Rough differential equations with unbounded drift term
aa r X i v : . [ m a t h . P R ] M a y ROUGH DIFFERENTIAL EQUATIONS WITH UNBOUNDED DRIFT TERM
S. RIEDEL AND M. SCHEUTZOW
Abstract.
We study controlled differential equations driven by a rough path (in the sense of T.Lyons) with an additional, possibly unbounded drift term. We show that the equation induces asolution flow if the drift grows at most linearly. Furthermore, we show that the semiflow existsassuming only appropriate one-sided growth conditions. We provide bounds for both the flowand the semiflow. Applied to stochastic analysis, our results imply strong completeness andthe existence of a stochastic (semi)flow for a large class of stochastic differential equations. Ifthe driving process is Gaussian, we can further deduce (essentially) sharp tail estimates for the(semi)flow and a Freidlin-Wentzell-type large deviation result.
Introduction
T. Lyons’ theory of rough paths can be used to solve controlled ordinary differential equations(ODE) of the form dy = b ( y ) dt + d X i =1 σ i ( y ) dx it ; t ∈ [0 , T ] y = ξ ∈ R m (0.1)for vector fields b, σ , . . . , σ d : R m → R m and non-differentiable, 1 /p -H¨older continuous paths x : [0 , T ] → R d . However, one of Lyons’ key insights was that the equation (0.1) as it stands isill-posed in the case of p ≥
2. Instead, one has to enhance the path x : [0 , T ] → R d with additionalinformation (which can be interpreted as its iterated integrals) to a path x taking values in a largerspace. Defining a suitable ( p -variation or H¨older-type) topology on this space of paths allows tosolve the corresponding “lifted” equation dy = b ( y ) dt + σ ( y ) d x t ; t ∈ [0 , T ] y = ξ ∈ R m (0.2)uniquely in the way that the solution map (also called It¯o-Lyons map ) x y is continuous. Thispaves way to a genuine pathwise stochastic calculus for a huge class of (not-necessarily martingale-type) driving signals (cf. e.g. [FV10, Chapter 13 - 20] and the references therein). Rough pathstheory is now well-established, and since Lyons’ seminal article [Lyo98], several monographs haveappeared (cf. [LQ02, LCL07, FV10, FH14]) which expose the theory and its various applications.Let us also briefly mention that rough paths ideas were used by M. Hairer to solve stochastic partialdifferential equations (SPDE) like the KPZ-equation ([Hai13]) and form an important part in his Mathematics Subject Classification.
Key words and phrases. controlled ordinary differential equations, rough paths, stochastic differential equations. More precisely, Lyons showed that the map assigning to each smooth path x the solution y to the ordinarydifferential equation (0.1) is not closable in the space of p -variation or 1 /p -H¨older continuous paths. theory of regularity structures (cf. [Hai14] and [FH14] where the link between rough paths andregularity structures is explained).In the present work, we aim to solve (0.2) for a general, possibly unbounded drift term b while weassume σ to be bounded and sufficiently smooth. In the literature about rough paths, a convenientway to take care of the drift part is to regard t t as an additional (smooth) component of therough path x , and b as another component of σ (cf. e.g. [FH14, Exercise 8.15]). However, thisyields unnecessary smoothness assumptions, and allowing b to be unbounded leads to the study ofgeneral unbounded vector fields for rough differential equations (which is a delicate topic, cf. [Lej12]for a discussion). Maybe more important, the bounds for the solution y which are available in thiscase (cf. e.g. [FV10, Exercise 10.56]) are bounds which grow exponentially in the rough path normof x , whereas bounded diffusion vector fields should yield polynomial bounds. The main theoremsin the present paper (Theorem 3.1 and Theorem 4.3) provide exactly the bounds expected.A rough differential equation can be seen as a special case of a non-autonomous ordinary differ-ential equation. Therefore, it should not come as a big surprise that such equations naturally inducecontinuous two parameter flows on the state space R m (at least if all vector fields are bounded,cf. [LQ98], [FV10, Section 11.2], [FH14, Section 8.9]). Note that this immediately implies that astochastic differential equation (SDE) induces a stochastic flow provided the driving process hassample paths in a rough paths space (which is the case, for instance, for a Brownian motion). Inparticular, the SDE is strongly complete which means that it can be solved globally on a set offull measure which does not depend on the initial condition. Note that an SDE may lack strongcompleteness while possessing strong solutions (in the It¯o-sense) for any initial condition. Indeed,this is even possible for b ≡ σ bounded and C ∞ (but with unbounded derivatives), cf. [LS11].However, using a pathwise calculus (like rough paths theory), strong completeness is immediate.We are interested in proving the existence of a (semi)flow induced by (0.2) for an unboundeddrift b . In Section 3, we first discuss the case of a proper flow, i.e. the case when (0.2) can be solvedforward and backward in time. In this case, it is natural to assume that b should be locally Lipschitzcontinuous with linear growth, and in Theorem 3.1 we prove the existence of the flow under theseassumptions and provide quantitative bounds. More interesting might be the case when we canonly expect to solve (0.2) in one time direction, say forward in time (a typical example would be b ( ξ ) = − ξ | ξ | ). In these situations, the best we can hope for is to prove existence of a semi flowinduced by (0.2). A classical condition to impose (both in the theory of ODE and SDE) is theone-sided growth condition h b ( ξ ) , ξ i ≤ C (1 + | ξ | ) for all ξ ∈ R m (0.3)together with a (one-sided) local Lipschitz condition. In the context of SDE driven by a d -dimensional Brownian motion, strong global existence and uniqueness under condition (0.3) wasproven in [PR07]. Recently, one of the authors showed in [SS16] the existence of a semiflow evenfor infinitely many Brownian motions under slightly stronger assumptions. Interestingly, if m ≥ σ being constant. There, the authors define an explicit vector field b : R → R with astrong (cubic is enough) growth in the tangential direction only. Then, they construct an (evensmooth!) path x : [0 , ∞ ) → R and show that the solution to (0.1) explodes in finite time. Thissuggests the need to impose an additional condition on b which controls the growth in tangential In fact, in [Bai15], the flow is even the central object of interest and it is constructed directly, skipping theintermediate step of defining the solution to (0.2) for a fixed initial datum ξ first. OUGH DIFFERENTIAL EQUATIONS WITH UNBOUNDED DRIFT TERM 3 direction. In the case of additive noise, it was shown in [SS16] that non-explosion can be assuredeven for quadratic tangential growth. In this work, we impose a linear growth of the form (cid:12)(cid:12)(cid:12)(cid:12) b ( ξ ) − h b ( ξ ) , ξ i ξ | ξ | (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (1 + | ξ | ) for all ξ ∈ R m . (0.4)Our second main result (Theorem 4.3) states that under the two growth conditions (0.3) and(0.4) and a suitable local Lipschitz condition, the semiflow to (0.2) exists. Moreover, we providequantitative bounds which are similar to those derived for the flow in Theorem 3.1.To illustrate our results, let us discuss some applications in stochastic analysis. Theorem 0.1.
