On a stochastic Camassa-Holm type equation with higher order nonlinearities

The subject of this paper is a generalized Camassa-Holm equation under random perturbation. We first establish local existence and uniqueness results as well as blow-up criteria for pathwise solutions in the Sobolev spaces $H^s$ with $s>3/2$. Then we analyze how noise affects the dependence of solutions on initial data. Even though the noise has some already known regularization effects, much less is known concerning the dependence on initial data, neither for general case nor for single instances. As a new concept we introduce the notion of stability of exiting times and construct an example showing that multiplicative noise (in It\^{o} sense) cannot improve the stability of the exiting time, and simultaneously improve the continuity of the dependence on initial data. Finally, we obtain global existence theorems and estimate associated probabilities.


Introduction
We consider a stochastic version of the generalized Camassa-Holm equation which is given for k ∈ N by In (1.1), h : R + × R → R is some nonlinear function andẆ is a cylindrical Wiener process. For h = 0 and k = 1, equation (1.1) reduces to the deterministic Camassa-Holm (CH) equation given by u t − u xxt + 3uu x = 2u x u xx + uu xxx . (1.2) Fokas & Fuchssteiner [23] introduced (1.2) when studying completely integrable generalizations of the Korteweg-de-Vries (KdV) equation with bi-Hamiltonian structure whereas Camassa & Holm [7] proposed (1.2) to describe the unidirectional propagation of shallow water waves over a flat bottom. Since then, the CH equation (1.2) has been studied intensively, and we cannot even attempt to survey the vast research history here. For the paper at hand it is important to mention the wave-breaking phenomenon which illustrates possible loss of regularity as a fundamental mechanism in the CH equations. In contrast to smooth soliton solutions to the KdV equation [42], solutions to the CH equation remain indeed bounded but their slope can become unbounded in finite time, cf. [13] and related work in [11,14,48]. Moreover, for a smooth initial profile, it is possible to predict exactly (by establishing a necessary and sufficient condition) whether wave-breaking occurs for solutions to the Cauchy problem for (1.2) [13,48]. The other essential feature of the CH equation is the occurrence of traveling waves with a peak at their crest, exactly like that the governing equations for water waves admit the so called Stokes waves of greatest height, see [12,15,16]. Bressan & Constantin proved the existence of dissipative as well as conservative solutions in [5,6]. Later, Holden & Raynaud [36,37] also obtained global conservative and dissipative solutions using Lagrangian transport ideas. When h = 0 and k = 2, equation (1.1) forms the cubic equation which has been derived by Novikov in [51]. It has been proven that (1.3) possesses a bi-Hamiltonian structure with an infinite sequence of conserved quantities and (1.3) admits peaked solutions of explicit form u(t, x) = ± √ ce −|x−ct| with c > 0 [25], as well as multipeakon solutions with explicit formulas [38]. For the study of other deterministic instances of (1.1) we refer to [8,31,62,63].
Equations like (1.2) are naturally embedded in high-order descriptions with energy-dissipative evolution. A weakly dissipative evolution is given for some parameter λ > 0 by u t − u xxt + (k + 2)u k u x − (1 − ∂ 2 xx )(λu) = (k + 1)u k−1 u x u xx + u k u xxx , k ≥ 1. (1.4) For instance, the weakly dissipative CH equation, i.e. (1.4) for k = 1, has been studied in [45,61], For the Novikov equation (1.3) with the same weakly dissipative term (1 − ∂ 2 xx )(λu), we can also refer to [45]. In this work, we assume that the energy exchange mechanisms are connected with randomness to account for external stochastic influence. We are interested in the case where in the term (1 − ∂ 2 xx )(λu) the deterministic parameter λ is substituted by a formal cylindrical Wiener process (cf. Section 2.1 for precise definitions), and where the previously linear dependency on u is replaced by a non-autonomous and nonlinear term h(t, u). Thus, we consider the Cauchy problem for (1.1) on the torus T = R/2πZ, with random initial data u 0 = u 0 (ω, x). Applying the operator (1 − ∂ 2 xx ) −1 to (1.1), we reformulate the problem as the stochastic evolution du + u k ∂ x u + F (u) dt = h(t, u)dW, x ∈ T, t > 0, k ≥ 1, u(ω, 0, x) = u 0 (ω, x), x ∈ T, (1.5) with F (u) = F 1 (u) + F 2 (u) + F 3 (u) and (1.6) Note that the operator (1 − ∂ 2 xx ) −1 in F (·) is understood as . (1.7) In (1.7) the symbol [x] stands for the integer part of x.
The first objective of this paper is to analyze the local existence and uniqueness of pathwise solutions to (1.5) as well as blow-up criteria (see Theorem 2.1). Here we remark that Chen et al. [10] have already studied the stochastic CH equation with additive noise. For linear multiplicative noise, we refer to the second author's work [55] that concerns the stochastic CH equation and to [9] for a stochastic modified CH equation.
The second major objective of the paper is to investigate whether stochastic perturbations can improve the dependence on initial data in the Cauchy problem (1.5). We notice that various instances of transporttype stochastic evolution laws have been studied with respect to the effect of noise on the regularity of their solutions. For example, we refer to [22,49,50,20] for linear stochastic transport equations and to what concerns nonlinear stochastic conservation laws, we refer to [46,47,26]. In [43,28,54,55] the dissipation of energy caused by linear multiplicative noise has been analyzed. In contrast to these works we focus in this paper on the initial-data dependence in the Cauchy problem (1.5). Actually, as far as we know, much less is known on the noise effect with respect to the dependence on initial data, neither for the general case nor for special examples. However, the question whether (and how) noise can affect initial-data dependence is interesting. In many cases, regularization produced by noise may be formally related to the regularization by adding a Laplacian; If one would add a real Laplacian to the governing equations, then by using techniques from the theory for semilinear parabolic equations, the dependence on initial data turns out to be more than continuous. For example, for the deterministic incompressible Euler equations, the dependence on initial data cannot be better than continuous [34], but for the deterministic incompressible Navier-Stokes equations, it is at least Lipschitz continuous, see pp. 79-81 in [30]. As a new setting to analyze initial data dependence, we introduce the concept of the stability of the exiting time (the first time that a solution escapes from a certain range, see Definition 2.2 below). Then we show under some conditions on h(t, u), that the multiplicative noise (in the Itô sense) cannot improve the stability of the exiting time, and, at the same time, improve the continuity of the map u 0 → u (see Theorem 2.2).
We conclude the paper with a discussion of time-global well-posedness for (1.5). For general noise terms, it is difficult to determine whether a local pathwise solution to (1.5) is globally defined. As shown in [28,43,44,54,55], the linear noise h(t, u)dW = βudW , with β ∈ R \ {0} and W being a standard 1-D Brownian motion, acts dissipatively for many SPDEs. Motivated by these works, we prove in Theorem 2.4 an existence result for the global dynamics of (1.5) with linear multiplicative noise, that is (1.8)

