A discrete approach to Rough Parabolic Equations

By combining the formalism of \cite{RHE} with a discrete approach close to the considerations of \cite{Davie}, we interpret and solve the rough partial differential equation $dy_t=A y_t \, dt+\sum_{i=1}^m f_i(y_t) \, dx^i_t$ ($t\in [0,T]$) on a compact domain $\mathcal{O}$ of $\R^n$, where $A$ is a rather general elliptic operator of $L^p(\mathcal{O})$ ($p>1$), $f_i(\vp)(\xi):=f_i(\vp(\xi))$ and $x$ is the generator of a $2$-rough path. The (global) existence, uniqueness and continuity of a solution is established under classical regularity assumptions for $f_i$. Some identification procedures are also provided in order to justify our interpretation of the problem.


Introduction
The rough paths theory introduced by Lyons in [17] and then refined by several authors (see the recent monograph [12] and the references therein) has led to a very deep understanding of the standard rough systems where σ i j : R → R is a smooth enough vector field and x : [0, T ] → R m is a so-called rough path, that is to say a function allowing the construction of iterated integrals (see Assumption (X) γ for the definition of a 2-rough path and [18] for a rough path of any order). The theory provides for instance a new pathwise interpretation of stochastic systems driven by very general Gaussian processes, as well as fruitful and highly non-trivial continuity results for the Itô solution of (1), i.e., when x is a standard Brownian motion.
One of the new challenges of the rough paths theory now consists in adapting the machinery to infinite-dimensional (rough) equations that involves a non-bounded operator, with, as a final objective, the possibility of new pathwise interpretations for stochastic PDEs. Some progresses have recently been made in this direction, with on the one hand the viscosity-solution approach due to Friz et al (see [2,3,10,9]) and on the other hand, the development of a specific algebraic formalism by Gubinelli et al (see [14,15,8]).
The present paper is a contribution to this global project. It aims at providing, in a concise and self-contained formulation, the analysis of the following rough evolution equation: where A is a rather general elliptic operator on a bounded domain of R n (see Assumptions (A1)-(A2)), f i (ϕ)(ξ) := f i (ϕ(ξ)) and x generates a m-dimensional 2-rough path (see Assumption (X) γ ).
Although the global form of (2) is quite similar to the equation treated in [8], several differences and notable improvements justify the interest of our study: (i) The equation is here analysed on a compact domain of R n . This allows to simplify the conditions relative to the vector field f i , which reduce to the classical assumptions of rough paths theory, ie k-times differentiable (k ∈ N * ) with bounded derivatives (see Assumption (F) k ).
(ii) The conditions on p are less stringent than in [8], where p has to be taken very large. It will here be possible to show the existence and uniqueness of a solution in L p ( ) (for a smooth enough initial condition ψ) as soon as p > n (see Theorem 2.11). In particular, we can go back to the Hilbert framework of [15] for the one-dimensional equation (n = 1, p = 2).
(iii) Last but not least, the arguments we are about to use lead to the existence of a global solution for (2), defined on any time interval [0, T ]. This is is a breakthrough with respect to [15,8], where only local solutions are obtained, on a time interval that depends on the data of the problem, namely x, f and ψ.
