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Review Special Issues

Dietary and orally-delivered miRNAs: are they functional and ready to modulate immunity?

  • Received: 29 November 2022 Revised: 08 February 2023 Accepted: 20 February 2023 Published: 03 March 2023
  • MicroRNAs (miRNAs), that is, short non-coding RNA molecules, have been found in different common foods, like fruits and vegetables, meat and its products, milk (including human breast milk) and dairy products, as well as honey and herbs. Moreover, they are isolated from supernatants from cultures of various mammalian cells. A growing amount of evidence appears to support the idea of using miRNAs as therapeutics. One possible and promising route of administration is oral, which is considered noninvasive and well-tolerated by patients. Association with extracellular vesicles (EVs), nanoparticles, RNA-binding proteins, lipoproteins, or lipid derivatives, protects miRNAs from an unfavorable gastrointestinal environment (including salivary and pancreatic RNases, low pH in the stomach, digestive enzymes, peristaltic activity and microbial enzymes). Such protection likely favors miRNA absorption from the digestive tract. Internalization of miRNA by gastric and intestinal cells as well as effects on the gut microbiota by orally delivered miRNA have recently been described. Furthermore, gene regulation by orally administered miRNAs and their immunomodulatory properties indicate the possibility of cross-species or cross-kingdom communication through miRNA. In addition to the local effects, these molecules may enter the circulatory system and reach distant tissues, and thus cell-free nucleic acids are promising candidates for future selective treatments of various diseases. Nonetheless, different limitations of such a therapy imply a number of questions for detailed investigation.

    Citation: Martyna Cieślik, Krzysztof Bryniarski, Katarzyna Nazimek. Dietary and orally-delivered miRNAs: are they functional and ready to modulate immunity?[J]. AIMS Allergy and Immunology, 2023, 7(1): 104-131. doi: 10.3934/Allergy.2023008

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  • MicroRNAs (miRNAs), that is, short non-coding RNA molecules, have been found in different common foods, like fruits and vegetables, meat and its products, milk (including human breast milk) and dairy products, as well as honey and herbs. Moreover, they are isolated from supernatants from cultures of various mammalian cells. A growing amount of evidence appears to support the idea of using miRNAs as therapeutics. One possible and promising route of administration is oral, which is considered noninvasive and well-tolerated by patients. Association with extracellular vesicles (EVs), nanoparticles, RNA-binding proteins, lipoproteins, or lipid derivatives, protects miRNAs from an unfavorable gastrointestinal environment (including salivary and pancreatic RNases, low pH in the stomach, digestive enzymes, peristaltic activity and microbial enzymes). Such protection likely favors miRNA absorption from the digestive tract. Internalization of miRNA by gastric and intestinal cells as well as effects on the gut microbiota by orally delivered miRNA have recently been described. Furthermore, gene regulation by orally administered miRNAs and their immunomodulatory properties indicate the possibility of cross-species or cross-kingdom communication through miRNA. In addition to the local effects, these molecules may enter the circulatory system and reach distant tissues, and thus cell-free nucleic acids are promising candidates for future selective treatments of various diseases. Nonetheless, different limitations of such a therapy imply a number of questions for detailed investigation.



    This paper deals with periodic measures of the following reaction-diffusion lattice systems driven by superlinear noise defined on the integer set Zk :

    dui(t)+λ(t)ui(t)dtν(t)(u(i11,i2,,ik)(t)+ui1,i21,,ik(t)++ui1,i2,,ik1(t)2ku(i1,i2,,ik)(t)+u(i1+1,i2,,ik)(t)+u(i1,i2+1,,ik)(t)++u(i1,i2,,ik+1)(t))dt=fi(t,ui(t))dt+gi(t)dt+j=1(hi,j(t)+δi,jˆσi,j(t,ui(t)))dWj(t), (1.1)

    along with initial conditions:

    ui(0)=u0,i, (1.2)

    where i=(i1,i2,,ik)Zk, λ(t),ν(t) are continuous functions, λ(t)>0, (fi)iZk and (ˆσi,j)iZk,jN are two sequences of continuously differentiable nonlinearities with arbitrary and superlinear growth rate from R×RR, respectively, g=(gi)iZk and h=(hi,j)iZk,jN are two time-dependent random sequences, and δ=(δi,j)iZk,jN is a sequence of real numbers. The sequence of independent two-sided real-valued Wiener processes (Wj)jN is defined on a complete filtered probability space (Ω,F,{Ft}tR,P). Furthermore, we assume that system (1.1) is a time periodic system; more precisely, there exists T>0 such that the time-dependent functions λ,ν,fi,g,h,σi,j(iZk,jN) in (1.1) are all T-periodic in time.

