The non-linear sewing lemma I : weak formulation
TThe non-linear sewing lemma I:weak formulation
Antoine Brault ∗† Antoine Lejay ‡ Feburary 24, 2019
We introduce a new framework to deal with rough differential equa-tions based on flows and their approximations. Our main result is toprove that measurable flows exist under weak conditions, even if solu-tions to the corresponding rough differential equations are not unique.We show that under additional conditions of the approximation, thereexists a unique Lipschitz flow. Then, a perturbation formula is given.Finally, we link our approach to the additive, multiplicative sewinglemmas and the rough Euler scheme.
Keywords: rough paths; rough differential equations; non uniqueness of solutions;flow approximations; measurable flows; Lipschitz flows; sewing lemma.
The theory of rough paths allows one to define the solution to a differential equationof type y t = a + (cid:90) t f ( y s ) d x s , (1) ∗ Université Paris Descartes, MAP5 (CNRS UMR 8145), 45 rue desSaints-Pères, 75270 Pariscedex 06, France. [email protected] † Institut de Mathématiques de Toulouse, UMR 5219; Université de Toulouse, UPS IMT, F-31062Toulouse Cedex 9, France. ‡ Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France, [email protected] a r X i v : . [ m a t h . P R ] M a y or a path x which is irregular, say α -Hölder continuous. Such an equation is thencalled a Rough Differential Equation (RDE) [20, 21, 25]. The key point of thistheory is to show that such a solution can be defined provided that x is extendedto a path x , called a rough path , living in a larger space that depends on the integerpart of /α . When α > / , no such extension is needed. This case is referredas the Young case , as the integrals are constructed in the sense given by L.C.Young [24, 28]. Provided that one considers a rough path, integrals and differentialequations are natural extensions of ordinary ones.The first proof of existence of a solution to (1) from T. Lyons relied on a fixed point[25, 26, 27]. It was quickly shown that RDE shares the same properties as ordinarydifferential equations, including the flow property. In [16], A.M. Davie gives analternative proof based on an Euler type approximation as well as counter-exampleto uniqueness. More recently, I. Bailleul gave a direct construction through theflow property [3, 4, 6].A flow in a metric space V is a family { ψ t,s } ≤ s ≤ t ≤ T of maps from V to V such that ψ t,s ◦ ψ s,r = ψ t,r for any ≤ r ≤ s ≤ t ≤ T . When y t ( s, a ) is a family of solutionsto differential equations with y s ( s, a ) = a , the element ψ t,s ( a ) can be seen as a mapwhich carries a to y t ( s, a ) . Flows are related to dynamical systems. They differfrom solutions. One of their interest lies in their characterization as lipeomorphims(Lipschitz functions with a Lipschitz inverse), diffeomorphisms...In this article, we develop a generic framework to construct flows from approxima-tions. We do not focus on a particular form of the solutions, so that our constructionis a non-linear sewing lemma , modelled after the additive and multiplicative sewinglemmas developed in [19, 20, 22, 26].In this first part, we study flows under weak conditions and prove existence of ameasurable flow even when the solutions of RDE are not necessarily unique. Thisis based on a selection theorem [11] due to J.E. Cardona and L. Kapitanski. Such aresult is new in the literature where existence of flows was only proved under strongerregularity conditions (the many approaches are summarized in [15]). Besides, ourapproach also contains the additive and multiplicative sewing lemmas [13, 19].The rough equivalent of the Duhamel formula for solving linear RDE [14] with aperturbative terms follows directly from our construction.In a second part [9], we provide conditions for uniqueness and continuity. Besides,we show that our construction encompass many of the previous approaches orresults: A.M. Davie [16], I. Bailleul [2, 3, 5] and P. Friz & N. Victoir [21].Our starting point in the world of classical analysis is the product formula whichrelates how the iterated product of an approximation of a flow converge to the flow[1, 12]. It is important both from the theoretical and numerical point of view.2n a Banach space V , let us consider a family ( (cid:15), a ) ∈ R + × V (cid:55)→ Φ( (cid:15), a ) , called an algorithm of class (cid:67) in ( (cid:15), a ) such that Φ(0 , a ) = a . The parameter (cid:15) is related tothe quality of the approximation.The algorithm Φ is consistent with a vector field f when f ( a ) = ∂ Φ ∂(cid:15) (0 , a ) , ∀ a ∈ V . (2)For a consider algorithm, when φ t ( a ) is the solution to φ t ( a ) = a + (cid:82) t f ( φ s ( a )) d s , Φ( t/n, Φ( t/n, · · · Φ( t/n, a ))) (cid:124) (cid:123)(cid:122) (cid:125) n times converges to φ t ( a ) as n → ∞ . (3)Eq. (3) is called the product formula.The Euler scheme for solving ODE is the prototypical example of such behavior.Set Φ( (cid:15), a ) := a + f ( a ) (cid:15) so that (2) holds. In this case, (3) expresses the convergenceof the Euler scheme.Using the product formula, one recovers easily the proof of Lie’s theorem onmatrices: If A is a matrix, then exp( tA ) is the solution to ˙ Y t = AY t with Y = Id and then exp( tA ) = lim n →∞ (cid:18) Id + tn A (cid:19) n . For two matrices A and B , exp( t ( A + B )) is given by exp( t ( A + B )) = lim n →∞ (cid:18) exp (cid:18) tAn (cid:19) exp (cid:18) tBn (cid:19)(cid:19) n . To prove the later statement, we consider Φ( (cid:15), a ) = exp( (cid:15)A ) exp( (cid:15)B ) a and we verifythat ∂ (cid:15) Φ(0 , a ) = ( A + B ) a for any matrix a .For unbounded operators, it is also related to Chernoff and Trotter’s results onthe approximation of semi-groups [18, 29]. The product formula also justifies theconstruction of some splitting schemes [8].In this article, we consider as an algorithm a family { φ t,s } ≤ s ≤ t ≤ T of functionsfrom V to V which is close to the identity map in short time and such that φ t,s ◦ φ s,r is close to φ t,r for any time s ≤ r ≤ t . For a path x of finite p -variation, ≤ p < with values in R d and a smooth enough function f : R m → L( R d , R m ) , such anexample is given by φ t,s ( a ) = a + f ( a ) x s,t .We then study the behavior of the composition φ π of the φ t i +1 ,t i along a partition π = { t i } i =0 ,...,n . Clearly, as the mesh of the partition π goes to , the limit, when it3xists is a candidate to be a flow. In the example given above, it will be the flowassociated to the family of Young differential equations y t ( a ) = a + (cid:82) t f ( y s ( a )) d x s ,which means according to A.M. Davie [16] that | y t ( a ) − φ t,s ( y s ( a )) | ≤ L ( a ) (cid:36) ( ω s,t ) . (4)We show that measurable flows may exist for Young or Rough Differential Equationseven when several paths satisfying (4) exist.In [9, 10], we exhibit a condition on almost flow that ensure existence of Lipschitzflows. Such an almost flow is called a stable almost flow . Besides, we study furtherthe connection between almost flows and solutions in the sense of (4). In particular,when an almost flow is stable, solutions exist and are unique. Stronger convergencerate of numerical approximations, as well as continuity results are then given.In order to present our main results, we introduce some necessary notations as wellas some central notions such as galaxies. The following notations and hypotheses will be constantly used throughout all thisarticle.
Let V and W be two metric spaces.The space of continuous functions from V to W is denoted by (cid:67) (V , W) .Let d (resp. d (cid:48) ) be a distance on V (resp. W ). For γ ∈ (0 , , we say that a function f : V → W is γ - Hölder if (cid:107) f (cid:107) γ := sup a,b ∈ V ,a (cid:54) = b d (cid:48) ( f ( a ) , f ( b )) d ( a, b ) γ < + ∞ . If γ = 1 we say that f is Lipschitz . We then set (cid:107) f (cid:107) Lip := (cid:107) f (cid:107) .For any integer r ≥ and γ ∈ (0 , , we denote by (cid:67) r + γ (V , W) the subspace offunctions from V to W whose derivatives d k f of order k ≤ r are continuous andsuch that d r f is γ -Hölder.We denote by (cid:67) r + γ b (V , W) the subset of (cid:67) r + γ (V , W) of bounded functions withbounded derivatives up to order r . 4 .2.2 Controls and remainders From now, V is a topological, complete metric space with a distance d . A distin-guished point of V is denoted by .We fix γ ∈ (0 , . Let N γ : V → [1 , + ∞ ) be a γ -Hölder continuous function withconstant (cid:107) N (cid:107) γ . The index γ in N γ refers to its regularity. If N γ is Lipschitzcontinuous ( γ = 1 ), then we simply write N .Let us fix a time horizon T . We write T := [0 , T ] as well as T +2 := { ( s, t ) ∈ T | s ≤ t } and T +3 := { ( r, s, t ) ∈ T | r ≤ s ≤ t } . (5)A control ω : T +2 → R + is a super-additive family, i.e. , ω r,s + ω s,t ≤ ω r,t , ∀ ( r, s, t ) ∈ T +3 with ω s,s = 0 for all s ∈ T , and for any δ > , there exists (cid:15) > such that ω s,t < δ whenever ≤ s ≤ t ≤ s + (cid:15) . A typical example of a control is ω s,t = C | t − s | for aconstant C ≥ .For p ≥ , we say that a path x ∈ (cid:67) ( T , V) is a path of finite p -variation controlledby ω if (cid:107) x (cid:107) p := sup ( s,t ) ∈ T +2 ,s (cid:54) = t d ( x s , x t ) ω /ps,t < + ∞ . A remainder is a function (cid:36) : R + → R + which is continuous, increasing and suchthat for some < κ < , (cid:36) (cid:18) δ (cid:19) ≤ κ (cid:36) ( δ ) , ∀ δ > . (6)A typical example for a remainder is (cid:36) ( δ ) = δ θ for any θ > .We consider that δ : R + → R + is a non-decreasing function with lim T → δ T = 0 .Finally, let η : R + → R + be a continuous, increasing function such that for all ( s, t ) ∈ T +2 , η ( ω s,t ) (cid:36) ( ω s,t ) γ ≤ δ T (cid:36) ( ω s,t ) . (7)Partitions of T are customary denoted by π = { t i } i =0 ,...,n . The mesh | π | of apartition π is | π | := max i =0 ,...,n ( t i +1 − t i ) . The discrete simplices π +2 and π +3 aredefined similarly to T +2 and T +3 in (5). 5 .2.3 Galaxies Notation 1.
