A study of backward stochastic differential equation on a Riemannian manifold

Suppose $N$ is a compact Riemannian manifold, in this paper we will introduce the definition of $N$-valued BSDE and $L^2(\mathbb{T}^m;N)$-valued BSDE for which the solution are not necessarily staying in only one local coordinate. Moreover, the global existence of a solution to $L^2(\mathbb{T}^m;N)$-valued BSDE will be proved without any convexity condition on $N$.


Introduction
Consider the following systems of backward stochastic differential equation (which will be written as BSDE for simplicity through this paper) in R n , Here {B s } s 0 is a standard m-dimensional Brownian motion defined on a probability space (Ω, F , P), ξ is a F T -measurable R n -valued random variable, {Y s } s∈[0,T ] , {Z s } s∈[0,T ] are R n -valued predictable process and R mn -valued predictable process respectively. We usually call the function f : Ω × [0, T ] × R n × R mn → R n the generator of BSDE (1.1). Bismut [2] first introduced the linear version of BSDE (1.1). A breakthrough was made by Pardoux and Peng [31] where the existence of a unique solution to (1.1) was proved under global Lipschitz continuity of generator f . Still under the global Lipschitz continuity of f , Pardoux and Peng [32] has established an equivalent relation between the systems of forward-backward stochastic differential equation (which will be written as FBSDE through this paper) and the solution of a quasi-linear parabolic system. Another important observation by [31,32] was that BSDE (1.1) could be viewed as a non-linear perturbation of martingale representation theorem or Feynman-Kac formula.
It is natural to ask what is the variant for BSDE (1.1) on a smooth manifold N. When an n-dimensional manifold N was endowed with only one local coordinate, Darling [11] introduced a kind of N-valued BSDE as follows Here Y t = (Y 1 t , · · · , Y n t ) denotes the components of Y t under (the only one) local coordinate, and {Γ k ij } n i,j,k=1 are the Christoffel symbols for a fixed affine connection Γ on N. The most important motivation to define (1.2) is to construct a Γ-martingale with fixed terminal value (we refer readers to [14] or [20] for the definition of Γ-martingale). In fact, with the special choice of generator in (1.2) (which depends on the connection Γ), the solution {Y t } t∈[0,T ] of (1.2) is a Γ-martingale on N with terminal value ξ. Moreover, Blache [3,4] investigated a more general N-valued BSDE as follows when {Y t } was restricted in only one local coordinate of N, where f : Ω × N × T m N → T N is uniformly Lipschitz continuous. Moreover, the Lie group valued BSDE has been studied by Estrade and Pontier [15], Chen and Cruzeiro [8].
On the other hand, the N-valued FBSDE is highly related to the heat flow of harmonic map with target manifold N. Partly using some idea of N-valued FBSDE, Thalmaier [38] studied several problems concerning about the singularity for heat flow of harmonic map by probabilistic methods. We also refer readers to [3,4,12,17,21,22,33,39] for various methods and applications for the subjects on Γ-martingale theory and its connection to the study of heat flow of harmonic map.
For the problems on N-valued BSDE mentioned above, there are two mainly difficulties. One is the quadratic growth (for the variable associated with Z) term in the generator of (1.2) and (1.3), for which the arguments in [31,32] may not be applied directly to prove global (in time) or local existence of a solution to (1.2) and (1.3). Kobylanski first proved the global existence of a unique solution to the scalar valued (i.e. n = 1) BSDE (1.1) with generator having quadratic growth and bounded terminal value. Briand and Hu [5,6] extended these results to the case where the terminal value may be unbounded. The problem for multi-dimensional BSDE is more complicated, Darling [11] introduced a kind of condition on the existence of some doubly convex function, under which the global existence of a unique solution to (1.2) or (1.3) has been obtained in [11,3,4]. Xing and Zitković [40] proved global existence of a unique Markovian solution of (1.1) based on the existence of a single convex function. We also refer readers to [19,18,24,37] and reference therein for various results concerning about the local existence of a solution to (1.1) in R n with generator having quadratic growth under different conditions, including the boundness for Malliavin derivatives of terminal value(see Kupper, Luo and Tangpi [24]), small L ∞ norm of terminal value (see Harter and Richou [18] or Tevzadze [37]) and the special diagonal structure of generators(see Hu and Tang [19]).
