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Rough differential equations containing path-dependent bounded variation terms

The existence and uniqueness of strong solutions to reflected stochastic differential equations on domains are established by Lions-Sznitman and Saisho under very general conditions on the boundary. We proved the existence of solutions to reflected rough differential equations under additional stronger conditions on the boundary in a previous paper by extending Davie's numerical scheme. In this paper, we prove the existence of the solutions under the same general condition as in Saisho {\it etal.} We prove the existence theorem for general path-dependent rough differential equations. We also study support theorem for the reflected processes.


Introduction
In [2], we studied reflected rough differential equations (=reflected RDEs, RRDEs) defined on a domain D in R d and proved the existence of solutions. This study depends on the important results of Skorohod equation which are due to Tanaka [28], Lions-Sznitman [22] and Saisho [27]. The reflected stochastic differential equations (=reflected SDEs) are uniquely solved in strong sense under the boundary conditions (A) and (B) ( [27]). The main strategy of the proof in [2] is a modification of Davie's approximation procedure of RDEs without reflection term ( [8]). Unfortunately, we need a stronger assumption (H1) in addition to (A) and (B) on the boundary to prove the existence theorem.
One of the aim of this paper is to prove the existence of solutions to RRDEs under conditions (A) and (B) only. Actually, we consider a certain class of path-dependent rough differential equations including RRDEs and prove the existence of solutions and give estimates of the solutions. We use Gubinelli's controlled paths to study the path-dependent RDEs. We note that the rate of strong convergence of Euler-Maruyama approximation and Wong-Zakai approximation for a class of path-dependent SDEs were studied in [1].
Another aim of this paper is to prove a support theorem for solutions to RRDEs. At the moment, although we do not know the uniqueness of solutions and the continuity of the solution mapping, we can prove the existence of universally measurable selection mapping of solutions as in [2]. We prove continuity property of the solution mappings at smooth rough paths under conditions (A) and (B). By using this, we prove a support theorem for path space measures determined by random rough paths. When the driving process is Brownian motion, Wong-Zakai type theorem is proved in [4,29,3,13,25]. Recently, under the conditions (A) and (B) with certain additional natural assumptions on the boundary, support theorem for reflected diffusion processes are proved by Ren and Wu [26] by using the Wong-Zakai type theorem. We give another proof of this theorem by using the continuity property of the solution mapping at smooth rough paths.
The structure of this paper is as follows. In Section 2, we introduce a Hölder continuous path spaces C θ with the exponent θ based on a control function ω of a rough path. In order to study RDEs containing path-dependent bounded variation terms, we need to consider continuous paths whose q-variation norms satisfies such a Hölder continuity. Hence we introduce a family of norms · q-var,θ and Banach spaces C q-var,θ and give a simple estimate of the integral defined by paths Φ ∈ C q-var,θ and w ∈ C θ which is important in this paper. We note that these kind of norms are studied in [18].
In Section 3, we introduce controlled path spaces based on ω. We next introduce a mapping L from a continuous path space to a continuous bounded variation path space. By using L, we introduce a path-dependent RDE. Typical example of L is given by a Skorohod mapping associated with a Skorohod problem. We give a related but different example also. In the case of reflected SDEs, we gave an estimate of the bounded variation norm of L(w) for a continuous w in [4] following the proof of Proposition 3.1 in [27] under the conditions (A) and (B). We note that this estimate can be improved as in Lemma 3.2. In Section 4, we prove the existence of solutions to the equations and give the estimate of them. For that purpose, we apply Schauder's fixed point theorem in the product Banach space of controlled paths and C q-var,α .
In Section 5, we consider the case of reflected RDEs on domains D under the conditions (A) and (B) on the boundary. In this case, the solution is unique when the driving rough path is smooth and the coefficient is Lipschitz. Although the solutions is not uniquely determined for generic rough paths, as in [2], we prove the existence of universally measurable selection solution mapping.
We prove the continuity of the solution mapping at the smooth (rough) paths and support theorems for the path space measure and reflected diffusions in Section 6.
In Section 7, we make a remark on path-dependent rough differential equations with drift term.
In [2], we proved the existence of solutions to an implicit Skorohod equation under the assumption (H1). In Section 8, we prove the existence of solutions under (A) and (B). It may be possible to prove the existence of solutions to RRDEs under the conditions (A) and (B) by using the proof in [2].