Preliminaries and Main Results
2.1. Stochastic setting, basic definitions and assumptions.
2.1.1. The stochastic setting. We begin by introducing some notations. Throughout the paper, (Ω, F , P) denotes a complete probability space, where P is the probability measure on Ω and F is a σ-algebra. Let t > 0 and τ ∈ [0, t]. Then σ{x(τ ), y(τ )} τ ∈[0,t] stands for the completion of the union σ-algebra generated by (x(τ ), y(τ )). All stochastic integrals are defined in the Itô sense and E· is the mathematical expectation of · with respect to P. For some separable Banach space X, B(X) denotes the Borel sets of X, and Pr(X) stands for the collection of Borel probability measures on X.
For function spaces on T, we will drop T if there is no ambiguity. We will use to denote estimates that hold up to some universal deterministic constant which may change from line to line but whose meaning is clear from the context. We briefly recall some aspects of the stochastic analysis theory which we use below. We refer the readers to [18,24,39] for an extended treatment of this subject.
We call S = (Ω, F , P, {F t } t≥0 , W) a stochastic basis, where {F t } t≥0 is a right-continuous filtration on (Ω, F ) such that {F 0 } contains all the P-negligible subsets and W(t) = W(ω, t), ω ∈ Ω is a cylindrical Brownian motion, defined on an auxiliary Hilbert space U , which is adapted to {F t } t≥0 . Formally, if {e k } is a complete orthonormal basis of U and {W k } k≥1 is a sequence of mutually independent standard one-dimensional Brownian motions, then one may define To guarantee the convergence of the (formal) summation above, we consider a larger separable Hilbert space U 0 such that the canonical embedding U ֒→ U 0 is Hilbert-Schmidt. Then we have that for any T > 0, cf. [18,24,40], To define the Itô stochastic integral on H s , it is required (see e.g. [18,52]) for the predictable stochastic process G to take values in the space of Hilbert-Schmidt operators from U to H s , denoted by L 2 (U, H s ). For such G, (2.1) is a well-defined continuous H s -valued square integrable martingale such that for all stopping times t and v ∈ H s , Moreover, the desirable Burkholder-Davis-Gundy inequality in our case turns out to be E sup or in terms of the coefficients, Here we remark that the stochastic integral (2.1) does not depend on the choice of the space U 0 , cf. [18,52]. For example, U 0 can be defined as

