Nonlinear Young Differential Equations: A Review

Nonlinear Young integrals have been first introduced in Catellier and Gubinelli (Stoch Process Appl 126(8):2323–2366, 2016) and provide a natural generalisation of classical Young ones, but also a versatile tool in the pathwise study of regularisation by noise phenomena. We present here a self-contained account of the theory, focusing on wellposedness results for abstract nonlinear Young differential equations, together with some new extensions; convergence of numerical schemes and nonlinear Young PDEs are also treated. Most results are presented for general (possibly infinite dimensional) Banach spaces and without using compactness assumptions, unless explicitly stated.


Introduction
The main goal of this article is to solve and study differential equations of the form where x is an α-Hölder continuous path taking values in a Banach space V and A : [0, T ] × V → V is a vector field with suitable space-time Hölder regularity. If A is sufficiently smooth in time, then A(ds, x s ) can be interpreted as ∂ t A(s, x s )ds, so that (1.1) can be regarded as an ODE in integral form; here however we are interested in the case ∂ t A does not exist, so that (1.1) does not admit a classical interpretation.
In the case A(t, z) = f (z)y t , where y is an U -valued α-Hölder continuous path and f maps V into the space of linear maps from U to V , equation (1.1) can be rewritten as which can be regarded as a rough differential equation driven by a signal y.
In the regime α ∈ (1/2, 1], for sufficiently regular f , equation (1.2) can be rigorously interpreted by means of Young integrals, introduced in [44]; wellposedness of Young differential equations (YDEs) was first studied in [34]. After that, several alternative approaches to (1.2) have been developed, either by means of fractional calculus [45] or numerical schemes [14]; see also the review [33] for a self-contained exposition of the main results for YDEs and the paper [13] for some recent developments. YDEs have found several applications in the study of SDEs driven by fractional Brownian motion (fBm) of parameter H > 1/2, see for instance [37].
Although equation (1.1) may be seen as a natural generalization of (1.2), its development is much more recent. Nonlinear Young integrals of the form t 0 A(ds, x s ) were first defined in [9] in applications to additively perturbed ODEs and subsequently rediscovered in [30], where they were employed to give a pathwise interpretation to Feynman-Kac formulas and SPDEs with random coefficients.
In this paper we will consider exclusively the time regularity regime α > 1/2, also known as the Young (or or level-1 rough path) regime. However it is now well known, since the pioneering work of Lyons [35], that it is possible to give meaning to equation (1.2) even in the case α ≤ 1/2 by means of the theory of rough paths, see the monographs [19], [18] for a detailed account on the topic. An analogue extesion of (1.1) to the case of nonlinear rough paths has been recently achieved in [12], [38]; so far however it hasn't found the same variety of applications, discussed below, as the nonlinear Young case. Let us finally mention that all of the above can also be seen as subcases of the theory of rough flows developed in [2], [4].
Nonlinear YDEs of the form (1.1) mostly present direct analogue results to their classical counterpart (1.2), but their importance and the main motivation for this work lies in their versatility. Indeed, many differential systems which a priori do not present such structure, may be recast as nonlinear YDEs; this allows to give them meaning in situations where classical theory breaks down.
This methodology seems seems particularly effective in applications to regularization by noise phenomena; to clarify what we mean, let us illustrate the following example, taken from [11], [10]. In these works the authors study abstract modulated PDEs of the form Under suitable assumption, even if w is not smooth (actually exactly because it is rough, as measured by its ρ-irregularity), it is possibile to rigorously define the field A, even if the integral appearing on the r.h.s. of (1.4) is not meaningful in the Lebesgue sense. As a consequence, the transformation of the state space given by ϕ → ψ allows to interpret the original PDE (1.3) as a suitable nonlinear YDE; the general abstract theory presented here can then be applied, immediately yielding wellposedness results.
A similar reasoning holds for additively perturbed ODEs of the form which were first considered in [9], in which case the transformation amounts to x → θ := x − w. This case has recently received a lot of attention and developed into a general theory of pathwise regularisation by noise for ODEs and SDEs, see [20], [21], [22], [28], [26] and on a related note [27].
Motivated by the above discussion, we collect here several results for abstract nonlinear YDEs which have appeared in the above references, together with some new extensions; they provide general criteria for existence, uniqueness and stability of solutions to (1.1), as well as convergence of numerical schemes and differentiability of the flow. This work is deeply inspired by the review [33], of which it can be partially regarded as an extension; all the theory is developed in (possibly infinte dimensional) Banach spaces and relies systematically on the use of the sewing lemma, a by now standard feature of the rough path framework. We hope however that the also reader already acquainted with RDEs can find the paper of interest due to later Sections 5-7, containing less standard results and applications to Young PDEs.
Structure of the paper. In Section 2, the nonlinear Young integral is constructed and its main properties are established. Section 3 is devoted to criteria for existence, uniqueness, stability and convergence of numerical schemes for nonlinear YDEs, Sections 3.4 and 3.5 focusing on several variants of the main case. Section 4 deals continuity of the solutions with respect to the data of the problem, giving conditions for the existence of a flow and differentiability of the Itô map. The results from Section 3.3 are revisited in Section 5, where more refined criteria for uniqueness of solutions are given; we label them as "conditional uniqueness" results, as they require additional assumptions which are often met in probabilistic applications, but are difficult to check by purely analytic arguments. Sections 6 and 7 deal respectively with Young transport and parabolic type of PDEs. We chose to collect in the Appendix some useful tools and further topics.
Notation. Here is a list of the most relevant and frequently used notations and conventions: • We write a b if a Cb for a suitable constant, a x b to stress the dependence C = C(x).
• We will always work on a finite time interval [0, T ]; the Banach spaces V , W appearing might be infinite dimensional but will be always assumed separable for simplicity.
• The above notation will be applied to several choice of E such as C α t V , C α t R d but also C α t C β,λ V,W or C α t C β V,W,loc , for which we refer to Definitions 2.3 and 2.5. • We denote by L(V ; W ) the set of all linear bounded operators from V to W , L(V ) = L(V ; V ).
• Whenever we will refer to differentiability this must be understood in the sense of Frechét, unless specified otherwise; given a map F : V → W we regard its Frechét differential D k F of order k as a map from V to L k (V ; W ), the set of bounded k-linear forms from V k to W . We will use indifferently DF (x, y) = DF (x)(y) for the differential at point x evaluated along the direction y.
• Given a linear unbounded operator A, Dom(A) denotes its domain, rg(A) its range.
• As a rule of thumb, whenever J(Γ) appears, it denotes the sewing of Γ : ∆ 2 → E; we refer to Section 2.1 for more details on the sewing map. Similarly, in proofs based on a Banach fixed point argument, I will denote the map whose constractivity must be established.
• As a rule of thumb, we will use C i , i ∈ N for the constants appearing in the main statements and κ i for those only appearing inside the proofs; the numbering restarts at each statement and is only meant to distinguish the dependence of the constants from relevant parameters.

The nonlinear Young integral
This section is devoted to the construction of nonlinear Young integrals and nonlinear Young calculus more in general, as a preliminary step to the study of nonlinear Young differential equations which will be developed in the next section. We follow the modern rough path approach to abstract integration, based on the sewing lemma as developed in [24] and [17], which is recalled first.

Preliminaries
This subsections contains an exposition of the sewing lemma and the definition of the joint space-time Hölder continous drifts A we will work with; the reader already acquainted with this concepts may skip it.
Remark 2.2 Let us stress two important aspects of the above result. The first one is that all the estimates do not depend on the Banach space V considered; the second one is that, even when the map J (Γ) is already known to exist, property (2.1) still gives non trivial estimates on its behaviour.
In particular, if f ∈ C α t V is a function such that Γ s,t − f s,t V κ|t − s| α for an unknown constant κ, then by the sewing lemma we can deduce that f = J (Γ) and that κ can be taken as C 1 δΓ β .
Next we need to introduce suitable classes of Hölder continuous maps on Banach spaces.
We say that f is locally β-Hölder continuous and write f ∈ C β V,W,loc if for any R > 0 the following quantities are finite: For λ ∈ (0, 1], we define the space C β,λ V,W as the collection of all f ∈ C(V ; W ) such that Finally, the classical Hölder space C β V,W is defined as the collection of all f ∈ C(V ; W ) such that Remark 2. 4 We ask the reader to keep in mind that although linked, f β,R and f β,λ denote two different quantities. Throughout the paper R will always denote the radius of an open ball in V and consequently all related seminorms are localised on such ball; instead the parameter λ measures the polynomial growth of · β,R as a function of R. C β V,W,loc is a Fréchet space with the topology induced by the seminorms { f β,R } R 0 , while C β,λ V,W and C β V,W are Banach spaces. Observe that if f ∈ C β,λ V,W , we have an upper bound on its growth at infinity, since for any In particular, if β + λ 1, then f has at most linear growth.
We can now introduce fields A : [0, T ] × V → W satisfying a joint space-time Hölder continuity. We adopt the incremental notation A s,t (x) := A(t, x) − A(s, x), as well as A t (x) = A(t, x); from now on, whenever A appears, it is implicitly assumed that A(0, x) = 0 for all x ∈ V . Definition 2.5 Given A as above, α, β ∈ (0, 1), we say that A ∈ C α t C β V,W,loc if for any R 0 it holds The definition can be extended to the cases α = 0 or β = 0 by interpreting the norm in the supremum sense: Given a smooth F : V → W , we regard its Frechét differential D k F of order k as a map from V to L k (V ; W ), the set of bounded k-linear forms from V k to W .
Analogue definitions hold for C α t C n+β V,W,loc and C α t C n+β,λ V,W .