We denote by (cid:70) (V) the set of functions { φ t,s } ( s,t ) ∈ T +2 from V to V which are continuous in ( s, t ) , i.e. for any a ∈ V , the map ( s, t ) ∈ T +2 (cid:55)→ φ t,s ( a ) iscontinuous. Notation 2 (Iterated products) . For any φ ∈ (cid:70) (V) , any partition π of T and any ( s, t ) ∈ T +2 , we write φ πt,s := φ t,t j ◦ φ t j ,t j − ◦ · · · ◦ φ t i +1 ,t i ◦ φ t i ,s , (8)where [ t i , t j ] is the biggest interval of such kind contained in [ s, t ] ⊂ T (possibly, t i = t j ). If no such interval exists, then φ πt,s = φ t,s .Clearly, for any partition, φ π ∈ (cid:70) (V) . A trivial but important remark is that fromthe very construction, φ πt,r = φ πt,s ◦ φ πs,r for any s ∈ π, r ≤ s ≤ t. (9)In particular, { φ πt,s } ( s,t ) ∈ π enjoys a (semi-)flow property when the times are re-stricted to the elements of π .The article is mainly devoted to study the possible limits of φ π as the mesh of π decreases to . Notation 3.
From a distance d on V , we define the distance ∆ N γ on the space offunctions from V to V by ∆ N γ ( f, g ) := sup a ∈ V d ( f ( a ) , g ( a )) N γ ( a ) , where N γ is defined in Section 1.2.This distance is extended on (cid:70) (V) by ∆ N γ ,(cid:36) ( φ, ψ ) := sup ( s,t ) ∈ T +2 s (cid:54) = t ∆ N γ ( φ t,s , ψ t,s ) (cid:36) ( ω s,t ) , where ω , (cid:36) are defined in Section 1.2. The distance ∆ N γ ,(cid:36) may take infinite values.If d is a distance for which (V , d ) is complete, then ( (cid:67) (V , V) , ∆ N ) and ( (cid:70) (V) , ∆ N,(cid:36) ) are complete. Definition 1 (Galaxy) . We define the equivalence relation ∼ on (cid:70) (V) by φ ∼ ψ if and only if there exists a constant C such that d ( φ t,s ( a ) , ψ t,s ( a )) ≤ CN γ ( a ) (cid:36) ( ω s,t ) , ∀ a ∈ V , ∀ ( s, t ) ∈ T +2 . In other words, φ ∼ ψ if and only if ∆ N γ ,(cid:36) ( φ, ψ ) < + ∞ . Each quotient class of (cid:70) (V) / ∼ is called a galaxy , which contains elements of (cid:70) (V) which are at finitedistance from each others. 6 .3 Summary of the main results The galaxies partition the space (cid:70) (V) . Each galaxy may contain two classes ofelements on which we focus on this article:1. The flows , that is the families of maps ψ : V → V which satisfy ψ t,r = ψ t,s ◦ ψ s,r , ∀ ( r, s, t ) ∈ T +3 , (10)or equivalently, ψ π = ψ (See (8)) for any partition π .2. The almost flows which we see as time-inhomogeneous algorithms. Besidessome conditions on the continuity and the growth given in Definition 2 below,an almost flow φ is close to a flow with the difference that d ( φ t,s ◦ φ s,r ( a ) , φ t,r ( a )) ≤ N γ ( a ) (cid:36) ( ω r,t ) , ∀ ( r, s, t ) ∈ T +3 , a ∈ V , for a suitable function N γ : V → [1 , + ∞ ) .Along with an almost flow φ comes the notion of manifold of D-solutions , that is afamily y := { y t ( a ) } t ≥ , a ∈ V of paths such that d ( y t ( a ) , φ t,s ( y s ( a ))) ≤ C(cid:36) ( ω s,t ) , ∀ ( s, t ) ∈ T +2 . (11)Each path y ( a ) that satisfies (11) is called a D-solution . This definition expandsnaturally the one introduced by A.M. Davie in [16].Clearly, a manifold of D-solutions associated to an almost flow φ is also associatedto any almost flow in the same galaxy as φ . Besides, if a flow ψ exists in thesame galaxy as φ , then z t ( a ) = ψ t, ( a ) defines a manifold of D-solutions z . Flowsare more constrained objects than solutions as (10) implies some compatibilityconditions, while it is possible to create new D-solutions by splicing two differentones. As it will be shown in [9], uniqueness of manifold of D-solutions is stronglyrelated to existence of a flow.Given an almost flow φ , it is natural to study the limit of the net { φ π } π as themesh of π decreases to . Any limit will be a good candidate to be a flow.Our first main result (Theorem 1) asserts that for any almost flow φ in a galaxy G ,any iterated map φ π belongs to G whatever the partition π although the map φ π isnot necessarily an almost flow. More precisely, one controls ∆ N γ ,(cid:36) ( φ π , φ ) uniformlyin the partition π .An immediate corollary is that any possible limit of { φ π } π as the mesh of thepartition decreases to also belongs to G .Our second main result (Theorem 2) is that when the underlying Banach space V isfinite-dimensional, there exists at least one measurable flow in a galaxy containing7n almost flow, even when several manifolds of D-solutions may exist. Our proofuses a recent result of J.E. Cardona and L. Kapitanski [11] on selection theorems.Our third result is to give conditions ensuring the existence of at most one flowin a galaxy. A sufficient condition is given for the galaxy G contains a Lipschitzflow ψ . In this case, { φ π } π converges to ψ whatever the almost flow φ in G . Therate of convergence is also quantified.Finally, we apply our results to the additive, multiplicative sewing lemmas [19]as well as to the algorithms proposed by A.M. Davie in [16] to show existence ofmeasurable flows even without uniqueness. In the sequel [9], we study in detailsthe properties of Lipschitz flows and give some conditions on almost Lipschitz flowsto generate them. In addition, we also apply our results to other approximations ofRDE, namely the one proposed by P. Friz & N. Victoir [21] and the one proposedby I. Bailleul [3] by using perturbation arguments. We show in Section 2 that a uniform control of the iterated product of approximationof flows with respect to the subdivision. In Section 3, we prove our main result:the existence of a measurable flow under weak conditions of regularity. Then, inSection 4 we show the existence and uniqueness of a Lipschitz flow under strongerassumptions. Moreover, we give a rate of the convergence of the iterated product tothe flow. In Section 5, we show that additive perturbations preserve the convergenceof iterated products of approximations of flows. Finally, we recover in Section 6the additive [26], multiplicative [14, 19] and Davie’s sewing lemmas [16].
In this section, we define almost flows which serve as approximations. The propertiesof an almost flow φ are weaker than the minimal condition to obtain the convergenceof the iterated product φ π as the mesh of the partitions π decreases to . However,we prove in Theorem 1 that we can control φ π uniformly over the partitions π .This justifies our definition. 8 .1 Definition of almost flows Definition 2 (Almost flow) . An element φ ∈ (cid:70) (V) is an almost flow if for any ( r, s, t ) ∈ T +3 , a ∈ V , φ t,t ( a ) = a, (12) d ( φ t,s ( a ) , a ) ≤ δ T N γ ( a ) , (13) d ( φ t,s ( a ) , φ t,s ( b )) ≤ (1 + δ T ) d ( a, b ) + η ( ω s,t ) d ( a, b ) γ , (14) d ( φ t,s ◦ φ s,r ( a ) , φ t,r ( a )) ≤ N γ ( a ) (cid:36) ( ω r,t ) . (15) Remark . A family φ ∈ (cid:70) (V) satisfying condition (14) with γ = 1 is said to be quasi-contractive . This notion plays an important role in the fixed point theory [7]. Definition 3 (Iterated almost flow) . For a partition π and an almost flow φ , wecall φ π an iterated almost flow , where φ π is the iterated product defined in (8). Definition 4 (A flow) . A flow ψ is a family of functions { ψ t,s } ( s,t ) ∈ T +2 from V to V (not necessarily continuous in ( s, t ) ) such that ψ t,t ( a ) = a and ψ t,s ◦ ψ s,r ( a ) = ψ t,r ( a ) for any a ∈ V and ( r, s, t ) ∈ T +3 .In this paper, we consider almost flows which are continuous although flows may apriori be discontinuous (See Theorem 2). Before proving our main result in Section 3, we prove an important uniform controlover φ π . Theorem 1.
Let φ be an almost flow. Then there exists a time horizon T smallenough and constants L T ≥ as well as K T ≥ that decrease to as T decreasesto such that d ( φ πt,s ( a ) , φ t,s ( a )) ≤ L T N γ ( a ) (cid:36) ( ω s,t ) , (16) N γ ( φ πt,s ( a )) ≤ K T N γ ( a ) , (17) for any ( s, t ) ∈ T , a ∈ V and any partition π of T . The choice of T , L T and K T depend only on δ , (cid:36) , ω , γ and (cid:107) N γ (cid:107) γ .Remark . The distance d may be replaced by a pseudo-distance in the statementof Theorem 1. 9he proof of Theorem 1 is inspired by the one of the Claim in the proof of Lemma 2.4in [16]. With respect to the one of A.M. Davie, we consider obtaining a uniformcontrol over a family of elements indexed by ( s, t ) ∈ T +2 which are also parametrizedby points in V . Definition 5 (Successive points / distance between two points) . Let π be apartition of [0 , T ] . Two points s and t of π are said to be at distance k if there areexactly k − points between them in π . We write d π ( t, s ) = k . Points at distance are called successive points in π . Proof.
