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Research article

Invariant measure of stochastic damped Ostrovsky equation driven by pure jump noise

  • Received: 22 July 2020 Accepted: 30 August 2020 Published: 10 September 2020
  • MSC : 60H15; 37A25

  • This paper is devoted to the stochastic damped Ostrovsky equation driven by pure jump noise. The uniformly bounded of solutions in H1(R) and L2(R) space are established respectively, which are the key tools to obtain the existence of invariant measure. By applying the convergence in measure in Hilbert space, we prove that the invariant measure is unique if the initial value is non-random. Some numerical simulation are provided to support the theoretical results.

    Citation: Shang Wu, Pengfei Xu, Jianhua Huang. Invariant measure of stochastic damped Ostrovsky equation driven by pure jump noise[J]. AIMS Mathematics, 2020, 5(6): 7145-7160. doi: 10.3934/math.2020457

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  • This paper is devoted to the stochastic damped Ostrovsky equation driven by pure jump noise. The uniformly bounded of solutions in H1(R) and L2(R) space are established respectively, which are the key tools to obtain the existence of invariant measure. By applying the convergence in measure in Hilbert space, we prove that the invariant measure is unique if the initial value is non-random. Some numerical simulation are provided to support the theoretical results.


    In this paper, we concentrate on the following stochastic damped Ostrovsky equation driven by pure jump noise

    {du(t)=[β3xu(t)12x(u2)+λu+γ1xu]dt+Zg(u(t),z)η(dt,dz),u(0)=u0. (1.1)

    where λ>0, β and γ are real numbers such that βγ0 and 1xf=((iξ)1ˆf(ξ)), and η(dt,dz) denotes the Poisson counting measure. The Ostrovsky equation was introduced by [17] to describe the weakly nonlinear long waves in a rotating liquid, where the parameter γ measures the effect of rotation which is supposed to be small. β represents the capillary waves on the surface of a liquid or for oblique magneto-acoustic waves in plasma. And λ>0 means the positive dispersion.

    For the stochastic Ostrvosky equation driven by white noise, Isaza [11,12,13] proved the well-posedness in Hs(R) with s>34. Yan et al. [21,22] obtained the local well-posedness for the Cauchy problem for (1.1) the initial data u0(,ω)Hs(R)(ωΩa.e.) with s>34, and the global well-posedness in L2(R)(a.e.ωΩ). Chen [4] established the existence of random attractor in ˜L2(R). Wu et al. [19] established the ergodicity of stochastic damped Ostrovsky equation with Gaussian noise, but they do not discuss the ergodicity of the equation with Lévy noise.

    The dispersive partial differential equation such as Schrödinger equation, KdV equation and Ostrovsky equation are integrable systems with infinite conservation laws. If λ=0, then (1.1) become a stochastic KdV equation. Bouard et al. [2,3] established the local well-posedness for stochastic KdV equation with additive noise and multiplicative noise respectively. The dissipative effect of damped dispersive systems are too weak to apply the theory for dissipative dynamical systems. There are two typical difficulties to establish the ergodicity for stochastic dispersive equation on R need to overtake, one is the non-compactness of the whole real line R, and the other one is the non-compactness of operator semigroup for stochastic linear dispersive equation. Recently, Ekren, Kukavica and Ziane [9,10] applied the stopping time technique to establish the boundedness of the uniform estimate, which can deduce the tightness of the measure sequences in local square integrable space L2loc(R), they also used the convergence in measure in Hilbert space in terms of the idea in [16] to obtain the existence of invariant measure, but they could not provide with the uniqueness and ergodicity, we refer it to [9] for details.

    Recently, Bouard et al. [1] studied the nonlinear Schrödinger eqution driven by jump process, and proved the existence of a martingale solution of the nonlinear Schrödinger equation with jump processes with infinite activity, but they could not study the ergodicity of invariant measure for stochastic Schrödinger equation with jump noise. In fact, there are a lot of paper on ergodicity for stochastic dissipative partial differential equation driven by jump process, Dong and Xie [7] studied ergodicity of stochastic 2D Navier-stokes equations with Lévy noise, Dong et.al.[6] study the existence of martingale solutions of stochastic 3D Navier-Stokes equations with jump noise. To the best of our knowledge, there is little paper on the ergodicity of stochastic damped dispersive PDE with Lévy noise due to the above mentioned difficulty.

