Distribution-Path Dependent Nonlinear SPDEs with Application to Stochastic Transport Type Equations

By using a regularity approximation argument, the global existence and uniqueness are derived for a class of nonlinear SPDEs depending on both the whole history and the distribution under strong enough noise. As applications, the global existence and uniqueness are proved for distribution-path dependent stochastic transport type equations, which are arising from stochastic fluid mechanics with forces depending on the history and the environment. In particular, the distribution-path dependent stochastic Camassa--Holm equation with or without Coriolis effect has a unique global solution when the noise is strong enough, whereas for the deterministic model wave-breaking may occur. This indicates that the noise may prevent blow-up almost surely.


Introduction
To describe the evolutions of stochastic systems depending on the history and micro environment, distribution-path dependent SDEs of the following type (1.1) dX(t) = b(t, Xt, LX t )dt + σ(t, Xt, LX t )dW (t), X(0 have been studied intensively investigated, see for instance [1,23,36,37,40] and references therein.However, the existing study in the literature does not cover distribution-path dependent nonlinear SPDEs containing a singular term which is not well-defined on the state space.The main purpose of this paper is to solve a class of such SPDEs including transport type fluid models.Nowadays there exists an abundant amount of literature concerning the stochastic fluid models under random perturbation which we do not attempt to survey here, and we recommend the lecture notes [10,14] and the monographs [2,29] for readers' references.On one hand, in the real world, it is natural that the random perturbation may rely on both the sample path due to inertia, and averaged stochastic interactions from the environment.On the other hand, to the best of our knowledge, almost nothing is known if the randomness in the stochastic fluid models also depends on the distribution and the path of unknown variables, i.e., distribution-path dependent stochastic fluid models.For such problems, the fundamental question on the well-posedness (even merely the existence) of solutions remains open.
Particularly, although the (distribution-path independent) stochastic transport equations have been intensively investigated (see for example [12,13,15,31,32,33]), there is no any study on distribution-path dependent stochastic transport type equations.
To study distribution-path dependent stochastic fluid models, we may need extend (1.1) to infinite dimensional case, i.e., assuming that X takes value in a separable Hilbert space H.If this is the case, a singular term, which is not well-defined on H, may occur and the existing study in the literature does not cover this case.More precisely, we consider the case that (1.1) contains one more singular drift term B taking value in a larger separable Hilbert space B such that H ֒→ B, i.e., (1.2) dX(t) = {B(t, X(t)) + b(t, Xt, LX t )} dt + σ(t, Xt, LX t )dW (t), X(0 Indeed, when we consider certain stochastic fluid models in Sobolev spaces H = H s , if B(t, X(t)) involves ∇X or some derivatives of X (see Examples 1.1 and 1.2), then B(t, X(t)) may not be expected to be in H = H s .Particularly, when B(t, X(t)) = −(X(t) • ∇)X(t), (1.2) reduces to the following transport type equaionts We refer to Sections 1.1 and 1.2 for the precise meaning of the notations and precise setting of (1.2) and (1.3), respectively.Before going further, we would like to explain that working with the abstract framework in (1.2) entails some difficulties: (a) Since we want to cover some stochastic fluid models in the abstract system (1.2), we only assume that the coefficients B, b and σ are locally Lipschitz in X.As a result, we do not a prior know that the solution exists globally in time.This bring us an essential difficulty.More precisely, we notice that the distribution, as a global object on the path space, does not exist for explosive stochastic processes whose paths are killed at the life time.As a result, to investigate distribution dependent SDEs/SPDEs, we have to either consider the non-explosive setting or modify the "distribution" by a local notion (for example, conditional distribution given by solution does not blow up at present time).
(b) Again, because the coefficients are only locally Lipschitz in X, we will have to localize them (by using stopping times) when we need to fix the changing Lipschitz constants.For instance, this happens when the uniqueness is considered.Then we will be confronted with the difficulty that distribution can not be controlled by any local condition, again.And we need to identify some appropriate topology under which the distribution can be measured locally.
(c) Because of the singular term B(t, X), compared to classical case, the Itô formula is no longer available.Indeed, to estimate X 2 H , to use the Itô formula in a Hilbert space (cf.[9,19]), the H inner product (B(t, X), X) H is required to be well-defined.But it is not because we only assume that B takes value in B ←֓ H.Likewise, to apply the Itô formula under a Gelfand triplet ( [28,35]), the dual product B(t, X), X B B * needs to be well-defined, where B * is the dual space of B with respect to H.Because H ֒→ B, we see that B * ֒→ H.However, we do not a prior know that the solution X takes value in B * because we only assume X(0) ∈ H.
