The non-linear sewing lemma II: Lipschitz continuous formulation
TThe non-linear sewing lemma II:Lipschitz continuous formulation
Antoine Brault * Antoine Lejay β October 21, 2018
We give an unified framework to solve rough differential equations.Based on flows, our approach unifies the former ones developed byDavie, Friz-Victoir and Bailleul. The main idea is to build a flow fromthe iterated product of an almost flow which can be viewed as a goodapproximation of the solution at small time. In this second article, wegive some tractable conditions under which the limit flow is Lipschitzcontinuous and its links with uniqueness of solutions of rough differentialequations. We also give perturbation formulas on almost flows whichlink the former constructions.
Keywords:
Rough differential equations; Lipschitz flows; Rough paths
The rough path theory was introduced to deal with differential equations driven byan irregular deterministic path multidimensional π₯ of the type d π¦ π‘ β π ` ΕΌ π‘π π p π¦ π q d π₯ π , (1)where π is an initial condition and π a regular vector field. Typically, the irregularityof π₯ is measured in πΌ -HΓΆlder ( πΌ Δ ) or in π -variation ( π Δ ) spaces. Such anequation is called Rough Differential Equation (RDE) [18, 25]. * Institut de MathΓ©matiques de Toulouse, UMR 5219; UniversitΓ© de Toulouse, UPS IMT, F-31062Toulouse Cedex 9, France; [email protected] β UniversitΓ© de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France, [email protected] a r X i v : . [ m a t h . P R ] O c t his theory was very fruitful to study stochastic equations driven by Gaussianprocess which is not covered by the ItΓ΄ framework, like the fractional Brownianmotion [13, 26]. More generally, the rough path framework allows one to separatethe probabilistic from the deterministic part in such equation and to overcomesome probabilistic conditions such as using adapted or non-anticipative processes.Recently, the ideas of the rough path theory were extended to stochastic partialdifferential equations (SPDE) with the works of [21, 22] which have led to significantprogress in the study of some SPDE. This theory also found applications in machinelearning and the recognizing of the Chinese ideograms [10, 24].Since the seminal article [25] by T. Lyons in 1998, several approaches emerged tosolve (1). They are based on two main technical arguments: fixed point theorems[20, 25] and flow approximations [2, 12, 14, 16, 19]. In particular, the rough flowtheory allows one to extend work about stochastic flows, which has been developedin β80s by Le Jan-Watanabe-Kunita and others, to a non-semimartinagle setting [4].The main goal of this article is to give a framework which unifies the approachesby flow and pursue further investigations on their properties and their relationswith families of solutions to (1).A flow is a family of maps t π π‘,π u from a Banach space to itself such that π π‘,π Λ π π ,π β π π‘,π for any π Δ π Δ π‘ . Typically, the map which associates the initial condition π to the solution of (1) has a flow property. The existence of a such flow heavilydepends on the existence and uniqueness of the solution. However, it was provedin [7, 8] and extended to the rough path case in [6] that when non-uniqueness holds,it is possible to build a measurable flow by a selection technique. In this article weare interested by the construction of a Lipschitz flows.The main idea to build the flow associated is to find a good approximation π π‘,π of π π‘,π when | π‘ Β΄ π | is small enough. We iterate this approximation on a subdivision π β t π Δ π‘ π Δ Β¨ Β¨ Β¨ Δ π‘ π Δ π‘ u of r π , π‘ s by setting π ππ‘,π : β π π‘,π‘ π Λ Β¨ Β¨ Β¨ Λ π π‘ π ,π . If π π does converges when the mesh of π goes to zero, π π , the limit is necessarily aflowThis computation is similar to the ones of numerical schemes as Eulerβs methods ofdifferent order [11]. Moreover, this idea is found among the Trotterβs formulas forbounded or unbounded linear operators which allows to compute the semi-group ofthe sum of two non-commutative operators only knowing the semi-groups associatedto each operator [15]. This property can be used to prove the Feynman-Kac formula.Rather than working with a particular choice for the almost flow π as in [14, 19], wegive here generic conditions on π . We generalize the multiplicative sewing lemma2f [16] and of [12], introduced to solve linear RDE to a non linear situation. In thisway, we construct directly some flows. In opposite to the additive and multiplicativesewing lemma, the limit is not necessarily unique. The approximations are assumedto be Lipschitz. This is not the case for the limit.Our framework is close to the one developed by I. Bailleul in [2, 5] with a notabledifference: in the previous article [6], we have shown that a flow may exist undersome weak conditions even if the iterated products π π are not uniformly Lipschitzcontinuous. Similarly to I. Bailleul, we have also shown that if π π is uniformlyLipschitz continuous for any partition π (UL Condition) then it converges to aLipschitz flow, which is necessarily unique.The present article gives a sufficient condition, called the , on thealmost flow π that ensures the UL Condition. We then study various consequencesof this condition: existence of an inverse, unique family of solutions, convergence ofthe Euler scheme, ... The can be checked on the almost flow, whichis then called a stable almost flow . This condition is weaker that the one givenby I. Bailleul in [2, 5]: there it should roughly be (cid:67) with a Lipschitz continuousspatial derivative while in our case, the spatial derivative may be only HΓΆldercontinuous. The question of the existence of a Lipschitz flow without the ULcondition, in relation with Stochastic Differential Equations, should be dealt within a subsequent work.We also study the relationship between almost flows and family of solutions to (1)in the sense of Davie [14] as they are two different objects. In particular, we showthat when an almost flow is stable, when the family of solutions to the RDE isunique and Lipschitz continuous. We also relate the distance between two familiesof solutions with respect to the distance between two almost flows when one isstable. Again, consequences of this result will be drawn in a subsequent work.We also give several conditions under which perturbations of almost flows, aconvenient tool to construct numerical schemes, converge to the same limit flow.These perturbative arguments are the key to unify expansions that are a priori ofdifferent nature.Finally, we apply our framework to recover the results of A.M. Davie [14], P. Friz &N. Victoir [17, 19] and I. Bailleul [2, 5] using various perturbation arguments.Although not done here, our framework could be applied to deal with branchedrough paths, that are high-order expansions indiced by trees, which are studiedin [9] and shown to fit the Bailleulβs framework [3]. Outline.
After introducing in Section 2 the main notations and general definitions,we recall in Section 3 the notion of almost flow which is introduced in our previousarticle [6]. In Section 4, we define the -point control as well as stable almost flow π .3e prove that under these conditions, π π converges to a Lipschitz flow. In Section 5,we give conditions to modify the almost flow π by adding a perturbation π whileretains the convergence to a flow. We prove that under suitable conditions, theinverse the approximation π and that π Β΄ is a good approximation of the inverse ofthe flow. The link between the uniqueness of the solution of (1) and the existenceof a flow is studied in Section 7. In Section 8, our formalism links the formerapproaches based on flow [2, 14, 19]. The following notations and hypotheses will be constantly used throughout all thisarticle.
Let us fix π Δ , a time horizon. We write T : β r , π s as well as T ` : β tp π , π‘ q P T | π Δ π‘ u and T ` : β tp π, π , π‘ q P T | π Δ π Δ π‘ u , T Β΄ : β tp π , π‘ q P T | π Δ π‘ u and T Β΄ : β tp π, π , π‘ q P T | π Δ π Δ π‘ u . We also set T β T ` Y T Β΄ .A control π is a family from T ` : β t Δ π Δ π‘ Δ π u to R ` which is super-additive ,that is π π,π ` π π ,π‘ Δ π π,π‘ , @p π, π , π‘ q P T ` , and continuous close to its diagonal with π π ,π β , π P T . For example π π ,π‘ β πΆ | π‘ Β΄ π | where πΆ is a non-negative constant.A remainder is a continuous, increasing function π : R ` Γ R ` such that for some Δ ΞΊ Δ , π Λ πΏ Λ Δ ΞΊ π p πΏ q , πΏ Δ . (2)A typical example for π is π p πΏ q β πΏ π for any π Δ .Let πΏ : R ` Γ R ` be non-decreasing function with lim π Γ πΏ π β .We fix πΎ P p , s . We also consider a continuous, increasing function π : R ` Γ R ` such that π p π π ,π‘ q π p π π ,π‘ q πΎ Δ πΏ π π p π π ,π‘ q , @p π , π‘ q P T ` . (3)4 .2 Functions spaces We denote by p V , |Β¨|q a Banach spaces. The space of continuous functions from V to V is denoted by (cid:67) p V q . We set } π₯ } : β sup π‘ Pr ,π s | π₯ π‘ | . Notation 1.
We denote by (cid:70) ` p V q the space of families t π π‘,π u p π ,π‘ qP T ` with π π‘,π P (cid:67) p V q for each p π , π‘ q P T ` . We also set (cid:70) Β΄ p V q the space of families t π π ,π‘ u p π ,π‘ qP T ` with π π ,π‘ P (cid:67) p V q for each p π , π‘ q P T ` (note the reversion of the indices).We now consider a partition π β t π‘ Δ Β¨ Β¨ Β¨ Δ π‘ π u of r , π s with a mesh denotedby | π | . Notation 2 (Iterated products) . For π P (cid:70) ` p V q , we write π ππ‘,π : β π π‘,π‘ π Λ π π‘ π ,π‘ π Β΄ Λ Β¨ Β¨ Β¨ Λ π π‘ π ` ,π‘ π Λ π π‘ π ,π , where r π‘ π , π‘ π s is the biggest interval of such kind contained in r π , π‘ s . We say that π ππ‘,π is the iterated product of π on a subdivision π . If no such interval exists, then π ππ‘,π β π π‘,π .For π P (cid:70) ` p V q , we define similarly π ππ ,π‘ : β π π ,π‘ Λ π π‘ ,π‘ Λ Β¨ Β¨ Β¨ Λ π π‘ π Β΄ ,π‘ π Λ π π‘ π ,π‘ . For any partition π , π π P (cid:70) Λ p V q when π P (cid:70) Λ p V q . A trivial but important remarkis that from the very construction, π ππ‘,π β π ππ‘,π Λ π ππ,π for any π P π. In particular, t π ππ‘,π u p π ,π‘ qP T Λ , π ,π‘ P π enjoys a (semi-)flow property (Definition 3). Anatural question is then to study the limit of π π as the mesh of π decreases to .Finally, for any p π, π , π‘ q P T Λ we write π π‘,π ,π : β π π‘,π Λ π π ,π Β΄ π π‘,π . Notation 3.
