In this paper, we first investigate a new asymmetric Huber regression (AHR) estimation procedure to analyze skewed data with partial functional linear models. To automatically reflect distributional features as well as bound the influence of outliers effectively, we further propose a weighted composite asymmetric Huber regression (WCAHR) estimation procedure by combining the strength across multiple asymmetric Huber loss functions. The slope function and constant coefficients are estimated through minimizing the combined loss function and approximating the slope function with principal component analysis. The asymptotic properties of the proposed estimators are derived. To realize the WCAHR estimation, we also develop a practical algorithm based on pseudo data. Numerical results show that the proposed WCAHR estimators can well adapt to the different error distributions, and thus are more useful in practice. Two real data examples are presented to illustrate the applications of the proposed methods.
Citation: Juxia Xiao, Ping Yu, Zhongzhan Zhang. Weighted composite asymmetric Huber estimation for partial functional linear models[J]. AIMS Mathematics, 2022, 7(5): 7657-7684. doi: 10.3934/math.2022430
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In this paper, we first investigate a new asymmetric Huber regression (AHR) estimation procedure to analyze skewed data with partial functional linear models. To automatically reflect distributional features as well as bound the influence of outliers effectively, we further propose a weighted composite asymmetric Huber regression (WCAHR) estimation procedure by combining the strength across multiple asymmetric Huber loss functions. The slope function and constant coefficients are estimated through minimizing the combined loss function and approximating the slope function with principal component analysis. The asymptotic properties of the proposed estimators are derived. To realize the WCAHR estimation, we also develop a practical algorithm based on pseudo data. Numerical results show that the proposed WCAHR estimators can well adapt to the different error distributions, and thus are more useful in practice. Two real data examples are presented to illustrate the applications of the proposed methods.
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)dt−ν(t)(u(i1−1,i2,…,ik)(t)+ui1,i2−1,…,ik(t)+…+ui1,i2,…,ik−1(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)i∈Zk and (ˆσi,j)i∈Zk,j∈N are two sequences of continuously differentiable nonlinearities with arbitrary and superlinear growth rate from R×R→R, respectively, g=(gi)i∈Zk and h=(hi,j)i∈Zk,j∈N are two time-dependent random sequences, and δ=(δi,j)i∈Zk,j∈N is a sequence of real numbers. The sequence of independent two-sided real-valued Wiener processes (Wj)j∈N is defined on a complete filtered probability space (Ω,F,{Ft}t∈R,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(i∈Zk,j∈N) 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 (p≥2) 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)i∈Zk:∑i∈Zk|ui|r<+∞} with norm ‖u‖r=(∑i∈Zk|ui|r)1r,∀r≥1. |
The norm and inner product of l2 are denoted by (⋅,⋅) and ‖⋅‖, respectively. For the nonlinear drift function fi∈C1(R×R,R) in the equation we assume that for all s∈R and i∈Zk,
fi(t,s)s≤−γ1|s|p+ϕ1,i, ϕ1={ϕ1,i}i∈Zk∈l1, | (2.1) |
|fi(t,s)|≤ϕ2,i|s|p−1+ϕ3,i, ϕ2={ϕ2,i}i∈Zk∈l∞, ϕ3={ϕ3,i}i∈Zk∈l2, | (2.2) |
|f′i(t,s)|≤ϕ4,i|s|p−2+ϕ5,i, ϕ4={ϕ4,i}i∈Zk∈l∞, ϕ5={ϕ5,i}i∈Zk∈l∞, | (2.3) |
where p>2 and γ1>0 are constants. For the sequence of continuously differentiable diffusion functions ˆσ=(ˆσi,j)i∈Zk,j∈N, we assume, for all s∈R and j∈N,
|ˆσi,j(t,s)|≤φ1,i|s|q2+φ2,i, φ1={φ1,i}i∈Zk∈l2pp−q, φ2={φ2,i}i∈Zk∈l2, | (2.4) |
|ˆσ′i,j(t,s)|≤φ3,i|s|q2−1+φ4,i, φ3={φ3,i}i∈Zk∈lq, φ4={φ4,i}i∈Zk∈l∞, | (2.5) |
where q∈[2,p) is a constant. For processes g(t)=(gi(t))i∈Zk and h(t)=(hi,j)i∈Zk,j∈N are both continuous in t∈R, which implies that for all t∈R,
‖g(t)‖2=∑i∈Zk|gi(t)|2<∞ and ‖h(t)‖2=∑i∈Zk∑j∈N|hi,j(t)|2<∞. | (2.6) |
In addition, we assume δ=(δi,j)i∈Zk,j∈N satisfies
cδ:=∑j∈N∑i∈Zk|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 t∈R for some T>0; that is, for all t∈R,i∈Zk and k∈N.
