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3. Rough pathsGuide for this sectionHölder p-rough paths, which control the rough diﬀerential equationsdxt F(xt )X(dt),dϕt F X(dt),and play the role of the control h in the model classical ordinary diﬀerential equationdxt Vi (xt ) dhit F(xt ) dhtare deﬁned in section 3.1.2. As R -valued paths, they are not regular enough for theformulaµts (x) x Xtsi Vi (x)to deﬁne an approximate ﬂow, as in the classical Euler scheme studied in exercice 1.The missing bit of information needed to stabilize the situation is a substitute of thetnon-existing iterated integrals s Xrj dXrk , and higher order iterated integrals, whichprovide a partial description of what happens to X during any time interval (s, t).A (Hölder) p-rough path is a multi-level object whose higher order parts provideprecisely that information. We saw in the introduction that iterated integrals appearnaturally in Taylor-Euler expansions of solutions to ordinary diﬀerential equations;they provide higher order numerical schemes like Milstein’ second order scheme. It isan important fact that p-rough paths take values in a very special kind of algebraicstructure, whose basic features are explained in section 3.1.1. A Hölder p-roughpath will then appear as a kind of p1 -Hölder path in that space. We shall then studyin section 3.2 the space of p-rough path for itself.3.1. Definitiona p-rough path.Iterated integrals, as they appear for instance t ofyt y rjkin the form s s dhr dhu or s s s (· · · ) are multi-indexed quantities. A usefulformalism to with such object is provided by the notion of tensor produc. Weﬁrst start our investigation by recalling some elementary facts about that notion.Eventually, all what will be used for practical computations on rough diﬀerentialequations will be a product operation very similar to the product operation onpolynomials. This abstract setting however greatly clariﬁes the meaning of thesecomputations.3.1.1. An algebraic prelude: tensor algebra over R and free nilpotent Lie group. Letﬁrst recall what the algebraic tensor product U V of any two Banach spaces Uand V is. Denote by V’ the set of all continuous linear forms on V. Given u Uand v V, we deﬁne a continuous linear map on V’ setting(u v)(v ) (v , v) u,for any v V . The algebraic tensor product U V is the set of all ﬁnite linearcombinations of such maps. Elementary elements u v) are 1-dimensional rankmaps. Note that an element of U V can have several diﬀerent decompositions asa sum of elementary elements; this has no consequences as they all deﬁne the samemap from V’ to U.1

2As an example, (R ) (R ) is the set of all linear maps from R to itself that isL(R ). We keep that interpretation for (R ) (R ) as R and (R ) are canonicallyidentiﬁed. To see which element of L(R ) corresponds to u v, it suﬃces to lookat the image of the j th vector j of the canonical basis by the map; it gives the j thcolumn of the matrix of u v in the canonical basis. We have (u v) j (v, j ) u.(N )For N N { }, write T that R(N )of T 0for the direct sumN r R , with the conventionr 0NNr 0r 0stands for R. Denote by a ar and b br two generic elements(N ). The vector space T is an algebra for the operationsNa b (ar br ),r 0(3.1)Nab c ,rr 0with c rr ak br k (R ) rk 0It is called the (truncated) tensor algebra of R (if N is ﬁnite). Note thesimilarity between these rules and the analogue rules for addition and product ofpolynomials.( )( )The exponential map exp : T T ( )T are deﬁned by the usual series(3.2)exp(a) ann 0n!,( )and the logarithm map log : T log(b) ( 1)nn 1( )n (1 b)n ,( )(N )with the convention bf a0 1 R T . Denote by πN : T T the natural(N )projection. We also denote by exp and log the restrictions to T of the maps(N ),1(N ),0πN exp and πN log respectively. Denote by T , resp. T , the elements(N )(N ),1a0 · · · cN of T such that a0 0, resp. a0 1. All the elements of T are(N ),0(N ),1(N ),1(N ),0 T and log : T T are smooth reciprocalinvertible, and exp : T bijections.(N ),1is naturally equipped with a norm deﬁned by the formulaThe set T a : i 1 a i ,Eucli 1 where ai Eucl stands for the Euclidean norm of ai (R ) i identiﬁed with anielement of R by looking at its coordinates in the canonical basis. The choice(N ),1is naturally equipped with a dilationof power 1i comes from the fact that T operation δλ (a) 1, λa1 , . . . , λN aN ,

