Transport and continuity equations with (very) rough noise
Carlo Bellingeri, Ana Djurdjevac, Peter K. Friz, Nikolas Tapia
aa r X i v : . [ m a t h . A P ] F e b Transport and continuity equations with (very) rough noise
February 25, 2020
Carlo Bellingeri , Ana Djurdjevac , Peter K. Friz , , Nikolas Tapia , TU Berlin, WIAS BerlinEmail: {bellinge,djurdjev,friz,tapia}@math.tu-berlin.de
Abstract
Existence and uniqueness for rough flows, transport and continuity equations driven by generalgeometric rough paths are established.
MSC (2020) Classification:
Keywords: rough transport equation, rough continuity equation, first order rough partial differentialequations.
Contents
We consider the transport equation, here posed (w.l.o.g.) as terminal value problem. This is, − ∂ t u ( t , x ) = d Õ i = f i ( x ) · D x u ( t , x ) Û W it ≡ Γ u t ( x ) Û W t in ( , T ) × R n , u = g on { T } × R n . (1.1)for fixed T > , with vector fields f = ( f , . . . , f d ) driven by a C -driving signal W = ( W , . . . , W d ) .The canonical pairing of Du = D x u = ( ∂ x u , . . . , ∂ x n u ) with a vector field is indicated by a dot, andwe already used the operator / vector notation Γ i = f i ( x ) · D x , Γ = ( Γ , . . . , Γ d ) . By the methods of characteristics, the unique (classical) C , transport solution u : [ , T ] × R n → R ,is given explicitly by u ( s , x ) = u ( s , x ; W ) : = g ( X s , xT ) , (1.2)provided g ∈ C and the vector fields f , . . . , f d are nice enough ( C b will do) to ensure a C solutionflow for the ODE Û X s , xt = d Õ i = f i ( X s , xt ) Û W it ≡ f ( X t ) Û W t , X s , xs = x . In turn, solving this ODE with random initial data induces a natural evolution of measures, given bythe continuity - or forward equation ∂ t ρ = d Õ i = div x ( f i ( x ) ρ t ) d W it in ( , T ) × R n ,ρ ( ) = µ on { } × R n . ntr oduction Well-posedness of the “trinity” transport/flow/continuity will depend on the regularity of the data.For W ∈ C we have an effective vector field b ( t , x ) = d Õ i = f i ( x ) Û W it which is continuous in t ∈ [ , T ] and inherits the regularity of f . In particular, f ∈ C will besufficient for a C , -flow. In a landmark paper, DiPerna–Lions [DL89] and then Ambrsosio [Amb04],showed that the transport problem (weak solutions) is well-posed under bounds on div b (ratherthan D x b ) which in turn leads to a generalized flow. Another fundamental direction may be called regularisation by noise , based on the observation that generically Û X = f ( X ) + ( noise ) is much betterbehaved than the noise-free problem, see e.g. [BFGM19, FGP09, FO18, CG16, Cat16, Mau11].Our work is not concerning with DiPerna-Lions type analysis, nor regularisation by noise. Infact, our driving vector fields will be very smooth , to compensate for the the irregularity of the noise,which we here assumed to be very rough . (This trade-off is typical in rough paths and regularitystructures.)Specifically, we continue a programme started independently by Bailleul–Gubinelli [BG17] (seealso [DGHT19]) and Diehl et al. [DFS17] and take W as rough path, henceforth called W . As inthese works, we are interested in an intrinsic notion of solution. (Rough path stability of transportproblems was already noted in [CF09]). The contribution of this article is a treatment of rough noiseof arbitrarily low regularity. Based on a suitable definition of solution, carefully introduced below,we can show Theorem 1.1.
Assume W is a weakly geometric rough path of Hölder regularity with exponent γ ∈ ( , ] . Assume f has ⌊ γ − ⌋ + bounded derivatives. Then there is a unique spatially regular(resp. measure-valued) solution to the rough transport (resp. continuity) equation with regularterminal data (resp. measure-valued initial data). This should be compared with [BG17, DFS17], which both treat the “level- case”, with Höldernoise of exponent γ > / . Treating the general case, i.e. with arbitrarily small Hölder exponent,requires us in particular to fully quantify the interaction of iterated integrals, themselves constrainedby shuffle-relations, and the controlled structure of the PDE problem at hand. In fact, the shufflerelations will be seen crucial to preserve the hyperbolic nature of the rough transport equation. Thisis different for (ordinary) rough differential equations where the shuffle relations can be discarded atthe price of working with branched (think: Ito-type) rough paths. For what it’s worth, our argumentsrestricted to the (well-known) level- -case still contain some worthwhile simplifications with regardto the existing literature, e.g. by avoiding the analysis of an adjoint equation [DFS17] and showinguniqueness for weak solutions of the continuity equations via a small class of test functions. On ourway we also (have to) prove some facts on (controlled) geometric rough paths of independent interest,not (or only in the branched setting [Gub10, HK15]) available in the literature. Relation to existing works:
Unlike the case of rough transport equation, when it comes tostochastic constructions it is impossible to mention all related works stretching over more than fourdecades, from e.g. Funaki [Fun79], Ogawa [Oga73] to recent works such as [OT15] with fractionalnoise and Russo–Valois integration.The many benefits of a robust theory of stochastic partial differential equations, by combining adeterministic RPDE theory with Brownian and more general noise, are now well documented andneed not be repeated in detail. Let us still recall one example of interest: multidimensional fractionalBrownian motion admits a canonical geometric rough path lift (see e.g. [FH14]) / < α < H, whichconstitutes an admissible rough noise for our rough transport and continuity equations. Variousauthors (see for example Unterberger [Unt13], Nualart and Tindel [NT11], etc.) have constructed“renormalised” canonical fractional Brownian rough paths for any H > , fully covered by Theorem1.1. Notations
We fix once and for all a time T > . In what follows we abbreviate estimates of the form |( a ) − ( b )| . | t − s | γ by writing ( a ) = γ ( b ) . Given γ ∈ ( , ) we denote by C γ the classical Hölderspace, i.e. consisting of functions f : [ , T ] → R such that sup t , s | f t − f s || t − s | γ < ∞ . ough paths Throughout the paper we say geometric rough path, when we really mean weakly geometric roughpath (since we only work with this type of rough path, the difference [FV06] will not matter to us).
Acknowledgement.
CB has received funding from DFG research unit FOR2402, AD and NThave received funding from Excellence Cluster MATH+ (AA4 and EF1, respectively). PKF hasreceived funding from the European Research Council (ERC) under the European Union’s Horizon2020 research and innovation programme (grant agreement No. 683164).