Construction and first properties
We are now ready to construct nonlinear Young integrals, following the line of proof from [30], [28]; other constructions are possible, see Appendix A.2.
Then for any [s, t] ⊂ [0, T ] and for any sequence of partitions of [s, t] with infinitesimal mesh, the following limit exists and is independent of the chosen sequence of partitions: The limit is usually referred as a nonlinear Young integral. Furthermore: it is locally δ-Hölder continuous in x for any δ ∈ (0, 1) such that δ < (α + βγ − 1)/γ and there exists Proof. In order to show convergence of the Riemann sums, it is enough to apply the sewing lemma to the choice Γ s,t := A s,t (x s ) = A(t, x s ) − A(s, x s ). Indeed we have In particular Γ ∈ C α,α+βγ 2 W with α + βγ > 1, therefore by the sewing lemma we can set It remains to show estimate (2.5). To this end, for fixed x, y ∈ C γ t V and R as above, we need to provide estimates for δΓ 1+ε forΓ s,t := A s,t (x s ) − A s,t (y s ) and suitable ε > 0. It holds for any θ ∈ (0, 1) such that (1 − θ)(α + βγ) + θα = 1 + ε > 1, namely such that The sewing lemma then implies that Dividing by |t − s| α and taking the supremum we obtain (2.5 for y ∈ C δ t R such that α + δ > 1 and A, x as above. This can be either interpreted as a more classical Young integral of the form · 0 y t d t 0 A(ds, x s ) = J (Γ) for Γ s,t = y s t s A(dr, x r ), or as the sewing of Γ s,t = y s A s,t (x s );it is immediate to check equivalence of the two definitions. This case can be further extended to consider a bilinear map G : W × U → Z, where U and Z are other Banach spaces, so that · 0 G(y s , A(ds, x s )) ∈ C α t Z is well defined for y ∈ C δ t U , A and x as above, as the sewing of Γ s,t = G(y s , A s,t (x s )) ∈ C α,α+δ Nonlinear Young integrals are a generalisation of classical ones, as the next example shows.
In particular, for any x ∈ C γ t R d with α + βγ > 1, we can consider · 0 A(ds, x s ); this corresponds to the classical Young integral · 0 f (x s )dy s , since both are defined as sewings of The previous example generalizes an infinite sum of Young integrals, i.e. considering sequences In this case we can define A(t, x) := n f n (x)y n t , which satisfies A α,β n f n β y n α and for any x ∈ C δ t R d it holds x ∈ C γ t and y ∈ C α t with α + βγ > 1, then one can define the Young integral · 0 f (s, x s )dy s . However, · 0 f (s, x s )dy s does not coincide with A(ds, x s ) for the choice A(t, x) := f (t, x)y t . This is partially because the domain of definition of the two integrals is different, since condition (2.6) (which is locally equivalent to f ∈ C βγ is not enough to ensure that A ∈ C α t C β x ; however, if we additionally assume f ∈ C α t C β x , then so does A, and the relation between the two integrals is given by

Nonlinear Young calculus
Theorem 2.7 establishes continuity of the map (A, x) → · 0 A(ds, x s ); if A is sufficiently regular, then we can even establish its differentiability. Proposition 2.11 Let α, β, γ ∈ (0, 1) such that α + βγ > 1, A ∈ C α t C 1+β V,W,loc . Then the nonlinear Young integral, seen as a map F : Proof. For notational simplicity we will assume A ∈ C α t C 1+β V,W . It is enough to show that, for any x, y ∈ C γ t V , the Gateaux derivative of F at x in the direction y is given by the expression above, i.e.
where the limit is in the C α t W -topology. Indeed, once this is shown, it follows easily from reasoning as in Theorem 2.7 that the map (x, y) → DA(ds, x s )y s is jointly uniformly continuous in bounded balls and linear in the second variable; Frechét differentiability then follows from existence and continuity of the Gateaux differential.
In order to show (2.9), setting for any ε > 0 Γ ε s,t := In particular by Lemma A.2 from the Appendix, we only need to check that Γ ε α → 0 as ε → 0 while δΓ ε α+βγ stays uniformly bounded. It holds which implies that δΓ α+βγ 1 uniformly in ε > 0. The conclusion the follows. ✷ Proposition 2.11 allows to give an alternative proof of Lemma 6 from [20].
the above formula meaningfully defines an element of C α t L(V, W ) which satisfies v α C DA α,β, where R x ∞ ∨ y ∞ and C = C(α, β, γ, T ).
Proof. It follows from the hypothesis on A that the map is well defined, the outer integral being in the Bochner sense, and it is linear in y; moreover estimate (2.3) combined with the trivial inequality 1 + In particular, if we define v t as the linear map appearing (2.13), it is easy to check that similar estimates yield v ∈ C α t L(V, W ). The fact that this definition coincide with the one from (2.11), i.e. that we can exchange integration in dλ and in "ds", follows from the Fubini theorem for the sewing map, see Lemma A.1 in the Appendix. Inequality (2.12) then follows from estimates analogue to the ones obtained above. Identity (2.10) is an application of the more abstract classical identity applied to F (x) = · 0 A(ds, x s ), for which the exact expression for DF is given by Proposition 2.11. ✷ The following Itô-type formula is taken from [30], Theorem 3.4.
In particular, if x = · 0 A(ds, y s ) for some A ∈ C γ t C δ V , y ∈ C η t V with γ + ηδ > 1, then (2.15) becomes Proof. Let 0 = t 0 < t 1 < · · · < t n = t, then it holds . Taking a sequence of partitions Π n with |Π n | → 0, by the above estimate we have I n 3 → 0 and by the sewing lemma we obtain which is exactly (2.14). If F ∈ C 0 t C 1+β ′ V,W,loc , then setting Γ 3 s,t := DF (s, x s )(x s,t ), it holds which under the assumption γ(1 + β ′ ) > 1 implies by the sewing lemma that J (Γ 2 ) = J (Γ 3 ) and thus (2.15). The proof of (2.16) is analogue, only this time consider Γ 4 s,t := DF (s, x s )(A s,t (y s )), then it's easy to check that Γ 3 s,t − Γ 4 Remark 2.14 The above formulas admit further variants. For instance for any F ∈ C α t C β V,W , x ∈ C γ t V and g ∈ C δ t R with α + βγ > 1, α + δ > 1 and βγ + δ > 1 it holds and we have the product rule formula Also observe that, whenever ∂ t F exists continuous, it holds

Existence, uniqueness, numerical schemes
This section is devoted to the study of nonlinear Young differential equations (YDE for short), defined below; it provides sufficient conditions for existence and uniqueness of solutions, as well as convergence of numerical schemes.
for some γ such that α + βγ > 1 and it satisfies (3.1) Before proceeding further, let us point out that by Example 2.9 any Young differential equation can be reinterpreted as a nonlinear YDE associated to A := f ⊗ y. Nonlinear YDEs therefore are a natural extension of the standard ones; most results regarding their existence and uniqueness which will be presented are perfect analogues (in terms of regularity requirements) to the well known classical ones (which can be found for instance in [33] or Section 8 of [18]). x s,t V |t − s| γ .

Existence
We provide here sufficient conditions for the existence of either local or global solutions to the YDE, under suitable compactness assumptions on A. The proof is based on an Euler scheme for the YDE, in the style of those from [14], [33]; its rate of convergence will be studied later on. Other proofs, based on a priori estimates and compactness techniques or an application of Leray-Schauder-Tychonoff fixed point theorem, are possible, see [9], [30].
where W is compactly embedded in V and α(1 + β) > 1. Then for any s > 0 and x s ∈ V there exists a solution to the YDE Proof. The proof is based on the application of an Euler scheme. Up to rescaling and shifting, we can assume for simplicity T = 1 and s = 0. Fix N ∈ N, set t n k = k/n for k ∈ {0, . . . , n} and define recursively (x n k ) n k=1 by x n 0 = x 0 and We can embed (x n k ) n k=1 into C 0 t V by setting note that by construction x n − x 0 is a path in C α t W . Using the identity we deduce that x n satisfies a YDE of the form By the properties of Young integrals, ψ n satisfies We first want to obtain a bound for ψ n γ,∆,W ; we can assume wlog ∆ > 1/n, since we want to take n → ∞. Estimates depend on whether s and t lie on the same interval [t n k , t n k+1 ] or not; assume first this is the case, then Next, given s < t such that |t − s| < ∆ which are not in the same interval, there are around n|t − s| intervals separating them, i.e. there exist l < m such that m − l ∼ n|t − s| and s t n l < · · · < t n m t. Therefore in this case we have where in the second line we used both (3.4) and the previous bound for ψ n s,t n l and ψ n t n m ,t , while in the last one the fact that −αβ 1 − α(1 + β). Overall we conclude that ψ n α,∆,W for a suitable constant κ 1 = κ 1 (α, β) independent of ∆ and n.
Our next goal is a uniform bound for x n α,∆,W . Since x n solves (3.3), it holds ) and so dividing by |t − s| and taking the supremum over all |t − s| < ∆, choosing ∆ such that ∆ αβ A α,β 1/4, then for all n big enough such that by the trivial bound a β 1 + a, which holds for all β ∈ [0, 1] and a 0. This implies the uniform bound x n α,∆,W 1 + A α,β for all n big enough.
The subspace {y ∈ C α ([0, 1]; W ) : y 0 = 0} is a Banach space endowed with the seminorm y α,∆,W , which in this case is equivalent to the norm y α,W ; {x n − x 0 } n∈N is a uniformly bounded sequence in this space. By Ascoli-Arzelà, since W compactly embeds in V , we can extract a subsequence (not relabelled for simplicity) such that Observe that ψ n satisfy (3.5) and x n β α,∆,V are uniformly bounded, therefore ψ n → 0 in C α t W as n → ∞; choosing ε small enough s.t. α + β(α − ε) > 1, by continuity of the non-linear Young integral it holds and therefore passing to the limit in (3.3) we obtain the conclusion. ✷