Preliminary
Let ω(s, t) (0 ≤ s ≤ t ≤ T ) be a control function. That is, (s, t) → ω(s, t) ∈ R + is a continuous function and ω(s, u) + ω(u, t) ≤ ω(s, t) (0 ≤ s ≤ u ≤ t ≤ T ) holds. We introduce a mixed norm by using ω and p-variation norm. Let E be a finite dimensional normed linear space. For a When ω(s, t) = |t − s|, w θ,[s,t] < ∞ is equivalent to that w u (s ≤ u ≤ t) is a Hölder continuous path with the exponent θ in usual sense. Hence we may say w is an ω-Hölder continuous path with the exponent θ ((ω, θ)-Hölder continuous path in short). For two parameter function F s,t (0 ≤ s ≤ t ≤ T ), we define F θ, [s,t] and F q-var,θ,[s,t] similarly.
We denote by C θ the set of ω-Hölder continuous paths w satisfying w θ = w θ,[0,T ] < ∞. C θ is a Banach space with the norm |w 0 | + w θ . Let C q-var,θ (E) denote the set of E-valued continuous paths of finite q-variation defined on [0, T ] satisfying w q-var,θ := w q-var,θ,[0,T ] < ∞. Note that C q-var,θ (E) is a Banach space with the norm |w 0 | + w q-var,θ . Obviously, any path w ∈ C q-var,θ satisfy |w s,t | ≤ w q-var,θ ω(s, t) θ . We next introduce the notation for mappings between normed linear spaces. Let E, F be finite dimensional normed linear spaces. For γ = n + θ (n ∈ N ∪ {0}, 0 < θ ≤ 1), Lip γ (E, F ) denotes the set of bounded functions f on E with values in F which are n-times continuously differentiable and whose derivatives up to n-th order are bounded and D n f is a Hölder continuous function with the exponent θ in usual sense. (2.5) Proof.
(1) We have The second inequality follows from the trivial bound (2) We need only to prove lim sup q→∞ w q-var,[s,t] ≤ w ∞-var, [s,t] . Suppose w q 0 -var,[s,t] < ∞. Then for q > q 0 , Taking the limit q → ∞, we obtain the desired estimate.
Finally, we give an estimate of the integral t s Φ s,r ⊗dw r , where ⊗ denotes the tensor product.
Let p be a positive number such that θp > 1. Let q be a positive number such that 1/p + 1/q ≥ 1 and Φ ∈ C q-var,θ ′ (R d ). Then the integral t s Φ s,r ⊗ dw r converges in the sense of Young integral and We write |P| = max i |t i − t i−1 |. The relation θp > 1 implies that w is a continuous path of finite p − ε-variation for sufficiently small ε. Since Φ is a continuous path of finite q-variation and 1/p + 1/q ≥ 1, the Young integral lim |P|→0 I(P) s,t converges. To estimate this integral, we consider a "dyadic partition" of [s, t] associated with ω. First, we choose t 1 1 in the following way. If ω(s, t) = 0, then set t 1 1 = (s + t)/2. Otherwise, let t 1 1 be a number such that ω(t 1 1 , t) = ω(s, t)/2. The superadditivity of ω implies ω(s, t 1 1 ) ≤ ω(s, t)/2. Also let t 1 0 = s, t 1 2 = t and set for some k. By the continuity of ω, we have lim N →∞ |P N | = 0 and t s Φ r ⊗ dw r = lim N →∞ I(P N ) s,t . Let p * be the number such that 1/p + 1/p * = 1. Then by using the notation {(u + v)/2} explained above, we have This completes the proof.
We denote by C β (R n ) the set of 1/β-rough paths. When ω(s, t) = |t − s|, X s,t is a β-Hölder rough path. If X s,t is a rough path with finite 1/β-variation, setting ω(s, t) = X [s,t] , X β ≤ 1 and X 2β ≤ 1 hold. We use the following quantity, We introduce a set of controlled paths D 2θ X (R d ) of X s,t , where 1/3 < θ ≤ β following [20,17]. A pair of ω-Hölder continuous paths (Z, is called a Gubinelli derivative of Z with respect to X. Note that Z ′ may not be uniquely determined by Z. It is easy to see that D 2θ The rough differential equation which we will study contains path dependent bounded variation term L(w) t . We consider the following condition on L.
and satisfy the following conditions.