We start by reviewing the definition of geometric rough paths of roughness γ ∈ ( , ) and controlledrough paths. We will do so in a Hopf-algebraic language following [HK15], but before we willintroduce some basic concepts.A word of length p ≥ over the alphabet { , . . . , d } is a tuple w = ( i , . . . , i p ) ∈ { , . . . , d } p ,and we set | w | ≔ p . We denote by ε the empty word , which is by convention the unique word withzero length. Given two non-empty words v = ( i , . . . , i p ) and w = ( i p + , . . . , i p + q ) , we denote by vw ≔ ( i , . . . , i p , i p + , . . . , i p + q ) their concatenation . By definition ε w = w ε = w . We observe thatin any case | vw | = | v | + | w | . The concatenation product is associative but not commutative.The symmetric group S p acts on words of length p by permutation of its entries, that is, σ. w ≔ ( i σ ( ) , . . . , i σ ( p ) ) . Given two integers p , q ≥ , a ( p , q ) -shuffle is a permutation σ ∈ S p + q such that σ ( ) < σ ( ) < · · · < σ ( p ) and σ ( p + ) < σ ( p + ) < · · · < σ ( p + q ) . We denote by Sh ( p , q ) the set of all ( p , q ) -shuffles. The shuffle product was introduced by Ree [Ree58] to study the combinatorial properties of iteratedintegrals, following K.-T. Chen’s work. Let d ≥ be fixed, and consider the tensor algebra H over R d , which is defined to be the direct sum H ≔ ∞ Ê p = ( R d ) ⊗ p . A linear basis for H is given by pure tensors e i ⊗ · · · ⊗ e i p , p ≥ where { e , . . . , e d } is a basis of R d ,and the additional element which generates R ⊗ ≔ R . In order to ease the notation we denote, fora word w = ( i , . . . , i p ) , e w ≔ e i ⊗ e i ⊗ · · · ⊗ e i p . By definition, the set { e w : | w | = p } is a linearbasis for ( R d ) ⊗ p for any p ≥ .The space H is endowed with a product (cid:1) : H ⊗ H → H , called the shuffle product , defined onpure tensors as e i ··· i p (cid:1) e i p + ··· i p + q = Õ σ ∈ Sh ( p , q ) e σ. ( i ,..., i p + q ) . There is also another operation, called the deconcatenation coproduct ∆ : H → H ⊗ H , defined by ∆ e w ≔ Õ uv = w e u ⊗ e v . (2.1)The shuffle product and the deconcatenation coproduct satisfy a compatibility relation (which will notplay any role in the sequel), turning the tripe ( H , (cid:1) , ∆ ) into a graded connected bialgebra. This impliesthe existence of a linear map S : H → H , called the antipode , turning ( H , (cid:1) , ∆ , S ) into a Hopf algebra.In our particular setting, S can be explicitly computed on basis elements by S ( e i ··· i p ) = (− ) p e i p ··· i .The coproduct endows the dual space H ∗ with an algebra structure via the convolution product given, for g , h ∈ H ∗ , by h g ⋆ h , x i ≔ h g ⊗ h , ∆ x i . On pure tensor this yields h g ⋆ h , e w i = Õ uv = w h g , e u ih h , e v i . A character is a linear map g ∈ H ∗ such that h g , x (cid:1) y i = h g , x ih g , y i for all x , y ∈ H . It is astandard result (see e.g. [Man08]) that the collection of all characters on H forms a group G under ough paths the convolution product whose identity is the function ∗ ∈ H ∗ , defined by ∗ ( e b ) = for everyword b and ∗ ( ) = . The inverse of an element g ∈ G can be computed by using the antipode: g − = g ◦ S .Given N ≥ , we consider the step- N truncated tensor algebra H N = N Ê p = ( R d ) ⊗ p . Definition 2.1. A step- N truncated character is a linear map g ∈ H ∗ N such that h g , x (cid:1) y i = h g , x ih g , y i (2.2) for all x ∈ ( R d ) ⊗ p and y ∈ ( R d ) ⊗ q with p + q ≤ N . It is not hard to show that the set G ( N ) of all step- N truncated characters is also a group underthe convolution product, whose identity is again ∗ . Denoting by e ∗ , . . . , e ∗ d the basis of R d dual to { e , . . . , e d } , we introduce the dual basis ( e ∗ a ) of H ∗ N in the canonical way, that is, for a word w wedenote by e ∗ w the unique linear map on H N such that h e ∗ w , e v i = δ w ( v ) . The convolution product of two of these basis elements can be explicitly computed. Indeed, bydefinition h e ∗ u ⋆ e ∗ v , e w i = Õ u ′ v ′ = w h e ∗ u , e u ′ ih e ∗ v , e v ′ i which is nonzero if and only if w = u v , in which case h e ∗ u ⋆ e ∗ v , e w i = . Therefore e ∗ u ⋆ e ∗ v = e ∗ uv .For this reason this product is also known as the concatenation product . We now recall the notion of geometric rough paths. The group G ( N ) can be endowed with a sub-additive homogeneous norm k · k N : G ( N ) → R + , see [LV07] for further details. This allows us todefine a left invariant metric on G ( N ) by setting d N ( g , h ) ≔ k h − g k N . Definition 2.2.
Let N γ ≔ ⌊ γ − ⌋ denote the integer part of γ − . A geometric rough path of regularity γ is a γ -Hölder path W : [ , T ] → ( G ( N γ ) , d N ) . The set of all geometric rough paths of regularity γ will be denoted by C γ . By definition of the increments W st ≔ W − s ⋆ W t satisfy the so-called Chen’s relations W st = W su ⋆ W ut (2.3)for all ≤ s , u , t ≤ T . Moreover, by construction of the homogeneous norm k · k N , for any word w such that | w | ≤ N γ one has sup t , s |h W st , e w i|| t − s | | w | γ < ∞ . (2.4) One of the main goals of rough paths theory is to give meaning to solutions of controlled equationsof the form d X t = d Õ i = f i ( X t ) d W it , (2.5)for some collection of sufficiently regular vector fields f , . . . , f d on R n and where the driving signals W , . . . , W d are very irregular. The general philosophy is that if the smoothness of the vector fieldscompensates the lack of regularity of the driving signals, then we can still have existence of solutionsgiven that we reinterpret the equation in the appropriate sense. The central ingredient for provingthis kind of results is the notion of controlled rough path which we now recall. ough paths ([Gub10, FH14]) . Let W ∈ C γ and ≤ N ≤ N γ + . A rough path controlled by W is a path X : [ , T ] → H N − if for any word w such that | w | ≤ N − the path t
7→ h e ∗ w , X t i ∈ C γ and h e ∗ w , X t i = ( N −| w |) γ h W st ⋆ e ∗ w , X s i . (2.6) for all s < t . We denote by D N γ W the (vector) space of paths X satisfying (2.6) .We say that a path X : [ , T ] → R is controlled by W if there exists a controlled path X ∈ D N γ W such that h ∗ , X t i = X t ; we call X a controlled rough path above (the controlled path) X . Remark 2.4.
The definition in [FH14] seems more restrictive in that one always take N = N γ , whichis the minimal value of N required for rough integration. The case N = N γ + is convenient to keeptrack of the additional information obtained by rough integration, see Remark 2.7. Remark 2.5.
Alternatively, by writing X and W as the sums X s = Õ | w |≤ N − h e ∗ w , X s i e w , W st = Õ | v |≤ N h W st , e v i e ∗ v , the condition in eq. (2.6) can be explicitly written h e ∗ w , X t i = ( N −| w |) γ Õ ≤ | v |≤ N −| w | h e ∗ wv , X s ih W st , e v i , (2.7) for any word w . By construction of the vector space D N γ W , the quantity k X k W ; N γ : = Õ ≤ | w | < N sup s < t |h e ∗ w , X t i − h W st ⋆ e ∗ w , X s i|| t − s | ( N −| w |) γ , is finite for any X ∈ D N γ W . We can easily show that k · k D N γ W is a seminorm and, it becomes a Banachspace under the norm k X k D N γ W ≔ max | w |≤ N − |h e ∗ w , X i| + k X k W ; N γ . We extend the notion of controlled rough path above a vector-valued path X : [ , T ] → R n . Inthis case, the path X takes values in ( H N − ) n , that is, each component path h e ∗ w , X i is a vector of R n ,which we denote by h e ∗ w , X i = (h e ∗ w , X i , . . . , h e ∗ w , X i n ) . Then we require the bound in eq. (2.6) to hold componentwise, or equivalently, we can replace theabsolute value of the left-hand side by any norm on R n . We denote this space by ( D N γ W ) n .Using the higher-order information contained in the controlled rough path X ∈ D N γ W , we recallthe rigorous notion of rough integral of X against W . For its proof see [FH14]. Theorem 2.6.