Remark 3.3
If V is finite dimensional, the compactness condition is trivially satisfied by taking V = W . The proof also works for non uniform partitions Π n of [0, T ], under the condition that their mesh |Π n | → 0 and that there exists

Remark 3.4
The proof provides several estimates, some of which are true even without the compactness assumption. For instance, by x n α,∆ 1 + A α,β and Exercise 4.24 from [18], choosing ∆ s.t.
Estimate (3.5) is true for any choice of ∆ > 0, in particular for ∆ = T , which gives a global bound; combining it with the above one, we deduce that . Also observe that from the assumptions on α and β it always holds Under the compactness assumption, since x n → x in C 0 t V , the solution x obtained also satisfies Finally observe that by going through the same proof of (3.5), for any T > 0 and α, β, γ such that This estimate is rather useful when A enjoys different space-time regularity at different scales, see the discussion at Section 3.4.
Then for any s ∈ [0, T ) and any x s ∈ V , there exists τ * ∈ (s, T ] and a solution to the YDE (3.2) defined on [s, T * ), with the property that either T * = T or Proof. As before it is enough to treat the case s = 0, We can now iterate this procedure, i.e. set x 1 := x τ1 and construct another solution to (3.2), defined on an interval [τ 1 , τ 2 ], and so on; by "gluing" these solutions together, we obtain an increasing sequence {τ n } ⊂ [0, 1] and a solution x · defined on [0, T * ), where T * = lim n τ n . Now suppose that T * < T and lim inf t→T * x t V < ∞, then we can find a sequence t n → T * such that x tn V M for some M > 0; but then starting from any of this x tn we can construct another solution y n defined on [t n , t n + ε], where ε is uniform in n since x tn M and ε can be estimated by (3.8) with R replaced by M . By replacing the solution x · on [t n , T * ) with y n , choosing n big enough, we can construct a solution defined on [0, T * + ε/2). Reiterating this procedure we obtain the conclusion. ✷

A priori estimates
A classical way to pass from local to global solutions is to establish suitable a priori estimates, which are also of fundamental importance for compactness arguments. Throughout this section, we assume that a solution x to the YDE is already given and focus exclusively on obtainig bounds on it; for simplicity we work on [0, T ], but all the statements immediately generalise to [s, T ].
x 0 ∈ V and x ∈ C α t V be a solution to the associated YDE. Then there exists C = C(α,β, T ) such that Proof. Let ∆ ∈ (0, T ] be a parameter to be chosen later. For any s < t such that |s − t| ∆, using the fact that x is a solution, it holds were we used the trivial inequality a β 1 + a. Dividing both sides by |t − s| α and taking the supremum over |s − t| ∆, we get Choosing ∆ small enough such that κ 1 ∆ αβ A α,β 1/2, we obtain If we can take ∆ = T , we get an estimate for x α , which gives the conclusion. If this is not the case, we can choose ∆ such that in addition κ 1 ∆ αβ A α,β 1/4 and then as before, by Exercise 4.24 from [18] it holds x α T ∆ α−1 x α,∆ , so that where we used the fact that α(1 + β) > 1 implies (1 − α)/(αβ) < 1. The conclusion follows by the standard inequality The assumption of a global bound on A of the form A ∈ C α t C β V is sometimes too strong for practical applications. It can be relaxed to suitable growth conditions, as the next result shows; it is taken from [30], Theorem 3.1 (see also Theorem 2.9 from [9]).
Then there exists a constant C = C(α, β, T ) such that any solution x on [0, T ] to the YDE associated to (x 0 , A) satisfies which implies, dividing by |r − u| α and taking the supremum, that By an application of Young's inequality, for any a, b 0 it holds a λ b β a β+λ + b β+λ ; moreover β + λ 1 so that a β+λ 1 + a for any θ ∈ [0, 1], therefore we obtain where in the second passage we used the estimate x ∞;s,t T x s V + x α;s,t . Overall we deduce the existence of a constant κ 1 = κ 1 (α, β, T ) such that If T satisfies κ 1 A α,β,λ T αβ 1, then we can take ∆ = T , which gives a global estimate and thus the conclusion. If this is not the case, then we can choose ∆ < T s.t. κ 1 A α,β,λ ∆ αβ = 1 and (3.11) implies that and thus . Therefore where again κ 2 = κ 2 (α, β, T ). In particular, in order to obtain the final estimate, we only need to focus on x ∞ . Let us consider, for ∆ as above, the intervals I n := [(n − 1)∆, n∆] and set J n := 1 + x ∞;In , with the convention J 0 = 1 + x 0 V . Then estimates analogue to (3.11) yield where we used the basic inequality 1 + x e x . Since [0, T ] is covered by N ∼ T ∆ −1 intervals and we chose ∆ −1 ∼ A 1/αβ , up to relabelling κ 1 into a new constant κ 3 we obtain Finally, combining this with the estimate for x α above we obtain where we used the inequality xe λx λ −1 e 2λx . The conclusion follows. ✷ up to possibly changing constant C = C(α, β, T ).
The dependence of C on T can be established by a rescaling argument: if x is a solution on [0, T ] to the YDE associated to (x 0 , A), then x t =x t/T wherex is a solution on [0, 1] to the YDE associated to (x 0 ,Ã),Ã(t, z) = A(T t, z). Therefore one can apply the estimates tox,Ã and T = 1 and then write explicitly how x α , A α,β,λ depend on x α , Ã α,β,λ . The same reasoning applies to several other estimates appearing later on, for which the dependence of C on T is not made explicit.
In classical ODEs, a key role in establishing a priori estimates (as well as uniqueness) is played by Gronwall's lemma; the following result can be regarded as a suitable replacement in the Young setting. One of the main cases of applicability is for A ∈ C α t L(V ; V ).
Then there exists a constant C = C(α) such that any solution x to the YDE satisfies the a priori bounds Proof. We can assume without loss of generality that T = 1, as the general case follows by rescaling. It is also clear that, up to changing constant C, inequality (3.17) follows from combining together (3.15) and (3.16) and using the fact that A 1/α α,1 1 + A 2 α,1 since α > 1/2. Up to renaming x 0 , we can also assume h 0 = 0. The proof is similar to that of Proposition 3.7, but we provide it for the sake of completeness.
Let ∆ > 0 to be chosen later, s < t such that |t − s| ∆, then by (3.14) it holds and so dividing both sides by |t − s| α , taking the supremum over s, t and choosing ∆ such that As usual, if κ 1 A α,1 1/2, then the conclusion follows from (3.18) with the choice ∆ = 1 and the trivial estimate In , then estimates similar to the ones done above show that which implies recursively that for a suitable constant κ 2 it holds J n e κ2n ( x 0 V + h α ). Since which gives (3.16); combined with ∆ −α ∼ A α,1 , estimate (3.18) and the basic inequality it also yields estimate (3.15). ✷ Another way to establish that solutions don't blow-up in finite time is to the show that the YDE admits (coercive) invariants. The next lemma gives simple conditions to establish their existence.
Proof. It follows immediately from the Itô-type formula (2.16), since it holds