The first major goal of this paper is to establish an abstract framework for (1.2).The second goal of this work is to apply the abstract theory for (1.2) to (1.3), which gives some new results for some ideal fluid systems.
• To achieve the first goal, we introduce the precise assumptions in Section 1.1 (see Assumption (A)).
Then we provide our main results for (1.2) in Theorem 1.1.The key requirements for the proof are the assumption on the existence of appropriate Lipschitz-continuous and monotone regularizations for the singular term B. For the difficulty (a), in this paper we restrict our attention to the nonexplosive case only.To this end, we assume that the noise grows fast enough (cf.(A3)), and then we will show that the blow-up of solutions can be prevented.For the difficulty (b), we introduce a "local" Wasserstein distance (see (1.7)) and assumption (A5) to measure the difference of two measures, which enables us to prove the uniqueness.By introducing a mollifier satisfying certain estimates (see assumption (A4)), we can overcome the difficulty (c).

A general framework
Let H, U be two separable Hilbert spaces, and let L2(U; H) be the space of Hilbert-Schmidt operators from U to H with Hilbert-Schmidt norm • L 2 (U;H) .Throughout the paper we fix a time T > 0. For a Banach space M, let P T,M be the set of probability measures on the path space C T,M := C([0, T ]; M).We also consider the weakly continuous path space Both C T,M and C w T,M are Banach spaces under the uniform norm Let P w T,M be the space of all probability measures on C w T,M equipped with the weak topology.Denote P T,M = {µ ∈ P w T,M : µ(C T,M ) = 1}.For any map ξ : [0, T ] → M and t ∈ [0, T ], the path πt(ξ) of ξ before time t is given by Then the marginal distribution before time t of a probability measure µ ∈ P w T,M reads Let L ξ stand for the distribution of a random variable ξ.When more than one probability measure are considered, we denote L ξ by L ξ|P to emphasize the reference probability measure P.
for an orthonormal basis {ei} i≥1 of U and a sequence of independent one-dimensional Brownian motions {β i } i≥1 on (Ω, {Ft} t≥0 , P).
Consider the following nonlinear distribution-path dependent SPDE on H: where, for some separable Hilbert space B with H ֒→֒→ B (" ֒→֒→ " means the embedding is compact), are progressively measurable maps.(2) A couple ( XT , WT ) = ( X(t), W (t)) t∈[0,T ] is called a weak solution of (1.2), if there exists a complete filtration probability space ( Ω, { Ft}t≥0, P) such that WT is a cylindrical Brownian motion on U and XT is a solution of (1.2) for ( WT , P) replacing (WT , P).
Since both X(t) and t 0 b(s, Xs, LX s )ds + t 0 σ(s, Xs, LX s )dW (s) are stochastic processes on H, so is t 0 B(s, X(s))ds, although B(s, X(s)) only takes values in B.
To ensure the non-explosion such that the distribution is well defined, we will take a Lyapunov type condition (A3) below.We write Consider the following "Wasserstein distance" induced by V ∈ V : where C (µ, ν) is the set of couplings of µ and ν.When Here and in the sequel, we set inf ∅ = ∞ by convention.We define the "local" L 2 -Wasserstein distance by , µ, ν ∈ P T,B .
We write µ ∈ P V T,H if µ ∈ P T,H and In general, • V may not be a norm, but we use this notation for simplicity.A subset Let T > 0 be arbitrary.For any N > 0, let Assumptions (A).Assume that H ֒→֒→ B is dense, and there exists a dense subset H0 of B * , the dual space of B with respect to H such that the following conditions hold for B, b and σ in (1.5).
and for any N ≥ 1 there exists a constant CN > 0 such that sup t∈[0,T ],η∈C .
(A4) There exists a sequence of continuous linear operators {Tn} n≥1 from B to H with (1.9) such that for any N ≥ 1, there exists a constant CN > 0 such that (1.10) sup x H ≤N,n≥1 | TnB(t, x), Tnx H | ≤ CN .