Another difficulty for N-valued BSDE is the lack of a linear structure for a general manifold N. In fact, the expression (1.2) and (1.3) only make sense in a local coordinate which is diffeomorphic to an open set of R n . If we want to extend (1.2) and (1.3) to the whole manifold N, the multiplication or additive operators (therefore the Itô integral term) may not be well defined because of the lack of a linear structure on N. Due to this reason, [11,3,4] gave the definition of an N-valued BSDE which was restricted in only one local coordinate. Meanwhile in [8] and [15], the left (or right) translation on a Lie group has been applied to provide a linear structure for associated BSDE.
By our knowledge, for a general N, how to define an N-valued BSDE which are not necessarily staying in only one local coordinate is still unknown. In this paper, we will solve this problem for the case that N is a compact Riemannian manifold, see Definition 3.1 and 3.5 in Section 2 below. Moreover, as explained above, the existence of a doubly convex or a single convex function is required to prove the global existence of BSDE whose generator has quadratic growth. The existence of these convex functions could be verified locally in N (in fact, at every small enough neighborhood), see e.g. [3,4,21]. But except for some special examples (such as Cartan-Hadamard manifold), it is usually difficult to check whether such a convex function exists globally or not in N. In this paper, we will also prove the global existence of an N-valued solution to some special BSDE without any convexity conditions mentioned above, see Theorem 3.4 and 3.6 in Section 3.
We also give some remarks on our results as follows (1) Given a Riemannian metric on N, in Definition 3.1 and 3.5 we view N as a submanifold of ambient space R L , so the linear structure on R L could be applied in BSDE (3.1) and (3.5). The key ingredient in (3.1) and (3.5) is that the term with quadratic growth is related to second fundamental form A. As illustrated in the proof of Theorem 3.2, it will ensure solution of R L -valued BSDE (3.1) to stay in N. The advantage of our definition is that it does not require the solution to be restricted in only one local coordinate as in [11,3,4], therefore we do not need any extra condition on the generator f in (3.1) (see e.g. condition (H) in [3,4]). Moreover, as explained in Remark 3.1, our definition will be the same as that in [3,4] when we assume that the solution of (3.1) is situated in only one local coordinate.
(2) The equation (3.5) could be viewed as an N-valued FBSDE with forward equation being x + B t in T m . In Definition 3.5, we study the FBSDE with a.e. initial point x ∈ T m . This kind of solution has been introduced in [1,28,41,42] to investigate the connection between FBSDE and weak solution of a quasi-linear parabolic system. The motivation of Definition 3.5 is to study the global existence of a solution to N-valued BSDE for more general N, especially for that without any convexity condition. Theorem 3.6 ensures us to find a global solution of (3.5) for any compact Riemannian manifold N. By the proof we know the result still holds for non-compact Riemannian manifold with suitable bounded geometry conditions. These results will also be applied to construct ∇-martingale with fixed terminal value in Corollary 3.5.
(3) Theorem 3.2 provides a systematic way to obtain the existence of a solution to Nvalued BSDE, based on which we can apply many results on the R L -valued BSDE whose generator has quadratic growth directly. By Theorem 3.4, for any compact Riemannian manifold N, there exists a unique global Markovian solution to (3.1) when the dimension m of filtering noise is equal to 1, which gives us another example about global existence of a solution to N-valued BSDE without any convexity condition. Meanwhile, it also illustrates that for some BSDE whose generator has quadratic growth, not only the dimension n of solution (see the difference between scalar valued BSDE and multi-dimensional BSDE), but also the dimension m of filtering noise, will have crucial effects.
