Wong–Zakai Approximation for Landau–Lifshitz–Gilbert Equation Driven by Geometric Rough Paths

We adapt Lyon’s rough path theory to study Landau–Lifshitz–Gilbert equations (LLGEs) driven by geometric rough paths in one dimension, with non-zero exchange energy only. We convert the LLGEs to a fully nonlinear time-dependent partial differential equation without rough paths term by a suitable transformation. Our point of interest is the regular approximation of the geometric rough path. We investigate the limit equation, the form of the correction term, and its convergence rate in controlled rough path spaces. The key ingredients for constructing the solution and its corresponding convergence results are the Doss–Sussmann transformation, maximal regularity property, and the geometric rough path theory.


Introduction
The stochastic Landau-Lifshitz-Gilbert equations (SLLGEs) describe the behaviour of the magnetisation under the influence of the randomly fluctuating effective field. In this work, we consider the SLLGEs with solutions taking values in the two-dimensional sphere S 2 in R 3  (1) The parameters λ 1 = 0, λ 2 > 0 are real constants. We assume that the material is saturated at the initial time, i.e.
If both anisotropy and the exchange energies are present (see Visintin [30] and Cimrák [11]), the total magnetic energy E of the LLGEs is given by where n is the unit outward normal vector field on the boundary ∂ D and ∂ M ∂n is the directional derivative of M in the direction n. Here we assume that the material is saturated at the initial time.
It is well-known that the stationary solutions of system (1) correspond to the equilibrium states of the ferromagnet and are not unique in general. An interesting question in the theory of ferromagnetism is to describe phase transitions between different equilibrium states induced by thermal fluctuations of the field H eff . Thus, conventionally, randomly fluctuating fields act responsible for magnetization fluctuations, see Neel [29]. It essentially characterizes deviations from the average magnetization trajectory in an ensemble of noninteracting nanoparticles, see Brown in [2]. According to Brown, the magnetization M evolves randomly, and for the stochastic version of system (1), one needs to modify the system in order to incorporate random fluctuations of the field H eff into the dynamics of M and to describe noise-induced transitions between equilibrium states of the ferromagnet. In this context, we refer to [3,4] for further clarification of adding noise to the effective field H eff .
The theory of rough paths was initiated by Terry Lyons in his seminal work [26] as an extension of the classical theory of controlled differential equations. Since its introduction, the theory of ordinary and partial differential equations driven by rough paths has developed intensively. We refer the reader to study the papers of Gubinelli et al. [13,[16][17][18], Friz et al. [9,10,14]. We quote the recent preprint [20], where using rough path formulation, existence, uniqueness and regularity for the SLLGE with Stratonovich noise on the one-dimensional torus has been studied.
In this paper, we are interested in the following form of the LLGEs equation: where X is a geometric rough path. We adapt Lyons' rough paths theory to study system (1) driven by an irregular noise. By proposing a suitable transformation, we convert system (1) to a fully nonlinear time-dependent partial differential equation without rough paths term. The primary interest is the regular approximation of the geometric rough path.
The rough path theory is informally connected to Wong-Zakai results as it necessarily allows to construct solutions as limits of Wong-Zakai type approximations, for references, see, e.g. the seminal paper by Lyons [26], and the recent monograph by Lyons et al. [27]. There are various works on rough PDEs where weak solutions are constructed by exploiting energy arguments. Here in our work, we applied this way of constructing a solution. For a quick survey, we refer to Bailleul et al. [1], Deya et al. [12], Hocquet et al. [21], Hofmanová et al. [22]. We refer the readers to Section 1.5 in [8] for detailed discussion.
In [19], the authors have studied equations similar to SLLGEs in R d , for any d > 0. However, the noise corresponds to a choice of g in equation (4) to be constant across the domain D, i.e., it can be independent of space and time variables. It is not clear how the concept of solution is defined. Brzeźniak et al. in [3] have proved existence of weak martingale solutions of SLLGEs taking values in a sphere S 2 . Furthermore, Li et al. in [5] have generalized the results in [3] with non-zero anisotropic energy E an and multidimensional noise. Finite dimensional analysis of this problem has been discussed in [23], [25]. Brzeźniak et al. have studied in [4] the one-dimensional case and prove the large deviations principle to the SLLGEs for small noise asymptotic. Brzeźniak et al. have proved in [6,7] existence of weak martingale solution for SLLGEs in three dimensions perturbed by jump noise in the Marcus canonical form with non-zero anisotropic energy E an , see [6] and non-zero exchange energy E ex only, see [7]. Recently, in [8,28], the authors have employed Wong-Zakai approximation technique to obtain the solvability and convergence of the time dependent transformed PDEs. The open questions framed in [8] have motivated us, as a first step to adapt Lyons' rough paths theory to study LLGEs driven by geometric rough paths in one dimension.
As a particular case, we can take Stratonovich Brownian rough paths as geometric rough paths. Furthermore, we plan to extend our techniques to a more general setting when the stochastic process is no longer one-dimensional (and nor the corresponding vector fields commute). The motivation to use geometric rough path comes from [24], where they discuss the heat equation with a geometric rough path.