Let σ = ( σ , . . . , σ d ) be a collection of infinitely often differentiable vector fields on R m where σ and all its derivatives are bounded. Consider the stochastic differential equation dY = ( Y − | Y | Y ) dt + d X i =1 σ i ( Y ) ◦ dX it ( ω ); t ∈ [0 , T ](0.5) Y = ξ ∈ R m (0.6) where X : [0 , T ] → R d is a continuous stochastic process which can be enhanced to a process withvalues in the space of weakly geometric rough paths on set of full measure (this can be a semi-martingale, a Gaussian process or a Markov processes, cf. [FV10] for a list of examples). Theequation (0.5) is understood in rough paths sense or in Stratonovich sense in case of X being asemimartingale and the lift is defined as in [FV10, Chapter 14] .Then the following holds: (i) The SDE (0.5) is strongly complete and induces a continuous stochastic semiflow. (ii) If X : [0 , T ] → R d is a centered Gaussian process with covariance of finite (1 , ρ ) -variationfor some ρ ∈ [1 , (cf. [FGGR16] for the precise definition), the random variable k Y k /ρ ∞ ;[0 ,T ] has Gaussian tails, i.e. there is a δ > such that E h exp( δ k Y k /ρ ∞ ;[0 ,T ] ) i < ∞ . Moreover, the random variables ( Y εt : 0 ≤ t ≤ T ) , Y ε being the solution to (0.5) whenwe replace dX it by εdX it , satisfy a Freidlin-Wentzell-type large deviation principle in thetopology of uniform convergene (cf. [FV10, Proposition 19.14] for the precise formulation).Proof. The vector field b ( ξ ) = ξ − | ξ | ξ satisfies the conditions of Theorem 4.3. Therefore, (0.5) canbe solved pathwise which implies (i). In case of X ( ω ) being a Gaussian process with lift X ( ω ), thequantity N ( X ( ω )) /ρ is a random variable with Gaussian tails; cf. [FGGR16, Theorem 1.1] and[FH14, Theorem 11.13], and the bound (4.17) implies the tail estimate in (ii). The large deviationresult follows by a Schilder-type large deviation result for X (cf. [FV10, Theorem 15.55]) and the contraction principle which can be used since X ( ω ) Y ( ω ) is continuous by Theorem 4.3. (cid:3) Let us remark that(i) the smoothness assumptions for σ can be relaxed and are linked to the “roughness” of thetrajectories of X , cf. Theorem 4.3.(ii) The uniform norm in Theorem 0.1 can be replaced by the p -variation norm for sufficientlylarge p (where p depends on the rough path trajectories).(iii) The large deviation principle also holds in p -variation topology (again, for p large enough).(iv) Fractional Brownian motion with Hurst parameter H falls into the framework of Theorem0.1 with H = 1 / (2 ρ ) (other examples of Gaussian processes may be found in [FGGR16]). S. RIEDEL AND M. SCHEUTZOW
The article is organized as follows: In Section 1, we quickly recall some basic facts about roughpaths and explain some notation. Section 2 introduces the flow decomposition (our main techniquefor proving our results) and some facts about flows induced by rough differential equations withbounded coefficients are proved. In Section 3, we prove our main result for b having linear growth,cf. Theorem 3.1. Finally, in Section 4 we study the case where b is assumed to satisfy only one-sidedgrowth conditions. Our main results here are formulated in Theorem 4.3.1. Notation, elements of rough path theory
We will now very briefly recall the elements of rough paths theory used in this paper. For moredetails we refer to [FV10], [LCL07], [LQ02] or [FH14]. Our notation coincides with the one used in[FV10].Let T N ( R d ) = R ⊕ R d ⊕ ( R d ⊗ R d ) ⊕ . . . ⊕ ( R d ) ⊗ N , be the truncated step- N tensor algebra, N ≥
1. We are concerned with T N ( R d )-valued paths, as naturally given by iterated integrationsof R d -valued smooth paths (“lifted smooth paths”). The projection of such a path x on the firstlevel is an R d -valued path and will be denoted by π ( x ), the projection to k th level is denoted by π k . Lifted smooth paths actually take values in G N ( R d ) ⊂ T N ( R d ), where ( G N ( R d ) , ⊗ ) denotesthe free step- N nilpotent Lie group with d generators (cf. [FV10, Theorem and Definition 7.30]).The group structure allows to define natural increments x s,t ≡ x − s ⊗ x t , s, t ∈ R , for paths x taking values in G N ( R d ). The (left-invariant) Carnot-Caratheodory metric turns ( G N ( R d ) , d ) intoa metric space ([FV10, Section 7.5.4]).Fix some time interval [0 , T ]. For p ≥ s, t ] ⊆ [0 , T ], we will use the p -variation and1 /p -H¨older “norm” k x k p -var;[ s,t ] = sup ( t i ) ⊂ [ s,t ] X i d (cid:0) x t i , x t i +1 (cid:1) p ! /p , k x k /p -H¨ol;[ s,t ] = sup s ≤ u 7→ | t − s | and ( s, t ) x k pp − var;[ s,t ] where x is any p -rough path. We say that ω controls the p -variation of x if d ( x s , x t ) p ≤ ω ( s, t ) holds for every s ≤ t . Note that this is equivalent to say that k x k pp − var;[ s,t ] ≤ ω ( s, t ) holdsfor every s < t . If x has finite p -variation, its p -variation is controlled by ω ( s, t ) = k x k pp − var;[ s,t ] .For a control ω and some δ > 0, we define a sequence ( τ n ) as follows: set τ := 0 and τ n +1 := inf { u : ω ( τ n , u ) ≥ δ, τ n < u ≤ T } ∧ T. Then we define N δ ( ω ) := sup { n ∈ N : τ n < T } . From superadditivity of ω , δN δ ( ω ) ≤ ω (0 , T ) < ∞ . If ω ( s, t ) = k x k pp − var;[ s,t ] for some rough path x , we will also write N δ ( x ) for N δ ( ω ). The quantity N δ ( ω ) first appeared in [CLL13] where theauthors observed that N δ ( X ) has significantly better integrability properties than k X k pp − var when X is the lift of a Gaussian stochastic process, cf. also [FH14, Section 11.2].A collection of vector fields σ = ( σ , . . . , σ d ) on R m is called γ -Lipschitz (in the sense of E. Stein) for γ > 0, denoted σ ∈ Lip γ , if all σ i are ⌊ γ ⌋ -times continuously differentiable, the vector fieldsand all derivatives up to order ⌊ γ ⌋ are bounded, and the ⌊ γ ⌋ -th derivatives are ( γ − ⌊ γ ⌋ )-H¨oldercontinuous. If γ is an integer, this means that the ( γ − ⌊ γ ⌋ -th derivatives is denoted by | σ | Lip γ .We will be interested in rough differential equations of the form dy = b ( y ) dt + σ ( y ) d x ; t ∈ [0 , T ](1.1)where x is a p -rough path in G ⌊ p ⌋ ( R d ), the solution y is a continuous path in R m , b and σ =( σ , . . . , σ d ) are vector fields in R m . In the following, we recall the definition of a solution to (1.1)in the sense of Friz-Victoir [FV10, Definition 10.17]: Definition 1.1. Let x be a p -rough path. A path y : [0 , T ] → R m is called a solution to (1.1) withinitial condition y = ξ ∈ R m in the sense of Friz-Victoir if the following holds:(i) y = ξ .