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Martingale solutions of stochastic nonlocal cross-diffusion systems

  • The work of Bendahmane is supported by the Fincome project

  • Received: 01 March 2022 Published: 16 June 2022
  • Primary: 60H15, 35K57; Secondary: 35M10, 35A05

  • We establish the existence of solutions for a class of stochastic reaction-diffusion systems with cross-diffusion terms modeling interspecific competition between two populations. More precisely, we prove the existence of weak martingale solutions employing appropriate Faedo-Galerkin approximations and the stochastic compactness method. The nonnegativity of solutions is proved by a stochastic adaptation of the well-known Stampacchia approach.

    Citation: Mostafa Bendahmane, Kenneth H. Karlsen. Martingale solutions of stochastic nonlocal cross-diffusion systems[J]. Networks and Heterogeneous Media, 2022, 17(5): 719-752. doi: 10.3934/nhm.2022024

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  • We establish the existence of solutions for a class of stochastic reaction-diffusion systems with cross-diffusion terms modeling interspecific competition between two populations. More precisely, we prove the existence of weak martingale solutions employing appropriate Faedo-Galerkin approximations and the stochastic compactness method. The nonnegativity of solutions is proved by a stochastic adaptation of the well-known Stampacchia approach.



    Notation: For two sequences of positive constants {an,n1} and {bn,n1}, symbols anbn, an=O(bn) and an=o(bn) stand for liman/bn=1, liman/bn(0,) and liman/bn=0, respectively. For simplicity, we shall write P, a.s. and Lp to express the convergence in probability, the almost certain convergence and p-mean convergence, respectively.

    The following concept of superadditive function was introduced in [1].

    Definition 1.1. A function ϕ:RnR is called superadditive if ϕ(xy)+ϕ(xy)ϕ(x)+ϕ(y) for all x,yRn, where is for componentwise maximum and is for componentwise minimum.

    Hu [2] introduced the concept of negatively superadditive-dependent (NSD) based on the above concept of superadditive function.

    Definition 1.2. A random vector X=(X1,X2,,Xn) is said to be NSD if

    Eϕ(X1,X2,,Xn)Eϕ(X1,X2,,Xn),

    where X1,X2,,Xn are independent such that Xi and Xi have the same distribution for each i and ϕ is a superadditive function such that the expectations in the above equation exists. A sequence {Xn,n1} of random variables is said to be NSD if for each n1, (X1,X2,,Xn) is NSD.

    Hu [2] established some basic properties and three structural theorems of NSD random variables. An interesting example was also presented in [2], which illustrated that NSD is not necessarily negatively associated (NA, [3]). Christofides and Vaggelatou [4] showed that NA is NSD. Eghbal et al. [5] derived two maximal inequalities and strong law of large numbers of quadratic forms of NSD random variables. Shen et al. [6] studied almost sure convergence and strong stability for weighted sums of NSD random variables. Wang et al. [7] studied complete convergence for arrays of rowwise NSD random variables, with applications to nonparametric regression. For more research of the limit theory for NSD random variables, the author can refer the reader to [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24].

    NA random variable has been studied many times and attracted extensive attention, so it is very significant to investigate the limit theorems of this wider NSD class, which is highly desirable and of considerable significance in theory and application.

    A random variable X is called to be a two-tailed Pareto distribution whose density is

    f(x)={qx2ifx1,0if1<x<1,px2ifx1, (1.1)

    where p+q=1.

    Let {Xn,n1} be independent Pareto-Zipf random variables satisfying P(Xn=0)=11/n,

    P(Xnx)=11x+nforallx>0, (1.2)

    and fXn(x)=1(x+n)2I(x>0).

    Obviously, if the random variable Xn satisfies Eq (1.1) or (1.2), then E|Xn|=, n1. Alder [25] considered independent and identically distributed (i.i.d.) random variables satisfying Eq (1.1) and studied the strong law of large numbers. Alder [26] obtained the weak law of large numbers for Pareto-Zipf random variables. For more research on laws of large numbers for i.i.d. random variables with infinite mean, the author can refer to works of Adler [27,28] and Matsumoto and Nakata [29,30,31].