In order to reach these three improvements, the strategy will combine elements of the formalism used in [8] with a discrete approach of the equation, close to the machinery developped in [6] for rough standard systems. A first step consists of course in giving some reasonable sense to Equation (2). We have chosen to work with an interpretation à la Davie, derived from the expansion of the ordinary solution (see Definition 2.6), and we have left aside the sewing map at the core of the constructions in [8]. Note however that the expansion under consideration here relies on the operator-valued paths X x,i , X ax,i , X x x,i j which were identified in [8] (see Subsection 2.3), and which plays the role of an infinite-dimensional rough path adapted to the problem. When applying the whole procedure to a differentiable driving path x (resp. a standard Brownian motion), the solution that we retrieve coincides with the classical solution (resp. the Itô solution), as reported in Subsection 2.4. Together with the continuity statement of Theorem 2.12, this identification procedure allows to fully justify our interpretation of (2) (see Corollary 2.13 and Remark 2.14).
Once endowed with this interpretation, our solving method is based on a discrete approach of the problem: as in [6], the solution is obtained as the limit of a discrete scheme the mesh of which tends to 0. Nevertheless, some fundamental differences arise when trying to mimic the strategy of [6].
To begin with, the middle-point argument at the root of the reasoning in the diffusion case (see the proof of [6, Lemma 2.4]) cannot take into account the space-time interactions that occur in the study of PDEs, i.e., the classical estimates (22) and (23). Therefore, the argument must here be replaced with a little bit more complex algorithm described in Appendix A, and which will be used throughout the paper. Let us also mention that the expansion of the vector field f i (ϕ)(ξ) := f i (ϕ(ξ)) is not as easy to control as in the standard finite-dimensional case, even if one assumes that the functions f i : R → R are very smooth. Observe for instance that if W α,p (α ∈ (0, 1)) stands for the fractional Sobolev space likely to accomodate the solution path, and if f i is assumed to be differentiable, bounded with bounded derivative, then one can only rely on the non-uniform estimate (see [23]) Consequently, more subtle patching arguments must be put forward so as to exhibit a global solution. The strategy involves in particular a careful examination of the dependence on the initial condition at each step of the procedure (see for instance the controls (45) and (46)).
The paper is structured as follows: In Section 2, we gather all the elements that allow to understand our interpretation of Equation (2), and we state the three main results of the paper, namely Theorems 2.10-2.12. The three sections that follow are dedicated to the proof of each of these results, with the existence theorem first (Section 3) and then the uniqueness (Section 4) and continuity (Section 5) results. Finally, Appendix A contains the description and the analysis of the algorithm at the root of our machinery, while Appendix B is meant to provide the details relative to the identification procedure in the Brownian case (see Proposition 2.9).
For the sake of clarity, we shall only consider Equation (2) on the generic interval [0, 1]. It is however easy to realize that the whole reasoning remains valid on any (fixed) finite interval [0, T ] at the price of very minor modifications.
Throughout the paper, we will denote by k,b (R; R l ) (k, l ∈ N * ) the set of R l -valued functions which are k-times differentiable with bounded derivatives.
Finally, we will use the classical convention for the summation over indexes x i y i = i x i y i , whenever the underlying index set is obvious from the context.