Definitions of the solutions and stability of the exiting times.
We now make precise the notions of a martingale and a pathwise solution to (1.5).
A major result of this paper is a (negative) dependence statement for the initial data of the solutions. Precisely, it refers to the stability of a point in time when the solution leaves a certain range. This point in time is called exiting time and we introduce For each n, let u and u n be the unique pathwise solutions to (1.5) with initial value u 0 and u 0,n , respectively. For any R > 0 and n ∈ N, we define the R-exiting times

2)
then the R-exiting time of u is said to be stable. (2) If u 0,n → u 0 in H s ′ for all s ′ < s almost surely implies that (2.2) holds true, the R-exiting time of u is said to be strongly stable.

2.1.3.
Assumptions. For our results on existence of pathwise solutions, on the stability of exiting times and global well-posedness of (1.1), we rely on generic but slightly different assumptions on the data in (1.1).
is also predictable. Furthermore, we assume the following: • There is an increasing and locally bounded function f (·) : [0, +∞) → [0, +∞) with f (0) = 0 such that for any t > 0 and u ∈ H s , Particularly, if h does not depend on u, i.e., the additive noise case, then the condition f (0) = 0 can be removed. • There is an increasing locally bounded function g(·) : [0, +∞) → [0, +∞), such that for any t > 0 and u ∈ H s , When we consider the initial-data dependence problem for (1.5) in Section 4, we need a similar assumption on h(t, ·). For s ≥ 0 and u ∈ H s , h :