We extend the norm |Β¨| on (cid:70) Λ p V q by } π } π : β sup p π ,π‘ qP T Λ π Ββ π‘ } π π‘,π } π p π π ,π‘ q , where π , π are defined in Section 2. Possibly, } π } π β 8 . Actually, this norm ismainly used to consider the distance between two elements of (cid:70) Λ p V q . With thisnorm, p (cid:70) Λ p V q , }Β¨} π q is a Banach space. 5 efinition 1. We define the equivalence relation β on (cid:70) Λ p V q by π β π if andonly if there exists a constant πΆ such that } π π‘,π Β΄ π π‘,π } Δ πΆπ p π π ,π‘ q , @p π , π‘ q P T . In other words, π β π if and only if } π Β΄ π } π Δ `8 . Each quotient class of (cid:70) Λ p V q{ β is called a galaxy , which contains elements of (cid:70) Λ p V q which are at finitedistance from each others. Notation 4 (Lipschitz semi-norm) . The Lipschitz semi-norm of a function π froma Banach space p V , | Β¨ |q to another Banach space p W , | Β¨ | q is } π } Lip : β sup π,π P V ,π β° π | π p π q Β΄ π p π q| | π Β΄ π | , whenever this quantity is finite. And if π΄ Δ V is a non-empty subset of V , we saythat π is Lipschitz continuous on π΄ when } π } Lip ,π΄ : β sup π,π P π΄,π β° π | π p π q Β΄ π p π q| | π Β΄ π | Δ `8 . Notation 5 (HΓΆlder spaces) . For πΎ P p , q and an integer π , we denote by (cid:67) π ` πΎπ p V q the space of bounded continuous functions from V to V with bounded derivativesup to order π and a π order derivative which is πΎ -HΓΆlder continuous. In this section, we recall some notions and results introduced in [6], which areuseful in next sections. As we are working on Banach spaces instead of metricspaces, we have a slightly stronger notion of almost flow than in [6].We denote by i the identity map from V to V . Definition 2 (Almost flow) . An element π P (cid:70) ` p V q is an almost flow if for any π Δ and any p π, π , π‘ q P T ` , π, π P V , π π‘,π‘ β i , (4) } π π‘,π Β΄ i } Δ πΏ π , (5) | π π‘,π p π q Β΄ π π‘,π p π q| Δ p ` πΏ π q| π Β΄ π | ` π p π π ,π‘ q| π Β΄ π | πΎ , (6) } π π‘,π ,π } Δ π π p π π,π‘ q , (7)where π Δ and π π‘,π ,π : β π π‘,π Λ π π ,π Β΄ π π‘,π . If we replace p π, π , π‘ q P T ` by p π, π , π‘ q P T Β΄ ,we say that π is a reverse almost flow . 6 efinition 3 (Semi-flow and Flow) . A semi-flow π is a family of functions p π π‘,π q p π ,π‘ qP T ` from V to V such that π π‘,π‘ β i and π π‘,π Λ π π ,π β π π‘,π (8)for any π P V and p π, π , π‘ q P T ` . It is a flow if each π π‘,π is invertible with an inverse π π ,π‘ for any p π , π‘ q P T ` and (8) holds for any p π, π , π‘ q P T : β T ` Y T Β΄ . Remark . A flow is invertible and for any p π , π‘ q P T , then π Β΄ π‘,π β π π ,π‘ . Indeed, π π‘,π Λ π π ,π‘ β π π‘,π‘ β i and π π ,π‘ Λ π π‘,π β π π ,π β i . Theorem 1 ([6]) . Let π be an almost flow (Definition 2) with π Δ and πΏ π , π defined in Section 2.1 Then there exists a time horizon π small enough and aconstant πΏ Δ π {p Β΄ p ` πΏ π q π Β΄ πΏ π q such that } π ππ‘,π Β΄ π π‘,π } Δ πΏπ p π π ,π‘ q (9) for any p π , π‘ q P T ` and any partition π of T . Definition 4 (Condition UL) . An almost flow π such that } π ππ ,π‘ } Lip Δ ` πΏ π forany p π , π‘ q P T ` whatever the partition π is said to satisfy the uniform Lipschitz (UL) condition.We give a sufficient condition on an almost flow to get a Lipschitz flow in a galaxy. Proposition 1.
Let π be an almost flow which satisfies the condition UL. Thenthere exists a Lipschitz flow π with } π π‘,π } Lip Δ ` πΏ π for any p π , π‘ q P T ` such that π ππ‘,π p π q converges to π π‘,π p π q for any π P V and any p π , π‘ q P T ` . On the other hand, there could be at most one flow in a galaxy if one is Lipschitz.
Proposition 2.
Assume that there is a Lipschitz flow π in a galaxy πΊ . Then π is the unique flow in πΊ . Besides, for any almost flow π β π , π ππ ,π‘ p π q converges to π π ,π‘ p π q for any p π , π‘ q P T ` and π P V . We then complete the results of [6] with the following ones.
Proposition 3.
Let π be an almost flow which satisfies the condition UL. Then π π is an almost flow for any partition π . The prototypical example for the next result is π p πΏ q β πΏ π for some π Δ . In thiscase, it slightly improves the rate of convergence as π Β΄ with respect to the onegiven in [6] which is π Β΄ Β΄ π for any π Δ .7 roposition 4 (Rate of convergence) . Let π be an almost flow in the same galaxyas a Lipschitz flow π with } π π‘,π } Lip Δ p ` πΏ π q and } π π‘,π Β΄ π π‘,π } Δ πΎπ p π π ,π‘ q forany p π , π‘ q P T ` . Let us assume that π is such that for a bounded function π , πΏ Β΄ π p πΏ q Δ π p πΏ q for any πΏ Δ . Then } π π‘,π p π q Β΄ π ππ‘,π p π q} Δ πΎπ p π q π ,π p ` πΏ π q with π p π q : β sup p π,π‘ q successivepoints in π π p π π,π‘ q . Proof.
The proof follows the one of Theorem 10.30 in [19, p. 238]. Let t π‘ π u π β ,...,π bethe points of π Y t π , π‘ u such that π‘ β π , π‘ π β π‘ and p π‘ π , π‘ π ` q are successive pointsin π Y t π , π‘ u . Set π§ π β π π‘,π‘ π p π π‘ π ,π‘ p π qq . Then | π π‘,π p π q Β΄ π ππ‘,π p π q| β | π§ π Β΄ π§ | Δ π Β΄ ΓΏ π β | π§ π ` Β΄ π§ π | . Since π π‘,π is Lipschitz and | π§ π ` Β΄ π§ π | Δ p ` πΏ π q| π π‘ π ` ,π‘ π p π π‘ π ` ,π‘ π p π qq Β΄ π π‘ π ` ,π‘ π p π π‘ π ` ,π‘ π p π qq| . The result follows easily.
In the previous section, we have recalled some results from [6] which endow theimportance of Lipschitz flows. However, the UL condition is not easy to verify. Inthis section, we give a sufficient condition on an almost flow π to ensure that itsatisfies the UL condition and then that its galaxy contains a unique flow which isLipschitz. In this section V , V , V , V are Banach spaces and we denote by |Β¨| their norms. Definition 5 (The -points control) . A function π : V Γ V is said to satisfy a -points control if there exists a non-decreasing, continuous function p π : R ` Γ R ` and a constant q π Δ such that | π p π q Β΄ π p π q Β΄ π p π q ` π p π q|Δ p π p| π Β΄ π | _ | π Β΄ π |q Λ p| π Β΄ π | _ | π Β΄ π |q ` q π | π Β΄ π Β΄ π ` π | (10)for any p π, π, π, π q P V . 8ny Lipschitz function π satisfies a -points control with p π β } π } Lip and q π β .However, for our purpose, we need to consider later more restrictive conditions on q π and p π .Let us start with a simple example that is found in [14]. Lemma 1.
Let π P (cid:67) ` πΎ p V , V q , Δ πΎ Δ , with a bounded derivative. Then π satisfies a -points control with p π p π₯ q β } β π } πΎ π₯ πΎ , π₯ Δ , and q π β } β π } . Proof.
For any π, π, π, π P V , π p π q Β΄ π p π q Β΄ π p π q ` π p π qβ p π Β΄ π q ΕΌ β π p ππ’ ` p Β΄ π’ q π q d π’ Β΄ p π Β΄ π q ΕΌ β π p ππ’ ` p Β΄ π’ q π q d π’ β p π Β΄ π q ΕΌ r β π p ππ’ ` p Β΄ π’ q π q Β΄ β π p ππ’ ` p Β΄ π’ q π qs d π’ ` p π Β΄ π Β΄ π ` π q ΕΌ β π p ππ’ ` p Β΄ π’ q π q d π’, which yields to | π p π q Β΄ π p π q Β΄ π p π q ` π p π q| Δ | π Β΄ π | } β π } πΎ p| π Β΄ π | _ | π Β΄ π |q πΎ ` } β π } | π Β΄ π Β΄ π ` π | . This concludes the proof.Here are a few properties of functions satisfying a -points control. Lemma 2.
Let π, π satisfying a -points control with π Lipschitz continuous. (i)
The function π is locally Lipschitz continuous. (ii) If π, π : V Γ V , then for any π, π P R , ππ ` ππ satisfies a -points control. (iii) If π : V Γ V and π : V Γ V , then π Λ π : V Γ V satisfies a -pointscontrol. (iv) If π : V Γ V with } π } Lip Δ , then i ` π is invertible and π : β p i ` π q Β΄ is Lipschitz and satisfies a -points control with p π p π₯ q β p π p} π } Lip π₯ q} π } Lip for } π } Lip Δ {p Β΄ } π } Lip q , and q π β {p Β΄ q π q .Proof. For showing (i), we choose π β π and π β π in (10) and then, | π p π q Β΄ π p π q| Δ β p π p| π Β΄ π |q ` q π Δ± | π Β΄ π | , π is locally Lipschitz continuous.Moreover, we can choose { ππ ` ππ β | π | p π ` | π | p π and Β ππ ` ππ β | π | q π ` | π | q π to obtaina -points control on ππ ` ππ . This proves (ii).To show (iii), we use the fact that π is Lipschitz according to (i). With β β π Λ π , | β p π q Β΄ β p π q Β΄ β p π q ` β p π q|Δ p π p| π p π q Β΄ π p π q| _ | π p π q Β΄ π p π q|q r| π p π q Β΄ π p π q| _ | π p π q Β΄ π p π q|s` q π p π p| π Β΄ π | _ | π Β΄ π |q r| π Β΄ π | _ | π Β΄ π |s ` q π q π | π Β΄ π Β΄ π ` π |Δ p π p} π } Lip | π Β΄ π | _ | π Β΄ π |q} π } Lip p| π Β΄ π | _ | π Β΄ π |q` q π p π p| π Β΄ π | _ | π Β΄ π |qr| π Β΄ π | _ | π Β΄ π |s ` q π q π | π Β΄ π Β΄ π ` π | . This proves that β satisfies the -points control.It remains to show (iv). From the Lipschitz inverse function theorem [1, p. 124], i ` π is invertible with an inverse π that satisfies } π } Lip Δ p Β΄ } π } Lip q Β΄ . Besides, | π Β΄ π Β΄ π ` π ` p π p π q Β΄ π p π q Β΄ π p π q ` π p π qq| Δ p Β΄ q π q| π Β΄ π Β΄ π ` π |Β΄ p π p| π Β΄ π | _ | π Β΄ π |q Λ r| π Β΄ π | _ | π Β΄ π |s , which yields to p Β΄ q π q| π p π q Β΄ π p π q Β΄ π p π q ` π p π q| Δ | π Β΄ π Β΄ π ` π |` p π p} π } Lip | π Β΄ π | _ | π Β΄ π |q} π } Lip r| π Β΄ π | _ | π Β΄ π |s . Hence, for π₯ P R ` , p π β p π p} π } Lip π₯ q} π } Lip and q π β p Β΄ q π q Β΄ . Therefore, π satisfies a4-points control.The reason for introducing the 4-points control lies in its good behavior with respectto composition. More precisely, if π satisfies a 4-points control while π and β areLipschitz continuous and bounded, } π Λ π Β΄ π Λ β } Lip Δ p π p} π Β΄ β } q} π } Lip _ } β } Lip ` Λ π } π Β΄ β } Lip , } π Λ π Β΄ π Λ β } Δ Β΄ p π p q ` Λ π Β― } π Β΄ β } . Definition 6.
A family π P (cid:70) ` p π q is said to satisfy a π -compatible -points control if there exists a family of functions p p π π‘,π q p π ,π‘ qP T ` and constants p q π π‘,π q p π ,π‘ qP T such that10or any r π , π‘ s Δ r , π s the estimation (10) holds. Moreover, for any r π , π‘ s Δ r π’, π£ s , p p π π‘,π q p π ,π‘ qP T ` is said π -compatible if p π π ,π‘ Δ p π π’,π£ , p π π‘,π p πΌπ p π π ,π‘ qq Δ π f p πΌ q π p π π ,π‘ q , where πΌ, π f p πΌ q are non-negative constants with π f p πΌ q which can depend on πΌ . Definition 7 (Stable almost flow) . We say that an almost flow π with πΎ β in(6) is a stable almost flow if β it satisfies a π -compatible -points control with q π π‘,π Δ ` πΏ π , β there a constant πΆ Δ such that for p π, π , π‘ q P T ` , } π π‘,π ,π } Lip Δ πΆπ p π π,π‘ q , (11)where π π‘,π ,π β π π‘,π Λ π π ,π Β΄ π π‘,π .We denote the family of stable almost flow (cid:83)(cid:65) πΏ π ,π p π q . If we replace assumption p π, π , π‘ q P T ` by p π, π , π‘ q P T Β΄ , we say that π is a reverse stable almost flow . The following proposition justifies Definition 7.