Let W ∈ C γ and X ∈ D N γ γ W . For every i ∈ { , . . . , d } there exists a unique realvalued path in C γ t ∫ t X u d W iu ≔ lim | π |→ Õ [ a , b ]∈ π Õ ≤ | w |≤ N γ − h e ∗ w , X a ih W ab , e wi i , (2.8) where π is a sequence of partitions of [ , t ] whose mesh | π | converges to . We call it the roughintegral of X with respect to W i . Moreover one has the estimate ∫ t X u d W iu − ∫ s X u d W iu ≕ ∫ ts X u d W iu = ( N γ + ) γ Õ < | w |≤ N γ h e ∗ w , X s ih W st , e wi i , (2.9) for any s < t . Introducing the function ∫ · X u d W iu : [ , T ] → H N γ given by (cid:28) ∗ , ∫ t X u d W iu (cid:29) ≔ ∫ t X u d W iu , (cid:28) e ∗ wi , ∫ t X u d W iu (cid:29) ≔ h e ∗ w , X t i (2.10) and zero elsewhere, one has ∫ · X u d W iu ∈ D ( N γ + ) γ W . ough paths Differently from the general definition of the D N γ W spaces, in order to define the roughintegral it is necessary to start from a controlled rough path X ∈ D N γ γ W . The operation of integrationon controlled rough path comes also with some quantitative bounds. Looking at the definition, it isalso possible to prove there exists a constant C ( T , γ, W ) > depending on T , γ , W such that (cid:13)(cid:13)(cid:13)(cid:13)∫ · X u d W iu (cid:13)(cid:13)(cid:13)(cid:13) D ( N γ + ) γ W ≤ C ( T , γ, W )k X k D N γγ W . Therefore the application X ∫ X d W i is a continuous linear map. The second operation we introduce is the composition of a controlled rough path and a smoothfunction. Given a smooth function φ : R n → R , its k -th derivative at x ∈ R n is the multilinear map D k φ ( x ) : ( R n ) ⊗ k → R such that for v , . . . , v k ∈ R n , D k φ ( x )( v , . . . , v k ) = n Õ α ,...,α k = ∂ k φ∂ x α · · · ∂ x α k ( x ) v α · · · v k α k . (2.11)To ease notation we define ∂ α φ ( x ) ≔ ∂ k φ∂ x α · · · ∂ x α k ( x ) = ∂ k φ∂ x i · · · ∂ x i n n ( x ) for a word α = ( α , . . . , α k ) ∈ { , . . . , n } k ; of course, such α induces a multi-index i = ( i , . . . , i n ) ∈ N n , where i j counts the number of entries of α that equal j .We note that D k φ ( x ) is symmetric, meaning that for any permutation σ ∈ S k we have that D k φ ( x )( v , . . . , v k ) = D k φ ( x )( v σ ( ) , . . . , v σ ( k ) ) . Remark 2.8.
Observe that we also use the notion of word in this case, albeit with a different alphabet.In order to avoid confusion we reserve latin letters such as u , v , w , etc for words on the alphabet { , . . . , d } , introduced in the beginning of Section 2, and greek letters such as α, β , etc for words onthe alphabet { , . . . , n } as above. With these notations, Taylor’s theorem states that if φ : R n → R m is of class C r + ( R n , R m ) thenfor any j = , . . . , m one has the identity φ j ( y ) = r Õ k = k ! D k φ j ( x ) (cid:16) ( y − x ) ⊗ k (cid:17) + O (| y − x | r + ) (2.12)In what follows, for any finite number of words u , . . . , u k we introduce the set Sh ( u , . . . , u k ) ≔ { w : h e ∗ w , e u (cid:1) . . . (cid:1) e u k i , } . Since the shuffle product is commutative, for any permutation σ ∈ S k we have that Sh ( u , . . . , u k ) = Sh ( u σ ( ) , . . . , u σ ( k ) ) . Thanks to this notation, we can prove Faà di Bruno’s formula (see also [Har06]). We denote by P ( m ) the collection of all partitions of { , . . . , m } . Given π = { B , . . . , B p } ∈ P ( m ) , we let π ≔ p denote the number of its blocks, and for each block we denote by | B | its cardinality. Lemma 2.9.
For any couple of functions g : R n → R n and f : R n → R sufficiently smooth andevery m ≥ , letting h ≔ f ◦ g one has the identity D m h ( x )( v , . . . , v m ) = Õ π ∈ P ( m ) D π f ( g ( x ))( D | B | g ( x )( v B ) , . . . , D | B p | g ( x )( v B p )) where v B ≔ ( v i , . . . , v i q ) for B = { i , . . . , i q } .In particular, for any word α = ( α , . . . , α m ) we have ∂ α h ( x ) = m Õ k = k ! Õ β ,...,β k α ∈ Sh ( β ,...,β k ) D k f ( g ( x ))( ∂ β g ( x ) , . . . , ∂ β k g ( x )) . (2.13) ough paths Proof.
We proceed by induction on m . For m = the formula reads Dh ( x ) v = D f ( g ( x )) D g ( x ) v which is the usual chain rule. Suppose the formula holds for some m ≥ . Then, applying the chainrule to each of the terms we get D m + h ( x )( v , . . . , v m + ) = Õ π ∈ P ( m ) k Õ l = D π + f ( g ( x )) (cid:16) D | B | g ( x ) v B , . . . , D | B l | + g ( x )( v B l , v m + ) , . . . , D | B k | g ( x ) v B k (cid:17) + Õ π ∈ P ( m ) D π + f ( g ( x ))( D | B | g ( x ) v B , . . . , D | B k | g ( x ) v B k , D g ( x ) v m + ) = Õ π ′ ∈ P ( m + ) D π ′ f ( g ( x )) (cid:16) D | B ′ | g ( x )( v B ′ ) , . . . , D | B ′ p | g ( x )( v B ′ k ′ ) (cid:17) where the last identity follows from the fact that every partition π ′ ∈ P ( m + ) can be obtained byeither appending m + to one of the blocks of some partition π ∈ P ( m ) or by adding the singletonblock { m + } to it.Given a word α = ( α , . . . , α m ) , we evaluate the previous formula in the canonical basis vectors v = e α , . . . , v m = e α m to obtain ∂ α h ( x ) = D m h ( x )( v , . . . , v m ) = Õ π ∈ P ( m ) D π f ( g ( x )) ( ∂ α B g ( x ) , . . . , ∂ α Bk g ( x )) where α B = ( α i , . . . , α i q ) if B = { i , . . . , i p } . It is now clear that for any choice of π ∈ P ( m ) thewords α B , . . . , α B k satisfy α ∈ Sh ( α B , . . . , α B k ) . Conversely, if α ∈ Sh ( β , . . . , β k ) , there is apartition π = { B , . . . , B k } with B j = is such that β j = α B j . Moreover, for any choice of such apartition, any of the k ! permutations of its blocks result in the same evaluation by symmetry of thedifferential. Thus ∂ α h ( x ) = m Õ k = k ! Õ β ,...,β k α ∈ Sh ( β ,...,β k ) D k f ( g ( x ))( ∂ β g ( x ) , . . . , ∂ β k g ( x )) . (cid:3) Remark 2.10.
This result should be well-known to experts, yet the closest reference we found in theliterature [Har06] only covers the scalar case (and does not immediately yield the multivariate case).
Using a similar technique we show a version of this identity for controlled rough paths.
Theorem 2.11.
Let W ∈ C γ , ≤ N ≤ N γ + , X ∈ ( D N γ W ) n , and φ ∈ C N ( R n , R m ) and set X t ≔ h , X t i . We introduce the path Φ ( X ) : [ , T ] → ( H N − ) m defined by h ∗ , Φ ( X ) t i j = φ j ( X t ) andfor any j = , . . . , m , and any non-empty word w by the identity h e ∗ w , Φ ( X ) t i j ≔ | w | Õ k = k ! Õ u ,..., u k w ∈ Sh ( u ,..., u k ) D k φ j ( X t )(h e ∗ u , X t i , . . . , h e ∗ u k , X t i) . (2.14) Then Φ ( X ) is also a controlled rough path belonging to ( D N γ W ) m . Remark 2.12.
A similar statement in the setting of branched rough paths [Gub10, Lemma 8.4] isknown and somewhat easier due to the absence of shuffle relations.