Interpretation of the equation
We first give some precisions about the setting of our study, as far as the operator A, the driving path x and the vector field f i are concerned (Subsection 2.1). Then we introduce the notation and the tools designed for our analysis (Subsections 2.2 and 2.3), and which enable us to interpret (2) (Subsection 2.4). We finally state the three main results of the paper (Subsection 2.5), and we discuss some possible extensions of the strategy to rougher driving paths (Subsection 2.6).

Assumptions
As it was announced in the introduction, we mean to tackle the equation ) and x is a Hölder path. More precisely, to be in a position to interpret and solve this equation, we will be led to assume that (some of) the following conditions are satisfied:

Assumption (A1):
A generates an analytic semigroup of contraction S on any L p ( ). Under this hypothesis, we will denote S ts := S t−s (s ≤ t), p := L p ( ), α,p := Dom(A α p ), and we endow the latter space with the graph norm ϕ α,p := A α p ϕ L p ( ) . We also assume that for any function g ∈ 1,b (R; R), there exists a constant c 1 g such that and for any function g ∈ 2,b (R; R), there exists a constant c 2 g such that where, in (3) and (4), g(ϕ) is just understood in the composition sense, i.e., g(ϕ)(ξ) := g(ϕ(ξ)).

Assumption (A2):
If 2αp > n, then α,p is a Banach algebra continuously included in the space ∞ of continuous functions on .
Assumption (X) γ : x allows the construction of a 2-rough path for some (fixed) coefficient γ ∈ (1/3, 1/2). In other words, we assume that x is a γ-Hölder path and that there exists a 2-variable path x 2 (also called a Lévy area) such that for any 0 ≤ s ≤ u ≤ t ≤ 1, We will then denote where Before pondering over the plausibility of these conditions, let us precise that we henceforth focus on the mild formulation of Equation (2) This is a standard change of perspective for the study of (stochastic) PDEs (see [5]), which allows to use the regularizing properties of the semigroup. In retrospect, owing to the regularity assumptions on f , it will however be possible to make a link between the mild and strong interpretations of the equation, see Remark 2.14.
Application: Properties (A1)-(A2) are satisfied by any elliptic operator on L p ((0, 1) n ) that can be written as where c ≥ 0 and the functional coefficients a i j are bounded, differentiable with bounded derivatives on [0, 1] n . Indeed, under these assumptions, it is proven in [7] that A generates an analytic semigroup of contraction. Then, thanks to [20], one can identify the domain (A α p ) with the complex interpolation [L p , (A p )] α and one can use the result of [22] to assert that .
As far as Condition (X) γ is concerned, the process that we have in mind in this paper is the fractional Brownian motion B H with Hurst index H > 1/3, for which the (a.s) existence of a Lévy area has been established in [4]. Condition (X) γ is in fact satisfied by a larger class of Gaussian processes, as reported in [12].
In brief, under the above-stated regularity assumptions, the results that we are about to state and prove can be applied to the stochastic equation

Hölder spaces
We suppose in this subsection that Assumption (A1) is satisfied. In order to introduce the functional framework of our analysis, let us focus on the following consideration: we know that one of the most appropriate space for the study of rough standard systems is the set of Hölder paths { y : [0, 1] → R d : y t − y s ≤ c |t − s| γ } (see [13]), and this is (among others) due to the convenient expression for the variations of the solution y of (1), namely y t − y s = t s σ i j ( y u ) d x j u . Here, if we denote by y the solution of (5) (assume for the moment that x is a differentiable path), it is readily checked that for all s < t, With this observation in mind, the following notation arises quite naturally: Notation. For all paths y : (δ y) ts := y t − y s , (δ y) ts := (δ y) ts − a ts y s = y t − S ts y s , (δz) tus := z ts − z tu − S tu z us .
The (ordinary) system (5) can now be written in the convenient form To make the notation (7)-(8) even more legitimate in this convolutional context, we let the reader observe the following elementary properties: Like with the standard finite-dimensional systems, the rough-paths treatment of Equation (9) leans on the controlled expansion of the convolutional integral t s S tu d x i u f i ( y u ). To express this control with the highest accuracy, we are naturally led to consider the following semi-norms, that can be seen as adapted versions of the classical Hölder seminorms: if y : Thenˆ λ 1 ([a, b]; V ) stands for the set of paths y :

Infinite-dimensional rough path
By anticipating the proof of Proposition 2.8, we know that, when x is a differentiable path, the expansion of t s S tu d x i u f i ( y u ) puts forward the three following operator-valued paths constructed from x: A priori, these expressions do not make sense for a non-differentiable γ-Hölder (rough)-path x. An integration by parts argument, retrospectively justified by Lemmas 2.3 and 2.4, leads here to the general definition:

Definition 2.2. Under Assumptions (A1) and (X) γ , we define the three operator-valued paths X x,i , X ax,i and X x x,i j by the formulas
X x x,i j ts

If in addition Assumption
and we associate to every path y : [0, 1] → p the two quantities Proof. As aforementioned, this is just a matter of integration by parts. For instance, one has Observe now that the three expressions contained in (17) can also be directly interpreted as Itô integrals when x stands for a standard Brownian motion. This interpretation remains consistent with Definition 2.2:

Lemma 2.4. Suppose that x is m-dimensional Brownian motion defined on a complete filtered probability space (Ω, , P), and let x 2 be its Lévy area, understood in the Itô sense as the first iterated integral of x. Then, under Assumption (A1), the three identifications of the previous lemma remain valid in this context.
Proof. It suffices to replace the integration by parts argument with Itô's formula, upon noticing that only Wiener integrals are involved here. For X x x , we know indeed that for any fixed s, the process To end up with this subsection, let us highlight the regularity properties that will be at our disposal throughout the study: We will denote by X γ,α,κ the norm attached to X := (X x , X ax , X x x ) through Properties (18)- (20), that is to say ( ( α,p , α+κ,p )) .