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:R→R is a continuous T-periodic function, we denote
¯m=max0≤t≤Tm(t),m_=min0≤t≤Tm(t). |
We want to reformulate problem (1.1)–(1.2) as an abstract one in l2. Given 1≤j≤k,u=(ui)i∈Zk∈l2 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),(B∗ju)i=u(i1,…,ij−1,…,ik)−u(i1,…,ij,…,ik),(Aju)i=−u(i1,…,ij+1,…,ik)+2u(i1,…,ij,…,ik)−u(i1,…,ij−1,…,ik), |
and
(Aku)i=−u(i1−1,i2,…,ik)−u(i1,i2−1,…,ik)−…−u(i1,i2,…,ik−1)+2ku(i1,i2,…,ik)−u(i1+1,i2,…,ik)−u(i1,i2+1,…,ik)−…−u(i1,i2,…,ik+1). |
For all 1≤j≤k,u=(ui)i∈Zk∈l2 and v=(vi)i∈Zk∈l2 we see
‖Bju‖≤2‖u‖,(B∗ju,v)=(u,Bjv),Aj=BjB∗j and Ak=k∑j=1Aj. | (2.8) |
Again, define the operators f,σj:R×l2→l2 by
f(t,u)=(fi(t,ui))i∈Zk and σj(t,u)=(δi,jˆσi,j(t,ui))i∈Zk,∀t∈R,∀u=(ui)i∈Zk∈l2. |
It follows from (2.3) that there exists θ∈(0,1) such that for p>2 and u,v∈l2,
∑i∈Zk|fi(t,ui)−fi(t,vi)|2=∑i∈Zk|f′i(θui+(1−θ)vi)|2|ui−vi|2≤∑i∈Zk(|ϕ4,i||θui+(1−θ)vi|p−2+|ϕ5,i|)2|ui−vi|2≤∑i∈Zk(22p−4|ϕ4,i|2(|ui|2p−4+|vi|2p−4)+2|ϕ5,i|2)|ui−vi|2≤(22p−4‖ϕ4‖2l∞(‖u‖2p−4+‖v‖2p−4)+2‖ϕ5‖2l∞)‖u−v‖2. | (2.9) |
This together with f(t,0)∈l2 by (2.2) yields f(t,u)∈l2 for all u∈l2, and thereby f:R×l2→l2 is well-defined. In addition, we deduce from (2.9) that f:R×l2→l2 is a locally Lipschitz continuous function, that is, for every n∈N, we can find a constant c1(n)>0 satisfying, for all u,v∈l2 with ‖u‖≤n and ‖v‖≤n,
‖f(u)−f(v)‖≤c1(n)‖u−v‖. | (2.10) |
For q∈[2,p) and u∈l2, one can deduce from(2.4), (2.7) and Young's inequality that for all ϖ>0,
ϖ∑j∈N‖σj(t,u)‖2=ϖ∑j∈N∑i∈Zk|δi,jˆσi,j(t,ui)|2≤2ϖ∑j∈N∑i∈Zk|δi,j|2(|φ1,i|2|ui|q+|φ2,i|2)≤2ϖcδ∑i∈Zk(|φ1,i|2|ui|q+|φ2,i|2)≤γ12∑i∈Zk|ui|p+p−qp(pγ12q)−qp−q(2ϖcδ)pp−q∑i∈Zk|φ1,i|2pp−q+2ϖcδ∑i∈Zk|φ2,i|2≤γ12‖u‖pp+p−qp(pγ12q)−qp−q(2ϖcδ)pp−q‖φ1‖2pp−q2pp−q+2ϖcδ‖φ2‖2, | (2.11) |
where γ1 is the same number as in (2.1). From (2.11) and l2⊆lp for p>2, we find that σj(t,u)∈l2 for all u∈l2. Then σj:R×l2→l2 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,v∈l2,