Main results.
In this section we summarize our major contributions providing proofs later in the remainder of the paper.
Theorem 2.1. Let s > 3/2, k ≥ 1 and let h(t, u) satisfy Assumption A 1 . For a given stochastic basis (2.6) Moreover, (u, τ ) can be extended to a unique maximal pathwise solution (u, τ * ) with Remark 2.1. The proof for Theorem 2.1 combines the techniques as employed in the papers [2,3,4,17,19,28,55]. However, the Faedo-Galerkin method used e.g. in [28,19] cannot be utilized directly since in (1.5), we do not have additional constraints like incompressibility, which guarantees the global existence of an approximate solution (see, e.g. [21,28]). Without this, we need to find a positive lower bound for the lifespan of the approximate solutions, which is generally not clear. Particularly, for our case, this difficulty will be overcome by constructing a suitable approximation scheme and establishing an appropriate blow-up criterium, which applies not only for u, but also for the approximate solution u ε . This idea is transferred from the recent work [17] on deriving blow-up criteria.
The next result addresses the dependence of the solution on initial data giving at least a partial answer. (2.11) To prove Theorem 2.2, we assume that for some R 0 ≫ 1, the R 0 -exiting time is strongly stable at the zero solution. Then we will construct an example to show that the solution map u 0 → u defined by (1.5) is not uniformly continuous. This example involves the construction (for each s > 3/2) of two sequences of solutions which are converging at time zero but remain far apart at any later time. Therefore we will first construct two sequences of approximation solutions u i,n (i ∈ {1, 2}) such that the actual solutions u i,n starting from u i,n (0) = u i,n (0) satisfy that as n → ∞, where u i,n exists at least on [0, τ i,n ]. If we obtain (2.12), then we can estimate the approximation solutions instead of the actual solutions and obtain (2.11). However, in order to obtain (2.12), due to the lack of life span estimate (see (4.7)-(4.8) in [58] and (3.8)-(3.9) in [59] for example) in the stochastic setting, we first connect the property inf n τ i,n > 0 with the stability property of the exiting time of the zero solution, then we estimate the error in H 2s−σ and H σ with suitable σ. Finally we use interpolation to derive (2.12).

Remark 2.3.
It is also worth noting that in the deterministic case, the question on the optimal dependence of solutions (for example, the solution map is continuous but not uniformly continuous) to transport type equations has been widely studied. For the incompressible Euler equation, we refer to [34,56], and for CH type equations, we refer to [32,33,57,58,59] and the references therein. Without the random noise, we have uniform lower bounds for the existence times of a sequence of solutions, see (4.7)-(4.8) in [58] and (3.8)-(3.9) in [59] for example. However, in the stochastic case, the existence time of the solution is also a random variable, and due to the lack of lifespan estimates, we will have to pay more attention to the existence time, which motivates us to introduce the definition on the stability of the exiting time (see Definition 2.2). Theorem 2.2 shows that one cannot expect too much for the issue of the dependence on initial data. More precisely, we cannot expect to improve the stability of the exiting time for the zero solution, and simultaneously to improve the continuous dependence of solutions on initial data.
Finally, we focus on (1.8). For the issue of global existence, we have , or equivalently, . Remark 2.4. Motivated by the recent papers [28,54,55], where the linear noise βudW with β ∈ R\{0} is considered, we focus on the non-autonomous linear multiplicative noise case, namely (1.8). We transform (1.8) to a non-autonomous random system (5.2). Although the stochastic integral is absent in (5.2), to extend the deterministic results to the stochastic setting, we need to overcome a few technical difficulties since the system is not only random but also non-autonomous. In this work, we manage to gain some estimates and asymptotic limits of the Girsanov type processes (see e.g., (5.6), (5.8), (5.10) and Lemma A.6), which enable us to apply the energy estimate pathwisely (namely for a.e. ω ∈ Ω) and obtain Theorem 2.3.  1], then the corresponding maximal pathwise solution (u, τ * ) to (1.8) satisfies That is to say, P {u exists globally} ≥ p + q.

Proof for Theorem 2.1
3.1. Blow-up criteria. Let us postpone the proof for existence and uniqueness of solutions to (1.5) to Section 3.2. We will first prove the blow-up criteria, since some estimates will be used later. Motivated by [17], we first consider the relationship between the blow-up time of u(t) H s and the blow-up time of u(t) W 1,∞ for (1.5). Even though one might expect that the u(t) Hs norm blow up earlier than u(t) W 1,∞ , the following result shows that this is not true.