    Yang et al. [24] investigated the law of large numbers for NSD random variables satisfying Pareto-type distributions with infinite means, and obtained the following theorems which extend and improve the corresponding ones in [25,26]:

    Theorem 1.1. Let {Xn,n1} be a nonnegative sequence of NSD random variables whose distributions are defined by P(Xn=0)=11/cn for n1 and the tail probability

    P(Xn>x)=1x+cnforallx>0andn1, (1.3)

    where {cn} is a nondecreasing constant sequence with cn1 and

    Cn=nj=11cj. (1.4)

    Then we have

    nj=1c1jXjCnlogCnP1. (1.5)

    Theorem 1.2. Let {Xn,n1} be a sequence of NSD random variables with the same distributions from a two-tailed Pareto distribution defined by Eq (1.1). Then for all β>0 we have

    1logβnnj=1logβ2jjXja.s.pqβ. (1.6)

    In the current work, the author studies the weak and strong laws of large numbers for NSD random variables. The obtained results in this article extend and improve Theorems 1.1 and 1.2. Meanwhile, the author investigates p-mean convergence for NSD random variables under some appropriate conditions, which was not considered in [24].

    Throughout this paper, the symbol C denotes a positive constant which may differ from one place to another. The symbol I(A) denotes the indicator function of the event A.

    To prove our main results, we first present some technical lemmas.

    Lemma 2.1. ([2]) If (X1,X2,,Xn) is NSD and f1,f2,,fn are all non decreasing, then (f1(X1),f2(X2),,fn(Xn)) is also NSD.

    As we know, moment inequalities are very important tools in establishing the limit theorems for sequences of random variables. Shen et al. [6] presented the following Marcinkiewicz-Zygmund inequality with exponent 2.

    Lemma 2.2. ([6]) Let {Xn,n1} be a sequence of NSD random variables with EXn=0 and EX2n< for n1. Then

    E(max1kn(ki=1Xi)2)2ni=1EX2i,n1.

    By means of similar methods in Shao [32], Wang et al. [7] established the following Rosenthal-type maximal inequality, which is very useful in establishing the convergence properties for NSD random variables:

    Lemma 2.3. ([7]) Let p>1. Let {Xn,n1} be a sequence of NSD random variables with E|Xi|p< for each i1. Then for all n1,

    E(max1kn|ki=1Xi|p)23pni=1E|Xi|pfor1<p2

    and

    E(max1kn|ki=1Xi|p)2(15plnp)p[ni=1E|Xi|p+(EX2i)p/2]forp>2.

    Lemma 2.4. ([6]) Let {Xn,n1} be a sequence of NSD random variables. If

    n=1Var(Xn)<,

    then n=1(XnEXn) almost certainly converges.

    Now we state our main results and the proofs will be presented in next section.

    Theorem 2.1. Let {Xn,n1} be a nonnegative sequence of NSD random variables whose distributions are defined by P(Xn=0)=11/cn for n1 and the tail probability

    P(Xn>x)=1x+cnforallx>0andn1, (2.1)

    where {cn,n1} is a nondecreasing constant sequence with cn1 and

    Cn=nj=11cj. (2.2)

    Let {Dn,n1} be a sequence of constants satisfying Dn and Cn=o(Dn). Then we have

    1Dnmax1kn|kj=1c1j(XjEXnj)|P0, (2.3)

    where Xnj=XjI(XjDncj)+DncjI(Xj>Dncj), 1jn.

    Take Dn=CnlogCn, then we can obtain the following corollary which extends Theorem 1.1.

    Corollary 2.1. Let {Xn,n1} be a nonnegative sequence of NSD random variables whose distributions are defined by P(Xn=0)=11/cn for n1 and the tail probability Eq (2.1), where {cn,n1} is a nondecreasing constant sequence satisfying cn1 and Eq (2.2). Then

    1CnlogCnmax1kn|kj=1c1j(XjEXnj)|P0. (2.4)

    Remark 2.1. Yang et al. [24] proved that

    1CnlogCnnj=1c1jEXjI(XjcjCnlogCn)1

    and

    1CnlogCnnj=1c1jE(cjCnlogCnI(Xj>cjCnlogCn))=nj=1P(Xj>cjCnlogCn)0,

    which yields

    1CnlogCnnj=1c1jEXnj1.