Before going into the proof, we introduce some more notation. If X is a controlled path, L ∈ L (( R n ) ⊗ k , R m ) , t ≥ and u , . . . , u k are words, we let L ( t ; u , . . . , u k ) ≔ L (h e ∗ u , X t i , . . . , h e ∗ u k , X t i) ough paths Proof.
It is sufficient to prove the result when m = . We first prove the result for the case of h ∗ , Φ ( X ) t i = φ ( X t ) . By Taylor expanding φ up to order N around X s we get that φ ( X t ) = N γ N − Õ k = k ! D k φ ( X s ) (cid:16) ( X t − X s ) ⊗ k (cid:17) Since X ∈ (cid:16) D N γ W (cid:17) n , according to Remark 2.5, we have h ∗ , X t − X s i = N γ h W st − ∗ , X s i = Õ < | u | < N h e ∗ u , X s ih W st , e u i . (2.15)Plugging this estimate into the above equation and using the character property of W st in (2.2) weobtain φ ( X t ) = N γ N − Õ k = k ! Õ u ,..., u k D k φ ( X s )( s ; u , . . . , u k )h W st , e u (cid:1) · · · (cid:1) e u k i = N − Õ k = k ! Õ u ,..., u k Õ | w |≤ N D k φ ( X s )( s ; u , . . . , u k )h e ∗ w , e u (cid:1) · · · (cid:1) e u k ih W st , e w i so the desired estimate follows.Now we show the bound (2.6) for all words w , . By fixing an integer ≤ k ≤ | w | and words u , . . . , u k such that w ∈ Sh ( u , . . . , u k ) we consider the term D k φ ( X t )( t ; u , . . . , u k ) . (2.16)Again, since X is controlled by W , plugging the estimate in Remark 2.5 into (2.16) and using themultilinearity :of the derivative we obtain D k φ ( X t )( t ; u , . . . , u k ) = ( N −| w |) γ Õ v ,..., v k D k φ ( X t )( s ; u v , . . . , u k v k )h W st , e v (cid:1) · · · (cid:1) e v k i . (2.17)Performing a Taylor expansion of D k φ up to order N − | w | between X t and X s , we obtain D k φ ( X t )( s ; u v , . . . , u k v k ) = ( N −| w |) γ N −| w |− Õ m = m ! D k + m φ ( X s ) (cid:16) ( X t − X s ) ⊗ m , h e ∗ u v , X s i , . . . , h e ∗ u k v k , X s i (cid:17) . (2.18)Combining the estimates (2.17) and (2.18) with (2.15) into the definition of h e ∗ w , Φ ( X ) t i , we obtainthe identity h e ∗ w , Φ ( X ) t i = ( N −| w |) γ | w | Õ k = N − −| w | Õ m = Õ u ,..., u k w ∈ Sh ( u ,..., u k ) Õ v ,..., v k z ,..., z m k ! m ! D k + m φ ( X s )( u v , . . . , u k v k , z , . . . , z m )× h W st , e v (cid:1) · · · (cid:1) e v k (cid:1) e z (cid:1) · · · (cid:1) e z m i . (2.19)Since the derivative D k + m φ ( X s ) is symmetric we can replace it with k ! m ! ( k + m ) ! Õ I k ⊔ J m = { ,..., m + k } D k + m φ ( X s )( u i v i , . . . , z j , . . . , u i k v i k , . . . ) . Replacing this expression in the right-hand side of (2.19), it is now an easy but tedious exercise toverify the resulting expression is equal to the sum Õ ≤ | u | < N −| w | | w | + | u | Õ l = Õ v ,..., v l wu ∈ Sh ( v ,..., v l ) l ! D l φ ( X s )( s ; v , . . . , v l )h W st , e c i . Thereby proving the result. (cid:3)
Remark 2.13.
A similar proof gives quantitative bounds on the application X → Φ ( X ) . Indeed forany φ ∈ C Nb ( R n , R m ) it is possible to prove that this application is locally Lipschitz on D N γ W . ough Differential Equations Now we come to the definition of solution of the RDE d X t = d Õ i = f i ( X t ) d W it , X = x . (3.1)We assume that the vector fields f , . . . , f d are of class at least C N γ , so that by Theorem 2.11 thecomposition f i ( X t ) can be lifted to a controlled path F i : (cid:16) D N γ γ W (cid:17) n → (cid:16) D N γ γ W (cid:17) n . Definition 3.1.
A path X : [ , T ] → R n is a solution of (3.1) if there exists a controlled path X ∈ (cid:16) D N γ γ W (cid:17) n satisfying h ∗ , X t i = X t such that X t − X s = d Õ i = ∫ ts F i ( X ) u d W iu . (3.2) for all s , t ∈ [ , T ] . Remark 3.2.
We stress that (3.2) is an equation in D N γ γ W , which in fact implies that h e ∗ w , X t i = F w ( X t ) for all words w with | w | ≤ N γ − . Remark 3.3. If X ∈ D N γ γ W satisfies eq. (3.2) , it can also be regarded as an element of D ( N γ + ) γ W , byeq. (2.8) . Therefore we freely treat solutions to RDEs as elements of either of these spaces. By solving a fixed point equation on (cid:16) D N γ γ W (cid:17) n (see e.g. [FH14]) of the form X t = X + d Õ i = ∫ t F i ( X ) u d W iu with (see below for the definition of the functions F w : R n → R n ) X = Õ | w |≤ N γ − F w ( x ) e w ∈ (cid:16) H N γ − (cid:17) n . We can prove that there exists a unique global solution of (3.2) if the vector fields are of class C N γ + b .We recall this interesting expansion of the solution. Proposition 3.4 (Davie’s expansion) . A path X : [ , T ] → R n is the unique rough path solution toeq. (2.5) in the sense of Definition 3.1 if and only if X t = ( N γ + ) γ Õ ≤ | w |≤ N γ F w ( X s )h W st , e w i (3.3) and the coefficients of its lift X ∈ ( D N γ + W ) n are given by h e ∗ w , X t i = F w ( X t ) where the functions F w : R n → R n are recursively defined by by F ε ≔ id and F iw ( x ) ≔ DF w ( x ) f i ( x ) . (3.4) Remark 3.5.
By eq. (2.7) this results actually implies the chain of estimates, for all words | w | ≤ N γ , F w ( X t ) = ( N γ + −| w |) γ Õ ≤ | u |≤ N −| w | F wu ( X s )h W st , e u i . Proof of Proposition 3.4.
Suppose that X is a rough solution to eq. (2.5) in the sense of Definition 3.1.We define the functions F w : R n → R n recursively by F i ( x ) ≔ f i ( x ) and F wi ( x ) ≔ | w | Õ k = k ! Õ u ,..., u k w ∈ Sh ( u ,..., u k ) D k f i ( x )( F u ( x ) , . . . , F u k ( x )) (3.5) ough Differential Equations Now it is an easy but tedious verification to show that these functions satisfy F iw ( x ) = DF w ( x ) f i ( x ) ;this identity essentially amounts to a reiterated use of the Leibniz rule. The form of the coefficientsof X is shown by induction, it being clear for a single letter i = , . . . , d . If w is any word with ≤ | w | ≤ N − and i ∈ { , . . . , d } by definition h e ∗ wi , X t − X i = * e ∗ wi , d Õ j = ∫ t F j ( X ) u d W ju + = h e ∗ w , F i ( X ) t i where, in the second identity we have used eq. (2.10). By Theorem 2.11, the last coefficient equals | w | Õ k = k ! Õ u ,..., u k w ∈ Sh ( u ,..., u k ) D k f i ( X t )( t ; u , . . . , u k ) = F wi ( X t ) by the induction hypothesis. Then we obtain eq. (3.3) from Definition 2.3 and Remark 2.5.Conversely, suppose that X admits the local expansion in eq. (3.3) and that the path X satisfies h e ∗ w , X t i = F w ( X t ) for all words w with | w | ≤ N . First we show that X is controlled by W withcoefficients given by X . For this we have to Taylor expand the difference F w ( X t ) − F w ( X s ) and collectterms as in the proof of Theorem 2.11. Then, by eq. (2.10) it is not difficult to see that in fact h e ∗ wi , X t i = F wi ( X t ) = * e ∗ wi , d Õ j = ∫ t F j ( X ) u d W ju + so that Definition 3.1 is satisfied. (cid:3) It is a standard result in classical ODE theory that given a regular enough vector field V , the equation Û X = V ( X ) induces a smooth flow on R d . Indeed, if we let X xt denote the unique solution of thisequation such that X x = x , then the map ( t , x ) 7→ X xt is a flow, in the sense that ( t , X xs ) 7→ X xt + s andthe mapping x X xt is a diffeomorphism for each fixed t . More precesily, if V is of class C k , thenthe application x X xt is also of class C k .Now we show that a similar statement is true in the case of RDEs. The statement is the following Theorem 3.6.