Interpretation of the equation
Let us now turn to the interpretation of (9) for a generic 2-rough paths x = (x, x 2 ). Like in [6], our approach is based on the Taylor expansion of the ordinary mild equation. We first give the general definition of a solution and then we clarify this definition by considering the two previously-known situations, namely when x is a differentiable path and when x is a standard Brownian motion. Remember that the notation J y has been introduced in Definition 2.2.
Definition 2.6. Under Assumptions (A1), (X) γ and (F) 1 , for all λ ≥ 0 and ψ ∈ λ,p , we will call a solution in λ,p of the equation any path y : [0, 1] → λ,p such that y 0 = ψ and there exists two coefficients µ > 1, ǫ > 0 for which Remark 2.7. The reader familiar with the strategy of [6] will not be surprised by the condition 1]; λ,p ) may be less expected. In fact, due to the property (23), the fractional spaces λ,p naturally arise from the controlled expansion of (27)). Proposition 2.8. Suppose that x is a m-dimensional differentiable path, and let x 2 be its Lévy area, understood in the Lebesgue sense. We suppose that Assumptions (A1) and (F) 1 are both satisfied. Then, for all η ∈ (0, 1) and ψ ∈ η,p , the (ordinary) solution of Equation (24) is also a solution in η,p in the sense of Definition 2.6.
Proof. Let y be the ordinary solution of (24), with initial condition ψ ∈ η,p . Then y ∈ . Now, notice that owing to the identification (17), we get and so, due to (23), one has To complete the proof, observe that by resorting to the identification (17) and thus where we have used the trivial relation X x,i us = X ax,i us + (δx j ) us . From this expression, it is easy to show that M i us p ≤ c y |u − s| η , which leads to (25) with µ = 1 + η, ǫ = 1. Proposition 2.9. Suppose that x is a m-dimensional standard Brownian motion defined on a complete filtered probability space (Ω, , P), and let x 2 be its Lévy area, understood in the Itô sense. Suppose also that Assumptions (A1) and (F) 2 are both satisfied. Then, for all η ∈ (1/2, 1) and ψ ∈ η,p , the Itô solution of Equation (24) is almost surely a solution in η,p in the sense of Definition 2.6.
Proof. For the sake of clarity, we have postponed the proof of this result to Appendix B.
Together with the forthcoming uniqueness result contained in Theorem 2.11, the above-stated properties allow to identify, in the two reference situations (i.e., when x is a differentiable path and when x is a standard Brownian motion), the solution in the sense of Definition 2.6 with the classical solution. We will then lean on the continuity Theorem 2.12 to fully justify our interpretation of (24) (see Remark 2.14).

Main results
With the tools and the definitions we have just introduced, we are in a position to state the three main results of this paper, which successively provide the existence, uniqueness and continuity of the solution to (24).  3 are all satisfied, then for all γ ′ ∈ (1−γ, γ+1/2) and ψ ∈ γ ′ ,p , the solution y in γ ′ ,p given by Theorem 2.10 is unique. Moreover, for any there exists a constant c x,ψ, f ,β such that for all n, where y n stands for the path given by the discrete scheme (35). Theorem 2.12. Under the assumptions of Theorem 2.11, the solution of (24) is continuous with respect to the initial condition and the driving rough path. More precisely, if y (resp.ỹ) is the solution in γ ′ ,p associated to (x, x 2 ) (resp. (x,x 2 )), with initial condition ψ (resp.ψ), then for some functions C : (R + ) 4 → R + growing with its arguments.
Together with the identification result established in Proposition 2.8, these three theorems offer another perspective on the solution of Equation (24), which may be more in accordance with the formalism used in [12] for rough standard systems: Corollary 2.13. Under the assumptions of Theorem 2.11, suppose that ψ ∈ γ ′ ,p and let (x n ) n be a sequence of differentiable paths such that x −x n γ + x 2 −x 2,n 2γ → 0 as n tends to infinity, wherẽ x 2,n stands for the standard Lévy area constructed fromx n . Letỹ n be the (ordinary) solution of (24) associated to eachx n . If y is the solution of (24) given by Theorem 2.11, then as n tends to infinity.
Remark 2.14. Through the latter result, one can see that the exhibited solution y is a solution in the rough paths sense, that is to say a limit of ordinary solutions with respect to some particular topology (compare with [12,Definition 10.17]). In this context, y can legitimately be called a mild solution of (2), as a limit of classical mild solutions. Furthermore, it is worth noticing that given the regularity assumptions on f i , if we suppose in addition that the initial condition ψ belongs to the domain (A p ), then each (ordinary) mild solutionỹ n is also a strong solution (see [19, Theorem 6.1.6]). Consequently, if ψ ∈ (A p ), y can also be considered as a strong solution of (2), keeping in mind the topology of the underlying convergence result (30).