(3) There exists a non-decreasing positive continuous function F on [0, ∞) such that The conditions (1), (2) are natural. In many cases, L is defined on continuous path spaces and is continuous with respect to the uniform norm. See the following examples. The condition (3) is strong assumption. This implies that the total variation of L(w) on [s, t] can be estimated by the norm of the path (w u ) on s ≤ u ≤ t. Note that this does not exclude the case where where (θ T ′ w) t = w T ′ +t . These enable us to solve the path-dependent rough differential equation on small intervals and obtain global solution on [0, T ] by concatenating the solutions. Actually, the condition (3) automatically implies the following stronger estimate. By this result, we may assume that the growth rate of F is at most of order x 1/β , that is, a polynomial order.
satisfies the condition (3) in Assumption 3.1. Then we have and N ≤ ω 1/β (s, t)/ε + 1. By the additivity property of the bounded variation norm, we have which implies the desired estimate. (1) A typical example of L appears in the study of reflected process. We refer the reader for the detail in the following statement to Section 5. Let D be a domain in R d satisfying conditions (A) and (B). Consider the Skorohod equation y t = w t + φ t , where w is a continuous path whose starting point is inD. Also y t ∈D (0 ≤ t ≤ T ) and φ t is the bounded variation term. The mapping L : w → φ satisfies the above conditions. ( Clearly this mapping is closely related to the example (1).
We now state our main theorem.
Further there exist positive constants K and κ which depend only on σ, β, γ such that

Remark 3.5.
(1) Since L(Z) t is continuous and bounded variation, it is easy to give the meaning of the integral t 0 σ(Z s , L(Z) s )dX s and it will be done below. The point of the proof of the main theorem is to find good estimates for the integral by which we apply Schauder's fixed point theorem.
(2) It is natural to conjecture that the uniqueness of the solutions hold under a certain stronger conditions on σ and L. By checking the proof of the main theorem, one may prove the uniqueness of solutions under the assumption that σ ∈ Lip γ and L is a Lipschitz map from C α to C q-var,α , where α andα are constants in (3.36). However, the assumption on L is too strong and cannot be applied to interesting case, that is, reflected rough differential equations, because we cannot expect the Lipschitz continuity of L in general, see [16,12] and the estimate in Lemma 5.4.
Falkowski and S lomiński [14,15] proved the Lipschitz continuity of the Skorohod mapping on a half space and proved the uniqueness of reflected SDE driven by fractional Brownian motions with the hurst parameter H > 1/2. Also we note that a recent preprint [9] studies the uniqueness of the solutions to 1-dimensional RRDEs driven by multidimensional rough paths.
(3) We assume L(w) is bounded variation in this paper. However, by checking the proof of Theorem 3.4, we may prove that similar results hold in the case where L(w) is q-variation path (q > 1) under suitable assumptions on σ, L, X. Assumption 3.6. Below we choose and fix p > 0 such that 1/β < p < γ.
If we write L(Z) t = Φ t , then the above equation reads We solve this equation by using Schauder's fixed point theorem. To obtain a compactness of the embedding of the set of continuous paths of finite q-variation, we need to consider the set C q-var,θ , q ≥ 1 and 1/3 < θ ≤ 1/2. We now give a meaning of the integral (3.8). We denote the derivative of σ = σ(x, y) (x ∈ R d , y ∈ R d ) with respect to x by D 1 σ and y by D 2 σ. Also we write Dσ(x, y)(u, v) = D 1 σ(x, y)u+ We assume that q, α,α satisfy the following condition.
By a simple calculation, we have for s < u < t, Thus, under the assumption on Z, Φ, using Lemma 2.
and · ∞ denotes the sup norm. Therefore, there exists a positive constant C which depends only on γ, β, p such that We write [u, v] ∈ P if u = t k−1 and v = t k for a 1 ≤ k ≤ N as before. Since β + (γ − 1)α > β + pα − α ≥ pα > 1, by a standard argument in the Young integration and rough path (see the proof of Lemma 2.2 and [24,19,17]), the following limit exists, We may denote this by simply if there are no confusion. This integral satisfies the additivity property is actually a controlled path of X. In fact, we have the following estimates.