    Lattice systems are gradually becoming a large and evolving interdisciplinary research field, due to wide range of applications in physics, biology and engineering such as pattern recognition, propagation of nerve pulses, electric circuits, and so on, see [1,2,3,4,5,6] and the references therein for more details. The well-posedness and the dynamics of these equations have been studied by many authors, [7,8,9,10] for deterministic systems and [11,12,13,14,15,16,17,18,19] for stochastic systems where the existence of random attractors and probability measures have been examined. Especially, the authors research the limiting behavior of periodic measures of lattice systems in [15].

    Nonlinear noise was proposed and studied for the first time in [19], the authors researches the long-term behavior of lattice systems driven by nonlinear noise in terms of random attractors and invariant measures. Before that, the research on noise was limited to additive noise and linear multiplicative noise, which can be transformed into a deterministic system. However, if the diffusion coefficients are nonlinear, then one cannot convert the stochastic system into a pathwise deterministic one, and thereby this problem cannot be studied under the frameworks of deterministic systems aforementioned. As an extension of [19], a class of reaction-diffusion lattice systems driven by superlinear noise, where the noise has a superlinear growth order q[2,p), is studied by taking advantage of the dissipativeness of the nonlinear drift function fi in (1.1) to control the superlinear noise in [20].

    In the paper, we will study the existence of periodic measures of reaction-diffusion lattice systems drive by superlinear noise. One of the main tasks in our analysis is to solve the superlinear noise terms. We remark that if the noise grows linearly, then the estimates we need can be obtained by applying the standard methods available in the literature. We adopt the ideas that take advantage of the nonlinear drift terms' the polynomical growth rate p (p2) to control the noise polynomical rate q[2,p). Furthermore, notice that l2 is an infinite-dimensional phase space and problem (1.1)–(1.2) is defined on the unbounded set Zk. The unboundedness of Zk as well as the infinite-dimensionalness of l2 introduce a major difficulty, because of the non-compactness of usual Sobolev embeddings on unbounded domains. We will employ the dissipativeness of the drift function in (1.1) as well as a cutoff technique to prove that the tails of solutions are uniformly small in L2(Ω,l2). Based upon this fact we obtain the tightness of distribution laws of solutions, and then the existence of periodic measures.

    In the next section, we discuss the well-poseness of solutions of (1.1) and (1.2). Section 3 is devoted to the uniform estimates of solutions including the uniform estimates on the tails of solutions. In Section 4, we show the existence of periodic measures of (1.1) and (1.2).

    In this section, we prove the existence and uniqueness of solutions to system (1.1) and (1.2). We first discuss the assumptions on the nonlinear drift and diffusion terms in (1.1).

    We begin with the following Banach space:

    lr={u=(ui)iZk:iZk|ui|r<+} with norm ur=(iZk|ui|r)1r,r1.

    The norm and inner product of l2 are denoted by (,) and , respectively. For the nonlinear drift function fiC1(R×R,R) in the equation we assume that for all sR and iZk,

    fi(t,s)sγ1|s|p+ϕ1,i, ϕ1={ϕ1,i}iZkl1, (2.1)
    |fi(t,s)|ϕ2,i|s|p1+ϕ3,i, ϕ2={ϕ2,i}iZkl, ϕ3={ϕ3,i}iZkl2, (2.2)
    |fi(t,s)|ϕ4,i|s|p2+ϕ5,i, ϕ4={ϕ4,i}iZkl, ϕ5={ϕ5,i}iZkl, (2.3)

    where p>2 and γ1>0 are constants. For the sequence of continuously differentiable diffusion functions ˆσ=(ˆσi,j)iZk,jN, we assume, for all sR and jN,

    |ˆσi,j(t,s)|φ1,i|s|q2+φ2,i, φ1={φ1,i}iZkl2ppq, φ2={φ2,i}iZkl2, (2.4)
    |ˆσi,j(t,s)|φ3,i|s|q21+φ4,i, φ3={φ3,i}iZklq, φ4={φ4,i}iZkl, (2.5)

    where q[2,p) is a constant. For processes g(t)=(gi(t))iZk and h(t)=(hi,j)iZk,jN are both continuous in tR, which implies that for all tR,

    g(t)2=iZk|gi(t)|2< and h(t)2=iZkjN|hi,j(t)|2<. (2.6)