Extension to rougher paths
Before we turn to the proof of Theorems 2.10-2.12, let us say a few words about the possibility of extending these results to a rougher path x, or otherwise stated when the Hölder coefficient γ is smaller than 1/3.
Remember that for standard finite-dimensional rough systems, the results obtained by Davie in [6] have been generalized to any γ ∈ (0, 1) by Friz and Victoir ( [11]): essentially, the system (1) can be interpreted and solved provided that (i) the vector field σ i j is smooth enough and (ii) one is able to construct the iterated integrals of x up to the k-th order, where 1 k+1 < γ ≤ 1 k . As far as Equation (24) is concerned, let us first consider the next step of the procedure, which corresponds to 1 4 < γ ≤ 1 3 . For more simplicity, we assume that f i is infinitely differentiable with bounded derivatives. Suppose for the moment that x is a differentiable path, and let us introduce, on top of (X x , X ax , X x x ), the two additional operator-valued paths Let us also define F 1 By looking closely at these expressions, it is not difficult to realize that the arguments displayed in the forthcoming sections 3-5 can be adapted to the decomposition (31) so as to handle the case where γ ∈ ( 1 4 , 1 3 ] (compare for instance (32)-(33) with (40)-(43)). This supposes that the intermediate paths J y , K y , L y should be controlled with the respective topologies 4γ 2 ( p ) ∩ ǫ 2 ( γ ′ ,p ), 3γ 2 ( p ), 2γ 2 ( p ), and that the space parameter γ ′ should be picked in the (non-empty) interval (1 − γ, γ + 1/2), as in Theorems 2.10-2.12. This also supposes, in order to extend the path X x x x , that x allows the construction of a 3-rough path x = (x, We know that this assumption covers in particular the case of a fractional Brownian motion with Hurst index H > 1/4, see [4]. The situation gets more complicated as soon as γ < 1/4, since we can no longer pick γ ′ in the (now empty) interval (1 − γ, γ + 1/2), and this assumption played a fundamental role in our estimates. Indeed, on the one hand, the condition γ ′ > 1 − γ ensures that the order of the terms derived from (32) or (41) is greater than γ + γ ′ > 1, or otherwise stated that these paths can be considered as residual terms. On the other hand, the condition γ ′ < γ + 1/2 is used in some estimates like (47) to go from γ ′ ,p to 1/2,p and thus take profit of the linear control (3) (instead of the quadratic control (4)). Therefore, when γ < 1/4, some sharpness is to be lost in our estimates and the method under consideration in this paper would only provide us with a local solution, on a time interval linked to x, f and ψ. To overcome this difficulty, it may be useful to modify the path (X x , X ax , X x x , X ax x , X x x x , . . .) into a more appropriate trajectory, which would for instance includes mixed operators such as X x a,i Observe however that the extension of (34) to a generic γ-Hölder path x (with γ < 1) can no longer be done via an integration by parts argument (as in Lemma 2.3), which considerably increases the difficulty of the study.

Existence of a solution
This section is devoted to the proof of Theorem 2.10. Thus, we henceforth suppose that the assumptions of the theorem, namely (A1), (X) γ and (F) 2 , are all satisfied. Besides, we fix a parameter γ ′ ∈ (1 − γ, γ + 1/2) and an initial condition ψ ∈ γ ′ ,p .