(1) This follows from the explicit form of (3.17) and Lemma 2.2.
(2) This follows from (3.20) and the standard argument explained above.
We prove the existence of solutions by using Schauder's fixed point theorem. To this end, we consider the product Banach space D 2θ 1 X × C q-var,θ 2 , where 1/3 < θ 1 ≤ 1/2 and 0 < θ 2 ≤ 1. The norm is defined by The solution of RDE could be obtained as a fixed point of the mapping, The following continuity property is necessary for the proof of the main result.
Lemma 3.8 (Continuity). Assume where β ′ is the number in Assumption 3.1. Then M is continuous.
We also have the following compactness result. This also is necessary for applying Schauder's fixed point theorem to the proof of existence theorem. Lemma 3.9. (1) Let 1 ≤ q ′ < q. Let θ, θ ′ be positive numbers such that qθ < q ′ θ ′ . Then the inclusion C q ′ -var,θ ′ ⊂ C q-var,θ is compact.
We prove Lemma 3.9 first.
Proof of Lemma 3.9. (1) By Lemma 2.1 (1), we have then by their equicontinuities, there exists a subsequence such that Φ n k converges to a certain function Φ ∞ in the uniform norm. By the above estimate, we can conclude that the convergence takes place with respect to the norm on C q-var,θ .
(2) Suppose s,t , {Z(n)} is also bounded and equicontinuous. Hence certain subsequence {Z(n k ), Z(n k ) ′ } converges uniformly. This and a similar calculation to (1) This follows from Lemma 3.7. We estimate Since β >α ≥ α, this shows the continuity of the mapping ((Z, (3.42) Hence this term is small in the ω-Hölder space C 2α on a bounded set of W T,α,α,q,ξ,η if N is large. We fix a partition so that this term is small. Although the partition number may be big, is a finite sum, and by the explicit form of δΞ as in (3.18), we see that this difference is small in C 2α if ((Z, Z ′ ), Φ) and ((Z,Z ′ ),Φ) are sufficiently close in W T,α,α,q,ξ,η . The estimate of the other terms are similar to the above and we obtain the continuity of the mapping We next prove the continuity of the mapping ((Z, Z ′ ), Φ)(∈ W T,α,α,q,ξ,η ) → L(ξ + I(Z, Φ))(∈ C q-var,α ).  First, we prove the existence of solutions on small interval [0, T ′ ]. Since the interval can be chosen independent of the initial condition, we obtain the global existence of solutions and the estimate for solutions. To this end, take 0 < T ′ < T such that ω(0, T ′ ) ≤ 1 and consider W T ′ ,α,α,q,ξ,0 . In this section, we assume that q, α,α satisfy where β ′ is the number in Assumption 3.1. We consider balls with radius 1 centered at ((ξ + σ(ξ, η)X t , σ(ξ, η)), η) (0 ≤ t ≤ T ′ ), Lemma 4.1 (Invariance and compactness). Assume (4.1) and let α < α <α <ᾱ < β. For sufficiently small T ′ , we have Moreover T ′ does not depend on ξ and η.
Proof. The second inclusion is immediate because ω(0, T ′ ) ≤ 1 and the definition of the norms. We prove the first inclusion. Recall that I(Z, Φ) ′ t = σ(Z t , Φ t ). From Lemma 3.7 (4), we have We next estimate R I(Z,Φ) . Let 0 < s < t < T ′ . By Lemma 3.7 (3), we have We turn to the estimate of L(ξ + I(Z, Φ)). By Assumption 3.1 (3) on L and applying Lemma 2.1 (1) and Lemma 3.7 (2), we have Here we have used ω(0, T ′ ) ≤ 1. This term is small if T ′ is small. All the estimates above imply the desired inclusion for small T ′ . To obtain explicit T ′ , let Proof of Theorem 3.4. In the proof of Lemma 3.8 and Lemma 4.1, technical constants α,α, α,ᾱ appeared. We fix them as follows, For these constants, let κ = κ 0 . By Lemma 3.8 and Lemma 4.1, applying Schauder's fixed point theorem, we obtain a fixed point for small interval [0, T ′ ] with That is, there exists a solution on [0, T ′ ]. We now consider the equation on [T ′ , T ]. We can rewrite the equation as For two paths, w 1 , w 2 ∈ C([0, T − T ′ ], R d ; w 0 = 0), note that w 1 −w 2 β is equal to the (ω(T ′ + ·, T ′ + ·), β)-Hölder norm of w 1 − w 2 . By the assumption on L, we see thatL T ′ also satisfies Assumption 3.1 with the same function F . Therefore, we can do the same argument as [0, T ′ ] for small interval. By iterating this procedure finite time, say, N -times, we obtain a controlled path (Z t , Z ′ t ) (0 ≤ t ≤ T ). This is a solution to (3.8) and (3.9). Clearly, We need to show (Z, Φ) ∈ W T,β,β,1,ξ,η and its estimate with respect to the norm · β . We give the estimate of the solution on [0, T ′ ]. The solution (Z, Z ′ ) which we obtained satisfies Therefore Z and L(Z) are (ω, β)-Hölder continuous paths. Thus, by combining these estimates and (3.31) (we need to replaceα by β and this is possible), we obtain for 0 ≤ u ≤ v ≤ T ′ , These local estimates hold on other small intervals. By the estimate (4.16), we obtain the desired estimate.