Uniqueness
We now turn to sufficient conditions for uniqueness of solutions; some of the results below also establish existence under different sets of assumptions than those from Section 3.1.
Then for any x 0 ∈ V there exists a unique global solution to the YDE associated to (x 0 , A).
Proof. The proof is based on an application of Banach fixed point theorem. Let M , τ be positive parameters to be fixed later and set which is complete metric space with the metric d(x, y) = x − y α ; define the map I by We want to show that I is a contraction from E to itself, for suitable choice of M and τ . It holds Choosing τ and M such that for any x ∈ V it holds By the hypothesis and Corollary 2.12, for any x, y ∈ V it holds which implies The same procedure allows to show existence and uniqueness of solutions and any x s ∈ V , where τ does not depend on (s, x s ); by iteration, global existence and uniqueness follows. ✷ then global existence and uniqueness holds.
Proof. We only sketch the proof, as it follows from classical ODE arguments and is similar to that of Corollary 3.5.
By localization, given any s ∈ [0, T ) and any x s ∈ V , there exists τ = τ (s, x s ) such that there exists a unique solution to the YDE associated to (x s , A) on the interval [s, s + τ ]. Therefore given two solutions x i defined on intervals [s, T i ] with x 1 s = x 2 s , they must coincide on [s, T 1 ∧ T 2 ]; in particular, any extension procedure of a given solution to a larger interval is consistent, which allows to define the maximal solution as the maximal extension of any solution starting from x 0 at t = 0.
The blow-up alternative can be established reasoning by contradiction as in Corollary 3.5. If A ∈ C α t C β,λ V , then by the a priori estimate (3.10) blow-up cannot occur and so global well-posedness follows. ✷ Once existence of solutions is established, their uniqueness can be alternatively shows by means of a Comparison Principle, which is the analogue of a Gronwall type estimate for classical ODEs. Such results are of independent interest as they also allow to compare solutions to different YDEs; they were first introduced in [9] and later revisited in [20].
; let x i be two given solutions associated respectively to (x i 0 , A i ). Then it holds for a constant C = C(α, β, T, R, M ) increasing in the last two variables.
Proof. Let x i be the two given solutions and set e t := x 1 t − x 2 t , then e satisfies where we applied Corollary 2.12. By the same result, combined with estimate (3.13), it holds similarly, by Point 4. of Theorem 2.7, Applying Theorem 3.9 to e, we have which combined with the previous estimates implies the conclusion. ✷ V and we consider solutions x i associated to (x i 0 , A), going through the same proof but applying instead estimate (3.9), we obtain which combined with (3.17) implies the existence of a constant C = C(α, β, T ) such that As a consequence, the solution map As a corollary, we obtain convergence of the Euler scheme introduced in Section 3.1, with rate 2α − 1. For simplicity we state the result in the case A ∈ C α t C 1+β V , but the same results follow for by the usual localization procedure.
V with α(1 + β) > 1 and x 0 ∈ V , denote by x n the element of C α t V constructed by the n-step Euler approximation from Theorem 3.2, and by x the unique solution associated to (x 0 , A). Then there exists a constant C = C(α, β, T ) such that Proof. Recall that by Theorem 3.2, x n satisfies the YDE where by Remark 3.4, for the choice β = 1, it holds Define e n := x − x n , then by Corollary 2.12 it satisfies Applying Theorem 3.9, we deduce the existence of which combined with the estimate for ψ n α yields the conclusion. ✷

The case of continuous ∂ t A
In this section we study how the well-posedness theory changes when, in addition to the regularity condition A ∈ C α t C β t , we impose ∂ t A : [0, T ] × V → V to exist continuous and uniformly bounded (we assume boundedness for simplicity, but it could be replaced by a growth condition).
The key point is that, by Point 2. from Theorem 2.7, any solution to the YDE is also a solution to the classical ODE associated to ∂ t A; as such, it is Lipschitz continuous with constant ∂ t A ∞ . We can exploit this additional time regularity, combined with nonlinear Young theory, to obtain well-posedness under weaker conditions than those from Theorem 3.12.
While the existence of ∂ t A is not a very meaningful requirement for classical YDEs, i.e. for A(t, x) = f (x)y t , as it would imply that y ∈ C 1 t , there are other situations in which it becomes a natural assumption. One example is for perturbed ODEsẋ = b(x) +ẇ, in which the associated A is the averaged field for which ∂ t A exists continuous as soon as b is continuous field; still classical wellposedness is not is not guaranteed under the sole continuity of b.
Then for any x 0 ∈ V there exists a unique global solution to the YDE associated to (x 0 , A).
Proof. Similarly to Theorem 3.12, the proof is by Banach fixed point theorem. For suitable values of M, τ > 0 to be fixed later, consider the space it is a complete metric space with the metric d(x, y) = x − y γ (the condition x Lip M is essential for this to be true). Define the map I by and observe that under the condition ∂ t A ∞ M it maps E into itself. By the hypothesis and Corollary 2.12, for any x, y ∈ E it holds as soon as we choose τ small enough such that κ 2 τ α A α,1+β (1+2M ) < 1. Therefore I is a contraction on E and for any x 0 ∈ V there exists a unique associated solution x ∈ C γ ([0, τ ]; V ). Global existence and uniqueness then follows from the usual iterative argument.
✷ We can also establish an analogue of Theorem 3.14 in this setting.
, and x i 0 ∈ V ; let x i be two given solutions associated respectively to (x i 0 , A i ). Then it holds for a constant C = C(α, β, T, M ) increasing in the last variable. A more explicit formula for C is given by (3.20).
Proof. The proof is analogous to that of Theorem 3.14, so we will mostly sketch it; it is based on an application of Corollary 2.12 and Theorem 3.9. Given two solutions as above, their difference e = x 1 − x 2 satisfies the affine YDE We have the estimates v α,1 α,β,T In particular, C can be taken of the form (3.20) ✷ Corollary 3.19 Given A as in Theorem 3.17, denote by x n the element of C α t V constructed by the n-step Euler approximation from Theorem 3.2 and by x the solution associated to (x 0 , A). Then there exists a constant C = C(α, β, T, A α,1+β , ∂ t A ∞ ) such that x − x n α Cn −α as n → ∞.
A more explicit formula for C is given by (3.21).
Proof. By Theorem 3.2, x n satisfies the YDE where A n (t, z) := A(t, z) + ψ n t and that by estimate (3.7), for the choice ∆ = T , β = γ = 1, we have Defining e n := x − x n , by the basic estimates A − A n α,β T ψ n α and ∂ t A n ∞ ∂ t A ∞ , going through the same proof as in Theorem 3.18 we deduce that and so finally that, for a suitable constant κ 2 = κ 2 (α, T ), it holds

Further variants
Several other kinds of differential equations involving a nonlinear Young integral term can be studied.
In this section we focus on two cases: nonlinear YDEs involving a classical drift term and fractional YDEs.

Mixed equations
Let us consider now an equation of the form where F : [0, T ] × V → V is continuous function; the first integral is meaningful as a classical one.
Proof. For simplicity we will use the notation A = A α,1+β ; the proof is analogue to that of Theorem 3.12. Let M , τ be positive parameters to be fixed later and define as usual A path x solves (3.22) if and only if it belongs to E and is a fixed point for the map We have the estimates which imply In order for I to map E into itself, it suffices to choose τ and M such that Next we check contractivity of I; given x, y ∈ E, it holds thus choosing τ small enough we deduce contractivity. Therefore existence and uniqueness of solutions holds on the interval [0, τ ]; as the choice of τ does not depend on x 0 , we can iterate the reasoning to cover the whole interval [0, T ]. ✷ Theorem 3.21 Let A ∈ C α t C 1+β V,loc with α(1 + β) > 1 and F be a continuous locally Lipschitz function, in the sense that for any R > 0 there exist a constant C R such that Then for any If in addition A ∈ C α t C β,λ V with β + λ 1 and F has at most linear growth, i.e. there exists C F > 0 s.t.
then global wellposedness holds. Moreover in this case there exists C = C(α, β, T ) such that, setting θ = 1 + 1−α αβ , any solution to (3.22) satisfies the a priori estimate Proof. The first part of the statement, regarding local wellposedness and the blow-up alternative, follows from the usual localisation arguments, so we omit its proof. The proof of a priori estimate (3.23) is analogue to that of Proposition 3.7, so we will mostly sketch it; as before A = A α,β,λ for simplicity. Let x be a solution to (3.22 Together with the estimates from the proof of Proposition 3.7 and the fact that |t − s| |t − s| αβ , this implies the existence of κ 1 = κ 1 (α, β, T ) such that any solution x to (3.22) satisfies The rest of the proof is identical, up to replacing A with C F + A in all the passages. Specifically, if T is such that κ 1 (C F + A )T αβ < 2, then we obtain a global estimate by choosing s = 0, t = T , which shows that T * = T and gives the conclusion in this case. Otherwise, taking ∆ < T such that κ 1 (C F + A )∆ αβ = 1 and defining J n as before, we obtain the recurrent estimate and going through the same reasoning the conclusion follows. ✷

Fractional Young equations
We restrict in this subsection to the finite dimensional case V = R d for some d ∈ N; as usual we work on a finite time interval [0, T ]. We are interested in studying a fractional type of equation of the form for a suitable parameter δ ∈ (0, 1). Here D δ 0+ denotes a Riemann-Liouville type of fractional derivative on [0, T ]; for more details on fractional derivatives and fractional calculus we refer the reader to [40]. In the case δ = 1, formally D δ x s = dx s and we recover the class of YDEs studied so far.
In order to study (3.24), it is more convenient to write it in integral form, using the fact that D δ