Let f , . . . , f d be a family of class C N γ + + kb vector fields in R d for some integer k ≥ ,and W ∈ C γ . Then1. the RDE d X t = d Õ i = f i ( X t ) d W it , X s = x has a unique solution X s , x ∈ D ( N γ + ) γ W ,2. the induced flow x X s , xt is a class C k + diffeomorphism for each fixed s < t , and3. the partial derivatives satisfy the system of RDEs d ∂ α X s , xt = d Õ i = | α | Õ k = k ! Õ α ∈ Sh ( β ,...,β k ) D k f i ( X s , xt )( ∂ β X s , xt , . . . , ∂ β k X s , xt ) d W it (3.6) with initial conditions X s , xs = x , ∂ i X s , xs = e i and ∂ α X s , xs = for all words with | α | ≥ .Proof. Point 1. and 2. are standard results in rough paths as found e.g. in Chapter 11 in [FV10]. Forthe algebraic identity in 3., it suffices to show the results in the case W is smooth. Indeed, by standardarguments W ∈ C γ can be approximated uniformly with uniform γ -Hölder rough path bound, andhence in C γ − η for any η > , while on the other hand the particular structure (cf. Chapter 11 in[FV10]) of the system of (rough) differential equations guarantees uniqueness and global existenceso that the limiting argument is justified. ough Differential Equations It remains to show point 3. for W smooth. We note that the integral representation of the solution X s , xt = x + d Õ i = ∫ ts f i ( X s , xu ) Û W iu d u holds. By Lemma 2.9, for any α = ( α , . . . , α m ) and s < u < t , we have ∂ α X s , xut = ∫ tu d Õ i = m Õ k = k ! Õ β ,...,β k D k f i ( X s , xr )( ∂ β X s , xr , . . . , ∂ β k X s , xr ) Û W ir d r (3.7)which is the smooth version of eq. (3.6). (cid:3) We aim now to obtain a Davie-type expansion of the partial derivatives ∂ α X s , x by making use ofpoint 3. above. We observe that the above system of equations has the form d X s , xt = d Õ i = f i ( X s , xt ) d W it d DX s , xt = d Õ i = D f i ( X s , xt ) DX s , xt d W it d D X s , xt = d Õ i = D f i ( X s , xt ) D X s , xt d W it + (· · · ) ... with initial conditions X s , xs = x , DX s , xs = I , D X s , xs = D X s , xs = · · · = , where the inhomogeneity (· · · ) is not important to spell out.The expansion is clear only for the first equation; it is just eq. (3.3). We would like to useProposition 3.4 to obtain an expansion of the second equation but the problem is that the vectorfield driving the equation depends on time, so the result does not directly apply. For the third andsubsequent equations the problem is not only that but also they are non-homogeneous.To solve this problem we extend our state space R n to (the still finite-dimensional space) S k ≔ R n ⊕ L ( R n , R n ) ⊕ · · · ⊕ L (cid:16) ( R n ) ⊗( k − ) , R n (cid:17) and define the vector fields (we give a more precise definition below in eq. (3.8)) f i : S k → S k by f i ( x ) ≔ ( f i ( x ) , D f i ( x )( y ) , D f i ( x )( y , y ) + D f i ( x )( y ) , . . . ) where x = ( x , y , y , . . . , y k − ) ∈ S k . The previous proposition shows that if X s , xt ≔ ( X s , xt , DX s , xt , . . . , D k − X s , xt ) then d X s , xt = d Õ i = f i ( X s , xt ) d W it , X s , xs ≔ x = ( x , I , , . . . , ) . This transformation turns the system of non-autonomous non-homogeneous RDEs into a singleautonomous homogeneous RDE in S k . Corollary 3.7.
For any word α , the partial derivatives of the solution flow X s , x have the followingDavie expansion: for any p = , . . . , k − , D p X s , xt = ( N γ + ) γ Õ ≤ | v |≤ N γ D p F w ( x )h W st , e w i . In particular, for a word α ∈ { , . . . , n } p we have that ∂ α X s , xt = ( N γ + ) γ Õ ≤ | v |≤ N γ ∂ α F w ( x )h W st , e w i . ough Differential Equations Proof.
The hypotheses on the vector fields f , . . . , f d imply that f , . . . , f d are of class C N γ + b on S k ,so this equation has a unique solution. Applying Proposition 3.4 in this extended space we obtain,for s < t , the expansion X s , xt = ( N + ) γ Õ ≤ | w | < N F w ( x )h W st , e w i . In order to deduce the result, we need to show that F w ( x ) p = D p F w ( x ) for all words w and p = , , . . . , k − . We do this by induction on the length of w . If w = i is a single letter, the p -thcomponent, p = , , . . . , k − , of the vector field f i is given by f i ( x ) = f i ( x ) and f i ( x ) p = p Õ j = Õ ( r ) j p ! r ! · · · r j ! ( ) r · · · ( j ! ) r k D p − j + f i ( x )( y r , . . . , y r j j ) (3.8)where the inner sum is over the set of indices ( r , . . . , r j ) such that r + · · · + r j = p − j + and r + r + · · · + jr j = p . For our particular initial condition, y j = for j = , , . . . , k − the formulasimplifies to f i ( x ) p = D p f i ( x ) ∈ L (cid:0) ( R n ) ⊗ p , R n (cid:1) since the only term left in (3.8) is the one with j = , r = p .We continue by induction on the length of the word. We compute the p -th derivative of x F iw ( x ) = DF w ( x ) f i ( x ) by recognizing that F iw = ϕ ◦ ϕ with ϕ ( x , h ) = DF w ( x ) h and ϕ ( x ) = ( x , f i ( x )) . A quick check gives that the higher order derivatives of ϕ and ϕ are given by D m ϕ ( x , h )(( u , v ) , . . . , ( u m , v m )) = D m + F w ( x )( u , . . . , u m , h ) + m Õ j = D m F w ( x )( u , . . . , ˆ u j , . . . , u m ) D m ϕ ( x )( h , . . . , h m ) = ( h δ m = , D m f i ( x )( h , . . . , h m )) where ˆ u j = v j . Thus, using Lemma 2.9 we get that D p F iw ( x )( h , . . . , h p ) = Õ π ∈ P ( p ) D π ϕ ( ϕ ( x ))( D | B | ϕ ( x ) h B , . . . , D | B q | ϕ ( x ) h B q ) . Now we have three cases, depending on the number of blocks of the partition in the abovesummation:1. q = p : there is a single partition with p blocks, and each block is a singleton. In this case theterm equals D p + F w ( x )( h , . . . , h p , f i ( x )) + p Õ j = D p F w ( x )( h , . . . , D f i ( x ) h j , . . . , h p ) . q = : there is a single partition with one block, namely π = { , . . . , p } . In this case the termequals DF w ( x )[ D p f j ( x )( h , . . . , h p )] . < q < p : there is at least one block of size greater than one, which means that the first termin the expression for D m ϕ vanishes since at least one of u , . . . , u m vanishes. For the rest ofthe terms, the exact result depends on whether there is a block with exactly one block or not: ifall blocks have more than one block then the whole expression vanishes; otherwise, we obtainone term for each of the blocks having size exactly one, and it is of the form D π F w ( x )( h B , D p −| B | f i ( x ) h π \ B ) . In either case, using the induction hypothesis it is possible to show that each of the terms appearingare of the form ∂ r F w ( x ) p f i ( x ) r , which then means that D p F iw ( x ) = [ D F w ( x ) f i ( x )] p as desired. Forexample, the term D p + F w ( h , . . . , h p , f i ( x )) corresponds to [ ∂ F w ( x ) p f i ( x ) ]( h , . . . , h p ) and so on. (cid:3) ough Differential Equations In particular for the first derivative, the first few terms of the expansion read DX s , xt = I + d Õ i = D f i ( x )h W st , e i i + d Õ i , j = (cid:16) D f j ( x ) D f i ( x ) + D f j ( x )( f i ( x ) , id ) (cid:17) h W st , e ij i + · · · The last ingredient to add in the study of the rough transport equation is to write down a change ofvariable formula for a solution of eq. (2.5) for some sufficiently smooth vector field f = ( f , . . . , f d ) .By analogy with terminology of stochastic calculus we call it an “Itô formula”. For any i = , . . . , n we denote by Γ i the differential operator f i ( x ) · D x and for any non-empty word w = i · · · i m we usethe shorthand notation Γ w : = Γ i ◦ · · · ◦ Γ i m . Moreover we adopt the convention Γ ε = id . Lemma 3.8.