Reflected rough differential equations
Let D be a connected domain. As in [27,22], we consider the following conditions (A), (B) on the boundary.
Definition 5.1. We write B(z, r) = {y ∈ R d | |y − z| < r}, where z ∈ R d , r > 0. The set N x of inward unit normal vectors at the boundary point x ∈ ∂D is defined by (B) There exist constants δ > 0 and 0 < δ ′ ≤ 1 satisfying: for any x ∈ ∂D there exists a unit vector l x such that (l x , n) ≥ δ ′ for any n ∈ ∪ y∈B(x,δ)∩∂D N y .
Let us recall the Skorohod equation. The Skorohod equation associated with a continuous path w ∈ C([0, ∞), R d ) with w 0 ∈D is given by Under the assumptions (A) and (B) on D, the Skorohod equation is uniquely solved. This is due to Saisho [27]. We write Γ(w) t = y t and L(w) t = φ t . By the uniqueness, we have the following flow property. where (θ s w)(τ ) = w(τ + s) − w(s).
We obtain the following estimate on L(w).
Lemma 5.3. Assume conditions (A) and (B) hold. Let w t be a continuous path of finite qvariation (q ≥ 1). Then we have the following estimate.
where C is a positive constant which depends on the constants δ, δ ′ , r 0 in conditions (A) and (B).
Proof. We proved the following estimate in [2,4] following the argument in [27]. where By combining this and Lemma 3.2, we complete the proof.
By applying main theorem, we obtain the following result. In this theorem, X is a 1/β-rough path on R n as in Theorem 3.4.
Theorem 5.5. Assume D satisfies conditions (A) and (B). Let σ ∈ Lip γ−1 (R d , L(R n , R d )) and ξ ∈D. Then there exists a controlled path (Z, Z ′ ) ∈ D 2β X (R d ) and a bounded variation path
Proof. We consider the following path-dependent equation: We called this assumption (H1) in [2]. This condition holds if D is convex and there exists a unit vector l ∈ R d such that inf {(l, n(x)) | n(x) ∈ N x , x ∈ ∂D} > 0. In [2], we proved the existence of solutions and gave estimates for them for rough path with finite 1/β-variation. That is, under the assumption σ ∈ C 3 b and the above assumptions on D, we proved for certain positive C. The above theorem shows that a similar estimate holds without the assumption (H1). The solution Y t we obtained in [2] is a 1st level path of a rough path which is a solution to the rough differential equation, the driving rough pathX is the rough path on R n ⊕ R d given bŷ This is proved by showing that there exists ε > 0 such that By the definition of the integral in (5.12) and Lemma 3.7 (2), Y t and Φ t in Theorem 3.4 also satisfies this estimate. Therefore, the solution obtained in this paper is also a 1st level path of a solution driven byX. Hence Theorem 5.5 is an extension of the main theorem in [2]. Actually, the condition (H1) is necessary to prove the existence of solutions to implicit Skorohod equations in [2]. However, we can prove the existence of solutions without (H1). We explain this in Section 8.