0+
is the inverse operator of the fractional integral I δ 0+ given by . From now on we will for simplicity drop the constant 1/Γ(δ), which can be incorporated in the drift A. We need the following lemma. For any α ∈ (0, 1) such that α + δ > 1 and any ε > 0, Ξ extends uniquely to a continuous linear map from C α ([0, T ]; R d ) to C α+δ−1−ε ([0, T ]; R d ); in particular, there exists C = C(α, δ, ε, T ), which will be denoted by Ξ , such that Proof. Up to multiplicative constant, Ξ = I α 0+ D. Recall that fractional integrals and fractional derivatives, on their domain of definition, satisfy the following properties, for α, β, α + β ∈ [0, 1]: Let f be a smooth function, then Ξ[f ] = I δ 0+ Df = D 1−δ 0+ f ; moreover for any γ < α, we can write The conclusion for general f follows from an approximation procedure. Indeed, since all inequalities are strict, we can replace α with α − ε and use the fact that functions in C α t can be approximated by smooth functions in the C α−ε t -norm. The fact that in (3.25) only the seminorm f appears is a consequence of the fact that by definition Ξ[1] = 0 and so we can always shift f in such a way that f 0 = 0. ✷ The general case follows from an approximation procedure.
Thanks to Lemma 3.22 we can give a proper meaning to the fractional YDE.
Definition 3. 24 We say that x is a solution to (3.24) if · 0 A(ds, x s ) is well defined as a nonlinear Young integral in C α t for some α > 1 − δ and x satisfies the identity Then for any x 0 ∈ R d and any γ < α + δ − 1 there exists a solution x ∈ C γ t to (3.24), in the sense of Definition 3.24.
Proof. Due to condition (3.26), we can find γ ∈ (0, 1), ε > 0 sufficiently small satisfying The existence of a solution is then equivalent to the existence of a fixed point in C γ t for the map The above conditions imply α + β(γ − ε) > 1, so by Theorem 2.7 the map x → A(ds, x s ), from C γ−ε t to C α t is continuous and satisfies which together with estimate (3.25) implies that I is continuous from C γ−ε t to C γ t with for suitable κ 1 = κ 1 (T, α + β(γ − ε)). It follows by Ascoli-Arzelà that I is compact from C γ−ε t to itself; for any λ ∈ (0, 1), if x solves x = λI(x), then . Since β < 1, any such solution x must satisfy (for instance) where the estimate is uniform in λ ∈ [0, 1]. We can thus apply Schaefer's theorem to deduce the existence of a fixed point for I in C γ−ε t , which also belongs to C γ t since I(x) does so. ✷ Theorem 3.26 Let A ∈ C α t C 1+β x with α, β, δ satisfying (3.26). Then for any x 0 ∈ R d there exists a unique solution x ∈ C γ t to (3.24), for any γ satisfying Proof. Existence is granted by Proposition 3.25, so we only need to check uniqueness. Let x and y be two solutions, say with x α , y α M for suitable M > 0; we are first going to show that they must coincide on an interval [0, τ ] with τ sufficiently small. It holds combined with the previous estimates we obtain Choosing τ small enough such that κ 3 Ξ A α,1+β (1 + M )τ γ < 1, we conclude that x ≡ y on [0, τ ]. As a consequence, (ds, y s+τ ) t whereÃ(t, x) = A(t + τ, x) has the same regularity properties of A. We can therefore iterate the previous argument, applied this time toÃ, x ·+τ and y ·+τ , to deduce that x and y also coincide on [τ, 2τ ]; repeating this procedure we can cover the whole interval [0, T ]. ✷

Flow
Having established sufficient conditions for the existence and uniqueness of solutions to the YDE associated to (x 0 , A), it is natural to study their dependence on the data of the problem. This section is devoted to the study of the flow, seen as the ensemble of all possible solutions, and its Frechét differentiability w.r.t. both (x 0 , A).
In order to avoid technicalities we will only consider the case of A ∈ C α t C 1+β V with global bounds, but everything extends easily by localisation arguments to A ∈ C α t C β,λ V ∩ C α t C 1+β V,loc ; similar results can also be established for the type of equations considered respectively in Sections 3.4 and 3.5.

Flow of diffeomorphisms
We start by giving a proper definition of a flow for the YDE associated to A; recall here that ∆ n denotes the n-simplex on [0, T ].

Definition 4.1 Given
we say that Φ : ∆ 2 × V → V is a flow of homeomorphisms for the YDE associated to A if the following hold: iv. Φ satisfies the group property, namely Φ(u, t, Φ(s, u, x)) = Φ(s, t, x) for all (s, u, t) ∈ ∆ 3 and x ∈ V ; v. for any (s, t) ∈ ∆ 2 , the map Φ(s, t, ·) is an homeomorphism of V , i.e. it is continuous with continuous inverse.
From now on, whenever talking about a flow Φ, we will use the notation Φ s→t (x) = Φ(s, t, x); we will denote by Φ s←t (·) the inverse of Φ s→t (·) as a map from V to itself. Definition 4.2 Given A as above, γ ∈ (0, 1), we say that it admits a locally γ-Hölder continuous flow Φ, Φ is C γ loc for short, if for any (s, t) ∈ ∆ 2 it holds Φ s→t , Φ s←t ∈ C γ loc (V ; V ); we say that Φ is a flow of diffeomorphisms if Φ s→t , Φ s←t ∈ C 1 loc (V ; V ) for any (s, t) ∈ ∆ 2 . Similar definitions hold for a locally Lipschitz flow, or a C n+γ loc -flow with γ ∈ [0, 1) and n ∈ N. If V = R d , we say that Φ is a Lagrangian flow if there exists a constant C such that where λ d denotes the Lebesgue measure on R d and B(R d ) the collection of Borel sets.

It follows from Remark 3.15 that, if
V with α(1 + β) > 1, then the solution map (x 0 , t) → x t is Lipschitz in space, uniformly in time. However we cannot yet talk about a flow, as we haven't shown the invertibility of the solution map, nor the flow property; this is accomplished by the following lemma.
x be a solution of the YDE associated to (x 0 , A). Then settingÃ(t, z) := A(T − t, z) andx t := x T −t ,x is a solution to the time-reversed YDEx Similarly, settingx t = x t−s ,Ã(t, x) = A(t − s, x) for t ∈ [s, T ], thenx is a solution to the time-shifted YDEx The proof is elementary but a bit tedious, so we omit it; we refer the interested reader to Lemma 2, Section 6.1 from [33] or Lemmas 11 and 12, Section 4.3.1 from [20].
As a consequence, we immediately deduce conditions for the existence of a Lipschitz flow.
Proof. The proof is a straightforward application of Remark 3.15 and Lemma 4.3. In both cases of time reversal and translation we have Ã α,1+β A α,1+β so that uniqueness holds also for the reversed/translated YDE, with the same continuity estimates; this provides respectively invertibility of the solution map and flow property. ✷ Actually, under the same hypothesis it is possible to prove that the YDE admits a flow of diffeomorphisms, which satisfies a variational equation.
is the unique solution to the variational equation where • denotes the composition of linear operators.
We postpone the proof of this result to Section 4.2, as the variation equation will follow from a more general result on the differentiability of the Itô map. Following [30], we give an alternative proof in the case of finite dimensional V , in which more precise information on Φ is known. ii. The Jacobian  s→t (x) := det(D x Φ s→t (x)) satisfies the identity and there exists a constant C = C(α, β, T, A α,1+β ) > 0 such that In particular, Φ is a Lagrangian flow of diffeomorphisms.
Proof. For simplicity we will prove all the statements for s = 0, the general case being similar. By Corollary 4.4, the existence of a locally Lipschitz flow Φ is known; to show differentiability, it is enough to establish existence and continuity of the Gateaux derivatives. Fix x, v ∈ R d and consider for any ε > 0 the map η ε t := ε −1 (Φ 0→· (x + ε n v) − Φ 0→· (x)); by estimate (4.1), the family {η ε } ε>0 is bounded in C α t R d . Thus by Ascoli-Arzelà we can extract a subsequence ε n → 0 such that η ε → η in C α−δ t for some η ∈ C α t and any δ > 0. Choose δ > 0 small enough such that (α − δ)(1 + β) > 1; using the fact that the map F (y) = · 0 A(ds, y s ) is differentiable from C α−δ t to itself by Proposition 2.11, with DF given by (2.8), by chain rule we deduce that namely, η satisfies the YDE whose meaning was defined in Remark 2.8. Equation (4.5) is an affine YDE, which admits a unique solution by Corollary 3.13; moreover it's easy to check that the unique solution must have the form whose global existence and uniqueness follows from Corollary 3.13 and Theorem 3.9. As the reasoning holds for any subsequence ε n we can extract and any v ∈ R d , we conclude that Φ 0→t (·) is Gateaux differentiable with DΦ 0→t (x) = J x 0→t which satisfies (4.3). A similar argument shows that J x 0→t depends continuously on x, from which Frechét differentiability follows. Part ii. can be established for instance by means of an approximation procedure; indeed by x and by Theorem 3.14, the solutions y n · = Φ n 0→· (x) associated to (x, A n ) converge to Φ 0→· (x) associated to (x, A). Moreover for A n the YDE is meaningful as the more classical ODE associated to ∂ t A n , so we can apply to it all the classical results from ODE theory; the Jacobian associated to A n is given by Passing to the limit as n → ∞, by the continuity of nonlinear Young integrals, we obtain (4.4). Moreover by equation (4.1) we have the estimate which gives Lagrangianity. ✷ It's possible to show that the flow inherits regularity from the drift, namely that to a spatially more regular A corresponds a more regular Φ.
We omit the proof, which follows similar lines to those of Theorems 4.5 and 4.6 and is mostly technical; we refer the interested reader to [20], [28] and the discussion at the end of Section 3 from [33].
, then it has a locally C n -regular flow, see the discussion in Section 4.3 from [20]. Similar reasonings allow to establish existence of a flow also for the equations treated in Section 3.5.