The rest of the paper is organized as follows. In Section 2 we will give a brief introduction on some preliminary knowledge and notations, including the theory of sub-manifold N in ambient space R L . In Section 3, we are going to summarise our main results and their applications. In Section 4, the proof of Theorem 3.2 and 3.4 will be given. And we will prove Theorem 3.6 in Section 5. Through this paper, suppose that N is an n-dimensional compact Riemannian manifold endowed with a Levi-Civita connection ∇. By the Nash embedding theorem, there exists an isometric embedding i : N → R L from N to an ambient Euclidean space R L with L > n. So we could view N as a compact sub-manifold of R L . We denote the Levi-Civita connection on R L by∇ (which is the standard differential on R L ). Let T N be the tangent bundles of N and let T p N be the tangent space at p ∈ N. For any m ∈ N + , we define

Preliminary knowledge and notations
as the tensor product of T N with order m.
For every p ∈ N ⊂ R L , by the Riemannian metric on N, we could split R L into direct sum as R L = T p N ⊕T ⊥ p N, where T ⊥ p N denotes orthonormal complement of T p N. Hence for every v ∈ R L and p ∈ N, we have a decomposition as follows, we usually call v T , v ⊥ the tangential projection and normal projection of v ∈ R L respectively. Given smooth vector fields X, Y on N, letX,Ȳ be the (smooth) extension of X, Y on R L (which satisfies thatX(p) = X(p),Ȳ (p) = Y (p) for any p ∈ N), then we have where X, Y are any smooth vector fields on N satisfying X(p) = u, Y (p) = v,X,Ȳ are any smooth vector fields on R L which are extension of X and Y respectively. The value of A(p)(u, v) is independent of the choice of extension X, Y ,X,Ȳ .
We define the distance from p ∈ R L to N as follows where |p − q| denotes the Euclidean distance between p and q in R L . Set Since N is compact, it is well known that there exists a δ 0 (N) > 0 such that dist 2 N (·) : B(N, 3δ 0 ) → R + and the nearest projection map P N : B(N, 3δ 0 ) → N are smooth, where for every p ∈ B(N, 3δ 0 ), P N (p) = q with q ∈ N being the unique element in N satisfying |p − q| = dist N (p). Moreover, for every p ∈ B(N, 3δ 0 ), suppose γ : [0, dist N (p)] → R L is the unique unit speed geodesic in R L (which is in fact a straight line) such that γ(0) = P N (p), γ(dist N (p)) = p, then for every p ∈ B(N, 3δ 0 ) it holds∇ Here we have used property γ ′ (0) = γ ′ (dist N (p)) since γ(·) is a straight line in R L . Moreover, we still have the following characterization for second fundamental form A, We choose a cut-off function φ ∈ C ∞ (R, R) such that It is easy to verify that p → φ(dist N (p)) is a smooth function on R L . Then we could extend the second fundamental form A defined by (2.2) toĀ : for all u ∈ R L . According to (2.4), (2.5) and the definition of φ, we know immediately thatĀ is a smooth map and We refer readers to [7,Section III.6], [13,Chapetr 6] or [26,Section 1.3] for detailed introduction concerning about various properties for sub-manifold N of R L .

Non-linear generator f
In this paper, we always make the following assumption for f .
T p N for every p ∈ N, u = (u 1 , · · · , u m ) ∈ T m p N. And there exists a C 0 > 0 such that for every p ∈ N, u ∈ T m p N, where ∇ p and ∇ u denote the covariant derivative with respect to the variables p in N and u in T m N respectively. Now we define a C 1 extensionf : Here φ : R → R, P N : B(N, 2δ 0 ) → N are the same as those in (2.5) and Π N (p) : R L → T p N denotes the projection map to T p N defined by (2.1) for every p ∈ N. Note that N is compact, combing (2.7) with (2.6) we obtain immediately following estimates for the extensionf : Here∇ p and∇ u denote the gradient in R L with respect to variables p and u respectively.

