Solving mean field rough differential equations

where W is a random rough path and LpXtq stands for the law of Xt, with mean field interaction in both the drift and diffusivity. We show that, in addition to the enhanced path of W , the underlying rough path-like setting should also comprise an infinite dimensional component obtained by regarding the collection of realizations of W as a deterministic trajectory with values in some L space. This advocates for a suitable notion of controlled path à la Gubinelli inspired from Lions’ approach to differential calculus on Wasserstein space, the systematic use of the latter playing a fundamental role in our study. Whilst elucidating the rough set-up is a key step in the analysis, solving the mean field rough equation requires another effort: the equation cannot be dealt with as a mere rough differential equation driven by a possibly infinite dimensional rough path. Because of the mean field component, the proof of existence and uniqueness indeed asks for a specific and quite elaborated localization-in-time argument.


Introduction
The first works on mean field stochastic dynamics and interacting diffusions/Markov processes have their roots in Kac's simplified approach to kinetic theory [28] and McKean's work [34] on nonlinear parabolic equations. They provide the description of evolutions pµ t q tě0 in the space of probability measures under the form of a pathspace random dynamics dX t pωq " V`X t pωq, µ t˘d t`F`X t pωq, LpX t q˘dW t pωq, (1.1) (where LpAq stands for the law of a random variable A) and relate it to the empirical behaviour of large systems of interacting dynamics. The main emphasis of subsequent works has been on proving propagation of chaos and other limit theorems, and giving stochastic representations of solutions to nonlinear parabolic equations under more and more general settings; see [36,37,25,17,18,35,27,7,8] for a tiny sample. Classical stochastic calculus makes sense of equation (1.1), in a probabilistic setting pΩ, F, Pq, only when the process W is a semi-martingale under P, for some filtration, and the integrand is predictable. However, this setting happens to be too restrictive in a number of situations, especially when the diffusivity is random. This prompted several authors to address equation (1.1) by means of rough paths theory. Indeed, one may understand rough paths theory as a natural framework for providing probabilistic models of interacting populations, beyond the realm of Itô calculus. Cass and Lyons [13] did the first study of mean field random rough differential equations and proved the well-posed character of equation (1.1), and propagation of chaos for an associated system of interacting particles, under the assumption that there is no mean field interaction in the diffusivity, i.e. Fpx, µq " Fpxq, and the drift depends linearly on the mean field interaction, i.e. V px, µq " ş V px, yq µpdyq, for some function V p¨,¨q on R dˆRd .
The method of proof of Cass and Lyons depends crucially on both assumptions. Bailleul extended partly these results in [3] by proving well-posedness of the mean field rough differential equation (1.1) in the case where the drift depends nonlinearly on the interaction term and the diffusivity is still independent of the interaction, and by proving an existence result when the diffusivity depends on the interaction. The naive approach to showing well-posedness of equation (1.1) in its general form consists in treating the measure argument as a time argument. However, this is of a rather limited scope since, in this generality, one cannot expect the time dependence in F to be better than 1 p -Hölder if the rough path W is itself 1 p -Hölder. Clearly, such a time regularity is not sufficient to make sense of the rough integral ş Fp¨¨¨q dW in the case p ě 2. This serious issue explains why, so far in the literature, the coefficient F has been assumed to be a function of the sole variable x. Including the time component as one of the components of W brings back the study of equation (1.1) to the study of equation dX t pωq " F`X t pωq, LpX t q˘dW t pωq ; (1.2) this is the precise purpose of the present paper. Treating the drift as part of the diffusivity has the drawback that we shall impose on V some regularity conditions stronger than needed. Our method accommodates the general case but we leave the reader the pleasure of optimizing the details and concentrate on the new features of our approach, working on equation (1.2). The raw driver`W t pωq˘t ě0 will be assumed to take values in some R m and to be 1 p -Hölder continuous, for p P r2, 3q, and the one form F will be an L pR m , R d q-valued function on R dˆP 2 pR d q, where L pR m , R d q is the space of linear mappings from R m to R d and P 2 pR d q is the so-called Wasserstein space of probability measures µ with a finite second-order moment. Inspired by Lions' approach [31,9,10] to differential calculus on P 2 pR d q, one of the key point in our analysis is to lift the function F into a function p F defined on the space R dˆL2`Ω , F, P; R d˘, given by the formula Mean field rough equations for x P R d and Z P L 2 pΩ, F, P; R d q, and then to use accordingly Lions' calculus in order to take care of the probability-measure valued mean field dependence of the dynamics. So, we may rewrite equation (1.2) as dX t pωq " p F`X t pωq, X t p¨q˘dW t pωq. (1.4) We used the notation X t p¨q to distinguish the realization X t pωq of the random variable X t at point ω from the random variable itself, seen as an element of the space L 2`Ω , F, P; R d˘. So, X t p¨q is a random variable, and thus an infinite-dimensional object, whilst X t pωq is a finite-dimensional vector. We feel that this writing is sufficiently explicit to remove the hat over F. Our main well-posedness result is stated below, in a preliminary form only. The precise statement requires additional ingredients that we introduce later on in the text. In this first formulation • the quantity wp¨,¨q "`wps, tq˘0 ďsăt is a random control function that is used to quantify the regularity of the solution path on subintervals rs, ts of a given finite interval r0, T s, using some associated notion of p-variation for the same p as above, see (2.10) for a more mathematical formulation, • the quantity N pr0, T s, αq is some local accumulated variation of the 'rough lift' of W that counts the increments of w of size α over a bounded interval r0, T s for a given α ą 0, see (2.14) for the mathematical formulation; We refer to Section 2 for a complete description of the set-up. The regularity assumptions on the diffusivity F are spelled-out in Section 3.3 and in Section 4, see Regularity assumptions 1 and Regularity assumptions 2 therein. Theorem 1.1. Let F satisfy the regularity assumptions Regularity assumptions 1 and Regularity assumptions 2. Assume there exists a positive time horizon T such that the random variables wp0, T q and`N`p0, T q, α˘˘α ą0 have sub and super exponential tails, respectively, • P`wp0, T q ě t˘ď c 1 exp`´t ε1˘, • P`N`r0, T s, α˘ě t˘ď c 2 pαq exp`´t 1`ε2pαq˘, α ą 0, for some positive constants c 1 and ε 1 and possibly α-dependent positive constants c 2 pαq and ε 2 pαq. Then for any d-dimensional square-integrable random variable X 0 , the mean field rough differential equation dX t " F`X t , LpX t q˘dW t has a unique solution defined on the whole interval r0, T s.
Results of that form seem out of reach of the methods used in [13,3]. Theorem 1.1 applies in particular to mean field rough differential equations driven by some fractional Brownian motion with Hurst parameter greater than 1 3 , other Gaussian processes or some Markovian rough paths; see Section 2. Importantly, the solution is shown to depend continuously on the driving 'rough path', in a quantitative sense detailed in Theorem 5.4. As an example that fits our regularity assumptions, one can solve the above mean field rough differential equation with Fpx, µq " ş f px, yqµpdyq, for some fuction f of class C 3 b (meaning that f is bounded and has bounded derivatives of order 1, 2 and 3), or with Fpx, µq " g`x, ş R d yµpdyq˘, for some function g of class C 3 b . The Curie-Weiss model, where F is of the form Fpx, µq " ∇U pxq`şpx´yqµpdyq, falls outside the scope of what is written here, because of the linear growth rate in x, but is within reach of our method.
One of the difficulties in solving equation (1.2) comes from the fact that it happens not to be sufficient to consider each signal W ‚ pωq as the first level of a rough path; one EJP 25 (2020), paper 21. somehow needs to consider the whole family`W ‚ pωq˘ω PΩ as an infinite-dimensional rough path. This leads us to defining in Section 2 a rough setting where`W t pωq, W t p¨q˘0 ďtďT is, for each ω, the first level of a rough path over R mˆLq`Ω , F, P; R m˘; seemingly, the natural choice for q, as dictated by the aforementioned lifting procedure of the Wasserstein space, is q " 2; we shall actually need a larger value. Unlike the seminal works [13,3] that set the scene in Davie's approach of rough differential equations, such as reshaped by Friz-Victoir and Bailleul respectively, we use here Gubinelli's versatile approach of controlled paths to make sense of equation (1.2). Our mixed finite/infinite dimensional setting introduces an interesting twist in the notion of controlled path presented in Section 3.1. Defining the rough integral of a controlled path with respect to a rough driver is done classically in Section 3.2 using the sewing lemma. We prove stability of a certain class of controlled paths by nonlinear mappings in Section 3.3, which is precisely the place where Lions' differential calculus on P 2 pR d q is of crucial use. One then has all the ingredients needed to formulate in Section 4 equation (1.2) as a fixed point problem in some space of controlled paths. It must be stressed here that solving rough differential equations driven by random rough paths and solving mean field rough differential equations are two different tasks. In the first setting, the solutions are constructed up to a random time, say ζ, yielding a random path px t q 0ďtďζ defined up to ζ, but, for such solutions, we can only make sense of Lpx t^ζ q rather than Lpx t q, for t ě 0.
Of course, this is a serious drawback for solving mean field rough equations, unless we know a priori that ζ is infinite, as is in fact the case in Cass and Lyons' work. However, we cannot hope to obtain for free ζ " 8 in the general case that we investigate here because the diffusivity is also mean field dependent. We are nonetheless able to prove local well-posedness, and sufficient conditions on the law of the driver are given to get well-posedness on any fixed time interval. As expected from any solution theory for rough differential equations, the solution depends continuously on all the parameters in the equation, most notably its law depends continuously on the law of the driving rough path, as shown in Section 5. This latter point is used in the companion paper [4] to provide a proof of propagation of chaos for an interacting particle system associated with equation (1.2) and quantify the convergence rate 1 . Among others, it recovers Sznitman' seminal work [36] on the case where the noise is a Brownian motion. Interestingly, the striking fact of the analysis performed in [4] is based upon an observation noticed first by Tanaka in his seminal work [38] on limit theorems for mean field type diffusions, and used crucially by Cass and Lyons in [13]. It says that, for a given ω P Ω, the aforementioned particle system associated with (1.2) may be interpreted as a mean field rough equation (in the sense of our Definition 4.1 below) but with respect to the empirical version of the rough setting. The fact that Tanaka's trick extends to the case under study sounds as an a posteriori justification of our construction and demonstrates that our approach to (1.2) is certainly the right one. In this regard, it is worth emphasizing that the proof of the identification of the particle system with an equation of the same type as (1.2) is entirely based upon the properties of Lions' derivatives, hence revealing again the contribution of Lions' calculus to our analysis. While Lyons formulated his theory in a Banach setting from the begining [32], the theory has mainly been explored for finite dimensional drivers, with the noticeable exception of the works of Ledoux, Lyons and Qian on Banach space valued rough paths [30,33], Dereich follow-up works [19,20], Kelly and Melbourne application to homogenization of fast/slow systems of ordinary differential equations [29], and Bailleul and Riedel's work on rough flows [2]. One can see the present work as another illustration of the strength of the theory in its full generality. However, although the underlying rough ( . • We denote by pΩ, F, Pq an atomless Polish probability space, F standing for the completion of the Borel σ-field under P, and denote by x¨y the expectation operator, by x¨y r , for r P r1,`8s, the L r -norm on pΩ, F, Pq and by ⟪¨⟫ and ⟪¨⟫ r the expectation operator and the L r -norm on`Ω 2 , F b2 , P b2˘. When r is finite, L r pΩ, F, P; Rq is separable as Ω is Polish. • As for processes X ‚ " pX t q tPI , defined on a time interval I, we often write X for X ‚ .