(A5) There exist constants K, ε > 0 and an increasing map 2) has a unique solution with initial value X0.Now we give some remarks regarding the proof of Theorem 1.1 and Assumption (A).Remark 1.1.Except for the difficulties (a), (b) and (c), we will be confronted with one additional technical obstacle.Indeed, we notice that the singular term B is in general not monotone in the sense of [34] (see also [35]).Therefore, even coming back to the distribution-path independent case, the Galerkin approximation under a Gelfand triple developed for quasi-linear SPDEs does not work for the present model.To overcome this obstacle, we will take a different regularization argument.The proof of Theorem 1.1 includes two main steps: Step 1: regular case We first establish the solvability of the regular case, i.e., B = 0 (see Proposition 2.1).In this step, we need (A1) as (A1) describes the local Lipschitz continuity of the regular coefficients b(t, ξ, µ) and σ(t, ξ, µ) in (ξ, µ) under the metric induced by • H and W 2,B .Recalling the difficulty (a) mentioned before, we restrict our attention to the non-explosive case.Hence we need the assumption (A3) which is a Lyapunov type condition ensuring the global existence of the solution.Furthermore, (A5) means that the dependence on the distribution of the coefficients is asymptotically determined by the distribution of local paths, and it will be used to prove the pathwise uniqueness.Actually, (A5) is proposed to overcome the difficulty (b).
Step 2: singular case Then we will propose a regularization argument to establish existence and uniqueness to (1.2).Therefore in (A2) we assume that the singular term B ∈ B can be approximated by a regular term Bn ∈ H with certain nice properties.The result in Step 1 guarantees that the approximation problem (see (3.1), where B in (1.2) is replaced by Bn) can be uniquely solved on [0, T ] for any given T > 0 and we refer to Proposition 2.1.Then we use the martingale approach to pass limit to the original problem (1.2), where we need the continuity of the coefficient in µ under the weak topology (see (A1)).Precisely speaking, by Prokhorov's theorem and Skorokhod's theorem, we can get almost sure convergence of the approximation solutions relative to a new probability space.Then by the martingale representation theorem, we can identify the limit of the stochastic integral.Finally we establish the uniqueness, which together with the Yamada-Watanabe type result gives the existence and uniqueness of a pathwise solution.Finally, as mentioned before, the Itô formula can not be applied to to X(t) 2 H directly (see difficulty (c)).Hence it is not obvious to obtain the time continuity of the solution in H.And we need to mollify the equation first by using some mollifiers.Hence (A4) provides certain properties of such mollifiers.

Distribution-path dependent stochastic transport type equations
Let d ≥ 1 and T d = (R/2πZ) d be the d-dimensional torus.Let ∆ be the Laplacian operator on T d , and let i denote the imaginary unit.Then {e i k,• } k∈Z d consists of an eigenbasis of the Laplacian ∆ in the complex L 2 -space of the normalized volume measure µ(dx For a function f ∈ L 2 (µ), its Fourier transform is given by and By the spectral representation, for any s ≥ 0, we have Now, we recall the stochastic transport SPDE (1.3) on H s : where W (t) is the cylindrical Brownian motion on are measurable.
To apply Theorem 1.1, we make the following assumptions on b and σ.
2, s ′ = s − 1 and T > 0 be arbitrary.We assume that the following conditions hold for H = H s and B = H s ′ .
(B3) There exist constants K, ε > 0 and an increasing map Then we have the following result: If, moreover, (B3) holds, then (1.3) has a unique solution.
Below we give some remarks concerning Theorem 1.2.Remark 1.2.We first notice that (1.3) does not contain the viscous term ∆X(t), which provides additional regularization effect to make the problem of existence easier, see [8,Chapter 5].Besides the existence and uniqueness, it is interesting to clarify the effect of noise on the properties of solutions.We notice that existing results on the regularization effects by noises for transport type equations are mainly for linear equations or for linear growing noises, see for instance [12,13,15,27,32,33] for linear transport equations, and [16,20,38,39] for linear noise.For nonlinear equations with nonlinear noise, there are examples with positive answers showing that noises can be used to regularize singularities caused by nonlinearity.For example, for the stochastic 2D Euler equations, coalescence of vortices may disappear [16].But there are also counterexamples such as the fact that noise does not prevent shock formation in the Burgers equation, see [14].Therefore, for nonlinear SPDEs, what kind of nonlinear noise can prevent blow-up is a question worthwhile to study.In the current work, the main idea is to use the stochastic part of the equation to avoid any blow-up phenomena that could arise under the presence of the singular drift.Hence we use the Lyapunov type condition (B2) to measure how strong the noise term needs to be (see also [26,Theorem III.4.1] for the finite dimensional case and [3] for the stochastic nonlinear beam equations).In this way, the noise effect given by the large enough noise is macroscopic and it is different from many previous works, where small noise can also bring regularization effect, see for example [15,16].Besides, the noise structure in [15,16] are transport noise in the Stratonovich sense.A priori, it is not clear how to interpret the equation (1.3).In the current work, our main interest are mainly mathematical and we believe that searching for nonlinear noise such that blow-up can be prevented is important because it helps us understand the regularizing mechanisms of noise.This in turn brings us one further step to find the really correct and physical noise which provides such regularization.