Additional notation
We consider the sequence (Π n ) n of dyadic partitions of [0, 1] (i.e., t n k = k 2 n ) and we define the discrete path y n following the iteration formula: The path y n is then extended on [0, 1] by linear interpolation. For the sake of clarity, we will denote in this section J n := J y n and K n := K y n . Observe that owing to the very definition of y n , one has J n t n k+1 t n k = 0.
In the rest of the paper, we will also appeal to the discrete versions of the generalized Hölder norms introduced in Subsection 2.2. Thus, for any n ∈ N, we denote a, b n := [a, b] ∩ Π n and

Preliminary results on J n
We fix t n p < t n q ∈ Π n and we apply the algorithm described in Appendix A to the uniform partition {t n p , t n p+1 , . . . , t n q }. Set N := q − p, and so, for any k ∈ {p, . . . , q}, t n k = t n p + (k−p)(t n q −t n p ) N . We also denote by (Π n,m )  Once endowed with this decomposition, we can show the following result, which turns out to be the starting point of our reasoning: Proof. We use the notation of Appendix A. By refering to Remark 3.1, one easily deduces Then, if C n := [δJ n ; κ 3 ( t n p , t n q n ; γ ′ ,p )] + [δJ n ; µ 3 ( t n p , t n q n ; p )], one has thanks to Proposition 6.2. The second control (37) can be obtained with the same arguments, upon noticing that (65) entails in particular and also (δJ y ) tus = I tus + I I tus + I I I tus + I V tus , with Proof. Those are only straightforward expansions. For (38), we use the fact that if m ts := g ts h s , then (δm) tus = (δg) tus h s − g tu (δh) us , together with the algebraic relations that can be readily checked from the expressions (12) and (14). The expansion of δ( f i ( y)) us − (δx j ) us F i j ( y s ) which then leads to (39) has already been elaborated on in the proof of Proposition 2.8, see (28).

Existence of a solution
Thanks to the above preliminary results, we are first able to control J n on successive time intervals independent of n: Then there exists a time T 0 = T 0 (x, f , γ, γ ′ , µ, ǫ) > 0, T 0 ∈ Π n , such that for any k, and Proof. This is an iteration procedure over the points of the partition, for which we first focus on the case k = 0 in (45) and (46). Assume that both estimates hold true on 0, t n q n . Then, for any t ∈ 0, t n q n , one has, thanks to (18), (20) and (3), and so [ y n ; 0 One can also rely on the estimate K n ts p ≤ J n ts p + X x x,i j ts Now, from the decomposition (39), we easily deduce, for all 0 ≤ s < u < t ∈ 0, t n q+1 n , Indeed, one has for instance where the constant c x, f may of course vary from line to line. Consequently, On the other hand, it is readily checked from (38) that and therefore By using the estimate (36), we get It only remains to pick T 0 such that We can follow the same lines to show (46) from the estimate (37).
It is now easy to realize that the same reasoning (with the same constants) can be applied on the interval [T 0 , 2T 0 ] by replacing ψ with y n T 0 , and then on the interval [2T 0 , 3T 0 ], etc.

Uniqueness of the solution
In this section, we mean to prove Theorem 2.11. Accordingly, we assume that p > n and that Conditions (A1), (A2), (X) γ and (F) 3 are all checked. Let y be a solution of (24) in γ ′ ,p , for some (fixed) parameter γ ′ ∈ (1 − γ, 1/2 + γ), with initial condition ψ ∈ γ ′ ,p , and let y n be the path described by the scheme (35), with the same initial condition ψ.
We introduce, for all s < t ∈ Π n , the quantity Proof. Going back to the notation of Subsection 3.
Lemma 4.2. Setμ := inf(γ + γ ′ , 3γ). Then for all s < t ∈ Π n , Proof. (58) is a consequence of the decomposition (39). Indeed, one has for instance, if y : where we have used the continuous inclusion γ ′ ,p ⊂ ∞ . As for (59), it suffices to observe, with the expression (38) in mind, that for instance, due to the assumption (A2) and the control (4), one has ≤ c x, f , y , y n |t − s| γ [ y − y n ; (I)] ≤ c x, f , y ,ψ |t − s| γ [ y − y n ; (I)], where, to get the last estimate, we have appealed to the uniform control y n ≤ c x, f ,ψ established in the proof of Theorem 2.10.
Proof of Theorem 2.11. Let T 1 ≤ 1 ∈ Π n . Writê and use the two previous lemmas to deduce first and secondly [ y − y n ; 0 and we have thus proved that Choose T 1 such that c 1 y,x, f ,ψ T γ 1 = 1 2 to obtain [ y − y n ; ( 0, T 1 n )] ≤ 2c 1 By using the same arguments on kT 1 , (k + 1)T 1 n , we get [ y − y n ; ( kT 1 , (k + 1)T 1 n )] ≤ 2c 1 and it is now easy to establish that [ y − y n ;ˆ γ 1 ( 0, 1 n ; p )]+ [ y − y n ; 0 This inequality clearly proves the uniqueness of the solution and therefore, it enables us to identify y with the solution constructed in Section 3. This identification allows in turn to choose µ and ǫ as in Proposition 3.4 and to assert that y ≤ c x, f ,ψ , which completes the proof.