3so the norm · is homogeneous with respect to this dilation, in the sense that onehas δλ (a) λ a(N ),1for all λ R, and all a T .(N )The formula [a, b] ab ba, deﬁnes a Lie bracket on T . Deﬁne inductively( )( )F F 1 R , considered as a subset of T , and F n 1 [F, F n ] T .(N )1NDefinition 1. The Lie algebra gNin T is generated by the F , . . . , Fcalled the N-step free nilpotent Lie algebra. As a consequenceof Baker-Campbell-Hausdorf-Dynkin formula, the subset (N ),1Nexp g of T is a group for the multiplication operation. It is called the(N )N-step nilpotent Lie group on R and denoted by G .As all ﬁnite dimensional Lie groups, the N-step nilpotent Lie group is equippedwith a natural (sub-Riemannian) distance inherited from its manifold structure. Its(N )(N )deﬁnition rests on the fact that the element au of T is for any a G and(N )(N )(N )u R T a tangent vector to G at point a (as u is tangent to G at theidentity and tnagent vectors are transported by left translation in the group). Sothe ordinary diﬀerential equationdat at ḣt(N )makes sense for any R -valued smooth control h, and deﬁnes a path in G from the identity. We deﬁne the size a of a by the formula 1 ḣt dt, a infstarted0where the inﬁmum is over the set of all smooth controls h such that a1 a. Thisset is non-empty as a exp gN can be written as a1 for some piecewise C 1 control,as a consequence of a theorem of sub-Riemannian geometry due to Chow; see forinstance the textbook [11] for a nice account of that theorem.The distance between (N ) 1 any two points a and b of G is then deﬁned as a b . It is homogeneous in the (N )sense that if a exp(u), with u R T , then exp(λu) λ a , for all λ R(N )and all u R T .(N )This way of deﬁning a distance is intrinsic to G and classical in geometry. From(N )(N )an extrinsic point of view, one can also consider G as a subset of T and use(N )the ambiantmetric to deﬁne the distance between any two points a and b of G as a 1 b . It can be proved (this is elementary, see e.g. proposition 10 in Appendix(N )A of [12], pp. 76-77) that the two norms · and · on G are equivalent, so onecan equivalently work with one or the other, depending on the context. This will beuseful in deﬁning the Brownian rough path for example.

43.1.2. Definition of a p-rough path. The relevance of the algebraic framework provided by the N-step nilpotent Lie group for the study of smooth paths was ﬁrstnoted by Chen in his seminal work [13]. Indeed, for any R -valued smooth path(xs )s 0 , the family of iterated integrals t s1 NXts : 1, xt xs ,dxs2 dxs1 , . . . ,dxs1 · · · dxsNss s1 ··· sN ts(N ),1deﬁnes for all 0 s t an element of T with the property that if x is scaled(N )NNinto λx then X becomes δλ X . We actually have XNts G . To see that, noticethat, as a function of t, the function XNts satisﬁes the diﬀerential equationNdXNts Xts dxt ,(N )(N )in T driven by the R -valued smooth contro x, so it deﬁnes a G -valued pathas an integral curve of a ﬁeld of tangent vectors. The above diﬀerential equationalso makes it clear the we have the following Chen relationsNNXNts Xus Xtu ,for all 0 s u t; they imply in particular the identityNXNts Xs0 1XNt0 ,which is here nothing but the "ﬂow" property for ordinary diﬀerential equationsolutions. Rough paths and weak geometric rough paths are somehow an abstractversion of this family of iterated integrals.([p]),1Definition 2. Let 2 p. A Hölder p-rough path on [0, T ] is a T [p]path X : t [0, T ] 1 Xt1 Xt2 · · · Xt such that i Xtsi X i : sup(3.3)i ,p0 s t T t s p-valuedfor all i 1 . . . [p], where we set Xts X 1s Xt . We define the norm of X to be (3.4)X : max X i i ,i 1.[p]pand a distance d(X, Y) X Y on the set of Hölder p-rough path. A Hölder([p])weak geometric p-rough path on [0, T ] is a G -valued p-rough path.(N ),1(orSo a (weak geometric) Hölder p-rough path is in a way nothing but a T (N )1G )-valued p -Hölder continuous path, for the · -norm introduced above and theuse of X 1s Xt in place of the usual Xt Xs . Note that the Chen relationXts Xus Xtuis granted by the deﬁnition of Xts X 1s Xt .For 2 p 3, Chen’s relation is equivalent to11 Xus,(i) Xts1 Xtu22112 Xus.(ii) Xts Xtu Xus Xtu