    The goal of this paper is to study the invariant measure of stochastic damped Ostrovsky equation with pure jump noise. The novelty of the current paper to obtain the uniformly bounded of solutions in H1(R) and L2(R) space respectively, which are the key tools to prove the tightness of the measure sequences in local square integrable space L2loc(R). By using the convergence in measure in Hilbert space in terms of the idea in [16], we can obtain the existence of invariant measure. Moreover, by applying the idea of Proposition 3.2.7 in [5], we prove that the invariant measure is unique if the initial value is non-random. The another novelty of the current paper is that the numerical simulation in the sense of Eu(t,)L2x reveals that stochastic damped Ostrovsky equation driven by pure jump may posses an unique ergodic invariant measure.

    We state our main result followed as:

    Theorem 1.1. Assume that u0X1, and the Hypothesis (H1)–(H3) are satisfied. Then stochastic damped Ostrovsky equation (1.1) admits an invariant measures. Moreover, it posses an ergodic invariant measure for the non-random initial data.

    The rest of paper is organized as follows. In Section 2, some function setting and useful lemmas or technique theorem are provided. In Section 3, the uniformly bounded of solutions in H1(R) and L2(R) space are established respectively, which are the key tools to obtain the ergodicity. By using the methods of [1,Appendix A], we prove the existence of invariant measure for stochastic damped Ostrovsky equation in Section 4. Moreover, by applying the idea of Proposition 3.2.7 in [5], we prove that the invariant measure is unique if the initial value is restricted to be non-random. In Section 5, some numerical simulation are provide to support the theoretical results.

    In this section, we introduce some basic concepts and some inequalities, which are from [1] and [18].

    Let (Ω,F,Ft,P) be a complete probability space, (Z,B(Z)) be a measurable space and ν(dz) be a σfinite measure on it. Let P=(p(t),tDp) be a stationary FtPoisson point process on Z with characteristic measure ν(dz), where Dp is a countable subset of [0,+) depending on random parameter ωΩ. Denote by η(dt,dz) the Poisson counting measure associated with p, that is η(dt,dz)=ΣsDp,st1A(p(s)), where 1A() is the indicator function with respect to A. Let ˜η(dt,dz)=η(dt,dz)dtν(dz) be the compensated Poisson measure. If ν(z)<, ˜η is a martingale with ˜η(Z)t=tν(z), and ν is the intensity measure of η.

    We impose the hypothesis (H1)(H3) on the Poisson processes just as [8]

    (H1) For any uX1, there exists a constant C< such that

    g(u,z)2X1ν(dz)C(1+u2X1),

    (H2) ν(0)=0, Zz2Zν(dz)< and ν(Z)=ρ<,

    (H3) Z is continuously embedded in H1(R).

    The measure ν describes the expected number of jumps of a certain height in a time interval of length 1. The hypothesis means there is an finite number of small jumps is expected.

    Let Y be a separable and complete metric space and T>0. The space X(T)=D([0,T];Y) denotes the space of all right continuous functions x:[0,T]Y with left limits, P(D([0,T];Y)) the space of Borel probability measures on D([0,T];Y). We equip D([0,T];Y) with the Skorohod topology such that D([0,T];Y) is both separable and complete.

    Let H be a Hilbert space, Hs(R) is the Sobolev space with norm

    fHs(R)=ξsFxfL2ξ(R),

    where ξs=(1+ξ2)s2 for any ξR,Fxu and F1xu denotes the Fourier transformation and the Fourier inverse transformation of u with respect to its space variable respectively. ˙Hs1,s2(R) is the Sobolev space with norm

    f˙Hs1,s2(R)=ξs1|ξ|s2FxfL2ξ(R).

    and ˙Hs=˙H0,s. With this choice of the antiderivative we have, 1xf=(ˆf(ξ)iξ), so it is natural to define the function space Xs as one in [15]

    Xs={fHs(R):1xfL2(R)},sR.

    Space S(R) is the Schwartz space and S(R) is its dual space. Fu and F1u denotes the Fourier transformation and the Fourier inverse transformation of u with respect to its all variables respectively.