    In addition, we assume δ=(δi,j)iZk,jN satisfies

    cδ:=jNiZk|δi,j|2<. (2.7)

    We will investigate the periodic measures of system (1.1)–(1.2) for which we assume that all given time-dependent functions are T-periodic in tR for some T>0; that is, for all tR,iZk and kN.

    λ(t+T)=λ(t),ν(t+T)=ν(t),h(t+T)=h(t),g(t+T)=g(t),f(t+T,)=f(t,),σ(t+T,)=σ(t,).

    If m:RR is a continuous T-periodic function, we denote

    ¯m=max0tTm(t),m_=min0tTm(t).

    We want to reformulate problem (1.1)–(1.2) as an abstract one in l2. Given 1jk,u=(ui)iZkl2 and i=(i1,i2,,ik)Zk. Let us define the operators from l2 to l2 by

    (Bju)i=u(i1,,ij+1,,ik)u(i1,,ij,,ik),(Bju)i=u(i1,,ij1,,ik)u(i1,,ij,,ik),(Aju)i=u(i1,,ij+1,,ik)+2u(i1,,ij,,ik)u(i1,,ij1,,ik),

    and

    (Aku)i=u(i11,i2,,ik)u(i1,i21,,ik)u(i1,i2,,ik1)+2ku(i1,i2,,ik)u(i1+1,i2,,ik)u(i1,i2+1,,ik)u(i1,i2,,ik+1).

    For all 1jk,u=(ui)iZkl2 and v=(vi)iZkl2 we see

    Bju2u,(Bju,v)=(u,Bjv),Aj=BjBj and Ak=kj=1Aj. (2.8)

    Again, define the operators f,σj:R×l2l2 by

    f(t,u)=(fi(t,ui))iZk and σj(t,u)=(δi,jˆσi,j(t,ui))iZk,tR,u=(ui)iZkl2.

    It follows from (2.3) that there exists θ(0,1) such that for p>2 and u,vl2,

    iZk|fi(t,ui)fi(t,vi)|2=iZk|fi(θui+(1θ)vi)|2|uivi|2iZk(|ϕ4,i||θui+(1θ)vi|p2+|ϕ5,i|)2|uivi|2iZk(22p4|ϕ4,i|2(|ui|2p4+|vi|2p4)+2|ϕ5,i|2)|uivi|2(22p4ϕ42l(u2p4+v2p4)+2ϕ52l)uv2. (2.9)

    This together with f(t,0)l2 by (2.2) yields f(t,u)l2 for all ul2, and thereby f:R×l2l2 is well-defined. In addition, we deduce from (2.9) that f:R×l2l2 is a locally Lipschitz continuous function, that is, for every nN, we can find a constant c1(n)>0 satisfying, for all u,vl2 with un and vn,

    f(u)f(v)c1(n)uv. (2.10)

    For q[2,p) and ul2, one can deduce from(2.4), (2.7) and Young's inequality that for all ϖ>0,

    ϖjNσj(t,u)2=ϖjNiZk|δi,jˆσi,j(t,ui)|22ϖjNiZk|δi,j|2(|φ1,i|2|ui|q+|φ2,i|2)2ϖcδiZk(|φ1,i|2|ui|q+|φ2,i|2)γ12iZk|ui|p+pqp(pγ12q)qpq(2ϖcδ)ppqiZk|φ1,i|2ppq+2ϖcδiZk|φ2,i|2γ12upp+pqp(pγ12q)qpq(2ϖcδ)ppqφ12ppq2ppq+2ϖcδφ22, (2.11)

    where γ1 is the same number as in (2.1). From (2.11) and l2lp for p>2, we find that σj(t,u)l2 for all ul2. Then σj:R×l2l2 is also well-defined. In addition, it yields from (2.5) and (2.7) that there exists η(0,1) such that for q[2,p) and u,vl2,