For a ∈ V , ( r, t ) ∈ T +2 and a partition π , we set U r,t ( a ) := d ( φ πt,r ( a ) , φ t,r ( a )) . We now restrict ourselves to the case ( r, t ) ∈ π +2 . To control U r,t ( a ) in a way thatdoes not depend on π , we use an induction in the distance between r and t .Our induction hypothesis is that there exist constants L T ≥ and K T ≥ independent from the partition π such that for any ( r, t ) ∈ π +2 at distance atmost m ≥ , U r,t ( a ) ≤ L T N γ ( a ) (cid:36) ( ω r,t ) , (18) N γ ( φ πt,s ( a )) ≤ K T N γ ( a ) , (19)with K T decreases to at T goes to .The induction hypothesis is true for m = 0 , since U r,r ( a ) = 0 and N γ ( φ πr,r ( a )) = N γ ( a ) .If r and t are successive points, φ πt,r = φ t,r so that U r,t ( a ) = 0 . Thus, (18) is truefor m = 1 . With (13) and since N γ ( a ) γ ≤ N γ ( a ) as N γ ( a ) ≥ by hypothesis, N γ ( φ t,s ( a )) ≤ N γ ( φ t,s ( a )) − N γ ( a ) + N γ ( a ) ≤ (cid:107) N γ (cid:107) γ d ( φ t,s ( a ) , a ) γ + N γ ( a ) ≤ (cid:107) N γ (cid:107) γ δ γT N γ ( a ) γ + N γ ( a ) ≤ ( (cid:107) N γ (cid:107) γ δ γT + 1) N γ ( a ) . For m = 2 , it is also true with L T = 1 as U r,t ( a ) = d ( φ t,s ◦ φ s,r ( a ) , φ t,r ( a )) where s is the point in the middle of r and t . This proves (18).In addition, using (18), N γ ( φ πt,s ( a )) ≤ (cid:107) N (cid:107) γ d ( φ πt,s ( a ) , φ t,s ( a )) γ + N γ ( φ t,s ( a )) − N γ ( a ) + N γ ( a ) ≤ K T N γ ( a ) with K T := (cid:107) N γ (cid:107) γ (cid:36) ( ω ,T ) γ + (cid:107) N γ (cid:107) γ δ γT + 1 . (20)10learly, K T decreases to at T decreases to . This proves (19) whenever (18), inparticular for m = 1 .Assume that (18)-(19) when the distance between r and t is smaller than m forsome m ≥ .For s ∈ π , r ≤ s ≤ t , with (9), U r,t ( a ) ≤ d ( φ πt,s ◦ φ πs,r ( a ) , φ t,s ◦ φ πs,r ( a ))+ d ( φ t,s ◦ φ πs,r ( a ) , φ t,s ◦ φ s,r ( a )) + d ( φ t,s ◦ φ s,r ( a ) , φ t,r ( a )) . With (14) and (15), U r,t ( a ) ≤ U s,t ( φ πs,r ( a )) + (1 + δ T ) U r,s ( a ) + η ( ω s,t ) U r,s ( a ) γ + N γ ( a ) (cid:36) ( ω r,t ) . (21)Using (21) on U s,t ( φ πs,r ( a )) by replacing ( r, s, t ) by ( s, s (cid:48) , t ) , where s and s (cid:48) are twosuccessive points of π leads to U s,t ( φ πs,r ( a )) ≤ U s (cid:48) ,t ( φ πs (cid:48) ,r ( a )) + N γ ( φ πs,r ( a )) (cid:36) ( ω s,t ) since φ s (cid:48) ,s ◦ φ πs,r ( a ) = φ πs (cid:48) ,r ( a ) and U s,s (cid:48) ( a ) = 0 .With (18)-(19) and our hypothesis on η , again since N γ ≥ and s, s (cid:48) are at distanceless than m provided that s is at distance at most from t , U r,t ( a ) ≤ N γ ( a ) ( K T L T (cid:36) ( ω s (cid:48) ,t ) + ( L T + δ T (1 + L γT )) (cid:36) ( ω r,s ) + (1 + K T ) (cid:36) ( ω r,t )) . (22)This inequality also holds true for r = s or s (cid:48) = t .We proceed as in [16] to split π in “essentially” two parts. We set s (cid:48) := min (cid:110) τ ∈ π (cid:12)(cid:12)(cid:12) ω r,τ ≥ ω r,t (cid:111) and s := max { τ ∈ π | τ < s (cid:48) } . Hence, s and s (cid:48) are successive points with r ≤ s < s (cid:48) ≤ t and ω r,s ≤ ω r,t / . Besides,since ω is super-additive, ω r,s (cid:48) + ω s (cid:48) ,t ≤ ω r,t . Therefore, ω r,s ≤ ω r,t and ω s (cid:48) ,t ≤ ω r,t . (23)With such a choice, since L T ≥ , (22) becomes with (6): U r,t ( a ) ≤ (cid:18) L T K T + 1 + 2 δ T κ + 1 + K T (cid:19) N γ ( a ) (cid:36) ( ω r,t ) . If T is small enough so that α := 1 + K T + 2 δ T κ < and L T α + 1 + K T ≤ L T , L T so that L T α + 1 + K T ≤ L T , that is L T ≥ max { , (1 + K T ) / (1 − α ) } . This choice ensures that U r,t ( a ) ≤ L T N γ ( a ) (cid:36) ( ω r,t ) when r and t are at distance m . Condition (19) follows from (20) and (18).The choice of T , K T and L T does not depend on π . In particular, d ( φ πt,s ( a ) , φ t,s ( a )) ≤ L T N γ ( a ) (cid:36) ( ω s,t ) becomes true for any ( s, t ) ∈ T +2 (it is sufficient to add the points s and t to π ). Corollary 1.
Let φ be an almost flow and π be a partition of T . Then φ π ∼ φ (wehave not proved that φ π is itself an almost flow). Notation 4.
For an almost flow φ , let us denote by S φ ( s, a ) the set of all thepossible limits of the net { φ π · ,s ( a ) } π in ( (cid:67) ([ s, T ] , V) , (cid:107)·(cid:107) ∞ ) for nested partitions.When V is a finite dimensional space with the norm |·| , S φ ( s, a ) (cid:54) = ∅ . We start witha lemma which will be useful to prove some equi-continuity. We denote ¯ B (0 , R ) the closed ball centered in of radius R > . Lemma 1.
Let
R > . We assume that V is a finite dimensional vector spaceand φ is an almost flow. Then, for all (cid:15) > there exists δ > such that for all ( s, t ) , ( s (cid:48) , t (cid:48) ) ∈ T +2 with | t − t (cid:48) | , | s − s (cid:48) | < δ and a ∈ ¯ B (0 , R ) , | φ t,s ( a ) − φ t (cid:48) ,s (cid:48) ( a ) | ≤ (cid:15). Proof.
In all the proof ( s, t ) , ( s (cid:48) , t (cid:48) ) ∈ T +2 . For any a ∈ ¯ B (0 , R ) , ( s, t ) ∈ T +2 (cid:55)→ φ t,s ( a ) is continuous, so uniformly continuous on the compact T +2 . Let (cid:15) > , there is δ a such that for all | t − t (cid:48) | , | s − s (cid:48) | < δ a , | φ t,s ( a ) − φ t (cid:48) ,s (cid:48) ( a ) | ≤ (cid:15) . (24)For a ρ > with (14), for all b ∈ B ( a, ρ ) , | φ t,s ( a ) − φ t,s ( b ) | ≤ (1 + δ T ) ρ + η ( ω ,T ) ρ γ , where B ( a, ρ ) denotes the open ball centred in a of radius ρ . We choose ρ ( (cid:15) ) > such that (1 + δ T ) ρ ( (cid:15) ) + η ( ω ,T ) ρ ( (cid:15) ) γ ≤ (cid:15)/ to obtain for all b ∈ B ( a, ρ ( (cid:15) )) , | φ t,s ( a ) − φ t,s ( b ) | ≤ (cid:15) . (25)We note that (cid:83) a ∈ ¯ B (0 ,R ) B ( a, ρ ( (cid:15) )) is a covering of ¯ B (0 , R ) which is a compact of V .There exist an integer N and a finite family of balls B ( a i , ρ ( (cid:15) )) for i ∈ { , . . . , N } such that ¯ B (0 , R ) ⊂ (cid:83) i ∈{ ,...,N } B ( a i , ρ ( (cid:15) )) .12t follows that for all b ∈ ¯ B (0 , R ) there exists i ∈ { , . . . , N } such that b ∈ B ( a i , ρ ( (cid:15) )) . From (24)-(25) there exists δ a i > such that for all | t − t (cid:48) | , | s − s (cid:48) | < δ a i , | φ t,s ( b ) − φ t (cid:48) ,s (cid:48) ( b ) | ≤ (cid:15). Taking δ := min i ∈{ ,...,N } δ a i , we obtain that for all | t − t (cid:48) | , | s − s (cid:48) | < δ and b ∈ ¯ B (0 , R ) , | φ t,s ( b ) − φ t (cid:48) ,s (cid:48) ( b ) | ≤ (cid:15), which achieves the proof. Proposition 1.