    The following Theorem is the key tool to prove the ergodicity for stochastic equation (1.1), which is from [1],

    Lemma 2.1 (Appendix A.1, [1]). Let {xn:nN} be a sequence of cˊadlˊag processes, each of the process defined on a probability space (Ωn,Fn,Pn). Then the sequence of laws of {xn:nN}is tight on D([0,T];Y) if

    (a) there exists a space Y1, with Y1Y compactly and some r>0, such that

    En|xn(t)|rY1C,nN.

    (b) there exist constants c>0, γ>0 and r>0 such that for all θ[0,T],t[0,Tθ], and n0, we have

    Ensuptst+θ|xn(t)xn(s)|rYcθγ.

    Denote

    ϕ(ξ)=βξ3+γξ,Sλ(t)u=eλtU(t),U(t)u0=eλtRei(xξtϕ(ξ))Fxu0(ξ)dξ,fLqtLpx=(R(R|f(x,t)|pdx)qpdt)1q,fLpxt=fLptLpx.

    Then the mild solution of equation (1.1) can be written as:

    u(t)=Sλ(t)u0t0Sλ(ts)uxuds+t0ZSλ(ts)g(u(s),z)η(ds,dz). (2.1)

    Definition 2.1. Assume that (X,T) is a polish space, Σ is a σ-algebra on X, M is a set of measures on Σ. M is said to be tight if for any ε>0, there exists a compact set KϵX such that

    |μ(Kcε)|<ε,foranyμM.

    The following Lemma is the key tool to prove the ergodicity for stochastic equation (1.1), which is from [5].

    Lemma 2.2 ([5], Proposition 3.2.7). An invariant probability measure for the semigroup Pt, t0, is ergodicity if and only if it is an extremal point of the set of all the invariant probability measures for the semigroup Pt,t0.

    In this section, we establish the uniform estimates of the solution in L2 norm and H1 norm respectively, which is the key tool to obtain the ergodicity for stochastic Ostrovsky equation (1.1).

    Lemma 3.1 ([22]). There exists some monotonely increasingly continous function ~C1(t) and a constant α>0, h,gX(T), such that

    t0Sλ(ts)h(s)xg(s)dsX(T)~C1(T)TαhX(T)gX(T).

    Here uX(T)=sups[0,T]u(s)H1.

    Lemma 3.2 ([22]). There exists some monotonely increasingly continuous function ~C2(t), such that

    Sλ(t)u0X(T)~C2(T)u0X(T).

    Lemma 3.3. If u(t)X1 solves the equation (2.1), and the Hypothesis (H1)–(H3) are satisfied. There exists a constant C>0 such that

    E[sup0tTu2L2p]C(1+E[u02L2p]), (3.1)
    E[sup0tTu2H1]C(1+E[u02H1]). (3.2)

    Proof. The proof of (3.1) can get by slightly modifying the proof of Lemma 5.1 in [22], we just prove (3.2). Let

    I(u)=Rβ(xu)2+γ2(1xu)2+13u3dx.

    Applying the Itô formula to I(u) yields

    I(u(t))I(u(0))+t0Z(xg(u(s),z)2L2+2xu(s),xg(u(s),z))˜η(ds,dz)+t0Z(xg(u(s),z)2L2+2xu(s),xg(u(s),z))dsν(dz)+t0Z(1xg(u(s),z)2L2+21xu(s),1xg(u(s),z))˜η(ds,dz)+t0Z(1xg(u(s),z)2L2+21xu(s),1xg(u(s),z))dsν(dz)+t0Zu(s)+g(u(s),z)3L3u(s)3L3˜η(ds,dz)+t0Zu(s)+g(u(s),z)3L3u(s)3L3dsν(dz)=I(u(0))+M1+I1+M2+I2+M3+I3.