Space of Malliavin differentiable random variables
Through this paper, we will fix a probability space (Ω, F , P) and an R m -valued standard Brownnian motion {B t = (B 1 t , · · · , B m t )} t 0 on (Ω, F , P) with some m ∈ Z + . Let {F t } t 0 denote the natural filtration associated with {B t } t 0 . For simplicity we call a process adapted (or predictable) when it is adapted (or predictable) with respect to the filtration {F t } t 0 . Set ; P) be the gradient operator such that for every ξ ∈ F C ∞ b (R L ) with expression (2.9) and non-random η ∈ L 2 ([0, T ]; R m ), where · denotes the inner product in R m . For every ξ ∈ F C ∞ b (R L ), we define with respect to norm · 1,2 . It is well known that (D, F C ∞ b (R L )) could be extended to a closed operator (D, D 1,2 (R L )).
We define the space of N-valued Malliavin differentiable random variables as follows We refer readers to the monograph [30] for detailed introduction on the theory of Malliavin calculus.

Other notations
We use := as a way of definition. Let T m = R m /Z m be the m-dimensional torus.
For every x ∈ T m , p ∈ R L and r > 0, set B T m (x, r) := {y ∈ T m ; |y − x| < r} and B(p, r) := {q ∈ R L ; |q − p| < r}. Let dt and dx be the Lebesgue measure on [0, T ] and T m respectively. We denote the derivative, gradient and Laplacian with respect to the variable x ∈ T m by ∂ x i , ∇ x and ∆ x respectively. The covariant derivative for the variable in N is denoted by ∇, while we use∇ and∇ 2 to represent the first and second order gradient operator in R L respectively. We use , to denote both the Riemannian metric on T N and the Euclidean inner product on R L (note that for every p ∈ N and u, v ∈ T p N, we have u, v TpN = u, v R L ). Meanwhile let · denote the inner product in R m (in the tangent space of T m ). Without extra emphasis, we use a.s. and a.e. to mean almost sure with respect to P and almost every where with respect to Lebesgue measure on T m respectively. Throughout the paper, the constant c i will be independent of ε. For any q 1 and k ∈ N + , set 3 Main theorems and their applications section2.1

N -valued BSDE
In this subsection we are going to give the definition of N-valued BSDE through the BSDE on ambient space R L . Fixing a time horizon T ∈ (0, ∞), m ∈ N + and q ∈ (1, ∞), We usually write the components of a Here ξ : Then by applying Itô formula to ϕ(Y t ) (by the same computation in the proof of Proposition 3.8 below) it is not difficult to verify that [3,4] with Γ k ij being the Christoffel symbols associated with Levi-Civita connection ∇, where dϕ : T N → R n denotes the tangential map of ϕ : U ⊂ N → R n .
Remark 3.2. Note that the fundamental form A in (3.1) will depend on the Riemmannian metric (due to the decomposition of tangential direction and normal direction) and associated Levi-Civita connection ∇ on N. But we are not sure whether Definition 3.1 could be extended to the case that N is only a smooth manifold endowed with an affine connection. Now we will give the following result about the relation between a solution of Nvalued BSDE and a general R L -valued solution of BSDE (3.1).
With Theorem 3.2, we can obtain the existence of a unique N-valued solution of (3.1) by several known results on the R L -valued solution of general BSDE whose generator has quadratic growth.
Then we can find a positive constant Proof. According to (2.5) and (2.8) we have for every y 1 , y 2 ∈ R L and z 1 ,  [24] where associated coefficients are required to be globally Lipschitz continuous with respect to variable y, following the same procedure in the proof of [24, Theorem 3.1] we can still obtain the desired conclusion here, see also the arguments in [24, Example 2.2]) we can find a T 0 > 0 such that there exists a unique solution 2) for some C 2 > 0. Then applying Theorem 3.2 we obtain the desired conclusion immediately.
Similarly, according to [40], under some condition on the existence of a Lyapunov function, we can also obtain the unique existence of a global Markovian solution of N-valued BSDE (3.1) by applying Theorem 3.2, and we omit the details here.
Moreover, without any convexity condition (such as the existence of Lyapunov function or doubly convex function), we also have the unique existence of global Markovian solution of N-valued BSDE (3.1) when m = 1.