∑j∈N∑i∈Zk|δi,jˆσi,j(t,ui)−δi,jˆσi,j(t,vi)|2=∑i∈Zk∑j∈N|δi,j|2|ˆσi,j(t,ui)−ˆσi,j(t,vi)|2=∑i∈Zk∑j∈N|δi,j|2|ˆσ′i,j(ηui+(1−η)vi)|2|ui−vi|2≤cδ∑i∈Zk(|φ3,i||ηui+(1−η)vi|q2−1+|φ4,i|)2|ui−vi|2≤cδ∑i∈Zk(2q−2|φ3,i|2(|ui|q−2+|vi|q−2)+2|φ4,i|2)|ui−vi|2≤cδ∑i∈Zk(2q−2(4q|φ3,i|q+q−2q|ui|q+q−2q|vi|q)+2|φ4,i|2)|ui−vi|2≤cδ(2q−1(‖φ3‖qq+‖u‖q+‖v‖q)+2‖φ4‖2l∞)‖u−v‖2. | (2.12) |
This implies that σj:R×l2→l2 is also locally Lipschitz continuous, more precisely, for every n∈N, one can find a constant c2(n)>0 satisfying, for all u,v∈l2 with ‖u‖≤n and ‖v‖≤n,
∑j∈N‖σj(u)‖2≤c22(n). | (2.13) |
and
∑j∈N‖σj(u)−σj(v)‖2≤c22(n)‖u−v‖2. | (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)=u0∈l2, | (2.16) |
in the present article, the solutions of system (2.15)–(2.16) are interpreted in the following sense.
Definition 2.1. Suppose u0∈L2(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 u∈L2(Ω,C([0,T],l2))∩Lp(Ω,Lp(0,T;lp)) for all T>0, and the following equation holds for all t≥0 and almost all ω∈Ω:
u(t)=u0+∫t0(−ν(s)Aku(s)−λ(s)u(s)+f(s,u(s))+g(s))ds+∞∑j=1∫t0(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 t≥0,
E(‖u(t,0,u0)‖2)+∫t0eα_(r−t)E(‖u(r,0,u0)‖pp)dr≤L1(E(‖u0‖2)+∞∑j=1¯‖hj‖2+¯‖g‖2+‖φ1‖2pp−q2pp−q+‖φ2‖2+‖ϕ1‖1), | (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)k∑j=1‖Bju(t)‖2dt+2λ(t)‖u(t)‖2dt=2(f(t,u(t)),u(t))dt+2(g(t),u(t))dt+∞∑j=1‖hj(t)+σ(t,u(t))‖2dt+2∞∑j=1u(t)(hj(t)+σj(t,u(t)))dWj(t). |
This implies
ddtE(‖u(t)‖2)+2ν(t)k∑j=1E(‖Bju(t)‖2)+2λ(t)E(‖u(t)‖2)≤2E(f(t,u(t)),u(t))+2E(g(t),u(t))+2∞∑j=1E(‖hj(t)‖2)+2∞∑j=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)|k∑j=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)+2‖ϕ1‖1. | (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
2∞∑j=1E(‖σj(t,u(t))‖2)≤γ12E(‖u(t)‖pp)+p−qp(pγ12q)−qp−q(4cδ)pp−q‖φ1‖2pp−q2pp−q+4cδ‖φ2‖2. | (3.7) |
By (3.3)–(3.7) we get
ddtE(‖u(t)‖2)+α_E(‖u(t)‖2)+32γ1E(‖u(t)‖pp)≤E(∞∑j=12‖hj(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)≤2∞∑j=1‖¯hj‖2+1λ_‖¯g‖2+C1, | (3.9) |
where C1=p−qp(pγ12q)−qp−q(4cδ)pp−q‖φ1‖2pp−q2pp−q+4cδ‖φ2‖2+2‖ϕ1‖1. Multiplying (3.9) by eα_t and integrating over (0,t) to obtain
E(‖u(t,0,u0)‖2)+32γ1∫t0eα_(r−t)E(‖u(r,0,u0)‖pp)dr≤e−α_tE(‖u0‖2)+C2∫t0eα_(r−t)dr, | (3.10) |