Probabilistic rough structure
We define in this section a notion of rough path appropriate for our purpose. It happens to be a mixed finite/infinite dimensional object. Throughout the section, we work on a finite time horizon r0, T s, for a given T ą 0. ‚ We define the first level of our rough path structure as an ω-indexed pair of paths W t pωq, W t p¨q˘0 ďtďT , where`W t p¨q˘0 ďtďT is a collection of q-integrable R m -valued random variables on the space pΩ, F, Pq, which we regard as a deterministic L q pΩ, F, P; R m q-valued path, for some exponent q ě 1, and`W t pωq˘0 ďtďT stands for the realizations of these random variables along the outcome ω P Ω; so the pair (2.1) takes values in R mˆLq pΩ, F, P; R m q. As we already explained, a natural choice would be to take q " 2, but for technical reasons that will get clear below, we shall require q ě 8. ‚ The second level of the rough path structure includes a two-index path W s,t pωq˘0 ďsďtďT with values in R mˆm , obtained as the ω-realizations of a collection of q-integrable random variables`W s,t p¨q˘0 ďsďtďT defined on Ω; importantly, this second level also comprises the sections`W K K s,t pω,¨q˘0 ďsďtďT and`W K K s,t p¨, ωq˘0 ďsďtďT of a collection of R mˆm -valued random variables defined on the product space`Ω 2 , F b2 , P b2˘a nd considered as a deterministic L q`Ω2 , F b2 , P b2 ; R mˆm˘valued path`W K K s,t p¨,¨q˘0 ďsďtďT . Each W K K s,t p¨,¨q, for ps, tq P S T 2 , belonging to the space L q`Ω2 , F b2 , P b2 ; R mˆm˘, we have for P-a.e. ω P Ω. Below, we shall assume (2.2) to be true for every ω P Ω. This is not such a hindrance since we can modify in a quite systematic way the definition of the rough path structure on the null event where (2.2) fails; this is exemplified in Proposition 2.3 below. Taken this assumption for granted, we can regard Ω Q ω Þ Ñ W K K s,t pω,¨q and Ω Q ω Þ Ñ W K K s,t p¨, ωq as random variables with values in L q pΩ, F, P; R mˆm q: Since L q pΩ, F, P; R mˆm q is separable, it suffices to notice from Fubini's theorem that, for any Z P L q pΩ, F, P; R mˆm q, Ω Q ω Þ Ñ @ W K K s,t pω,¨q´Z D q is measurable, and similarly for W K K s,t p¨, ωq.
EJP 25 (2020), paper 21. Hence, the entire second level has the form of an ω-dependent two-index path with values in`R mˆLq pΩ, F, P; R m q˘b 2 and is encoded in matrix form aŝ Here, s,t p¨,¨q is in L q`Ωb2 , F b2 , P b2 ; R mˆm˘, the realizations of which read in the form Ω 2 Q pω, ω 1 q Þ Ñ W K K s,t pω, ω 1 q P R mˆm and the two sections of which are precisely given by W K K s,t pω,¨q : Ω Q ω 1 Þ Ñ W K K s,t pω, ω 1 q, and W K K s,t p¨, ωq Q ω 1 Þ Ñ W K K s,t pω 1 , ωq, for ω P Ω.
Below, we formulate several additional assumptions on the rough path structure, the introduction of which is rather lengthy and is, for that reason, split into three distinct subsections.

Algebraic conditions
As usual with rough paths, algebraic consistency requires that Chen's relations W r,t pωq " W r,s pωq`W s,t pωq`W r,s pωq b W s,t pωq, r,t p¨,¨q " W K K r,s p¨,¨q`W K K s,t p¨,¨q`W r,s p¨q b W s,t p¨q, (2.4) hold for any 0 ď r ď s ď t ď T . We used here the very convenient notation f r,s :" f s´fr , for a function f from r0, 8q into a vector space. In (2.4) and throughout, we denote by Xp¨q b Y p¨q, for any two X and Y in L q pΩ, F, P; R m q, the random variable`ω, ω 1 q Þ Ñ X i pωqY j pω 1 q˘1 ďi,jďm defined on Ω 2 . It is in L q`Ω2 , F b2 , P b2 ; R mˆm˘.
Remark 2.1. The last three lines in Chen's relations (2.4) are somewhat redundant. Assume indeed that we are given a collection of random variables`W K K s,t p¨,¨q˘0 ďsďtďT satisfying the last line of (2.4). Then, for all 0 ď r ď s ď t ď T and for P b2 -a.e. pω, ω 1 q P Ω 2 , Clearly, for P-almost every ω P Ω, the second and third lines in (2.4) hold true as well.
This is slightly weaker than the formulation (2.4) as, therein, the second and third lines are required to hold for all ω P Ω. As exemplified in the proof of Proposition 2.3, one may modify the definition of W K K on a null event so that the second and third lines in (2.4) hold true for all ω and for all 0 ď r ď s ď t ď T .

Definition 2.2.
We shall denote by W pωq the rough set-up specified by the ω-dependent collection of maps given by (2.1) and (2.3).
As for the component W K K of W pωq, the notation K K is used to indicate, as we shall make it clear below, that W K K s,t p¨,¨q should be thought of as the random variable pω, ω 1 q Þ Ñ ż t s´W r pωq´W s pωq¯b dW r pω 1 q.
Page 6/51 http://www.imstat.org/ejp/ Since Ω 2 Q pω, ω 1 q Þ Ñ pW t pωqq 0ďtďT and Ω 2 Q pω, ω 1 q Þ Ñ pW t pω 1 qq 0ďtďT are independent under P b2 , we then understand W K K s,t as an iterated integral of two independent copies of the noise. While such a construction is elementary for a random C 1 path, the well-defined character of this integral needs to be proved for more general probability measures P.
Example 2.1. Let W be an R m -valued Brownian motion, defined on pΩ, F, Pq. Denote by W t p¨q the equivalence class of Ω Q ω Þ Ñ W t pωq in L q`Ω , F, P; R m˘, and extend W t on the product space`Ω 2 , F b2 , P b2˘, setting W t pω, ω 1 q :" W t pωq. Define also on the product space the random variable W 1 t pω, ω 1 q :" W t pω 1 q. Then, W and W 1 are two independent m-dimensional Brownian motions under P b2 , and one can construct the time-indexed stochastic integral (say of Stratonovich or Itô type, but this does not really matter here since W and W 1 are independent) The stochastic integral is uniquely defined up to an event of zero measure under P b2 . Up to an exceptional event (of pΩ 2 , F b2 , P b2 q), we then let We can specify the definition of W K K on the remaining exceptional event and then modify the definition of W on a null event of pΩ, F, Pq in such a way that Chen's relations (2.4) hold everywhere -see the end of the proof of Proposition 2.3 below for a detailed proof of this fact. The process`W s,t pωq˘0 ďsďtďT is defined in a standard way as a Stratonovich or Itô (depending on the choice performed for the rough path) integral outside a set of null measure: W s,t pωq :"ˆż t s pW r´Ws q b dW r˙p ωq, 0 ď s ď t ď T.
The principle underpinning the above example may be put in a more general framework which will be useful to prove continuity of the Itô-Lyons solution map to the equation (1.2). We state it in the form of a proposition that provides a quite systematic way for constructing rough set-ups in practice. We advise the reader to come back to this proposition later on. Proposition 2.3. Let pΞ, G, Qq be a probability space, and W 1 :"`W 1 t˘0 ďtďT and W 2 :" W 2 t˘0 ďtďT be two independent and identically distributed R m -valued processes defined on Ξ. Assume they have continuous trajectories and E Q " sup 0ďtďTˇW i,j s,tˇq ı ă 8, for i, j " 1, 2, and`W 1 , W 1,1˘i s independent of W 2 . Last, assume that, for a.e. ξ P Ξ, the pair´W 1 pξq W 2 pξq¯,´W 1,1 pξq W 1,2 pξq W 2,1 pξq W 2,2 pξq¯ṡ atisfies Chen's relation in the sense that W i,j r,t pξq " W i,j r,s pξq`W i,j s,t pξq`W i r,s pξq b W j s,t pξq for any i, j P t1, 2u and 0 ď r ď s ď t ď T . Set Ω :" Ξˆr0, 1s with r0, 1s equipped with its Borel σ-algebra B`r0, 1s˘, and denote by Leb the Lebesgue measure on r0, 1s. Then we can find a triple of random variables`W, W, W K K˘, the first two components being defined on`Ω, F b Bpr0, 1sq, Q b Leb˘, the last component being constructed on the product space Ω 2 , and the whole family satisfying all the above requirements for a rough set-up, such that P´!pξ, uq :`W, W˘pξ, uq "`W 1 , W 1,1˘p ξq )¯" 1, and, for P-a.e. ω " pξ, uq, the law of W K K p¨, ωq is the same as the conditional law of W 2,1 given`W 1 pξq, W 2 pξq, W 1,1 pξq˘.
The reader may worry about the fact that, in the statement, we only appeal to W 1,1 and W 2,1 , and not to W 2,2 and W 1,2 . The reason is that, in our construction of the rough set-up, the processes W K K pω,¨q, W K K p¨, ωq and W K K p¨,¨q are intrinsically connected. As made clear by the proof below, the relationships that hold true between W K K pω,¨q, W K K p¨, ωq and W K K p¨,¨q must transfer to pW i q i"1,2 and pW i,j q i,j"1,2 . In short, everything works as if the pair pW 2 , W 2,2 q was a mere independent copy of pW 1 , W 1,1 q and the conditional law of W 1,2 given pW 2 , W 1 , W 2,2 q was the same as the conditional law of W 2,1 given pW 1 , W 2 , W 1,1 q, in which case the only needed ingredients are W 1 , W 1,1 , W 2 and W 2,1 . The latter is consistent with the statement.
Proof. Recall first from [6] the following form of Skorokhod representation theorem.
There exists a function Ψ : ‚ for every probability µ on CpS T 2 q, equipped with its Borel σ-field, r0, 1s Q u Þ Ñ Ψpu, µq is a random variable with µ as distribution -r0, 1s being equipped with Lebesgue measure, ‚ the map Ψ is measurable.
Since the law of`W, W 1 , W˘under P b2 is the same as the law of`W 1 , W 2 , W 1,1˘u nder Q, we deduce that the law of`W, W 1 , W, W J J˘u nder P b2 , with W J J pω, ω 1 q :" W K K pω 1 , ωq, is the same as the law of`W 1 , W 2 , W 1,1 , W 2,1˘u nder Q. In particular, with probability 1 under P b2 , for all 0 ď r ď s ď t ď T , Call now A P F the set of those ω's in Ω for which the above relation fails for ω 1 in a set of positive probability measure under P. Clearly, PpAq " 0. Define in a similar way A 1 by exchanging the roles of ω and ω 1 . For ω P A Y A 1 , set W pωq " 0; and whenever ω P A Y A 1 or ω 1 P A Y A 1 , set W K K pω, ω 1 q " 0. If ω R A Y A 1 , we have, by definition of A and A 1 , the third identity in (2.4) -pay attention that we use the fact that the identity is understood as an equality between classes of random variables that are P-a.e. equal. If ω P A Y A 1 , it is also true since all the terms are zero. The second identity in (2.4) is checked in the same way. As for the first one, it holds on the complementary B A of a null event B. We then replace A by A Y B and A 1 by A 1 Y B in the previous lines and set W p¨q " 0 and EJP 25 (2020), paper 21.