Problem Description
We introduce some notations and summarize the most important definition of rough paths which are taken from [14].
Notations For a domain D ⊂ R d with d = 1, 2, 3, we will use the notation L p for the space L p (D; R 3 ) and W m, p for the Sobolev space W m. p (D; R 3 ). We will often write H m instead of W m,2 . We will also denote, for a Banach space For simplicity we write the semi-norm as | f | α, [0,T ] A rough path on an interval [0, T ] with values in a Banach space V then consists of a continuous function X : [0, T ] → V , as well as a continuous "second order process" X : [0, T ] 2 → V × V , subjected to certain conditions which are given by Definition 3. Generically, we write C α ([0, T ], V ) for the space of α-Hölder rough paths and C α g ([0, T ], V ) for the space of α-Hölder geometric rough paths over a Banach space V . For a Banach space H and α ∈ ( 1 3 , 1 2 ], we denote the space of controlled rough paths by D 2α . Furthermore, we recall that for any α ∈ (1/3, 1/2) with probability one, the corresponding Stratonovich lift is For more details about rough paths and their integration, we refer to Appendix 2 and the book [14].
In this paper, we consider the case d = 1 and α ∈ ( 1 3 , 1 2 ) and assume that D to be a bounded open interval in R. In particular, we take D = (0, 1). The LLGEs in consideration in this paper is of the form where T > 0 is fixed, g : D → R 3 is given function such that g ∈ W 4,∞ , and . Without loss of generality we assume that X 0 = 0.
One of the most fundamental questions related to problems similar to the above is whether the solutions depend in a continuous way of the coefficients (the geometric rough path in our case). Let us describe our approach to this question. We first recall Let us now consider the corresponding system (7) approximating (5) We note that if X ∈ C 0,α g ([0, T ], R) ⊂ C α g ([0, T ], R) then one can choose X (n) to be a sequence of piecewise smooth path or (piecewise) C 1 paths and X (n) r , see page 17 of [14]. Moreover, in the stochastic case one can take Stratonovich Brownian rough paths B strat as X and also sequence B (n) defined in Proposition 3.6 of [14] as X (n) . For more details we refer to Sect. 4.1 and the book [14].
Our primary goal is to prove that the solution to the system (5) exists and is a unique strong solution in controlled rough path spaces, which indeed is a limit as n → ∞ of the solutions of sequence of the corresponding system (7) approximating (5). In particular, we show the convergence with respect to the distance d X ,X (n) ,2α,[0,T ] (see Definition 4 in Appendix 2).
We now briefly describe the content of the paper. In Sect. 2, we introduce an auxiliary ODE and state some auxiliary facts necessary for the transformation of system (7) to a deterministic PDE without rough paths term. Sections 3 and 4 are devoted to the proof of the main results. In Sect. 3, we show how a unique weak solution to (7) can be obtained from a unique weak solution of the reformulated equation in controlled rough path spaces; see Theorem 1. Section 4 is devoted to the proof of the convergence of solutions to the weak solution of the reformulated form; see Theorem 4 . Finally, in the Appendix, for the reader's ease, we list several facts that are used in the course of the analysis. We split the Appendix into four subsections. First, in Appendix 1, we present several auxiliary lemmata which are essential to prove Theorem 1. Next, in Appendix 2 we handle some technical issues required in this paper, i.e., we introduce few results (see Lemmas 10,11 and Corollary 5). These results, another contribution of the paper and have their own interest. In the last two subsections, we recall some simple results and algebraic identities used in this paper.