Remark 1.3.We remark here that there is a gap between the index s > d 2 + 2 in Theorem 1.2 and the critical value s > d 2 + 1 such that H s ֒→ W 1,∞ .Formally speaking, on one hand, because the transport term (u • ∇)u loses one order of regularity, we have to consider uniqueness in H s ′ with s ′ ≤ s − 1, i.e., we ask B = H s ′ in (B3).One the other hand, since (u H s for smooth u, to verify (B2), we have to pick s ′ ≤ s − 1 such that B = H s ′ ֒→ W 1,∞ .Therefore we have to require s − 1 > d 2 + 1.However, if we only consider local solutions in H s without assuming (B2) (as is explained before, in this case the distribution has to be modified), then s > d 2 + 1 will be enough.To conclude this section, we present below two examples to illustrate Theorem 1.2.
Example 1.1.Let s, s ′ = s − 1 be in assumption (B) and take U = H s .Let µ(F ) = F dµ for F ∈ L 1 (µ), and take where α, β > 0 are constants to be determined, and (1) x0 ∈ H s with x0 H s = 1 is a fixed element; (2) F b , Fσ : C T,H s ′ → R m are bounded and Lipschtiz continuous for some m ≥ 1; 2 and β is large enough, then for any probability measure µ0 on H s with µ0( 2) has a weak solution ( XT , WT ) with L X(0)| P = µ0, which is continuous in H s and satisfies 2) has a unique solution, which is continuous in H s and satisfies Proof of Example 1.1.Let α ≥ 1 2 , and take V (r) = log(1 + r) ∈ V .By Theorem 1.2, we only need to verify conditions (A1), (B2) with H = U = H s , B = H s ′ , H0 = H s+1 and large enough β > 0, and finally prove (B3) with m = 1 and F b (ξ) = Fσ(ξ) = ξ T,H s ∧ R.
To begin with, it is easy to see that the weak convergence in P T,B is equivalent to that in the metric Then (1)-( 4) and H ֒→ B imply that for any N ≥ 1 there exists a constant CN > 0 such that for all Therefore, (A1) holds.
Example 1.2.Now we consider a family of stochastic models which are more physical relevant.Let s, s ′ be in assumption (B) with d = 1 and take U = H s .We focus on the following PDE (1.17) where ai (i = 0, 1, 2, 3, 4) are some constants.Before we consider their stochastic versions, we briefly recall some background of (1.17).Due to the abundance of literature on (1.17), here we only mention a few related results.If a1 = 1, a2 = 1 2 and a0 = a3 = a4 = 0, (1.17) becomes the Camassa-Holm equation Equation (1.18) models the unidirectional propagation of shallow water waves over a flat bottom and it appeared initially in the context of hereditary symmetries studied by Fuchssteiner and Fokas [17] as a bi-Hamiltonian generalization of KdV equation.Later, Camassa and Holm [4] derived it by approximating directly in the Hamiltonian for Euler equations in the shallow water regime.It is well known that (1.18) exhibits both phenomena of (peaked) soliton interaction and wave-breaking.When a1 = b 2 , a2 = 3−b 2 with b ∈ R and a0 = a3 = a4 = 0, (1.17) reduces to the so-called b-family equations, cf.[18,7], When a0 ∈ R, a1 = 1 , a2 = 1 2 and a3 = a4 = 0, (1.17) is a dispersive evolution equation derived by Dullin et al. in [11] as a model governing planar solutions to Euler's equations in the shallow-water regime.Finally, when ai (i = 0, 1, 2, 3, 4) are suitably chosen, (1.17) becomes the recently derived rotation-Camassa-Holm equation describing the motion of the fluid with the Coriolis effect from the incompressible shallow water in the equatorial region, cf.[21, equation (4.9)].In this case, a3 = 0 and a4 = 0 so that the equation has a cubic and quartic nonlinearities.