Analytical conditions
We use in this work the notion of p-variation to handle the regularity of the various trajectories in hand. The choice of the p-variation, instead of the simplest Hölder (semi-) norm, is dictated by the arguments we use below to prove well-posedness of (1.4). We shall indeed invoke some integrability results from [12], which are explicitly based upon the notion of p-variation and are not proved in Hölder (semi-) norm. Several types of p-variations are needed to handle differently the finite and infinite dimensional components of a rough set-up W . Throughout, p is taken in the interval r2, 3q. For a continuous function G from the simplex S T 2 into some R , we set, for any p 1 ě 1, and define for any function g from r0, T s into R , }g} p r0,T s,p´var :" }G} p r0,T s,p´var where G s,t :" g t´gs . Similarly, for a random variable Gp¨q on Ω with values in CpS T 2 ; R q, and p 1 ě 1, we define its p 1 -variation in L q as xGp¨qy p 1 q;r0,T s,p 1´v ar :" sup and define for a random variable Gp¨q on Ω, with values in Cpr0, T s; R q, @ Gp¨q D p q;r0,T s,p´var :" @ Gp¨q D p q;r0,T s,p´var , as the p-variation semi-norm in L q of S T 2 Q ps, tq Þ Ñ G s,t p¨q " G t p¨q´G s p¨q. Last, for a random variable Gp¨,¨q from pΩ 2 , F b2 q into CpS T 2 ; R q, we set ⟪Gp¨,¨q⟫ p q;r0,T s,p´var :" sup 0"t0ăt1¨¨¨ătn"T n ÿ i"1 ⟪G ti´1,ti p¨,¨q⟫ p q . (2.6) Given these definitions, we require from the rough set-up W that • For any ω P Ω, the path W pωq is in the space Cpr0, T s; R m q, and the map W : Ω Q ω Þ Ñ W pωq P Cpr0, T s; R m q is Borel-measurable and q-integrable (meaning that the supremum of W over r0, T s is q-integrable). • For any ω P Ω, the two-index path Wpωq is in CpS T 2 ; R mˆm q, and the map W : Ω Q ω Þ Ñ Wpωq P CpS T 2 ; R mˆm q is Borel-measurable and q-integrable (i.e., the supremum of W over S T 2 has a finite q-moment). • For any pω, ω 1 q P Ω 2 , the two-index path W K K pω, ω 1 q is an element of CpS T 2 ; R mˆm q, and the map W K K : Ω 2 Q pω, ω 1 q Þ Ñ W K K pω, ω 1 q P CpS T 2 ; R mˆm q is Borel-measurable and q-integrable. In particular, for a.e. ω P Ω, the two-index path W K K pω,¨q belongs to C`S T 2 ; L q pΩ, F, P; R mˆm q˘, and the map Ω Q ω Þ Ñ W K K pω,¨q is Borel-measurable and q-integrable, and similarly for W K K p¨, ωq; as before, we assume the latter to be true for every ω P Ω. Also, the two-index deterministic path W K K p¨,¨q is a continuous mapping from S T 2 into L q`Ω2 , F b2 , P b2 ; R mˆm˘.
2 is a random variable with values in CpS T 2 ; R`q. Throughout the analysis, we assume xvp0, T,¨qy q ă 8, for any rough set-up considered on the interval r0, T s. By Lebesgue's dominated convergence theorem, the function S T 2 Q ps, tq Þ Ñ xvps, t,¨qy q is continuous. We shall actually assume that it is of bounded variation on r0, T s, i.e., xvp¨qy q;rs,ts,1´var :" sup 0ďt1ă¨¨¨ătnďT n ÿ i"1 xvpt i´1 , t i ,¨qy q ă 8.
Below, we call a control any family of random variables pω Þ Ñ wps, t, ωqq ps,tqPS T 2 that is jointly continuous in ps, tq and that satisfies, wps, t, ωq ě vps, t, ωq`xvp¨qy q;rs,ts,1´var , together with xwps, t,¨qy q ď 2 wps, t, ωq, wpr, t, ωq ě wpr, s, ωq`wps, t, ωq, r ď s ď t. (2.9) Of course, a typical choice to get (2.8) and (2.9) is to choose wps, t, ωq :" vps, t, ωq`xvp¨qy q;rs,ts,1´var . where the supremum is taken over all dissections pt i q i and ps j q j of the interval rs, ts. Without any loss of generality, we may assume that the process W is constructed on the canonical space pΩ, F, Pq, where Ω " W, with W :" Cpr0, T s; R m q, F is the Borel σ-field, and W is the coordinate process. We then denote by pΩ " W, H, Pq the abstract Wiener space associated with W , see [24,Appendix D], where H is a Hilbert space, which is automatically embedded in the subspace C ´var`r 0, T s; R m˘o f C`r0, T s; R mc onsisting of continuous paths of finite -variation. By Theorem 15.33 in [24], we know that, for ω outside an exceptional event, the trajectory W pωq may be lifted into a rough path pW pωq, Wpωqq with finite p-variation for any p P p2ρ, 3q, namely W pωq has a finite p-variation and Wpωq has a finite p{2-variation. We lift arbitrarily (say onto the zero path) on the null set where the lift is not automatic. The pair pW, Wq, indexed by ω is part of our rough set-up. In this regard, we recall from Theorem 15.33 in [24] that the random variables have respectively Gaussian and exponential tails, and thus have a finite L q -moment. One can proceed as follows to construct the other elements`W K K pω,¨q˘ω PΩ , W K K p¨, ωq˘ω PΩ , W K K p¨,¨q of our rough set-up. We extend the space into pΩ 2 , F b2 , P b2 q, with Ω embedded in the first component say, and denote by pW, W 1 q the canonical coordinate process on Ω 2 . They are independent and have independent Gaussian components under P 2 . The associated abstract Wiener space is nothing but`Ω 2 , H ' H, P b2˘. The process pW, W 1 q also satisfies Theorem 15.33 in [24] for the same exponent ρ as before, so, we can enhance pW, W 1 q into a Gaussian rough path, with some arbitrary extension EJP 25 (2020), paper 21. outside the P b2 -exceptional event on which we cannot construct the enhancement. To ease the notations, we merely write W pωq for W pω, ω 1 q as it is independent of ω; similarly, we write W 1 pω 1 q for W 1 pω, ω 1 q. Proceeding as before, we call`W K K pω, ω 1 q˘ω ,ω 1 PΩ , the upper off-diagonal mˆm block in the decomposition of the second-order tensor of the rough path in the form of a p2mqˆp2mq-matrix with four blocks of size mˆm. Chen's relationship then yields, for P b2 -a.e. pω, ω 1 q, for all r ď s ď t. As before, we know from Theorem 15.33 in [24] that the 1{p-Hölder seminorm of W pωq, which we denote by }W pωq › › r0,T s,p1{pq´Höl , and the 2{p-Hölder semi-norm of W K K pω, ω 1 q, which we denote by r0,T s,p2{pq´Höl , have respectively Gaussian and exponential tails, when considered as random variables on the spaces pΩ, F, Pq and Ω 2 , F b2 , P b2˘. In particular, for a.e. ω P Ω, we may consider`W K K s,t pω,¨q˘p s,tqPS T 2 as a continuous process with values in L q . Moreover, which shows that the left-hand side has finite moments of any order. Arguing in the same way for`W K K p¨, ωq˘ω PΩ and for W K K , we deduce that v in (2.7) is almost surely finite and q-integrable. Obviously, by replacing r0, T s by rs, ts Ă r0, T s, we obtain that the q-moment of v is Lipschitz (and thus of finite 1-variation), as required. All these properties (that hold true on a full event) may be extended to the full set Ω 2 by arguing as in the proof of Proposition 2.3.

Local accumulation
To use that rough set-up in our machinery, we need a version of an integrability result of [12] whose proof is postponed to Appendix A. Given a nondecreasing 2 continuous positive valued function on S 2 " tps, tq P r0, 8q 2 : s ď tu, a parameter s ě 0 and a threshold α ą 0, we define inductively a sequence of times τ 0 ps, αq :" s, and τ n`1 ps, αq :" inf ! u ě τ n ps, αq : `τ n ps, αq, u˘ě α ) , (2.13) with the understanding that inf H "`8. For t ě s, set N `r s, ts, α˘:" sup ! n P N : τ n ps, αq ď t ) . (2.14) Below, we call N the local accumulation of (of size α if we specify the value of the threshold): N prs, ts, αq is the largest number of disjoint open sub-intervals pa, bq of rs, ts on which pa, bq is greater than or equal to α. When ps, tq " wps, t, ωq 1{p with w a control satisfying (2.8) and (2.9) and when the framework makes it clear, we just write N prs, ts, ω, αq for N prs, ts, αq. Similarly, we also write τ n ps, ω, αq for τ n ps, αq when ps, tq " wps, t, ωq 1{p . We will also use the notation τ n ps, t, αq :" τ n ps, αq^t.
The proof of the following statement is given in Appendix A. Recall that a positive random variable A has a Weibull tail with shape parameter 2{ if A 1{ρ has a Gaussian tail.
Theorem 2.4. Let W be a continuous centred Gaussian process, defined over some finite interval r0, T s. Assume it has independent components, and denote by pW, H, Pq its associated Wiener space. Suppose that the covariance function satisfies the Lipschitz estimate (2.11). Then, for p P p2 , 3q and α ą 0, the process N p¨, αq :" pN pr0, T s, ω,αqq ωPΩ associated to the rough-set up built from W , with w being defined as in (2.10), has a Weibull tail with shape parameter 2{ .
As a corollary, we deduce that the estimate on N required in Theorem 1.1 is satisfied in the above setting. For the same value of p, the quantity wp0, T q in (2.10) also satisfies the integrability statement of Theorem 1.1; the latter then applies in the above Gaussian setting. Building on the work [14] on Markovian rough paths one can prove a similar result as Theorem 2.4 for Markovian rough paths.

Controlled trajectories and rough integral
Following [26], we now define a controlled path and the corresponding rough integral.
Throughout the section, we are given a control w satisfying (2.8) and (2.9).

Controlled trajectories
We first define the notion of controlled trajectory for a given outcome ω P Ω.
The value 4{3 is somewhat arbitrary here. Our analysis could be managed with another exponent strictly greater than 1, but this would require higher values for the exponent q than that one we use in the definition of the rough set-up -recall q ě 8. It seems that the value 4{3 is pretty convenient, as 4{3 is the conjugate exponent of 4. It follows from the fact that~Xpωq~‹ ,r0,T s,p is finite that an ω-controlled path is controlled in the usual sense by the first level`W t pωq, W t p¨q˘0 ďtďT of our rough set-up, provided the latter is considered as taking values in an infinite dimensional space, see Section 3.2 below.
We now define the notion of random controlled trajectory, which consists of a collection of ω-controlled trajectories indexed by the elements of Ω.

Definition 3.2.
A family of ω-controlled paths pXpωqq ωPΩ such that the maps are measurable and satisfy is called a random controlled path on r0, T s.
Note from (2.9) the following elementary fact, whose proof is left to the reader.

Lemma 3.3.
Let`pX t pωqq˘0 ďtďT q ωPΩ be a random controlled path on a time interval r0, T s. Then, for any 0 ď s ă t ď T , we have A straightforward consequence of Lemma 3.3 is that a random controlled trajectory induces a continuous path from r0, T s to L 2 pΩ, F, P; R d q.