L 2 (T m ; N )-valued BSDE
Still for a given time horizon T ∈ (0, ∞), we define We call a pair of process (Y, Z) is a solution of L 2 (T m ; N)-valued BSDE (3.5) if we can find an equivalent version of (Y, Z) ∈ S ⊗ M 2 (T m ; N) (still denoted by (Y, Z) for simplicity of notation) such that for a.e. x ∈ T m the following equation holds for every t Here h : T m → N is an N-valued non-random function,Ā : R L → L(R L × R L ; R L ) and f : R L × R mL → R L are defined by (2.5) and (2.7) respectively. Now we will give the following results concerning about the global existence of a solution of L 2 (T m ; N) valued BSDE (3.5) for an arbitrarily fixed compact Riemannian manifold N.
} is a Lebesgue-null set in T m (which could be seen in the proof of Theorem 3.6).
Meanwhile, due to the lack of monotone condition on the generator (see the corresponding monotone conditions in [1,28,41,42]), it seems difficult to prove the uniqueness of the solution of L 2 (T m ; N)-valued BSDE (3.5).
Remark 3.4. The exceptional Lebesgue null set for x ∈ T m in (3.5) may depend on the choice of h. We do not know whether we can find a common null set Ξ which ensures (3.5) valid for every h ∈ C 1 (T m ; N) and x / ∈ Ξ.
We also have the following characterization for solution of Proof. If (3.5) holds for a.e. x ∈ T m , obviously we can verify (3.6). Now we assume that there exists a (Y, Z) ∈ S ⊗ M 2 (T m ; N) such that (3.6) holds a.s. for every ψ ∈ C 2 (T m ; R L ) and t ∈ [0, T ]. Since there exists a countable dense This, along with (2.8) implies immediately that for every ω / ∈ Π and x / Then by definition it is easy to verify that (Ỹ x , Z x ) satisfies (3.5) for every This along with the definition ofỸ implies immediately that given any ω / ) for a.e. x ∈ T m (the exceptional set for x ∈ T m may depend on t). Combing all the properties above we arrive at Hence Y andỸ is the same element in S 2 (T m ; N), so we can find an equivalent version (Ỹ , Z) of (Y, Z) which satisfies (3.5) a.s. for each x / ∈ Ξ 1 ∪ Ξ 2 .

Existence of ∇-martingale with fixed terminal value
In this subsection we will give an application of Theorem 3.4 and 3.6 on the construction of ∇-martingales, which also illustrates that Definition 3.1 and 3.5 are natural for an N-valued BSDE.
where Hess denotes the Hessian operator on N associated with Levi-Civita connection ∇. Then {M g t } t∈[0,T ] is a local martingale.
(2) Suppose (Y, Z) is a solution of L 2 (T m ; N)-valued BSDE (3.5). Given some g ∈ C 2 (N; R) and x ∈ T m we define {M g,x t } t∈[0,T ] by the same way of (3.7) with (Y t , Z t ) replaced by (Y x t , Z x t ). Then there exists a Lebesgue-null set Ξ ⊂ T m such that {M g,x t } t∈[0,T ] is a local martingale for every g ∈ C 2 (N; R) and x / ∈ Ξ.
Here in the third step above we have applied the property that ∇ḡ (p),X(p) = ∇g(p), X(p) for every p ∈ N due to (∇ḡ(p)) T = ∇g(p). Similarly for every p ∈ N and u ∈ T m p N (note that f (p, u) ∈ T p N) we obtain Combing all above properties with the fact that Y t ∈ N a.s. for every t ∈ [0, T ], s. for every t ∈ [0, T ]. Therefore we know immediately that M g t is a local martingale. Recall that we call the adapted process {X t } t∈[0,T ] a ∇-martingale if it is an Nvalued semi-martingale and for every g ∈ C 2 (N; R), is a local martingale. Here (dX t , dX t ) denotes the quadratic variation for X t . Then taking f ≡ 0, combing Theorem 3.4, Theorem 3.6 and Proposition 3.8 together we could obtain the following results concerning about the existence of ∇martingale on N with fixed terminal value in arbitrary time interval immediately. c2-2 Corollary 3.5. Suppose h ∈ C 1 (T m ; N) and T > 0, then the following statements hold.