    jNiZk|δi,jˆσi,j(t,ui)δi,jˆσi,j(t,vi)|2=iZkjN|δi,j|2|ˆσi,j(t,ui)ˆσi,j(t,vi)|2=iZkjN|δi,j|2|ˆσi,j(ηui+(1η)vi)|2|uivi|2cδiZk(|φ3,i||ηui+(1η)vi|q21+|φ4,i|)2|uivi|2cδiZk(2q2|φ3,i|2(|ui|q2+|vi|q2)+2|φ4,i|2)|uivi|2cδiZk(2q2(4q|φ3,i|q+q2q|ui|q+q2q|vi|q)+2|φ4,i|2)|uivi|2cδ(2q1(φ3qq+uq+vq)+2φ42l)uv2. (2.12)

    This implies that σj:R×l2l2 is also locally Lipschitz continuous, more precisely, for every nN, one can find a constant c2(n)>0 satisfying, for all u,vl2 with un and vn,

    jNσj(u)2c22(n). (2.13)

    and

    jNσj(u)σj(v)2c22(n)uv2. (2.14)

    By above notations one is able to rewrite (1.1)–(1.2) as the following system in l2 for t>0 :

    du(t)+ν(t)Aku(t)dt+λ(t)u(t)dt=f(t,u(t))dt+g(t)dt+j=1(hj(t)+σj(t,u(t)))dWj(t), (2.15)

    with initial condition:

    u(0)=u0l2, (2.16)

    in the present article, the solutions of system (2.15)–(2.16) are interpreted in the following sense.

    Definition 2.1. Suppose u0L2(Ω,l2) is F0-measurable, a continuous l2-valued Ft-adapted stochastic process u is called a solution of equations (2.15) and (2.16) if uL2(Ω,C([0,T],l2))Lp(Ω,Lp(0,T;lp)) for all T>0, and the following equation holds for all t0 and almost all ωΩ:

    u(t)=u0+t0(ν(s)Aku(s)λ(s)u(s)+f(s,u(s))+g(s))ds+j=1t0(hj(s)+σj(s,u(s)))dWj(s) in l2. (2.17)

    Similar to Ref.[20], we can get (2.15) and (2.16) exist global solutions in the sense of Definition 2.1.

    In this section, we derive the uniform estimates of solutions of (2.15)–(2.16). These estimates will be used to establish the tightness of a set of probability distributions of u in l2.

    We assume that

    α(t)=λ(t)16k|ν(t)|>0. (3.1)

    Lemma 3.1. Let (2.1)–(2.7) and (3.1) hold. Then the solutions u(t,0,u0) of system (2.15) and (2.16) with initial data u0 at time 0 satisfy, for all t0,

    E(u(t,0,u0)2)+t0eα_(rt)E(u(r,0,u0)pp)drL1(E(u02)+j=1¯hj2+¯g2+φ12ppq2ppq+φ22+ϕ11), (3.2)

    where L1>0 is a positive constant which depends on α_,p,q,γ,cδ,t, but indepentent of u0.

    Proof. Applying Ito's formula to (2.15) we get

    d(u(t)2)+2ν(t)kj=1Bju(t)2dt+2λ(t)u(t)2dt=2(f(t,u(t)),u(t))dt+2(g(t),u(t))dt+j=1hj(t)+σ(t,u(t))2dt+2j=1u(t)(hj(t)+σj(t,u(t)))dWj(t).

    This implies

    ddtE(u(t)2)+2ν(t)kj=1E(Bju(t)2)+2λ(t)E(u(t)2)2E(f(t,u(t)),u(t))+2E(g(t),u(t))+2j=1E(hj(t)2)+2j=1E(σ(t,u(t))2). (3.3)

    For the second term on the left-hand side of (3.3), we have

    2|ν(t)|kj=1E(Bju(t)2)8k|ν(t)|E(u(t)2). (3.4)

    For the first term on the right-hand side of (3.3), we get from (2.1) that

    2E(f(t,u(t)),u(t))2γ1E(u(t)pp)+2ϕ11. (3.5)

    For the second term on the right-hand side of (3.3), we have

    2E(g(t),u(t))λ(t)E(u(t)2)+1λ(t)E(g(t)2). (3.6)

    For the last term on the right-hand side of (3.3), we infer from (2.11) with ω=2 that

    2j=1E(σj(t,u(t))2)γ12E(u(t)pp)+pqp(pγ12q)qpq(4cδ)ppqφ12ppq2ppq+4cδφ22. (3.7)