    Direct computation shows that

    [MτN1,MτN1]12t=[M1,M1]12τNt=(sDp,sτNt(xg(u(s),z)2L2+2u(s),g(u(s),p(s)))2)12(2sDp,sτNt(xg(u(s),z)4L2+8sDp,sτNtxu(s)2L2xg(u(s),p(s))2L2)12C(sDp,sτNtxg(u(s),z)4L2)12+C(sDp,sτNtxu(s)2L2xg(u(s),p(s))2L2)12CsDp,sτNtxg(u(s),z)2L2+C(sDp,sτNtxu(s)2L2xg(u(s),p(s))2L2)12CsDp,sτNtxg(u(s),z)2L2+CsupstτNxu(s)L2(sDp,sτNtxg(u(s),p(s))2L2)12CsDp,sτNtxg(u(s),z)2L2+12supstτNxu(s)2L2.

    Taking the expectation and using (H1)–(H3) gives

    E[sup0st|MτN1|]CE([MτN,MτN]12t)CE[sDp,sτNtxg(u(s),z)2L2]+12E[supstτNxu(s)2L2]=CE[tτN0Zxg(u(s),z)2L2dsν(dz)]+12E[supstτNxu(s)2L2]CE[tτN0(1+xu(s)2L2)ds]+12E[supstτNxu(s)2L2]C(1+N)t+12N<.

    We can deduce that for I1(tτN),

    E[I1(tτN)]=E[tτN0Z(xg(u(s),z)2L2+2u(s),g(u(s),z))dsν(dz)]E[tτN0Z(xg(u(s),z)2L2+2xu(s)2L2xg(u(s),z)2L2)dsν(dz)]2E[tτN0Zxg(u(s),z)2L2dsν(dz)]+E[tτN0Zxu(s)2L2dsν(dz)]=2E[tτN0Zxg(u(s),z)2L2dsν(dz)]+E[tτN0Zxu(s)2L2dsν(dz)]CE[tτN0(1+xu(s)2L2)ds]+ρE[tτN0xu(s)2L2ds]Ct+(C+ρ)Nt<.

    Similarly,

    E[sup0st|MτN2|]CE([MτN,MτN]12t)CE[tτN0(1+1xu(s)2L2)ds]+12E[supstτN1xu(s)2L2]C(1+N)t+12N<,

    and

    E[I2(tτN)]=E[tτN0Z(1xg(u(s),z)2L2+2u(s),g(u(s),z))dsν(dz)]CE[tτN0(1+1xu(s)2L2)ds]+ρE[tτN01xu(s)2L2ds]Ct+(C+ρ)Nt<.

    For M3, by Agmon's inequality, uLu1/2L2xu1/2L2, we have

    [M,τN3,M,τN3]12t=[M3,M3]12τNt=(sDp,sτNt(g(u(s),z)3L3+2u2(s),g(u(s),p(s))+2u(s),g2(u(s),p(s)))2)12C(sDp,sτNt(g(u(s),z)6L3+sDp,sτNtxu(s)4L2g(u(s),p(s))2L2+sDp,sτNtu(s)4L2xg(u(s),p(s))2L2)12C(sDp,sτNtxu(s)2L2+g(u(s),p(s))4L2+u(s)4L2+xg(u(s),p(s))2L2),

    and by (3.1)

    E[sup0st|MτN3|]CE[sDp,sτNtxg(u(s),z)2L2]+12E[supstτNxu(s)2L2]CE[tτN0(1+xu(s)2L2)ds]+12E[supstτNxu(s)2L2]C(1+N)t+12N<,E[I3(tτN)]CE[tτN0(1+xu(s)2L2)ds]+12E[supstτNxu(s)2L2]C(1+N)t+12N<.

    Combining the above arguments, we can obtain

    E[sup0stτNI(u(s))2L2]=E[sup0stI(u(sτN))2L2]E[I(u0)2L2]+Ct+CE[tτN0(1+u(s)2L2)ds]+12E[supstτNu(s)2L2]+CE[tτN0(1+xu(s)2L2)ds]+ρE[tτN0xu(s)2L2ds]+CE[tτN0(supstτN1xu(s)2L2)ds]+12E[supstτN1xu(s)2L2].

    Applying Gronwall's inequality leads to

    E[sup0stτNI(u(s))2L2]C(Ct+E[I(u0)2L2])eCt.

    Since τNTT as N,P.a.s.. It follows from Young's inequality that

    u3(x,t)dxu(t)Lu02L22u05/2L2xu(t)1/2L2β2xu(t)2L2+C. (3.3)

    which implies that

    βxu(t)2L2+γ21xu(t)2L2=I2(u0)13u(t)3L3|I2(u0)|+β2xu(t)2L2+C. (3.4)

    Then, we can prove that

    E[sup0tTu2H1]C(1+E[u02H1]).