Assume that V is finite-dimensional and that φ is an almost flow.Then S φ ( r, a ) is not empty for any ( r, a ) ∈ T × V .Proof. Let us show for an almost flow φ , { φ π · ,r ( a ) } π is equi-continuous and bounded.The result is then a direct consequence of the Ascoli-Arzelà theorem.Let ( π n ) n ∈ N be an increasing sequence of partitions of T such that | π n | → when n → + ∞ and (cid:83) n ∈ N π n dense in T . Let R > and R (cid:48) := N γ ( R )( L T (cid:36) ( ω ,T )+ δ T )+ R .From Theorem 1, for any a ∈ ¯ B (0 , R ) and ( s, t ) ∈ T +2 , | φ π n t,s ( a ) | ≤ | φ π n t,s ( a ) − φ t,s ( a ) | + | φ t,s ( a ) − a | + | a |≤ N γ ( a )( L T (cid:36) ( ω ,T ) + δ T ) + | a | ≤ R (cid:48) . Let ( r, s, t ) ∈ T +3 and s, t ∈ (cid:83) n ∈ N π n , let (cid:15) > and δ > given by Lemma 1 for (cid:15)/ .For | t − s | < δ , let m an integer such that s, t ∈ π m .We differentiate two cases. If m ≤ n , then π m ⊂ π n , which implies that φ π n t,r = φ π n t,s ◦ φ π n s,r . From Theorem 1 and Lemma 1, for all a ∈ ¯ B (0 , R ) , for all | t − s | < δ | φ π n t,r ( a ) − φ π n s,r ( a ) | ≤ | φ π n t,s ◦ φ π n s,r ( a ) − φ π n s,r ( a ) | ≤ (cid:15) L T N γ ( φ π n s,r ( a )) (cid:36) ( ω s,t ) . Since, ω s,s = 0 and ω is continuous close to diagonal, there exists δ (cid:48) > such thatfor all | t − s | < δ (cid:48) , L T N γ ( φ π n s,r ( a )) (cid:36) ( ω s,t ) < (cid:15)/ . Thus for all | t − s | < min ( δ, δ (cid:48) ) , | φ π n t,r ( a ) − φ π n s,r ( a ) | ≤ (cid:15) .In the case m > n , let s − , s + be successive points in π n such that [ s, t ] ⊂ [ s − , s + ] .Then, φ π n t,r ( a ) = φ t,s − ◦ φ π n s − ,r ( a ) and φ π n s,r ( a ) = φ s,s − ◦ φ π n s − ,r ( a ) . According to Lemma 1,for | t − s | < δ with s, t ∈ π m , and all a ∈ ¯ B (0 , R ) , | φ π n t,r ( a ) − φ π n s,r ( a ) | = | φ t,s − ◦ φ π n s − ,r ( a ) − φ s,s − ◦ φ π n s − ,r ( a ) | ≤ (cid:15). (26)By continuity of t (cid:55)→ φ π n t,r ( a ) , and density of (cid:83) m ∈ N π m in T , we obtain (26) for all ( s, t ) ∈ T +2 with r ≤ s and | t − s | < δ . This proves that { t (cid:55)→ φ π n t,r ( a ) } n is uniformlyequi-continuous for all a ∈ V . We conclude the proof with the Ascoli-Arzelàtheorem. 13 The non-linear sewing lemma
We now show that in the finite dimensional case, we can build a flow from φ π using a selection principle [11]. In this section, we consider that almost flows φ aredefined for T := [0 , + ∞ ) . Definition 6 (Solution in the sense of Davie or D-solution) . For an almost flow φ ,a time r ∈ T and a point a ∈ V , a solution in the sense of Davie (or a D-solution)is a path y ∈ (cid:67) ( S , V) with S = [ r, r + T ] ⊂ T such that d ( y t , φ t,s ( y s )) ≤ KN γ ( a ) (cid:36) ( ω s,t ) , ∀ ( s, t ) ∈ S , (27)where K ≥ is a constant. Remark . Our definition of D-solutions extends the one of Davie in [16] to a metricspace V and a general almost flow φ . Remark . When φ is only an almost flow, it is not guaranteed that a D-solutionexists or is unique. When S φ ( r, a ) (cid:54) = ∅ (see Notation 4), we prove below in Lemma 3that a D-solution exists. However, even if for all ( r, a ) ∈ T × V , S φ ( r, a ) (cid:54) = ∅ thisdoes not imply the existence of families of solutions { ψ · ,r ( a ) } r ∈ T ,a ∈ V which satisfiesthe flow property. Notation 5.
We denote Ω( r, a ) the set of continuous paths such that y ∈ (cid:67) ( S , V) verifies y r = a . We denote by G Kφ ( r, a ) the set of paths in Ω( r, a ) verifying (27) forthe constant K . Definition 7 (Splicing of paths) . For r ≤ s , let us consider y ( r, a ) ∈ Ω( r, a ) and z ( s, b ) ∈ Ω( s, b ) with b = y s ( r, a ) . Their splicing is ( y (cid:46)(cid:47) s z ) t ( r, a ) = (cid:40) y t ( r, a ) if t ≤ s,z t ( s, y s ( r, a )) if t ≥ s. We restate here, the definition of a family of abstract integral local funnels (Defini-tion in [11]), which leads to the existence of a measurable flow. Definition 8.
A family F ( r, a ) with r ∈ [0 , + ∞ ) , a ∈ V , will be called a family ofabstract local integral funnels with terminal time T ( r, a ) ∈ (0 , + ∞ ) if H0 The map ( r, a ) ∈ [0 , + ∞ ) × V (cid:55)→ T ( r, a ) is lower semi-continuous in the sensethat if ( r n , a n ) → ( r, a ) , then T ( r, a ) ≤ lim inf n T ( r n , a n ) . H1 Every set F ( r, a ) is a non-empty compact in the space (cid:67) ([ r, r + T ( r, a )] , V) and every path y ( r, a ) ∈ F ( r, a ) is a continuous map from [ r, r + T ( r, a )] to V with y r ( r, a ) = a . 14 For all r ≥ , the map a ∈ V (cid:55)→ F ( r, a ) is measurable in the sense that forany closed subset (cid:75) ⊂ (cid:67) ([0 , , V) , { a ∈ V | ˜ F ( r, a ) (cid:84) (cid:75) (cid:54) = ∅} is Borel, where ˜ F ( r, a ) is the set of re-parametrizations of paths of F ( r, a ) on [0 , . H3 If y ( r, a ) ∈ F ( r, a ) and τ < T ( r, a ) , then T ( r + τ, y r + τ ( r, a )) ≥ T ( r, a ) − τ and t ∈ [ r + τ, r + τ + T ( r + τ, y r + τ ( r, a ))] (cid:55)→ y t ( r, a ) belongs to F ( r + τ, y r + τ ( r, a )) . H4 If y ( r, a ) ∈ F ( r, a ) and τ < T ( r, a ) , and z ∈ F ( r + τ, y t + τ ( r, a )) then the splicedpath (Definition 7) x := y (cid:46)(cid:47) r + τ z belongs to F ( r, a ) . Definition 9 (Lipschitz almost flow) . A Lipschitz almost flow is an almost flowfor which (14) is satisfied with η = 0 , and N γ = N = N is Lipschitz continuous.A flow is constructed by assigning to each point of the space a particular D-solution,in a sense which is compatible. Hypothesis 1.
Let V be a finite dimensional vector space. Let φ := { φ t,s } ≤ s ≤ t< + ∞ be a Lipschitz almost flow (Definition 9) with N bounded. We fix a time horizon T > such that κ (1 + δ T ) < . Remark . When N bounded, we can choose K T = 1 and L T = 2 / (1 − κ (1 + δ T )) where K T and L T are the constants of Theorem 1.The main theorem of this paper is the following one. Theorem 2 (Non-linear sewing lemma, weak formulation) . Under Hypothesis 1,there exists ψ ∈ (cid:70) (V) in the same galaxy as φ satisfying the flow property and suchthat ψ t,s is Borel measurable for any ( s, t ) ∈ T .Remark . Proving such a result with a general Banach space V is false as evenexistence of solutions to ordinary differential equations may fail [17, 23]. Remark . To prove Theorem 2, we show that ( G L T ( r, a )) r ∈ [0 , + ∞ ) ,a ∈ V is a family ofabstract local integral funnels in the sense of Definition 8. Then, we use Theorem of [11].Lemmas 2-7 prove that G L T ( r, a ) is a family of local abstract funnels in the senseof the Definition 2 in [11]. Then we apply Theorem 2 in [11] to obtain the abovetheorem. Lemma 2.
Under Hypothesis 1, the terminal time T ( r, a ) := T is independent ofthe starting time r and the starting point a . In particular, H0 holds for F = G L T φ .Proof. It is sufficient to notice that the constants κ and δ T do not depend on a ∈ V neither on r . 15e recall that S φ ( r, a ) is defined in Notation 4. Our first result is that when S φ ( r, a ) (cid:54) = ∅ , then there exists at least one D-solution in G L T φ ( r, a ) . Lemma 3.
Assume that K ≥ K T L T in Definition 6, where K T and L T areconstants in Theorem 1. For any ( r, a ) ∈ T × V , S φ ( r, a ) ⊂ G Kφ ( r, a ) for an almostflow φ (note that S φ ( r, a ) may be empty).Proof. If y ∈ S φ ( r, a ) when S φ ( r, a ) (cid:54) = ∅ , then there exists a sequence { π k } k ∈ N ofpartitions such that y t = lim φ π k t,r ( a ) uniformly in t ∈ [ r, T ] . We note that y r = a .For k ∈ N and s k ∈ π k , with (9) and Theorem 1, d ( φ π k t,r ( a ) , φ t,s k ◦ φ π k s k ,r ( a )) = d ( φ π k t,s k ◦ φ π k s k ,r ( a ) , φ t,s k ◦ φ π k s k ,r ( a )) ≤ L T N γ ( φ π k s k ,r ( a )) (cid:36) ( ω s k ,t ) ≤ L T K T N γ ( a ) (cid:36) ( ω s k ,t ) . (28)moreover, fixing s ∈ [ r, T ] and using (13), d ( φ t,s k ◦ φ π k s k ,r ( a ) , φ t,s ( y s ( a ))) ≤ d ( φ t,s k ◦ φ π k s k ,r ( a ) , φ t,s k ◦ y s ( a )) + d ( φ t,s k ◦ y s ( a ) , φ t,s ◦ y s ( a )) ≤ (1 + δ T ) d ( φ π k s k ,r ( a ) , y s ( a )) + η ( ω ,T ) d ( φ π k s k ,r ( a ) , y s ( a )) γ + d ( φ t,s k ◦ y s ( a ) , φ t,s ◦ y s ( a )) . (29)Choosing { s k } k ∈ N so that s k decreases to s and passing to the limit, we obtainwith (29) that φ t,s k ◦ φ π k s k ,r ( a ) converges uniformly to φ t,s ◦ y s ( a ) . Thus, when k → + ∞ , (28) shows that y is a D-solution. Lemma 4.
Under Hypothesis , G L T φ ( r, a ) is a non-empty compact subset of theset of paths y ∈ (cid:67) ( S , V) such that y r = a for any r ∈ T and a ∈ V . It shows that H1 holds for F := G L T φ .Proof. It follows directly from Proposition 1 and Lemma 3 (with K T = 1 and K = L T ) that G Kφ ( r, a ) is not empty. Now, if { y k } k is a sequence in G Kφ ( r, a ) then { y k } k is equi-continuous with the same argument as in the proof of Proposition 1.The subsequence of { y k } k converges in G Kφ ( r, a ) because a ∈ V (cid:55)→ φ t,s ( a ) iscontinuous for any ( s, t ) ∈ T +2 .Let us denote ˜ G L T φ ( r, a ) , the set of paths y ∈ G L T φ ( r, a ) reparametrised on [0 , as t ∈ [0 , (cid:55)→ ˜ y t := y r + t ( T − r ) . Lemma 5.