For this family of PDEs, if distribution-path dependent noise is involved, which can be explained as the weakly random dissipation, cf.[ and v ∈ H s is a fixed element such that v H s = 1 and σ0 satisfies condition (4) with m = 1 as in Example 1.1.Let It is easy to show that there is a constant C > 0 such that satisfies the the estimates for drift part as in (B1) and (B3).Going along the lines as in the proof of Example 1.1 with minor modification, we can see that if β > 1 is large enough and then for any u(0) ∈ L 2 (Ω → H s , F0, P), (1.20) has a unique solution with continuous path in H s and Therefore, in contrast to the deterministic case where wave-breaking phenomenon may occur in finite time, see [5,6,41], the blow-up is prevented when the growth of the noise coefficient in (1.20) is faster enough.
The remainder of the paper is organized as follows.In Section 2, we consider the regular case where B = 0. Then we prove Theorem 1.1 and Theorem 1.2 in Section 3 and Section 4 respectively.
Then assumption (A) for B = 0 implies the following assumption (C): Assumptions (C).With the same notation as in (1.8), we assume the following, for some Hilbert space B with dense and compact embedding H ֒→֒→ B: (C1) For any N ≥ 1, there exists a constant CN > 0 such that for any ξ, η ∈ C T,H,N and µ, ν ∈ P V T,H , we have that P-a.s.
(C4) There exist constants K, ε > 0, an increasing map C• : N → (0, ∞),such that for any ξ, η The main result of this section is the following.
To prove this result, we first consider the global monotone situation, and then extend to the local case.
Proof.By (2.3), the uniqueness follows from Itô's formula and Grönwall's inequality.Below we only prove the existence by using the procedure as in [40].Let X 0 (t) ≡ X0 and µ If for some n ≥ 1 we have a continuous adapted process By (2.3) and induction, we can construct a sequence of continuous adapted processes {X T } n≥1 is a Cauchy sequence in L 2 (Ω → C T,H ; P), and hence has a limit XT in this space as n → ∞, so that due to (2.3) and the continuity of b(t, ξ, µ) and σ(t, ξ, µ) in (ξ, µ), we may let n → ∞ in (2.4) for t ∈ [0, T ] to conclude that XT is a solution of (2.1).By (2.3) and Itô's formula, for where Then for λ > 0, (2.5) We observe that T,H .
Taking λ = 6(K + c2), we arrive at Hence, for any n ≥ 2 we have T } n≥1 is a Cauchy sequence as desired.Lemma 2.3.Assume (C1)-(C3).For any T > 0, X(0) ∈ L 2 (Ω → H, F0, P), and any Proof.By (C1), we see that this equation has a unique solution up to the life time τ.Now we prove that τ > T (i.e. the solution is non-explosive) and (2.8).To this end, with the convention inf ∅ = ∞ we set By (C3) and Itô's formula, we obtain This gives rise to Thus, P(τ > T ) = 1.Moreover, by (2.9) and BDG inequality, we obtain that for all n ≥ 1, Combining this with (2.10), we arrive at As C does not depend on n, letting n → ∞ and noting (2.10) give rise to (2.8).
Proof of Proposition 2.1.The estimate (2.2) is implied by Lemma 2.3 with µt = LX t once existence has been established.So, it remains to prove the existence and uniqueness.
(a) Existence.To construct a solution using Lemma 2.2, we make a localized approximation of b and σ as follows.For τ ξ n in (1.6), let and define By (C1), we see that for each n ≥ 1, b n and σ n satisfy (2.3) for γ = 1 and some constant K depending on n.Therefore, by Lemma 2.2, the equation (2.12) has a unique solution.By the definition of φn, we have (2.13) Moreover, for any measurable set A ⊂ C T,H , we have So, by (C3) and applying Itô's formula to V ( X n (t) 2 H ) up to time T ∧ τ n , as in (2.11), we derive Consequently, the stopping times Next, by (C1) and (2.12), we find a constant CN > 0 such that for any n ≥ N , (2.17) Indeed, for any l ≥ 1, by (C1), (2.12) and BDG inequality, there exists a constant C N,l > 0 such that Let k ∈ N such that kε ∈ [T, T + ε).We find some constant c(l) > 0 such that Therefore, by Jensen's inequality, we obtain This implies the pathwise uniqueness up to time t0 := {ε/C} ∧ T.