    Then we can find that Theorem 1.1 is a special case of Corollary 2.1 for k=n. Therefore, Theorem 2.1 and Corollary 2.1 extend and improve Theorem 1.1.

    Theorem 2.2. Let {Xn,n1} be a sequence of identically distributed NSD random variables. Let {dn,n1} be a sequence of positive constants satisfying dn, and {cn,n1} be a sequence of positive constants such that φ(n)cndn satisfies φ(n) as n,

    m=n1φ2(m)=O(nφ2(n)) (2.5)

    and

    n=1P(|X1|>φ(n))<. (2.6)

    Then

    1dnnj=1c1j(XjE~Xj)0a.s., (2.7)

    where ~Xj=φ(j)I(Xj<φ(j))+XjI(|Xj|φ(j))+φ(j)I(Xj>φ(j)), 1jn.

    Remark 2.2. We will show that Theorem 1.2 is a special case of Theorem 2.2. In fact, if we assume that {Xn,n1} is a sequence of NSD random variables with the same distributions from a two-tailed Pareto distribution defined by Eq (1.1), and take cn=nlog2βn and dn=logβn (β>0), then φ(n)=cndn=nlog2n. We can verify that φ(n)=nlog2n satisfies the conditions stated in Theorem 2.2.

    First, it is clear that φ(n)=nlog2n satisfies φ(n) as n.

    Second, we have by standard calculations that

    m=n1φ2(m)n1x2log4xdx=O(n1log4(n))=O(nφ2(n)),

    which shows that Eq (2.5) is verified.

    Next, we have by Eq (1.1) and φ(n)=nlog2n that

    n=1P(|X1|>φ(n))=n=1P(|X1|>nlog2n)=n=1(nlog2nqx2dx+nlog2npx2dx)=n=1p+qnlog2n=n=11nlog2n<

    and then Eq (2.6) is verified.

    Finally, we also obtain by Eq (1.1) and φ(j)=jlog2j that

    1dnnj=1c1jE~Xj=1dnnj=1(djP(Xj<φ(j))+c1jEXjI(|Xj|<φ(j))+djP(Xj>φ(j)))=pqlogβnnj=1logβ2jj+pqlogβnnj=1logβ1jj=:J1+J2.

    By similar argument as in the proof of H0 in [24], we can obtain J10. By similar argument as in the proof of Eq (3.5) in [24], we can prove J2pqβ. Then we obtain by Eq (2.7) that

    1logβnnj=1logβ2jjXja.s.pqβ.

    To sum up, Theorem 1.2 is a special case of Theorem 2.2 and then Theorem 2.2 extends Theorem 1.2.

    Next, we present a new theorem of p-mean convergence for NSD random variables under some appropriate conditions, which was not considered in [24,25,26].

    Theorem 2.3. Let {Xn,n1} be a sequence of NSD random variables satisfying

    limxsupj1xαP(|Xj|>x)<,α(1,2). (2.8)

    Let {dn,n1} be a sequence of positive constants satisfying dn, and {cn,n1} be a sequence of positive constants such that cj1 and

    nj=1cαj=o(dαn). (2.9)

    Then for p(1,α),

    1dnmax1kn|kj=1c1j(XjE^Xnj)|Lp0, (2.10)

    where ^Xnj=dncjI(Xj<dncj)+XjI(|Xj|dncj)+dncjI(Xj>dncj), 1jn.

    Proof of Theorem 2.1. We first observe that for every ε>0,

    P(1Dnmax1kn|kj=1c1j(XjEXnj)|>2ε)P(max1kn|kj=1c1j(XjXnj)|>Dnε)+P(max1kn|kj=1c1j(XnjEXnj)|>Dnε)=:H1+H2.