(1) For a.e. x ∈ T m , there exists a ∇- By the choice of δ 0 and χ we have where δ ij denotes the Kronecker delta function (i.e. δ ij = 0 if i = j and δ ij = 1 when i = j), P k N (p) means the k-th components of P N (p), thus P N (p) = P 1 N (p), · · · P L N (p) .
According to definition ofĀ in (2.5) we have for every p ∈ B(N, δ 0 ), u = (u 1 , · · · , u L ) ∈ R L , Here in the last step above we have used the following equality which is due to the property ∂P N ∂p l (p) ∈ T p N and p − P N (p) ∈ T ⊥ p N. Combing above estimates together yields that Meanwhile for every p ∈ B(N, δ 0 ) and u = (u 1 , · · · , u L ) ∈ R L we have where the last step is due to the fact that ∂P N ∂p l (p) ∈ T p N,f (p, u) ∈ T p N and p−P N (p) ∈ T ⊥ p N. By all these estimates we arrive at Still by the definition of G,Ā andf we know that for every p ∈ R L /B(N, δ 0 ) and u ∈ R L , where in the second inequality above we have used the fact that G(p) δ 2 0 for every p ∈ R L /B(N, δ 0 ).
Combing above two estimates yields that Hence by (3.1), (4.2) and applying Itô's formula we get for every t ∈ [0, T ], Here we have applied (3.2) and the fact that G(ξ) = 0 a.s. (since ξ ∈ N a.s.). Taking the expectation in above inequality we arrive at So by Grownwall's inequality we obtain E[G(Y t )] = 0 which implies G(Y t ) = 0 and Y t ∈ N a.s. for every t ∈ [0, T ]. As explained in the proof of [24, Theorem 3.1] (which is due to the original idea in [32]), it holds that Y t ∈ D 1,2 (N) and we can find an equivalent version of Z i t and DY t (ω)(t) such that where (P) lim ε→0 denotes limit under the convergence in probability and e r i (t) := (t ∧ r)e i . Based on this and the property that Y t ∈ N a.s. we deduce that for every t, r ∈ [0, T ], r 0 DY t (s) · e i ds ∈ T Yt N, a.s..

Therefore we can find a version of Z i t such that
Now we have proved the desired conclusion.
Proof of Theorem 3.4. Now we assume that m = 1. In this proof we use the notation ∂ x , ∂ 2 xx to represent the first order and second order derivative with respect to x ∈ T 1 respectively.
According to standard theory of quasi-linear parabolic equation (see e.g. [ By the same arguments in the proof of Theorem 3.2 we will deduce that v(t, ·) ∈ N for every t ∈ [0, T 1 ). So we can replace the termsĀ,f by A and f in (4.3) respectively. At the same time, by (4.3) we have for every t ∈ (0, T 1 ), By direct computation we obtain Note that by (4.3) there is an orthogonal decomposition for ∂ 2 xx v as follows So we obtain Here the fourth step above follows from the fact xx v) T and the last step is due to Young's inequality.
Combing all above estimates together for I 1 , I 2 and I 3 we arrive at So for e(t, x), ∀ t ∈ (0, T 1 ). Applying Itô's formula to e(t − s, B s + x) directly we obtain for every δ ∈ (0, T 1 ) and t ∈ (δ, T 1 ), where ρ (0,x) (t, y) is the heat kernel defined by (5.4) below. This implies immediately that t2-2-6 t2-2-6 (4.4) sup So we have lim t↑T 1 sup x∈T 1 |∂ x v(t, x)| 2 < ∞, hence by standard theory of quasi-linear parabolic equation, we could extend the solution v of (4.3) to time interval (0, T 2 ] for some T 2 > T 1 . By the same arguments above we can prove that (4.4) holds with T 1 replaced by T 2 . Therefore repeating this procedure again, we can extend the solution v of (4.3) to time interval [0, T ] for any T > 0. Then for any fixed  In this section we will partly use the idea of [9,35] (with some essential modification for the appearance of termf ) to construct a solution to L 2 (T m ; N)-valued BSDE (3.5).