    By (3.3)–(3.7) we get

    ddtE(u(t)2)+α_E(u(t)2)+32γ1E(u(t)pp)E(j=12hj(t)2+1λ(t)g(t)2)+C1, (3.8)

    implies that

    ddtE(u(t)2)+α_E(u(t)2)+32γ1E(u(t)pp)2j=1¯hj2+1λ_¯g2+C1, (3.9)

    where C1=pqp(pγ12q)qpq(4cδ)ppqφ12ppq2ppq+4cδφ22+2ϕ11. Multiplying (3.9) by eα_t and integrating over (0,t) to obtain

    E(u(t,0,u0)2)+32γ1t0eα_(rt)E(u(r,0,u0)pp)dreα_tE(u02)+C2t0eα_(rt)dr, (3.10)

    where C2=2j=1¯hj2+1λ_¯g2+C1. This completes the proof.

    Lemma 3.2. Let (2.1)–(2.7), and (3.1) be satisfied. Then for compact subset K of l2, one can find a number N0=N0(K)N such that the solutions u(t,0,u0) of (2.15) and (2.16) satisfy, for all nN0 and t0,

    E(in|ui(t,0,u0)|2)+t0eα_(rt)E(in|ui(r,0,u0)|p)drε, (3.11)

    where u0K and i:=maxijk|ij|.

    Proof. Define a smooth function ξ:R[0,1] such that

    ξ(s)=0 for |s|1 and ξ(s)=1 for |s|2. (3.12)

    Denote by

    ξn=(ξ(in))iZk and ξnu=(ξ(in)ui)iZk,u=(ui)iZk,nN. (3.13)

    Similar notations will also be used for other terms. It follows from (2.15) that

    d(ξnu(t))+ν(t)ξnAku(t)dt+λ(t)ξnu(t)dt=ξnf(t,u(t))dt+ξng(t)dt+j=1(ξnhj(t)+ξnσj(t,u(t)))dWj(t). (3.14)

    By Ito's formula and (3.14) we have

    dξnu(t)2+2ν(t)(Ak(u(t)),ξ2nu(t))dt+2λ(t)ξnu(t)2dt=2(f(t,u(t)),ξ2nu(t))dt+2(g(t),ξ2nu(t))dt+j=1ξnhj(t)+ξnσj(t,u(t))2dt+2j=1(hj(t)+σj(t,u(t)),ξ2nu(t))dWj. (3.15)

    This yields

    ddtE(ξnu(t)2)+2ν(t)E(Ak(u(t)),ξ2nu(t))+2λ(t)E(ξnu(t)2)=2E(f(t,u(t)),ξ2nu(t))+2E(g(t),ξ2nu(t))+2j=1E(ξnhj(t)2)+2j=1E(ξnσj(t,u(t))2)dt. (3.16)

    For the second term on the left-hand side of (3.16), we have

    2ν(t)E(Ak(u(t)),ξ2nu(t))=2ν(t)kj=1E(Bju(t),Bj(ξ2nu(t)))=2ν(t)E(kj=1iZk(ui1,,ij+1,,ikui)×(ξ2((i1,,ij+1,,ik)n)u(i1,,ij+1,,ik)ξ2(in)ui))=2ν(t)E(kj=1iZkξ2(in)(ui1,,ij+1,,ikui)2)+2ν(t)E(kj=1iZk(ξ2((i1,,ij+1,,ik)n)ξ2(in))×(u(i1,,ij+1,,ik)ui)u(i1,,ij+1,,ik)). (3.17)

    We first deal with the first term on the right-hand side of (3.17). Notice that

    2|ν(t)|E(kj=1iZkξ2(in)(ui1,,ij+1,,ikui)2)=2|ν(t)|E(kj=1iZk|ξ(in)u(i1,,ij+1,,ik)ξ(in)ui|2)4|ν(t)|E(kj=1iZk|(ξ(in)ξ((i1,,ij+1,,ik)n))u(i1,,ij+1,,ik)|2)+4|ν(t)|E(kj=1iZk|ξ((i1,,ij+1,,ik)n)u(i1,,ij+1,,ik)ξ(in)ui|2). (3.18)

    By the definition of function ξ, there exists a constant C3>0 such that |ξ(s)|C3 for all sR. Then the first term on the right-hand side of (3.18) is bounded by