(ii) There exists a sequence ( x n ) of continuous paths in R d with finite variation such that thelifted paths x n satisfysup n ∈ N k x n k p − var < ∞ and lim n →∞ sup ≤ s Let x : [0 , T ] → R d be smooth and σ : R m → Lin( R d , R m ) be smooth and bounded with boundedderivatives. Let ψ : [0 , T ] × [0 , T ] × R m → R m be the solution flow to the (non-autonomous) ordinarydifferential equation ˙ y t = σ ( y t ) ˙ x t . (2.1)For given b : R m → R m , assume that we can make sense of the ordinary differential equation˙ z u = ( D ξ ψ ( s, u, ξ ) | ξ = z u ) − b ( ψ ( s, u, z u )); u ∈ [0 , T ] z s = ξ (2.2)for any s, t ∈ [0 , T [ and any ξ ∈ R m . Let χ s ( t, ξ ) denote the value of the solution to (2.1) at timepoint t . Then an easy application of the chain rule shows that φ ( s, t, ξ ) := ψ ( s, t, χ s ( t, ξ )) coincideswith the solution flow to the equation ˙ y t = b ( y t ) + σ ( y t ) ˙ x t . In the cases we will consider, solutions to (2.2) will only exist on small time intervals and possiblyonly forward in time. Taking this into account, we make the following definition: Definition 2.1. Let I ⊆ { [ s, t ] : s ≤ t, s, t ∈ [0 , T ] } be a subset of the set of all intervals contained in [0 , T ] for which there exists a finite subset I ⊆ I such that [ [ u,v ] ∈I [ u, v ] = [0 , T ]holds and for which [ u, v ] ⊂ [ s, t ] and [ s, t ] ∈ I implies that [ u, v ] ∈ I . Let x be a weak geometric p -rough path with values in G ⌊ p ⌋ ( R d ), p ∈ [1 , ∞ ), b a vector field and σ = ( σ , . . . , σ d ) a collectionof vector fields on R m . Assume that the rough differential equation dy t = σ ( y t ) d x t ; t ∈ [0 , T ](2.3)induces a continuously differentiable solution flow ψ x : [0 , T ] × [0 , T ] × R m → R m and that theordinary differential equation˙ z u = ( D ξ ψ x ( v, u, ξ ) | ξ = z u ) − b ( ψ x ( v, u, z u )); u ∈ [ s, t ] z v = ξ (2.4)has a unique solution (forward and backward in time) for every [ s, t ] ∈ I , v ∈ [ s, t ] and ξ ∈ R m .We denote this solution by [ s, t ] ∋ u χ x v ( u, ξ ). Set φ x ( s, t, ξ ) = ψ x ( s, t, χ x s ( t, ξ )) and φ x ( t, s, ξ ) = ψ x ( t, s, χ x t ( s, ξ )) OUGH DIFFERENTIAL EQUATIONS WITH UNBOUNDED DRIFT TERM 7 for [ s, t ] ∈ I and φ x ( s, t, ξ ) := φ x ( t n − , t n , · ) ◦ · · · ◦ φ x ( t , t , ξ ) and φ x ( t, s, ξ ) := φ x ( t , t , · ) ◦ · · · ◦ φ x ( t n , t n − , ξ )for arbitrary s ≤ t where s = t < . . . < t n = t and [ t i , t i +1 ] ∈ I for every i = 0 , . . . , n − 1. Then wecall the map φ x : [0 , T ] × [0 , T ] × R m → R m the solution flow to (1.1). If the solution to (2.3) existsonly forward in time, we define φ ( s, t, ξ ) for s ≤ t and ξ ∈ R m as above and call it the solutionsemiflow to (1.1).To simplify notation, we will sometimes drop the upper index x and just write φ , ψ and χ .We have to check that φ is well defined and does not depend on the choice of I . This is done inthe next lemma. Lemma 2.2. Under the conditions stated in Definition 2.1, φ : [0 , T ] × [0 , T ] × R m → R m is welldefined, does not depend on the choice of I and satisfies the (semi-)flow property.Proof. We first check that φ is well defined as a semiflow. Note that it is enough to prove that φ ( u , t, φ ( s, u , ξ )) = φ ( u , t, φ ( s, u , ξ ))holds for s ≤ u ≤ u ≤ t , [ s, u ], [ s, u ], [ u , t ], [ u , t ] ∈ I and ξ ∈ R m . From the flow property of ψ , this is equivalent to ψ ( u , t, ψ ( u , u , χ u ( t, φ ( s, u , ξ )))) = ψ ( u , t, χ u ( t, φ ( s, u , ξ )) . Therefore, it is enough to check that ψ ( u , u , χ u ( t, φ ( s, u , ξ ))) = χ u ( t, φ ( s, u , ξ ) . We do this by showing that both objects, seen as functions in t , solve the differential equation˙ z u = ( D ζ ψ ( u , u, ζ ) | ξ = z u ) − b ( ψ ( u , u, z u )); u ∈ [ u , t ](2.5) z u = φ ( s, u , ξ ) . (2.6)The function u χ u ( u, φ ( s, u , ξ ) solves this equation by definition, and it remains to show thatalso the function on the left hand side solves the same equation. We first check that it has the sameinitial condition. We need to show that ψ ( u , u , χ u ( u , φ ( s, u , ξ ))) = φ ( s, u , ξ ) = ψ ( s, u , χ s ( u , ξ )) = ψ ( u , u , ψ ( s, u , χ s ( u , ξ ))) . Thus, it is enough to establish the identity χ u ( u , φ ( s, u , ξ )) = ψ ( s, u , χ s ( u , ξ )) . This is done by showing that both expressions, seen as functions in u , solve the differential equation˙ z u = ( D ζ ψ ( u , u, ζ ) | ζ = z u ) − b ( ψ ( u , u, z u )); u ∈ [ u , u ] z u = φ ( s, u , ξ ) . The function on the left hand side solves the equation by definition. For the function on the righthand side, we have ψ ( s, u , χ s ( u , ξ )) | u = u = ψ ( s, u , χ s ( u , ξ )) = φ ( s, u , ξ ) , S. RIEDEL AND M. SCHEUTZOW thus the initial condition is satisfied. Differentiating this function, using the chain rule, gives ddu ψ ( s, u , χ s ( u, ξ )) = D ζ ψ ( s, u , ζ ) | ζ = χ s ( u,ξ ) ddu χ s ( u, ξ )= D ζ ψ ( s, u , ζ ) | ζ = χ s ( u,ξ ) ( D ζ ψ ( s, u, ζ ) | ζ = χ s ( u,ξ ) ) − b ( ψ ( s, u, χ s ( u, ξ )))= D ζ ψ ( s, u , ζ ) | ζ = χ s ( u,ξ ) ( D ζ ψ ( s, u, ζ ) | ζ = χ s ( u,ξ ) ) − b ( ψ ( u , u, ψ ( s, u , χ s ( u, ξ )))) . It remains to show that D ζ ψ ( s, u , ζ ) | ζ = χ s ( u,ξ ) ( D ζ ψ ( s, u, ζ ) | ζ = χ s ( u,ξ ) ) − = ( D ζ ψ ( u , u, ζ ) | ζ = ψ ( s,u ,χ s ( u,ξ )) ) − . This identity follows by differentiating both sides of ψ ( s, u, θ ) = ψ ( u , u, ψ ( s, u , θ )) with respectto θ and substituting θ = χ s ( u, ξ ). Going back our proof, we see that we still have to show that u ψ ( u , u , χ u ( u, φ ( s, u , ξ ))) satisfies (2.5), but this is done exactly as above. It follows that φ is indeed well defined. The semiflow property follows by definition.Next, we show that φ does not depend on I . Let φ and φ be two semiflows associated to I resp. I . Note that I := I ∪ I satisfies the same conditions as I and I . The semiflow φ associated to I can be constructed by using only elements in I , therefore it coincides with φ . Bythe same argument, it also coincides with φ , thus φ = φ .Proving that φ is well defined as a flow follows exactly in the same way. (cid:3) In the next lemma, we collect some properties of the flow ψ and the inverse of its derivative. Lemma 2.3. Let x be a weak geometric p -rough path on [0 , T ] and let ω be a control function whichcontrols its p -variation. Let σ ∈ Lip γ for some γ > p and choose ν ≥ | σ | Lip γ . Consider the equation dy t = σ ( y t ) d x t ; t ∈ [0 , T ] . (2.7)(i) The solution flow ψ to (2.7) exists and there is a constant C = C ( γ, p ) such that | ψ ( s, v, ξ ) − ψ ( s, u, ξ ) | ≤ C ( νω ( u, v ) /p ∨ ν p ω ( u, v )) and | ψ ( s, v, ξ ) − ψ ( s, u, ξ ) − ψ ( s, v, ζ ) + ψ ( s, u, ζ ) | ≤ Cνω ( u, v ) /p | ξ − ζ | exp( Cν p ω ( s, t )) holds for every s ≤ u ≤ v ≤ t , s, t ∈ [0 , T ] and ξ, ζ ∈ R m . (ii) For every s, t ∈ [0 , T ] , J ( s, t, ξ ) := ( D ξ ψ ( s, t, ξ )) − exists and satisfies the bound | J ( s, t, ξ ) − I m | ≤ Cνω ( s, t ) /p exp( Cν p ω ( s, t )) for every s ≤ t , s, t ∈ [0 , T ] and ξ ∈ R m where I m denotes the identity matrix in R m × m and C as in (i).Proof. Claim (i) is a slight generalization of [FV10, Theorem 10.14 and Theorem 10.26] when westart the equation at time point s instead of 0.Concerning claim (ii), note first that the derivative D ξ ψ ( s, t, ξ ) and the derivative of the inversemap D ξ ψ − ( s, t, ξ ) exist by [FV10, Proposition 11.11]. Fix s and t . From (i), note that for every h > | ψ ( s, t, ξ + he i ) − ψ ( s, t, ξ ) − he i | ≤ hCνω ( s, t ) /p exp( Cν p ω ( s, t )) . Dividing the equation by h and sending h → | ∂ ξ i ψ ( s, t, ξ ) − e i | ≤ Cνω ( s, t ) /p exp( Cν p ω ( s, t )) OUGH DIFFERENTIAL EQUATIONS WITH UNBOUNDED DRIFT TERM 9 for every i = 1 , . . . , m , thus | D ξ ψ ( s, t, ξ ) − I m | ≤ Cνω ( s, t ) /p exp( Cν p ω ( s, t )) . The inverse flow ψ − is given by solving (2.7) where we let the rough path x run “backwards