(b2) If t0 = T , then the proof is finished.Otherwise, since Zt 0 = 0, (2.25) implies Using Fatou's lemma and Grönwall's inequality as before, we arrive at Thus, the uniqueness holds up to time (2t0) ∧ T .Repeating the procedure for finite many times, we prove the uniqueness up to time T .
3 Proof of Theorem 1.1 and Consequently, the stopping times , n, N ≥ 1.
Due to this and (3.2), one can use the Arzelá-Ascoli theorem for measures to find that {µ , where WT is a continuous process on a separable Hilbert space Ũ such that the embedding U ⊂ Ũ is Hilbert-Schmidt, and is a continuous process on B. By the Prokhorov theorem, there exists a subsequence {n k } k≥1 such that µ (n k ) → µ weakly in P T,B , and Λ n k → Λ weakly in the probability space on P(C 2 T,B × Ũ).Then the Skorokhod theorem guarantees that there exists a complete filtration probability space ( Ω, { Ft}t≥0, P) and a sequence ( Xn Summarizing this, (A1), (A2), (3.2), (3.6) and (3.9), and then letting k → ∞, we derive It is easy to see that (A1), (A2) and (3.7) imply that for some constant CN > 0, is a continuous process on B as well.On account of (3.7) and (3.8), we identify that ( XT , WT ) is a weak solution of (1.2).
Proof of (ii) in Theorem 1.1.Now, assume (A4).We aim to prove the continuity of X(t) in H. Since X(t) is an adapted continuous process on B, and hence weak continuous in H, it suffices to prove the continuity of [0, T ] ∋ t → X(t) H .By (3.8), we only need to prove the continuity up to time τN for each N ≥ 1, where τN is given in (3.8).If X ∈ H, then B(t, X) ∈ B and B(t, X), X H does not make sense, therefore we can not use the Itô formula to X 2 H directly.To overcome this difficulty, we consider Tm X 2 H firstly, where Tm is the operator as in (A4).Applying Tm to (1.2) with noting (A4), we see that Tm X(t ∧ τN ) =Tm( X(0)) + Combining this with (A1), (A4) and the Itô's formula, we find a constant CN > 0 such that Therefore, Kolmogorov's continuity theorem ensures the continuity of t → X(t ∧ τN ) H as desired.
As is explained in step (b2) in the proof of Proposition 2.1, we assume that CN → ∞ as N → ∞ and it suffices to prove the pathwise uniqueness up to a time t0 > 0 independent of the initial value X(0).Let τn be defined by (2.21).As is shown in (b1) in the proof of Proposition 2.1, it follows from (A5), Itô's formula and BDG inequality that there is a constant K0 > 1 such that Jnf, g H s = f, Jng H s , Tnf, g H s = f, Tng H s , f, g ∈ H s , (4.4) where for two sequences of positive numbers {an, bn} n≥1 , an bn means that an ≤ cbn holds for some constant c > 0 and all n ≥ 1.Moreover, we write an = o(bn) if limn→∞ b −1 n an = 0. Then (4.6) X − JnX H r = o(n r−s ), 0 ≤ r ≤ s, X ∈ H s , and for any r ≥ s, (4.7) JnX H r n r−s X H s uniformly in X ∈ H s .
To verify conditions in Theorem 1.1, we need more properties of Jn, Tn and D s .In general, the commutator for two operators P, Q is given by [P, Q] := P Q − QP.
Lemma 4.1.There exists a constant C > 0 such that Proof.It is worth noticing that in 1-D case, the above commutator estimate has been established for a different mollifier on the whole space, see [22].In current setting, periodicity is required and the mollifier is different, so we present also the proof here. .
Hence, it suffices to find a constant c > 0 such that Noting that , by Tn = (I − 1 n ∆) −1 , (1.12), and (1.13), we find a constant c > 0 such that Let Z = X − Y .By H s ֒→ H s ′ ֒→ W 1,∞ and Lemma 4.2, we find constants c1, c2 > 0 such that which is the desired estimate.
(see Lemma 4.1), from which we can verify the assumptions introduced Section 1.1 and obtain global existence and uniqueness of solutions in Sobolev spaces.This results is stated in Theorem 1.2.Two examples of Theorem 1.2 are given.The first one, cf.Example 1.1, is a general nonlinear stochastic transport equation, and the second one is the the distribution-path dependent stochastic Camassa-Holm equation with or without Coriolis effect, cf.Example 1.2.