Proof.
To begin with, we see that u(· ∧ τ ) ∈ C([0, ∞); H s ) means that for any t ∈ [0, τ ], Therefore u(t), as a W 1,∞ -valued process, is also F t -adapted. We then infer from the embedding H s ֒→ W 1,∞ for s > 3/2 that for some M > 0 and m ∈ N, where [M ] means the integer part of M . Therefore we have τ 1,m ≤ τ 2,([M]+1)m ≤ τ 2 P − a.s., which means that τ 1 ≤ τ 2 P − a.s. Now we only need to prove τ 2 ≤ τ 1 P − a.s. It is easy to see that for all n 1 , n 2 ∈ Z + , sup t∈[0,τ2,n 1 ∧n2] holds true, then for all n 1 , n 2 ∈ Z + , P {τ 2,n1 ∧ n 2 ≤ τ 1 } = 1 and Since (3.2) requires the assumption (3.1), it suffices to prove (3.1). Apply the Itô formula for u 2 H s = D s u 2 L 2 to deduce that for any n 1 , n 2 ≥ 1 and t ∈ [0, τ 2,n1 ∧ n 2 ], where {e k } is the complete orthonormal basis of U . On account of the Burkholder-Davis-Gundy inequality, we arrive at Then (2.3) and the stochastic Fubini theorem [18] lead to For L 2 , we commute the operator D s with u to derive, Using integrating by parts, Lemmas A.1 and A.2, we find that Similarly, it follows from Lemma A.4 and the assumption (2.3) that Therefore we combine the above estimates to have where C = C(n 1 ) through n k 1 and n k 1 + f 2 (n 1 ). Then Gronwall's inequality shows that for each n 1 , n 2 ∈ Z + , there is a constant C = C(n 1 , n 2 , u 0 ) > 0 such that E sup t∈[0,τ2,n 1 ∧n2] u(t) 2 H s < C(n 1 , n 2 , u 0 ), which gives (3.1). Let us assume the existence and uniqueness first. We notice that for fixed m, n > 0, even if P{τ 1,m = 0} or P{τ 2,n = 0} is larger than 0, for a.e. ω ∈ Ω, there is m > 0 or n > 0 such that τ 1,m , τ 2,n > 0. By continuity of u(t) H s and the uniqueness of u, it is easy to check that τ 1 = τ 2 is actually the maximal existence time τ * of u in the sense of Definition 2.1. Consequently, we obtain the desired blow-up criteria.