Differentiability of the Itô map
Denote by Φ A s→· (x) the solution to the YDE associated to (x, A); the aim of this section is to study the dependence of the flow Φ A as a function of A ∈ C α t C 1+β V , namely to identify D A Φ A s→· (x). For simplicity we will restrict to the case s = 0; we will actually fix A ∈ C α t C 1+β V , consider Φ A+εB with B varying and set X x t := Φ A 0→t (x).
and is given explicitly by where J x 0→· is the unique solution to (4.2) and (J x 0→s ) −1 denotes its inverse as an element of L(V ). The proof requires the following preliminary lemma.
moreover M t is invertible for any t ∈ [0, T ] and N · := (M · ) −1 ∈ C α t L(V ) is the unique solution to Finally, for any y 0 ∈ V and any ψ ∈ C α t V , the unique solution to the affine YDE is given by ,loc and so existence and uniqueness of a global solution to (4.8) follows from Corollary 3.13 and Theorem 3.9; similarly for (4.9) withÃ(t, N ) = N • L t . Let M · , N · ∈ C α t L(V ) be solution respectively to (4.8), (4.9), we claim that they are inverse of each other. Indeed by the product rule for Young integrals it holds t V be the unique solution to (4.10), whose global existence and uniqueness follows as above, and set z t = N t y t ; then again by Young product rule it holds dz t = N t dψ t and thus and that it is a solution to (4.6). Once this is shown, we can apply Lemma 4.10 for the choice L t = t 0 D x A(ds, X x s ), y 0 = 0 and ψ t = t 0 B(ds, X x s ) to deduce that the limit is given by formula (4.7), which is meaningful since J x 0→· is defined as the solution to (4.8) for such choice of L and is therefore invertible. The explicit formula (4.7) for the Gateaux derivatives readily implies existence and continuity of the Gateux differential D A Φ A 0→· (x) and thus also Frechét differentiability. In order to prove the claim, let Y x ∈ C α t V be the solution to (4.6), which exists and is unique by Lemma 4.10; then we need to show that Set X ε,x · := Φ A+εB 0→· (x); recall that by the Comparison Principle (Theorem 3.14), we have where ψ ε is given by In order to conclude, it is enough to show that ψ ε α → 0 as ε → 0, since then we can apply the usual a priori estimates from Theorem 3.9 to e ε , which solves an affine YDE starting at 0. We already know that X ε,x → X x as ε → 0, which combined with the continuity of nonlinear Young integrals implies that ψ ε,2 which by virtue of (4.12) satisfies which implies that Γ ε α → 0 as ε → 0. On the other hand we have We can therefore apply Lemma A.2 from the Appendix to conclude. ✷

Remark 4.11 Although
as given by formula (4.7) is well defined and continuous for any and by linearity it's easy to check that for any h ∈ V , Y h t := J x 0→t (h) is the unique solution to Therefore in order to conclude it suffices to show that the directional derivatives exist in C α t V and are solutions to (4.13), as this implies that D x Φ A 0→· (x) = J x 0→· . Now fix x, h ∈ V and let y ε = Φ A 0→· (x + εh), then z ε := y ε − εh solves . It's easy to see that, if the first limit below exists, then By the Frechét differentiability of A → Φ A 0→· (x) and the chain rule, it holds which is characterized as the unique solution Z h to i. Consider the simple case of an additive perturbation, i.e. for fixed (x 0 , A) we want to understand how the solution x of

This implies by linearity that
ii. Consider the classical Young case, namely V = R d , with for regular vector fields σ i : R d → R d and ω ∈ C α t R m , α > 1/2; assume σ i are fixed and we are interested in the dependence on the drivers ω, namely the map Φ ω (4.14) The above formula uniquely extends by continuity to the case ψ ∈ W 1,1 t , in which case we can write it in compact form as 0→t (x)(ψ) = T 0 K(t, r)ψ r dr, K(t, r) = 1 r t J x 0→t (J x 0→r ) −1 σ(X x r ). Formulas (4.14) and (4.15) are well known by Malliavin calculus, mostly in the case ω is sampled as an fBm of parameter H > 1/2, see Section 11.3 from [18]; formula (4.7) can be regarded as a generalisation of them.

Conditional uniqueness
This section provides several criteria for uniqueness of the YDE, under additional assumptions on the properties of the associated solutions. Typically such properties can't be established directly, at least not under mild regularity assumptions on A; yet the criteria are rather useful in application to SDEs, where the analytic theory can be combined with more probabilistic techniques.

A Van Kampen type result for YDEs
The following result is inspired by the analogue results for ODEs in the style of van Kampen and Shaposhnikov, see [42], [41].
Proof. Let x 0 ∈ V and x be a given solution to the YDE starting at x 0 . By the a priori estimate (3.10), we can always find R = R(x 0 ) big enough such that therefore in the following computations, up to a localisation argument, we can assume without loss of generality that A ∈ C α t C β V and that Φ is globally γ-Hölder. It suffices to show that f t := Φ(t, T, x t ) − Φ(0, T, x 0 ) satisfies f s,t V |t − s| 1+ε for some ε > 0; if that's the case, then f ≡ 0, Φ(t, T, x t ) = Φ(0, T, x 0 ) for all t ∈ [0, T ] and so inverting the flow x t = Φ(0, t, x 0 ), which implies that Φ(0, ·, x 0 ) is the unique solution starting from x 0 .
By the flow property Since both x and Φ(s, ·, x s ) are solutions to the YDE starting from x s , it holds and so overall we obtain f s,t V |t − s| γα(1+β) , which implies the conclusion. ✷

Remark 5.2
The assumption can be weakened in several ways. For instance, the existence of a γ-Hölder regular semiflow is enough to establish that Φ(t, T, x t ) = Φ(0, T, x 0 ), even when Φ is not invertible. Uniqueness only requires Φ(t, T, ·) to be invertible for t ∈ D, D dense subset of [0, T ]; indeed this implies x t = Φ(0, t, x 0 ) on D and then by continuity the equality can be extended to the whole [0, T ]. Similarly, it is enough to require sup t∈D Φ(t, T, ·) γ,R < ∞ for all R 0 for D dense subset of [0, T ] as before.

Averaged translations and Conditional Comparison Principle
The concept of averaged translation has been introduced in [9], Definition 2.13. We provide here a different construction based on the sewing lemma (although with the same underlying idea).
Proof. Observe that τ y A corresponds to the sewing of Γ : ∆ 2 → C n+β V given by Γ s,t := A s,t ( · + y s ) .
Since α + γη > 1, by the sewing lemma we deduce that J (Γ) = τ y A ∈ C α t C n+β−η V , together with estimate (5.1). ✷ Young integrals themselves can indeed be regarded as averaged translations evaluated at z = 0. Moreoveor iterating translations is a consistent procedure, as the following lemma shows.
Proof. The statement follows immediately from the observation that for any s t it holds so that the two integrals must coincide. ✷ The main reason for introducing averaged translations is the following key result.
Theorem 5.6 (Conditional Comparison Principle) Let A 1 , A 2 ∈ C α t C β V with α(1 + β) > 1 for some α, β ∈ (0, 1) and let x i ∈ C α t V be given solutions respectively to the YDE associated to (x i 0 , A i ). Suppose in addition that x 1 is such that τ x 1 A 1 ∈ C α t Lip V . Then there exists C = C(α, β, T ) s.t.
In particular, uniqueness holds in the class C α t V to the YDE associated to (x 1 0 , A 1 ).
Proof. The final uniqueness claim immediately follows from inequality (5.2), since in that case we can consider A 1 = A 2 , x 1 0 = x 2 0 . Now let x i be two solutions as above, then their difference where in the third line we applied Lemma 5.5 and we take By the hypothesis, B ∈ C γ t Lip V with B(t, 0) = 0 for all t ∈ [0, T ], while ψ ∈ C α t V . Therefore from Theorem 3.9 applied to v we deduce the existence of a constant κ 1 = κ 1 (α, T ) such that . On the other hand, estimates (2.4) and (3.6) imply that α,β (1 + A 2 2 α,β ) for some κ 2 = κ 2 (α, β, T ). Combining the above estimates the conclusion follows.
✷ Remarkably, the hypothesis τ x A ∈ C α t Lip V allows not only to show that this is the unique solution starting at x 0 , but also that any other solution will not get too close to it. In the next lemma, in order to differentiate · V , we assume for simplicity V to be a Hilbert space, but a uniformly smooth Banach space would suffice.
x, y ∈ C α t V solutions respectively to the YDEs associated to (x 0 , A), (y 0 , A) and assume that τ Proof. The first inequality is an immediate consequence of Theorem 5.6, so we only need to prove the second one. By the same computation as in Theorem 5.6, the map v = y − x satisfies V is in C α t R and by Young chain rule We are going to show that z satisfies a bound from above which does not depend on the interval [0, T ε ]; as a consequence, for all ε > 0 small enough it must hold T ε = T , which yields the conclusion. For dividing by |u − r| α and taking the supremum we obtain Since T ε T , it takes at most N ∼ T /∆ intervals of size ∆ to cover [0, T ε ], and ∆ ∼ τ x A 1/α α,1 , therefore overall we have found a constant C = C(α, T ) such that As the estimate does not depend on ε, the conclusion follows. ✷

Conditional rate of convergence for the Euler scheme
Remarkably, under the assumption of regularity of τ x A, convergence of the Euler scheme to the unique solution can be established, with the same rate 2α − 1 as in the more regular case of A ∈ C α t C 1+β V . The following results are direct analogues of Corollaries 3.16 and 3.19.
t Lip V with α > 1/2, x 0 ∈ V and suppose there exists a solution x associated to (x 0 , A) such that τ x A ∈ C α t Lip V (which is therefore the unique solution); denote by x n the element of C α t V constructed by the n-step Euler approximation from Theorem 3.2. Then there exists C = C(α, T ) such that x Proof. As in the proof of Corollary 3.16, recall that x n satisfies the YDE Applying Theorem 3.9 we obtain that, for suitable κ = κ(α, T ) it holds x − x n α κ exp(κ τ x A 1/α α,1 ) ψ n α which combined with the above inequality for ψ n α gives the conclusion. ✷ Corollary 5.9 Let A be such that A ∈ C α t C β V and ∂ t A ∈ C 0 ([0, T ] × V ; V ) with α(1 + β) > 1, x 0 ∈ V and suppose there exists a solution x associated to (x 0 , A) such that τ x A ∈ C α t Lip V (which is therefore the unique solution); denote by x n the element of C α t V constructed by the n-step Euler approximation from Theorem 3.2. Then there exists C = C(α, T ) such that Proof. Recall that x n satisfies the YDE The rest of the proof is mostly identical to that of Corollary 5.8. ✷