    Thus, the proof of Lemma is completed.

    Next, we will prove the global well-posdness of (1.1) in D([0,T],H1) by the method developed in [1].

    Theorem 3.1. Assume the conditions (H1)–(H3) are satisfied. Then, the equation (1.1) admits a unique global mild solution in X1, which is cˊadlag.

    Proof. Let {τn:nN} be a family of independent exponential distributed random variables with parameter ρ, and set

    Tn=nj=1τj,nN.

    Define {N(t):t0} as the counting process

    N(t)=Σj=11[Tj,)(t),t0.

    Then for any fixed t>0, N(t) is a Poisson distributed random variable with parameter ρt. Let {Yn:nN} be a family of independent, ν/ρ distributed random variables. Then

    t0Zg(u(s),z)˜η(ds,dz)={t0Zg(u(s),z)ν(dz)ds,ifN(t)=0,ΣN(t)j=1g(u(t),Yn)t0Zg(u(s),z)ν(dz)ds,ifN(t)>0,

    Notice that N(t)=0 on the interval [0,T1), then the equation (1.1) can be rewritten as the following equations

    {du(t)=[β3xu(t)12x(u2)+λu+γ1xu]dt,u(0)=u0. (3.5)

    Thus, the mild solution of equation (3.5) can be represented as follows

    u(t)=Sλ(t)u0t0Sλ(ts)uxuds.

    Define the operator F by

    Fu=Sλ(t)u0t0Sλ(ts)uxuds.

    Then

    FuX1(T1)Sλ(t)u0X1(T1)+t0Sλ(ts)uxudsX1(T1)

    for uX1(T1)<. Lemma 3.1 and 3.2 implies that there exists some constant C>0 such that

    FuX1(T1)C(u0X1(T1)+Tα1u2X1(T1))<.

    Thus, F maps X1(T1) into itself. Next, we will verify that the operator F is contractive. In fact, for any u1,u2X1(T1), we have

    Fu1Fu2X1(T1)=t0Sλ(ts)(u1xu1u2xu2)dsX1(T1)t0Sλ(ts)u1x(u1u2)dsX1(T1)+t0Sλ(ts)(u1u2)xu2dsX1(T1)C1(T1)Tα1(u1X1(T1)+u2X1(T1))u1u2X1(T1).

    Hence, there exists a sufficient small T1>0 such that equation (3.5) posses an unique local solution on [0,T1]. In the sequel, we extend the local solution to the global one by Lemma 3.3. To the end, denote this local solution on [0,T1) by u1. We will consider the solution on [T1,T2). Notice that a jump with size g(u(T1),Yn) happens at time T1, let u01=u1(T1)+g(u1(T1),Yn), and we consider a second process on [T1,T2) as following

    {du(t)=[β3xu(t)12x(u2)+λu+γ1xu]dt,u(0)=u0, (3.6)

    Similarly, we can deduce that equation (3.6) posses a unique mild solution u2 on [T1,T2). Repeating the above arguments, we can obtain that the equation (1.1) has a unque global mild solution in X(T).

    Since P(N(t)<)=1, the solution u(t) is almost surely defined on [0,T]. It is clearly that the jumps take place at each Tj, and limtTju(t)=u0j, and limtTju(t) exists. Then u is cˊadlag. Thus the proof of Theorem 3.1 is complete.

    In this section, we first prove the tightness of semigroup P(t),t0 according to Lemma 2.1, then prove the ergodicity of stochastic equation (1.1).

    Theorem 4.1. Assume that u0H1(R), and the conditions (H1)–(H3) are satisfied. Then for any sequence of deterministic initial conditions {un0} with R=supnN{un0H1}<, {Ptn(un0,):nN} is tight on H1 for any tn>0 with tn as n.

    Proof. Without loss of generality, we assume that tn is an increasing sequence. Denote by un(t) the solution with the initial condition un(0). We claim that {un(t)}n=1 converges in H1(R). In fact, since

    E[sup0tTu2H1]C(1+E[u02H1]),

    then we have

    supt0E[un(tn)2H1]C(supn(un(0)2H1)+supn(u02L2)2+1):=C(R).