Rough integral
Set U :" R mˆLq pΩ, F, P; R m q and note that U b U can be canonically identified with We take as a starting point of our analysis the fact that W pωq may be considered as a rough path with values in U ' U b2 , for any given ω. Indeed the first level W p1q pωq :" W t pωq, W t p¨q˘t ě0 of W pωq is a continuous path with values in U and its second level ωq as an element of L q pΩ, F, P; R m q b R m , and W K K 0,t p¨,¨q as an element of L q pΩ, F, P; R m q b L q pΩ, F, P; R m q. Condition (2.4) then reads as Chen's relation for W pωq.
We can then use sewing lemma [22], in the form given in [15,16], to construct the rough integral of an ω-controlled path and a Banach-valued rough set-up. from the space of ω-controlled trajectories equipped with the norm~¨~‹ ,r0,T s,p , onto the space of continuous functions from S T 2 into R d b R m (that are equal to zero on the diagonal) with finite norm }¨} r0,T s,w,p{2 , with w in the latter norm being evaluated along the realization ω, that satisfies for any 0 ď r ď s ď t ď T the identity Here, δ x X s pωq W s,t pωq is the product of two dˆm and mˆm matrices, so it gives back a dˆm matrix, with components`δ x X s pωqW s,t pωq˘i ,j " ř m k"1`δ x X i s pωq˘k`W s,t pωq˘k ,j , for i P t1,¨¨¨, du and j P t1,¨¨¨, mu. We stress that the notation E " δ µ X s pω,¨qW K K s,t p¨, ωq ‰ , which reads as the expectation of a matrix of size dˆm, can be also interpreted as a contraction product between an element of R d b L 4{3 pΩ, F, P; R m q and an element of L q pΩ, F, P; R m q b R m . This remark is important for the proof below.
Proof. The proof is a consequence of Proposition 2 in Coutin and Lejay's work [15], except for one main fact. In order to use Coutin and Lejay's result, we consider W pωq as a rough path with values in U ' U b2 and`Xpωq, δ x Xpωq, δ µ Xpωq, R X pωq˘as a controlled path; this was explained above. When doing so, the resulting integral is constructed as a process with values in R d b U , whilst the integral given by the statement of Theorem 3.4 takes values in R d . We denote the R d b U -valued integral by pI t s X s,u pωq b dW u pωqq ps,tqPS T 2 . We use a simple projection to pass from the infinite dimensionalvalued quantity I t s X s,u pωqbdW u pωq to the finite dimensional-valued quantity ş t s X s,u pωqb dW u pωq. Indeed, we may use the canonical projection from As usual, we define an additive process setting for 0 ď t ď T . We can thus consider the integral process`ş t 0 X s pωq b dW s pωq˘0 ďtďT as an ω-controlled trajectory with values in R dˆm , with x-derivative a linear map from R m into R dˆm , and entrieŝ for i P t1,¨¨¨, du and j, k P t1,¨¨¨, mu, where δ j,k stands for the usual Kronecker symbol, and with null µ-derivative, namely ď~Xpωq~‹ ,r0,T s,w,p´1`w p0, T, ωq 1{p¯w ps, t, ωq 2{p , so that, with the notation of Definition 3.1, When Xpωq is given as the ω-realization of a random controlled path pXpω 1 qq ω 1 PΩ , the integral may be defined for any ω 1 P Ω. For the integral ş¨0 X s pωq b dW s pωq to define a random controlled path, its~¨~r 0,T s,w,p -semi-norm needs to have finite 8-th moment, see (3.2) (we give later on more precise estimates to guarantee that this may be indeed the case). In this respect, it is worth noticing that the measurability properties of the integral with respect to ω can be checked by approximating the integral with compensated Riemann sums, see once again (3.3). This gives measurability of Ω Q ω Þ Ñ ş t 0 X s pωq b dW s pωq for any given time t P r0, T s. Measurability of the functional We then set for i P t1,¨¨¨, dû and consider ş t 0 X s pωqdW s pωq as an element of R d .

Stability of controlled paths under nonlinear maps
We show in this section that controlled paths are stable under some nonlinear, sufficiently regular, maps and start by recalling the reader about the regularity notion used when working with functions defined on Wasserstein space. We refer the reader to Lions' lectures [31], to the lecture notes [9] of Cardaliaguet or to Carmona and Delarue's monograph [10,Chapter 5] for basics on the subject. ‚ Recall that pΩ, F, Pq stands for an atomless probability space, with Ω a Polish space and F its Borel σ-algebra. Fix a finite dimensional space E " R k and denote by L 2 : " L 2 pΩ, F, P; Eq the space of E-valued random variables on Ω with finite second moment. We equip the space P 2 pEq :" LpZq ; Z P L 2 ( with the 2-Wasserstein distance ) .
An R k -valued function u defined on P 2 pEq is canonically extended into L 2 by setting, for any Z P L 2 , U pZq :" u`LpZq˘.
‚ The function u is then said to be differentiable at µ P P 2 pEq if its canonical lift is Fréchet differentiable at some point Z such that LpZq " µ; we denote by ∇ Z U P pL 2 q k the gradient of U at Z. The function U is then differentiable at any other point Z 1 P L 2 such that LpZ 1 q " µ, and the laws of ∇ Z U and ∇ Z 1 U are equal, for any such Z 1 .
‚ The function u is said to be of class C 1 on some open set O of P 2 pEq if its canonical lift is of class C 1 in some open set of L 2 projecting onto O. It is then of class C 1 in the whole fiber in L 2 above O. If u is of class C 1 on P 2 pEq, then ∇ Z U is σpZq-measurable and given by an LpZq-dependent function Du from E to E k such that ∇ Z U " pDuqpZq; (3.6) we have in particular Du P L 2 µ pE; E k q:" L 2 pE, BpEq, µ; E k q, where BpEq is the Borel σ-field on E. In order to emphasize the fact that Du depends upon LpZq, we shall write DupLpZqqp¨q instead of Dup¨q. Sometimes, we shall put an index µ and write D µ upLpZqqp¨q in order to emphasize the fact that the derivative is taken with respect to the measure argument; this will be especially useful for functionals u depending on additional variables. Importantly, this representation is independent of the choice of the probability space pΩ, F, Pq; in fact, it can be easily transported from one probability space to another. (Simpler proofs of the structural equation (3.6) can be found in [1,39].) ‚ As an example, take u of the form upµq " ş R d f pyqdµpyq for a continuously differentiable function f : R d Ñ R such that ∇f is at most of linear growth. The lift Z Þ Ñ U pZq " Erf pZqs has differential pd Z U qpHq " Er∇f pZqHs and gradient ∇f pZq.
‚ Back to controlled paths. Let F stand here for a map from R dˆL2 pΩ, F, P; F should be thought of as the lift of the coefficient driving equation (1.2), or, with the same notation as in (1.3), as p F itself, with the slight abuse of notation that it requires to identify F and p F. Our goal now is to expand the image of a controlled trajectory by F.

Regularity assumptions 1.
Assume that F is continuously differentiable in the joint variable px, Zq, that B x F is also continuously differentiable in px, Zq and that there is some positive finite constant Λ such that and Importantly, the L 2 -Lipschitz bound required in the second line of (3.7) may be formulated as a Lipschitz bound on P 2 pR d q equipped with d 2 . Moreover, notice that the space L 2`Ω , F, P; L pR d , R d b R m q˘can be identified with L 2 pΩ, F, P; R d q dˆm ; also, B x Fpx, Zq and ∇ Z Fpx, Zq will be considered as random variables with values in L pR d , As an example, the functions Fpx, µq " ş R d f px, yqµpdyq for some function EJP 25 (2020), paper 21.
f of class C 2 b , and Fpx, µq " g`x,

both satisfy
Regularity assumptions 1. A counter-example is the function Fpx, µq " ş R d |z| 2 dµpzq. We expand below the path`FpX t pωq, Y t p¨qq˘0 ďtďT , which we write FpXpωq, Y p¨qq, where Xpωq is an ω-controlled path and Y p¨q is an R d -valued random controlled path, both of them being defined on some finite interval r0, T s. Identity (3.4) tells us that a fixed point formulation of (1.2) will only involve pairs pXpωq, Y p¨qq such that which prompts us to restrict ourselves to the case when Xpωq and Y have null µderivatives in the expansion (3.1).

Proposition 3.5. Let
Xpωq be an ω-controlled path and Y p¨q be an R d -valued random controlled path. Assume that condition (3.8) hold together with the ω-independent ,¨¨¨, du and j, k P t1,¨¨¨, mu, and (with a similar interpretation for the product) and one can find a constant C Λ,M , depending only on Λ and M , such that )`! (4)` (5) ) , :" :" :" we used here the fact that Xpωq and Y p¨q have null µ-derivative and where we let X pλq s;ps,tq pωq " X s pωq`λX s,t pωq, Y pλq s;ps,tq p¨q " Y s p¨q`λY s,t p¨q. We read on (3.9) the formulas for the x and µ-derivatives of FpXpωq, Y p¨qq. The remainder R FpX,Y q s,t in the controlled decomposition of the path FpXpωq, Y p¨qq is We now compute F`Xpωq, Y p¨q˘ ‹,r0,T s,w,p .
• We have first from the assumptions on F that the initial conditions for the quanti- Using the Lipschitz property of F and Lemma 3.3, we and, applying Hölder inequality with exponents 3{2 and 3, pωq~r 0,T s,w,p`x~Y p¨q~r 0,T s,w,p y 4¯w ps, t, ωq 1{p Λ xδ x Y s,t p¨qy 4 ď 2ΛM´~Xpωq~r 0,T s,w,p`x~Y p¨q~r 0,T s,w,p y 4¯w ps, t, ωq 1{p
Collecting the various terms, we complete the proof.