where C2=2∑∞j=1‖¯hj‖2+1λ_‖¯g‖2+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 n≥N0 and t≥0,
E(∑‖i‖≥n|ui(t,0,u0)|2)+∫t0eα_(r−t)E(∑‖i‖≥n|ui(r,0,u0)|p)dr≤ε, | (3.11) |
where u0∈K and ‖i‖:=maxi≤j≤k|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(‖i‖n))i∈Zk and ξnu=(ξ(‖i‖n)ui)i∈Zk,∀u=(ui)i∈Zk,n∈N. | (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
d‖ξnu(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=1‖ξnhj(t)+ξnσj(t,u(t))‖2dt+2∞∑j=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))+2∞∑j=1E(‖ξnhj(t)‖2)+2∞∑j=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)k∑j=1E(Bju(t),Bj(ξ2nu(t)))=2ν(t)E(k∑j=1∑i∈Zk(ui1,…,ij+1,…,ik−ui)×(ξ2(‖(i1,…,ij+1,…,ik)‖n)u(i1,…,ij+1,…,ik)−ξ2(‖i‖n)ui))=2ν(t)E(k∑j=1∑i∈Zkξ2(‖i‖n)(ui1,…,ij+1,…,ik−ui)2)+2ν(t)E(k∑j=1∑i∈Zk(ξ2(‖(i1,…,ij+1,…,ik)‖n)−ξ2(‖i‖n))×(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(k∑j=1∑i∈Zkξ2(‖i‖n)(ui1,…,ij+1,…,ik−ui)2)=2|ν(t)|E(k∑j=1∑i∈Zk|ξ(‖i‖n)u(i1,…,ij+1,…,ik)−ξ(‖i‖n)ui|2)≤4|ν(t)|E(k∑j=1∑i∈Zk|(ξ(‖i‖n)−ξ(‖(i1,…,ij+1,…,ik)‖n))u(i1,…,ij+1,…,ik)|2)+4|ν(t)|E(k∑j=1∑i∈Zk|ξ(‖(i1,…,ij+1,…,ik)‖n)u(i1,…,ij+1,…,ik)−ξ(‖i‖n)ui|2). | (3.18) |
By the definition of function ξ, there exists a constant C3>0 such that |ξ′(s)|≤C3 for all s∈R. Then the first term on the right-hand side of (3.18) is bounded by
4|ν(t)|E(k∑j=1∑i∈Zk|(ξ(‖i‖n)−ξ(‖(i1,…,ij+1,…,ik)‖n))u(i1,…,ij+1,…,ik)|2)=4|ν(t)|E(k∑j=1∑i∈Zk|ξ(‖i‖n)−ξ(‖(i1,…,ij+1,…,ik)‖n)|2|u(i1,…,ij+1,…,ik)|2)≤4C23n2|ν(t)|E(k∑j=1∑i∈Zk|u(i1,…,ij+1,…,ik)|2)≤4C23kn2|ν(t)|E(‖u‖2). | (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(k∑j=1∑i∈Zk|ξ(‖(i1,…,ij+1,…,ik)‖n)u(i1,…,ij+1,…,ik)−ξ(‖i‖n)ui|2)≤4|ν(t)|E(k∑j=1‖Bj(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(k∑j=1∑i∈Zkξ2(‖i‖n)(u(i1,…,ij+1,…,ik)−ui)2)≤16k|ν(t)|E(‖ξnu(t)‖2)+4C23kn2|ν(t)|E(‖u‖2). | (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(k∑j=1∑i∈Zk(ξ2(‖(i1,…,ij+1,…,ik)‖n)−ξ2(‖i‖n))×(u(i1,…,ij+1,…,ik)−ui)u(i1,…,ij+1,…,ik))|≤2|ν(t)|E(k∑j=1∑i∈Zk|ξ2(‖(i1,…,ij+1,…,ik)‖n)−ξ2(‖i‖n)|×|u(i1,…,ij+1,…,ik)−ui||u(i1,…,ij+1,…,ik)|)≤4|ν(t)|E(k∑j=1∑i∈Zk|ξ(‖(i1,…,ij