The Auxiliary Equations
In this section, we present some basic results on the the spaces and operators involved in the course of analysis. We also introduce new processes m and m (n) gained by the Doss-Sussman transformation from the corresponding processes M and M (n) . We refer to [4,8,15] for further details about the properties of these processes.

Preliminaries
We define the Laplacian with the Neumann boundary conditions acting on R 3 -valued functions by We know that the unbounded operator A is self-adjoint in L 2 and A −1 1 is compact for A 1 := I + A. Therefore, there exists an orthonormal basis {e n } ∞ n=1 of L 2 consisting of eigenvectors of A. Furthermore, we know that, if V := D(A 1/2 1 ) is endowed with the graph norm, then V coincides with H 1 . Later on, we will write V → L 2 → V which is a Gelfand triple.
We now present the following interpolation inequality which will be used in later sections.
Next, let us we recall an elementary result from [4].
Let us define the map G : L 2 → L 2 by Lemma 2 Suppose g ∈ L ∞ . Then, G is a bounded linear map and well defined.
In Appendix 3, Lemma 12, we list the properties of G. Let H 2 (D; S 2 ) be the set of all R 3 -valued functions defined on the domain D belonging to the Sobolev space H 1 := H 1 (D; R 3 ) and satisfy the saturation condition (2). In particular, let In other words, H 2 (D; S 2 ) is the set of all functions belonging to the Sobolev space H 2 whose values are in the sphere. Since D is one-dimensional, H 2 is embedded in C(D; R 3 ) and the 'a.e.' condition in (11) can be substituted by 'all'.

The Doss-Sussmann Transformation and the Corresponding New Processes m and m (n)
Following the discussions in [8,15], we define a new process m from M by where the operator G is introduced in (10). We know, by identity (91) and Lemma 12, that e ·G , Ge ·G and G 2 e ·G are bounded functions. Therefore, by Proposition 7.6 of [14], we obtain and e X (·)G ∈ D 2α X ([0, T ], L(L 2 )). In Lemma 5, we will show that if a process M solves (5), then m defined by (12) solves the following non-linear time dependent PDE with random coefficients given below. Later on, in Sect. 3, we will show that there exists indeed a process m solving (14). Let us consider where F is given by Here,C is an operator coming up in Lemma 15 and is given bỹ where s ∈ R, v ∈ W 1,∞ 0 and C is an operator coming up in Lemma 14 and is given by From the assumption X (0) = 0, we note that the initial condition (2) is equivalent to an analogous one for m 0 , i.e.
Note that, we could consider the transformation (12) for all t ≥ 0. Moreover, as one can easily prove |[e tG (u)](x)| R 3 = |u(x)| R 3 for all t ∈ [0, T ], for a.a. x ∈ D, we see that the following saturation conditions for M and m are equivalent: For proof of the saturation condition (20), we refer to Lemma 3.15 of [8]. Repeating the same algebraic calculations as done in [15], i.e. using the definition of G and the identity (17), we obtain Substituting (21) in (16), we get whereC(X (t), m) has now the following form Applying identity (91), we obtain the following representatioñ where we define the following abbreviations Remark 1 Using the representation in (23) and the fact that g ∈ W 2,∞ (D), we know that there exists a constant K > 0 such that Using the following inequalities we obtain Finally, by straightforward calculations we can estimate the nonlinearity F defined in (22). In particular, there exists a constant K > 0 such that using (20), we have Let us now consider the corresponding system (36) approximating (14). Again, let us define a new auxiliary process m (n) from M (n) and X (n) by By the same calculations as above and Lemma 7, one can show that if M (n) is a solution to (7), then m (n) is a solution to and vice versa. Here, Moreover, in Theorem 3, we will show that system (36) has indeed a unique solution.