Existence and uniqueness.
(1) (Approximate solutions) The first step is to construct a suitable approximation scheme. For any R > 1, we let χ R (x) : [0, ∞) → [0, 1] be a C ∞ function such that χ R (x) = 1 for x ∈ [0, R] and χ R (x) = 0 for x > 2R. Then we consider the following cut-off problem on T, From Lemma A.4, we see that the nonlinear term F (u) preserves the H s -regularity of u ∈ H s for any s > 3/2. However, to apply the theory of SDEs in Hilbert space to (3.3), we will have to mollify the transport term u k ∂ x u since the product u k ∂ x u loses one regularity. To this end, we consider the following approximation scheme: where J ε is the Friedrichs mollifier defined as J ε f (x) = j ε * f (x). Here * stands for the convolution, j ε (x) = k∈Z j(εk)e ixk and j(x) is a Schwartz function satisfying 0 ≤ j(ξ) ≤ 1 for all the ξ ∈ R and j(ξ) = 1 for any ξ ∈ [−1, 1]. From the construction, we see that J ε enjoys some useful estimates, see [55] for example. With a uniform L ∞ (Ω; W 1,∞ ) condition provided by the cut-off function χ R ( u W 1,∞ ), we can split the expectation E( u ε 2 H s u ε W 1,∞ ). Otherwise, the a priori L 2 (Ω; H s ) estimate for u ε is not closed. After introducing the mollifying of the transport term u k u x , for a fixed stochastic basis S = (Ω, F , P, {F t } t≥0 , W ) and for u 0 ∈ L 2 (Ω; H s ) with s > 3, according to the existence theory of SDE in Hilbert space (see for example [52, Theorem 4.2.4 with Example 4.1.3]), (3.4) admits a unique solution u ε ∈ C([0, T ε ), H s ) P − a.s. In the same way as we prove Lemma 3.1, we see that for each fixed ε, if T ε < ∞, then lim sup t→Tε u ε (t) W 1,∞ = ∞. Due to the cut-off in (3.4), for a.e. ω ∈ Ω, u ε (t) W 1,∞ is always bounded and hence u ε is actually a global in time solution, that is, u ε ∈ C([0, ∞), H s ) P − a.s. We remark here that the global existence of u e is necessary in our framework due to the lack of life-span estimate in the stochastic setting. Otherwise we will have to prove P{inf ε>0 T ε > 0} = 1, which is not clear in general.
(2) (Pathwise solution to the cut-off problem in H s with s > 3) We pass to the limit ε → 0.
By applying the stochastic compactness arguments from Prokhorov's and Skorokhod's theorems we obtain the almost sure convergence for a new approximate solution ( u ε , W ε ) defined on a new probability space. By virtue of a refined martingale representation theorem [35, Theorem A.1], we may send ε → 0 in ( u ε , W ε ) to build a global martingale solution in H s with s > d/2 + 3 to the cut-off problem. Finally, since F satisfies the estimates as in Lemma A.4 and h satisfies Assumption A 1 , one can obtain the pathwise uniqueness easily. Then the Gÿongy-Krylov characterization [29] of the convergence in probability can be applied here to prove the convergence of the original approximate solutions. For more details, we refer to [55]. We remark that even though the Yamada-Watanabe type result in infinite dimensional space has been established in [53] for SPDEs with variational structure, however, the conditions therein are unverifiable for our problem. (3) (Remove the cut-off and extend the range of s to s > 3/2) When u 0 ∈ L ∞ (Ω, H s ) with s > 3/2, by mollifying the initial data, we obtain a sequence of regular solutions {u n } n∈N to (1.5). Motivated by [27,55], one can prove that there is a subsequence (still denoted by u n ) such that for some almost surely positive stopping time τ , Then we can pass the limit n → ∞ to prove that (u, τ ) is a solution to (1.5). Besides, using a cutting argument, as in [28,27,2], enables us to remove the L ∞ (Ω) assumption on u 0 . More precisely, when E u 0 2 H s < ∞, we consider the decomposition Ω m = {m − 1 ≤ u 0 H s < m}, m ≥ 1.
Since E u 0 2 H s < ∞, m≥1 Ω m is a set of full measure and 1 = m≥1 1 Ωm P − a.s. Therefore we have Taking expectation in the above inequality, we obtain (2.6). Since the passage from (u, τ ) to a unique maximal pathwise solution (u, τ * ) in the sense of Definition 2.1 can be carried out as in [17,28,27,54], we omit the details. The proof for Theorem 2.1 is finished.

Proof for Theorem 2.2
Now we are going to prove Theorem 2.2. To this end, it suffices to show that if for some R 0 ≫ 1, the R 0 -exiting time is strongly stable at the zero solution, then the solution map u 0 → u is not uniformly continuous.