    To prove Eq (2.3), we need only to show that Hi0 as n, i=1,2. For H1, we have by the definition of Xnj, Cn=o(Dn), Eqs (2.1) and (2.2) that

    H1P(nj=1(XjXnj))nj=1P(Xj>Dncj)=nj=11Dncj+cj=1Dn+1nj=1c1j=CnDn+10.

    For fixed n1, Xnj is the nondecreasing function of Xj. Hence, it follows by Lemma 2.1 that {Xnj,1jn} is a sequence of NSD random variables. Hence we have by Markov's inequality and Lemma 2.3 with 1<p2,

    H2CDpnE(max1kn|kj=1c1j(XnjEXnj)|)pCDpnnj=1cpjE|Xnj|p=CDpnnj=1cpjE|Xj|pI(XjDncj)+Cnj=1P(Xj>Dncj)=CDpnnj=1cpj(Dncj)p0P(|Xj|pI(XjDncj)t)dt+Cnj=11Dncj+cj=CDpnnj=1cpj(Dncj)p0P(|Xj|pt)dt+CCnDn+1=CDpnnj=1cpj(Dncj)p01t1/p+cjdt+CCnDn+1(by(2.1))CDpnnj=1cpj(Dncj)p01t1/pdt+CCnDn+1=CCnDn+CCnDn+10.

    The proof is completed.

    Proof of Theorem 2.2. Obviously, to prove Eq (2.7), we need only to show

    1dnnj=1c1j(Xj~Xj)0a.s. (3.1)

    and

    1dnnj=1c1j(~XjE~Xj)0a.s.. (3.2)

    By Eq (2.6), dn and the Borel-Cantelli lemma, we obtain

    1dnnj=1c1j|Xj|I(|Xj|>φ(j))0a.s..

    Noting that

    |Xj+φ(j)|I(Xj<φ(j))+|Xjφ(j)|I(Xj>φ(j))|Xj|I(|Xj|>φ(j)).

    Then

    |1dnnj=1c1j(Xj~Xj)|=|1dnnj=1c1jXjI(|Xj|>φ(j))+(Xj+φ(j))I(Xj<φ(j))+(Xjφ(j))I(Xj>φ(j))|2dnnj=1c1j|Xj|I(|Xj|>φ(j))0a.s.,

    which yields Eq (3.1).

    It follows by the definition of ~Xj that

    j=11φ2(j)E(~XjE~Xj)2Cj=11φ2(j)EX2jI(|Xj|φ(j))+Cj=1P(|Xj|>φ(j))=:I1+I2.

    We obtain directly by Eq (2.6) that I2<. Let F(x) be the distribution of X1, then

    I1=Cj=11φ2(j)EX21I(|X1|φ(j))=Cj=11φ2(j)x2I(|X1|φ(j))dF(x)
    =Cx2j:φ(j)|x|1φ2(j)dF(x). (3.3)

    Define N(|x|)={j:φ(j)<|x|} and j=inf{j:φ(j)|x|}. Hence we can obtain N(|x|)j1 and

    j:φ(j)|x|1φ2(j)j=j1φ2(j)Cjφ2(j)(by Eq (2.5))Cjx2
    CN(|x|)+1x2. (3.4)

    It follows by Eqs (2.6), (3.3) and (3.4) that

    I1C(N(|x|)+1)dF(x)=CEN(|X1|)+C=CE[j=1I(|X1|>φ(j))]+C=Cj=1P(|X1|>φ(j))+C<.

    Now we obtain by I1< and I2< that

    j=11φ2(j)E(~XjE~Xj)2<. (3.5)

    Consequently, by Lemma 2.4 and Eq (3.5), we get

    j=11φ(j)(~XjE~Xj)convergesa.s.,

    which implies Eq (3.2) by Kronecker's lemma, together with the condition dn.

    The proof is completed.