Theorem 2. If π P (cid:83)(cid:65) πΏ π ,π p V q is a stable almost flow then for any partition π , } π ππ‘,π Β΄ π π‘,π } Lip Δ πΏπ p π π ,π‘ q , @p π , π‘ q P T ` , (12) where πΏ is a constant that depends on π , π ΓΓ πΏ π , ΞΊ , π , q π , π f and πΆ in (11) . Inparticular, the almost flow π satisfies the condition UL, up to changing πΏ π .Remark . When π is a stable almost flow, we assume that πΎ β in Definition 2.This implies that (6) becomes | π π‘,π p π q Β΄ π π‘,π p π q| Δ p ` πΏ π q| π Β΄ π | . (13) Proof.
Let us choose a partition π . Let π P T be fixed and p π , π‘ q P T ` , such that π Δ π , π ππ ,π‘ : β } π ππ‘,π Β΄ π π‘,π Λ π ππ ,π } Lip . Δ π Δ π Δ π‘ Δ π’ Δ π , π ππ ,π’ Δ π ππ‘,π’ ` } π π’,π‘ Λ π ππ‘,π Β΄ π π’,π‘ Λ π π‘,π Λ π ππ ,π } Lip ` } π π’,π‘ Λ π π‘,π Λ π ππ ,π Β΄ π π’,π Λ π ππ ,π } Lip . (14)With the -points control on π π’,π‘ , } π π’,π‘ Λ π ππ‘,π Β΄ π π’,π‘ Λ π π‘,π Λ π ππ ,π } Lip Δ p π π’,π‘ ` } π ππ‘,π Β΄ π π‘,π Λ π ππ ,π } Λ Λ ` } π ππ‘,π } Lip _ p ` πΏ π q} π ππ ,π } Lip Λ ` q π π’,π‘ π ππ ,π‘ . According to (9) (Theorem 1) for π small enough, } π ππ‘,π Β΄ π π‘,π Λ π ππ ,π } Δ πΆπ p π π ,π‘ q . Since y π π’,π‘ is π -compatible and with the control (6) with πΎ β of the Definition 2of almost flow, } π π’,π‘ Λ π ππ‘,π Β΄ π π’,π‘ Λ π π‘,π Λ π ππ ,π } Lip Δ π f p πΆ q π p π π,π‘ q ` } π ππ‘,π } Lip _ p ` πΏ π q} π ππ ,π } Lip Λ ` q π π’,π‘ π ππ ,π‘ . For bounding the last term of (14), (11) yields } π π’,π‘ Λ π π‘,π Λ π ππ ,π Β΄ π π’,π Λ π ππ ,π } Lip Δ πΆπ p π π ,π’ q} π ππ ,π } Lip . Assuming that π , π , π‘ and π’ belong to π and combining these inequalities and thefact the π is stable (see Definition 7), π ππ ,π’ Δ π ππ‘,π’ ` p ` πΏ π q π ππ ,π‘ ` πΏ π p π f p πΆ qp ` πΏ π q ` πΆ q π p π π ,π’ q where πΏ π : β sup p π ,π‘ qP T ` π ,π‘ P π } π ππ‘,π } Lip . For two successive points π and π‘ Δ π of π (See Definition 13), π ππ‘,π β π π‘,π Λ π ππ ,π sothat π ππ ,π‘ β .We assume that π is small enough so that ΞΊ p ` πΏ π ` πΏ π q Δ . From the DavieLemma (Lemma 9 in Appendix), π ππ ,π‘ Δ πΏ π πΌ π π p π π ,π‘ q with πΌ π : β p ` πΏ π q p π f p πΆ qp ` πΏ π q ` πΆ q Β΄ ΞΊ p ` πΏ π ` πΏ π q . (15)12n particular, for π β π , π π,π‘ β } π ππ‘,π Β΄ π π‘,π } Lip .Let us bound πΏ π . For this, with (15) and (6) with πΎ β , πΏ π Δ sup p π ,π‘ qP T ` } π ππ‘,π Β΄ π π‘,π } Lip ` max p π ,π‘ qP T ` } π π‘,π } Lip Δ πΏ π πΌ π π p π ,π q ` ` πΏ π . For π small enough so that πΌ π π p π ,π q Δ { , πΏ π is uniformly bounded. Injectingthis control of πΏ π in (15), } π ππ‘,π Β΄ π π‘,π } Lip Δ πΎπ p π π,π‘ q , @p π, π‘ q P T ` , π, π‘ P π, (16)for some constant πΎ that does not depend on the partition π .It remains to establish (16) for any pair of time p π, π‘ q P T ` .For this, let π‘ π (resp. π π ) be the greatest (resp. smallest) point of π below (resp.above) π‘ (resp. π ). Then with the definition of π π (Notation 2), π ππ‘,π Β΄ π π‘,π β π π‘,π‘ π Λ π ππ‘ π ,π Β΄ π π‘,π‘ π Λ π π‘ π ,π ` π π‘,π‘ π ,π . With (11) and (6) of Definition 2 with πΎ β , } π ππ‘,π Β΄ π π‘,π } Lip Δ p ` πΏ π q} π ππ‘ π ,π Β΄ π π‘ π ,π } Lip ` πΆπ p π π,π‘ q . (17)Similarly, π ππ‘ π ,π Β΄ π π‘ π ,π β π ππ‘ π ,π π Λ π ππ π ,π Β΄ π π‘ π ,π π Λ π π π ,π ` π π‘ π ,π π ,π . Using (9) and (16), } π ππ‘ π ,π Β΄ π π‘ π ,π } Lip Δ } π ππ‘ π ,π π Β΄ π π‘ π ,π π } Lip Β¨ } π π π ,π } Lip ` πΆπ p π π,π‘ qΔ p ` πΏ π q πΎπ p π π,π‘ q ` πΆπ p π π,π‘ q . (18)Inequality (12), which is (16) applied for any p π, π‘ q P T ` , stems from (16), (17)and (18). Corollary 1. If π P (cid:83)(cid:65) πΏ π ,π p V q is a stable almost flow then there exists a uniqueLipschitz flow π in the galaxy containing π . Moreover, there is a constant πΏ Δ such that for all p π , π‘ q P V , } π π‘,π Β΄ π π‘,π } Lip Δ πΏπ p π π ,π‘ q . (19) Proof.
According to Theorem 2, π satisfies the UL condition of Proposition 4.Hence, it converges in the sup-norm to a Lipschitz flow π β π . According toProposition 2, π is the only flow in the galaxy of π .Passing to the limit in (12) leads to (19).13 Perturbations
In [6], we have introduced the notion of perturbation of almost flow. This notionstill gives an almost flow.We recall that π , πΏ π and πΎ are defined in Section 2.1. Definition 8 (Perturbation) . A perturbation is an element π P (cid:70) p V q such that forany p π , π‘ q P T ` and π, π P V , π π‘,π‘ β , (20) } π π‘,π } Δ πΆπ p π π ,π‘ q , (21) | π π‘,π p π q Β΄ π π‘,π p π q| Δ πΏ π | π Β΄ π | ` π p π π ,π‘ q| π Β΄ π | πΎ , (22)where π is defined by (3) and πΆ Δ is a constant. Proposition 5 ([6, Proposition 1]) . If π P (cid:70) p V q is an almost flow and π P (cid:70) p V q is a perturbation, then π : β π ` π is an almost flow in the same galaxy as π . We introduce now the notion of Lipschitz perturbation, which is a perturbation onwhich a control stronger than (22) holds.
Definition 9 (Lipschitz perturbation) . A Lipschitz perturbation is a perturbation π P (cid:70) ` p V q with satisfies for a constant πΆ Δ } π π‘,π } Lip Δ πΆπ p π π ,π‘ q , @p π , π‘ q P T ` . (23)Stable almost flows remain stable almost flows under Lipschitz perturbations. Proposition 6 (Stability of stable almost flow under Lipschitz perturbation) . If π P (cid:83)(cid:65) πΏ π ,π is a stable almost flow (see Definition 7) and π is a Lipschitz perturbation,then π : β π ` π is also a stable almost flow.Proof. It is proved in Proposition 5 that π ` π is an almost flow. Here we showthat π ` π is a stable almost flow.First, for any π, π, π, π P π , | π π‘,π p π q Β΄ π π‘,π p π q Β΄ π π‘,π p π q ` π π‘,π p π q| Δ πΆπ p π π ,π‘ q| π Β΄ π | _ | π Β΄ π | , so that π π‘,π satisfies a π -compatible -points control (see Definition 6) with p π π‘,π : β πΆπ p π π ,π‘ q and q π π‘,π : β . Thus, π ` π satisfies a π -compatible -points control with z π ` π β p π ` πΆπ p π π ,π‘ q and Β π π‘,π ` π π‘,π β | π π‘,π Δ ` πΏ π .14t remains to show that for any p π, π , π‘ q P T ` , } π π‘,π ,π } Lip Δ πΆπ p π π,π‘ q , with π π‘,π ,π : β π π‘,π Λ π π ,π Β΄ π π‘,π . For any π P π , we write π π‘,π ,π p π q β r π π‘,π Λ p π π ,π ` π π ,π qp π q Β΄ π π‘,π Λ π π ,π p π qs looooooooooooooooooooooomooooooooooooooooooooooon I π,π ,π‘ p π q ` r π π‘,π Λ p π π ,π ` π π ,π qp π q Β΄ π π‘,π Λ π π ,π p π qs loooooooooooooooooooooomoooooooooooooooooooooon II π,π ,π‘ p π q ` π π‘,π ,π p π q ` π π‘,π ,π p π q looooooooomooooooooon III π,π ,π‘ p π q . On the one hand, using the π -compatible -points control of π π‘,π , (13), (21) and(23) we write, } I π,π ,π‘ } Lip Δ x π π‘,π p} π π ,π } qp} π π ,π } Lip ` } π π ,π } Lip q ` | π π‘,π } π π ,π } Lip Δ x π π‘,π p πΆπ p π π,π‘ qqp ` πΏ π ` πΆπ p π ,π qq ` p ` πΏ π q πΆπ p π π,π‘ qΔ p πΆ p ` πΏ π ` πΆπ p π ,π qq ` ` πΏ π q π p π π,π‘ q , where πΆ is a constant.On the other hand, with (6), (23) } II π,π ,π‘ } Lip Δ } π π‘,π } Lip p} π π ,π } Lip ` } π π ,π } Lip q ` } π π‘,π } Lip } π π ,π } Lip Δ πΆπ p π π,π‘ qp ` πΏ π ` πΆπ p π ,π qq ` πΆ π p π ,π q π p π π,π‘ q Δ πΎ π π p π π,π‘ q , where πΎ π Γ when π Γ .Finally, with (11) and (23), } III π,π ,π‘ } Lip Δ } π π‘,π ,π } Lip ` } π π‘,π } Lip ` } π π‘,π } Lip } π π ,π } Lip Δ p πΆ ` πΆ π p π ,π qq π p π π,π‘ q . This concludes the proof.Now, we prove another perturbation formula which is useful in Subsection 8.3.We recall the πΏ π , π and πΎ are defined in Section 2.1. Proposition 7.
Let π be a flow which may be decomposed as π π‘,π p π q β π π‘,π p π q ` π π ,π‘ p π q , π P V , p π , π‘ q P T with for any p π , π‘ q P T ` and π, π P V , π π‘,π‘ β i , π π‘,π‘ β , (24) | π π‘,π p π q Β΄ π π‘,π p π q| Δ p ` πΏ π q| π Β΄ π | ` π p π π ,π‘ q| π Β΄ π | πΎ , p π, π q P V , (25) } π π‘,π Β΄ i } Δ πΏ π , (26) } π π ,π‘ } Δ π π p π π ,π‘ q . (27)15 hen π is an almost flow in the same galaxy as π . Besides, for any partition π of T , } π ππ‘,π Β΄ π π‘,π } Δ πΏπ p π π ,π‘ q (28) where πΏ Δ rp ` πΏ π q π ` πΏ π π πΎ s {p Β΄ p ` πΏ π q π Β΄ πΏ π q .Proof. To show that π is an almost flow, it is sufficient to consider (24)-(27) aswell as controlling π π‘,π ,π . For p π, π , π‘ q P T ` , π π‘,π Λ π π ,π p π q β I π,π ,π‘ hkkkkkkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkkkkkkj π π‘,π p π π ,π p π q ` π π ,π p π qq Β΄ π π‘,π p π π ,π p π qq ` π π‘,π p π π ,π p π qq ` π π‘,π p π π ,π p π qq . Since π is a flow, π π,π ,π‘ β and then π π‘,π ,π p π q β I π,π ,π‘ ` π π‘,π p π π ,π p π qq Β΄ π π‘,π p π q . With (25), | I π,π ,π‘ | Δ p ` πΏ π q π π p π π,π q ` π p π π ,π‘ q π πΎ π p π π,π q πΎ . With (3), | I π,π ,π‘ | Δ π΄π p π π,π‘ q with π΄ : β p ` πΏ π q π ` πΏ π π πΎ . It follows that } π π‘,π ,π } Δ p π ` π΄ q π p π π,π‘ q . This proves that π is an almost flowfollowing Definition 3. The control (28) follows from Theorem 1. Corollary 2.