+1,…,ik)‖n)−ξ(‖i‖n)|×|u(i1,…,ij+1,…,ik)−ui||u(i1,…,ij+1,…,ik)|)≤4C3n|ν(t)|E(k∑j=1∑i∈Zk|u(i1,…,ij+1,…,ik)−ui||u(i1,…,ij+1,…,ik)|)≤8kC3n|ν(t)|E(‖u‖2). | (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(‖u‖2)+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(∑i∈Zkξ2(‖i‖n)|ui(t)|p)+2E(∑i∈Zkξ2(‖i‖n)|ϕ1,i|)≤−2γ1E(∑i∈Zkξ2(‖i‖n)|ui(t)|p)+2∑‖i‖≥n|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(∑i∈Zkξ2(‖i‖n)|gi|2)≤λ_E(‖ξnu(t)‖2)+1λ_∑‖i‖≥n|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
2∞∑j=1E(‖ξnσj(t,u(t))‖2)=2∞∑j=1E(∑i∈Zk|ξ(‖i‖n)δi,jˆσi,j(t,ui(t))|2)≤4∞∑j=1E(∑i∈Zkξ2(‖i‖n)|δi,j|2(|φ1,i|2|ui(t)|q+|φ2,i|2))≤4cδE(∑i∈Zkξ2(‖i‖n)(|φ1,i|2|ui(t)|q+|φ2,i|2))≤γ1E(∑i∈Zkξ2(‖i‖n)|ui(t)|p)+p−qp(pγ1q)−qp−q(4cδ)pp−q∑i∈Zkξ2(‖i‖n)|φ1,i|2pp−q+4cδ∑i∈Zkξ2(‖i‖n)|φ2,i|2≤γ1E(∑i∈Zkξ2(‖i‖n)|ui(t)|p)+p−qp(pγ1q)−qp−q(4cδ)pp−q∑‖i‖≥n|φ1,i|2pp−q+4cδ∑‖i‖≥n|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(∑i∈Zkξ2(‖i‖n)|ui(t)|p)≤C4|ν|(1n+1n2)E(‖u‖2)+C5(∑‖i‖≥n(¯|gi|2+|φ1,i|2pp−q+|φ2,i|2+|ϕ1,i|)+∑‖i‖≥n∞∑j=1¯|hi,j|2), | (3.27) |
where C5=2+1λ_+p−qp(pγ1q)−qp−q(4cδ)pp−q+4cδ. One can multiply (3.27) by eα_t and integrate over (0,t) in order to obtain
E(‖ξnu(t,0,u0)‖2)+γ1∫t0eα_(r−t)E(∑i∈Zkξ2(‖i‖n)|ui(r,0,u0)|p)dr≤e−α_tE(‖ξnu0‖2)+C4|ν|(1n+1n2)∫t0eα_(r−t)E(‖u(r,0,u0)‖2)dr+C5α_(∑‖i‖≥n(¯|gi|2+|φ1,i|2pp−q+|φ2,i|2+|ϕ1,i|)+∑‖i‖≥n∞∑j=1¯|hi,j|2). | (3.28) |
Since \mathcal K is a compact subset of l^2 we infer from (3.1) that
\begin{equation} \lim\limits_{n\to\infty}\sup\limits_{u_0\in\mathcal K}\sup\limits_{t\ge 0}e^{-\underline{\alpha}t}E(\|\xi_nu_0\|^2)\le \lim\limits_{n\to\infty}\sup\limits_{u_0\in\mathcal K}E(\sum\limits_{\|i\|\ge n}|u_{0,i}|^2) = 0. \end{equation} | (3.29) |
By Lemma 3.1, we find that for all u_0\in\mathcal K and t\ge 0 , as n\to\infty ,