Existence and Uniqueness of the Solutions to the Systems (5) and (7)
We begin with the definition of (strong) solution to the system (5).
is said to be a (strong) solution of the system (5) if the following properties hold: (i) M satisfies the following saturation condition (ii) M satisfies the second equation of (5) and sup (iv) M satisfies the rough differential equation given by where the third integral is interpreted in the sense of (77).
In this section, we prove the following theorem which shows that the solution to the system (5) (resp. (7)) exists and is unique. .
, then there exist unique (strong) solutions M and M (n) to the respective systems (5) and (7). In addition, we have Before we present the proof of Theorem 1, we introduce Theorem 2 and Theorem 3. In the first theorem, we show the existence, uniqueness and regularity of the solution to the system (14). In the second theorem, we obtain the same for system (36). Then, using the Doss-Sussmann transformation, we show that from the existence of a unique solution to systems (14) and (36), respectively, follows the existence of the solution of systems (5) and (7), respectively. Similarly, the convergence is proven.
Let us start with introducing the definition of (weak) solution m to the system (14) and the theorem for existence and regularity of m.
is said to be a (weak) solution to the system (14) if the following properties hold: (i) m satisfies the following saturation condition (ii) m satisfies the second equation of (14) and . Then there exists a weak solution m to the system (14) in the sense of Definition 2 satisfying the following: and m is also a strong solution of the system (14).
Proof For proof, we refer to Theorem 3.2 and Lemma 3.13 of [8]. In addition, for the regularity property of m, we refer to Lemma 8 and Lemma 9 in Appendix 2.
In the same way as above (see Definition 2 and Theorem 2), we can define weak solutions for the system of equations (36) and obtain the following theorem.
. Then there exists a weak solution m (n) to the system (36), i.e., it satisfies the following: Moreover, we obtain that m (n) ∈ H 1 (0, T ; L 2 ) ∩ L ∞ (0, T ; H 2 ) and m (n) is the strong solution of the system (36). (14). Hence, applying Lemma 5, we conclude that there exists a unique strong solution M ∈ L ∞ (0, T ; H 2 ) of the system (5) and

Proof of Theorem 1 Thanks to Theorem 2, there exists a unique strong solution
. Finally, using Lemma 10, we conclude that the integration against rough paths in (5) is well defined. Proceeding in similar lines, one can observe that there exists a unique strong solution M (n) ∈ L ∞ (0, T ; H 2 ) to the system (7) and , thus completing the proof.

Convergence of Solution in Controlled Rough Path Spaces
In this section we prove the convergence result in the space D 2α (5) and (7), respectively. Then we have the following convergence

Theorem 4 Let M and M (n) be the solutions to the systems
Before going into the proof of Theorem 4, we introduce and prove some lemmata and corollaries. These lemmata are essential to show Theorem 4. (n) be the solutions to the systems (14) and (36), respectively. Then we have the following convergence

Lemma 3 Let m and m
Proof By similar arguments as with the proof of Lemma 5.2 in [8], one can show that there exists a constant C > 0 and an integrable function ϕ C , so that we have the following estimate Using (6), we obtain (5) and (7), respectively. Then we have the following convergence Proof We note that

Corollary 1 Let M and M (n) be the solutions to the systems
Since Finally, from Lemma 3, we obtain which is the assertion.
In the next lemma, we investigate the convergence of sequence m (14) and (36), respectively. Then we have the following convergence