    4|ν(t)|E(kj=1iZk|(ξ(in)ξ((i1,,ij+1,,ik)n))u(i1,,ij+1,,ik)|2)=4|ν(t)|E(kj=1iZk|ξ(in)ξ((i1,,ij+1,,ik)n)|2|u(i1,,ij+1,,ik)|2)4C23n2|ν(t)|E(kj=1iZk|u(i1,,ij+1,,ik)|2)4C23kn2|ν(t)|E(u2). (3.19)

    By the definition of |Bju|i, the last term on the right-hand side of (3.18) is bounded by

    4|ν(t)|E(kj=1iZk|ξ((i1,,ij+1,,ik)n)u(i1,,ij+1,,ik)ξ(in)ui|2)4|ν(t)|E(kj=1Bj(ξnu(t))2)16k|ν(t)|E(ξnu(t)2). (3.20)

    Then we find from (3.18) to (3.20) that the first term on the right-hand side of (3.17) is bounded by

    2|ν(t)|E(kj=1iZkξ2(in)(u(i1,,ij+1,,ik)ui)2)16k|ν(t)|E(ξnu(t)2)+4C23kn2|ν(t)|E(u2). (3.21)

    In addition, we find that the last term on the right-hand side of (3.17) can be bounded by

    2|ν(t)E(kj=1iZk(ξ2((i1,,ij+1,,ik)n)ξ2(in))×(u(i1,,ij+1,,ik)ui)u(i1,,ij+1,,ik))|2|ν(t)|E(kj=1iZk|ξ2((i1,,ij+1,,ik)n)ξ2(in)|×|u(i1,,ij+1,,ik)ui||u(i1,,ij+1,,ik)|)4|ν(t)|E(kj=1iZk|ξ((i1,,ij+1,,ik)n)ξ(in)|×|u(i1,,ij+1,,ik)ui||u(i1,,ij+1,,ik)|)4C3n|ν(t)|E(kj=1iZk|u(i1,,ij+1,,ik)ui||u(i1,,ij+1,,ik)|)8kC3n|ν(t)|E(u2). (3.22)

    By (3.21), (3.22) and (3.17), we infer that the second term on the left-hand side of (3.16) satisfied

    2|ν(t)E(Ak(u(t)),ξ2nu(t))|C4|ν(t)|(1n+1n2)E(u2)+16k|ν(t)|E(ξnu(t)2), (3.23)

    where C4=4kC3(2+C3). For the first term on the right-hand side of (3.16), we find from (2.1) that

    2E(f(t,u(t)),ξ2nu(t))2γ1E(iZkξ2(in)|ui(t)|p)+2E(iZkξ2(in)|ϕ1,i|)2γ1E(iZkξ2(in)|ui(t)|p)+2in|ϕ1,i|. (3.24)

    For the second term on the right-hand side of (3.16), we infer from Young's inequality that

    2E(g,ξ2nu(t))λ_E(ξnu(t)2)+1λ_E(iZkξ2(in)|gi|2)λ_E(ξnu(t)2)+1λ_in|gi|2. (3.25)

    For the last term on the right-hand side (3.16), we infer from (2.4) and Young's inequality that

    2j=1E(ξnσj(t,u(t))2)=2j=1E(iZk|ξ(in)δi,jˆσi,j(t,ui(t))|2)4j=1E(iZkξ2(in)|δi,j|2(|φ1,i|2|ui(t)|q+|φ2,i|2))4cδE(iZkξ2(in)(|φ1,i|2|ui(t)|q+|φ2,i|2))γ1E(iZkξ2(in)|ui(t)|p)+pqp(pγ1q)qpq(4cδ)ppqiZkξ2(in)|φ1,i|2ppq+4cδiZkξ2(in)|φ2,i|2γ1E(iZkξ2(in)|ui(t)|p)+pqp(pγ1q)qpq(4cδ)ppqin|φ1,i|2ppq+4cδin|φ2,i|2. (3.26)