\begin{align} \begin{split} &\Big(\frac{1}{n}+\frac{1}{n^2}\Big)\int_{0}^{t}e^{\underline{\alpha}(r-t)}E(\|u(r,0,u_0)\|^2)dr\\ &\le \frac{L_1}{\underline{\alpha}}\Big(\frac{1}{n}+\frac{1}{n^2}\Big)\Big(E(\|u_0\|^2)+\sum\limits_{j = 1}^{\infty}\overline{\|h_j\|}^2+\overline{\|g\|}^2+\|\varphi_1\|^{\frac{2p}{p-q}}_{\frac{2p}{p-q}}+\|\varphi_2\|^2+\|\phi_1\|_1\Big)\\ &\le \frac{L_1}{\underline{\alpha}}\Big(\frac{1}{n}+\frac{1}{n^2}\Big)\Big(C_{6}+\sum\limits_{j = 1}^{\infty}\overline{\|h_j\|}^2+\overline{\|g\|}^2+\|\varphi_1\|^{\frac{2p}{p-q}}_{\frac{2p}{p-q}}+\|\varphi_2\|^2+\|\phi_1\|_1\Big)\to0, \end{split} \end{align} | (3.30) |
where L_1 is the same number of (3.1) and C_{6} > 0 is a constant depending only on u_0 .By \varphi_1\in l^{\frac{2p}{p-q}}, \varphi_2\in l^2, \phi_1\in l^1 , (2.6) and (3.1), we infer that
\begin{equation} \sum\limits_{\|i\|\ge n}\Big(\overline{|g_i|}^2+|\varphi_{1,i}|^{\frac{2p}{p-q}}+|\varphi_{2,i}|^2+|\phi_{1,i}|\Big)+\sum\limits_{\|i\|\ge n}\sum\limits_{j = 1}^{\infty}\overline{|h_{i,j}|}^2\to 0\ \mathit{\text{as}}\ n\to\infty. \end{equation} | (3.31) |
It follows from (3.28) to (3.31) that as n\to\infty ,
\begin{equation} \sup\limits_{u_0\in\mathcal K}\sup\limits_{t\ge 0}\bigg(E(\|\xi_nu(t,0,u_0)\|^2)+\int_{0}^{t}e^{\underline{\alpha}(r-t)}E\Big(\sum\limits_{i\in\mathbb Z^k}\xi^2\Big(\frac{\|i\|}{n}\Big)|u_i(r,0,u_0)|^p\Big)dr\bigg)\to0. \end{equation} | (3.32) |
Then for every \varepsilon > 0 we can find a number N_0 = N_0(\mathcal K)\in\mathbb N satisfying, for all n\ge N_0 and t\ge 0 ,
\begin{align} \begin{split} &\bigg(E\Big(\sum\limits_{\|i\|\ge 2n}|u_i(t,0,u_0)|^2\Big)+\int_{0}^{t}e^{\underline{\alpha}(r-t)}E\Big(\sum\limits_{\|i\|\ge 2n}|u_i(t,0,u_0)|^p\Big)dr\bigg)\\ &\le \bigg(E\Big(\|\xi_nu(t,0,u_0)\|^2\Big)+\int_{0}^{t}e^{\underline{\alpha}(r-t)}E\Big(\sum\limits_{i\in\mathbb Z^k}\xi^2\Big(\frac{\|i\|}{n}\Big)|u_i(t,0,u_0)|^p\Big)dr\bigg)\le\varepsilon, \end{split} \end{align} | (3.33) |
uniformly for u_0\in\mathcal K and t\ge 0 . This concludes the proof.
In the sequel, we use \mathcal L(u(t, 0, u_0)) to denote the probability distribution of the solution u(t, 0, u_0) of (2.15)–(2.16) which has initial condition u_0 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 \{\mathcal L(u(t, 0, u_0)):t\ge 0\} of the distributions ofthe solutions of (2.15)–(2.16) is tight on l^2 .