Lemma 4 Let m and m (n) be the solutions to the systems
We proceed in the same way as we advance in the proof of Lemma 9 in Appendix 2. Substituting z in system (14), we get the following identity m).
Testing with z and integrating over the interval (0, 1), we obtain Now we estimate each I j 's for j = 1, 2, 3. Estimate of I 1 : Integrating by parts and using the fact that z x = 0 and z (n) x = 0 at x = 0 and x = 1, we obtain Using Young's inequality, we obtain for any ε > 0, Above, we introduced the abbreviation ψ 1 := ε|λ 1 |, Estimate of I 2 : By a simple vector algebraic identity and equation (20), we obtain We now derive estimates I 2, j with j = 1, 2, · · · , 5 for a fixed ε. Proceeding in a similar manner as in I 1 (t), we get Now we estimate I 2,2 . Applying Young's inequality, we have Proceeding in similar manner as in I 2,2 , we get for I 2,3 and I 2,4 Now, by similar argument as applied to I 2,1 , we obtain Combining the estimates of I 2, j , j = 1, 2, · · · , 5 and substituting back in (44), we get Estimate of I 3 : To simplify the notation, we introduce y = m (n) −m. By the identities (24) and (26), it can be observed that C, C x , C x x , S, S x and S x x are Lipschitz continuous functions. In particular, there exists a constant K = K (|g| W 4,∞ ) > 0 such that the following holds: Furthermore, an elementary calculation using the identities (25) and (27) yields that Hence using the estimates (46), (47) and the identity (28) we have Now we are ready to estimate I 3 . Using the identity (22) and a simple vector algebraic identity, we obtain: We now derive estimates I 3, j with j = 1, 2, · · · , 9 for fixed ε. Using (48a), (29) and Young's inequality, we have Again using (48a), (29) and Young's inequality, we get Proceeding in similar manner as in I 3,4 , we have Again, proceeding as in I 3,4 , we obtain Now applying (48b), (30) and Young's inequality, we obtain Again using (48b), (30) and Young's inequality, we get Proceeding in similar manner as in I 3,7 , we have Now using Young's inequality, (48b) and (30), we get Again using (48c), (31) and Young's inequality, we get Thus combining the estimates of I 3, j , j = 1, 2, · · · , 9 and substituting back in (49), we have Substituting (43), (45) and (50) in (42) we have We note by (20) and Corollary 4, we have m (n) , m (n) x ∈ L ∞ (0, T ; L ∞ ) for all n ∈ N. Therefore we can choose ε such that x x ∈ L ∞ (0, T ; L 2 ) and u x , u (n) x ∈ L 2 (0, T ; L 2 ). Thus ϕ and χ are integrable on [0, T ]. Using Gronwall's inequality we get

Now by Lemma 3 and (8), we note that
As This completes the proof of Lemma 4. (7), respectively. Then we have the following convergence

Corollary 3 Let M and M (n) be the solutions to the systems (5) and
Proof This is a direct consequence of Corollaries 1 and 2.
Now we are ready to prove Theorem 4.

Proof of Theorem 4 Since
sup Let 0 < T 1 ≤ min{1, T }. In particular, we assume that C α,N ,g T α 1 ≤ 1 2 , where C α,N ,g > 0 is a generic constant depending only on α, N , and g and popping up in estimate (59). In the next lines we proof the following estimate Next, using the identities M s,t = M s X s,t + R M s,t and M (n) Using (54)-(57), we have This implies . (58) We now start to estimate |R M − R M (n) | 2α,[0,T 1 ],L 2 . Using system (5), we obtain where Ξ s,t := (M s × g)X s,t + (M s × g)X s,t and In similar way, we also obtain the same identity as above for R M (n) s,t , i.e. where Setting Ψ = Ξ − Ξ (n) , we use equation (4.11) of [14] with β = 3α and replaced Ξ by Ψ , so that we can write for 0 For 0 ≤ s < u < t ≤ T 1 , we get now Continuing, we obtain This implies Finally, using (58) and (59), we obtain the estimate Observe, we have taken C α,N ,g T α 1 ≤ 1 2 , from which it follows that Noting that the choice T 1 does not depend on the initial condition, we can extend the estimate by gluing techniques to the whole time interval [0, T ] and get for j = 1, · · · , K where K ∈ N and K < ∞ such that K T 1 < T and (K + 1)T 1 ≥ T . Also, we obtain Therefore, combining (61) and (62), we achieve and B (n) = B (n) , B (n) be the corresponding geometric rough path. Here, the integral t 0 B (n) d B (n) is understood as classical Riemann-Stieltjes. Since B (n) is a piecewise linear function and continuous function, the Riemann-Stieltjes is well defined. From Proposition 3.6 of [14], we know that B (n) is a sequence in C α g ([0, T ], R) and we have with probability one Let us now consider the following approximation of system (64)

Equivalence of Systems (5) and (14) and Systems (7) and (36)
In this section, we present several auxiliary lemmas to deal with the equivalence and the regularity properties between m (m n resp.) and M (M n resp.). These results are essential to prove Theorem 1.