Through this section, let G : R L → R be defined by (4.1) and we define g : Inspired by [9,35], we are going to give several estimates for v ε . l3-1 Lemma 5.1. Suppose that v ε is the solution to (5.1), then for every ε > 0, it holds that where C 4 > 0 is a positive constant independent of ε and T .
Proof. We multiple both side of (5.1) with ∂ t v ε to obtain that for every s ∈ [0, T ], Since (t, x)) , it holds Here the last equality is due to the fact that G(p) = 0 for every p ∈ N and h(x) ∈ N for a.e. x ∈ T m . Meanwhile by (2.8) and Young inequality we have for every s ∈ [0, T ], where the positive constants c 1 , c 2 are independent of ε. Therefore combing all above estimates together yields that for every s ∈ [0, T ] Hence applying Grownwall lemma we can prove (5.2).
Here ϕ x 0 ∈ C ∞ (T m ; R) is a cut-off function which satisfies that ϕ x 0 (x) = 1 for every and in the last equality of (5.5) and (5.6) we extend ϕ x 0 to a function defined on R m with compact supports.
where C 5 is a positive constant independent of ε and z 0 = (t 0 , x 0 ).
Proof. In the proof, all the constants c i are independent of ε, z 0 and R. For every 1 < t < 4 and 0 < R R 0 min(1/2, x), applying integration by parts formula we obtain Meanwhile according to integration by parts formula we have, l3-2-4 l3-2-4 (5.10) Note that and ∇ x ρ (0,0) (t, x) = − x t ρ (0,0) (t, x), putting these estimates into (5.10) we arrive at By the same way we obtain Here the second step from the change of variable and the fact that ∇ x ϕ x 0 (x) = 0 only if x ∈ B(x 0 , 1/2)/B(x 0 , 1/4) (note that we still denote the extension of ϕ x 0 to a function on R m with compact support by ϕ x 0 ) , in the third step we have applied the property that sup R∈(0,1/2),t∈ [1,4] sup x∈B(x 0 ,1/2)/B(x 0 ,1/4) the last step is due to (5.2).
Handling J ε 4 (R) by the same way of that for J ε 3 (R) we arrive at Combing all above estimates for J ε i (R), i = 1, 2, 3, 4 together yields that Applying Grownwall's lemma we obtain (5.8) immediately.
The proof for (5.7) is similar with that for (5.8), so we omit the details here.
l3-3 Lemma 5.3. Given a ε ∈ (0, 1) and R > 0, suppose that v ε,R ∈ C 2 ((0, T ] × T m ; R L ) satisfies the following equation x) . Then there exists a positive constant C 6 > 0 such that for every ε ∈ (0, 1) and R > 0, Proof. By (2.3) (see e.g. [7, Section III.6]) we know that l3-3-2 l3-3-2 (5.13)∇dist N (p) ∈ T ⊥ P N (p) N, ∀ p ∈ B(N, 3δ 0 ), l3-3-3 l3-3-3 (5.14) Note that G(u) = χ dist 2 N (p) , by direct computation we have Here we use the notation to · to denote the total inner product for all the components in R m and R L . (For example, According to (5.14) and (5.11) we find that Here in the last step we have used the property that which is due to the fact thatf R (p, u) ∈ T P N (p) N (see the definition (2.7) off ) and ∇ dist 2 N (p) ∈ T ⊥ P N (p) N. Note that for every p ∈ B(N, 3δ 0 ), dist N (p) 2 = |p − P N (p)| 2 , hence for every p ∈ B(N, 3δ 0 ), Here δ ij denotes the Kronecker delta function (i.e. δ ij = 0 if i = j and δ ij = 1 when i = j), P k N (p) means the k-th components of P N (p), i.e. P N (p) = P 1 N (p), · · · P L N (p) . Based on this we obtain that when dist N (v ε ) 2δ 0 , where in the last step we have used the fact |v k ε, This along with the fact χ ′ 0 yields that when dist N (v ε,R ) 2δ 0 , Here second inequality follows from Young's inequality and the fact χ ′′ (s) = 0 only when s δ 2 0 , in the third inequality we have applied the property that By (5.11) again we have According to (2.8) we obtain immediately that where in the second inequality above we have used Young's inequality. Combing all above estimates for I ε,R 1 , I ε,R 2 , I ε,R 3 together we can prove the desired conclusion (5.12).