Solving the equation
We now have all the tools to formulate the equation (1.4) (or (1.2)) as a fixed point problem and solve it by Picard iteration. Our definition of the fixed point is given in the form of a two-step procedure: The first step is to write a frozen version of the equation, in which the mean field component is seen as an exogenous collection of ω-controlled trajectories; the second step is to regard the family of exogenous controlled trajectories as an input and to map it to the collection of controlled trajectories solving the frozen version of the equation. In this way, we define a solution as a collection of ω-controlled trajectories. In order to proceed, recall the generic notation`Xpωq; δ x Xpωq; B µ Xpω,¨qf or an ω-controlled path and its derivatives; we sometimes abuse notations and talk of Xpωq as an ω-controlled path. In all the following, W and its enhancement W are assumed to form a rough-set up as defined in Section 2 and to satisfy all the conditions prescribed in this section. Definition 4.1. Let W together with its enhancement W satisfy the assumption of Section 2 on a finite interval r0, T s, and let Y p¨q stand for some R d -valued random controlled path on r0, T s, with the property that δ µ Y p¨q " 0 and sup 0ďtďT xδ x Y t p¨qy 8 ă 8. For a given ω P Ω, let Xpωq be an R d -valued ω-controlled path on r0, T s, with the properties that δ µ Xpωq " 0 and sup 0ďtďT |δ x X t pωq| ă 8. We associate to ω and Xpωq an ω-controlled path by setting Γ`ω, Xpωq, Y p¨q: A solution to the mean field rough differential equation dX t " F`X t , LpX t q˘dW t , on the time interval r0, T s, with given initial condition X 0 p¨q P L 2 pΩ, F, P; R d q is a random controlled path Xp¨q starting from X 0 p¨q and satisfying the same prescription as Y p¨q, such that for P-a.e. ω the path Xpωq and Γ`ω, Xpωq, Xp¨q˘coincide.
We should more properly replace Xpωq in Γ`ω, Xpωq, Y p¨q˘by`Xpωq ; δ x Xpωq ; 0˘and Y p¨q by`Y p¨q ; δ x Y p¨q ; 0˘, but we stick to the above lighter notation. Observe also that our formulation bypasses any requirement on the properties of the map Γ itself. To make it clear, we should be indeed tempted to check that, for a random controlled path Xp¨q, the collection`Γpω, Xpωq, Y p¨qq˘ω PΩ , for Y p¨q as in the statement, is also a random controlled path. Somehow, our definition of a solution avoids this question; however, we need to check this fact in the end; below, we refer to it as the stability properties of Γ, see Section 4.1.
What remains of the above definition when W is the Itô or Stratonovich enhancement of a Brownian motion? The key point to connect the above notion of solution with the standard notion of solution to mean field stochastic differential equation is to observe that the rough integral therein should be, if a solution exists, the limit of the compensated Riemann sums n´1 ÿ j"0ˆF`X tj pωq, X tj p¨q˘W tj ,tj`1 pωq`B x F`X tj pωq, X tj p¨q˘F`X tj pωq, X tj p¨q˘W tj ,tj`1 pωq A D µ F`X tj pωq, X tj p¨q˘`X tj p¨q˘F`X tj pωq, X tj p¨q˘W K K tj ,tj`1 p¨, ωq E˙, as the step of the dissection 0 " t 0 ă¨¨¨ă t n " t tends to 0. When the solution is constructed by a contraction argument, such as done below, the process pX t p¨qq 0ďtďT is adapted with respect to the completion of the filtration pF t q 0ďtďT generated by the initial condition X 0 p¨q and the Brownian motion W p¨q. Returning if necessary to Example 2.2, EJP 25 (2020), paper 21. we then check that E " W K K tj ,tj`1 p¨, ωq | F tj ‰ " 0, whatever the interpretation of the rough integral, Itô or Stratonovich. Pay attention that the conditional expectation is taken with respect to "¨", while ω is kept frozen. This implies that, for any j P t0,¨¨¨, n´1u, we have A D µ F`X tj pωq, X tj p¨q˘`X tj p¨q˘F`X tj pωq, X tj p¨q˘W K K tj ,tj`1 p¨, ωq  • The function B x F is differentiable in px, µq in the same sense as F itself.
• For each px, µq P R dˆP • The three functions px, Zq Þ Ñ B x D µ F`x, LpZq˘pZp¨qq, px, Zq Þ Ñ D µ B x F`x, LpZq˘pZp¨qq, and px, Zq Þ Ñ B z D µ F`x, LpZq˘pZp¨qq from R dˆL2 pΩ, F, P; R d q to L 2`Ω , F, P; R d b R m bR d bR d˘, are bounded by Λ and Λ-Lipschitz continuous. (By Schwarz' theorem, the transpose of B x D µ F i,j is in fact equal to D µ B x F i,j , for any i P t1,¨¨¨, du and j P t1,¨¨¨, mu.) • For each µ P P 2 pR d q, we denote by D 2 µ Fpx, µqpz,¨q the derivative of D µ Fpx, µqpzq with respect to µ -which is indeed given by a function. For Denote by`r Ω, r F, r P˘a copy of pΩ, F, Pq, and given a random variable Z on pΩ, F, Pq, write r Z for its copy on p r Ω, r F, r Pq. We assume that px, Zq The two functions Fpx, µq " ş f px, yqµpdyq for some fuction f of class C 3 b , and Fpx, µq " g`x, ş yµpdyq˘for some function g of class C 3 b , both satisfy Regularity assumptions 2. We refer to [10,Chapter 5] and [11,Chapter 5] for other examples of functions that satisfy the above assumptions and for sufficient conditions under which these assumptions are satisfied. We feel free to abuse notations and write Zp¨q for LpZq in the argument of the functions B x D µ F, B z D µ F and D 2 µ F. We prove in Section 4.1 that the map Γ sends some large ball of its state space into itself for a small enough T . The contractive character of Γ is proved in Section 4.2, and Section 4.3 is dedicated to proving the well-posed character of (1.4). EJP 25 (2020), paper 21.

Stability of balls by Γ
Recall Λ was introduced in Regularity assumptions 1 and 2 as a bound on F and some of its derivatives. Recall also from (2.14) the definition of N`r0, T s, ω; α˘. We also use below the notations~¨~r a,bs,w,p and~¨~‹ ,ra,bs,w,p , for some interval ra, bs, to denote the same quantity as in Definition 3.2 but for paths defined on ra, bs rather than on r0, T s (the initial condition is then taken at time a). Proposition 4.2. Let F satisfy Regularity assumptions 1 and w be a control satisfying (2.8) and (2.9). Consider an ω-controlled path Xpωq together with a random controlled path Y p¨q, both of them satisfying (3.8) together with sup 0ďtďT´ˇδ   for all 0 ď i ď N , with N :" N pr0, T s, ω, 1{p4Lqq, and for the sequence of times`t i :" τ i p0, T, ω, 1{p4Lqq˘i "0,¨¨¨,N`1 given by (2.13) with ps, tq " wps, t, ωq 1{p . Then: ‚ There exists a constant c ą 1, which depends only on Λ, such that (4.2) and (4.3) remain true if we replace L by L 1 , provided that L 1 ě cL and the partition pt i q i"0,¨¨¨,N`1 is recomputed accordingly (since L enters the definition of the partition). Also, we can find a constant L 1 0 , only depending on L, such that for the same constant c and for L 1 ě L 1 0 , the path Γ`ω, Xpωq, Y p¨q˘satisfies for each ω the size estimate (4.3), L being replaced by c in the right-hand side and the partition pt i q i"0,¨¨¨,N`1 in the left-hand side being defined with respect to L 1 instead of L. ‚ Moreover, there exist two constants L 0 and C, only depending on Λ, such that, if L in (4.2) and (4.3) is greater than L 0 , the following estimates hold for each ω: ‚ Lastly, if Xpωq is the ω-realization of a random controlled path Xp¨q "`Xpω 1 q˘ω 1 PΩ 1 such that the estimate Xpω 1 q 2 rti,ti`1s,w,p ď L holds for all ω 1 , for the ω 1 -dependent partition`t i :" τ i p0, T, ω 1 , 1{p4Lqq˘i "0,¨¨¨,N`1 of r0, T s, with L in (4.2) satisfying L ě L 0 and with N :" N pr0, T s, ω 1 , 1{p4Lqq, and if T is small enough to have Following the discussion after (3.5), the measurability properties of the map ω Þ Ñ Γ`ω, Xpωq, Y p¨q˘implicitly required above can be checked by approximating the integral EJP 25 (2020), paper 21. in the definition of Γ`ω, Xpωq, Y p¨q˘, using (3.3). We also notice that the constraint L ě L 0 required in the second and third bullet points may be easily circumvented. Indeed, the first claim in the statement guarantees that, for L satisfying (4.2) and (4.3), L 1 ě cL also satisfy (4.2) and (4.3), see footnote 4 . In particular, we can always apply the second and third bullet points with L 1 ě cL 0 instead of L itself, which is a good point since L 1 is here a free parameter while the value of L is prescribed by the statement.
Proof. We first explain the reason why (4.3) remains true for possibly larger values of L provided that the right-hand side is multiplied by a universal multiplicative constant. Take L 1 ą L and call pt 1 j q j"0,¨¨¨,N 1`1 the corresponding dissection. It is clear that any interval rt 1 j , t 1 j`1 s must be included in an interval of the form rt i , t i`2^T s. If rt 1 j , t 1 j`1 s Ă rt i , t i`1 s, the proof is done. If t i`1 P pt 1 j , t 1 j`1 q, it is an easy exercise 5 to check that¨~r j`1 s,w,p ď γ~¨~r t 1 j ,ti`1s,w,p`γ~¨~rti`1,ti`2^T s,w,p , for some universal constant γ.
Given this preliminary remark, the proof proceeds in three steps.
‚ For ω P Ω, consider a subdivision pt i q 0ďiďN`1 of r0, T s such that wpt i , t i`1 , ωq ď 1 for all i P t0,¨¨¨, N u, for some integer N ě 0. Then, following [16,Proposition 4] (rearranging the terms therein), we know that 6 żẗ i F`X r pωq, Y r p¨q˘dW r pωq rti,ti`1s,w,p ď γ`γwpt i , t i`1 , ωq 1{p F`Xpωq, Y p¨q˘ rti,ti`1s,w,p , for a universal constant γ that may depend on Λ. By Proposition 3.5 and (4.1), we deduce 4 While the reader may find it obvious, she/he must be aware of the fact that, in (4.3), t i and t i`1 themselves depend on L, which forces to recompute the subdivision when L is changed. 5 The proof is as follows. By the super-addivitiy of w, see (2.9), and the inequality a 1{p`b1{p ď 2 1´1{p pab q 1{p , the terms }Xpωq} rt 1 j ,t 1 j`1 s,w,p , }δxXpωq} rt 1 j ,t 1 j`1 s,w,p and xδµXpω,¨qy rt 1 j ,t 1 j`1 s,w,p,4{3 are easily handled.
So, the only difficulty is to handle }R X } rt 1 j ,t 1 j`1 s,w,p . By (3.1), we have, for any 0 ď r ď s ď t ď T , R X r,t pωq " R X r,s pωq`R X s,t pωq`δxXr,spωqWs,tpωq`E " δµXr,spω,¨qWs,tp¨q ‰ , which suffices for our purpose. 6 In fact, the inequality may be checked directly. Identity for a constant γ that may depend on Λ. This permits to handle R ş F . As the Gubinelli derivative of ş¨t i F`Xrpωq, Yrp¨q˘dW r pωq is exactly given by FpX¨pωq, Y¨p¨qq itself, we get from (3.1) with X " F that where R F is the remainder in the expansion of F. We conclude as for R ş F . In order to control the variation of ş¨t i F`Xrpωq, Yrp¨q˘dW r pωq itself, it suffices to invoke (3.1) again, but with X " ş F, which yields The conclusion is the same.
that (for a new value of C Λ,Λ ) żẗ i F`X r pωq, Y r p¨q˘dW r pωq rti,ti`1s,w,p ď γ`C Λ,Λ γ wpt i , t i`1 , ωq 1{p´1`~X pωq~2 rti,ti`1s,w,p`@~Y p¨q~r 0,T s,w,p D 2 8¯. (4.5) By the first conclusion in the statement (see also the discussion after the statement itself), we can assume that L differs from the value prescribed in the statement and is as large as needed. So, for the time being, we take L ě 1 and we assume that wpt i , t i`1 , ωq 1{p ď 1{p4Lq ď 1 and @~Y p¨q~r 0,T s,w,p D 2 8 ď L,   ‚ We now use a concatenation argument to get an estimate on the whole interval r0, T s. For all s ă t in r0, T s, we havě " Γ`ω, Xpωq, Y p¨q˘‰ s,tˇ( 4.9) where we let t 1 i " maxps, minpt, t i qq and where used the super-additivity of w in the last line. In the same way, δ x " Γ`ω, Xpωq, Y p¨q˘‰ s,tˇď γ wps, t, ωq 1{p`N`1˘p p´1q{p . (4.10) Setting, abusively, Fpω,¨q :"`F r pω,¨q˘0 ďrďT :"`FpX r pωq, Y r p¨qq˘0 ďrďT , we have R Γ s,t pωq " ż t s F r pω,¨qdW r pωq´F s pω,¨qW s,t pωq ) .
‚ Assume now that Xpωq is the ω-realization of a random controlled path Xp¨q " pXpω 1 qq ω 1 PΩ 1 satisfying (4.3) for any ω 1 , for the ω 1 -dependent partition pt i q i"0,¨¨¨,N`1 . Then, taking the fourth moment with respect to ω in the conclusion of the second point we get We get the conclusion of the statement if one assumes that @ N`r0, T s,¨, 1{p4Lq˘D 8 ď 1, by choosing L such that 2 C γ ď L.
Proof. We get the conclusion after four steps. Following the statement, we are given a subdivision pt i q i"0,¨¨¨,N`1 of r0, T s such that wpt i , t i`1 , ωq 1{p ď 1{p4Lq, for a frozen ω P Ω and for L ě L 0 . We assume that pt i q i"0,¨¨¨,N`1 refines the subdivision`t 0 i " τ i p0, T, ω, 1{p4L 0 qq˘i "0,¨¨¨,N 0`1 , where N 0 pωq " N`r0, T s, ω, 1{p4L 0 q˘. Like in the first step of the proof of Proposition 4.2 (see in particular footnote 6 ), we start from the estimate for a universal constant γ ě 1. Modifying the constant γ if necessary, we may easily change s into t i in the first three lines of the right-hand side. We obtain wpt i , t i`1 , ωq 1{p F`Xpωq, Y p¨q˘´F`X 1 pωq, Y 1 p¨q˘ ,rti,ti`1s,w,p . The first point is to bound the quantity F`Xpωq, Y p¨q˘´F`X 1 pωq, Y 1 p¨q˘ ‹,rti,ti`1s,w,p , which contains all the terms that appear in the above inequality.
As for the second term in (4.19), we write where the symbol " is used to denote independent copies of the various random variables and where, as before, we used the notation (3.10), with an obvious analogue for the processes tagged with a prime or a tilde. By using Hölder inequality with exponents 3 and 3/2, we get , where, to get the first line, we used the boundedness and continuity assumptions of the functions B x D µ F, B z D µ F and D 2 µ F. Up to the exponent 4 appearing on the first and last lines of the right-hand side, we end up with the same bound as in the analysis of r∆F pω,¨qs s,t in the first step, namely @ δ µ r∆Fpω,¨qs D rti,ti`1s,w,p,4{3 ď γ´~∆Xpωq~r ti,ti`1s,w,p`@~∆ Y p¨q~r ti,ti`1s,w,p D 8γ wp0, t i , ωq 1{p´~∆ Xpωq~r 0,tis,w,p`@~∆ Y p¨q~r 0,tis,w,p D 8¯.
Step 4. We use (3.11) to write the remainder term R ∆F in the form and similarly for (2'), (3') and (5'), putting a prime on all the occurrences of X and Y . We start with the first four lines in R ∆F . Doing as before, the first line is less thaň .
This completes the proof.