Proof. For simplicity, we will write the solution u (t, 0, u_0) as u(t) from now on. It follows from Lemma 3.1 that there exists a constant c_1 > 0 such that
\begin{equation} E\left( {\left\| {u(t) } \right\|^2 } \right) \le c_1 ,\quad\text{for all} \quad t\geq0. \end{equation} | (4.1) |
By Chebyshev's inequality, we get from (4.1) that for all t\geq0 ,
P\left( {\left\| {u(t) } \right\|^2\ge R } \right) \le \frac{{c_1 }}{{R^2 }}\rightarrow 0\quad\text{as}\quad R\rightarrow \infty. |
Hence for every \epsilon > 0 , there exists R_1 = R_1(\epsilon) > 0 such that for all t\geq0 ,
\begin{equation} P\left\{ {\left\| {u(t) } \right\|^2 \ge R_1 } \right\} \le \frac{1}{2}\epsilon. \end{equation} | (4.2) |
By Lemma 3.2, we infer that for each \epsilon > 0 and m\in \mathbb{N} , there exists an integer n_m = n_m(\epsilon, m) such that for all t\geq 0 ,
E\left( { \sum\limits_{\left| i \right| > n_m } {\left| {u_i \left( t \right)} \right|^2 } } \right) < \frac{\epsilon }{{2^{2m + 2} }}, |
and hence for all t\geq 0 and m\in \mathbb{N} ,
\begin{equation} P\left( {\left\{ { \sum\limits_{\left| i \right| > n_m } {\left| {u_i \left( r \right)} \right|^2 } \ge \frac{1}{{2^m }}} \right\}} \right) \le 2^m E\left( { \sum\limits_{\left| i \right| > n_m } {\left| {u_i \left( r \right)} \right|^2 } } \right) < \frac{\epsilon }{{2^{m + 2} }}. \end{equation} | (4.3) |
It follows from (4.3) for all t\geq 0 ,
P\left( {\mathop \cup \limits_{m = 1}^\infty \left\{ { \sum\limits_{\left| i \right| > n_m } {\left| {u_i \left( t \right)} \right|^2 } \ge \frac{1}{{2^m }}} \right\}} \right) \le \sum\limits_{m = 1}^\infty {\frac{\epsilon }{{2^{m + 2} }} \le \frac{1}{4}\epsilon ,} |
which shows that for all t\geq 0 ,
\begin{equation} P\left( {\left\{ { \sum\limits_{\left| i \right| > n_m } {\left| {u_i \left( t \right)} \right|^2 } \le \frac{1}{{2^m }}\,\,\text{for all}\,\, m\in \mathbb{N}} \right\}} \right) > 1 -\frac{\epsilon}{2}. \end{equation} | (4.4) |
Given \epsilon > 0 , set
\begin{align} Y_{1,\epsilon } & = \left\{ {v \in l^2:\left\| {v } \right\| \le R_1 \left(\epsilon \right)} \right\}, \end{align} | (4.5) |
\begin{align} Y_{2,\epsilon } & = \left\{ {v \in l^2: \sum\limits_{\left| i \right| > n_m } {\left| {v_i \left( r \right)} \right|^2 } \le \frac{1}{{2^m }}\,\,\text{for all}\,\, m\in \mathbb{N}} \right\}, \end{align} | (4.6) |
and
\begin{equation} Y_\epsilon = Y_{1,\epsilon} \cap Y_{2,\epsilon }. \end{equation} | (4.7) |
By (4.2) and (4.4) we get, for all t\geq0 ,
\begin{equation} P\left( {\left\{ {u(t) \in Y_\epsilon } \right\}} \right) > 1 - \epsilon . \end{equation} | (4.8) |
Now, we show the precompactness of \left\{ {v:v \in Y_\epsilon } \right\} in l^2 . Given \kappa > 0 , choose an integer m_0 = m_0 \left(\kappa \right) \in \mathbb{N} such that 2^{m_0 } > \frac{8}{{\kappa ^2 }} . Then by (4.6) we obtain
\begin{equation} \sum\limits_{\left| i \right| > n_{m_0 } } {\left| {v_i } \right|^2 } \le \frac{1}{{2^{m_0 } }} < \frac{{\kappa ^2 }}{8},\quad \forall v \in Y_\epsilon. \end{equation} | (4.9) |
On the other hand, by (4.5) we see that the set \left\{ {(v_i)_{|i|\leq m_0}:v \in Y_\epsilon } \right\} is bounded in the finite-dimensional space R^{2m_0+1} and hence precompact. Consequently, \left\{ {v:v \in Y_\epsilon } \right\} has a finite open cover of balls with radius \frac{\kappa }{2} , which along with (4.9) implies that the set \left\{ {v:v \in Y_\epsilon } \right\} has a finite open cover of balls with radius \kappa in l^2 . Since \kappa > 0 is arbitrary, we find that the set \left\{ {v: v\in Y_\epsilon } \right\} is precompact in l^2 . This completes the proof.