Remark 5.7. When D = R d and σ ∈ Lip γ , the mapping M is contraction mapping for small T ′ and we can apply contraction mapping theorem. Gubinelli [20] showed that the existence can be checked by applying Schauder's fixed point theorem when D = R d even if σ ∈ Lip γ−1 . When ∂D = ∅, as in Lemma 5.4, the Skorohod mapping Γ is 1/2-Hölder continuous in the uniform norm under the condition (A) and (B). If D is a half space of R d , Γ can be written down explicitly and it is easy to check that Γ is Lipschitz in the uniform norm. However, in general, the Lipschitz continuity does not hold any more. We refer the reader to [12] for this.

A continuity property of the solution map and support theorems
In this section, we continue the study of reflected rough differential equations on a domain D ⊂ R d which satisfies the conditions (A) and (B). Let h be a Lipschitz path on R n starting at 0. If σ is Lipschitz continuous, there exists a unique solution (Y (h, ξ) t , Φ(h, ξ) t ) to the reflected ODE in usual sense, We may omit denoting h, ξ. Moreover, Z(h) t = ξ + t 0 σ(Y s (h))dh s , Z t (h) ′ = σ(Y t (h)) and Φ(h) t are a unique pair of solution to the equation in Theorem 5.5 for the smooth rough path h s,t = (h s,t ,h 2 s,t ) defined by h, wherē Hence the solution (Z(h), R Z(h) , Φ(h)) satisfies the estimate (5.14) with the same constant C 1 , C 2 . It may not be clear to see the unique existence for the driving Lipschitz path h when we try to prove it via the (transformed) path-dependent ODE because the functional w → L(w) is not Lipschitz in C([0, T ]) with the uniform norm in general. However, by using some idea due to [28,22,27], the unique existence naturally follows from the view point of original reflected ODEs. We use the uniqueness in the argument below.
In general path-dependent setting, the trick for reflected case cannot be applied to prove the uniqueness of the solution. Of course, if we assume the Lipschitz continuity of L with respect to the uniform norm, we have the uniqueness of the solution although such an assumption does not hold for L associated with general reflected RDEs. By finding suitable assumptions on L, especially the uniqueness, the results in this section probably may be extended to more general path-dependent setting.
We consider the case where ω(s, t) = |t − s|. This means we consider Hölder rough paths. Let us denote the set of β-Hölder geometric rough path (1/3 < β ≤ 1/2) by C β g (R n ) which is the closure of the set of smooth rough paths in the topology of C β (R n ). In this paper, smooth rough path means the rough path defined by a Lipschitz path and its iterated integral.
(1) Let X ∈ C β g (R n ). Let {h n } be a sequence of smooth rough paths associated with Lipschitz paths h n such that lim n→∞ |||X − h n ||| β = 0. Let ((Z(h n ), σ(Y (h n )), Φ(h n ))) be the unique solution to (6.1). There exists a subsequence is a solution to (5.12) and satisfies the estimate (5.14) with the same constant.
Note that we do not mean by Sol β− (X) all solutions to (5.12) because we do not prove a priori estimates of the solutions.
Proof. For simplicity, we write Z(h n ) = Z n , Φ(h n ) = Φ n , Y n = Z n + Φ n . By Theorem 5.5, This implies there exists a subsequence {Z n k } and {Φ n k } which converges in C β− and C 1-var+,β− respectively and the limit point (Z, Φ) also belongs to C β × C β,1 . Moreover, we see that Z, R Z , Φ satisfies the estimate (5.14). Hence (Z, σ(Y )) ∈ D 2β X (R d ). By (3.27) in Lemma 3.7, there exists C > 0 such that for any n, Thus, we get which implies that Y t = Z t + Φ t and Φ t is a solution to (5.11) and (5.12). We need to prove (5.13). It holds that Since Y n k and Φ n k converges uniformly to Y and Φ respectively, · 0 σ(Y n k (s))dh n k (s) also converges to · 0 σ(Y s )dX s uniformly. Hence the continuity of L implies that (5.13) holds. The statement (3) follows from the uniqueness of the solution at h. This completes the proof.
By a similar argument to the proof of Theorem 4.9 in [2], we can prove the existence of universally measurable selection mapping of solutions as follows. Proposition 6.2. Assume the same conditions on D and σ in Lemma 6.1. Then there exists a universally measurable mapping satisfying the following.