    Substituting (3.23)–(3.26) into (3.16) we get

    ddtE(ξnu(t)2)+α_E(ξnu(t)2)+γ1E(iZkξ2(in)|ui(t)|p)C4|ν|(1n+1n2)E(u2)+C5(in(¯|gi|2+|φ1,i|2ppq+|φ2,i|2+|ϕ1,i|)+inj=1¯|hi,j|2), (3.27)

    where C5=2+1λ_+pqp(pγ1q)qpq(4cδ)ppq+4cδ. One can multiply (3.27) by eα_t and integrate over (0,t) in order to obtain

    E(ξnu(t,0,u0)2)+γ1t0eα_(rt)E(iZkξ2(in)|ui(r,0,u0)|p)dreα_tE(ξnu02)+C4|ν|(1n+1n2)t0eα_(rt)E(u(r,0,u0)2)dr+C5α_(in(¯|gi|2+|φ1,i|2ppq+|φ2,i|2+|ϕ1,i|)+inj=1¯|hi,j|2). (3.28)

    Since K is a compact subset of l2 we infer from (3.1) that

    limnsupu0Ksupt0eα_tE(ξnu02)limnsupu0KE(in|u0,i|2)=0. (3.29)

    By Lemma 3.1, we find that for all u0K and t0, as n,

    (1n+1n2)t0eα_(rt)E(u(r,0,u0)2)drL1α_(1n+1n2)(E(u02)+j=1¯hj2+¯g2+φ12ppq2ppq+φ22+ϕ11)L1α_(1n+1n2)(C6+j=1¯hj2+¯g2+φ12ppq2ppq+φ22+ϕ11)0, (3.30)

    where L1 is the same number of (3.1) and C6>0 is a constant depending only on u0.By φ1l2ppq,φ2l2,ϕ1l1, (2.6) and (3.1), we infer that

    in(¯|gi|2+|φ1,i|2ppq+|φ2,i|2+|ϕ1,i|)+inj=1¯|hi,j|20 as n. (3.31)

    It follows from (3.28) to (3.31) that as n,

    supu0Ksupt0(E(ξnu(t,0,u0)2)+t0eα_(rt)E(iZkξ2(in)|ui(r,0,u0)|p)dr)0. (3.32)

    Then for every ε>0 we can find a number N0=N0(K)N satisfying, for all nN0 and t0,

    (E(i2n|ui(t,0,u0)|2)+t0eα_(rt)E(i2n|ui(t,0,u0)|p)dr)(E(ξnu(t,0,u0)2)+t0eα_(rt)E(iZkξ2(in)|ui(t,0,u0)|p)dr)ε, (3.33)

    uniformly for u0K and t0. This concludes the proof.

    In the sequel, we use L(u(t,0,u0)) to denote the probability distribution of the solution u(t,0,u0) of (2.15)–(2.16) which has initial condition u0 at initial time 0. Then we have the following tightness of a family of distributions of solutions.

    Lemma 4.1. Suppose (2.1)–(2.7) and (3.1) hold. Then the family {L(u(t,0,u0)):t0} of the distributions ofthe solutions of (2.15)–(2.16) is tight on l2.

    Proof. For simplicity, we will write the solution u(t,0,u0) as u(t) from now on. It follows from Lemma 3.1 that there exists a constant c1>0 such that

    E(u(t)2)c1,for allt0. (4.1)

    By Chebyshev's inequality, we get from (4.1) that for all t0,

    P(u(t)2R)c1R20asR.

    Hence for every ϵ>0, there exists R1=R1(ϵ)>0 such that for all t0,

    P{u(t)2R1}12ϵ. (4.2)

    By Lemma 3.2, we infer that for each ϵ>0 and mN, there exists an integer nm=nm(ϵ,m) such that for all t0,

    E(|i|>nm|ui(t)|2)<ϵ22m+2,

    and hence for all t0 and mN,

    P({|i|>nm|ui(r)|212m})2mE(|i|>nm|ui(r)|2)<ϵ2m+2. (4.3)

    It follows from (4.3) for all t0,

    P(m=1{|i|>nm|ui(t)|212m})m=1ϵ2m+214ϵ,

    which shows that for all t0,

    P({|i|>nm|ui(t)|212mfor allmN})>1ϵ2. (4.4)

    Given ϵ>0, set

    Y1,ϵ={vl2:vR1(ϵ)}, (4.5)
    Y2,ϵ={vl2:|i|>nm|vi(r)|212mfor allmN}, (4.6)

    and

    Yϵ=Y1,ϵY2,ϵ. (4.7)

