are some remarks. 14. In proving the convergence of the Picard iterations, we proved the convergence in operator norm oftheir associated one-forms. In particular, we proved that the one-form associated with the differencebetween the n th and ( n + 1)th Picard iterations decays factorially on [0 , T ] as n tends to infinity.2. Let ρ : [0 , T ] → W be a path dominated by g . Then the integral of ρ against y n is well defined: R t ρ u ⊗ dy nu = R t ρ u ⊗ f (cid:0) y n − u (cid:1) dx u , ∀ t ∈ [0 , T ] . In particular, since y n is a dominated path, there exists a canonical enhancement of y n to a group-valued path, which could take values in nilpotent Lie group or Butcher group.3. When treated as a Banach space-valued path, the group-valued enhancement is again a dominatedpath. Since the one-form associated with the enhancement is continuous with respect to the one-form associated with the base dominated path, the one-forms of the enhancement of y n also convergein operator norm, which implies the uniform convergence of the group-valued enhancements.4. When f is Lip ( γ ) for γ > p −
1, the one-forms associated with the Picard iterations are uniformlybounded. When the dimension is finite, based on Arzel`a-Ascoli theorem, there exists a subsequenceof the one-forms which converges, so the associated paths (and their enhancements) converge to asolution.5. When f is locally Lipschitz and the dimension is finite, the solution exists (uniquely) up to explosion.Indeed, by Whitney’s extension theorem, the restriction of f to any compact set can be extended toa global Lipschitz function without increasing its Lipschitz norm, so the solution exists up to exittime of that compact set. For similar reason, when f is locally Lip ( γ ) for γ > p , any two solutionsmust agree on any compact set, so the solution exists uniquely up to explosion. Proof.
Suppose { y n } n are the Picard iterations in Definition 15. Since z i +1 = y i +1 − y i and β ,i is theintegrable one-form associated with z i +1 , if we define β n : [0 , T ] → B (cid:0) G [ p ] , U (cid:1) , n ≥
1, by β ns ( a, b ) = P n − i =0 β ,is ( a, b ) , ∀ a, b ∈ G [ p ] , ∀ s ∈ [0 , T ] ,then β n is integrable and y nt = ξ + R t β nu ( g u ) dg u , ∀ t ∈ [0 , T ] , ∀ n ≥ y ≡ ξ , we set β ≡ y · = ξ + R · β ( g ) dg .) Based on Lemma 21, we have (17) holds and β n converge in operator norm as n tends to infinity (denote the limit by β ), so y n · = ξ + R · β n ( g ) dg convergeuniformly to y · := ξ + R · β ( g ) dg . Moreover, by using the division property of f (i.e. f ( x ) − f ( y ) = h ( x, y ) ( x − y ) for all x, y in U and k h k Lip( γ − ≤ C k f k Lip( γ ) ), we have y n +1 t − y nt = z n +1 t = R t h (cid:0) y nu , y n − u (cid:1) z nu dx u = R t h (cid:0) y nu , y n − u (cid:1) (cid:0) y nu − y n − u (cid:1) dx u = R t (cid:0) f ( y nu ) − f (cid:0) y n − u (cid:1)(cid:1) dx u , ∀ t ∈ [0 , T ] .Hence, y n +1 t = ξ + R t f ( y nu ) dx u , ∀ t ∈ [0 , T ] , ∀ n ≥
0, with y ≡ ξ .Since both y n and y n +1 are dominated paths and their associated one-forms converge to β as n tendsto infinity, by letting n → ∞ on both sides, we have β is the fixed point of the mapping β ˆ β whereˆ β is the one-form associated with the dominated path t R t f ( y ) dx . Hence, y is a dominated pathsatisfying the integral equation and y is a solution.Then we prove that the solution is unique. Suppose ˆ y is another solution. By using the divisionproperty of f , we have y t − ˆ y t = R t ( f ( y u ) − f (ˆ y u )) dx u = R t h ( y u , ˆ y u ) ( y u − ˆ y u ) dx u , ∀ t ∈ [0 , T ] .15y iterating this process, we have, for any integer n ≥ y t − ˆ y t = R ··· R
1) function, we can define based on Lemma 11 thedominated paths ρ n : [0 , T ] → L ( U , U ), n ≥
1, recursively by ρ n +1 t = R t h ( y u , ˆ y u ) ρ nu dx u with ρ t = R t h ( y u , ˆ y u ) dx u , ∀ t ∈ [0 , T ] ,and we have y t − ˆ y t = R t ρ nu ( y u − ˆ y u ) dx u , ∀ t ∈ [0 , T ] , ∀ n ≥ ρ n decays fac-torially. Since y − ˆ y is another dominated path, the one-form associated with the dominated path R · ρ nu ( y u − ˆ y u ) dx u also decays factorially, which implies that y = ˆ y .It is clear that for any integer n ≥
1, the mapping g β n is continuous. Suppose g m → g in p -variation norm, then by uniform convergence of the mapping β n β with respect to the p -variation of g (based on Lemma 21), we have g β is continuous, which implies that the mapping g y is continuouswith respect to g in p -variation norm. The Oxford-Man Institute, University of Oxford References [1] J. C. Butcher. An algebraic theory of integration methods.
Mathematics of Computation , 26(117):79–106, 1972.[2] A. Connes and D. Kreimer. Hopf algebras, renormalization and noncommutative geometry. In
Quantum field theory: perspective and prospective , pages 59–109. Springer, 1999.[3] M. Gubinelli. Controlling rough paths.
Journal of Functional Analysis , 216(1):86–140, 2004.[4] M. Gubinelli. Ramification of rough paths.
Journal of Differential Equations , 248(4):693–721, 2010.[5] K. Hara and M. Hino. Fractional order taylor’s series and the neo-classical inequality.
Bulletin ofthe London Mathematical Society , 42(3):467–477, 2010.[6] T. Lyons. Differential equations driven by rough signals. i. an extension of an inequality of lc young.
Math. Res. Lett , 1(4):451–464, 1994.[7] T. J. Lyons. Differential equations driven by rough signals.
Rev. Mat. Iberoamericana , 14(2), 1998.[8] T. J. Lyons, M. Caruana, and T. L´evy.
Differential equations driven by rough paths . Springer, 2007.[9] T. J. Lyons and D. Yang. Integration of time-varying cocyclic one-forms against rough paths. arXivpreprint arXiv:1408.2785 , 2014.[10] C. Reutenauer. Free lie algebras.
Handbook of algebra , 3:887–903, 2003.[11] L. Schwartz. La convergence de la s´erie de picard pour les eds (equations diff´erentielles stochastiques).In
S´eminaire de Probabilit´es XXIII , pages 343–354. Springer, 1989.[12] E. M. Stein.
Singular integrals and differentiability properties of functions , volume 2. Princetonuniversity press, 1970.[13] L. C. Young. An inequality of the h¨older type, connected with stieltjes integration.