Well-posedness
We first prove a well-posedness result in small time from which Theorem 1.1 follows.
Recall from Definition 4.1 the fact that the map Γ depends on X 0 pωq.  A" γ´1`wp0, T,¨q 1{p¯ı N pr0,Ss,¨,1{p4Lqq E 32 ď η, (4.24) and for any d-dimensional random square-integrable variable X 0 , there exists a random controlled path Xp¨q " pXpωqq ωPΩ defined on the time interval r0, Ss satisfying @ δ x Xp¨q D 8 ď Λ, and @~X p¨q~r 0,Ss,w,p D 8 ă 8 (the bound for the latter only depending on Λ and the parameters in (4.22)), such that, for every ω P Ω, the paths Xpωq and Γpω, Xpωq, Xp¨qq coincide on r0, Ss. Any other random controlled path X 1 p¨q with X 1 0 " X 0 almost surely, and such that the paths X 1 pωq and Γ`ω, X 1 pωq, X 1 p¨q˘coincide almost surely, satisfies P´~Xp¨q´X 1 p¨q~‹ ,r0,Ss,w,p " 0¯" 1. Proof. We construct a fixed point of Γ, see Definition 4.1, as the limit of the Picard sequence`X n`1 pωq; δ x X n`1 pωq; 0: " Γ´ω,`X n pωq; δ x X n pωq; 0˘,`X n pω 1 q; δ x X n pω 1 q; 0˘ω 1 PΩ¯, (4.25) started from`X 0 pωq; B x X 0 pωq; 0˘"`X 0 pωq; 0; 0˘, for each ω P Ω. By induction, for any n ě 0, the pair pXpωq, Y p¨qq " pX n pωq, X n p¨qq satisfies (4.1) in the statement of Xpωq " X n pωq satisfies (4.3) for any n ě 1, provided that L therein is taken large enough (independently on n). By (4.4) and from the tail estimates (4.22), we deduce that, for any n ě 0,~X n p¨q~r 0,T s,w,p has finite moments of any order: According to Definition 3.2, each X n p¨q " pX n pωqq ωPΩ , n ě 1, is a random controlled trajectory.
Step 1. Instead of working with S such that @ N pr0, Ss¨, 1{p4L 0 qq D 8 ď 1, we directly assume that @ N pr0, T s,¨, 1{p4L 0 qq D 8 ď 1, with L 0 as in Proposition 4.2. Recalling that we may take L 0 large enough so that (4.3) holds true with L " L 0 and X " X n for any n ě 0, we deduce that, for any n ě 1, both X n and X n´1 satisfy (4.13) and (4.14): (4.13) follows from the third item in the conclusion of Proposition 4.2, whilst (4.14) follows from the first item. Hence, by Proposition 4.3, ∆X n pωq rti,ti`1s,w,p , with ∆X n :" X n`1´X n is bounded above by γ wp0, t i , ωq 1{p´1`1 4L¯! ∆X n´1 pωq r0,tis,w,p`A~∆ X n´1 p¨q~r 0,T s,w,p for any n ě 1, where γ depends on L 0 and Λ, L is greater than L 0 , and the sequence pt i q i"0,¨¨¨,N is as in the statement of Proposition 4.3. The precise value of L will be fixed later on; the key fact is that it may be taken as large as needed. We start with the case i " 0. The above bound yields, for all n ě 1, ∆X n pωq r0,t1s,w,p ď 3γ 4L ! ∆X n´1 pωq r0,t1s,w,p`A~∆ X n´1 p¨q~r 0,T s,w,p ) .
Clearly, A S ď p3γ 2 ζ T q 2N pr0,Ss,¨,1{p4Lqq , since γ and ζ T are greater than 1. Since the term N`r0, Ss,¨, 1{p4Lq˘tends to 0 with S, we have lim SOE0 @`3 where δpSq ą 0 tends to 0 with S. So, we have Assuming that 3γ{p4Lq ď 1{16 and choosing S small enough, we may assume that a :" 1`δpSq 1´a3γ{p4Lq´3 we can find a positive constant C such that In order to conclude, we notice the following two facts. First, the above estimate remains true if we replace ∆X n pωq r0,Ss,w,p by ∆X n pωq ‹,r0,Ss,w,p in the left-hand side. Second, Proposition 4.2 guarantees that @~X 1 p¨q~r 0,Ss,w,p D 16 ă 8. Using a Cauchy like argument, we deduce that, for any ω P Ω, the sequence`X n pωq, B x X n pωq, R X n pωq˘n ě0 is convergent for the norm~¨~‹ ,r0,Ss,w,p . Using Proposition 4.3, the limit is a fixed point of Γ.
Uniqueness. Let`X 1 p¨q; δ x X 1 p¨q; 0˘stand for another fixed point of Γ, with δ x X 1 pωq " F`X 1 pωq, X 1 p¨q˘, for almost every ω P Ω, together with x~X 1 p¨q~r 0,T s,w,p D 8 ă 8. In particular, we have @ δ x X 1 p¨q D 8 ď Λ. Allowing the value of the constant L 0 to increase, we can assume that @~X 1 p¨q~r 0,T s,w,p D 2 8 ď L 0 . We can also assume that, for P-a.e. ω, X 1 pωq Therefore, we can apply Proposition 4.3 in order to compare X and X 1 and then duplicate the analysis of the convergence sequence, replacing ∆X n by ∆X :" X´X 1 . Similar to (4.27) (recalling that X 1 therein is understood as ∆X 0 ), ∆Xpωq r0,T s,w,p is bounded above bý Letting n tend to 8, this yields ∆Xpωq r0,T s,w,p ď`3γ 2 ζpωq˘2 N pω,1{p4Lqq 3γ{p4Lq 1´3γ{p4Lq Taking the L 8 norm, replacing T by S as in the third step and recalling from (4.29) that ? 3γ{p4Lq 1´?3γ{p4Lq @`3 γ 2 ζ T˘2 N pr0,Ss,¨,1{p4Lqq D 16 ă 1, we get uniqueness in small time.
Application to the proof of Theorem 1.1. Applying iteratively Theorem 4.4 along a sequence pS 0 " 0,¨¨¨, S " T q (shifting in an obvious way r0, S 1 s into rS 1 , S 2 s,¨¨¨) we get existence and uniqueness on the whole interval r0, T s. We notice that, at each node pS j q j"1,¨¨¨, of the subdivision, xX Sj p¨qy 2 ď xX Sj´1 p¨qy 2`2 x~X~r Sj´1,Sj s,w,p y 4 xwp0, T,¨qy 4 , which is finite by a straightforward induction. By sticking the paths constructed on each subinterval of the subdivision, we indeed obtain a random controlled path on the entire r0, T s. This is Theorem 1.1. Importantly, uniqueness holds whatever the choice of w in (2.8) and (2.9): If X and X 1 are two solutions, driven by different w and w 1 , then we may easily work with w`w 1 , which also satisfies (2.8) and (2.9). The control w`w 1 and the accumulation N pw`w 1 q 1{p also satisfy (4.22), see for instance (A.1) for a simple bound on the local accumulation associated to the sum of two different controls w and w 1 .

Uniqueness in law on strong rough set-ups
Since the solution given by Theorem 4.4 is constructed by Picard iteration on each interval rS j´1 , S j s, for j " 1,¨¨¨, , we should expect its law to be somehow independent of the probability space used to build the rough set-up W . Recall indeed from (3.3) the following expansion, which holds true for any rank n in the Picard iteration (4.25) and for any subdivision 0 " t 0 ă¨¨¨ă t K " T , pωq, X n tj´1 p¨q˘´F`X n tj´1 pωq, X n tj´1 p¨q˘W tj´1,tj pωqī ÿ j"1 A D µ F`X n tj´1 pωq, X n tj´1 p¨q˘`X n tj´1 p¨q˘´F`X n tj´1 p¨q, X n tj´1 p¨q˘W K K tj´1,tj p¨, ωq¯È i ÿ j"1 S n`1 tj´1,tj pωq; the last term converging to 0 as the step size of the subdivision tends to 0. In the second line, the matrix product B x F`X n s pωq, X n s p¨q˘`F`X n s pωq, X n s p¨q˘W s,t pωq˘should be understood as`ř d "1 ř m j,k"1 B x F i,j`X n s pωq, X n s p¨q˘`F ,k`X n s pωq, X n s p¨q˘W k,j s,t pωq˘˘i "1,¨¨¨,d and similarly for the term on the third line. Our guess is that the above expansion should permit to identify the law of X n`1 and, passing to the limit, to express in a somewhat canonical manner the law of the solution of the mean field rough equation in terms of the law of the rough set-up. However, although it seems to be a relevant concept in our context, uniqueness in law requires some care as the rough set-up explicitly depends upon the underlying probability space pΩ, F, Pq; recall indeed that the random variables Ω Q ω Þ Ñ W K K pω,¨q and Ω Q ω Þ Ñ W K K p¨, ωq are not only defined on pΩ, F, Pq but also take values in L q pΩ, F, P; R m q.
The fact that the arrival spaces of both random variables explicitly depend upon the probability space is a serious drawback to get a form of weak uniqueness. It is thus relevant to identify the canonical information in the rough set-up that is needed to determine the law of the solution. Somehow, we encountered a similar problem in the example of a rough set-up given by Proposition 2.3. The difficulty therein is indeed to reconstruct the iterated integral W K K pω 1 , ωq from the observation of W pωq, W pω 1 q and Wpωq; in the proof of Proposition 2.3, this is made at the price of an extra source of randomness. Interestingly, things become trivial when W K K pω 1 , ωq can be (almost surely) written as the image of`W pωq, W pω 1 q˘by a measurable function. Fortunately, all the examples we may have in mind in practice enter in fact this simpler setting. For instance, both Examples 2.1 and 2.2 fall within this case. More generally, in the framework of Proposition 2.3, we can write W 2,1 as the almost sure image of`W 1 , W 2b y a measurable function from C`r0, T s; R m˘2 into C`S T 2 ; R m b R m˘, when, for a.e. ξ P Ξ, the quantity W 2,1 pξq can be approximated by the iterated integral of mollified versions of W 1 pξq and W 2 pξq, provided the mollification procedure defines a measurable map from Cpr0, T s; R m q into itself. The following proposition makes it clear. If for Q-a.e. ξ P Ξ, for all ps, tq P S T 2 , W 2,1 s,t pξq " lim nÑ8 ż t s´W 2,n r pξq´W 2,n s pξq¯b dW 1,n r pξq, then there exists a measurable function I from Cpr0, T s; R m q 2 into C`S T 2 ; R m b R m˘s uch that Q´!ξ P Ξ : W 2,1 pξq " I`W 2 pξq, W 1 pξq˘)¯" 1.
The scope of Proposition 5.1 is limited to so-called geometric rough paths, but the underlying principle is actually more general. This prompts us to introduce the following definition.