Assume that V is a finite-dimensional Banach space. Let t π π u π P N bea family of flows with decomposition π π β π π ` π π where p π π , π π q satisfy (24) - (27) uniformly in π . Assume moreover that } π ππ ,π‘ } Δ πΏ π‘ Β΄ π , @p π , π‘ q P T ` . (29) Then any possible limit π of π ππ‘,π (at least one exists) satisfies (24) - (27) as wellas (29) with the same constants.Proof. With (29), Lemma 1 in [6] can be applied uniformly. As (25) is also uniformin π , this proves that for any π Δ , t π π p π q π ,π‘ u p π ,π‘ qP T ` , π P π΅ p ,π q is equi-continuouswhere π΅ p , π q is the closed ball of center and some radius π Δ . The Ascoli-ArzelΓ shows that at least one limit of π π exists. Clearly, this limit satisfies thesame properties as π π . 16 Inversion of the flow
In this section, we prove that our definition of stable almost (Definition 7) flow isstable respect to inversion.
Proposition 8.
Let π P (cid:83)(cid:84) πΏ π ,π be a stable almost flow and π the unique flow inthe same galaxy as π (Corollary 1). We assume that π : β π Β΄ i satisfies a -pointcontrol such that @p π , π‘ q P T ` , | π π‘,π β | π π‘,π Β΄ , x π π‘,π β x π π‘,π , and } π π‘,π } Lip Δ πΏ π . Then, π is invertible and p π π ,π‘ q p π ,π‘ qP T ` : β p π Β΄ π‘,π q p π ,π‘ q T ` is a stable reverse almost flowwhich galaxy contains a unique flow which equal to π Β΄ .Proof. According to item (iv) of Lemma 2 and because } π π‘,π } Lip Δ πΏ π we knowthat for π Δ such that πΏ π Δ , π π‘,π is invertible and that π Β΄ π‘,π satisfies a -pointscontrol with } π Β΄ π‘,π β {p Β΄ | π π‘,π q and y π Β΄ π‘,π p π₯ q β x π π‘,π p} π π‘,π } Lip π₯ q} π π‘,π } Lip for any π₯ P R ` .It follows that } π Β΄ π‘,π Δ ` πΏ π , with πΏ π : β πΏ π {p Β΄ πΏ π q and that π π‘,π satisfies a π -compatible -points control.Moreover, } π Β΄ π‘,π } Lip Δ {p Β΄ } π π‘,π } Lip q and we assume } π π‘,π } Lip Δ πΏ π . It follows that } π Β΄ π‘,π } Lip Δ ` πΏ π which proves that (6) holds for π Β΄ . In substituting π by π Β΄ π‘,π p π q in (5) we show that (5) holds for π Β΄ .To prove that p π π ,π‘ q p π ,π‘ qP T ` : β p π Β΄ π‘,π q p π ,π‘ qP T ` is a reverse stable almost flow, it remainsto show that the conditions (7) and (11) hold for any p π, π , π‘ q P T ` . Firstly, wecompute with (7), since π π‘,π Λ π π ,π is one-to-one, } π Β΄ π ,π Λ π Β΄ π‘,π Λ π π‘,π Λ π π ,π Β΄ π Β΄ π‘,π Λ π π‘,π Λ π π ,π } β } π Β΄ π‘,π Λ π π‘,π Β΄ π Β΄ π‘,π Λ π π‘,π Λ π π ,π } Δ p ` πΏ π q} π π‘,π Β΄ π π‘,π Λ π π ,π } Δ π π p π π,π‘ q , which yields with to } π π,π Λ π π ,π‘ Β΄ π π,π‘ } Δ π π p π π,π‘ q .Secondly, for any π, π P V and p π, π , π‘ q P T ` , we set π : β π Β΄ π ,π Λ π Β΄ π‘,π p π q , π : β π Β΄ π ,π Λ π Β΄ π‘,π p π q ,and Ξ¦ π,π ,π‘ : β p π Β΄ π ,π Λ π Β΄ π‘,π Β΄ π Β΄ π‘,π q Λ π π‘,π Λ π π ,π p π q Β΄ p π Β΄ π ,π Λ π Β΄ π‘,π Β΄ π Β΄ π‘,π q Λ π π‘,π Λ π π ,π p π qβ π Β΄ π‘,π Λ π π‘,π p π q Β΄ π Β΄ π‘,π Λ π π‘,π Λ π π ,π p π q Β΄ π Β΄ π‘,π Λ π π‘,π p π q ` π Β΄ π‘,π Λ π π‘,π Λ π π ,π p π q . We know that π Β΄ π‘,π satisfies a π -compatible -points control and we use (11), | Ξ¦ π,π ,π‘ | Δ y π Β΄ π‘,π p} π π‘,π ,π } q r} π π‘,π } Lip _ } π π‘,π Λ π π ,π } Lip s | π Β΄ π | ` } π Β΄ π‘,π } π π‘,π ,π } Lip | π Β΄ π |Δ π Β΄ , f p π q π p π π,π‘ qp ` πΏ π q | π Β΄ π | ` p ` πΏ π q πΆπ p π π,π‘ q| π Β΄ π | . π and π by π Β΄ π ,π Λ π Β΄ π‘,π p π q and π Β΄ π ,π Λ π Β΄ π‘,π p π q , } π Β΄ π ,π Λ π Β΄ π‘,π Β΄ π Β΄ π‘,π } Lip Δ r π Β΄ , f p π q π p π π,π‘ qp ` πΏ π q `p ` πΏ π q πΆπ p π π,π‘ qs} π Β΄ π ,π Λ π Β΄ π‘,π } Lip Δ β π Β΄ , f p π qqp ` πΏ π q ` p ` πΏ π q πΆ β° p ` πΏ π q π p π π,π‘ q . Hence π is a stable reverse almost flow. According to Corollary 1, π π converges to aunique Lipschitz flow π in (cid:70) p V q . But, π ππ ,π‘ β p π ππ‘,π q Β΄ , which yields to π ππ ,π‘ Λ π ππ‘,π β i and passing to limit π π ,π‘ Λ π π‘,π β i . This concludes the proof. Almost flows approximates of flows, similarly to numerical algorithms. In classicalanalysis, flows are strongly related to solutions of ordinary differential equations(ODE). Rough differential equations (RDE) were solve first using fixed pointtheorems on paths [25]. The technical difficulty with this approach is that thesolution itself should be a rough path.Later, A.M. Davie introduced in [14] another notion of solution of RDE which nolonger involves solutions as rough paths, but only as paths. We abstract here thisapproach in order to relate almost flows and paths.
Definition 10 (Generalized solution in the sense of Davie) . Let π be an almostflow. Let π P V and π P T . A π -valued path t π¦ π Γ± π‘ u p π,π‘ qP T is said to be a solutionin the sense of Davie if π¦ π Γ± π β π and there exists a constant πΆ such that | π¦ π Γ± π‘ Β΄ π π‘,π p π¦ π Γ± π q| Δ πΆπ p π π ,π‘ q , @ π Δ π Δ π‘ Δ π. (30) Definition 11 (Manifold of solutions) . A family t π¦ π Γ± Β¨ p π qu π P T , π P V of solutionssatisfying (30) and π¦ π Γ± π p π q β π is called a manifold of solutions . We write π¦ β π p π¦ q to denote the whole family of solutions. Definition 12 (Lipschitz manifold of solutions) . If π ΓΓ π¦ π Γ± Β¨ p π q is uniformlyLipschitz continuous from p V , π q to p (cid:67) pr π, π s , V q , }Β¨} q , then we say that themanifold of solutions is Lipschitz . Remark . When π π‘,π β i ` π π‘,π , then (30) may be written | π¦ π ,π‘ Β΄ π π‘,π p π¦ π q| Δ πΆπ p π π ,π‘ q with π¦ π ,π‘ : β π¦ π Γ± π‘ Β΄ π¦ π Γ± π . for p π, π , π‘ q P T ` This is the form used by A.M. Daviein [14].Flows and manifold of solutions are of closely related. Besides, a manifold ofsolutions is in relation with a whole galaxy. The proof of the next lemma isimmediate so that we skip it. 18 emma 3.
A flow π generates a manifold of solutions to π¦ β π p π¦ q through π¦ π Γ± π‘ p π q : β π π‘,π p π q , p π, π‘ q P T ` , π P V . Besides, π¦ is also solution to π¦ β π p π¦ q forany almost flow π in the galaxy containing π . In a first time, we show how to construct a flow from a suitable family of paths.
Proposition 9.
Consider an almost flow π . Assume that there exists a family t π¦ Γ± π‘ p π qu π‘ P T ,π P V of V -valued paths, continuous in time and Lipschitz continuous inspace such that such that π¦ Γ± β i , } π¦ Γ± π‘ Β΄ π π‘,π p π¦ Γ± π q} Δ πΆπ p π π ,π‘ q , @p π , π‘ q P T ` , sup π‘ P T t} π¦ Γ± π‘ Β΄ i } Lip ` } π¦ Γ± π‘ Β΄ i } u Δ πΎ π where πΎ π ΓΓΓΓ π Γ . Then the family t π¦ Γ± π‘ p π qu π‘ P T ,π P V may be extended to a manifold of solutions to π¦ β π p π¦ q . Besides, if π π ,π‘ p π q : β π¦ π Γ± π‘ p π q , then π is an invertible, Lipschitz flow inthe same galaxy as π .Proof. The Lipschitz inverse mapping shows that π¦ Γ± π‘ is invertible with a Lipschitzcontinuous inverse π¦ Β΄ Γ± π‘ when πΎ π Δ ([1] p. 124).Assuming that π is small enough, we define π π‘,π p π q : β π¦ Γ± π‘ Λ π¦ Β΄ Γ± π p π q for any p π , π‘ q P T and π P V . Clearly, π is a flow which is invertible. Besides, for any p π , π‘ q P T ` , π π ,π‘ is Lipschitz continuous since both π¦ Γ± π‘ and π¦ Β΄ Γ± π are Lipschitz continuous.It remains to prove that π β π . For π P V , let us set π : β π¦ Β΄ Γ± π p π q . Thus, | π π‘,π p π q Β΄ π π‘,π p π q| β | π π‘,π p π¦ Γ± π p π qq Β΄ π π‘,π p π¦ Γ± π p π qq|Δ | π π‘,π p π¦ Γ± π p π qq Β΄ π¦ Γ± π‘ p π q| Δ πΆπ p π π ,π‘ q . Thus, π β π . A stable almost flow π satisfies the condition UL (see Theorem 2), so that thereexists a unique flow π in the same galaxy as π . Furthermore, π is Lipschitz.The flow π generates a manifold of solutions. We show that there exists onlyone such manifold with a Lipschitz continuity result. Note that in the followingproposition, π is not assumed to be stable.19 roposition 10. Let π be a stable almost flow and π be an almost flow.Let π¦ and π§ be two paths from r , π s to V such that | π¦ π‘ Β΄ π π‘,π p π¦ π q| Δ πΎπ p π π ,π‘ q and | π§ π‘ Β΄ π π‘,π p π§ π q| Δ πΎπ p π π ,π‘ q , @p π , π‘ q P T ` . Let us write πΌ π‘,π : β π π‘,π Β΄ π π‘,π and πΌ π‘,π ,π : β π π‘,π ,π Β΄ π π‘,π ,π . Let π , π , π Δ be such thatfor any p π, π , π‘ q P T ` , | πΌ π‘,π ,π p π§ π q| Δ π π p π π,π‘ q , } πΌ π ,π‘ } Lip Δ π and | πΌ π ,π‘ p π§ π q| Δ π . Then there exists a time π small enough and a constant πΆ that depends only on π , πΎ , π ΓΓ πΏ π , ΞΊ and sup p π,π ,π‘ qP T } π π‘,π ,π } Lip { π p π π,π‘ q such that | π¦ π‘ Β΄ π§ π‘ | Δ πΆ p π ` π ` π ` | π¦ Β΄ π§ |q and | π¦ π‘ Β΄ π π‘,π p π¦ π q Β΄ π§ π‘ ` π π‘,π p π§ π q| Δ πΆ p π ` π ` π ` | π¦ Β΄ π§ |q π p π π ,π‘ q for all p π , π‘ q P T . The following corollary is then immediate by applying π β π to Proposition 10. Corollary 3. If π is a stable almost flow, there exists one and only one manifoldof solutions to π¦ β π p π¦ q . Besides, this manifold of solutions is Lipschitz.Remark . As seen in Lemma 3, the notion of manifold of solution is associatedto a galaxy. Hence, a galaxy with a stable almost flow is associated to a uniquemanifold of solutions (actually, we have not proved that if π is a stable almost flow,then the associated flow is also stable). Proof of Proposition 10.