Let f , . . . , f d ∈ C N γ + ( R n ; R n ) be vector fields on R n . If φ : R n → R is a smoothfunction and w is a nonempty word, then Γ w φ ( x ) = | w | Õ k = k ! Õ u ,..., u k w ∈ Sh ( u ,..., u k ) D k φ ( x )( F u ( x ) , . . . , F u k ( x )) . (3.9) Proof.
Before commencing we introduce some notation. If φ : R n → R and g , . . . , g k : R n → R n are smooth functions, we define D k φ ( x ) : ( g , . . . , g k ) ≔ D k φ ( x )( g ( x ) , . . . , g k ( x )) where the right-hand side was defined in eq. (2.11). The Leibniz rule then gives that for any h ∈ R n we have h · D x (cid:16) D k φ ( x ) : ( g , . . . , g k ) (cid:17) = D k + φ ( x ) : ( h , g , . . . , g k ) + k Õ i = D k φ ( x ) : ( g , . . . , ( D x g i ) h , . . . , g k ) . We now prove the result by induction on the word’s length | w | . If w = i is a single letter then Γ i φ ( x ) = f i ( x ) · ∇ φ ( x ) = D φ ( x ) f i ( x ) which is exactly eq. (3.9). Supposing the identity true for anyword w ′ such that | w ′ | ≤ | w | , we prove it for j w where j ∈ { , . . . , d } . By induction one has Γ w φ ( x ) = | w | Õ k = k ! Õ u ,..., u k w ∈ Sh ( u ,..., u k ) D k φ ( x )( F u ( x ) , . . . , F u k ( x )) . By the above form of Leibiz’s rule, with g i = F u i and h = f j ( x ) , and noticing that by definition D x F u i ( x ) f j ( x ) = F ju i ( x ) we obtain that Γ j (cid:16) D k φ ( x ) : ( F u , . . . , F u k ) (cid:17) = D k + φ ( x ) : ( f j , F u , . . . , F u k ) + k Õ i = D k φ ( x ) : ( F u , . . . , F ju i , . . . , F u k ) . Summing this expression over words u , . . . , u k , we can rewrite it as k Õ r = Õ u ,..., u k jw ∈ Sh ( u ,..., ju r ,..., u k ) D k φ ( x ) : ( F u , . . . , F ju r , . . . , F u k ) + k + k + Õ r = Õ u ,..., u k jw ∈ Sh ( u ,..., j ,..., u k ) D k + φ ( x ) : ( F u , · · · , r th place z}|{ f j , . . . , F u k ) , ough Differential Equations the factor /( k + ) is introduced because of the symmetry of D k + φ ( x ) . Summing finally over k , wecan express the final expression as Γ jw φ ( x ) = | w | Õ k = k ! k Õ r = Õ u ,..., u k jw ∈ Sh ( u ,..., ju r ,..., u k ) D k φ ( x ) : ( F u , · · · , F ju r , . . . , F u k ) + | w | Õ k = ( k + ) ! k + Õ r = Õ u , ··· , u k jw ∈ Sh ( u ,..., j ,..., u k ) D k + φ ( x ) : ( F u , · · · , r th place z}|{ f j , . . . , F u k ) . Since the letter j may appear as a single word or concatenated at the right with some word, we finallyidentify the whole expression above with | w | + Õ k = k ! Õ u ,..., u k ja ∈ Sh ( u ,..., u k ) D k φ ( x ) : ( F u , . . . , F u k ) . (cid:3) Now we show a formula for the composition of the solution to the RDE (2.5) and a sufficientlysmooth function.
Theorem 3.9 (Itô formula for RDEs) . Let f i ∈ C N γ + and let X ∈ D ( N γ + ) γ W be the unique solutionof eq. (2.5) and X t = h ∗ , X t i . Then for any real valued function φ ∈ C N γ + b ( R n ) one has the identity φ ( X t ) = φ ( X s ) + d Õ i = ∫ ts ( Γ i φ )( X r ) d W ir . (3.10) More generally, one has the following estimates at the level of controlled rough paths h e ∗ w , Φ ( X ) t i = ( N γ + −| w |) γ h e ∗ w , Φ ( X ) s i + * e ∗ w , d Õ i = ∫ ts ( Γ i Φ )( X ) r d W ir + , (3.11) where Γ i Φ ( X ) is the controlled lift of composition of X with the function Γ i φ ∈ C N γ and anynon-empty word such that | w | ≤ N γ .Proof of Theorem 3.9. The theorem is obtained by comparing the coefficients of the controlled roughpaths Φ ( X ) t and ∫ t ( Γ i Φ )( X r ) d W ir for every i = , . . . , d . Using Lemma 3.8 and Proposition 3.4, forevery non-empty word a one has h e ∗ w , Φ ( X ) t i = | w | Õ k = k ! Õ u ,..., u k w ∈ Sh ( u ,..., u k ) D k φ ( X t )( t ; u , . . . , u k ) = | w | Õ k = k ! Õ u ,..., u k w ∈ Sh ( u ,..., u k ) D k φ ( X t ) : ( F u , . . . , F u k ) = Γ w φ ( X t ) . (3.12)Using the same identities we also deduce for any word w , * e ∗ wj , d Õ i = ∫ t ( Γ i Φ )( X ) r d W ir + = h e ∗ w , ( Γ j Φ )( X ) t i = Γ w ( Γ j φ )( X t ) = Γ wj φ ( X t ) . (3.13)Since Í ni = ∫ t ( Γ i Φ )( X ) r d W ir and Φ ( X ) t belong both to D ( N γ + ) γ W for any word w one has both h e ∗ w , Φ ( X ) t i − h e ∗ w , Φ ( X ) s i = ( N γ + −| w |) γ Õ < | v |≤ N γ −| w | h e ∗ wv , Φ ( X ) s ih W st , e v i , ough transport and continuity and * e ∗ w , n Õ i = ∫ ts ( Γ i Φ )( X ) r d W ir + = ( N γ + −| w |) γ Õ < | v |≤ N γ −| w | * e ∗ wv , n Õ i = ∫ s ( Γ i Φ )( X ) r d W ir + h W st , e v i . The identities (3.12) and (3.13) imply that the right-hand sides of the above estimates are the samequantities. Thus we obtain eq. (3.11) by simply subtracting one side from the other. In case w = one has φ ( X t ) − φ ( X s ) − d Õ i = ∫ ts ( Γ i φ )( X r ) d W ir = ( N γ + ) γ . Since ( N γ + ) γ > and the right hand side is the increment of a path, one has the identity (3.10). (cid:3) Using the identities (3.12) we can rewrite the Itô formula using only the operators Γ w . Corollary 3.10 (Itô-Davie formula for RDEs) . Let X : [ , T ] → R n be the unique solution of eq. (2.5) .Then for any real valued function φ ∈ C N γ + b ( R n ) and any word w one has the estimate Γ w φ ( X t ) = ( N γ + −| w |) γ Õ ≤ | v |≤ N γ −| w | Γ wv φ ( X s )h W st , e v i . (3.14) We now consider the rough transport equation ( − d u s = Í di = Γ i u s d W is , u ( T , ·) = g (·) (4.1)where we recall the differential operator Γ i ≔ f i · D x for some vector fields f , . . . , f d on R n .We now prepare the definition of a regular solution to the rough transport equation. Since we arein the fortunate position to have an explicit solution candidate we derive a graded set of rough pathestimates that provide a natural generalisation of the classical transport differential equation. Definition 4.1.