(2) There exists a sequence of Lipschitz paths h n such that |||X − h n ||| β → 0 and I(h n , ξ) Proof. Below, we omit writing ξ. We consider the product space, and its subset We consider the closureĒ 0 in E. ThenĒ 0 is a separable closed subset of E. The separability follows from the continuity of the mapping h → (Z(h), σ(Y (h)), Φ(h)). Note that Sol β− (X) coincides with the projection of the subset ofĒ 0 whose first component is X. Hence by the measurable selection theorem (See 13.2.7. Theorem in [11]), there exists a universally measurable mapping I : C β g (R n ) → E such that I(X) ∈ {X} × Sol β− (X). By Lemma 6.1, this mapping satisfies the desired properties in (1) and (2).
Let X −h s,t be the translated rough path of X ∈ C β (R n ) by h. That is, the 1st level path and the second level path are given by, 14) These imply that if |||h − X||| β ≤ ε ≤ 1, then |||X −h ||| β ≤ 1 + 1 + 2 1+β T 1−β h 1 ε. By the definition of controlled paths, we immediately obtain the following.
and Φ ∈ C q-var,α (R d ) with Φ 0 = 0 and q, α,α satisfy the assumptions in Lemma 2.2. By the above lemma, we can define the integral t s σ(Y u )dX −h u and the estimates in Lemma 3.7 hold for this integral. Here Since |Ξ s,t | ≤ C(t − s) 1+α , the sum of these terms converges to 0. Thus we obtain We now consider the following condition on the boundary.
Definition 6.4 (Condition (C)). There exists a C γ function f on R d and a positive constant k such that for any x ∈ ∂D, y ∈D, n ∈ N x it holds that Usually, the function f is assume to be C 2 b in the condition (C). Here, we assume f ∈ C γ to make use of estimates in Lemma 3.7.
Under additional condition (C), we can prove the following explicit modulus of continuity.
Lemma 6.5. Assume that D satisfies the conditions (A), (B), (C) and σ ∈ Lip γ−1 . Let Y t (X, ξ), Z t (X, ξ), Φ t (X, ξ), Y t (h, ζ), Φ t (h, ζ) be a solution as in Lemma 6.1. Assume |||h−X||| β ≤ ε ≤ 1. Then there exists a positive constant C which depends only on σ, f, k such that Condition (C) implies that the fourth integral on the right-hand side of the equation (6.23) is always negative. By the estimates of the solution Y,Ỹ , Φ,Φ in the main theorem and the estimates in Lemma 3.7 and the Gronwall inequality, we obtain the desired estimate.
Let µ be a probability measure on C β g (R n ). Here we restrict the map I in Proposition 6.2 as It is now immediate to determine the support of the image measure of µ by I under a certain assumption on µ.
Theorem 6.6. Assume that the set of smooth rough path is included in the topological support of µ. Then where Supp (I * µ) denotes the topological support of I * µ.

Proof. The inclusion Supp (I
follows from Proposition 6.2 (2). The converse inclusion follows from the continuity in Lemma 6.1 (3) and the assumption on µ.
Finally, we consider the case of Brownian rough path. Let B t be the standard Brownian motion on R n . Let B N t be the dyadic polygonal approximation, that is, It is known that µ W (Ω 1 ) = 1 and the limit point of (B N s,t , B N s,t ) for B ∈ Ω 1 is called the Brownian rough path. Here µ W is the Wiener measure. We identify the element of B ∈ Ω 1 and the associated Brownian rough path B. For application to the support theorem of diffusion processes, it is important to obtain the support of B. The following is due to Ledoux-Qian-Zhang [21]. More general results can be found in [19]. Theorem 6.7. Let β < 1/2. The topological support of the probability measure of B on C β (R n ) is equal to the closure of the set of smooth rough paths in the topology of C β (R n ).
and consider a Stratonovich reflected SDE, The corresponding solution Y N t which is obtained by replacing B t by B N t is called the Wong-Zakai approximation of Y t . It is proved in [4,29,3] that the Wong-Zakai approximations of the solution to a reflected Stratonovich SDE under (A), (B) and (C) converge to the solution in the uniform convergence topology almost surely. Note that a similar statement under the conditions (A) and (B) is proved in [3]. By using the result in [4,29], a support theorem for the reflected diffusion under the conditions (A), (B) and (C) is proved by Ren and Wu [26]. We now prove a support theorem for the reflected diffusion under (A) and (B) by using the estimates in rough path analysis in this paper and in [3,4]. In [3,4], it is proved that the following hold. Let Then µ W (Ω 2 ) = 1.