Definition 5.2.
A rough set-up, as defined in Section 2, is called strong if there exists a measurable mapping I from C`r0, T s; R m˘2 into C`S T 2 ; R m b R m˘s uch that P b2´ pω, ω 1 q P Ω 2 : W K K pω, ω 1 q " I`W pωq, W pω 1 q˘(¯" 1. may not fall within the scope of Proposition 5.1, since the latter is limited to geometric rough paths, see footnote 8 . Proposition 2.3 sheds a light on the rationale for the word strong in Definition 5.2. Here strong has the same meaning as in the theory of strong solutions to stochastic differential equations: The second level W 2,1 of the rough-path is a measurable function of pW 2 , W 1 q. In contrast, the general set-up considered in the statement of Proposition 2.3 may not be strong as W 2,1 may carry, in addition to pW 1 , W 2 q, an additional external independent randomization. If this additional randomization is not trivial, the set-up should be called weak, see again footnote 8 for a typical instance. Also, we refer the reader to Deuschel et al. [21] for a related use of the notion of strong set-up, although the terminology strong does not appear therein.
We now have all the ingredients to formulate a weak uniqueness property.
As the two set-ups have the same law, we can use the same mapping I in the representations (5.2) of W K K and of W K K,1 . Iterating on n in (5.1), the result easily follows by proving, at each rank, that the law of pW, W, X n q is uniquely determined.

Continuity of the Itô-Lyons map
As expected from a robust solution theory of differential equations, we have continuity of the solution with respect to the parameters in the equation, most notably the rough set-up itself. The next statement quantifies that fact. Theorem 5.4. Let F satisfy the same assumptions as in Theorem 4.4. Given a time interval r0, T s and a sequence of probability spaces pΩ n , F n , P n q, indexed by n P N, let, for any n, X n 0 p¨q :" pX n 0 pω n qq ωnPΩn be an R d -valued square-integrable initial condition and W n p¨q :"´W n pω n q, W n pω n q, W n,K K pω n , ω 1 n q¯ω n ,ω 1 n PΩn be an m-dimensional rough set-up with corresponding control w n , as given by (2.10), and local accumulated variation N n , for fixed values of p P r2, 3q and q ą 8. Assume that ‚ the collection`P n˝p |X n 0 p¨q| 2 q´1˘n ě0 is uniformly integrable; ‚ for positive constants ε 1 , c 1 and pε 2 pαq, c 2 pαqq αą0 , the tail assumption (4.22) holds for w n and N n , for all n ě 0; 8 A trivial example of rough set-up is given by the collection of real-valued rough paths W 1 pξq " W 2 pξq " 0, W 1,1 pξq " 0, W 2,1 s,t pξq " apξqpt´sq, ps, tq P S T 2 , for ξ in a probability space pΞ, G, Qq, where a is a real-valued random variable on pΞ, G, Qq. If a is deterministic and non-zero, the set-up is strong but is not geometric. If the support of a does not reduce to one point, then the set-up induced by`W 1 p¨q, W 2 p¨q, W 1,1 p¨q, W 2,1 p¨q˘is not strong. EJP 25 (2020), paper 21.
‚ associating a control v n with each W n p¨q as in (2.7), the functions`S T 2 Q ps, tq Þ Ñ xv n ps, t,¨qy 2q˘n ě0 are uniformly Lipschitz continuous, in the sense that, uniformly in n ě 0, sup ps,tqPS T 2 ,s "t xv n ps, t,¨qy 2q {pt´sq is finite. Assume also that there exist, on another probability space pΩ, F, Pq, a square integrable initial condition X 0 p¨q with values in R d and a strong rough set-up W p¨q :"´W pωq, Wpωq, W K K pω, ω 1 q¯ω ,ω 1 PΩ with values in R m , such that the law under the probability measure P b2 n of the random variable Ω 2 n Q pω n , ω 1 n q Þ Ñ`X n 0 pω n q, W n pω n q, W n pω n q, W n,K K pω n , ω 1 n q˘, seen as a random variable with values in the space R dˆC pr0, T s; R m qˆ CpS T 2 ; R m b R m q ( 2 , converges in the weak sense to the law of Ω 2 Q pω, ω 1 q Þ Ñ`X 0 pωq, W pωq, Wpω n q, W K K pω, ω 1 q˘. Then, W p¨q satisfies the requirements of Theorem 4.4 for some p 1 P pp, 3q and q 1 P r8, qq, with w therein being given by (2.10). Moreover, if X n p¨q, resp. Xp¨q, is the solution of the mean field rough differential equation driven by W n p¨q, resp. W p¨q, then X n p¨q converges in law to Xp¨q on Cpr0, T s; R d q.
The rationale for the framework and the assumptions used in the statement of Theorem 5.4 is two-fold. First, it allows for a proof based on compactness arguments; in particular, the proof completely bypasses any lengthy stability estimate of the paths with respect to the rough structure, which, in our extended framework, would be especially cumbersome. Also, this compactness argument is pretty interesting in itself and complements quite well Section 5.1 on weak uniqueness; noticeably, it allows the set-ups to be supported by different probability spaces. Second, our formulation of the continuity of the Itô-Lyons map turns out to be well-fitted to the applications addressed in our companion paper [4], see also Section 4 in the earlier version [5].
The assumption that the limiting rough set-up is strong is tailored-made to the compactness arguments we use below as it permits to pass quite simply to the weak limit along the laws of the rough set-ups pW n p¨qq ně0 and to identify the limiting law.
Proof. Throughout the proof, we call p P r2, 3q and q ą 8 the fixed indices used to define the set-ups and, in particular, to control the variations in the definition (4.22) of each w n , n ě 0, w n being associated with v n through (2.10). This is important because, at some points of the proof, we will use other values p 1 ą p and q 1 ă q.
Step 1. We prove key properties on the tightness of the sequence pW n p¨qq ně0 . 1a. For any n ě 0, we introduce the modulus of continuity of pW n p¨q, W n p¨q, W n,K K p¨qq, namely we let, for any δ ą 0, ς n`δ , ω n , ω 1 n˘: " sup |s´t|ďδ |W n t pω n q´W n s pω n q| sup |s´s 1 |`|t´t 1 |ďδˇW n s 1 ,t 1 pω n q´W n s,t pω n qˇˇ`sup |s´s 1 |`|t´t 1 |ďδˇW n,K K s 1 ,t 1 pω n , ω 1 n q´W n,K K s 1 ,t 1 pω n , ω 1 n qˇˇ, where pω n , ω 1 n q P Ω 2 n . Since the laws of the processes pW n p¨q, W n p¨q, W n,K K p¨,¨qq ně0 are tight in the space Cpr0, T s; R m qˆ CpS T 2 ; R m b R m q ( 2 , we deduce that @ε ą 0, lim δOE0 sup ně0 P b2 n´ pω n , ω 1 n q P Ω 2 n : ς n`δ , ω n , ω 1 n˘ě ε (¯" 0.
1b. We now prove that, for any q 1 P r8, qq, the laws of the processes`Ω n Q ω n Þ Ñ xW n,K K pω n ,¨qy q 1˘n ě0 are tight 9 , and similarly for the laws of the processes`Ω n Q ω n Þ Ñ 9 In the notation x¨y q 1 , the expectation is implicitly taken under Pn.
xW n,K K p¨, ω n qy q 1˘n ě0 . By (2.10), we have, for any ω n P Ω n , sup ps,tqPS T 2 @ W n,K K s,t pω n ,¨q D q ď`w n p0, T, ω n q˘2 {p .
By the second bullet point in the assumption, the tails of the right-hand side are uniformly dominated. So, lim AÑ8 sup ně0 P n´ ω n P Ω n : sup which is one first step in the proof of tightness. For any a ą 0, we now consider the event E n pδ, aq :" ! ω n P Ω n : P n´ ω 1 n P Ω n : ς n pδ, ω n , ω 1 n q ě ε (¯ě a ) .
Take now ω n P E n pδ, a ε pδqq A such that sup ps,tqPS T 2 @ W n,K K s,t pω n ,¨q D q ď A, for a given A ą 0. Then, for any q 1 P r8, qq and ps, tq, ps 1 , t 1 q P S T 2 with |s´s 1 |`|t´t 1 | ď δ,ˇˇ@ A W n,K K s 1 ,t 1 pω n ,¨q´W n,K K s,t pω n ,¨q For A fixed and δ small enough, the right-hand side is less than 2ε. We easily deduce that, for any ε ą 0, lim δOE0 sup ně0 P nˆ! ω n P Ω n : sup which, together with (5.3), proves tightness. Clearly, the same holds for the family`Ω n Q ω n Þ Ñ xW n,K K p¨, ω n qy q 1˘n ě0 . Similarly, the two deterministic functions`xW n p¨qy q 1˘n ě0 and ⟪W n,K p¨,¨q⟫ q 1˘n ě0 are relatively compact in Cpr0, T s; Rq and CpS T 2 ; Rq.
1c. For each coordinate of the family of processeś Ω n Q ω n Þ Ñ`|W n s,t pω n q|, |W n s,t pω n q|,