5111dCondition 1 (i) means that Xts Xt0 Xs0 represents the increment of the R -valuedpath Xr0 0 r T . Condition (ii) is nothing but the analogue of the elementary t r u r t u t rproperty s s s s u s u u , satisﬁed by any reasonable notion of integralon R that satisﬁes the Chasles relation t u t ssu This remark justiﬁes thinking of the R R -part of a rough path as a kindof iterated integral of X 1 against itself, although this hypothetical iterated integraldoes not make sense in itself for lack of an integration operation for a general Hölderpath in R . In that setting, a p-rough path X is a weak geometric p-rough path iﬀthe symmetric part of Xts2 is 12 Xts1 Xts1 , for all 0 s t T .Note that the space of Hölder p-rough paths is not a vector space; this preventsthe use of the classical Banach space calculus.It is clear that considering the iterated integrals of any given smooth path deﬁnesa p-rough path above it, for any p 2. This lift is not unique, as if we are givena p-rough path X X 1 , X 2 , with 2 p 3 say, and any 2p -Hölder continuous (R ) 2 -valued path Mt 0 t 1 , we deﬁne a new rough path setting Mts Mt Ms ,and X ts Xts1 , Xts2 Mtsfor all 0 s t 1. Relations (i) and (ii) above are indeed easily checked.Last, note that a Hölder p-rough path is also a Hölder q-rough path for anyp q [p] 1.3.2. The metric space of p-rough paths. The distance d deﬁned in deﬁnition 2is actually not a distance since only the increments Xts Yts are taken into account.We deﬁne a proper metric on the set of all Hölder p-rough paths setting d(X, Y) X 1 Y 1 d(X, Y).00Proposition 3. The metric d turns the set of all Hölder p-rough paths into a(non-separable) complete metric space.Proof – Given a Cauchy sequence of Hölder p-rough paths (n) X, there is no lossof generality in supposing that their ﬁrst level starts point in from the same (n) i (m) i R . It follows from the uniform Hölder bounds for X ts X ts i , and (anpeasily proved version of) Ascoli-Arzela theorem (for 2-parameter maps) that (n) Xconverges uniformly to some Hölder p-rough path X. To prove the convergenceof (n) X to X in d-distance, it suﬃces to send m to inﬁnity in the inequality i (n) i(m) i X ts X ts t s p ,which holds for all n, m bigger than some N , uniformly with respect to 0 s t 1.An uncountable family of R -valued 1p -Hölder continuous functions at pairwise 1p Hölder distance bounded below by a positive constant is constructed in example

65.28 of [3]. As the set of all ﬁrst levels of the set of Hölder p-rough paths is asubset of the set of R -valued 1p -Hölder paths, this examples implies the nonseparability of set of all Hölder p-rough paths. The following interpolation result will be useful in several places to prove roughpaths convergence results at a cheap price.Proposition 4. Assumebounds(n)X is a sequence of Hölder p-rough paths with uniform sup (n) X C ,(3.5)nwhich converge pointwise, in the sense that (n) Xts converges to some Xts for each0 s t 1. Then the limit object X is a Hölder p-rough path, and (n) X convergesto X as a Hölder q-rough path, for any p q [p] 1.Proof – (Following the solution of exercice 2.9 in [5]) The fact that X is a Hölderp-rough path is a direct consequence of the uniform bounds (3.5) and pointwiseconvergence: i Xts lim (n) X its C t s pi .nWould the convergence of (n) X to X be uniform, we could ﬁnd a sequence ndecreasing to 0, such that, uniformly in s, t, ii i i Xts (n) X ts n , Xtsi (n) X ts 2C t s p .Using the geometric interpolation a b a1 θ bθ , with θ i1 pi i Xts (n) X ts n q t s p ,pq 1, we would havewhich entails the convergence result as a Hölder q-rough path.We proceed as follow to see that pointwise convergence suﬃces to get the result.Given a partition π of [0, 1] and any 0 s t 1, denote by s, t the nearestpoints in π to s and t respectively. Writing (3.6)d Xts , (n) Xts d Xts , Xts d Xts , (n) Xts d (n) Xts , (n) Xtsand the fact thatXts Xss Xts Xtt ,(n)Xts (n) Xss (n) Xts (n) Xttand the uniform estimate (3.5) to see that the ﬁrst and third terms in theabove upper bound can be made arbitrarily small by choosing a partition witha small enough mesh, uniformly in s, t and n. The second term is dealt with thepointwise convergence assumption as it involves only ﬁnitely many points oncethe partition π has been chosen as above.

73.3. Exercices. 7. Lyons’ extension theorem [10]. Let n be a positive integer. An(n),1-valuedn-truncated multiplicativefunctional over R in the sense of Chen-Lyons is a T kmap X Xst 0 s t 1 , with components Xst , such that we haveXst Xsu Xutfor all 0 s u t 1, that isiXts i i kkXsuXutk 0for all 0 i n. Aevery we have(n),1T -valuedmap X is an n-almost-multiplicative functional if for k Xts Xus Xtuk c t s afor all 0 s t 1 and 0 k n, for some control ω and some constant a 1. Provethat if X is an n-truncated multiplicative functional and Ytsn 1 is a c

Rough paths Guide for this section Hölder p-rough paths, which control the rough diﬀerential equations dxt F(xt)X(dt),d ϕt F X(dt), and play the role of the controlhin the model classical ordinary diﬀerential equation dxt Vi(xt)dh i t F(xt)dht are deﬁned in section 3.1.2. As R -valued paths, they are not regular enough for the formula µts(x) x Xi ts Vi(x) to deﬁne an .

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