Let us assume Hypothesis 1. Let r ≥ , for any closed subset (cid:75) ⊂ (cid:67) ([0 , , V) , the set S (cid:48) ( r ) := { a ∈ V | ˜ G L T φ ( r, a ) (cid:84) (cid:75) (cid:54) = ∅} is closed in V , inparticular it is a Borel set in V . It shows that H2 holds for F = G L T φ . roof. Let { a k } k ∈ N be a convergent sequence of S (cid:48) ( r ) . For each k ∈ N , we choosea path ˜ y k ∈ ˜ G L T φ ( r, a k ) (cid:84) (cid:75) (which is not empty by definition). Then, for every s, t ∈ [0 , , s ≤ t , d (˜ y kt , ˜ y ks ) ≤ d (˜ y kt , φ ˜ t, ˜ s (˜ y ks )) + d ( φ ˜ t, ˜ s (˜ y ks ) , ˜ y ks ) ≤ [ L T (cid:36) ( ω ˜ s, ˜ t ) + δ ˜ t − ˜ s ] (cid:107) N (cid:107) ∞ , (30)where ˜ t := r + t ( T − r ) and ˜ s := r + s ( T − r ) . Since ˜ t − ˜ s goes to zero when t − s → , it follows that { t ∈ [0 , (cid:55)→ ˜ y kt } k ∈ N is equi-continuous.The sequence { a k } k ∈ N converges, so it is bounded by a constant A ≥ . Apply-ing (30) between s = 0 and t , we get | ˜ y kt | ≤ ( L T (cid:36) ( ω , ) + δ T ) (cid:107) N (cid:107) ∞ + A , whichproves that t ∈ [0 , (cid:55)→ y k is uniformly bounded.By Ascoli-Arzelà theorem, there is a convergent subsequence { ˜ y k i } i ∈ N in ( (cid:67) ([0 , , V) , (cid:107)·(cid:107) ∞ ) to a path y . This path belongs to (cid:75) since (cid:75) is closed. Because φ t,s is continuous, ˜ y ∈ ˜ G L T φ ( r, a ) . Hence S (cid:48) ( r ) is closed and then Borel.The proof of the next lemma is an immediate consequence of the definition ofD-solutions. Lemma 6. If t ∈ [ r, r + T ] (cid:55)→ y t ( r, a ) belongs to G L T φ ( r, a ) , then for any r (cid:48) ≥ , itsrestriction t ∈ [ r + r (cid:48) , r + r (cid:48) + T ] (cid:55)→ y t ( r, a ) belongs to G L T φ ( r + r (cid:48) , y r + r (cid:48) ( r, a )) . Itshows that H3 holds for F = G L T φ . Lemma 7.
We assume that Hypothesis 1 hold. For r (cid:48) ≥ , if y ∈ G L T φ ( r, a ) and z ∈ G L T φ ( r + r (cid:48) , y r + r (cid:48) ( r, a )) , then y (cid:46)(cid:47) r + r (cid:48) z ∈ G L T φ ( r, a ) . It shows that H4 holds for F := G L T φ Proof.
Let us write x := y (cid:46)(cid:47) s z where s := r + r (cid:48) and U τ,t := d ( x t , φ t,τ ( x τ )) for τ ≤ t . On the one hand, for any r ≤ τ ≤ s ≤ t with (27), (14) and (15), U τ,t ≤ d ( x t , φ t,s ( x s )) + d ( φ t,s ( x s ) , φ t,s ◦ φ s,τ ( x τ )) + d ( φ t,s ◦ φ s,τ ( x τ ) , φ t,τ ( x τ )) ≤ (cid:107) N (cid:107) ∞ (2 + δ T ) L T (cid:36) ( ω τ,t ) . (31)On the other hand, for s ≤ τ ≤ t or τ ≤ t ≤ s with (27) U τ,t ≤ (cid:107) N (cid:107) ∞ L T (cid:36) ( ω τ,t ) . (32)Thus, combining (31) and (32), for any r ≤ τ ≤ t ≤ T , U τ,t ≤ (cid:107) N (cid:107) ∞ (2 + δ T ) L T (cid:36) ( ω τ,t ) . Besides, for any r ≤ τ ≤ u ≤ t ≤ T with (14) and (15), U τ,t ≤ U u,t + (1 + δ T ) U τ,u + (cid:107) N (cid:107) ∞ (cid:36) ( ω τ,t ) . (33)17et λ ∈ (0 , such that (cid:36) λ satisfies (6) with κ λ := 2 − λ κ λ < . Let T ( λ ) > be areal number such that κ λ (1 + δ T ( λ ) ) < . For any, two successive points τ, t of asubdivision π , U τ,t ≤ D L T ( π, λ ) (cid:36) λ ( ω τ,t ) , (34)where D L T ( π, λ ) := (cid:107) N (cid:107) ∞ (2 + δ T ) L T sup τ,t successive points of π (cid:36) − λ ( ω τ,t ) .Let us show by induction over the distance m between points τ and t in π ∩ [0 , T ( λ )] that U τ,t ≤ A L T ( π, λ ) (cid:36) λ ( ω τ,t ) , (35)where A L T ( π, λ ) := D L T ( π, λ )(1 + δ T ( λ ) ) + 2 (cid:107) N (cid:107) ∞ (cid:36) λ ( ω ,T ( λ ) )1 − κ λ (1 + δ T ( λ ) ) . When m = 0 , U τ,τ = 0 so that (35) holds. For m = 1 , τ and t are successivepoints then (35) holds with (34). Now, we assume that (35) holds for any twopoints at distance m . Let τ and t be two points at distance m + 1 in π ∩ [0 , T ( λ )] .Since ω is super-additive, one may choose two successive points s and s (cid:48) in π with τ < s < s (cid:48) < t such that ω τ,s ≤ ω τ,t / and ω s (cid:48) ,t ≤ ω τ,t / , as in the proof ofTheorem 1. Then, by applying (33) between ( τ, s, s (cid:48) ) and ( s, s (cid:48) , t ) we obtain, U τ,t ≤ U s,t + (1 + δ T ( λ ) ) U τ,s + (cid:107) N (cid:107) ∞ (cid:36) − λ ( ω ,T ( λ ) ) (cid:36) λ ( ω τ,t ) ≤ U s (cid:48) ,t + (1 + δ T ( λ ) ) U s,s (cid:48) + (1 + δ T ( λ ) ) U τ,s + 2 (cid:107) N (cid:107) ∞ (cid:36) − λ ( ω ,T ( λ ) ) (cid:36) λ ( ω τ,t ) ≤ [ A L T ( π, λ ) κ λ (1 + δ T ( λ ) ) + (1 + δ T ( λ ) ) D L T ( π, λ ) + 2 (cid:107) N (cid:107) ∞ (cid:36) − λ ( ω ,T ( λ ) )] (cid:36) λ ( ω τ,t ) ≤ A L T ( π, λ ) (cid:36) λ ( ω τ,t ) , with our choice of A L T ( π, λ ) . This concludes the induction, so (35) holds for any τ, t ∈ π ∩ [0 , T ( λ )] .Clearly, D L T ( π, λ ) → when the mesh of π goes to zero. Then, A L T ( π, λ ) → A ( λ ) := 2 (cid:107) N (cid:107) ∞ (cid:36) λ ( ω ,T ( λ ) ) / (1 − κ λ (1 + δ T ( λ ) )) when the mesh of π goes to zero.By continuity of ( τ, t ) (cid:55)→ U τ,t , considering finer and finer partitions leads to U τ,t ≤ A ( λ ) (cid:36) λ ( ω τ,t ) for any r ≤ τ ≤ t ≤ T ( λ ) .Finally, choosing T ( λ ) so that T ( λ ) increases to T defined in Hypothesis 1 when λ goes to , we conclude that for any r ≤ τ ≤ t ≤ T , U τ,t ≤ (cid:107) N (cid:107) ∞ L T (cid:36) ( ω τ,t ) , where L T is defined in Hypothesis 1. This proves that z ∈ G L T φ ( r, a ) .18 roof of Theorem 2. Lemma 2-7 prove that conditions H0-H4 of Definition 8 holdfor F = G L T φ . This means that G L T ( r, a ) is a family of abstract local integralfunnels. We apply Theorem 1 in [11]. For any ( r, a ) ∈ T × V , there exists ameasurable map a (cid:55)→ ( t (cid:55)→ ψ t,r ( a )) with respect to the Borel subsets of (cid:67) ( T , V) with the property that ψ r,r ( a ) = a and ψ t,s ◦ ψ s,r ( a ) = ψ t,r ( a ) , t ≥ r . A Lipschitz almost flow which has the flow property is said to be a Lipschitz flow.We recast the definition.
Definition 10 (Lipschitz flow) . A flow ψ ∈ (cid:70) (V) is said to be a Lipschitz flow iffor any ( s, t ) ∈ T +2 , ψ t,s is Lipschitz in space with (cid:107) ψ t,s (cid:107) Lip ≤ δ T .In this section, we consider galaxies that contain a Lipschitz flow.We prove that such a Lipschitz flow ψ is the only possible flow in the galaxy(Theorem 5), and that the iterated almost flow φ π of any almost flow φ convergesto ψ (Theorem 3). We also characterize the rate of convergence (Theorem 4).Let us choose λ ∈ (0 , such that (cid:36) λ satisfies the same properties as (cid:36) upto changing κ to κ λ := 2 − λ κ λ , provided κ λ < . This is possible as soon as λ > / (1 − log ( κ )) with (6).Clearly, if for ψ, χ ∈ (cid:70) (V) , ∆ N,(cid:36) ( ψ, χ ) < + ∞ , then ∆ N,(cid:36) λ ( ψ, χ ) ≤ ∆ N,(cid:36) ( ψ, χ ) (cid:36) − λ ( ω ,T ) < + ∞ , (36)where (cid:36) is defined by (6). Hence, the galaxies remain the same when (cid:36) is changedto (cid:36) λ . We define Θ( π ) := sup d π ( s,s (cid:48) )=1 (cid:36) − λ ( ω s,s (cid:48) ) . (37) Theorem 3.
Let φ be an almost flow such that (cid:107) φ π (cid:107) Lip ≤ δ T whatever thepartition π , we say that φ satisfies the uniform Lipschitz (UL) condition. Thenthere exists a Lipschitz flow ζ ∈ (cid:70) (V) with (cid:107) ζ s,t (cid:107) Lip ≤ δ T such that { φ π } converges to ζ as | π | → . Theorem 4.