T¯ně0
, we know that the corresponding family of laws is tight in CpS T 2 ; Rq and that the associated family of p-variations over r0, T s has tight laws in R (because of the second item in the assumption). Hence, we can apply Lemma 5.5 below, with any p 1 P pp, 3q instead of p itself, and with Z n s,t pωq equal to one of the coordinate of the above process.
We proceed in the same way with the coordinates of the deterministic sequencè z n s,t "`@W n s,t p¨q D q 1 , ⟪W n,K K s,t p¨,¨q⟫ q 1˘p s,tqPS T 2˘n ě0 . We deduce that, for any p 1 P pp, 3q, the sequence of probability measures´P˝pS T 2 Q ps, tq Þ Ñ v n,1 ps, t,¨qq´1¯n where v n,1 is associated with W n p¨q through (2.7) using the pair of parameters pp 1 , q 1 q instead of pp, qq. 1d. Obviously, v n,1 ps, t,¨q ď pv n ps, t,¨qq p 1 {p . Since p 1 {p ď 2 and the function S T 2 Q ps, tq Þ Ñ xv n ps, t,¨qy 2q is Lipschitz continuous, uniformly in n ě 0, we deduce that ps, tq Þ Ñ xv n,1 ps, t,¨qy q is Lipschitz continuous, uniformly in n ě 0. Hence, @ε ą 0, lim δÑ0 sup ně0 P nˆs up ps,tqPS T 2 :t´sďδ w n,1 ps, t,¨q ą ε˙" 0, where, as above, w n,1 is associated with v n,1 and pp 1 , q 1 q through (2.10). Importantly, we deduce from the bound pv n,1 p0, T,¨qq 1{p 1 ď pv n p0, T,¨qq 1{p that, similar to w n and N n (the latter is associated with w n through (2.14)), the function w n,1 and the corresponding local accumulated variation N n,1 (given by (2.14) with " w n,1 ) satisfy the tail assumption (4.22), uniformly in n ě 0. The bound on the tails of N n,1 is easily obtained by comparison with the tails of N n .
Step 2. 2a. The next step is to observe, as a corollary of the proof of Theorem 4.4, see (4.30), that there exist a constant C and a real S ą 0 such that, for all n ě 0, A~X n p¨q~r 0,Ss,w n,1 ,p 1 E 8 ď C.
The fact that C and S can be chosen independently of n is a consequence of the fact that the tails of N n and w n are controlled uniformly in n ě 0. Here S is chosen small enough so that (4.23) and (4.24) in the statement of Theorem 4.4 are satisfied, uniformly in n ě 0.
2b. Arguing as in the derivation of Theorem 1.1 from the statement of Theorem 4.4, we can iterate the argument and construct a sequence of deterministic times 0 " S 0 ă S " S 1 ă . . . ă S K " T , for some deterministic K ě 1, such that, for all n ě 0 and all j P t0,¨¨¨, K´1u, @~X n p¨q~r Sj ,Sj`1s,w n,1 ,p 1 D 8 ď C. Up to a modification of the constant C, we deduce that, for all n ě 1, @~X n p¨q~r 0,T s,w n,1 ,p 1 D 8 ď C. Recalling that`P n˝p |X n 0 p¨q| 2 q´1˘n ě0 is uniformly integrable, it is easily checked that P n˝p sup 0ďtďT |X n t p¨q| 2 q´1˘n ě0 is also uniformly integrable. 2c. As another result of the previous step, for any ε ą 0, we can find a ą 0 such that sup ně0 P n´~X n p¨q~r 0,T s,w n,1 ,p 1 ą a¯ď ε, from which, we deduce that @a ą 0, Dε ą 0 : sup ně0 P n´@ ps, tq P S T 2 , |X n s,t | p 1 ą aw n,1 ps, tq¯ď ε.
From the conclusion of 2b, the sequence`P n˝p X n p¨qq´1˘n ě0 is tight in C`r0, T s; R d˘.
Step 3. 3a. As a consequence of the assumptions of Theorem 5.4 and of Step 2, we have the following tightness properties: ‚`P n˝p W n p¨qq´1˘n ě0 and`P n˝p X n p¨qq´1˘n ě0 are tight in the spaces C`r0, T s; R mȃ nd C`r0, T s; R d˘; EJP 25 (2020), paper 21.
‚`P n˝p W n q´1p¨q˘n ě0 is tight in C`S T 2 ; R m b R m˘; ‚´P b2 n˝´Ω 2 n Q pω n , ω 1 n q Þ Ñ W n,K K pω n , ω 1 n q P CpS T 2 ; R m b R m q¯´1¯n ě0 is tight in C`S T 2 ; R m b R m˘; ‚´P n˝´v n,1 pω n q : Ω n Q ω n Þ Ñ`S T 2 Q ps, tq Þ Ñ v n,1 ps, t, ω n q˘P CpS T 2 ; Rq¯´1¯n ě0 is tight in C`S T 2 ; R˘; 3b. By Skorokhod's representation theorem, we can find an auxiliary Polish probability space`p Ω, p F, p P˘, such that, up to a subsequence, for p P-a.e. p ω P p Ω, lim nÑ8´x W n,1 pp ωq, x W n,2 pp ωq, x W n,1,1 pp ωq, x W n,2,1 pp ωq, p v n,1,1 pp ωq, p v n,2,1 pp ωq, p X n,1 pp ωq, p X n,2 pp ωq"´x W 1 pp ωq, x W 2 pp ωq, x W 1,1 pp ωq, x W 2,1 pp ωq, p v 1,1 pp ωq, p v 2,1 pp ωq, p X 1 pp ωq, p X 2 pp ωq¯, (5.4) where`x W n,1 , x W n,2 , x W n,1,1 , x W n,2,1 , p v n,1,1 pp ωq, p v n,2,1 pp ωq, p X n,1 pp ωq, p X n,2 pp ωq˘has the same law as the random variable Ω 2 n Q pω n , ω 1 n q Þ Ñ´W n pω n q, W n pω 1 n q, W n pω n q, W n,K K pω 1 n , ω n q, v n,1 pω n q, v n,1 pω 1 n q, X n pω n q, X n pω 1 n q¯, which takes values in the space Cpr0, T s; Cpr0, T s; R d q ( 2 , and where`x W 1 p¨q, x W 2 p¨q, x W 1,1 p¨q, x W 2,1 p¨q, X 1 0 p¨q˘has the same law as the random variable Ω 2 Q pω, ω 1 q Þ Ñ´W pωq, W pω 1 q, Wpωq, W K K pω 1 , ωq, X 0 pωq¯. (5.5) 3c. At this point of the proof, the difficulty is that`x W 1 p¨q, x W 2 p¨q, x W 1,1 p¨q, x W 2,1 p¨q˘does not form a rough set-up. Still, we have the following two properties. First, using the fact that the limiting set-up is strong, we have p P´!p ω P p Ω : x W 2,1 pp ωq " I`W 2 pp ωq, W 1 pp ωq˘)¯" 1, for a measurable mapping I : Cpr0, T s; R m q 2 Ñ CpS T 2 ; R m b R m q, which follows from the identification with the law of (5.5). Also, passing to the limit in Chen's relations satisfied by each W n , we have, for p P-a.e. p ω P p Ω, and all 0 ď r ď s ď t ď T , x W 1,1 r,t pp ωq " x W 1,1 r,s pp ωq`x W 1,1 s,t pp ωq`x W 1 r,s pp ωq b x W 1 s,t pp ωq, x W 2,1 r,t pp ωq " x W 2,1 r,s pp ωq`x W 2,1 s,t pp ωq`x W 2 r,s pp ωq b x W 1 s,t pp ωq.
Obviously, p x W 2 , p X 2 q is independent of`x W 1 , x W 1,1 , p X 1 , p v 1,1˘. Following the proof of Proposition 2.3, but in a simpler setting here since the limiting rough set-up is strong, we can find ‚ four random variables x W p¨q, x Wp¨q, p v 1 p¨q and p Xp¨q from`p Ω, p F, p P˘into Cpr0, T s; R m q, C`S T 2 ; R m b R m˘, C`S T 2 ; R˘and Cpr0, T s; R d q such that p P´!p ω P p Ω :`x W , x W, p v 1 , p X˘pp ωq "`W 1 , W 1,1 , p v 1,1 , p X 1˘p p ωq )¯" 1; ‚ a random variable x W K K p¨,¨q from`p Ω 2 , p F b2 , p P b2˘i nto C`S T 2 ; R m b R m˘s uch that p P b2´! pp ω, p ω 1 q P p Ω 2 : x W K K pp ω, p ω 1 q " I`x W pp ωq, x W pp ω 1 q˘)¯" 1; D q 1 is σt x W p¨qu-measurable, we get, for any ps, tq P S T 2 and for a.e. p ω P p Ω, x x W K K s,t pp ω,¨q D p 1 q 1 ď p v 1 ps, t, p ωq. Obviously, the latter is true for a.e.ω, for any ps, tq P S T 2 X Q 2 . By almost sure (in pp ω, p ω 1 q) continuity of the paths S T 2 Q ps, tq Þ Ñ x W K K s,t pp ω, p ω 1 q and by Fatou's lemma, we deduce that it holds true for a.e. p ω, for any ps, tq P S T 2 . The same holds for x x W K K s,t p¨, p ωq D q 1 .
Associating with the rough set-up x W a (random) control function s v 1 through the definition (2.7) with pp, qq replaced by pp 1 , q 1 q, we deduce that, for p P-a.e. p ω P p Ω, for all ps, tq P S T 2 , s v 1 ps, t, p ωq is less than p v 1 ps, t, p ωq.
accumulation associated with each of these six terms. To make it clear, we have the following property. For a given threshold α ą 0 and for any two nondecreasing continuous functions v 1 : S T 2 Ñ R`and v 2 : S T 2 Ñ R`, set N i pαq :" N vi`r 0, T s, α˘, for 1 ď i ď 2, and N pαq :" N v1`v2`r 0, T s, α˘; see (2.14) for the original definition. Then max´N 1´α 2¯, N 2´α 2¯¯ě N pαq. (A.1) For sure, the result is true with the first and third terms in (2.7) as this fits the original property established in [12]. Also, it is obviously true for the second and sixth terms since they are completely deterministic. Hence, the only difficulty is to control the local accumulation associated with the fourth and fifth terms. The strategy is as follows. As we work with Gaussian rough paths, the set-up, as defined in Section 2, is strong. So, we can transfer it to any arbitrarily fixed probability space (provided that the letter is rich enough). Hence, we can choose Ω as the path space W, see the notation in the statement of Theorem 2.4.
We denote by W pω, ω 1 q the enhanced Gaussian rough path associated to`W pωq, W 1 pω 1 q˘along the lines of Example 2.2, for P b2 -a.e. pω, ω 1 q P Ω 2 . The second level of W pω, ω 1 q reads W r2s pω, ω 1 q :"ˆW pωq I`W pωq, W 1 pω 1 qȊ`W 1 pω 1 q, W pωq˘Wpω 1 q˙, where I is as in Definition 5.2, and where we used the same symbol W as in Section 2 for the enhanced path although the meaning here is not exactly the same. Here, W pω, ω 1 q is a function of both ω and ω 1 and takes values in R 2m ' pR 2m q b2 . Following Section 3 in [12], see also (11.5) in [23], we define, for h ' k P H ' H the translated rough path pT h'k W qpω, ω 1 q, where, as in Example 2.2, H is the underlying Cameron-Martin space. We then recall that, with probability 1 under P b2 , T h'k W pω, ω 1 q " W pω`h, ω 1`k q.
Following the argument given in Proposition 6.2 in [12], see also Lemma 11.4 in [23], we have, for any h P H and any ps, tq P S T 2 , W pω, ω 1 q p rs,ts,p´var ď c´ T h'0 W pω, ω 1 q p rs,ts,p´var`} h} p rs,ts, ´var¯, where we recall that 1{p`1{ ą 1 and c only depends on p and , and where W pω, ω 1 q rs,ts,p´var :" }pW, W 1 qpω, ω 1 q} rs,ts,p´var`b }W r2s pω, ω 1 q} rs,ts,pp{2q´var , and similarly for T h'0 W pω, ω 1 q rs,ts,p´var . Taking the power q, allowing the constant c to depend on q and integrating with respect to ω 1 , we get We now let W pω, ω 1 q rs,ts,p1{pq´Höl :" }pW, W 1 qpω, ω 1 q} rs,ts,p1{pq´Höl`b }W r2s pω, ω 1 q} rs,ts,p2{pq´Höl , for the standard Hölder semi-norm of the rough path, see Theorem 11.9 in [23]. Observe that if the left-hand side is equal to or less than α, the above statement remains true even if }h} rs,ts, ´var ą 1; it suffices to change the constant c accordingly. Define now N pr0, T s, ω, αq :" N pr0, T s, αq, when ps, tq " @ W where }¨} H is the standard norm on the reproducing Hilbert space H, see again for instance Appendix D in [24]. We conclude by recalling that the quantity ⟪ W p¨,¨q p r0,T s,p1{pq´Höl ⟫ q is finite, by observing that E :" ! pω, ω 1 q P Ω 2 : T h'0 W pω, ω 1 q " W pω`h, ω 1 q, h P H ) , is of full P b2 -probability measure, see Theorems 11.5 and 11.9 in [23], and then by invoking Theorem 11.7 in [23].