Let γ ∈ ( , ) , W ∈ C γ a weakly-geometric rough path of roughness γ and g ∈ C N γ + .A C γ, N γ + -function u : [ , T ] × R n → R such that u ( T , ·) = g (·) is said to be a regular solution to therough transport equation (4.1) if one has the estimates Γ w u s ( x ) = ( N γ + −| w |) γ Õ ≤ | v |≤ N γ −| w | Γ wv u t ( x )h W st , e v i , (4.2) for every s < t ∈ [ , T ] , uniformly on compact sets in x and any word w . Remark 4.2.
Since each application of the vector fields Γ i ··· i n amounts to take n derivatives,these estimates have the interpretation that time regularity of Γ i ··· i n u , can be traded against spaceregularity in a controlled sense. Theorem 4.3.
Let f ∈ C N γ + b , g ∈ C N γ + and consider the rough solution X s , x to eq. (2.5) . Then u ( s , x ) ≔ g ( X s , xT ) is a solution to the rough transport equation in the sense of Definition 4.1.Proof. We first note that by Theorem 3.6 the map ( s , x ) 7→ X s , xT belongs to C γ, N γ + . Since g ∈ C N γ + then u ( s , x ) = g ( X s , xT ) ∈ C γ, N γ + . Let us show that u is a solution by proving the estimates given inDefinition 4.1 for some fixed times s < t < T and x in compact set. By uniqueness of the RDE flowone has X s , xT = X t , yT where y = X s , xt . Thus we deduce from the definition of u the identity u s ( x ) = u t ( X s , xt ) . (4.3)Let X denote the controlled rough path such that X s , xt = h ∗ , X t i . Since g ∈ C N γ + b , we can apply therough Itô formula in eq. (3.14) to the function x → u t ( x ) obtaining u t ( X s , xt ) = ( N γ + ) γ Õ | w |≤ N Γ w u t ( x )h W st , w i ough transport and continuity obtaining (4.2) for the case of w = ε . To show the estimates on Γ i ··· i l u s , we apply Lemma 3.8 to thefunction x → u s ( x ) Γ w u s ( x ) = | w | Õ k = k ! Õ u ,..., u k w ∈ Sh ( u ,..., u k ) D k u s ( x )( F u ( x ) , . . . , F u k ( x )) . (4.4)Using again the identity (4.3), for any word α we apply eq. (2.13) obtaining ∂ α ( u s ( x )) = k Õ l = l ! Õ β ,...,β l α ∈ Sh ( β ,...,β l ) D l u t ( X s , xt )( ∂ β X s , xt , · · · , ∂ β l X s , xt ) . Since the vector field f ∈ C N γ + b and every β i such that α ∈ Sh ( β , . . . , β l ) satisfies | β i | ≤ | a | wecan apply Corollary 3.7 to get ∂ β i X s , xt = ( N γ + −| w |) γ Õ ≤ | v |≤ N γ −| w | ∂ β i F v ( x )h W st , e v i Plugging these estimates in D l u t ( X s , xt ) and one has D l u t ( X s , xt )( ∂ β X s , xt , . . . , ∂ β l X s , xt ) = ( N γ + −| w |) γ Õ ≤ | v |···| v l |≤ N γ −| a | D l u t ( X s , xt ) (cid:16) ∂ β F v ( x ) , . . . , ∂ β l F v l ( x ) (cid:17) h W st , e v (cid:1) · · · (cid:1) e v l i . (4.5)Plugging this expression into (4.4) and we obtain Γ w u s ( x ) = ( N γ + −| w |) γ | w | Õ k = Õ u ,..., u k w ∈ Sh ( u ,..., u k ) n Õ α , ··· ,α k = k Õ l = l ! 1 k ! F α u ( x ) · · · F α k u k ( x ) Õ ≤ | d |···| d l |≤ N γ −| a | Õ β ,...,β l α ∈ Sh ( β ,...,β l ) D l u t ( X s , xt )( ∂ β F v ( x ) , . . . , ∂ β l F v l ( x ))h W st , e v (cid:1) · · · (cid:1) e v l i . (4.6)Rearranging the sums and applying the definition of the functions F w we obtain the identity | w | Õ k = l k ! Õ u ,..., u k w ∈ Sh ( u ,..., u k ) Õ α ∈ Sh ( β ,...,β l )| α | = k D l u t ( X s , xt )( ∂ β F v ( x ) , . . . , ∂ β l F v l ( x )) F α u ( x ) · · · F α k u k ( x ) = Õ u ′ ,..., u ′ l w ∈ Sh ( u ′ ,..., u ′ l ) D l u t ( X s , xt )( F u ′ v ( x ) , · · · , F u ′ l v l ( x )) . Therefore the right-hand side of (4.6) becomes | w | Õ l = Õ ≤ | v |···| v l |≤ N γ −| w | Õ u ′ ,..., u ′ l w ∈ Sh ( u ′ ,..., u ′ l ) l ! D l u t ( X s , xt )( F u ′ v ( x ) , · · · , F u ′ l v l ( x ))h W st , e v (cid:1) · · · (cid:1) e v l i . (4.7)We perform now a Taylor expansion of D l u t ( X s , xt ) up to order N − | w | between X s , xt and x , yieldingfor any words u ′ , · · · , u ′ k D l u t ( X s , xt ) (cid:18) F u ′ v ( x ) , · · · , F u ′ l v l ( x ) (cid:19) = ( N γ + −| w |) γ N −| w | Õ m = m ! D l + m u t ( x ) (cid:16) ( X s , xt − x ) ⊗ m , F u ′ v ( x ) , · · · , F u ′ l v l ( x ) (cid:17) . (4.8) ough transport and continuity Plugging now the Davie expansion (3.3) truncated at order N γ − | w | into (4.7) we have the followingestimate Γ w ( u s ( x )) = ( N γ + −| w |) γ | w | Õ l = N γ −| w | Õ m = l ! 1 m ! Õ u ′ ,..., u ′ l w ∈ Sh ( u ′ ,..., u ′ l ) Õ ≤ | v |···| v l |≤ N −| w | < | z |···| z m |≤ N −| w | D l + m u t ( x ) : ( F u ′ v , . . . , F z , . . . )h W st , e v (cid:1) · · · (cid:1) e z (cid:1) · · · i . (4.9)Using the symmetry of D l + m u t ( x ) , we deduce l ! m ! ( l + m ) ! Õ I l ⊔ J m = { , ··· , m + l } D m + l φ ( x ) : ( F u ′ i v i , · · · , F z j , · · · , F u ′ il v il , · · · ) . Replacing this expression in the right-hand side of (4.9), we can easily verify that the resultingexpression is equal to the sum Õ ≤ | v |≤ N −| w | | w | + | v | Õ n = Õ u ,..., u n wv ∈ Sh ( u ,..., u n ) n ! D n u t ( x ) : ( F u , · · · , F u n )h W st , e v i . Thereby proving the result. (cid:3)
We can now show that solutions in the sense of Definition 4.1 are unique.
Theorem 4.4.