intime” from t to s (cf. [FV10, Section 11.2]). Note that the p -variation is invariant under timereversion. Therefore, the same estimate holds for ψ replaced by ψ − . From the chain rule, J ( s, t, ξ ) = ( D ξ ψ ( s, t, ξ )) − = D ζ ψ − ( s, t, ζ ) | ζ = ψ ( s,t,ξ ) from which we can deduce claim (ii). (cid:3) RDE flows for a drift with linear growth The next theorem is our first main result. Theorem 3.1. Let x be a weak geometric p -rough path with values in G ⌊ p ⌋ ( R d ) , p ∈ [1 , ∞ ) , and let ω be a control function which controls its p -variation. Assume that σ = ( σ , . . . , σ d ) is a collectionof Lip γ +1 -vector fields on R m for some γ > p and choose ν ≥ | σ | Lip γ . Moreover, assume that b isa locally Lipschitz continuous vector field with linear growth on R m , i.e. there are some constants κ , κ ≥ such that | b ( ξ ) | ≤ κ + κ | ξ | for all ξ ∈ R m . Then the following holds true: (1) The solution flow φ to (1.1) exists and is continuous. (2) There is a constant C depending on p , γ , κ , κ and ν such that k φ (0 , · , ξ ) k ∞ ;[0 ,T ] ≤ C exp(2 κ T )(1 + N ( ω ) + | ξ | + T )(3.1) and k φ (0 , · , ξ ) k p − var ;[0 ,T ] ≤ C exp( Cκ T )(1 + N ( ω ) + κ | ξ | + T )(3.2) hold for every ξ ∈ R m . If x is a /p -H¨older rough path, sup ≤ s In the following proof, C will be a constant which may depend on p , γ , κ , κ and ν but whose actual value may change from line to line.From our assumptions on σ , we know from [FV10, Proposition 11.11] that the solution flow ψ to (2.3) exists, is twice differentiable and has a twice differentiable inverse. For s, t ∈ [0 , T ] and ξ ∈ R m , set J ( s, t, ξ ) := ( D ξ ψ ( s, t, ξ )) − ∈ R m × m . The same proposition states that the maps ( t, ξ ) D ξ ψ (0 , t, ξ ) and ( t, ξ ) D ξ ψ (0 , t, ξ ) arebounded, and the same is true for the inverse ψ − . By the flow property, also the two time parameterflow maps ( s, t, ξ ) D ξ ψ ( s, t, ξ ) and ( s, t, ξ ) D ξ ψ ( s, t, ξ ) are bounded, and the same holds for ψ − . From the chain rule, ( s, t, ξ ) J ( s, t, ξ ) and ( s, t, ξ ) D ξ J ( s, t, ξ ) are also bounded. Hencethe map ( s, t, ξ ) J ( s, t, ξ ) b ( ψ ( s, t, ξ )) is continuous, locally Lipschitz continuous in space andgrows at most linearly in space. Thus the ordinary differential equation (2.4) has unique solutionson every time interval, forward and backward in time. Therefore χ (defined as in Definition 2.1)exists and the flow φ is well defined by Lemma 2.2.We proceed by proving the bound (3.1) for the sup-norm of φ . From Lemma 2.3, we know thatthere is a constant C = C ( p, γ ) such thatsup ξ ∈ R m | ψ ( s, t, ξ ) − ξ | ≤ C ( νω ( s, t ) /p ∨ ν p ω ( s, t ))and sup ξ ∈ R m | J ( s, t, ξ ) − I m | ≤ Cνω ( s, t ) /p exp( Cν p ω ( s, t ))hold for every s < t . Choose δ ∈ (0 , 1] small enough such that2( κ + κ ) δ ∨ [ Cνδ /p exp( Cν p δ )] ∨ [ C ( νδ /p ∨ ν p δ )] ≤ ω ( s, t ) := ω ( s, t ) + | t − s | . Define a sequence ( τ n ) as follows: set τ := 0 and τ n +1 := inf { u : ˜ ω ( τ n , u ) ≥ δ, τ n < u ≤ T } ∧ T. Moreover, set N := N δ (˜ ω ) = sup { n ∈ N : τ n < T } . It follows that | τ n +1 − τ n | ≤ δ and ω ( τ n , τ n +1 ) ≤ δ for every n = 0 , . . . , N . Lemma 2.3 implies thatfor every t ∈ [ τ n , τ n +1 ], n = 0 , . . . , N , sup ξ | ψ ( τ n , t, ξ ) − ξ | ≤ ξ | J ( τ n , t, ξ ) | ≤ . Now let y : [0 , τ ] → R m be the solution to˙ y t = J (0 , t, y t ) b ( ψ (0 , t, y t )); t ∈ [0 , τ ](3.5)with initial condition y = ξ . Integrating the equation, we obtain for every t ∈ [0 , τ ] | y t | ≤ | ξ | + 2 κ t + 2 κ Z t | ψ (0 , s, y s ) | ds ≤ | ξ | + 2( κ + κ ) t + 2 κ Z t | y s | ds. Gronwall’s Lemma implies that | χ ( t, ξ ) | = | y t | ≤ exp(2 κ t )( | ξ | + 2( κ + κ ) t ) ≤ exp(2 κ τ )( | ξ | + 1)for every t ∈ [0 , τ ]. Repeating the same argument shows that | χ τ n ( t, ξ ) | ≤ exp(2 κ | τ n +1 − τ n | )( | ξ | + 1) OUGH DIFFERENTIAL EQUATIONS WITH UNBOUNDED DRIFT TERM 11 holds for every t ∈ [ τ n , τ n +1 ]. Fix some n ∈ { , . . . , N } and some t ∈ [ τ n , τ n +1 ]. From the flowproperty of φ , | φ (0 , t, ξ ) | = | φ ( τ n , t, φ (0 , τ n , ξ )) | = | ψ ( τ n , t, χ τ n ( t, φ (0 , τ n , ξ ))) |≤ | ψ ( τ n , t, χ τ n ( t, φ (0 , τ n , ξ ))) − χ τ n ( t, φ (0 , τ n , ξ )) | + | χ τ n ( t, φ (0 , τ n , ξ )) |≤ κ | τ n +1 − τ n | ) + exp(2 κ | τ n +1 − τ n | ) | φ (0 , τ n , ξ ) | . Set φ n := sup t ∈ [ τ n ,τ n +1 ] | φ (0 , t, ξ ) | and C n := exp(2 κ | τ n +1 − τ n | ). The estimate above reads φ ≤ C (1 + | ξ | ) and φ n +1 ≤ C n +1 + C n +1 φ n for every n = 0 , . . . , N . By induction, φ n ≤ C n + C n C n − + . . . + C n · · · C ) + C n · · · C C (1 + | ξ | ) ≤ C . . . C N (1 + 2 N + | ξ | ) . This implies that sup t ∈ [0 ,T ] | φ (0 , t, ξ ) | ≤ κ T )(1 + 2 N + | ξ | ) . Next, N δ (˜ ω ) ≤ N δ ( ω ) + 2 T /δ + 2 ≤ N ( ω ) /δ + 2 /δ + 2 T /δ + 2where the first inequality follows from [BFRS16, Lemma 5], the second one uses [FR13, Lemma 3].This implies the bound (3.1).We proceed with proving (3.2). Choose δ ∈ (0 , 1] as in (3.4) with the additional condition δ ≤ ν − p . We define ˜ ω , ( τ n ) and N = N δ (˜ ω ) as above. If y : [0 , τ ] → R m denotes the solution to(3.5) with initial condition y = ξ , we have for u < v , u, v ∈ [0 , τ ], | y v − y u | ≤ κ + κ ) | v − u | + 2 κ sup t ∈ [0 ,τ ] | y t || v − u | + 2 κ Z vu | y s − y u | ds ≤ κ + κ ) | v − u | + 2 κ exp(2 κ τ )(1 + | ξ | ) | v − u | + 2 κ Z vu | y s − y u | ds. From Gronwall’s Lemma, | χ ( v, ξ ) − χ ( u, ξ ) | = | y v − y u | ≤ | v − u | [2( κ + κ ) + 2 κ exp(2 κ τ )(1 + | ξ | )] exp(2 κ τ ) ≤ κ + κ ) + κ (1 + | ξ | )] | v − u | . Similarly, one can show that for every n = 0 , . . . , N , u, v ∈ [ τ n , τ n +1 ] and u < v , | χ τ n ( v, ξ ) − χ τ n ( u, ξ ) | ≤ κ + κ ) + κ (1 + | ξ | )] | v − u | . (3.6)Fix n ∈ { , . . . , N } and u, v ∈ [ τ n , τ n +1 ] with u < v . Using the flow property, | φ (0 , u, ξ ) − φ (0 , v, ξ ) | = | ψ ( τ n , u, χ τ n ( u, φ (0 , τ n , ξ ))) − ψ ( τ n , v, χ τ n ( v, φ (0 , τ n , ξ ))) |≤ | ψ ( τ n , u, χ τ n ( u, φ (0 , τ n , ξ ))) − ψ ( τ n , v, χ τ n ( u, φ (0 , τ n , ξ ))) | + | ψ ( τ n , v, χ τ n ( u, φ (0 , τ n , ξ ))) − ψ ( τ n , v, χ τ n ( v, φ (0 , τ n , ξ ))) | . For the first term, we use Lemma 2.3 to see that | ψ ( τ n , u, χ τ n ( u, φ (0 , τ n , ξ ))) − ψ ( τ n , v, χ τ n ( u, φ (0 , τ n , ξ ))) | ≤ Cνω ( u, v ) /p . For the second term, we use again Lemma 2.3, the triangle inequality and the estimate (3.6) to seethat | ψ ( τ n , v, χ τ n ( u, φ (0 , τ n , ξ ))) − ψ ( τ n , v, χ τ n ( v, φ (0 , τ n , ξ ))) |≤ | ψ ( τ n , v, χ τ n ( u, φ (0 , τ n , ξ ))) − ψ ( τ n , v, χ τ n ( v, φ (0 , τ n , ξ ))) − χ τ n ( u, φ (0 , τ n , ξ )) + χ τ n ( v, φ (0 , τ n , ξ )) | + | χ τ n ( u, φ (0 , τ n , ξ )) − χ τ n ( v, φ (0 , τ n , ξ )) |≤ | χ τ n ( u, φ (0 , τ n , ξ )) − χ τ n ( v, φ (0 , τ n , ξ )) |≤ κ + κ ) + κ (1 + | φ (0 , τ n , ξ ) | )] | v − u | Putting these estimates together, we have shown that for any u, v ∈ [ τ n , τ n +1 ], | φ (0 , u, ξ ) − φ (0 , v, ξ ) | ≤ C [1 + κ (1 + | φ (0 , τ n , ξ ) | )] | v − u | + Cω ( u, v ) /p (3.7)which implies that k φ (0 , · , ξ ) k p − var;[ τ n ,τ n +1 ] ≤ C [1 + κ (1 + | φ (0 , τ n , ξ ) | )] | τ n +1 − τ n | + Cδ /p . Now k φ (0 , · , ξ ) k p − var;[0 ,T ] ≤ N X n =0 k φ (0 , · , ξ ) k p − var;[ τ n ,τ n +1 ] ≤ C (cid:2) κ (1 + k φ (0 , · , ξ ) k ∞ ;[0 ,T ] ) (cid:3) T + C ( N δ (˜ ω ) + 1) δ /p The claim follows by using the bound (3.1) for the sup-norm and again [BFRS16, Lemma 5] and[FR13, Lemma 3].Next we prove the bound (3.3) in the H¨older case. Here, we may choose ω ( s, t ) = k x k p /p − H¨ol | t − s | which implies N ( ω ) ≤ k x k p /p − H¨ol T . As for (3.7), we can show that for every u ≤ v for which k x k p /p − H¨ol | v − u | = ω ( u, v ) ≤ δ holds, we have | φ (0 , u, ξ ) − φ (0 , v, ξ ) | ≤ C (cid:2) κ (1 + k φ (0 , · , ξ ) k ∞ ;[0 ,T ] ) (cid:3) | v − u | + C k x k /p − H¨ol | v − u | /p . We claim that for every u ≤ v , u, v ∈ [0 , T ], we have | φ (0 , u, ξ ) − φ (0 , v, ξ ) | ≤ C (cid:2) κ (1 + k φ (0 , · , ξ ) k ∞ ;[0 ,T ] ) (cid:3) | v − u | + C h ( k x k /p − H¨ol | v − u | /p ) ∨ ( k x k p /p − H¨ol | v − u | ) i . We prove this similarly to [FH14, Exercise 4.24]. First, there is nothing to show for v − u ≤ h := δ k x k − p /p − H¨ol . If this is not the case, we define t i := ( u + ih ) ∧ v and observe that t M = v for M ≥ ( v − u ) /h and that t i +1 − t i ≤ h . Then | φ (0 , u, ξ ) − φ (0 , v, ξ ) | ≤ X ≤ i< ( v − u ) /h | φ (0 , t i +1 , ξ ) − φ (0 , t i , ξ ) |≤ C (cid:2) κ (1 + k φ (0 , · , ξ ) k ∞ ;[0 ,T ] ) (cid:3) | v − u | + C k x k /p − H¨ol h /p (1 + | v − u | /h ) ≤ C (cid:2) κ (1 + k φ (0 , · , ξ ) k ∞ ;[0 ,T ] ) (cid:3) | v − u | + 2 C k x k /p − H¨ol h /p − | v − u | which shows the claim by definition of h . This also implies (3.3) by using the sup-bound of oursolution and the bound for N ( ω ) we saw above. OUGH DIFFERENTIAL EQUATIONS WITH UNBOUNDED DRIFT TERM 13 Next we show that φ is continuous. From the flow property, for s ≤ u ≤ v ≤ t , φ ( s, t, ξ ) = φ ( v, t, φ ( u, v, φ ( s, u, ξ ))) , therefore it suffices to prove that for some given δ > 0, [ s, t ] × R m ∋ ( u, ξ ) φ ( u, v , ξ ) and[ s, t ] × R m ∋ ( v, ξ ) φ ( u , v, ξ ) are continuous for fixed v resp. u where u , v ∈ [ s, t ] and s < t satisfy | t − s | ≤ δ . By continuity of ψ , it suffices to show that ( u, ξ ) χ u ( v , ξ ) and ( v, ξ ) χ u ( v, ξ )are continuous. This, however, follows by standard arguments for ordinary differential equations(or see the proof of the forthcoming Theorem 4.3 where this is carried out in more detail in evenmore generality).It remains to show that for fixed ξ ∈ R m , the path y t := φ x (0 , t, ξ ) is a solution to (1.1) in thesense of Friz-Victoir. We will give the proof in even more generality in the forthcoming Theorem4.3. (cid:3) RDE semiflows Next, our goal is to further relax the assumptions on b which will yield a semi flow of (1.1). Wefirst prove an a priori estimate for ordinary differential equations. Lemma 4.1. Consider an ordinary differential equation of the form ˙ z t = J ( t, z t ) b ( ψ ( t, z t )); t ∈ [0 , T ](4.1) where (1) b : R m → R m is continuous and satisfies the following conditions: (i) There exists a constant C such that h b ( ξ ) , ξ i ≤ C (1 + | ξ | ) for every ξ ∈ R m . (4.2) (ii) There exists a constant C such that (cid:12)(cid:12)(cid:12)(cid:12) b ( ξ ) − h b ( ξ ) , ξ i ξ | ξ | (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (1 + | ξ | ) for every ξ ∈ R m \ { } . (4.3)(2) J : [0 , T ] × R m → R m × m is continuous and satisfies sup t ∈ [0 ,T ]; ξ ∈ R m | J ( t, ξ ) − I m | ≤ where I m denotes the identity matrix in R m × m . (3) ψ : [0 , T ] × R m → R m is continuous and there exists a constant C such that sup t ∈ [0 ,T ]; ξ ∈ R m | ψ ( t, ξ ) − ξ | ≤ C . Then any solution z : [0 , T ] → R m to (4.1) with initial condition z = ξ ∈ R m satisfies the bounds k z k ∞ ;[0 ,T ] ≤ ( CT + | ξ | ) e CT and k z k − var ;[0 ,T ] ≤ C (1 + k z k ∞ ;[0 ,T ] ) T + ( | ξ | − ˆ C ) + − ( | z T | − ˆ C ) + where ˆ C = ( C + 1) ∨ (4 C ) and C is a constant depending on C i , i = 1 , , , where C := sup {| b ( ξ ) | : | ξ | ≤ (2 C + 1) ∨ (5 C ) + 1 } . Proof. Let z : [0 , T ] → R m be a solution to (4.1) with initial condition z = ξ ∈ R m . For t ∈ [0 , T ],set h t := ψ ( t, z t ) − z t . Note that by assumption, | h t | ≤ C for every t . From the chain rule, ddt | z t | = 2 h z t , ˙ z t i = 2 h z t , J ( t, z t ) b ( z t + h t ) i . Fix t ∈ [0 , T ]. To simplify notation, set z = z t and h = h t . We aim to show that there exists aconstant C depending on C i , i = 1 , , , 4, but independent of t such that h z, J ( t, z ) b ( z + h ) i ≤ C (1 + | z | ) . (4.4)Note that the bound clearly holds for | z | ≤ ( C + 1) ∨ (4 C ) since J is bounded and | z + h | ≤ (2 C + 1) ∨ (5 C ) in this case. From now on, we assume that | z | ≥ ( C + 1) ∨ (4 C ). Let b ( z + h ) = α ( z + h ) + βv where α, β ∈ R and v ⊥ ( z + h ), | v | = 1. From (4.2), we see that C (1 + | z + h | ) ≥ h b ( z + h ) , z + h i = α | z + h | . Since | z | ≥ C + 1, we have | z + h | ≥ α ≤ C . The bound (4.3) implies that | β | ≤ C (1 + | z + h | ). We have h z, J ( t, z ) b ( z + h ) i = α h z, J ( t, z )( z + h ) i + β h z, J ( t, z ) v i . For the second term, we use the Cauchy Schwarz inequality to see that | β h z, J ( t, z ) v i| ≤ | β || z || J ( t, z ) | ≤ C | z | . Concerning the first term, note that | h | ≤ C ≤ | z | / 4, thus h z, J ( t, z )( z + h ) i = h z, z + h i + h z, ( J ( t, z ) − Id)( z + h ) i ≥ | z | − | z || h | − | z | / − | z || h | / ≥ | z | − | z | / − | z | / − | z | / | z | / > . (4.5)Therefore, we obtain the bound α h z, J ( t, z )( z + h ) i ≤ C | z | . This shows that indeed (4.4) holds for every z . Gronwall’s Lemma implies the claim for the sup-norm.We proceed with the bound for the total variation norm of t z t . Let [ a, b ] be a subinterval of[0 , T ] on which | z t | ≤ ( C + 1) ∨ (4 C ) + 1 for all t ∈ [ a, b ]. In this case, for every a ≤ u ≤ v ≤ b , | z v − z u | ≤ Z vu J ( s, z s ) b ( z s + h s ) ds ≤ C ( v − u )which implies that k z k − var;[ a,b ] ≤ C ( b − a ) . (4.6)Now assume that [ a, b ] is a subinterval of [0 , T ] on which | z t | ≥ ( C + 1) ∨ (4 C ) for all t ∈ [ a, b ]. Ina first step, we show that the total variation of t 7→ | z t | has a good bound on [ a, b ]. As above, wecan show that ddt | z t | = h z t , ˙ z t i| z t | ≤ C (1 + | z t | ) ≤ C (1 + k z k ∞ ;[ a,b ] )(4.7) OUGH DIFFERENTIAL EQUATIONS WITH UNBOUNDED DRIFT TERM 15 for t ∈ ( a, b ). Since t 7→ | z t | =: f ( t ) has finite total variation, there is a decomposition f = f + − f − where f + and f − are increasing functions, and k f k − var;[ a,b ] = k f + k − var;[ a,b ] + k f − k − var;[ a,b ] . The estimate (4.7) implies that k f + k − var;[ a,b ] ≤ C (1 + k z k ∞ ;[ a,b ] )( b − a ) . Since f is nonnegative, k f − k − var;[ a,b ] = k f + k − var;[ a,b ] + f ( a ) − f ( b ), therefore k| z |k − var;[ a,b ] = Z ba (cid:12)(cid:12)(cid:12)(cid:12) ddt | z t | (cid:12)(cid:12)(cid:12)(cid:12) dt ≤ C (1 + k z k ∞ ;[ a,b ] )( b − a ) + | z a | − | z b | . (4.8)We proceed with proving a bound for the total variation of t z t on [ a, b ]. By the triangleinequality, | ˙ z t | ≤ (cid:12)(cid:12)(cid:12)(cid:12) h ˙ z t , z t | z t | i (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ˙ z t − h ˙ z t , z t | z t | i z t | z t | (cid:12)(cid:12)(cid:12)(cid:12) . (4.9)We