Technical Lemmas
Now we present the following lemma, which states that by using the transformation (12) we obtain that if M is a solution to the system (5), then m is a solution to the system (14) and vice versa.  (14); (ii) M ∈ L ∞ (0, T ; H 2 ) satisfies system (5) and . Before going to the proof of Lemma 5, we give the following lemma which is essential to the proof of Lemma 5.

Lemma 6 Let X be any continuous function on [0, T ] and
Assume that g ∈ W 2,∞ (D). If m belongs to L ∞ (0, T ; H 2 ), then M belongs to L ∞ (0, T ; H 2 ) and vice versa.
By similar arguments we also have the converse.
We are now ready to prove Lemma 5.
Proof of Lemma 5 We first assume that the statement (i) is correct. Using Lemma 6, we know M ∈ L ∞ (0, T ; H 2 ). Using (8) and the fact that m ∈ L ∞ (0, T ; H 2 ) and d m dt Thus by (32), we have F(·, m(·)) ∈ L ∞ (0, T ; Since m ∈ C 2α ([0, T ], L 2 ) and e X G ∈ D 2α X ([0, T ], L(L 2 )) we have Therefore, we conclude It follows that R M ∈ C 2α ([0, T ], L 2 ) and we have We now prove that M satisfies system (5). It can be clearly observed that M satisfies the last two equations of (5). Next, using (13a), Corollary 5 and the fact that m ∈ Using the property of geometric rough paths Sym(X) u,v = 1 2 X u,v ⊗ X u,v we have This implies Multiplying both sides by a test function ψ ∈ C ∞ c (D) and integrating over D we obtain Recalling m = e −X (r )G M and using (96), we obtain Thus by (104) and (93), we get Integrating by part and applying (100), we have Again using m = e −X (r )G M, (104) and integrating by part, we obtain Since it is satisfied for all ψ ∈ C ∞ c (D), using integration by parts, we have for all where F(t, m) is given by (15). By standard argument, one can conclude that m ∈ H 1 (0, T ; L 2 ), which completes the proof.
We consider the systems (7) and (36) and provide a lemma similar to Lemma 5 which establishes the equivalence of these two systems. We note that the proof of Lemma 7 is completely analogous to the proof of Lemma 5. We use the fact that X (n) = X (n) , X (n) is a geometric rough path, i.e. Sym(X (n)

Regularity Properties
In this subsection, we first precisely give the following lemma. For proof we refer to Lemma 3.10, Lemma 3.13 and Corollary 6.2 of [8].
Lemma 8 Let m 0 ∈ H 2 (D; S 2 ) and suppose that m and m (n) satisfy the systems (14) and ( Proof We show that m x x ∈ L ∞ (0, T ; L 2 ). Denoting z := m x x , system (14) can be written as Thus one can obtain in the sense of distribution Using the identity z × z = 0 we have By (107) we have Our goal is to obtain the existence of the fourth derivative of m in a.e. sense. Let us approximate z by a Faedo-Galerkin approximation. For k ∈ N, let z k be the solution of Note, since the system is finite dimensional, a unique solution exists by standard arguments. Taking the dot product with z k in R 3 , we get , m), z k .
Using the saturation condition, i.e. (20), identity (105) and integrating over (0, 1), we obtain Here, ·, · denotes the inner product in R 3 . Using standard calculation, we obtain Integration by parts and the fact that that z x = 0 at x = 0 and x = 1 gives Choosing ε = λ 2 5 , we get This yields Lemma 8 yields |ϕ 2 | L ∞ (0,T ;R) < ∞, and by the interpolation inequality (8), we get Using that z 0 ∈ L 2 and the estimate (73), we can show that (71) has a solution in L 2 (0, T ; H 1 ) ∩ L 2 (0, T ; H −1 ). Therefore, the weak limit of z k , denoted by z, exists and satisfies (70). It follows the existence of z x x , i.e., that the fourth derivative of m exists. Passing to the limit as k → ∞ in (73), it can be clearly observed that z satisfies This completes the proof.

Corollary 4
Let m 0 ∈ H 2 (D; S 2 ) and suppose that m and m (n) satisfy systems (14) and x x ) belong to L ∞ (0, T ; L 2 ). Furthermore, we also have m x and m (n) x belong to L ∞ (0, T ; L ∞ ).
Proof This is a direct consequence from Lemmas 8, 9 and inequality (8).