We define π π ,π‘ β | π§ π‘ Β΄ π π‘,π p π§ π q Β΄ π¦ π‘ ` π π‘,π p π¦ π q| , @p π , π‘ q P T . Clearly, π π ,π‘ Δ πΎπ p π π ,π‘ q .For any p π, π , π‘ q P T , π§ π‘ Β΄ π π‘,π p π§ π q Β΄ π¦ π‘ ` π π‘,π p π¦ π qβ π§ π‘ Β΄ π π‘,π p π§ π q Β΄ π¦ π‘ ` π π‘,π p π¦ π q ` π π‘,π p π§ π q Β΄ π π‘,π p π π ,π p π§ π qqΒ΄ π π‘,π p π¦ π q ` π π‘,π p π π ,π p π¦ π qq ` π π‘,π ,π p π§ π q Β΄ π π‘,π ,π p π¦ π qβ π§ π‘ Β΄ π π‘,π p π§ π q Β΄ π¦ π‘ ` π π‘,π p π¦ π q ` πΌ π‘,π p π§ π q Β΄ πΌ π‘,π p π π ,π p π§ π qq` π π‘,π p π§ π q Β΄ π π‘,π p π π ,π p π§ π qq Β΄ π π‘,π p π¦ π q ` π π‘,π p π π ,π p π¦ π qq` πΌ π‘,π ,π p π§ π q ` π π‘,π ,π p π§ π q Β΄ π π‘,π ,π p π¦ π q . πΏ : β sup p π,π ,π‘ qP T } π π‘,π ,π } Lip . With the -points control of π π ,π‘ , π π‘,π Δ π π‘,π ` q π π‘,π π π,π‘ ` πΏ | π§ π Β΄ π¦ π | π p π π,π‘ q` p π π‘,π p| π§ π Β΄ π π ,π p π§ π q| _ | π¦ π Β΄ π π ,π p π¦ π q|q Β¨ p| π§ π Β΄ π¦ π | _ | π π ,π p π§ π q Β΄ π π ,π p π¦ π q|q` | πΌ π‘,π ,π p π§ π q| ` } πΌ π‘,π } Lip | π§ π Β΄ π π ,π p π§ π q| . Since q π π‘,π Δ ` πΏ π and } π π‘,π } Lip Δ ` πΏ π , π π‘,π Δ π π‘,π ` p ` πΏ π q π π,π‘ ` πΏ } π§ Β΄ π¦ } π p π π,π‘ q` p π π‘,π p πΎπ p π π ,π qq ` p ` πΏ π q} π§ Β΄ π¦ } ` π q ` p π ` π πΎ q π p π π,π‘ q . Since p π π‘,π is π -compatible (see Defintion 6), p π π‘,π p πΎπ p π π,π qq Δ Ξ¦ p πΎ q π p π π,π‘ q so that π π,π‘ Δ π π ,π‘ ` p ` πΏ π q π π,π‘ ` π΅π p π π,π‘ q (31)with π΅ : β p πΏ ` p ` πΏ π q Ξ¦ p πΎ qq} π¦ Β΄ π§ } ` π ` π πΎ ` π Ξ¦ p πΎ qq . (32)Owing to (31)-(32), from the continuous time version of the Davieβs Lemma 10, for π small enough (depending only on ΞΊ and π ΓΓ πΏ π ), for all p π, π‘ q P T , π π,π‘ Δ π΅πΆπ p π π,π‘ q with πΆ : β ` πΏ π Β΄ p ΞΊ p ` πΏ π q ` πΏ π q . For any π‘ P r , π s , since } π π‘, } Lip Δ ` πΏ π , | π¦ π‘ Β΄ π§ π‘ | Δ | π¦ π‘ Β΄ π§ π‘ Β΄ π π‘, p π¦ q ` π π‘, p π§ q| ` | π π‘, p π§ q Β΄ π π‘, p π§ q| ` | π π‘, p π§ q Β΄ π π‘, p π¦ q|Δ π ,π‘ ` π ` p ` πΏ π q| π¦ Β΄ π§ | . With the expression of π΅ in (32), for any π‘ P r , π s , we see that there existsconstants π΄ and π΄ that depend only on ΞΊ , πΏ , πΏ π , πΎ and Ξ¦ p πΎ q such that | π¦ π‘ Β΄ π§ π‘ | Δ π΄ } π¦ Β΄ π§ } π p π ,π q ` π΄ p π ` π ` π q π p π ,π q ` π ` p ` πΏ π q| π¦ Β΄ π§ | . Choosing π small enough so that π΄π p π ,π q Δ { implies that } π¦ Β΄ π§ } Δ π΄ p π ` π ` π q π p π ,π q ` π ` p ` πΏ π q| π¦ Β΄ π§ | . This concludes the proof. 21
Application to Rough differential equation
In this section, we show how our framework allows us to link the different flowbased approaches. The key is to show that Friz-Victoirβs and Bailleulβs almostflows are different perturbation of the Davieβs almost flow.We start by giving some notations. We did not recall notions of the rough paththeory. The reader can find a clear introduction in [23] and in [18].In this section, the remainder introduced in Section 2.1 is of the type π p πΏ q β πΏ p ` πΎ q{ π , @ πΏ Δ . with πΎ P p , s and a real number π Δ satisfying ` πΎ Δ π . Before showing the link between the different based flow approaches, we set notationsof classical objects of the rough path theory.Given another Banach space p U , |Β¨|q and a real number π Δ , let us denote by (cid:67) π Β΄ π p T , U q the space of { π -HΓΆlder paths controlled by π , which we equip we thesemi-norm } π₯ } π : β sup p π ,π‘ qP T ` ,π β° π‘ | π₯ π ,π‘ | π { ππ ,π‘ , this quantity being bounded by definition.We define also (cid:67) π Β΄ var pr π , π‘ s , U q the space of bounded π -variation paths from r π , π‘ s to U which we equip with the π -variation semi-norm on r π , π‘ s denoted by } π₯ } r π ,π‘ s ,π .Moreover, if π₯ P (cid:67) π Β΄ π pr , π s , U q , then π₯ P (cid:67) π Β΄ var pr π , π‘ s , U q and } π₯ } r π ,π‘ s ,π Δ } π₯ } π π { ππ ,π‘ . We denote by t Β¨ u the floor function.For an integer π Δ t π u , let (cid:84) π,π p U q be the space of { π -HΓΆlder rough path controlledby π of order π . If x P (cid:84) π,π p U q we denote by x p π q the component of x in U b π with Δ π Δ π an integer and π π p x q : β Ε ππ β x p π q . Obviously, x β π π p x q . We denotethe homogeneous semi-norm } x } π : β sup π Δ π sup r π ,π‘ sP T ,π β° π‘ | x p π q π ,π‘ | π π { ππ ,π‘ , (cid:84) π p π q : β (cid:84) π, t π u p π q .For π Δ , we denote πΊ π p U q the free nilpotent group (Chapter in [19]).Let (cid:71) π p U q : β (cid:67) π Β΄ π pr , π s , πΊ t π u p U qq be the set of weak-geometric rough paths offinite { π -HΓΆlder rough path controlled by π with values in U .When U β R β ( β Δ an integer). For any multi-indice πΌ : β p π , . . . , π π q P t , . . . , β u π we set | πΌ | : β π and π πΌ : β π π b Β¨ Β¨ Β¨ b π π π where t π , . . . , π β u is the canonical basis of R β . If x P (cid:84) π p R β q . If x P (cid:84) π p R β q , then x πΌ denote the coordinates of x p π q in thebasis p π πΌ q | πΌ |β π . It follows that π π p x q β Ε | πΌ |Δ π x πΌ π πΌ . If π₯ P (cid:67) Β΄ var p T , R π q then forany integer π Δ , π π p π₯ q β ΓΏ | πΌ |Δ π π₯ πΌ π πΌ , where π₯ πΌπ ,π‘ : β Ε π Δ π‘ π ΔΒ¨Β¨Β¨Δ π‘ Δ π‘ d π₯ π π π‘ π . . . d π₯ π π‘ . Let us consider now a π -rough path x P (cid:84) π p U q with Δ π Δ for a Banach space U .A Rough Differential Equation (RDE) is a solution π¦ taking its values in anotherBanach space V to π¦ π‘ β π ` ΕΌ π‘π π p π¦ π’ q d x π’ , @p π , π‘ q P T ` , (33)provided that π : V Γ πΏ p U , V q is regular enough.Existence of solution to (33) was proved first by T. Lyons using a Picard fixedpoint theorem [25]. In [14], A.M. Davie provided an alternative approach based onEuler-type schemes. Over the approach of T. Lyons, it has the advantage that thesolution is sought as a path with values in V and not in the tensor space T p U β V q .For π P (cid:67) π p V , πΏ p U , V qq , we define π π‘,π p π q β π ` π p π q x p q π ,π‘ ` d π p π q Β¨ π p π q x p q π ,π‘ , (34)where d π p π q is the differential of π in π . Definition 10 coincides with the onedefined by A. M. Davie in [14] for the notion of solution to (33).The following lemma is a generalization in an infinite dimensional setting ofTheorems 3.2 and 3.3 in [14] with a bounded function π . Lemma 4.