    Proof of Theorem 2.3. Noting that

    E{1dnmax1kn|kj=1c1j(XjE^Xnj)|}p1dpnE{max1kn|kj=1c1j(^XnjE^Xnj)|}p+1dpnE{max1kn|kj=1c1j(Xj^Xnj)|}p1dpn{E(max1kn|kj=1c1j(^XnjE^Xnj)|)2}p/2+1dpnE{max1kn|kj=1c1j(Xj^Xnj)|}p=:J1+J2.

    To prove Eq (2.10), it is sufficient to prove J10 and J20. By Lemma 2.1 and the fact that ^Xnj is the nondecreasing function of Xj, {^Xnj,1jn} is also a sequence of NSD random variables.

    We have by Lemma 2.2 that

    J2/p1=1d2nE{max1kn|kj=1c1j(^XnjE^Xnj)|}2Cd2nnj=1c2jE(^XnjE^Xnj)2Cd2nnj=1c2jEX2jI(|Xj|dncj)+Cnj=1P(|Xj|>dncj)=:J3+J4.

    By dn, Eqs (2.8) and (2.9), we have

    J4C1dαnnj=1cαj0asn. (3.6)

    Now we will show J30. Observing

    J3=Cd2nnj=1c2j(dncj)20P(X2jI(|Xj|dncj)t)dtCd2nnj=1c2j(dncj)20P(X2jt)dt.

    Let t=u2, then

    J3Cd2nnj=1c2jdncj0uP(|Xj|u)du.

    From Eq (2.8), we know that, there exists M>0 and N0N such that

    P(|Xj|u)Muαforu>N0. (3.7)

    Since dn and cj1, while n is sufficiently large, we can obtain dncj>N0. Hence

    J3Cd2nnj=1c2jN00uP(|Xj|u)du+CMd2nnj=1c2jdncjN0u1αdu=:J3+J3.

    By α<2, cj1 and Eq (2.9), we have

    J3Cd2nnj=1c2jN00uduCd2nnj=1c2jCd2αn1dαnnj=1cαj0asn

    and

    J3Cd2nnj=1c2j[(dncj)2αN2α0]Cdαnnj=1cαj0asn.

    Finally, we need only to show J20 as n. Let

    Znj=Xj^Xnj=(Xj+dncj)I(Xj<dncj)+(Xjdncj)I(Xj>dncj).

    We first prove that

    EZnj0asn. (3.8)

    Observing

    |EZnj|E|Znj|E|Xj|I(|Xj|>dncj)=(dncj0+dncj)P(|Xj|I(|Xj|>dncj)t)dt=dncj0P(|Xj|>dncj)dt+dncjP(|Xj|t)dt=dncjP(|Xj|>dncj)+dncjP(|Xj|t)dt=:J5+J6.

    By Eq (3.7) and α>1, we have

    J5M(dncj)α10asn

    and

    J6MdncjtαdtCM(dncj)α10asn,

    which yields Eq (3.8). Therefore, we obtain by Lemma 2.3 that

    J21dpnE{max1kn|kj=1c1j(ZnjEZnj)|}pCdpnnj=1cpjE|Znj|pCdpnnj=1cpjE|Xj|pI(|Xj|>dncj).(bythedefinitionofZnj)

    By similar arguments as in the proof of Eq (3.8), we can obtain

    E|Xj|pI(|Xj|>dncj)=(dncj)pP(|Xj|>dncj)+(dncj)pP(|Xj|pt)dt.

    Then

    J2Cnj=1P(|Xj|>dncj)+Cdpnnj=1cpj(dncj)pP(|Xj|pt)dt=:J2+J2.

    By similar arguments as the proof of J40, we obtain J20. We also have by Eq (3.7), p<α and Eq (2.9) that

    J2Cdpnnj=1cpj(dncj)ptα/pdtCdαnnj=1cαj0asn.

    The proof is completed.

    In this work the author investigated the limit theorems for negatively superadditive-dependent random variables, and obtained some new results on the law of large numbers and mean convergence under some appropriate conditions. As a future work, we propose to consider some other strong convergence for sequence of negatively superadditive-dependent random variables.

    The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.

    This work is supported by the Social Sciences Planning Project of Anhui Province (AHSKY2018D98).

    The author declares that he has no conflict of interest.



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