Rough Paths
In this section we summarize the most important definitions, lemma and corollary which are necessary for the existence theory and the convergence results. In order to prove the equivalence between m and M which is given in Lemma 5, some technical issues require us to introduce Lemmas 10, 11 and Corollary 5. These results, another contribution of the paper, have its own interest.
and X s,t = X s,u +X u,t + X s,u ⊗ X u,t for all s, t.u ∈ [0, T ] (Chen identity). Moreover, if Sym(X s,t ) = 1 2 X s,t ⊗ X s,t then we say that X is α-Hölder geometric rough paths and we denote the space by C α g ([0, T ], V ). Also, we define C 0,α g ([0, T ], V ) as the closure of lifts of smooth paths in C α ([0, T ], V ). In addition, given two rough paths Y 1 , Y 2 ∈ C α ([0, T ], V ), we define the (inhomogenous) α-Hölder rough path metric In order to present an integration against rough paths, we also need to give the following definition. H )) (we call Gubinelli derivative) so that the reminder term R Y given implicitly through the relation Y s,t = Y s X s,t + R Y s,t , satisfies |R Y | 2α,H < ∞. This defines the space of controlled rough paths denoted by D 2α

Definition 4 Given a path
In addition, consider X := (X , X),X : We are now ready to extend Young's integral to that of a path controlled by X against X = (X , X). The definition of the rough integral Y d X in terms of compensated Riemann sums then immadeately suggests to define the integral of Y against X by  [14]). Now we introduce the following lemma which gives us the required conditions for the integration against rough paths in (5) and (7) to be well defined.
Lemma 10 Let g be as before and for any α ∈ (1/3, 1/2) we consider X = (X , Proof We will prove that Y × g, Y × g ∈ C α ([0, T ], L 2 ) and R Y ×g s,t = Y s,t × g − (Y s × g)X s,t ∈ C 2α . In order to prove first term we use (108), i.e.
Since Y ∈ C α ([0, T ], L 2 ), we have Y × g ∈ C α ([0, T ], L 2 ). The similar way, we can prove that Y × g ∈ C α ([0, T ], L 2 ) and R Y ×g s,t ∈ C 2α . Furthermore, we have To prove the equivalence of weak solution between m and M, we have to apply Itô formula, see [15]. Therefore, we present the following lemma to handle our case.

Lemma 11
Let X = (X , X) ∈ C α ([0, T ], V ) and let (Y , Y ), (Z , Z ) ∈ D 2α X are controlled rough paths of the form for some controlled rough paths (Y , Y ), (Z , Z ) ∈ D 2α X and some paths Γ , Λ ∈ C 2α . Then, which we interpret the integrals T 0 (dΓ r Z r ), T 0 Y r dΛ r and T 0 Y r Z r dX r as Young integrals. Furthermore, the last integral is given by where the bracket [·, ·] is defined in [14,Definition 5.8,p. 73].
Proof To show (80), note first that a consequence of (78), (79) and Theorem 4.10 of [14], the increments of Y and Z are of the form Z s,t = Z s X s,t + Z s X s,t + Λ s,t + o(|t − s|).
Thanks to (81) and (82), we have By taking the limit |P| → 0, also nothing that [u,v]∈P o(|v − u|) → 0, we have Thanks to Remark 5.11 and Corollary 7.4 of [14], we obtain By assumption (Y , Y ), (Z , Z ) ∈ D 2α X and Γ , Λ ∈ C 2α , we have Now using the Chen identity, we get Applying Lemma 5.9 of [14], we obtain The convergence to the Young integral in the last three integrals follows from (Y , Y ), (Z , Z ) ∈ D 2α X and Γ , Λ ∈ C 2α . The proof is complete. and where Lemma 15 Assume that g ∈ H 2 . For any s ∈ R, u ∈ H 1 and v ∈ W 1,∞ 0 , whereC (s, v) = e −sG (sin s)C + (1 − cos s)(GC + CG) v, and C is given by (99).

Some Algebraic Identities
Here, we present list all algebraic identities will be used in this paper. Suppose that a, b, c, d ∈ R 3 . Then a × (b × c) = a, c b − a, b c, (107) |a × b| ≤ |a| |b|.
In particular, if a, b = 0, then