(2) Assume (A), (B) hold. Then there exists an increasing subsequence of natural numbers N k such that µ W (Ω 3 ) = 1 holds where, It is proved in [3] that max 0≤t≤T |Y N t − Y t | converges to 0 in probability under (A) and (B). This and the moment estimate in [4] implies the almost sure convergence in Theorem 6.8 (2).
The following is a support theorem for reflected diffusion Y t .
Theorem 6.9. Assume D satisfies (A) and (B) and σ ∈ C 2 b . Let P Y be the law of Y . Let 0 < β < 1/2. The law of P Y is a probability measure on C β (R d ; Y 0 = ξ) and (6.30) The converse inclusion follows from Lemma 6.1 and Theorem 6.7.

Path-dependent RDE with drift
We consider path-dependent rough differential equations with drift term. It is necessary to study such kind of equations for the study of diffusion operators with drift. In the case of n-dimensional Brownian motion B t = (B 1 t , . . . , B n t )), one possible approach to include the drift term is to consider the geometric rough path defined byB t = (B t , t) ∈ R n+1 . By considering the geometric rough path which is naturally defined by Brownian rough path B s,t , we may extend all results in previous sections to the corresponding results for the solutions to the equation, However, we need to assume b ∈ Lip γ−1 (R d × R d , R d ) (2 < γ ≤ 3) to do so and the assumption on b is too strong. Hence, we explain different approach to deal with the drift term. Let us consider β-Hölder rough path, that is, the case where ω(s, t) = |t − s|. Let b ∈ C 1 b (R d × R d , R d ) and consider the equation, The meaning of this equation is as follows. The controlled path (Z, Z ′ ) and Φ are elements as in the definition of Ξ s,t and I(Z, Φ) s,t . In the present case, we consider Ξ s,t := σ(Y s )X s,t + (D 1 σ)(Y s )Z ′ s X s,t + (D 2 σ)(Y s ) t s Φ s,r ⊗ dX r + b(Y s )(t − s). (7.4) Then, (δΞ) s,u,t = (δΞ) s,u,t + (b(Y s ) − b(Y u )) (t − u) for s < u < t and By using this, we define For this, (I(Z, Φ) 0,t , σ(Z t , Φ t )) ∈ D 2β X and similar estimates to Lemma 3.7 holds. Moreover, Lemma 3.8 and Lemma 4.1 hold. Thus, Theorem 3.4 holds for suitable constants which depend only on σ, b, β, p, γ. In the case of reflected rough differential equation, all statements can be extended to differential equations with drift term b ∈ C 1 b . In particular, this results include the support theorem for reflected diffusions which is proved in [26].

Implicit Skorohod equation
We prove the existence of solutions to implicit Skorohod equations under the conditions (A) and (B). Proposition 8.1. Let ω(s, t) be a control function. Let η t be a continuous path on R d with η 0 = 0 and x t be a continuous path on R n with x 0 = 0 such that |η s,t | + |x s,t | ≤ ω(s, t) 1/q for some q ≥ 1. Let F be a linear mapping from R d ⊗ R n to R d . We consider the following implicit Skorohod equation: where y t ∈D (0 ≤ t ≤ T ) and Φ t is a continuous bounded variation path which satisfies Assume D satisfies (A) and (B). Then there exists a solution (y t , Φ t ) to (8.1). Further, this solution satisfies the following estimate.
This implies that there exists ε > 0 which depends only on C, q, F such that ω(0, T ′ ) ≥ ε.
If Φ ∆ 1-var,[0,T ] ≤ 1, then this estimate clearly holds for all 0 ≤ s, ≤ t ≤ T . Consequently, by choosing a sequence {∆ n } with |∆ n | → 0, we can conclude that Φ ∆n converges to a certain continuous bounded variation path Φ t on [0, T ] uniformly. By the continuity of the Skorohod mapping and the Young integration, y ∆n also converges to a certain continuous y t and (y t , Φ t ) is a desired solution. This proves the statement (1). The assertion (2) follows from (1).