Continuity of the solution
It remains to prove Theorem 2.12. In accordance with the statement of this result, we suppose that p > n and that Assumptions (A1), (A2), (X) γ and (F) 3 are all satisfied. We fix γ ′ ∈ (1−γ, γ+1/2) and the two initial conditions ψ,ψ ∈ γ ′ ,p . We denote by X = (X x , X ax , X x x ) (resp.X = (X x ,X ax ,X x x )) the path constructed from (x, x 2 ) (resp. (x,x 2 )) through Definition 2.2. With this notation, we define y n as the path described by the scheme (35) andỹ n as the path obtained by replacing (ψ, X x , X x x ) with (ψ,X x ,X x x ) in the latter scheme.
Besides, we defineJ andK by replacing (X x , X x x ) with (X x ,X x x ) in Formulas (15) and (16). For the sake of clarity, we also set J n := J y n , K n := K y n ,J n :=J y n ,K n =Kỹ n , and as in the previous section, we introduce the intermediate quantity Remember that owing to the results of Section 3, we can rely on the uniform control Proof. It suffices to follow the lines of the proof of Lemma 3.2.

Appendix A: a useful algorithm
We give here the description and an analysis of the algorithm used in the proofs of Lemmas 3.2, 4.1 and 5.1.
Consider a generic partition {0, 1, 2, . . . , N }. We remove the inner points of this partition ({1, 2, . . . , N − 1}) one by one according to the following procedure (see Figure 1): • At step 1, we successively remove, from the right to the left, every two points, starting from N (excluded) until 0 (also excluded). Then, still at step 1, we take off the point of the (updated) partition between 0 (excluded) and the last removed point, if such a middle point exists.
• We repeat the procedure with the remaining points (steps 2,3,...) until the partition is empty.
We denote by: • M the number of steps necessary to empty the partition.
• k + m the point of the partition (at 'time' m of the algorithm) that follows k m (when reading from the left to the right), k − m the point that precedes it.
• A r the total number of points that have been taken off at the end of step r. We also set A 0 := 0. In particular, A r − A r−1 − N 2 r ≤ 1 and 2 M −1 ≤ N ≤ 2 M +1 .
Proof. Actually, we use the following explicit expressions: at step r (r ∈ {1, . . . , M − 1}), if N − A r−1 is even, one has, for every m ∈ A r−1 + 1, A r − 1 , and k + A r = N − 2 r (A r − A r−1 ) + 2 r , k − A r = 0, while if N − A r−1 is odd, Formulas (66) and (67) remain true for m ∈ A r−1 + 1, A r − 1 , but k − From these expressions, we first deduce according to Lemma 6.1. Then, if N − A r−1 is even, one has A r m=A r−1 +2 since A r − A r−1 ≥ 2. In the same way, if N − A r−1 is odd, one has ) µ . since, in that case, A r − A r−1 ≥ 3. Thanks to Lemma 6.1, we now easily deduce

Appendix B
This section is devoted to the proof of Proposition 2.9. To this end, we will resort to the two following lemmas, respectively borrowed from [15]  . Together with Lemma 7.1, these results clearly provide the expected regularity, i.e., J y ∈ µ 2 ([0, 1]; p ) a.s, with µ = γ + η > 1. The control of the regularity of J y as a process with values in η,p stems from the same reasoning. Indeed, we first deduce from (38)