Let π P r , q and πΎ Δ π Β΄ . (i) If π P (cid:67) ` πΎπ , then the family π defined by (34) is an almost flow. If π is of class (cid:67) ` πΎπ , then π is a stable almost flow.Proof. According to Proposition 5 in [6], π is an almost flow as soon as π P (cid:67) ` πΎπ with ` πΎ Δ π .We now assume that π P (cid:67) ` πΎπ . We show that π verifies a π -compatible -pointscontrol (see Definition 6). For any π, π, π, π P V , p π , π‘ q P T , we compute π π‘,π p π q Β΄ π π‘,π p π q Β΄ π π‘,π p π q ` π π‘,π p π q β π Β΄ π Β΄ π ` π ` r π p π q Β΄ π p π q Β΄ π p π q ` π p π qs x p q π ,π‘ looooooooooooooooooomooooooooooooooooooon I π ,π‘ ` r d π Β¨ π p π q Β΄ d π Β¨ π p π q Β΄ d π Β¨ π p π q ` d π Β¨ π p π qs x p q π ,π‘ loooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooon II π ,π‘ . (35)Then, we apply Lemma 1 to π P (cid:67) π and to d π Β¨ π P (cid:67) ` πΎπ to obtain | I π ,π‘ | Δ } x p q } π π { ππ ,π‘ } d π } p| π Β΄ π | _ | π Β΄ π |q | π Β΄ π |` } x p q } π π { π ,π } d π } | π Β΄ π Β΄ π ` π | , (36)and | II π ,π‘ | Δ } x p q } π π { ππ ,π‘ Β« } d p d π Β¨ π q} πΎ p| π Β΄ π | _ | π Β΄ π |q πΎ | π Β΄ π |` } d p d π Β¨ π q} | π Β΄ π Β΄ π ` π | ο¬ . (37)Combining (35), (36) and (37) we set for all π¦ P R ` , x π π‘,π p π¦ q : β π β π { ππ ,π‘ π¦ ` π { ππ ,π‘ π¦ πΎ Δ± where π : β } x p q } π } d π } ` } x p q } π } d p d π Β¨ π q} πΎ and | π π‘,π : β ` } x p q } π π { π ,π ` } d p d π Β¨ π q} } x p q } π π { π ,π . It follows that | π π‘,π Δ ` πΏ π and that π π‘,π is π -compatible. Indeed, for πΌ P R ` , x π π‘,π Β΄ πΌπ p ` πΎ q{ ππ ,π‘ Β― Δ π p πΌ _ πΌ πΎ q Β΄ π p ` πΎ q{ ππ ,π‘ ` π p ` πΎ ` πΎ q{ ππ ,π‘ Β― Δ π p πΌ _ πΌ πΎ q Β΄ π p ` πΎ q{ π ,π ` π p πΎ ` πΎ q{ π ,π Β― π p ` πΎ q{ ππ ,π‘ Δ πΏ π π p π π ,π‘ q . It remains to show that (11) holds. As π π‘,π P (cid:67) ` πΎ , thus the two semi-norms } π π‘,π } Lip and } d π π‘,π } are equivalent. We recall that π π‘,π ,π β π π‘,π Λ π π ,π Β΄ π π‘,π . For24ny π P V and p π, π , π‘ q P T ` , d π π‘,π ,π p π q βp d π π ,π p π q d π Λ π π ,π p π q Β΄ d π p π qq x p q π ,π‘ Β΄ d p d π Β¨ π qp π qp x p q π ,π‘ Β΄ x p q π,π b x p q π ,π‘ q` d π π ,π p π q d p d π Β¨ π q Λ π π ,π p π q x p q π ,π‘ β ` Β΄ d π p π q ` d π π ,π p π q d π Λ π π ,π p π q Β΄ d p d π Β¨ π qp π q x π,π Λ x p q π ,π‘ loooooooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooooooon I π,π ,π‘ Β΄ p d p d π Β¨ π qp π q Β΄ d π π ,π p π q d p d π Β¨ π q Λ π π ,π p π qq x π ,π‘ looooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooon II π,π ,π‘ . (38)Each term is estimated separately. For the first one, | I π,π ,π‘ | Δ| d π Λ π π ,π p π q Β΄ d π p π q Β΄ d π p π qp π π ,π p π q Β΄ π q|| x p q π ,π‘ |` | d π p π q d π p π q x p q π,π x π ,π‘ | ` | d p d π Β¨ π qp π q d π Λ π π ,π p π q x p q π,π x p q π ,π‘ |` | d π p π qr d π Λ π π ,π p π q Β΄ d π p π qs x π,π b x π ,π‘ Δ} d π } πΎ | π π ,π p π q Β΄ π | ` πΎ | x p q π ,π‘ | ` p} d π } ` } d p d π Β¨ π q d π } q| x p q π,π || x p q π ,π‘ |` } d π } } d π } | π π ,π p π q Β΄ π || x π,π || x π ,π‘ |Δ π p π p ` πΎ q{ ππ,π‘ ` π { ππ,π‘ q Δ π p ` π p Β΄ πΎ q{ π ,π q π p ` πΎ q{ ππ,π‘ , (39)where π is a constant which depends on π , x , πΎ .The second one is more simple, | II π,π ,π‘ | Δ| d p d π Β¨ π q Λ π π ,π p π q Β΄ d p d π Β¨ π qp π q|| x p q π ,π‘ | ` | d π π ,π p π q Β΄ | d p d π Β¨ π q Λ π π ,π p π q x π ,π‘ Δ} d p d π Β¨ π q} | π π ,π p π q Β΄ π | πΎ | x p q π ,π‘ | ` } d p d π Β¨ π q} | d π π ,π p π q Β΄ || x p q π ,π‘ |Δ π p π p ` πΎ q{ ππ,π‘ ` π { ππ,π‘ q Δ π p ` π p Β΄ πΎ q{ π ,π q π p ` πΎ q{ ππ,π‘ . (40)Finally, combining (38), (39) and (40), } d π π‘,π ,π } Δ p π ` π qp ` π p Β΄ πΎ q{ π ,π q π p π π,π‘ q with π p π π,π‘ q β π p ` πΎ q{ ππ,π‘ . This proves (11) and that π is a stable almost flow.Combining Lemma 4, Proposition 2, Corollaries 1 and 3 leads to the followingresult. Corollary 4. If π is of class (cid:67) ` πΎπ , then π π converges to a unique Lipschitz flow π .Moreover, if π is an almost flow in a galaxy containing π , then, π π converges to π .Besides, there exists a unique manifold of solutions to π¦ β π p π¦ q which is Lipschitz. .3 Almost flows constructed from sub-Riemanniangeodesics, as in P. Friz and N. Victoir In [19], P. Friz and N. Victoir proposed an approach based on the use of geodesics.The following proposition is one of the fundamental result of their framework.Now, we assume that U β R β . Proposition 11 (Remark 10.10, [19, p. 216]) . Let π Δ a real number and aninteger π Δ t π u . For any x P (cid:67) π Β΄ π pr , π s , πΊ π p R β qq and any p π , π‘ q P T , there existsa path π₯ π ,π‘ P (cid:67) Β΄ var p R β q defined on r π , π‘ s such that π π p π₯ π ,π‘ q π ,π‘ β x π ,π‘ and } π₯ π ,π‘ } r π ,π‘ s , Δ πΎ } x } π π { ππ ,π‘ Δ πΎ } π t π u p x q} π π { ππ ,π‘ . for some universal constant πΎ (resp. πΎ ) that depends only on π ( π and π ). Wesay that π₯ π ,π‘ is a geodesic path associated to x .Remark . If π₯ P (cid:67) Β΄ var pr π , π‘ s , R β q , then } π₯ } r π ,π‘ s , β Ε π‘π | d π₯ π | .For notational convenience, we prefer now to express differential equations withrespect to vector fields, that is a family of functions ΓΓ π : β pΓΓ π , . . . , ΓΓ π β q that actson (cid:67) p V , πΏ p U , V qq . Therefore for π₯ P (cid:67) Β΄ var p R ` , R β q , the equation π§ π‘ β π ` Ε π‘ ΓΓ π i p π§ π q d π₯ π is equivalent to π¦ π‘ β π ` Ε π‘ π p π§ π q d π₯ π with π β ΓΓ π i . For a multi-indice πΌ : β p π , . . . , π π q P t , . . . , β u π and p π , π‘ q P T we denote ΓΓ π πΌ : β ΓΓ π π . . . ΓΓ π π π . Byconvention, ΓΓ π H i : β i . We employ the Einstein convention of summation.Let us fix π Δ in ΓΓ π i P (cid:67) π Β΄ for π Δ π . Let x be a rough path with values in T π p U q and π be a vector field such that ΓΓ π i P (cid:67) π Β΄ . Let us define π p π q π‘,π r x , ΓΓ π sp π q : β π ΓΏ π β ΓΏ | πΌ |β π ΓΓ π πΌ i p π q x πΌπ ,π‘ , @p π , π‘ q P T ` . (41)The next proposition summarizes various results on RDEs (Theorem 10.26 in [19,p. 233], Theorem 10.30 in [19, p. 238]). When ΓΓ π i P (cid:67) ` πΎ b but no in (cid:67) ` πΎ b with ` πΎ Δ π , several solutions to the RDE (33) may exist (See Example 2 in [14]). Theorem 3.
Assume that U and V are finite dimensional Banach spaces. Choose π Δ as well as an integer π with Δ π Δ t π u . Let x be a π -rough paths withvalues in T π p V q . Let us assume that ΓΓ π i P (cid:67) π Β΄ for a vector field ΓΓ π : V Γ πΏ p U , V q .It holds that (i) When π Δ π , there exists a flow π r x , ΓΓ π s in the same galaxy as π p π q r x , ΓΓ π s , When ` π Δ π , then there exists a unique flow as well as a unique Lipschitzmanifold of solutions to π¦ β π p π q1 p π¦ q ( π p π q is defined in (41) ).In addition, for any partition π β t π‘ π u ππ β , | π π‘,π r x , ΓΓ π sp π q Β΄ π p π q ππ‘,π r x , ΓΓ π s| Δ πΆ } x } π sup π β ,...,π Β΄ π π ` πΎπ Β΄ π‘ π ,π‘ π ` , (42) with πΎ : β min t π Β΄ π, u and a constant πΆ that depends on π ,π , } x } π and π . From the next lemma, we obtain that of ΓΓ π i P (cid:67) π b with π Δ π , then there exists aunique flow associated to π p π q r x , ΓΓ π s . Lemma 5.
For any π Δ , π p π q r x , ΓΓ π s belongs to the same galaxy as π p q r x , ΓΓ π s ,which the Davie expansion given by (34) with π β ΓΓ π i .Proof. Let us write for π Δ , π p π q π ,π‘ r x , ΓΓ π s : β π p π q π ,π‘ r x , ΓΓ π s Β΄ π p q π ,π‘ r x , ΓΓ π s β π ΓΏ π β ΓΏ | πΌ |Δ π ΓΓ π πΌ i p π q x πΌπ ,π‘ , @p π , π‘ q P T ` . Clearly, π p π q r x , ΓΓ π s is a perturbation with π p πΏ q β πΏ { π . Moreover, as ΓΓ π i P (cid:67) π Β΄ , ΓΓ π πΌ i P (cid:67) π Β΄ π b for any word πΌ with | πΌ | β π so that when π Β΄ π Δ , π p π q r x , ΓΓ π s is aLipschitz perturbation (Definition 5).This is however not sufficient to obtain the rate in (42).For a path π₯ P (cid:67) Β΄ var , we denote for π P V by π Β¨ ,π r π₯, ΓΓ π sp π q the unique solution to π π‘,π r π₯, ΓΓ π sp π q β π ` ΕΌ π‘π ΓΓ π i p π π,π r π₯, ΓΓ π sp π qq d π₯ π , π‘ Δ π . (43)The family π r π₯, ΓΓ π s satisfies the flow property.We set for any p π , π‘ q P T ` , π p π q π‘,π r π₯, ΓΓ π sp π q : β π p π q π‘,π r π π p π₯ q , ΓΓ π sp π q ,π p π q π‘,π r π₯, ΓΓ π sp π q β ΓΏ | πΌ |β π ΕΌ π Δ π‘ π Β΄ ΔΒ¨Β¨Β¨Δ π‘ Δ π‘ Β΄ ΓΓ π πΌ i p π π‘ π ,π r π₯, ΓΓ π sp π qq Β΄ ΓΓ π πΌ i p π q Β― d π₯ πΌπ‘ π , where d π₯ πΌπ‘ π β d π₯ π π π‘ π . . . d π₯ π π‘ . Using iteratively the Newton formula on (43), π π‘,π r π₯, ΓΓ π sp π q β π p π q π‘,π r π₯, ΓΓ π sp π q ` π p π q π‘,π r π₯, ΓΓ π sp π q .
27e denote by } π } πΎ the πΎ -HΓΆlder semi-norm of a function π . For πΎ β , this is theLipschitz semi-norm. Thus, πΎ : β min t π Β΄ π, u is the HΓΆlder indice of ΓΓ π πΌ i with | πΌ | β π .From Proposition 10.3 in [19, p. 213], for a constant πΆ that depends on π and on } π } Λ : β max | πΌ |Δ π }ΓΓ π πΌ i } ` max | πΌ |Δ π Β΄ }ΓΓ π πΌ i } Lip ` max | πΌ |β π }ΓΓ π πΌ i } πΎ , it holds that | π p π q π‘,π r π₯, ΓΓ π sp π q| Δ πΆ } π₯ } π ` πΎ r π ,π‘ s , . (44) Lemma 6.