    Due to Lemma 2.2, it suffices to prove there exist two constants c>0 and γ>0 and a real number r>0 such that for all θ>0,t[0,Tθ], and n0

    Ensuptst+θ|un(t)un(s)|rXcθγ.

    By the Definition 2.1, we get

    Eun(t+θ)un(s)2H1Et+θtSλ(ts)uxuds2+Et+θtZSλ(ts)g(u(s),z)η(ds,dz)2.

    It follows from Lemma 3.1 and assumptions (H1)–(H3) that there exist γ>0 such that

    Eun(t+θ)un(s)2θγ+Et+θt1+Sλ(ts)g(u(s),z)2ds,

    which implies that the tightness of {Ptn(un0,):nN} holds, and the proof of Theorem 4.1 is complete.

    Theorem 4.2. Assume that u0H1(R) and the conditions (H1)–(H3) are satisfied. If K is a compact set of {H1}, then the sequence of measures {Ps(v,):s[0,1],vK} is tight on H1.

    Proof. We need to prove that {Ps(v,):s[0,1],vK} posses a convergent subsequence. Without loss of generality, we assume that (sn,vn)[0,1]×K converges to (s,v)[0,1]×K, because K is a compact set. Let un(t) be the solution with the initial value vn(0), and u(t) the solution with the initial value v. For uD([0,1];H1), then we can derive that

    limnu(sn)u(s)H1=0,P.a.s.

    Next, we will prove that there exists a subsequence (snk,vnk) of sequence (sn,vn) such that

    limk(u(snk)u(s)H1+unk(s)u(s)H1)=0.

    Finally, set R0=supvKvH1+1, and choose R>0 such that C(R0)Rε2, then we have

    P{max{sups[0,t]u(s)2H1,sups[0,t]un(s)2H1}R}P{sups[0,t]u(s)2H1+sups[0,t]un(s)2H1R}1RE[sups[0,t]u(s)2H1+sups[0,t]un(s)2H1]C(R0)Rε2.

    Define τ=inf{s0:4˜C(t)sα(˜C(t)R+ˉuX(s)+s0U(sr)xu2drX(s))>1}. is a stopping time. Let τ0=τ, and

    τk+1=inf{sτk:4˜C(t)(sτk)α(˜C(t)R+¯uτkX(τk,s)+sτkU(sr)xu2drX(τk,s))>1},An={max{sups[0,t]u(s)2H1,sups[0,t]un(s)2H1}R},

    and choose N such that P(τN1)ε2, we obtain

    sups[0,1]u(s)un(s)H1(2˜C(t))N+1vvnH10

    on the interval An{τN1}.

    Choosing n be large enough such that (2˜C(t))N+1vvnH1<δ, we have

    P(sups[0,1]u(s)un(s)H1δ)P(An{τN1})1ε,

    which implies that

    limnsups[0,1]u(s)un(s)H10,P.a.s.

    Therefore, there exists a sequence nk such that

    limku(s)unk(s)H10,P.a.s.

    We can deduce that

    |Psnkξ(vnk)Psξ(v)|E[|ξ(unk(snk))ξ(u(snk))|]+E[|ξ(u(snk))ξ(u(s))|]0

    for any real valued uniformly continuous function ξH1(R). Therefore, {Ps(v,):s[0,1],vK} has a convergent subsequence. The proof of Theorem 4.2 is complete.

    Theorem 4.3. Assume that u0H1(R) and the conditions (H1)–(H3) are satisfied. Then μn()=1nn0Pt(0,)dt,n=1,2,... is tight on H1(R).

    Proof. Since for any ε>0, {Pn(0,):n0} is tight, we can choose a compact set KεH1(R) such that supn{Pn(0,Kcε)ε2, and {Ps(v,):s[0,1],vK} is tight on H1. We can also choose a compact set AεHm(R) such that sups[0,1],vKε{Pn(0,Acε)ε2. Therefore, we have

    μn(Acε)=1nn0Pt(0,Acε)dt=1nn1i=0i+1iH1Pi(0,dy)Pti(y,Acε)dt=1nn1i=0i+1i[KεPi(0,dy)Pti(y,Acε)+KcεPi(0,dy)Pti(y,Acε)]dt1nn1i=0i+1i[ε2KεPi(0,dy)+KcεPi(0,dy)]dt1nn1i=0i+1iεdt=ε.