Young transport equations
This section is devoted to the study of Young transport equations of the form which we will refer to as the YTE associated to (A, c).
We restrict here to the case V = R d ; as in Section 4 for simplicity we will assume on A global bounds like A ∈ C α t C 1+β x , but slightly more tedious localisation arguments allow to relax them to growth conditions and local regularity requirements.
Classical results on weak solutions to (6.1) in the case A dt = b t dt, c dt =c t dt can be found in [16], [1]. Our approach here mostly follows the one given in [20], although slightly less based on the method of characteristics and more on a duality approach; other works concerning transport equations in the Young (or "level-1") regime are given by [8], [30] and Chapter 9 from [36]. Let us also mention on a different note the works [3] [15], [5] which treat with different techniques and in various regularity regimes rough trasnport equations of "level-2" or higher (namely corresponding to a time regularity α ≤ 1/2).
Before explaining the meaning of (6.1), we need some preparations. Given any compact K ⊂ R d , we denote by is the set of all compactly supported β-Hölder continuous functions. C β c is a direct limit of Banach spaces and thus it is locally convex; we denote its topological dual by (C β c ) * . Given γ, β ∈ (0, 1), we say that We will use the bracket ·, · to denote both the classical L 2 -pairing and the one between C β c and its dual. Finally, M loc denotes the space of Radon measures on R d , M K the space of finite signed measure supported on K; observe that the above notation is consistent with M loc = (C 0 c ) * . We are now ready to give a notion of solution to the YTE.
Observe that under the above assumptions, for any ϕ ∈ C ∞ c , A · ∇ϕ and (div A − c)ϕ belong to C α t C β c ; since u ∈ C αβ t (C β c ) * with α(1 + β) > 1, the integral appearing in (6.2) is meaningful as a functional Young integral. Remark 6.2 For practical purposes, it is useful to consider the following equivalent characterization of solutions: under the above regularity assumptions on u, A, c, u is a solution if and only if for any compact K ⊂ R d and ϕ ∈ C ∞ K it holds Clearly in the l.h.s. above one can replace u s with u t to get a similar estimate.

Remark 6.3
The presence of c in (6.1) allows to also consider nonlinear Young continuity equations (YCE for short) of the form v dt + ∇ · (A dt v t ) + c dt v t = 0; weak solutions to the above equation must be understood as weak solutions to the YTE associated to (A,c) withc = c + ∇ · A.
Let us quickly recall some results from Section 4: given A ∈ C α t C 1+β x , the YDE admits a flow of diffeomorphisms Φ s→t (x) and there exists C = C(α, β, T, A α,1+β ) such that for all x, y ∈ R d , (s, t) ∈ ∆ 2 , together with similar estimates for Φ ·←t . Moreover and similarly Then for any µ 0 ∈ M loc , a solution to the YTE is given by the formula If µ 0 (dx) = u 0 (x)dx for u 0 ∈ L p loc , then u t corresponds to the measurable function If in addition c ∈ C α t C 1+β x , then for any T α , it is always possible to find R 0 big enough such that supp ϕ(Φ 0→t (·)) ⊂ supp ϕ + B R for all t ∈ [0, T ]; by estimates (2.4) and (3.9), it holds It is therefore clear that u t defined as in (6.4) belongs to L ∞ t (C 0 c ) * . Similarly, combining the estimates it is easy to check that u ∈ C αβ t (C β c ) * . Let us show that it is a solution to the YTE in the sense of Definition 6.1. Given ϕ ∈ C ∞ K and x ∈ R d , define By Itô formula, z satisfies By the properties of Young integrals and the above estimates, which are uniform in x, it holds in the sense that the two quantities differ by O(|t − s| α(1+β) ), uniformly in x ∈ R d . Therefore where the two quantities differ by O( ϕ C 1+β K |t − s| α(1+β) ). By Remark 6.2 we deduce that u is indeed a solution.
The statements for u 0 ∈ L p loc are an easy application of formula (4.4); it remains to prove the claims for u 0 ∈ C 1 loc , under the additional assumption c ∈ C α t C 1+β x . First of all observe that, for any (s, t) ∈ ∆ 2 , it holds as a consequence, the map (t, x) → u 0 (Φ 0←t (x)) belongs to C α t C 0 loc . Consider now the map

It holds
by Corollary 2.12 and estimate (6.6) we have As a consequence, g ∈ C α t C 0 loc and so does u. The verification that u ∈ C 0 t C 1 loc is similar and thus omitted. ✷ Remark 6.5 Analogous computations show that a solution to the YTE with terminal condition u(T, ·) = µ T (·) is given by This solution satisfies the same space-time regularity as in Proposition 6.4. Moreover by the properties of the flow, if µ 0 (resp. µ T ) has compact support, then it's possible to find K ⊂ R d compact such that supp u t ⊂ K uniformly in t ∈ [0, T ]. In particular if c ∈ C α t C 1+β x and u 0 ∈ C 1 c (resp. u T ∈ C 1 c ), then the associated solution belongs to C α t C 0 c ∩ C 0 t C 1 c . The following result is at the heart of the duality approach and our main tool to establish uniqueness.
c be a solution of the YTE u dt + A dt · ∇u t + c dt u t = 0 (6.7) Then it holds v t , u t = v s , u s for all (s, t) ∈ ∆ 2 . A similar statement holds for u ∈ C α t C 0 loc ∩ C 0 t C 1 loc and v as above and compactly supported uniformly in time.
The proof requires some preparations. Let {ρ ε } ε>0 be a family of standard spatial mollifiers (say ρ 1 supported on B 1 for simplicity) and define the R ε , for sufficiently regular g and h, as the following bilinear operator: the following commutator lemma is a slight variation on Lemma 16, Section 5.2 from [20], which in turn is inspired by the general technique first introduced in [16].
Lemma 6.7 The operator R ε : C 1+β loc × C 1 loc → C β loc defined by (6.9) satisfies the following. i. There exists a constant C independent of ε and R such that ii. For any fixed g ∈ C 1+β loc , h ∈ C β loc it holds R ε (g, h) → 0 in C β ′ loc as ε → 0, for any β ′ < β. Proof. It holds Thus claim i. follows from (h div g) ε β,R h 1,R+1 g 1+β,R+1 and where the estimate is uniform in x, y ∈ B R and in ε > 0. Claim ii. follows from the above uniform estimate, the fact that R ε (g, h) → 0 C 0 loc by Lemma 16 from [20] and an interpolation argument. ✷ Proof.[of Proposition 6.6] We only treat the case u ∈ C α t C 0 the other one being similar. Applying a mollifier ρ ε on both sides of (6.7), it holds u ε dt + A dt · ∇u ε t + (c dt u t ) ε + R ε (A dt , u t ) = 0 where we used the definition of R ε ; equivalently by Remark 6.2, the above expression can be interpreted as u ε s,t + A s,t · ∇u ε s + (c s,t u s ) ε + R ε (A s,t , u s ) C 0 ε |t − s| α(1+β) uniformly in (s, t) ∈ ∆ 2 to (6.8) with terminal condition v τ = µ, up to taking a suitable compact set K. By Proposition 6.6 it follows that u τ , µ = u τ , v τ = u 0 , v 0 = 0; as the reasoning holds for any compactly supported µ ∈ M , u τ ≡ 0 and thus u ≡ 0.