Let π₯ be a path of finite -variation with } π₯ } r π ,π‘ s , Δ π΄π π ,π‘ for any p π , π‘ q P T ` . Let π₯ π ,π‘ be the geodesic path given by Proposition 11. Assume that thereexists a constant πΎ such that } π₯ π ,π‘ } r π ,π‘ s , Δ πΎπ { ππ ,π‘ . Then there exists a time π Δ small enough and a constant π· that depend only on π , πΎ and }ΓΓ π i } βΉ such that } π π‘,π r π₯ π ,π‘ , ΓΓ π s Β΄ π π‘,π r π₯, ΓΓ π s} Δ π·π π ` πΎπ π ,π‘ , @p π , π‘ q P T ` . In particular, the choice of π and π· does not depend on π΄ .Proof. Let us assume that } π₯ } r π ,π‘ s , Δ π΄π π ,π‘ and } π₯ π ,π‘ } r π ,π‘ s , Δ πΎπ { ππ ,π‘ .Let π₯ π,π ,π‘ be the concatenation of π₯ π,π and π₯ π ,π‘ . Then π π‘,π r π₯, ΓΓ π sp π q Β΄ π π‘,π r π₯ π,π‘ , ΓΓ π sp π q β π π‘,π r π₯, ΓΓ π sp π π ,π r π₯, ΓΓ π sp π qq Β΄ π π‘,π r π₯ π,π ,π‘ , ΓΓ π sp π q loooooooooooooooooooooooooomoooooooooooooooooooooooooon I π,π ,π‘ ` π π‘,π r π₯ π,π ,π‘ , ΓΓ π sp π q Β΄ π π‘,π r π₯ π,π‘ , ΓΓ π sp π q looooooooooooooooooooomooooooooooooooooooooon II π,π ,π‘ . Since π₯ π,π ,π‘ is the concatenation between two paths, π π‘,π r π₯ π,π ,π‘ , ΓΓ π sp π q β π π‘,π r π₯ π ,π‘ , ΓΓ π sp π π ,π r π₯ π,π , ΓΓ π sp π qq . Thus, | I π,π ,π‘ | Δ | π π‘,π r π₯, ΓΓ π sp π π ,π r π₯, ΓΓ π sp π qq Β΄ π π‘,π r π₯ π ,π‘ , ΓΓ π sp π π ,π r π₯, ΓΓ π sp π qq|` | π π‘,π r π₯ π ,π‘ , ΓΓ π sp π π ,π r π₯, ΓΓ π sp π qq Β΄ π π‘,π r π₯ π ,π‘ , ΓΓ π sp π π ,π r π₯ π,π , ΓΓ π sp π qq| . Writing π π,π‘ : β } π π‘,π r π₯, ΓΓ π s Β΄ π π‘,π r π₯ π,π‘ , ΓΓ π s} , it holds that | I π ,π,π‘ | Δ π π‘,π ` } π π‘,π r π₯ π ,π‘ , ΓΓ π s} Lip π π,π . π‘ Δ π , } π π‘,π r π₯ π ,π‘ , ΓΓ π s} Lip Δ ` }ΓΓ π i } Lip ΕΌ π‘π } π π,π r π₯ π ,π‘ , ΓΓ π s} Lip | d π₯ π ,π‘π |Δ ` }ΓΓ π i } Lip ΕΌ π‘π } π π,π r π₯ π ,π‘ , ΓΓ π s} Lip | π₯ π ,π‘π | d π, (45)where the derivative π₯ π ,π‘ is almost everywhere defined because π₯ π ,π‘ P (cid:67) Β΄ var p R β q .Then, using the GrΓΆnwallβs inequality with (45) and Proposition 11, there is constant πΆ that depends only on πΎ (defined in Proposition 11), } x } π and }ΓΓ π i } Lip such that } π π‘,π r π₯ π ,π‘ , ΓΓ π s} Lip Δ exp p πΆπ { π ,π q . Besides, π π p π₯ π,π ,π‘ q π,π‘ β π π p π₯ π,π q π,π b π π p π₯ π ,π‘ q π ,π‘ β x π,π b x π ,π‘ β x π,π‘ β π π p π₯ π,π‘ q π,π‘ . It follows that π p π q r π₯ π,π ,π‘ , ΓΓ π s β π p π q r π₯ π,π‘ , ΓΓ π s . Thus, | II π,π ,π‘ | β | π p π q π‘,π r π₯ π,π ,π‘ , ΓΓ π sp π q Β΄ π p π q π‘,π r π₯ π,π‘ , ΓΓ π sp π q|Δ } π p π q π‘,π r π₯ π,π ,π‘ , ΓΓ π s} ` } π p π q π‘,π r π₯ π,π‘ , ΓΓ π s} Δ πΆ π π ` πΎπ π ,π‘ , where πΆ Δ is a new constant and using (44) and Proposition 11 for the lastestimation.Thus, π π,π‘ Δ π π ,π‘ ` exp p πΆπ { π ,π q π π,π ` πΆ π π ` πΎπ π ,π‘ . On the other hand, when π π ,π‘ Δ , π π ,π‘ β } π π‘,π r π₯, ΓΓ π s Β΄ π π‘,π r π₯ π ,π‘ , ΓΓ π s} Δ } π π‘,π r π₯, ΓΓ π s Β΄ π p π q π‘,π r π₯, ΓΓ π s} ` } π π‘,π r π₯ π ,π‘ , ΓΓ π s Β΄ π p π q π‘,π r π₯ π ,π‘ , ΓΓ π s} Δ πΆ max t π΄ π ` πΎ π π ` πΎπ ,π‘ , πΎ π ` πΎπ π π ` πΎπ π ,π‘ uΔ π΅π π ` πΎπ π ,π‘ with π΅ : β πΆ max t π΄ π ` πΎ , πΎ π ` πΎπ u . From the continuous time Davie lemma (Lemma 10 in Appendix), there exists aconstant π· that does not depend on π΅ (hence on π΄ ) such that π π ,π‘ Δ π·π π ` πΎπ π ,π‘ . Lemma 7.
Let π₯ P (cid:67) Β΄ var be as in Lemma 7. Then } π p π q π‘,π r π₯, ΓΓ π s} β } π π‘,π r π₯, ΓΓ π s Β΄ π p π q π‘,π r π₯, ΓΓ π s} Δ πΈπ π ` πΎπ π ,π‘ , where πΈ depends only on π , π , and πΎ . roof. Since π p π q π‘,π r π₯, ΓΓ π s β π p π q π‘,π r π₯ π ,π‘ , ΓΓ π s , π p π q π‘,π r π₯, ΓΓ π s : β π π‘,π r π₯, ΓΓ π s Β΄ π p π q π‘,π r π₯, ΓΓ π s β π π‘,π r π₯, ΓΓ π s Β΄ π p π q π‘,π r π₯, ΓΓ π s ` π p π q π‘,π r π₯, ΓΓ π s . The results is then an immediate consequence of (44) and Lemma 6.
Proof of Theorem 3.
We recall that we assume that U and V are finite dimensionalBanach spaces. For any p π, π q P V , any p π , π‘ q P T ` , | π p π q π‘,π r π₯, ΓΓ π sp π q Β΄ π p π q π‘,π r π₯, ΓΓ π sp π q|Δ π Β΄ ΓΏ π β ΓΏ | πΌ |β π }ΓΓ π πΌ i } Lip | π₯ πΌπ ,π‘ | Β¨ | π Β΄ π | ` ΓΏ | πΌ |β π }ΓΓ π πΌ i } πΎ Β¨ | π₯ πΌπ ,π‘ || π Β΄ π | πΎ Δ π Β΄ ΓΏ π β }ΓΓ π π i } Lip } π₯ } π r π ,π‘ s , Β¨ | π Β΄ π | ` }ΓΓ π π i } πΎ } π₯ } π r π ,π‘ s , | π Β΄ π | πΎ . It then follows from Proposition 7 that π p π q r π₯, ΓΓ π s is an almost flow with βΊβΊβΊ π p π q π‘,π r π₯, ΓΓ π sp π p π q π ,π r π₯, ΓΓ π sp π qq Β΄ π p π q π‘,π r π₯, ΓΓ π sp π q βΊβΊβΊ Δ πΏπ π ` πΎπ π ,π‘ , @p π, π , π‘ q P T ` , for a constant πΏ that depends only on } π₯ } r π ,π‘ s , and of } π } Λ .Let p π₯ π q π P N be a sequence of bounded variation paths such that π π p π₯ π q ΓΓΓΓ π Γ8 x uniformly on r , π s and such that sup π P N } π π p π₯ π q} π Δ π } x } π for a uniform constant π in π . Such a sequence exists according to Remark 10.32 in [19]. It consists inconcatenating the geodesic approximations given by Proposition 11. From this, | π π p π₯ π q π ,π‘ | Δ πΎπ { ππ ,π‘ with πΎ β π } x } π .Clearly, π p π q π‘,π r π₯ π , ΓΓ π sp π q converges to π p π q π‘,π r x , ΓΓ π s for any p π , π‘ q P T ` and any π P V .The result follows from Corollary 2 and Lemma 7. For π P V , the solution to the ordinary differential equation π¦ π‘ p π q β π ` ΕΌ π‘ ΓΓ π i p π¦ π p π qq d π is a path from r , π s to V such that π p π¦ π‘ p π qq β π p π q ` ΕΌ π‘ π π p π¦ π p π qq d π (46)30or any π P (cid:67) p V , V q . Assuming enough regularity on both π and ΓΓ π , we iterate (46)so that π p π¦ π‘ p π qq β π p π q ` π‘π π p π q ` Β¨ Β¨ Β¨ ` π‘ π π ! π π π p π q ` π p π π π, π ; π‘ q with ΓΓ π π β π , ΓΓ π π ` π β ΓΓ π pΓΓ π π π q , π β , , . . . and π π p π, π ; π‘ q β ΕΌ π‘ ΕΌ π‘ Β¨ Β¨ Β¨ ΕΌ π‘ π Β΄ ` π p π¦ π‘ π p π qq Β΄ π p π q Λ d π‘ π Β¨ Β¨ Β¨ d π‘ . for a function π : V Γ V . Lemma 8. If ΓΓ π i is uniformly Lipschitz, then for any π P V and any π‘ Δ . | π¦ π‘ p π q Β΄ π | Δ π‘ |ΓΓ π i p π q| exp p}ΓΓ π i } Lip π‘ q . (47) Moreover, if ΓΓ π i satisfies a 4-points control, then for any π, π P V and any π‘ Δ , β π‘ p π, π q : β | π¦ π p π q Β΄ π Β΄ π¦ π p π q ` π |Δ π‘ x ΓΓ π i p πΌ π‘ p π, π qq exp Λ p x ΓΓ π i p πΌ π‘ p π, π qq ` | ΓΓ π i q π‘ Λ | π Β΄ π | (48) with πΌ π‘ p π, π q : β sup π Pr ,π‘ s | π¦ π p π q Β΄ π | _ | π¦ π p π q Β΄ π | Δ π‘ ` p|ΓΓ π i p π q| _ |ΓΓ π i p π q|q exp p}ΓΓ π i } Lip π‘ q Λ . (49) In particular, if ΓΓ π i satisfies a 4-points control and is bounded, then π ΓΓ π¦ p π q isLipschitz from V to p (cid:67) pr , π s , V q , }Β¨} q .Proof. Let us write π£ : β ΓΓ π i P (cid:67) p V , V q . Since π¦ π‘ p π q Β΄ π β ΕΌ π‘ p π£ p π¦ π p π qq Β΄ π£ p π qq d π ` π‘π£ p π q , (50)an immediate application of the Gronwall lemma gives (47).Since π£ satisfies a 4-points control, for π, π P V , with β π p π, π q : β | π¦ π p π q Β΄ π Β΄ π¦ π p π q ` π | , | π£ p π¦ π p π qq Β΄ π£ p π q Β΄ π£ p π¦ π p π qq ` π£ p π q|Δ p π£ p| π¦ π p π q Β΄ π | _ | π¦ π p π q Β΄ π |q| π¦ π p π q Β΄ π¦ π p π q| _ | π Β΄ π | ` q π£ β π p π, π q . (51)Besides, | π¦ π p π q Β΄ π¦ π p π q| Δ | π Β΄ π | ` β π p π, π q . Injecting (51) into (50) shows that β π‘ p π, π q Δ πΌ π‘ p π, π q π‘ p π£ p πΌ π‘ p π, π qq| π Β΄ π | ` p p π£ p πΌ π‘ p π, π qq ` q π£ q ΕΌ π‘ β π p π, π q d π with πΌ π‘ p π, π q given by (49). Again, the Gronwall lemma yields (48).31 .4.2 Bailleulβs approach by truncated logarithmic series Here U β R β for a dimension β Δ .Let π : β pΓΓ π , . . . , ΓΓ π β q be a family of vector fields which acts on πΆ p V , V q and x P (cid:71) π p R β qq be a weak-geometric π -rough path with Δ π Δ . By definition of theweak geometric rough paths, x π,π ` x π,π β x π x π for any π, π P t , . . . , β u . We denoteby rΓΓ π π , ΓΓ π π s : β ΓΓ π π ΓΓ π π Β΄ ΓΓ π π ΓΓ π π , the Lie bracket of vector fields ΓΓ π π and ΓΓ π π . The Liebracket is itself a vector field.Assuming that π is smooth, we define for any p π , π‘ q P T , πΌ P T and π P V , thesolution p πΌ, π q ΓΓ π¦ π ,π‘ p πΌ, π q of the ODE, π¦ π ,π‘ p πΌ, π q β π ` ΕΌ πΌ ΓΓ π π i p π¦ π ,π‘ p π½, π qq x ππ ,π‘ d π½ ` ΕΌ πΌ rΓΓ π π , ΓΓ π π s i p π¦ π ,π‘ p π½, π qq x π,ππ ,π‘ d π½, (52)where we omit the summation over all indice π, π P t , . . . , β u . We write π π‘,π p π q : β π¦ π ,π‘ p , π q β π π ,π‘ ` π π‘,π , (53)where, by iterating (52), π π‘,π p π q : β π ` ΓΓ π π i p π q x ππ ,π‘ ` ΓΓ π π ΓΓ π π i p π q x ππ ,π‘ x ππ ,π‘ ` rΓΓ π π , ΓΓ π π s i p π q x π,ππ ,π‘ ,π π‘,π p π q : β ΕΌ ΕΌ π½ ΓΓ π π ΓΓ π π p i p π¦ π ,π‘ p πΎ, π qq Β΄ i p π qs x ππ ,π‘ x ππ ,π‘ d πΎ d π½ ` ΕΌ ΕΌ π½ rΓΓ π π , ΓΓ π π sr i p π¦ π ,π‘ p πΎ, π qq Β΄ i p π qs x π,ππ ,π‘ d πΎ d π½ ` ΕΌ ΕΌ π½ ΓΓ π π rΓΓ π π , ΓΓ π π s i p π¦ π ,π‘ p πΎ, π qq x ππ ,π‘ x π,ππ ,π‘ d πΎ d π½. With the weak geometric property of x : x π,ππ ,π‘ ` x π,ππ ,π‘ β π₯ ππ ,π‘ π₯ ππ ,π‘ , we simplify theexpression of π such that π π‘,π p π q β π ` ΓΓ π π i p π q x ππ ,π‘ ` ΓΓ π π ΓΓ π π i p π q ` x π,ππ ,π‘ ` x π,ππ ,π‘ Λ ` rΓΓ π π , ΓΓ π π s i p π q x π,ππ ,π‘ β π ` ΓΓ π π i p π q x ππ ,π‘ ` ΓΓ π π ΓΓ π π i p π q x π,ππ ,π‘ . So π corresponds to the Davieβs almost flow defined in (34). Proposition 12.