Let φ be an almost flow and ψ be a Lipschitz flow with ψ ∼ φ . Thenthere exists a constant K that depends only on λ , ∆ N,(cid:36) ( φ, ψ ) , κ and T (assumedto be small enough) so that ∆ N,(cid:36) λ ( ψ, φ π ) ≤ K Θ( π ) . In particular, { φ π } π converges to ψ as | π | → . emark . In [9, 10], we develop the notion of stable almost flow around a necessarycondition for an almost flow to be associated to a Lipschitz flow. Under such acondition, a stronger rate of convergence may be achieved by taking Θ( π ) :=sup d π ( s,s (cid:48) )=1 (cid:36) ( ω s,s (cid:48) ) /ω s,s (cid:48) [10]. Theorem 5 (Uniqueness of Lipschitz flows) . If ψ is a Lipschitz flow and χ is aflow (not necessarily Lipschitz a priori) in the same galaxy as ψ , that is χ ∼ ψ ,then χ = ψ . Hypothesis 2.
Let us fix a partition π . We consider ψ and χ in (cid:70) (V) such that ψ ∼ χ and for any ( r, s, t ) ∈ π , (cid:107) ψ t,s (cid:107) Lip ≤ δ T , (38) N ( χ t,s ( a )) ≤ (1 + δ T ) N ( a ) , ∀ a ∈ V , (39) ∆ N ( ψ t,s ◦ ψ s,r , ψ t,r ) ≤ β ψ (cid:36) ( ω r,t ) and ∆ N ( χ t,s ◦ χ s,r , χ t,r ) ≤ β χ (cid:36) ( ω r,t ) , (40)for some constant β χ , β φ ≥ . Remark . In Hypothesis 2, the role of ψ and χ are not exchangeable: ψ is assumedto be Lipschitz, there is no such requirement on χ . The reason of this dissymmetrylies in (43). Remark . If ψ is a Lipschitz almost flow and χ is an almost flow, then ( ψ, χ ) satisfies Hypothesis 2 for any partition π . The condition (39) is a particular caseof (17).We choose λ and T so that − log ( κ ) < λ < and δ T + δ T < − κ λ κ λ . We define (recall that Θ( π ) is given by (37)), ρ T := (cid:36) ( ω ,T ) − λ ,γ ( π ) := sup d π ( s,s (cid:48) )=1 ∆ N ( ψ, χ ) (cid:36) λ ( ω s,s (cid:48) ) ≤ ∆ N,(cid:36) λ ( ψ, χ )Θ( π ) ,β ( π ) := (2 + 3 δ T + δ T )( β ψ + β χ ) ρ T + (1 + δ T ) γ ( π ) ≥ γ ( π ) , and L ( π ) := 2 β ( π )2 − κ λ (2 + 3 δ T + δ T ) ≥ γ ( π ) . Here, Θ( π ) and thus γ ( π ) converge to zero when the mesh of π tends to zero. Lemma 8.
Let φ, χ ∈ (cid:70) (V) and π be satisfying Hypothesis 2. With the abovechoice of λ and T , it holds that d ( φ t,r ( a ) , χ t,r ( a )) ≤ L ( π ∪ { t, r } ) N ( a ) (cid:36) λ ( ω r,t ) , ∀ ( r, t ) ∈ T . (41)20 roof. We set F r,t := ∆ N ( ψ t,r , χ t,r ) , where ∆ N is defined in Notation 3. FromDefinition 1, F r,t ≤ ∆ N,(cid:36) ( ψ, χ ) (cid:36) ( ω r,t ) < + ∞ since ψ ∼ χ .In particular, for ( r, s, t ) ∈ π , with (39) in Hypothesis 2, d ( ψ t,s ◦ χ s,r ( a ) , χ t,s ◦ χ s,r ( a )) ≤ F s,t N ( χ s,r ( a )) ≤ (1 + δ T ) N ( a ) F s,t . (42)For any ( r, s, t ) ∈ π +3 , the fact that φ , χ are almost flow combined with (38)-(40)and (42) imply that for any a ∈ V , d ( ψ t,r ( a ) , χ t,r ( a )) ≤ d ( ψ t,s ◦ ψ s,r ( a ) , ψ t,s ◦ χ s,r ( a ))+ d ( ψ t,s ◦ χ s,r ( a ) , χ t,s ◦ χ s,r ( a ))+( β φ + β χ ) N ( a ) (cid:36) ( ω r,t ) ≤ (1 + δ T ) N ( a ) F r,s + (1 + δ T ) N ( a ) F s,t + ( β χ + β φ ) N ( a ) (cid:36) ( ω r,t ) . (43)Thus, dividing by N ( a ) , F r,t ≤ (1 + δ T )( F r,s + F s,t ) + ( β χ + β ψ ) ρ T (cid:36) λ ( ω r,t ) . (44)We proceed by induction. Our hypothesis is that F r,t ≤ L ( π ) (cid:36) λ ( ω r,t ) , ∀ ( r, t ) ∈ π +2 , at distance at most m. (45)When m = 0 , F r,r = 0 since ψ r,r ( a ) = χ r,r ( a ) = a for any a ∈ V . Thus (18) is truefor m = 0 . When m = 1 , r and t are successive points. From the very definition of γ ( π ) , F r,t ≤ γ ( π ) (cid:36) λ ( ω r,t ) . (46)The induction hypothesis (18) is true for m = 1 since L ( π ) ≥ γ ( π ) .Assume that the induction hypothesis is true at some level m ≥ . Let ( r, s, t ) ∈ π +3 with r < s < t and d π ( r, t ) = m + 1 . Let s (cid:48) be such that s and s (cid:48) are successivepoints in π (possibly, s (cid:48) = t ). Clearly, d π ( r, s ) ≤ m and d π ( s (cid:48) , t ) ≤ m . Using (44)to decompose F s,t using s (cid:48) and using (46), F r,t ≤ (1 + δ T ) F r,s + (1 + δ T ) F s (cid:48) ,t + (1 + δ T ) γ ( π ) (cid:36) λ ( ω s,s (cid:48) )+ (2 + 3 δ T + δ T )( β ψ + β χ ) ρ T (cid:36) λ ( ω r,t ) ≤ (1 + δ T ) F r,s + (1 + δ T ) F s (cid:48) ,t + β ( π ) (cid:36) λ ( ω r,t ) . With the induction hypothesis, since r and s (resp. s (cid:48) and t ) are at distance atmost m , F r,t ≤ L ( π )(1 + δ T ) (cid:36) λ ( ω r,s ) + L ( π )(1 + δ T ) (cid:36) λ ( ω s (cid:48) ,t ) + β ( π ) (cid:36) λ ( ω r,t ) . s and s (cid:48) to satisfy (23), our choice of L ( π ) and (6) imply that F r,t ≤ (cid:18) L ( π ) 2 + 3 δ T + δ T κ λ + β ( π ) (cid:19) (cid:36) λ ( ω r,t ) ≤ L ( π ) (cid:36) λ ( ω r,t ) . The induction hypothesis (45) is then true at level m + 1 , and then whatever thedistance between the points of the partition.Finally, (41) is obtained by replacing π by π ∪ { r, t } . Proof Theorem 3.
Let σ and π be two partitions with π ⊂ σ . We set ψ := φ σ and χ := φ π .With Theorem 1, ∆ N,(cid:36) ( φ σ , φ π ) ≤ ∆ N,(cid:36) ( φ σ , φ ) + ∆ N,(cid:36) ( φ π , φ ) ≤ L T . With (36), ∆ N,(cid:36) λ ( ψ, χ ) ≤ L T ρ T , so that { ∆ N,(cid:36) λ ( ψ, χ ) } π,σ is bounded.Again with Theorem 1, ( ψ, χ ) satisfies Hypothesis 2 for the subdivision π (up tochanging δ T ) with β ψ = β χ = 0 .Hence, L ( π ) = Cγ ( π ) where C := 2(1 + δ T ) (2 − κ λ (2 + 3 δ T + δ T ) . (47)We may then rewrite (41) as d ( φ σt,r ( a ) , φ πt,r ( a )) ≤ Cγ ( π ∪ { r, t } ) N ( a ) (cid:36) λ ( ω r,t ) . (48)Since γ ( π ) decreases to as | π | decreases to and | π ∪ { r, t }| ≤ | π | , it is easilyshown that { φ πt,s } π forms a Cauchy net with respect to the nested partitions.Then, it does converges to a limit ζ s,t ( a ) . By Theorem 8 and the continuity of N , N ( ζ s,r ) ≤ K T N ( a ) . From the UL condition, a (cid:55)→ ζ t,s ( a ) is Lipschitz continuouswith (cid:107) ζ t,s (cid:107) Lip ≤ δ T .Moreover ζ does not depend on the subdivision π . Indeed, if ˜ π is another subdivision,we obtain with (48), that { φ ˜ π } ˜ π converges to ζ when | ˜ π | → .Finally, if if { π k } k ≥ is a family of nested partitions, and ( r, s, t ) ∈ T +3 , φ π k ∪{ s } t,r = φ π k t,s ◦ φ π k s,r . | π k ∪{ s }| ≤ | π k | and (48), φ π k ∪{ s } converges to ζ when k → + ∞ . Moreover,for any a ∈ V , d ( ζ t,s ◦ ζ s,r ( a ) , φ π k t,s ◦ φ π k s,r ( a )) ≤ d ( ζ t,s ◦ ζ s,r ( a ) , φ π k t,s ◦ ζ s,r ( a )) + d ( φ π k t,s ◦ ζ s,r ( a ) , φ π k t,s ◦ φ π k s,r ( a )) ≤ Cγ ( π k ∪ { r, t } ) N ( ζ s,r ( a )) (cid:36) λ ( ω r,t ) + (1 + δ T ) d ( ζ s,r ( a ) , φ π k s,r ( a )) ≤ Cγ ( π k ∪ { r, t } )(1 + K T ) N ( a ) (cid:36) ( ω r,t ) , because N ( ζ s,r ( a )) ≤ K T N ( a ) . So, { φ π k t,s ◦ φ π k s,r } π k converges uniformly to ζ t,s ◦ ζ s,r when m → + ∞ . Then, the flow property ζ t,s ◦ ζ s,r = ζ t,r holds. Proof Theorem 4.
For a partition π , the pair ( ψ, φ π ) satisfies Hypothesis 2 for thesubdivision π with β ψ = β χ = 0 . As in the proof of Theorem 3 (we have assumedfor convenience that ∆ N,(cid:36) ( φ, ψ ) ≤ L T ), ∆ N,(cid:36) λ ( ψ, φ π ) ≤ Cγ ( π ) ≤ CL T ρ T Θ( π ) for C given by (47). This proves the result. Proof Theorem 5.