Parabolic nonlinear Young PDEs
We present in this section a generalization to the nonlinear Young setting of some of the results contained in [25]. Specifically, we are interested in studying a parabolic nonlinear evolutionary problem of the form where −A is the generator of an analytical semigroup.
In order not to create confusion, in this section the nonlinear Young term will be always denoted by B. As we will use a one-parameter family of spaces {V α } α∈R , the regularity of B will be denoted by B ∈ C γ t C β W,U , with W and U being taken from that family; whenever it doesn't create confusion, we will still denote the associated norm by B γ,β .
Let us first recall the functional setting from [25], Section 2.1. It is based on the theory of analytical semigroups and infinitesimal generators, see [39] for a general reference, but the reader not acquainted with the topic may consider for simplicity Let (V, · V ) be a separable Banach space, (A, Dom(A)) be an unbounded linear operator on V , rg(A) be its range; suppose its resolvent set is contained in Σ = {z ∈ C : | arg(z)| > π/2 − δ} ∪ U for some δ > 0 and some neighbourhood U of 0 and that there exist positive constants C, η such that its resolvent R α satisfies Under these assumptions, −A is the infinitesimal generator of an analytical semigroup (S(t)) t 0 and there exist positive constants M, λ such that Moreover, −A is one-to-one from Dom(A) to V and the fractional powers (A α , Dom(A α )) of A can be defined for any α ∈ R; if α < 0, then Dom(A α ) = V and A α is a bounded operator, while for α 0 (A α , Dom(A α )) is a closed operator with Dom(A α ) = rg(A −α ) and A α = (A −α ) −1 . For α 0, let V α be the space Dom(A α ) with norm x Vα = A α x V ; for α = 0 it holds A 0 = Id and V 0 = V . For α < 0, let V α be the completion of V w.r.t. the norm x Vα = A α x V , which is thus a bigger space than V . The one-parameter family of spaces {V α } α∈R is such that V δ embeds continuously in V α whenever δ α and A α A δ = A α+δ on the common domain of definition; moreover A −δ maps V α onto V α+δ for all α ∈ R and δ 0.
The operator S(t) can be extended to V α for all α < 0 and t > 0 and maps V α to V δ for all α ∈ R, δ 0, t > 0; finally, it satisfies the following properties: Remark 7.1 It follows from the statements above and the semigroup property of S(t) that for any α ∈ R, δ > 0, x ∈ V α and any s t it holds which implies that S(t)− S(s) L(V α+δ ;Vα) |t− s| δ , equivalently S(·) ∈ C δ t L(V α+δ ; V α ). It also follows that for any given x 0 ∈ V α+δ , the map t → S(t)x 0 belongs to C δ t V α with S(·)x 0 δ,Vα α,δ x 0 V α+δ . (7.4) The following result shows that the mild solution formula for the linear equation which is formally given by can be extended by continuity to suitable non differentiable functions y ∈ C([0, T ]; V ).
Theorem 7.2 Let α ∈ R and consider the map Ξ defined for any y ∈ C 1 t V −α by Then for any γ > α, Ξ extends uniquely to a map Ξ ∈ L(C γ t V −α ; C κ t V δ ) for all δ ∈ (0, γ − α) and all κ ∈ (0, (γ − α − δ) ∧ 1). Moreover there exists a constant C = C(α, κ, δ, γ) such that We omit the proof, for which we refer to Theorem 1 from [25]. Let us only provide an heuristic derivation of the relation between the parameters α, κ, δ, γ based on a regularity counting argument. It follows from Remark 7.1 that S(t − s) L(V−α;V δ ) |t − s| −δ−α ; if it's possible to define the map Ξ(y) taking values in V δ , then we would expect its time regularity to be analogue to that of where now f, g are real valued functions, f ∈ C γ t ; indeed, considering a fixed y 0 ∈ V −α , the result should also apply to y t := f t y 0 . The integral in (7.6) is a type of fractional integral of order 1 − δ − α and by hypothesis df ∈ C γ−1 t , therefore g should have regularity γ − δ − α, which is exactly the threshold parameter for κ (this is because Hölder spaces do not behave well under fractional integration and one must always give up an ε of regularity by embedding them in nicer spaces). Definition 7.3 Given A as above and B ∈ C γ t C β V δ ,Vρ , ρ δ, we say that x ∈ C κ t V δ is a mild solution to equation (7.1) with initial data x 0 ∈ V δ if γ + βκ > 1, so that · 0 B(ds, x s ) is well defined as a nonlinear Young integral, and if x satisfies where Ξ is the map defined by Theorem 7.2 and the equality holds in V α for suitable α.
We are now ready to prove the main result of this section.
Proof. The basic idea is to apply a Banach fixed point argument to the map defined on a suitable domain. By Remark 7.1, if x 0 ∈ V δ+κ , then S(·)x 0 ∈ C κ t V δ ; moreover B ∈ C γ t C 1 V δ ,Vρ , so under the condition γ + κ > 1 the nonlinear Young integral in (7.9) is well defined for x ∈ C κ t V δ , y t = t 0 B(ds, x s ) ∈ C γ t V ρ and then Ξ(y) ∈ C κ t V δ under the condition κ < γ + ρ − δ. So under our assumptions I maps C κ t V δ into itself; our first aim is to find a closed bounded subset which is invariant under I.
We now want to findτ ∈ [0, τ ] such that I is a contraction onẼ,Ẽ being defined as E in terms of τ , M . Given x 1 , x 2 ∈Ẽ, it holds Choosingτ small enough such that κ 5 B γ,1+β (1 + M )τ κ < 1, we deduce that there exists a unique solution to (7.1) defined on [0,τ ]. Since we have the uniform estimate (7.10), we can iterate the contraction argument to construct a unique solution on [0, τ ]; but since the choice of τ does not depend on x 0 and x τ ∈ V δ+κ , we can iterate further to cover the whole interval [0, T ] with subintervals of size τ .
To check the Lipschitz dependence on (x 0 , B), one can reason using the Comparison Principle as usual, but let us give an alternative proof; we only check Lipschitz dependence on B, as the proof for x 0 is similar.
Given B i , i = 1, 2 as above, denote by I B i the map associated to B i defined as in (7.9); we can chooseτ and M such that they are both strict contractions of constant κ 6 < 1 on E defined as before. Observe that for any z ∈ E it holds d E (I B 1 (z), I B 2 (z)) = Ξ γ,β . Denote by x i the unique solutions on E associated to B i , then by the above computation we get γ,β which implies that γ,β which shows Lipschitz dependence on B i on the interval [0,τ ]. As before, a combination of a priori estimates and iterative arguments allows to extend the estimate to a global one. ✷ By the usual localization and blow-up alternative arguments, we obtain the following result. Corollary 7.6 Assume A as above, B ∈ C γ t C 1+β V δ ,Vρ,loc with ρ > δ −1 and suppose there exists κ ∈ (0, 1) satisfying (7.8). Then for any x 0 ∈ V δ+κ there exists a unique maximal solution x starting from x 0 , defined on an interval [0, T * ) ⊂ [0, T ], such that either T * = T or lim t↑T * x t V δ+κ = +∞.
Remark 7.7 For simplicity we have only treated here uniqueness results, but if the embedding V δ ֒→ V α for δ > α is compact, as is often the case, one can use compactness arguments to deduce existence of solutions under weaker regularity conditions on B, in analogy with Theorem 3.2. Once can also consider equations of the form dx t = −Ax t dt + F (x t )dt + B(dt, x t ), in which case uniqueness can be achieved under the same conditions on B as above and a Lipschitz condition on F , see also Remark 1 from [25].
Lemma A.2 Let {Γ n } n ⊂ C α,β 2 V be a sequence such that sup n δΓ n β R and lim n Γ n α → 0. Then J Γ n → 0 in C α t V and for all n big enough it holds The conclusion follows choosing θ = (1 − α)/(β − α). ✷ The following basic result was used in Section 5.2.
Lemma A.3 Let f ∈ C n+β V , z 1 , z 2 ∈ V . Then for any η ∈ (0, 1) with η < n + β it holds Proof. It is enough to prove the claim in the cases n = 0 and n = 1, the others being similar. Assume first n = 0, then we have the elementary estimates which interpolated together give the conclusion. Now consider n = 1 and η ∈ (β, 1 + β), then inverting the roles of z 1 and x (respectively z 2 and y) we also obtain Interpolating the two inequalities again yields the conclusion. ✷

A.2 Alternative constructions of Young integrals
We collect in this appendix several other constructions of the nonlinear Young integral, although mostly equivalent to the one from Section 2.
In Section 2 we constructed the nonlinear Young integral following the modern approach based on an application of the sewing lemma, but this is not how it was first introduced in [9]. The approach therein was instead based on combining property 4. of Theorem 2.7 with estimate (2.3); namely, the classical integral · 0 ∂ t A(s, x s )ds can be controlled by A α,β and x γ , and thus its definition can be extended by an approximation procedure, as the following lemma shows.
Lemma A.4 Any A ∈ C α t C β V,W can be approximated in C α− t C β− V,W by a sequence A n such that ∂ t A n exists and is continuous. it's immediate to check that sup (t,x) A − A ε → 0 as ε → 0 by the uniform continuity of A (which is granted from the fact that A ∈ C α t C β V,W ). We also have the uniform bound A ε α,β A α,β , since as well as similar uniform bounds for A s,t β , etc. Interpolating these estimates together, convergence of A ε to A in C α−δ t C β−δ V,W as ε → 0, for any δ > 0, immediately follows. ✷ Observe that in the above giving up a δ of regularity is not an issue in terms of defining · 0 A(ds, x s ), since we can always find δ > 0 small enough such that it still holds α − δ + (β − δ)γ > 1.
Another more functional way to define nonlinear Young integrals is the following one: for any β > 0, consider the map J : V → L(C β V,W ; W ) given by x → δ x ; such a map is trivially β-Hölder regular, since where we denoted by ·, · the pairing between L(C β V,W ; W ) and C β V,W . Therefore for any x ∈ C γ t V , the map t → Jx t = δ xt is now an element of C γβ t L(C β V,W ; W ). If on the other hand A ∈ C α t C β V,W and α + γβ > 1, then we can define the (linear) Young integral   which immediately shows that it coincides with the definition from Section 2.
While this construction might seem unnecessarily abstract, it shows that nonlinear Young integrals can be regarded as linear ones, after the nonlinear transformation x → δ x has been applied. It also allows to give intuitive derivations of several integral relations: for instance by Young product rule it must hold δ xt , A t − δ x0 , A 0 = t 0 δ xs , A ds + t 0 dδ xs , A s Proof. It follows immediately from Theorem A.6 and the fact that lower semicontinuous maps are measurable. ✷ Remark A.10 For simplicity we have only treated the case V = R d , but it's clear that Theorem A.6 admits several extensions; for instance it can be readapted to the case of equations of the form (3.22) with A ∈ C α t C β,λ x and F continuous of linear growth. In alternative, one can consider a general Banach space V and A ∈ C α t C β,λ V,W with W compactly embedded in V ; this is enough to grant global existence by Corollary 3.5 and the usual a priori estimates.