Assume that π P (cid:67) ` πΎπ with ` πΎ Δ π . Then, π defined by (53) is an almost flow which generates a Lipschitz manifold of solutions. Moreover π π converges to the Davieβs flow π of the Corollary 4. roof. We proved in Lemma 4 that π is a stable almost flow. We shall show that π π‘,π is a perturbation in the sense of Definition 8 and then we use Proposition 5 toconclude that π is an almost flow which is in the galaxy of π . We use Corollary 4and Remark 4 to conclude the proof.It is straightforward that π π‘,π‘ β . We start by computing an a priori estimate of p πΌ, π q ΓΓ π¦ π ,π‘ p πΌ, π q , for any p π , π‘ q P T , π P V and πΌ P r , s , | π¦ π ,π‘ p πΌ, π q Β΄ π | Δ πΌ }ΓΓ π π i } } x π } π π { ππ ,π‘ ` πΌ }rΓΓ π π , ΓΓ π π s i } } x π,π } π π { ππ ,π‘ Δ Β΄ }ΓΓ π π i } ` }rΓΓ π π , ΓΓ π π s i } π { π ,π Β― } x } π π { ππ ,π‘ (54) Δ πΆ } x } π π { ππ ,π‘ , (55)where πΆ : β }ΓΓ π π i } ` }rΓΓ π π , ΓΓ π π s i } π { π ,π . With (55), we control the remainder π π‘,π , } π π‘,π } Δ β } x π } π } x π } π }ΓΓ π π ΓΓ π π i } Lip ` } x π,π } π }rΓΓ π π , ΓΓ π π s i } Lip Δ± πΆ } x } π π { ππ ,π‘ ` }ΓΓ π π rΓΓ π π , ΓΓ π π s i } } x π } π } x π,π } π π { ππ ,π‘ , which proves (21).To show the last estimation (22), we compute for any p π , π‘ q P T ` and any π, π P V , π π‘,π p π q Β΄ π π‘,π p π q β ΕΌ ΕΌ π½ ΓΓ π π ΓΓ π π p i p π¦ π ,π‘ p πΎ, π qq Β΄ i p π q Β΄ i p π¦ π ,π‘ p πΎ, π qq ` i p π qs x ππ ,π‘ x ππ ,π‘ d πΎ d π½ loooooooooooooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooooooooooooon I ` ΕΌ ΕΌ π½ rΓΓ π π , ΓΓ π π sr i p π¦ π ,π‘ p πΎ, π qq Β΄ i p π q Β΄ i p π¦ π ,π‘ p πΎ, π qq ` i p π qs x π,ππ ,π‘ d πΎ d π½ loooooooooooooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooooooooooooon II ` ΕΌ ΕΌ π½ ΓΓ π π rΓΓ π π , ΓΓ π π sr i p π¦ π‘,π p πΎ, π qq Β΄ i p π¦ π‘,π p πΎ, π qqs x ππ ,π‘ x π,ππ ,π‘ d πΎ d π½ loooooooooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooooooooon III . We assume that ΓΓ π i P (cid:67) ` πΎπ , so ΓΓ π π ΓΓ π π i P (cid:67) ` πΎπ . It follows from Lemma 1 that ΓΓ π π ΓΓ π π satisfies a -points control such that z ΓΓ π π ΓΓ π π i p π₯ q β πΆ | π₯ | πΎ where πΆ a positive constantwith depends on the πΎ -HΓΆlder norm of the derivative of ΓΓ π π ΓΓ π π . It follows that, | I | Δ πΆ sup πΎ Pr , s ,π P V | π¦ π ,π‘ p πΎ, π q Β΄ π | πΎ Β« sup πΎ Pr , s | π¦ π ,π‘ p πΎ, π q Β΄ π¦ π ,π‘ p πΎ, π q| ` | π Β΄ π | ο¬ } x p q } π π { ππ ,π‘ ` }ΓΓ π π ΓΓ π π } Lip sup πΎ Pr , s | π¦ π ,π‘ p πΎ, π q Β΄ π Β΄ π¦ π ,π‘ p πΎ, π q ` π |} x p q } π π { ππ ,π‘ , | I | Δ πΆπΆ πΎ ,π } x } πΎπ π πΎ { ππ ,π‘ p ` πΆ π q| π Β΄ π |} x p q } π π { ππ ,π‘ ` }ΓΓ π π ΓΓ π π } Lip πΆ π | π Β΄ π |} x p q } π π { ππ ,π‘ , where πΆ π is a constant which is compute in (48). And finally, | I | Δ πΏ π | π Β΄ π | where πΏ π is a constant depending on the norms of π , x which decreases to when π Γ . Similarly, we obtain the same estimation for II . To estimate III , wenote that ΓΓ π π rΓΓ π π , ΓΓ π π s i P (cid:67) πΎπ . Then with (48) it follows that | III | Δ πΆ π π { ππ ,π‘ | π Β΄ π | πΎ ,where πΆ π is another constant which has the same dependencies as πΆ π . Thus | π p π q Β΄ π p π q| Δ πΏ | π Β΄ π | ` πΆ π π { ππ ,π‘ | π Β΄ π | πΎ . This concludes the proof. Remark . This results can be extend to the case U is a Banach case. It is anadvantage compare to the Friz-Victoirβs approach of Subsection 8.3. A The Davie lemma
In this section, we introduce our main tool to control the iterated products (Nota-tion 2) on a partition.
Definition 13 (Successive points) . Let π be a partition of r , π s . Two points π and π‘ of π are said to be at distance π if there are π Β΄ points between them in π .Points at distance are then successive points in π .We now state the Davie lemma . Lemma 9 (The Davie lemma, discrete time version) . Let us consider a family π : β t π π ,π‘ u π ,π‘ P π,π Δ π‘ with values in R ` satisfying for any p π, π , π‘ q P π Ε T , π π,π β , π π,π Δ π·π p π π,π q when π and π are successive points ,π π,π‘ Δ p ` πΌ π q π π,π ` p ` πΌ π q π π ,π‘ ` π΅π p π π,π‘ q , (56) for some constants π· Δ , π΅ Δ and πΌ π Δ that decreases to as π Γ .Then for all π Δ such that ΞΊ p ` πΌ π q ` πΌ π Δ , π π,π‘ Δ π΄π p π π,π‘ q , @p π, π‘ q P r , π s X π , (57) with π΄ : β π· p ` πΌ π qp ` πΌ π q ` π΅ p ` πΌ π q Β΄ p ΞΊ p ` πΌ π q ` πΌ π q . (58) In particular, π΄ does not depend on the choice of the partition. What we call here the Davie lemma differs from the Davie lemma A and B in [19], and alsofrom the one in [14]. However, they all share the same key idea which is due to A.M. Davie. roof. We perform an induction on the distance π between points in π .If π β or π β , (57) is true since π΄ Δ π· .Let us assume that this is true for any two points at distance π . Fix two points π and π‘ at distance π ` in π . Hence, there exists two successive points π and π in π such that π π,π Δ π π,π‘ and π π ,π‘ Δ π π,π‘ with possibly π β π or π β π‘ .Applying (56) twice with p π, π , π‘ q and p π , π , π‘ q , π π,π‘ Δ p ` πΌ π q π π,π ` p ` πΌ π q π π ,π‘ ` π΅π p π π,π‘ qΔ p ` πΌ π q π π,π ` p ` πΌ π q p π π ,π ` π π ,π‘ q ` p ` πΌ π q π΅π p π π,π‘ q . Both π π,π and π π ,π‘ satisfy the induction property. With (2), π π,π‘ Δ π΄ p ` πΌ π q π Β΄ π π,π‘ Β― ` p ` πΌ π q π·π p π π ,π q ` p ` πΌ π q π΅π p π π,π‘ qΔ β π΄ ` ΞΊ p ` πΌ π q ` πΌ π Λ ` π· p ` πΌ π q p ` πΌ π q ` p ` πΌ π q π΅ β° π p π π,π‘ q . Our choice of π΄ Δ π· in (58) ensures the results at level π ` . The control (57) isthen true whatever the partition.We could now state a continuous time version of the Davie lemma. Lemma 10 (The Davie lemma, continuous time version) . Let us consider a family π : β t π π ,π‘ u π ,π‘ P , T with values in R ` satisfying for any p π, π , π‘ q P T ,π π,π Δ πΈπ p π π,π q , (59) π π,π‘ Δ p ` πΌ π q π π,π ` p ` πΌ π q π π ,π‘ ` π΅π p π π,π‘ q , for some constants πΈ Δ , π΅ Δ and πΌ π Δ that decreases to as π Γ . Thenfor any π such that ΞΊ p ` πΌ π q ` πΌ π Δ , π π,π‘ Δ π΄π p π π,π‘ q , @p π, π‘ q P T , with π΄ : β π΅ p ` πΌ π q Β΄ p ΞΊ p ` πΌ π q ` πΌ π q . (60) In particular, the choice of π΄ in (60) does not depend on the bound πΈ in (59) .Proof. The proof is similar as the one of Lemma 7 in [6] from Eq. (31).35 cknowledgments
The authors are very grateful to Laure Coutin for her numerous valuable remarksand corrections.
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