Let f i ∈ C N γ + b with associated differential operators Γ i , and W ∈ C γ . Given regularterminal data g ∈ C N γ + , there exists a unique regular solution to the rough transport equation (4.1) .Proof. Existence is clear, since Proposition 4.3 exactly says that ( t , x ) 7→ g ( X t , xT ) gives a regularsolution. Let now u be any solution to the rough transport equation. We show that, whenever X = X ¯ s , ¯ y for every ¯ s , ¯ y one has the estimate u ( t , X t ) − u ( s , X s ) = ( N γ + ) γ . (4.10)Since ( N γ + ) γ > this entails that t u ( t , X t ) is constant, and so we recover the uniqueness fromthe identities u ( s , x ) = u ( s , X s , xs ) = u ( T , X s , xT ) = g ( X s , xT ) . To prove (4.10) we show that for every k = , · · · , N γ and any choice of indexes i , · · · , i k (if k = we do not consider indexes) one has the estimates Γ i ··· i k u t ( X t ) = ( N γ + − k ) γ Γ i ··· i k u s ( X s ) . Let us prove this estimate by reverse induction on the indices length. The case when the indices i · · · i N γ have length N γ comes easily from the algebraic manipulation Γ i ··· i N γ u t ( X t ) − Γ i ··· i N γ u s ( X s ) = (cid:18) Γ i ··· i N γ u t ( X t ) − Γ i ··· i N γ u s ( X t ) (cid:19) + (cid:18) Γ i ··· i N γ u s ( X t ) − Γ i ··· i N γ u s ( X s ) (cid:19) . Using the defining property of a solution in the estimates (4.2), the first difference on the right-handside is of order γ . Moreover by hypothesis on u one has Γ i ··· i N γ u s (·) ∈ C , always uniformly in s ∈ [ , T ] , therefore the second difference is also of order γ , as required. Supposing the estimate truefor every indices of length k we will prove it on every indices i · · · i k − of length k − . By repeatingthe same procedure as before we obtain Γ i ··· i k − u t ( X t ) − Γ i ··· i k − u s ( X s ) = (cid:18) Γ i ··· i k − u t ( X t ) − Γ i ··· i k − u s ( X t ) (cid:19)| {z } I + (cid:18) Γ i ··· i k − u s ( X t ) − Γ i ··· i k − u s ( X s ) (cid:19)| {z } II . ough transport and continuity Using the definition of a solution, the first difference on the right-hand side satisfies I = ( N γ + − k ) γ − N γ + − k Õ k = Õ | w | = k Γ i ··· i k − w u t ( X t )h W st , w i . On the other hand, using Lemma 3.8 two times we write Γ i ··· i k − u s ( X t ) = h e ∗ i ··· i k − , U s ( X ) t i so thatthe second difference can be replaced by the usual remainder I I = ( N γ + − k ) γ N γ + − k Õ k = Õ | w | = k h e ∗ i ··· i k − w , U s ( X ) s ih W st , w i = ( N γ + − k ) γ N γ + − k Õ k = Õ | w | = k Γ i ··· i k − w u s ( X s )h W st , w i . Combining the two estimates we obtain I + I I = − N γ + − k Õ k = Õ | w | = k (cid:18) Γ i ··· i k − w u t ( X t ) − Γ i ··· i k − w u s ( X s ) (cid:19) h W st , w i . Since the terms in the sum involve the increment Γ σ u t ( X t ) − Γ σ u s ( X s ) where σ has length bigger orequal than k we apply the recursive hypothesis obtaining that each term satisfies Γ i ··· i k − w u t ( X t ) − Γ i ··· i k − w u s ( X s ) = ( N γ + − k −| w |) γ and the multiplication with h W st , w i gives the desired estimate. (cid:3) Given a finite measure ρ ∈ M ( R n ) and a continuous bounded function φ ∈ C b ( R n ) , we write ρ ( φ ) = ∫ φ ( x ) ρ ( dx ) for the natural pairing. We are interested in measure-valued (forward) solutionsto the continuity equation d t ρ t = d Õ i = div x ( f i ( x ) ρ t ) d W it in ( , T ) × R n ,ρ = µ on { } × R n when W is again a weakly geometric rough path. As before we use the notation Γ i = f i ( x ) · D x ,whose formal adjoint is Γ ⋆ i = − div x ( f i ·) . Definition 4.5.
Let γ ∈ ( , ) , W ∈ C γ g and µ ∈ M ( R n ) . Any function ρ : [ , T ] → M ( R n ) such that ρ = µ is called a weak or measure-valued solution to the rough continuity equation d ρ t = d Õ i = div x ( f i ( x ) ρ t ) d W it (4.11) if for every φ bounded in C N γ + b and any word w with | w | ≤ N γ one has the estimates ρ t ( Γ w φ ) = ( N γ + −| w |) γ Õ ≤ | v | < N γ + −| w | ρ s ( Γ wv φ )h W st , e v i (4.12) for every s < t ∈ [ , T ] and uniformly in φ . Theorem 4.6.
Let f ∈ C N γ + b and W ∈ C γ g . Given initial data µ ∈ M ( R n ) , there exists a uniquesolution to the measure-valued rough continuity equation, explicitly given for φ ∈ C N γ + b by ρ t ( φ ) = ∫ φ ( X , xt ) µ ( dx ) , where X , x is the unique solution of the RDE d X t = Í di = f i ( X t ) d W it such that X , x = x . ough transport and continuity Proof. (Existence)
Using the composition of the controlled rough path X , x with φ ∈ C N γ + b and theshorthand notation X , xt = X t we can write φ ( X t ) = ( N γ + ) γ φ ( X s ) + N γ Õ k = Õ | w | = k Γ w φ ( X s )h W st , w i , Γ i ··· i n φ ( X t ) = ( N γ + − n ) γ Γ i ··· i n φ ( X s ) + N γ − n Õ k = Õ | w | = k Γ i ··· i n w φ ( X s )h W st , w i . This showing the existence when µ = δ x thanks to Proposition 4.3. Since we are dealing withbounded vector fields, all these estimates are uniform in X = x . Thus we can integrate both sideswith respect to the measure µ , obtaining the existence. (Uniqueness) To prove the uniqueness, we will show that for any < t ≤ T , any function g ∈ C N γ + b and any solution u : [ , t ] × R n → R of the RPDE d u r = d Õ i = Γ i u ( r , x ) d W ir , u t = g , the function r ∈ [ , t ] 7→ α ( r ) ≔ ρ r ( u r ) is constant. This property implies that for any function g ∈ C N γ + b and t > one has the identity ρ t ( g ) = ρ t ( u t ) = ρ ( u ) = µ ( u ) which uniquely determines the measure ρ t for any < t ≤ T . Since the parameter T was alsoarbitrary it is not restrictive to prove the result when t = T . Then α is constant if and only if one hasthe estimate α ( r ) = ( N γ + ) γ α ( s ) . (4.13)Writing u s , r = u r − u s and similarly for ρ one has ρ r ( u r ) − ρ s ( u s ) = ρ s , r ( u r ) + ρ s ( u s , r ) . By construction of regular solution with φ = u r ∈ C N γ + b the first summand expands as ρ s , r ( u r ) = ( N γ + ) γ N γ Õ k = Õ | w | = k ρ s ( Γ w u r )h W sr , w i . (4.14)On the other hand, we expand the second summand on the right-hand using the very definition ofregular backward RPDE obtaining u s , r ( x ) = ( N γ + ) γ − N γ Õ k = Õ | w | = k Γ w u r ( x )h W sr , w i , where the remainder is uniform on x . By integrating this estimate on ρ s , we obtain ρ s ( u s , r ) = ( N γ + ) γ − N γ Õ k = Õ | w | = k ρ s ( Γ w u r )h W sr , w i . (4.15)Combining the two estimates (4.15) and (4.14) we obtain (4.13) and the theorem is proven. (cid:3) References [Amb04] Luigi Ambrosio,
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