will first estimate the second term. Fix t ∈ ( a, b ) and define h as above. As before, we decompose b ( z + h ) = α ( z + h ) + βv with v ⊥ ( z + h ), | v | = 1. Then, (cid:12)(cid:12)(cid:12)(cid:12) ˙ z t − h ˙ z t , z t | z t | i z t | z t | (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) J ( t, z ) b ( z + h ) − h J ( t, z ) b ( z + h ) , z | z | i z | z | (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) αJ ( t, z )( z + h ) + βJ ( t, z ) v − h J ( t, z )( α ( z + h ) + βv ) , z | z | i z | z | (cid:12)(cid:12)(cid:12)(cid:12) ≤ | α | (cid:12)(cid:12)(cid:12)(cid:12) J ( t, z )( z + h ) − h J ( t, z )( z + h ) , z | z | i z | z | (cid:12)(cid:12)(cid:12)(cid:12) + 3 | β | . Note that the vector h J ( t, z )( z + h ) , z | z | i z | z | is the orthogonal projection of J ( t, z )( z + h ) on the spacespan { z } . Since the distance between J ( t, z )( z + h ) and its orthogonal projection is minimal amongall elements in span { z } , we obtain in particular (cid:12)(cid:12)(cid:12)(cid:12) J ( t, z )( z + h ) − h J ( t, z )( z + h ) , z | z | i z | z | (cid:12)(cid:12)(cid:12)(cid:12) ≤ | J ( t, z )( z + h ) − z | = | ( J ( t, z ) − I m ) z + J ( t, z ) h |≤ | z | + 32 | h |≤ | z | . In (4.5) we have seen that |h J ( t, z )( z + h ) , z/ | z |i| ≥ | z | . Putting these estimates together implies that (cid:12)(cid:12)(cid:12)(cid:12) ˙ z t − h ˙ z t , z t | z t | i z t | z t | (cid:12)(cid:12)(cid:12)(cid:12) ≤ | β | + 7 | α | |h J ( t, z )( z + h ) , z/ | z |i|≤ | β | + 7 |h J ( t, z )( α ( z + h ) + βv ) , z/ | z |i| + 7 | β || J ( t, z ) |≤ | β | + 7 (cid:12)(cid:12)(cid:12)(cid:12) h ˙ z t , z t | z t | i (cid:12)(cid:12)(cid:12)(cid:12) . Going back to (4.9), we obtain the estimate | ˙ z t | ≤ | β | + 8 (cid:12)(cid:12)(cid:12)(cid:12) h ˙ z t , z t | z t | i (cid:12)(cid:12)(cid:12)(cid:12) for every t ∈ ( a, b ). We have already seen that | β | ≤ C (1 + | z t | ) for some constant C . From (4.8), Z ba (cid:12)(cid:12)(cid:12)(cid:12) h ˙ z t , z t | z t | i (cid:12)(cid:12)(cid:12)(cid:12) dt ≤ C (1 + k z k ∞ ;[ a,b ] )( b − a ) + | z a | − | z b | . Therefore, we can conclude that there is a constant C such that k z k − var;[ a,b ] = Z ba | ˙ z t | dt ≤ C (1 + k z k ∞ ;[ a,b ] )( b − a ) + | z a | − | z b | . (4.10)Now define S := { t ∈ (0 , T ) : | z t | < ˆ C + 1 } and U := { t ∈ (0 , T ) : | z t | > ˆ C } . Note that both sets are open in R . Hence, there are countable sets I and J such that { ( a k , b k ) } k ∈ I and { ( a k , b k ) } k ∈ J are disjoint families of open intervals for which S = [ k ∈ I ( a k , b k ) and U = [ k ∈ J ( a k , b k ) . Clearly, k z k − var;[0 ,T ] ≤ X k ∈ I ∪ J k z k − var;[ a k ,b k ] . From (4.6), we have X k ∈ I k z k − var;[ a k ,b k ] ≤ C T. Note that for any k ∈ J , by continuity, lim t ց a k | z t | > ˆ C implies that a k = 0, therefore | z a k | = | ξ | .By the same reasoning, lim t ր b k | z t | > ˆ C implies b k = T and | z b k | = | z T | . In particular, there are atmost two elements k , k ∈ J for which | z a ki | 6 = | z b ki | , i = 1 , 2. Using (4.10), this implies that X k ∈ J k z k − var;[ a k ,b k ] ≤ C (1 + k z k ∞ ;[ a,b ] ) T + ( | ξ | − ˆ C ) + − ( | z T | − ˆ C ) + and we can conclude our assertion. (cid:3) The next Lemma states similar conditions which will imply uniqueness. Lemma 4.2. Consider two solutions z , z : [0 , T ] → R m to the equations ˙ z it = J i ( t, z it ) b ( ψ i ( t, z it )); t ∈ [0 , T ] z i = ξ i ∈ R m ; i = 1 , . (4.11) Let R > be such that R ≥ sup t ∈ [0 ,T ] | z it | for i = 1 , . We assume the following: (1) b : R m → R m is continuous and satisfies the following conditions: (i) There exists a constant C = C ( R ) such that h b ( ξ ) − b ( ζ ) , ξ − ζ i ≤ C | ξ − ζ | for every ξ, ζ ∈ B (0 , R ) . (4.12) OUGH DIFFERENTIAL EQUATIONS WITH UNBOUNDED DRIFT TERM 17 (ii) There exists a constant C = C ( R ) such that (cid:12)(cid:12)(cid:12)(cid:12) b ( ξ ) − b ( ζ ) − h b ( ξ ) − b ( ζ ) , ξ − ζ i ( ξ − ζ ) | ξ − ζ | (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | ξ − ζ | for every ξ, ζ ∈ B (0 , R ) with ξ − ζ = 0 . (4.13)(2) Both J , J : [0 , T ] × R m → R m × m are continuous and there exists a constant C = C ( R ) such that sup t ∈ [0 ,T ] | J ( t, ξ ) − J ( t, ζ ) | ≤ C | ξ − ζ | for every ξ, ζ ∈ B (0 , R ) . Moreover, we assume that sup t ∈ [0 ,T ]; | ξ |≤ R | J ( t, ξ ) − I m | ≤ where I m denotes the identity matrix in R m × m . (3) Both ψ , ψ : [0 , T ] × R m → R m are continuous and sup t ∈ [0 ,T ] | ψ ( t, ξ ) − ξ − ψ ( t, ζ ) + ζ | ≤ | ξ − ζ | for every ξ, ζ ∈ B (0 , R ) . (4) There exists an ε > such that sup t ∈ [0 ,T ]; ξ ∈ B (0 ,R ) | J ( t, ξ ) − J ( t, ξ ) | ≤ ε and sup t ∈ [0 ,T ]; ξ ∈ B (0 ,R ) | b ( ψ ( t, ξ )) − b ( ψ ( t, ξ )) | ≤ ε. Let C ≥ sup s ∈ [0 ,T ] , | ξ |≤ R | b ( ψ ( s, ξ )) | . Then there is a constant C depending on C i , i = 1 , , , ,on R and on T such that sup t ∈ [0 ,T ] | z t − z t | ≤ C ( | ξ − ξ | + √ ε ) . In particular, if b satisfies the conditions (4.12) and (4.13) locally on every compact set and if ψ = ψ =: ψ and J = J =: J satisfy the stated spatial conditions globally, solutions to (4.1) withthe same initial condition are unique.Proof. We have | z t − z t | = | ξ − ξ | + 2 Z t h z s − z s , J ( s, z s ) b ( ψ ( s, z s )) − J ( s, z s ) b ( ψ ( s, z s )) i ds = | ξ − ξ | + 2 Z t h z s − z s , ( J ( s, z s ) − J ( s, z s )) b ( ψ ( s, z s )) i ds + 2 Z t h z s − z s , J ( s, z s )( b ( ψ ( s, z s )) − b ( ψ ( s, z s ))) i ds for every t ∈ [0 , T ]. By the Cauchy-Schwarz inequality, (cid:12)(cid:12)(cid:12)(cid:12)Z t h z s − z s , ( J ( s, z s ) − J ( s, z s )) b ( ψ ( s, z s )) i ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ C C Z t | z s − z s | ds + 2 εC T R for every t ∈ [0 , T ]. We aim to prove that a similar estimate holds for the second integral Z t h z s − z s , J ( s, z s )( b ( ψ ( s, z s )) − b ( ψ ( s, z s ))) i ds (4.14) = Z t h z s − z s , J ( s, z s )( b ( ψ ( s, z s )) − b ( ψ ( s, z s ))) i ds (4.15) + Z t h z s − z s , J ( s, z s )( b ( ψ ( s, z s )) − b ( ψ ( s, z s ))) i ds (4.16)The integral (4.15) can be estimated by (cid:12)(cid:12)(cid:12)(cid:12)Z t h z s − z s , J ( s, z s )( b ( ψ ( s, z s )) − b ( ψ ( s, z s ))) i ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ T Rε. We proceed with the integral (4.16). Fix s ∈ [0 , T ] and set h is := ψ ( s, z is ) − z is . To simplify notation,set z i := z is and h i := h is , i = 1 , 2. Choose α , β ∈ R such that b ( z + h ) − b ( z + h ) = α ( z + h − z − h ) + βv where v ⊥ ( z + h − z − h ), | v | = 1. From our conditions on b , α ≤ C and | β | ≤ C | z + h − z − h | . Note that | h − h | = | ψ ( s, z ) − z − ψ ( s, z ) + z | ≤ | z − z | and h z − z , J ( s, z )( b ( ψ ( s, z )) − b ( ψ ( s, z ))) i = α h z − z , J ( s, z )( z + h − z − h ) i + β h z − z , J ( s, z ) v i . The second term can be estimated using Cauchy-Schwarz and our bound for β : | β h z − z , J ( s, z ) v i| ≤ C | z − z | where C can be chosen uniformly over s ∈ [0 , T ]. For the first term, note that h z − z , J ( s, z )( z + h − z − h ) i = h z − z , z + h − z − h i + h z − z , ( J ( s, z ) − Id)( z + h − z − h ) i≥ | z − z | − | z − z || h − h | − | z − z | − | z − z || h − h |≥ | z − z | − (cid:18) 14 + 12 + 18 (cid:19) | z − z | ≥ . This implies that α h z − z , J ( s, z )( z + h − z − h ) i ≤ C | z − z | for some C which does not depend on s . This shows that there is a constant C such that theintegral (4.14) can be bounded by 3 T Rε + C Z t | z s − z s | ds OUGH DIFFERENTIAL EQUATIONS WITH UNBOUNDED DRIFT TERM 19 for every t ∈ [0 , T ]. Together with our former estimates, this shows that | z t − z t | ≤ | ξ − ξ | + εT R (3 + 2 C ) + C Z t | z s − z s | ds for every t ∈ [0 , T ]. Applying Gronwall’s Lemma shows the claim. (cid:3) The next theorem contains the second main result of our work. Theorem 4.3. Let x be a weak geometric p -rough path for some p ≥ and let ω be a controlfunction which controls its p -variation. Assume that σ = ( σ , . . . , σ d ) is a collection of Lip γ +1 -vector fields for some γ > p and choose ν ≥ | σ | Lip γ . Assume that b : R m → R m is continuous,satisfies the local conditions (4.12) , (4.13) on every compact set and the growth conditions (4.2) and (4.3) .Then the following holds true: (1) The solution semiflow φ to (1.1) exists and is continuous. (2) There is a constant C depending on p , γ , ν , the constants C and C from (4.2) and (4.3) and on C := sup {| b ( ξ ) | : | ξ | ≤ } such that k φ (0 , · , ξ ) k ∞ ;[0 ,T ] ≤ C exp( CT )(1 + N ( ω ) + | ξ | )(4.17) and k φ (0 , · , ξ ) k p − var ;[0 ,T ] ≤ C exp( CT )(1 + N ( ω ) + | ξ | )(4.18) hold for every ξ ∈ R m . (3) For every initial condition ξ ∈ R m , the path t φ (0 , t, ξ ) is a solution to (1.1) in the senseof Friz–Victoir.Remark . (i) For m = 1, the growth condition (4.3) is always satisfied. For m ≥ 2, assumingonly condition (4.2) does in general not prevent explosion of the solution in finite time; seethe discussion in the introduction.(ii) The local conditions (4.12) and (4.13) are satisfied in the case when b is locally Lipschitzcontinuous. In the proof, it will become clear that they are used in order to prove uniquenessand the continuity statements. Dropping them would still imply a priori estimates forsolutions to (1.1).(iii) Note that we can in general not expect to obtain a bound similar to (4.18) for the H¨oldernorm, not even for x being a H¨older rough path. Indeed, let σ ≡ m = 1 and b ( v ) = −| v | .Let y be the solution to (1.1) with initial condition ξ . Then k y k − H¨ol;[0 ,T ] ≥ lim t ց | y t − y | t = | b ( ξ ) | = | ξ | which shows that the H¨older norm can not grow at most linearly in | ξ | . Proof of Theorem 4.3. In the following proof, C will be a constant which may depend on p , γ , ν and the constants C , C , C , but whose actual value may change from line to line. As already seen in Theorem 3.1, our assumptions on σ imply that the solution flow ψ to (2.3)exists, is twice differentiable and has a twice differentiable inverse. Define J to be the inverse of itsderivative and χ as in Definition 2.1. From [FV10, Proposition 11] and the chain rule,sup s,t ∈ [0 ,T ]; ξ ∈ R m | D ξ J ( s, t, ξ ) | < ∞ and in particular, there exists a constant C such thatsup s,t ∈ [0 ,T ] | J ( s, t, ξ ) − J ( s, t, ζ ) | ≤ C | ξ − ζ | holds for every ξ, ζ ∈ R m . Using 2.3, we can choose δ > ω ( s, t ) ≤ δ implies that sup u ∈ [ s,t ] | J ( s, u, ξ ) − I m | ≤ , sup u ∈ [ s,t ]; ξ ∈ R m | ψ ( s, u, ξ ) − ξ | ≤ u ∈ [ s,t ] | ψ ( s, u, ξ ) − ξ − ψ ( s, u, ζ ) + ζ | ≤ | ξ − ζ | for every ξ, ζ ∈ R m . Continuity of ψ , J and b , Lemma 4.1 and Lemma 4.2 imply that [ s, t ] ∋ u χ s ( u, ξ ) is well defined for such [ s, t ] and every ξ ∈ R m . The set I := { [ s, t ] : ω ( s, t ) ≤ δ } satisfies the assumptions in Definition 2.1, and by Lemma 2.2, φ is well defined as the semiflow to(1.1).We proceed with the bound for the sup-norm. Set ˜ ω ( s, t ) = | t − s | + ω ( s, t ), define ( τ n ) by setting τ := 0 and τ n +1 := inf { u : ˜ ω ( τ n , u ) ≥ δ, τ n < u ≤ T } ∧ T, and set N := N δ (˜ ω ) = sup { n ∈ N : τ n < T } . Choosing δ smaller if necessary, Lemma 4.1 shows thatsup t ∈ [ τ n ,τ n +1 ] | χ τ n ( t, ξ ) | ≤ (1 + | ξ | ) exp( C | τ n +1 − τ n | )(4.19)holds for every n = 0 , . . . , N . As seen in the proof of Theorem 3.1, this bound implies thatsup t ∈ [0 ,T ] | φ (0 , t, ξ ) | ≤ CT )(1 + 2 N + | ξ | )and we can conclude as seen in Theorem 3.1.Next, we prove the bound (4.18). Let τ n ≤ u < v ≤ τ n +1 . As seen in the proof of Theorem 3.1,we can use the semiflow property of φ and the triangle inequality to obtain the estimate | φ (0 , v, ξ ) − φ (0 , u, ξ ) | ≤ Cνω ( u, v ) /p + 2 | χ τ n ( u, φ (0 , τ n , ξ )) − χ τ n ( v, φ (0 , τ n , ξ )) | . Using the total variation bound and the bound for the sup-norm in Lemma 4.1, we see that k φ (0 , · , ξ ) k p − var;[ τ n ,τ n +1 ] ≤ Cνδ /p + 2 k χ τ n ( · , φ (0 , τ n , ξ )) k − var;[ τ n ,τ n +1 ] ≤ Cνδ /p + Ce CT (1 + | φ (0 , τ n , ξ ) | ) | τ n +1 − τ n | + 2( φ (0 , τ n , ξ ) − + − χ τ n ( τ n +1 , φ (0 , τ n , ξ )) − + . OUGH DIFFERENTIAL EQUATIONS WITH UNBOUNDED DRIFT TERM 21 Therefore, k φ (0 , · , ξ ) k p − var;0 ,T ] ≤ N X n =0 k φ (0 , · , ξ ) k p − var;[ τ n ,τ n +1 ] ≤ Cν ( N δ (˜ ω ) + 1) δ /p + Ce CT (1 + k φ (0 , · , ξ ) k ∞ ;[0 ,T ] ) T + 2 N X n =0 ( φ (0 , τ n , ξ ) − + − ( χ τ n ( τ n +1 , φ (0 , τ n , ξ )) − + . For the last sum, we can estimate N X n =0 ( φ (0 , τ n , ξ ) − + − ( χ τ n ( τ n +1 , φ (0 , τ n , ξ )) − + ≤ | ξ | + N − X n =0 | φ (0 , τ n +1 , ξ ) − χ τ n ( τ n +1 , φ (0 , τ n , ξ )) | and by the semiflow property of φ , | φ (0 , τ n +1 , ξ ) − χ τ n ( τ n +1 , φ (0 , τ n , ξ )) | = | ψ ( τ n , τ n +1 , χ τ n ( τ n +1 , φ (0 , τ n , ξ ))) − χ τ n ( τ n +1 , φ (0 , τ n , ξ )) |≤ N X n =0 ( φ (0 , τ n , ξ ) − + − ( χ τ n ( τ n +1 , φ (0 , τ n , ξ )) − + ≤ | ξ | + N δ (˜ ω ) . As seen in Theorem 3.1, this implies the claim.Next, we show that φ is continuous. As in Theorem 3.1, we can use the semiflow property to seethat it is enough to prove that [ s, t ] × R m ∋ ( u, ξ ) φ ( u, v , ξ ) and [ s, t ] × R m ∋ ( v, ξ ) φ ( u , v, ξ )are continuous for fixed u , v ∈ [ s, t ] where [ s, t ] is any subinterval of [0 , T ] with the property that | t − s | ≤ δ . Fix such an interval and choose sequences u n → u , v n → v and ξ n → ξ for n → ∞ .We first prove that χ u ( v n , ξ n ) → χ u ( v , ξ )for n → ∞ . By the triangle inequality, | χ u ( v n , ξ n ) − χ u ( v , ξ ) | ≤ | χ u ( v n , ξ n ) − χ u ( v n , ξ ) | + | χ u ( v n , ξ ) − χ u ( v , ξ ) | . The second term converges to 0 for n → ∞ by time continuity of the solution. From Lemma 4.1,sup n ≥ k χ u ( · , ξ n ) k ∞ ;[ u ,t ] + k χ u ( · , ξ ) k ∞ ;[ u ,t ] < ∞ and therefore | χ u ( v n , ξ n ) − χ u ( v n , ξ ) | ≤ k χ u ( · , ξ n ) − χ u ( · , ξ ) k ∞ ;[ u ,t ] → n → ∞ by Lemma 4.2. From continuity of ψ , this implies that [ s, t ] × R m ∋ ( v, ξ ) φ ( u , v, ξ )is continuous. Next, we use again the triangle inequality to see that | χ u n ( v , ξ n ) − χ u ( v , ξ ) | ≤ | χ u n ( v , ξ n ) − χ u n ( v , ξ ) | + | χ u n ( v , ξ ) − χ u ( v , ξ ) | . The first term converges to 0 again by Lemma 4.2. It remains to show that χ u n ( v , ξ ) → χ u ( v , ξ ) as n → ∞ . To do so, we first claim that J ( u n , · , · ) → J ( u , · , · ) and b ( ψ ( u n , · , · )) → b ( ψ ( u , · , · ))converge uniformly on compact sets as n → ∞ . Indeed: For the second claim, since b is continuous,it is enough to prove that ψ ( u n , · , · ) → ψ ( u , · , · ) converges uniformly on compact sets as n → ∞ .Choose t ∈ [0 , T ] and ξ ∈ R m . From the flow property, ψ ( u n , t, ξ ) = ψ (0 , t, ψ ( u n , , ξ )) , and Lemma 2.3 shows that it is enough to prove that ψ ( u n , , · ) → ψ ( u , , · ) converges uniformly oncompact sets as n → ∞ . By the flow property, ψ ( u n , , · ) = ψ − (0 , u n , · ), seen as homeomorphismson R m . The inverse flow ψ − is generated by a rough differential equation where we let the roughpath run backwards in time (cf. [FV10, Section 11.2]), therefore the second claim follows by thestandard estimates for solutions to rough differential equations, see Lemma 2.3. Concerning thefirst claim, the chain rule shows that J ( u n , t, ξ ) = D ζ ψ − ( u n , t, ζ ) | ζ = ψ ( u n ,t,ξ ) , therefore it is enough to show that D ξ ψ − ( u n , · , ξ ) → D ξ ψ − ( u , · , ξ ) converges uniformly on com-pact sets as n → ∞ . From D ξ ψ − ( u n , t, ξ ) = D ξ ψ ( t, u n , ξ ) and the identity D ξ ψ ( t, u n , ξ ) = D ζ ψ (0 , u n , ζ ) | ζ = ψ ( t, ,ξ ) D ξ ψ ( t, , ξ ) , we see that it is sufficient to prove that D ξ ψ (0 , u n , ξ ) → D ξ ψ (0 , u , ξ ) converges uniformly oncompact sets as n → ∞ . Assume first that 0 ≤ u n ≤ u for all n ∈ N . Using 2.3, we can deducethe estimate | D ξ ψ (0 , u n , ξ ) − D ξ ψ (0 , u , ξ ) | ≤ Cνω ( u n , u ) /p exp( Cν p ω (0 , T ))and the claim follows in this case. If 0 ≤ u ≤ u n for all n ∈ N , the same estimate holds with ω ( u n , u ) replaced by ω ( u , u n ) which again implies the claim.Now set y nw := χ u n ( w, ξ ) for n ≥ R := sup n ≥ k y n k ∞ ;[ u n ,t ] . Note that R is finite by Lemma 4.1. We first prove right-continuity, i.e. we assume that u ≤ u n for all n ≥ 0. Note that for every n ≥ y v = y u n + Z vu n J ( u , w, y w ) b ( ψ ( u , w, y w )) dw. By uniform convergence of J ( u n , · , · ) and b ( ψ ( u n , · , · )), we can use Lemma 4.2 to see that for anygiven ε > n large enough such that | y v − y nv | ≤ C ( ε + | y u n − ξ | )holds for every n ∈ N where C does not depend on n or ε . By continuity of y , we see that theright hand side can be made arbitrary small for large n , therefore y nv → y v as n tends to infinity.Now we prove left-continuity, i.e. we assume that u n ≤ u . Noting that for every n ≥ y nv = y nu + Z vu J ( u n , w, y nw ) b ( ψ ( u n , w, y nw )) dw, we can argue as before to see that for any given ε > 0, we can find large n such that | y v − y nv | ≤ C ( ε + | ξ − y nu | ) . OUGH DIFFERENTIAL EQUATIONS WITH UNBOUNDED DRIFT TERM 23 Therefore, it suffices to prove that y nu → ξ for n → ∞ , but this follows immediately from theestimate | y nu − ξ | ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z u u n J ( u n , w, y nw ) b ( ψ ( u n , w, y nw )) dw (cid:12)(cid:12)(cid:12)(cid:12) ≤ | u n − u | sup u,v ∈ [ s,t ] , | ξ |≤ R | J ( u, v, ξ ) b ( ψ ( u, v, ξ )) | . Thus we have proven left- and right continuity of u χ u ( v , ξ ) which implies that indeed φ iscontinuous.We finally show that t φ (0 , t, ξ ) is a solution to (1.1) in the sense of Friz–Victoir. Fix a roughpath x = x , an initial condition ξ ∈ R m and let ( x n ) be a sequence of smooth paths for whichthe lifts x n satisfy (1.2). Fix some p ′ ∈ ( p, γ ). Set ω n ( s, t ) := k x n k p ′ p ′ − var , [ s,t ] . The ω n are controlfunctions which control the p ′ -variation of x n for every n ≥ 0. By interpolation [FV10, Lemma8.16], d p ′ − var;[0 ,T ] ( x , x n ) → n → ∞ . Choose δ > d p ′ − var;[0 ,T ] ( x , x n ) ≤ δ /p ′ / n ≥ 1. Let s ≤ t such that ω ( s, t ) ≤ δ/ p ′ . It follows that ω n ( s, t ) = k x n k p ′ p ′ − var , [ s,t ] ≤ δ/ p ′ − k x k p ′ p ′ − var , [ s,t ] ≤ δ. Therefore, I := { [ s, t ] : ω ( s, t ) ≤ δ/ p ′ } is a family of intervals for which ω n ( s, t ) ≤ δ for every [ s, t ] ∈ I and every n ≥ 0. We set φ n := φ x n and use a similar notation for ψ , J and χ . We have to show that φ n (0 , · , ξ ) → φ (0 , · , ξ ) uniformy as n → ∞ . Fix a sequence 0 = τ < τ < . . . < τ N < τ N +1 = T with [ τ i , τ i +1 ] ∈ I for each i = 0 , . . . , N . Fix t ∈ [0 , τ ]. Then | φ (0 , t, ξ ) − φ n (0 , t, ξ ) | ≤ k ψ (0 , · , χ ( · , ξ )) − ψ n (0 , · , χ ( · , ξ )) k ∞ ;[0 ,τ ] + k ψ n (0 , · , χ ( t, ξ )) − ψ n (0 , · , χ n ( t, ξ )) k ∞ ;[0 ,τ ] The first term converges to 0 for n → ∞ by [FV10, Theorem 11.12]. Concerning the second term,we can use Lemma 2.3 to see that k ψ n (0 , · , χ ( t, ξ )) − ψ n (0 , · , χ n ( t, ξ )) k ∞ ;[0 ,τ ] ≤ C | χ ( t, ξ ) − χ n ( t, ξ ) | . Therefore it is enough to show that χ n ( · , ξ ) → χ ( · , ξ ) as n → ∞ uniformly on [0 , τ ]. From [FV10,Theorem 11.12 and Theorem 11.13], we can deduce that J n → J and ψ n → ψ converge uniformlyas n → ∞ , thus the claimed convergence follows from Lemma 4.2. This proves that k φ (0 , · , ξ ) − φ n (0 , · , ξ ) k ∞ ;[0 ,τ ] → n → ∞ . Now assume that we have shown that k φ (0 , · , ξ ) − φ n (0 , · , ξ ) k ∞ ;[0 ,τ i ] → for some i = 1 , . . . , N . Let t ∈ [ τ i , τ i +1 ]. By the semiflow property, | φ (0 , t, ξ ) − φ n (0 , t, ξ ) | = | φ ( τ i , t, φ (0 , τ i , ξ )) − φ n ( τ i , t, φ n (0 , τ i , ξ )) |≤ k ψ ( τ i , · , χ τ i ( · , φ (0 , τ i , ξ ))) − ψ n ( τ i , · , χ τ i ( · , φ (0 , τ i , ξ ))) k ∞ ;[ τ i ,τ i +1 ] + k ψ n ( τ i , · , χ τ i ( · , φ (0 , τ i , ξ ))) − ψ n ( τ i , · , χ nτ i ( · , φ (0 , τ i , ξ ))) k ∞ ;[ τ i ,τ i +1 ] + k ψ n ( τ i , · , χ nτ i ( · , φ (0 , τ i , ξ ))) − ψ n ( τ i , · , χ nτ i ( · , φ n (0 , τ i , ξ ))) k ∞ ;[ τ i ,τ i +1 ] The first term converges to 0 again by [FV10, Theorem 11.12]. The other two terms converge to 0if we can show that k χ τ i ( · , φ (0 , τ i , ξ )) − χ nτ i ( · , φ (0 , τ i , ξ )) k ∞ ;[ τ i ,τ i +1 ] → k χ nτ i ( · , φ (0 , τ i , ξ )) − χ nτ i ( · , φ n (0 , τ i , ξ )) k ∞ ;[ τ i ,τ i +1 ] → n → ∞ . This follows again by using Lemma 4.2 together with our induction hypothesis. Thisshows that k φ (0 , · , ξ ) − φ n (0 , · , ξ ) k ∞ ;[ τ i ,τ i +1 ] → n → ∞ , and by induction hypothesis the convergence also holds uniformly on [0 , τ i +1 ] whichfinishes the induction. This finally proves uniform convergence on the whole time interval whichshows that indeed t φ (0 , t, ξ ) is a solution in the sense of Friz-Victoir. (cid:3) Acknowledgements. SR would like to thank Peter Friz for valuable discussions about flow de-compositions for rough differential equations. Financial support by the DFG via Research UnitFOR 2402 is gratefully acknowledged. References [Bai15] Isma¨el Bailleul. Flows driven by rough paths. Rev. Mat. Iberoam. , 31(3):901–934, 2015.[BFRS16] Christian Bayer, Peter K. Friz, Sebastian Riedel, and John Schoenmakers. From rough path estimates tomultilevel Monte Carlo. to appear in SIAM Journal on Numerical Analysis , 2016+.[CHJ13] Sonja Cox, Martin Hutzenthaler, and Arnulf Jentzen. 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Sebastian Riedel, Institut f¨ur Mathematik, Technische Universit¨at Berlin, Germany E-mail address : [email protected] Michael Scheutzow, Institut f¨ur Mathematik, Technische Universit¨at Berlin, Germany E-mail address ::