If \phi:l^2\to\mathbb R is a bounded Borel function, then for 0\le r\le t and u_0\in l^2 , we set
(p_{r,t}\phi)(u_0) = E(\phi(u(t,r,u_0))) |
and
p(r,u_0;t,\Gamma) = (p_{r,t}1_\Gamma)(u_0), |
where \Gamma\in\mathcal B(l^2) and 1_\Gamma is the characteristic function of \Gamma . The operators p_{s, t} with 0\le s\le t are called the transition operators for the solutions of (2.15)–(2.16). Recall that a probability measure \nu on l^2 is periodic for (2.15)–(2.16) if
\int_{l^2}(p_{0,t+T}\phi)(u_0)d\nu(u_0) = \int_{l^2} (p_{0,t}\phi)(u_0)d\nu(u_0),\qquad\forall t\ge0. |
Lemma 4.2. [21]Let \varrho(\psi, \omega) be a scalar bounded measurable randomfunction of \psi , independent of \mathcal F_s . Let \varsigma be an \mathcal F_s -measurable random variable. Then
E\left( {\varrho \left( {\varsigma ,\omega } \right)|\mathcal F_s } \right) = E\left( {\varrho \left( {\varsigma ,\omega } \right)} \right). |
The transition operators \{p_{r, t}\}_{0\le r\le t} have the following properties.
Lemma 4.3. Assume that (2.1)–(2.7) and (3.1) hold. Then:
(i) \{p_{r, t}\}_{0\le r\le t} is Feller; that is, for every bounded andcontinuous \phi: l^2\to\mathbb R , the function p_{r, t}\phi: l^2\to\mathbb R is also bounded and continuous for all 0\le r\le t.
(ii) The family \{p_{r, t}\}_{0\le r\le t} is T-periodic; that is, for all 0\le r\le t ,
p(r, u_0;t,\cdot) = p(r+T,u_0;t+T,\cdot),\qquad\forall u_0\in l^2. |
(iii) \{u(t, 0, u_0)\}_{t\ge 0} is a l^2 -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 l^2 .
Proof. We apply Krylov-Bogolyubov's method to prove the existence of periodic measures of (2.15)–(2.16), define a probability measure \mu_n by
\begin{equation} \mu_n = \frac{1}{n}\sum\limits_{l = 1}^{n}p(0,0;lT,\cdot). \end{equation} | (4.10) |
By Lemma 4.1 we see the sequence \{\mu_n\}^\infty_{n = 1} is tight on l^2 , and hence there exists a probability measure \mu on l^2 such that, up to a subsequence,
\begin{equation} \mu_n\to\mu,\qquad\text{as}\ n\to\infty. \end{equation} | (4.11) |
By (4.10)–(4.11) and Lemma 4.3, we infer that for every t\ge0 and every bounded and continuous function \phi:l^2\to\mathbb R,
\begin{align} \begin{split} &{\int_{l^2} {\left( {p_{0,t} \phi } \right)\left( u_0 \right)d\mu \left( u_0 \right)} } = \int_{l^2} {\left( {\int_{l^2} {\phi \left( y \right) p\left( {0, u_0 ;t,dy} \right)} } \right)} d\mu \left( u_0 \right) \\ & = \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{l = 1}^n {\int_{l^2} {\left( {\int_{l^2} {\phi \left( y \right)p\left( {0,u_0 ;t,dy} \right)} } \right)} p\left( {0,0;lT,du_0 } \right)} \\ & = \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{l = 1}^n {\int_{l^2} {\left( {\int_{l^2} {\phi \left( y \right)p\left( {kT,u_0 ;t + lT,dy} \right)} } \right)} p\left( {0,0;kT,du_0 } \right)} \\ & = \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{l = 1}^n {\int_{l^2} {\phi \left( y \right)p\left( {0,0;t + lT,dy} \right)} } \\ & = \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{l = 1}^n {\int_{l^2} {\phi \left( y \right)p\left( {0,0;t + lT + T,dy} \right)} } \\ & = \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{k = 1}^n {\int_{l^2} {\left( {\int_{l^2} {\phi \left( y \right)p\left( {0,u_0 ;t + T,dy} \right)} } \right)} p\left( {0,0;lT,du_0 } \right)} \\ & = \int_{l^2} {\left( {\int_{l^2} {\phi \left( y \right)p\left( {0,u_0;t + T,dy} \right)} } \right)} d\mu \left( u_0\right)\\ & = {\int_{l^2} {\left( {p_{0,t + T} \phi } \right)\left( u_0 \right)d\mu \left( u_0 \right)} } , \end{split} \end{align} | (4.12) |
which shows that \mu is a periodic measure of (2.15)–(2.16), as desired.
The author declares there is no conflict of interest.
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