For any partition π , ψ and χ satisfy Hypothesis 2 with β ψ = β χ = 0 . Thus, ∆ N,(cid:36) λ ( ψ, χ ) ≤ Cγ ( π ) with C given by (47). As γ ( π ) decreases to when | π | decreases to , we obtainthat ψ = χ . Corollary 2.
Let ψ and χ be two almost flows with ψ ∼ χ and ψ be Lipschitz.Then for T small enough (in function of some λ < , κ and δ ) ∆ N,(cid:36) ( ψ, χ ) ≤ δ T + δ T )( β ψ + β χ )2 − κ λ (2 + 3 δ T + δ T ) . Proof.
With Remark 10, ( ψ, χ ) satisfies Hypothesis 2. Letting the mesh of thepartition decreasing to as the in proof of Theorem 5, and then letting λ increasingto leads to the result. In this section, we consider the construction of an almost flow by perturbations ofexisting ones. We assume that V is a Banach space.Let φ ∈ (cid:70) (V) be an almost flow with respect to a function N γ such that N γ ( a ) ≥ N γ (0) ≥ . 23 otation 6. For φ ∈ (cid:70) (V) when V is a Banach space, we write φ t,s,r ( a ) := φ t,s ( φ s,r ( a )) − φ t,r ( a ) . Definition 11.
Let (cid:15) ∈ (cid:70) (V) such that for any ( s, t ) ∈ T +2 , a, b ∈ V , (cid:15) t,t ≡ , (49) | (cid:15) t,s ( a ) | ≤ λN γ ( a ) (cid:36) ( ω s,t ) , (50) | (cid:15) t,s ( b ) − (cid:15) t,s ( a ) | ≤ η ( ω s,t ) | b − a | γ (51)for some λ ≥ . We say that (cid:15) is a perturbation . Proposition 2. If φ ∈ (cid:70) (V) is an almost flow and (cid:15) ∈ (cid:70) (V) is a perturbation,then ψ := φ + (cid:15) is an almost flow. Besides, ψ ∼ φ .Proof. Let ( r, s, t ) ∈ T +3 and a, b ∈ V . From (49), (12) is satisfied. With δ (cid:48) T := δ T + λ(cid:36) ( ω ,T ) , (13) is also true. In addition, with (51), | ψ t,s ( b ) − ψ t,s ( a ) | ≤ (1 + δ T ) | b − a | + 2 η ( ω s,t ) | b − a | γ . Thus, ψ satisfies (14).To show (15), we write ψ t,s,r ( a ) = φ t,s ◦ ψ s,r ( a ) + (cid:15) t,s ◦ ψ s,r ( a ) − φ t,r ( a ) − (cid:15) t,s ( a )= φ t,s,r ( a ) (cid:124) (cid:123)(cid:122) (cid:125) I r,s,t + φ t,s ◦ ( φ s,r + (cid:15) s,r )( a ) − φ t,s ◦ φ s,r ( a ) (cid:124) (cid:123)(cid:122) (cid:125) II r,s,t + (cid:15) t,s ◦ ( φ s,r ( a ) + (cid:15) s,r ( a )) − (cid:15) t,s ◦ φ s,r ( a ) (cid:124) (cid:123)(cid:122) (cid:125) III r,s,t + (cid:15) t,s ◦ φ s,r ( a ) − (cid:15) t,s ( a ) (cid:124) (cid:123)(cid:122) (cid:125) IV r,s,t . (52)We control the first term with (15), | I r,s,t | ≤ N γ ( a ) (cid:36) ( ω r,t ) . For the second one, weuse (7), (14) and (50), | II r,s,t | ≤ (1 + δ T ) | (cid:15) s,r ( a ) | + η ( ω s,r ) | (cid:15) s,r | γ ≤ (1 + δ T ) λN γ ( a ) (cid:36) ( ω r,s ) + η ( ω s,t ) λ γ N γγ ( a ) (cid:36) ( ω r,s ) γ ≤ [1 + (1 + λ γ ) δ T ] N γ ( a ) (cid:36) ( ω r,t ) , because N γ ( a ) ≥ implies that N γγ ( a ) ≤ N γ ( a ) for γ ∈ (0 , .With (50) and (51), we obtain for the third term, | III r,s,t | ≤ η ( ω s,t ) | (cid:15) s,r ( a ) | γ ≤ λ γ N γ ( a ) γ η ( ω s,t ) (cid:36) ( ω r,s ) γ ≤ λ γ N γ ( a ) δ T (cid:36) ( ω s,t ) , | IV r,s,t | ≤ | (cid:15) t,s ◦ φ s,r ( a ) | + | (cid:15) t,s ( a ) | ≤ ( N γ ( φ s,r ) + N γ ( a )) λ(cid:36) ( ω s,t ) ≤ ( K T + 1) N γ ( a ) λ(cid:36) ( ω s,t ) . Thus, combining estimations for each four terms of (52), we obtain (15) whichproves that ψ is an almost flow.Besides, | ψ t,s ( a ) − φ t,s ( a ) | = | (cid:15) t,s ( a ) | ≤ λN γ ( a ) (cid:36) ( ω s,t ) , which proves that ψ ∼ φ and concludes the proof. On this section, we show that our framework covers former different sewing lemmas.
The additive sewing lemma is the key to construct the Young integral [28] and therough integral [22, 26].We consider that V is a Banach space with a norm |·| . The distance d is d ( a, b ) := | b − a | . Definition 12 (Almost additive functional) . A family { α s,t } ( s,t ) ∈ T +2 is an almostadditive functional if α r,s,t := α r,s + α s,t − α r,t satisfies | α r,s,t | ≤ (cid:36) ( ω r,t ) , ∀ ( r, s, t ) ∈ T +3 . It is an additive functional if α r,s,t = 0 for any ( r, s, t ) ∈ T +3 . Proposition 3 (The additive sewing lemma [19, 25]) . If { α s,t } ( s,t ) ∈ T +2 is an almostadditive functional with | α s,t | ≤ δ T , there exists an additive functional { γ s,t } ( s,t ) ∈ T +2 which is unique in the sense that for any constant C ≥ and any additive functional { β s,t } ( s,t ) ∈ T +2 , | β s,t − α s,t | ≤ C(cid:36) ( ω s,t ) implies that β = γ .Proof. Clearly, φ t,s ( a ) = a + α s,t is an almost flow which satisfies the UL condition.Hence the result. 25 .2 The multiplicative sewing lemma Here we recover the results of [14, 19, 26]. We consider now that the metric space V has a monoid structure: there exists a product ab ∈ V of two elements a, b ∈ V .We also assume that there exists a Lipschitz function N : V → [1 , + ∞ ) such that d ( ac, bc ) ≤ N ( c ) d ( a, b ) and d ( ca, cb ) ≤ N ( c ) d ( a, b ) for all a, b, c ∈ V . Definition 13.
A family { α s,t } ( s,t ) ∈ T +2 is said to be an almost multiplicative func-tional if d ( α r,s α s,t , α r,t ) ≤ (cid:36) ( ω r,t ) , ∀ ( r, s, t ) ∈ T . It is a multiplicative functional if α r,s α s,t = α r,t . Proposition 4 (The multiplicative sewing lemma [19]) . If { α s,t } ( s,t ) ∈ T is an almost multiplicative functional then there exists a unique multiplicative functional { γ s,t } ( s,t ) ∈ T such that any other multiplicative functional { γ s,t } ( s,t ) ∈ T such that d ( β s,t , α s,t ) ≤ C(cid:36) ( ω s,t ) for any ( s, t ) ∈ T satisfies β = γ .Proof. For this, it is sufficient to consider φ t,s ( a ) = aα s,t which is an almost flowwhich satisfies the UL condition. Remark . Actually, as for the additive sewing lemma (which is itself a subcaseof the multiplicative sewing lemma), we have a stronger statement: No (non-linear) flow satisfies d ( ψ t,s ( a ) , aα s,t ) ≤ C(cid:36) ( ω s,t ) except { a (cid:55)→ aβ s,t } ( s,t ) ∈ T whichis constructed as the limit of the products of the α s,t over smaller and smallerintervals. Consider now that V has a Banach algebra structure with a norm |·| such that | ab | ≤ | a | × | b | and a unit element (the product of two elements is still denotedby ab ).A typical example is the Banach algebra of bounded operators over a Banachspace X .This situation fits in the multiplicative sewing lemma with d ( a, b ) = | a − b | and N ( a ) = | a | , a, b ∈ V . As seen in [14], we have many more properties: continuity,existence of an inverse, Dyson formula, Duhamel principle, ...In particular, this framework is well suited for considering linear differential equa-tions of type y t,s = a + (cid:90) ts y r,s d α r , a ∈ V , t ≥ s ≥ α : T +2 → V . If α is γ -Hölder with γ > / , then φ t,s ( a ) = a (1 + α s,t ) defines an almost flow which satisfies the condition UL (at theprice of imposing some conditions on α , this could be extended to γ < / ).Defining an “affine flow” φ t,s ( a ) = a (1 + α s,t ) + β s,t where both α and β are γ -Hölderwith γ > / , the associated flow ψ is such that ψ t,s ( a ) is solution to the perturbedequation ψ t,s ( a ) = a + (cid:90) ts ψ r,s ( a ) d α r + β s,t . This gives an alternative construction to the one of [14] where a backward integral between β and α was defined in the style of the Duhamel formula. All these resultsare extended to the rough case / < γ ≤ / . Example . With the tensor product ⊗ as product anda suitable norm, for any integer k , the tensor algebra T k (X) := R ⊕ X ⊕ X ⊗ ⊕ · · · ⊕ X ⊗ k is a Banach algebra. Chen series of iterated integrals (and then rough paths) taketheir values in some space T k (X) . The Lyons extension theorem states that anyrough path x of finite p -variation with values in T k (X) for some k ≥ (cid:98) p (cid:99) is uniquelyextended to a rough path with values in T (cid:96) (X) for any (cid:96) ≥ k , which leads to theconcept of signature [25, 26]. This follows a (cid:55)→ a ⊗ x s,t as an almost flow whichsatisfies the UL condition (see also [19] and also [14]). Now, we show that our construction is related to the one of A.M. Davie [16]. Themain idea of Davie was to construct solutions as paths y : T → V that satisfies (27)for a suitable “algorithm” φ t,s of the solution between s and t . Solutions passingthrough a in are then constructed as limit of using { φ πt, ( a ) } π (See Proposition 1).The algorithm φ t,s is given by a truncated Taylor expansion of the solution of (1).The number of terms to consider in the Taylor expansion depends directly on theregularity of x . In the Young case one term is needed whereas in the rough casetwo terms are required.In this section, we show that the algorithms provided in [16] are almost flowsunder the same regularity on the vector field f and the path x . Not only werecover existence of D-solutions, but we also show that measurable flows exist whenthe vector fields f are (cid:67) (Young case) or (cid:67) (rough case) in situation in whichnon-uniqueness of solutions is known to hold, again due to [16], unless f is of class (cid:67) γ b (Young case) or (cid:67) γ b (rough case).27ere U and V are two Banach spaces, where we use the same notation |·| for theirnorms. We denote by (cid:76) (U , V) the continuous linear maps from U to V . Let f be amap from V to (cid:76) (U , V) . If f is regular, we denote its Fréchet derivative in a ∈ V , d f ( a ) ∈ (cid:76) ( V, (cid:76) ( U, V )) .Moreover, for any a ∈ W and ( r, s, t ) ∈ T +3 , we set φ t,s,r ( a ) := φ t,s ◦ φ s,r ( a ) − φ t,r ( a ) . Let x : T → U be a path of finite p -variation controlled by ω with ≤ p < .We define a family ( φ t,s ) ( s,t ) ∈ T +2 in (cid:70) (V) such that for all a ∈ V and ( s, t ) ∈ T +2 , φ t,s ( a ) := a + f ( a ) x s,t , (53)where x s,t := x t − x s . Proposition 5.