Remark 5.4. Due to the appearance of termf , the solution v ε to (5.1) is no longer scaling invariant. Therefore compared with the method in [9] and [35], in Lemma 5.2 and Lemma 5.3 above we could not only consider the situation for R = 1.
l3-4 Lemma 5.5. Suppose that Ψ ε (R) is defined (5.6). There exist positive constants θ 0 and R 0 ∈ (0, 1/2) such that if for some Here κ is a positive constant depending only on E 0 := T m |∇ x h(x)| 2 dx, R (but independent of ε), and C 7 is a positive constant independent of ε and R.
Proof. According to formula (5.7), (5.8) (and the comments in the proof [9,Lemma 4.4]), the proof is exactly the same as that of [35, Theorem 6.1], so we do not include the details here.
Now we start to prove Theorem 3.6 Proof of Theorem 3.6.
Meanwhile by standard approximation procedure it is easy to verify that for every fixed t ∈ [0, T ], lim y→0 T m |v(t, x + y) − v(t, x)| 2 dx = 0.
Combing all estimates above we deduce that t → Y · t (ω) is continuous in L 2 (T m ; R L ) a.s.. According to this and the property that sup t∈[0,T ] v(t, ·) L 2 (T m ;R L ) < ∞ we can prove Hence by all the properties above we have verified that Y ∈ S 2 (T m ; R L ). At the same time, since ∇ x v ∈ L ∞ ([0, T ]; L 2 (T m ; R L )), we have immediately that Z ∈ M 2 (T m ; R L ).
Moreover, for every ψ ∈ C 2 (T m ; R L ) it holds that By Lemma 5.6 we know that Σ has locally finite m-dimensional Hausdorff with respect tod, so under the Lebesgue measure on [0, T ]×T m , Σ is a null set. Therefore according to (5.27) it holds that G v(t, x) = 0, dt × dx − a.e. (t, x) ∈ [0, T ] × T m , which implies that v(t, ·) ∈ L 2 (T m ; N) for a.e. t ∈ [0, T ]. Combing this with (5.26) we know that for every fixed t ∈ [0, T ].
and v(t, ·) ∈ L 2 (T m ; N). By this we know for every fixed t ∈ [0, T ] and 1 i m, Hence Y · t = v(T − t, B t + ·) ∈ L 2 (T m ; N) for every t ∈ [0, T ] and Z i,x x, ω). Note that it has been proved that Y ∈ S 2 (T m ; R L ), Z ∈ M 2 (T m ; R L ) in Step (i) above, so we have (Y, Z) ∈ S ⊗M 2 (T m ; N).
Note that we could find a collection of countable open neighborhoods {Q i (z 0 )} ∞ i=1 as above to cover [0, T ] × T m /Σ, by diagonal principle there exists a subsequence {v ε k } such that (since the measure of Σ is zero under dt × dx) This together with (5.27) implies immediately that (taking a subsequence if necessary) for t3-1-11a t3-1-11a (5.35) lim Meanwhile by (2.8) and (5.2) it is easy to verify thatf (v ε k (T − t, x), ∇ x v ε k (T − t, x)) is uniformly integrable with respect to dt × dx since where {ν i (p)} L−n j=1 is an orthonormal basis of T ⊥ p N at p ∈ N, in the second equality above we have used the factf (v, ∇ x v) ∈ T v N, ∂ t v ∈ T v N for a.e. x ∈ T m , and the last step follows from the standard property of sub-manifold (see e.g. [26, Section 1.3]).
Given (5.37) and applying the same procedures in the proof of [9, Theorem 3.1(Page 95)] (using again the fact that the measure of Σ is zero under dt × dx) we obtain that for everyψ ∈ L ∞ ([0, T ] × T m ; R L ),