Approximate solutions and associated estimates. Define the approximate solutions as
Substituting u l,n into (1.5), we see that the error E l,n (t) can be defined as Now we analyze the error as follows.
Here r s is a parameter with Proof. Direct computation with using(4.1) shows that Then we have From Lemmas A.5 and A.2, we get the following estimate, Therefore we arrive at Recall that F (·) is given by (1.6). Since (1 − ∂ 2 xx ) −1 is bounded from H δ to H δ+2 , we can use Lemma A.5 to estimate F i (u l,n ) H δ (i = 1, 2, 3) as follows. (4.5) In the above estimates, we used the fact that F 3 (·) appears only for k ≥ 2. Combining (4.5), (4.6) and (4.7), we have Then, for any T > 0 and t ∈ [0, T ], by virtue of the Itô formula, we arrive at Taking the supremum with respect to t ∈ [0, T ], using the Burkholder-Davis-Gundy inequality and using (2.5) and (4.8) yield sup 4.2. Construction of actual solutions. Now we consider the following periodic boundary value problem with deterministic initial data u l,n (0, x), i.e., Since h satisfies (2.5), we see that (2.3) and (2.4) are also verified. Then Theorem 2.1 yields that for each n ∈ N, (4.9) has a uniqueness maximal pathwise solution (u l,n , τ * l,n ). 4.3. Estimates on the errors. Proof. We first notice that by Lemma A.5, for l satisfying (4.1), Let q = q l,n = k i=0 u l,n k−i (u l,n ) i and v = v l,n = u l,n − u l,n . In view of (4.2), (4.3) and (4.9), we see that For any T > 0, we use the Itô formula on [0, T ∧ τ R l,n ], take the supremum over t ∈ [0, T ∧ τ R l,n ] and use the Burkholder-Davis-Gundy inequality with noticing (2.5) to find

We can first infer from Lemma A.4 that
Therefore, applying Lemmas A.3 and A.4, H δ ֒→ L ∞ , integrating by part and (4.4), we have If for some R 0 ≫ 1, the R 0 -exiting time of the zero solution to (1.5) is strongly stable, then for l satisfying (4.1) and τ R0 l,n given in (4.10), we have lim n→∞ τ R0 l,n = ∞ P − a.s. (4.14) Proof. We notice that for all s ′ < s, lim n→∞ u l,n (0) H s ′ = lim n→∞ u l,n (0) − 0 H s ′ = 0. Since the unique solution with zero initial data to (1.5) is zero and the R 0 -exiting time of the zero solution is ∞, we see that (4.14) holds true provided the R 0 -exiting time of the zero solution to (1.5) is strongly stable.
Proof for Theorem 2.2. For each n > 1, for l satisfying (4.1) and for the fixed R 0 ≫ 1, Lemma A.5 and (4.10) give us P{τ R0 l,n > 0} = 1 and Lemma 4.3 implies (2.8). Besides, Theorem 2.1 and (4.10) show that u l,n ∈ C([0, τ R0 l,n ]; H s ) P − a.s. and (2.9) holds true. By virtue of the interpolation inequality, we have E sup When k is odd, we use (4.15) to find that for any T > 0, which is (2.11). Similarly, we can also prove (2.11) when k is even. The proof is completed.

Global existence
In this section, we study the global existence and the blow-up of solutions to (1.8), and estimate the associated probabilities. Motivated by [28,54,55], we introduce the following Girsanov type transform is the corresponding unique maximal pathwise solution to (1.8), then for t ∈ [0, τ * ), the process v given in Consequently, Theorem 2.1 implies that (1.8) has a unique maximal pathwise solution (u, τ * ). Via the Itô formula, we have Notice that F (u) = F (βv) = β k+1 F (v). Then we have Since v(0) = u 0 (ω, x), we see that v satisfies (5.2). Moreover, Theorem 2.1 implies u ∈ C ([0, τ * ); H s ) P − a.s., so v ∈ C([0, τ * ); H s ) P − a.s. Besides, from Lemma A.4 and (5.2) 1 , we see that for a.e. ω ∈ Ω, Multiplying both sides of the above equation by v and then integrating the resulting equation on x ∈ T with noticing that (k Using Lemmas A.1 and A.4, we conclude that there is a C = C(s) > 1 such that for a.e. ω ∈ Ω, where β is given in (5.1).
Since v = G T * V with G T > 0 given in (1.7), we have sign(v) = sign(V ) P − a.s.
Via (5.1), we obtain the desired estimate.

Appendix A. Auxiliary results
In this section we formulate and prove some estimates employed in above proofs.
We apply the Itô formula to X p with p > 0 to obtain that dX p = pλb 2 (t) + b 2 (t)p(p − 1) 2 X p dt + pb(t)X p dW.