Assume that f ∈ (cid:67) γ (V , (cid:76) ( U, V )) , with γ > p . Then φ is analmost flow.Proof. We check that assumptions of Definition 2 hold. Let ( r, s, t ) be in T +3 andlet a, b be in V . First, φ t,t ( a ) = a because x t,t = 0 . Second, | φ t,s ( a ) − a | ≤ | f ( a ) | · | x s,t | ≤ | f ( a ) | · (cid:107) x (cid:107) p ω /ps,t , which proves (13). Third, | φ t,s ( a ) − φ t,s ( b ) | ≤ | a − b | + | f ( a ) − f ( b ) || x s,t | ≤ | a − b | + (cid:107) f (cid:107) γ (cid:107) x (cid:107) p ω /ps,t | a − b | γ , which proves (14). It remains to prove (15). Since φ t,s,r ( a ) = f ( φ s,r ( a )) x s,t − f ( a ) x s,t , we obtain | φ t,s,r ( a ) | ≤ (cid:107) f (cid:107) γ (cid:107) x (cid:107) p ω /ps,t | φ s,r ( a ) − a | γ ≤ (cid:107) f (cid:107) γ | f ( a ) | γ (cid:107) x (cid:107) p ω (1+ γ ) /pr,t ≤ (cid:107) f (cid:107) γ (1 + | f ( a ) | ) (cid:107) x (cid:107) p ω (1+ γ ) /pr,t . Setting (cid:36) ( ω r,t ) := ω (1+ γ ) /pr,t , η ( ω s,t ) := (cid:107) f (cid:107) γ (cid:107) x (cid:107) p ω /ps,t and N γ ( a ) := (1 + | f ( a ) | ) (cid:0) (cid:107) x (cid:107) p + (cid:107) f (cid:107) γ (cid:107) x (cid:107) p (cid:1) , it proves that φ is an almost flow.This concludes the proof. 28et ψ be a flow in the same galaxy as the almost flow φ . For any a ∈ V and any ( r, t ) ∈ T , we set y t ( r, a ) := ψ t,r ( a ) so that y r ( r, a ) = a. Clearly, ( r, a ) (cid:55)→ ( t ∈ [ r, T ] (cid:55)→ y t ( r, a )) is a family of continuous paths whichsatisfies | y t ( r, a ) − φ t,r ( y s ( r, a )) | ≤ CN γ ( y s ( r, a )) ω /ps,t , ∀ ( s, t ) ∈ T , ∀ a ∈ V since y t ( r, a ) = ψ t,s ( y s ( r, a )) . Besides, s ∈ [ r, T ] (cid:55)→ N γ ( y s ( r, a )) is bounded. There-fore, with our choice of the almost flow φ , ψ · ,r ( a ) = y ( r, a ) is a solution in the sensedefined by A.M. Davie [16] for the Young differential equation z t = a + (cid:82) tr f ( z s ) d x s .Even if several solutions may exist for a given ( r, a ) , the flow corresponds to aparticular choice of a family of solutions which is constructed thanks to a selectionprinciple. This family of solution is stable under splicing (see Definition 7). Corollary 3.
We assume that V is a finite-dimensional vector space and f ∈ (cid:67) (V , (cid:76) (U , V)) . Then there exists a flow ψ ∈ (cid:70) (V) in the same galaxy as φ suchthat ψ t,s is Borel measurable for any ( s, t ) ∈ T .Remark . When f ∈ (cid:67) γ b , several D-solutions to the Young differential equation y = a + (cid:82) · f ( y s ( a )) d x s may exist (Example 1 in [16]). Uniqueness arises when f ∈ (cid:67) γ b with γ > p . Hence, a measurable flow may exist even when severalD-solution may exist. Proof.
According to Proposition 5, φ is an almost flow. Here γ = 1 , so φ is Lipschitz.Then, we conclude the proof in applying Theorem 2 to φ . When the regularity of x is weaker than in the Young case, we need more terms inthe Taylor expansion to obtain an almost flow.Let T (U) := R ⊕ U ⊕ (U ⊗ U) be the truncated tensor algebra (with addition + and tensor product ⊗ ). A distance is defined on the subset of elements of T (U) onthe form a = 1 + a + a with a i ∈ U ⊗ i by d ( a, b ) = | a − ⊗ b | where |·| is a normon T (U) such that | a ⊗ b | ≤ | a | · | b | for any a, b ∈ U .Let x = (1 , x , x ) be a rough path with values in T (U) of finite p -variation, ≤ p < , controlled by ω (see e.g. , [20, 25] for a complete definition).We define a family ( φ t,s ) ( s,t ) ∈ T +2 in (cid:70) (V) such that for all a ∈ V and ( s, t ) ∈ T +2 , φ t,s ( a ) := a + f ( a ) x s,t + d f ( a ) · f ( a ) x s,t . (54)29 roposition 6. Assume that f ∈ (cid:67) γ b (V , (cid:76) (U , V)) , with γ > p . Then φ isan almost flow.Proof. We check that the assumptions of Definition 2 hold. The proofs of (12), (13)and (14) are very similar to the ones in the proof of Proposition 5. The computationto show (15) is a bit more involved.Indeed, for any a ∈ V , ( r, s, t ) ∈ T +3 , φ t,s,r ( a ) = − f ( a ) x s,t + f ( φ s,r ( a )) x s,t − d f ( a ) · f ( a )( x s,t + x r,s ⊗ x s,t )+ d f ( φ s,r ) · f ( φ s,r ( a )) x s,t =[ f ( φ s,t ( a )) − f ( a ) − d f ( a ) · f ( a ) x r,s ] ⊗ x s,t + [ d f ( φ s,r ( a )) · f ( φ s,r ( a )) − d f ( a ) · f ( a )] x s,t = f ( φ s,t ( a )) − f ( a ) − d f ( a ) · ( φ s,r ( a ) − a ) (cid:124) (cid:123)(cid:122) (cid:125) I r,s,t + d f ( a ) · f ( a ) x r,s ⊗ x s,t (cid:124) (cid:123)(cid:122) (cid:125) II r,s,t + [ d f ( φ s,t ( a )) · f ( φ s,r ( a )) − d f ( a ) · f ( a )] x s,t (cid:124) (cid:123)(cid:122) (cid:125) III r,s,t . For the first term, | I r,s,t | ≤ (cid:107) d f (cid:107) γ (cid:107) x (cid:107) p ω /ps,t | φ s,r ( a ) − a | γ ≤ (cid:107) d f (cid:107) γ (cid:107) x (cid:107) p ω /ps,t [ (cid:107) f (cid:107) ∞ (cid:107) x (cid:107) p + (cid:107) d f · f (cid:107) ∞ (cid:107) x (cid:107) p ω /p ,T ] γ ω (1+ γ ) /pr,s . For the two last terms, | II r,s,t | ≤ (cid:107) d f · f (cid:107) ∞ (cid:107) x (cid:107) p (cid:107) x (cid:107) p ω /pr,t ≤ (cid:107) d f · f (cid:107) ∞ (cid:107) x (cid:107) p (cid:107) x (cid:107) p ω (1 − γ ) /p ,T ω (2+ γ ) /pr,t and | III r,s,t | ≤ (cid:107) x (cid:107) p ω /pr,t [ (cid:107) d f (cid:107) γ (cid:107) f (cid:107) ∞ | φ s,r ( a ) − a | γ + (cid:107) d f (cid:107) ∞ (cid:107) f (cid:107) Lip | φ s,r ( a ) − a | ] ≤ Cω /pr,t , where C is a constant which depends on f , d f , ω , γ , x . It proves that φ is aLipschitz almost flow.This concludes the proof.As for the Young case, any flow ψ in the same galaxy as the almost flow φ givenby (54) gives rise to a family of solutions to the RDE z t = a + (cid:82) t f ( z s ) d x s . Corollary 4.
We assume that V is a finite-dimensional vector space and f ∈ (cid:67) (V , (cid:76) (U , V)) . Then there exists a flow ψ ∈ (cid:70) (V) in the same galaxy as φ suchthat ψ t,s is Borel measurable for any ( s, t ) ∈ T . emark . When f ∈ (cid:67) γ b , several D-solutions to the RDE y = a + (cid:82) · f ( y s ( a )) d x s may exist (Example 2 in [16]). Uniqueness requires f to be (2+ γ ) -Hölder continuouswith γ > p . Hence, Corollary 4 shows that a measurable flow exists even whenseveral D-solutions may exist. Proof.
According to Proposition 6, φ is an almost flow. Here γ = 1 , so φ is Lipschitz.Then, we conclude the proof in applying Theorem 2 to φ . Acknowledgments
The authors are very grateful to Laure Coutin for her numerous valuable remarksand corrections. We are also grateful to the CIRM (Marseille, France) for its kindhospitality with the Research-in-Pair program. Finally, we thank the referees fortheir careful reading.
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