    Then we get that μn()=1nn0Pt(0,)dt, n=1,2,... is tight on H1(R)˙H1(R). The proof of Theorem 4.3 is complete.

    Theorem 4.4. Assume that u0H1(R) and the conditions (H1)–(H3) are satisfied. Then equation (1.1) admits an invariant measures. Moreover, equation (1.1) with the deterministic initial condition posses an ergodic invariant measure.

    Proof. Using Krylov-Bogoliubov proposition, Theorem 4.1 and Theorem 4.3, we can prove there exist at least an invariant measures for equation (1.1). Denote K be the set of all the invariant measure, then it is easy to check that K is convex. Let {μn}nN+ be a sequence of invariant measures in K. Then there exists some constant C such that

    supnu2H1μn(du)Candsupnu4H1μn(du)C.

    For any deterministic initial condition, {(μnPtn)():nN+}={μn():nN+} is tight. Since K is a self sequentially compact set, then K is compact. It follows from Krein-Milman theorem, a convex compact set posses extremal point. Theorem 2.2 yields that this extremal point is ergodic. Therefore, stochastic equation (1.1) admits an ergodic invariant measure. Thus, the proof of Theorem 4.4 is complete.

    In this section, we will take the numerical simulation of the invariant measure. To the end, we give the distribution of the solution 1TMTMm=0E[Φ(u(tm))] by using the so-called the Monte Carl method as following. One can prove theoretically that it does have a unique invariant measure, which derive to the ergodicity[5].

    We do the numerical simulation to find what happen if the Ostrovsky equation perturbed by pure jump noise. We firstly use the the norm conservative finite difference scheme introduced by [20,scheme 1] to simulate the equation (1.1) as

    U(n+1)jU(n)jΔt+βδ(3)(U(n)j+U(n+1)j2)+13(δ(1)(U(n)j+U(n+1)j2)2+U(n)j+U(n+1)j2δ(2)(U(n)j+U(n+1)j2))=λ(U(n)j+U(n+1)j2)+γδ1FD(U(n)j+U(n+1)j2)+tisti+1g(ΔLs)1A(ΔLs(ω)) (5.1)

    where

    δ1FDU(m)j=Δx(U(n)12+j1k=2U(n)k+U(n)j2)(Δx)2LNk=1(U(n)12+k11=2U(n)1+U(n)k2),δ(1)U(n)j=U(n)j+1U(n)j12Δx,δ(3)U(n)j=U(n)j+22U(n)j+1+2U(n)j1U(n)j22(Δx)3

    and st1A(ΔLs(ω)) is a possion process with the parameter PΔt.

    Now we set β=0.01,γ=1, λ=0.5, and u0(x)=sin(x). The simulation of (1.1) driven by Poisson noise with g(u(t),z)=0.2u(t)z is given in Figure 1. Figure 2 gives time changes of u(t,)L2x using different sample trajectories.

    Figure 1.  Solution of u with Poisson Process.
    Figure 2.  Changes of u(t,)L2x.

    It can be clearly seen from the Figure 2 that the decay rate of the equation is slowed down under the influence of noise. At the same time, as shown in Figure 3, it can be seen that for Φ(y)=exp(|y|2), the distribution of the solution 1N+1Nn=0E[Φ(U(n))] tends to a measure μ as T.

    Figure 3.  Solution of u with Poisson Process.

    The above numerical simulation of stochastic damped Ostrovsky equation (Figure 3) in the sense of Eu(t,)L2x reveals that stochastic damped Ostrovsky equation driven by pure jump posses an unique ergodic invariant measure.

    In this paper, the global well-posedness of stochastic damped Ostrovsky equation driven by pure jump noise is given at first. Then we use the tightness criterion to investigate the existence of invariant measures, and give the ergodicity under determined initial condition. A numerical experiment is also given to support our results.

    The paper is Supported by the National Natural Science Foundation of China (No.11771449) and China Postdoctoral Science Foundation(No. 2019T120966).



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