    By (4.2) and (4.4) we get, for all t0,

    P({u(t)Yϵ})>1ϵ. (4.8)

    Now, we show the precompactness of {v:vYϵ} in l2. Given κ>0, choose an integer m0=m0(κ)N such that 2m0>8κ2. Then by (4.6) we obtain

    |i|>nm0|vi|212m0<κ28,vYϵ. (4.9)

    On the other hand, by (4.5) we see that the set {(vi)|i|m0:vYϵ} is bounded in the finite-dimensional space R2m0+1 and hence precompact. Consequently, {v:vYϵ} has a finite open cover of balls with radius κ2, which along with (4.9) implies that the set {v:vYϵ} has a finite open cover of balls with radius κ in l2. Since κ>0 is arbitrary, we find that the set {v:vYϵ} is precompact in l2. This completes the proof.

    If ϕ:l2R is a bounded Borel function, then for 0rt and u0l2, we set

    (pr,tϕ)(u0)=E(ϕ(u(t,r,u0)))

    and

    p(r,u0;t,Γ)=(pr,t1Γ)(u0),

    where ΓB(l2) and 1Γ is the characteristic function of Γ. The operators ps,t with 0st are called the transition operators for the solutions of (2.15)–(2.16). Recall that a probability measure ν on l2 is periodic for (2.15)–(2.16) if

    l2(p0,t+Tϕ)(u0)dν(u0)=l2(p0,tϕ)(u0)dν(u0),t0.

    Lemma 4.2. [21]Let ϱ(ψ,ω) be a scalar bounded measurable randomfunction of ψ, independent of Fs. Let ς be anFs-measurable random variable. Then

    E(ϱ(ς,ω)|Fs)=E(ϱ(ς,ω)).

    The transition operators {pr,t}0rt have the following properties.

    Lemma 4.3. Assume that (2.1)–(2.7) and (3.1) hold. Then:

    (i) {pr,t}0rt is Feller; that is, for every bounded andcontinuous ϕ:l2R, the function pr,tϕ:l2Ris also bounded and continuous for all 0rt.

    (ii) The family {pr,t}0rt is T-periodic; that is, for all 0rt,

    p(r,u0;t,)=p(r+T,u0;t+T,),u0l2.

    (iii) {u(t,0,u0)}t0 is a l2-valued Markov process.

    Finally, we present our main result on the existence of periodic measures for problem (2.15)–(2.16).

    Theorem 4.4. Assume that (2.1)–(2.7) and (3.1) hold. Then problem (2.15)–(2.16) has a periodic measure on l2.

    Proof. We apply Krylov-Bogolyubov's method to prove the existence of periodic measures of (2.15)–(2.16), define a probability measure μn by

    μn=1nnl=1p(0,0;lT,). (4.10)

    By Lemma 4.1 we see the sequence {μn}n=1 is tight on l2, and hence there exists a probability measure μ on l2 such that, up to a subsequence,

    μnμ,as n. (4.11)

    By (4.10)–(4.11) and Lemma 4.3, we infer that for every t0 and every bounded and continuous function ϕ:l2R,

    l2(p0,tϕ)(u0)dμ(u0)=l2(l2ϕ(y)p(0,u0;t,dy))dμ(u0)=limn1nnl=1l2(l2ϕ(y)p(0,u0;t,dy))p(0,0;lT,du0)=limn1nnl=1l2(l2ϕ(y)p(kT,u0;t+lT,dy))p(0,0;kT,du0)=limn1nnl=1l2ϕ(y)p(0,0;t+lT,dy)=limn1nnl=1l2ϕ(y)p(0,0;t+lT+T,dy)=limn1nnk=1l2(l2ϕ(y)p(0,u0;t+T,dy))p(0,0;lT,du0)=l2(l2ϕ(y)p(0,u0;t+T,dy))dμ(u0)=l2(p0,t+Tϕ)(u0)dμ(u0), (4.12)

    which shows that μ is a periodic measure of (2.15)–(2.16), as desired.

    The author declares there is no conflict of interest.


    Acknowledgments



    The language editing service was supported by grant number N41/DBS/001026 from the Polish Ministry of Education and Science.

    Conflict of interest



    The authors declare no conflict of interest.

    Author contributions



    M.C.—original draft preparation; K.B. and K.N.—review and editing. All authors have read and agreed to the published version of the manuscript.

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