(1.Department of Basic Curriculums,North China Institute of Science and Technology, Beijing,101601)

(2.Collage of Science,China University of Mining and Technology,Xuzhou, Jiangsu,221008)

Approximation Theoretic Aspects of Probabilistic Representations for Bi-continuous C-semigroups

CANG DING-BANG1,CHEN CANG1AND SONG XIAO-QIU2

(1.Department of Basic Curriculums,North China Institute of Science and Technology, Beijing,101601)

(2.Collage of Science,China University of Mining and Technology,Xuzhou, Jiangsu,221008)

Communicated by Ji You-qing

By means of Riemann-Stieltjes stochastic process,moment-generating functions and operator-valued mathematical expectation,the problem of probabilistic approximation for bi-continuousC-semigroups is studied and the general probabilistic approximation of exponential formulas and the generation theorems are given.

bi-continuousC-semigroup,probabilistic approximation,generation theorem

1 Introduction

In 1940s,Hille[1]and Yosida[2]studied the theory of strongly continuous semigroups on Banach spaces.They were devoted to a qualitative and quantitative analysis for the solutionsu(t)=T(t)xof the initial value problemu′(t)=Au(t)withu(0)=x∈D(A),whereAis the generator of the strong continuous semigroup andD(A)is the domain ofA.Since then on,the theory of strongly continuous semigroup has been matured and applied to di ff erent areas of science.However,it was clear that not every semigroup is strongly continuous and that a comprehensive theory requires a more general set-up.For this reason,many other classes of semigroups were studied,such as distribution semigroups,semigroups of growthof orderα,integrated semigroups,convolution semigroups,and so on.It is then natural to look for suitable locally convex topologies weaker than the norm topology to treat the lack of strong continuity.

In order to treat one-parameter semigroups of linear operators on Banach spaces which are not strongly continuous,Kuhnemund[3]introduced a new class of semigroups called bicontinuous semigroups which are de fi ned on a Banach space with an additional locally convex topologyτ.This class of semigroups has two remarkable properties.There is variety of interesting semigroups belonging this class,such as Feller semigroups,Ornstein-Uhlenbeck semigroups,as well as certain evolutions semigroups.Albanese and Mangino[4]got some results on the convergence of bi-continuous semigroups,obtained a Lie-Trotter product formula,and applied it to Feller semigroups generated by second order elliptic di ff erential operators with unbounded coefficients.Patricio[5]gave some introduction to bi-continuous semigroups and some remark on the Riemann-Stieltjes integral and discussed the Hille-Phillips functional calculus for generators of bi-continuous semigroups.

In the last decades,mathematicians have begun to use the powerful tool of probability theory to solve the semigroup approximation problem and achieved fruitful results,and related works can be found in[6–12].The plan of this paper is as follows.In Sections 2 and 3,by means of Riemann-Stieltjes stochastic process,moment-generating functions, operator-valued mathematical expect andCbi-continuous modi fi ed modulus,the problem of probabilistic approximation for bi-continuousC-semigroups is studied,and the general probabilistic approximation of exponential formulas and the generating theorems are given.

2 Probabilistic Representations

We assume that the spaceXsatis fi es the following conditions.Let(X,τ)′be a Banach space with topological dualX′,andτa locally convex topology onXwith the following properties:

(1)The space(X,τ)is sequentially complete on‖·‖-bounded sets,i.e.,every‖·‖-boundedτ-Cauchy sequence converges in(X,τ);

(2)If the topologyτis coarser,then the‖·‖-topology is a Hausdor fftopology;

(3)The space(X,τ)′has a norm(X,‖·‖),i.e.,

LetPτdenote a family of seminorms inducing from the locally convex topologyτonXand without loss of generality one can assume thatp(x)≤‖x‖for allx∈Xandp∈Pτ.In this paper,all operators are linear andD(A)denotes the domain ofA.NBV[0,R]denotes the normalized functions of bounded variation in[0,R].

De fi nition 2.1[5]Let X be a Banach space with a topology τ and α∈NBV[0,R].A function f:[0,R]→X is τ Riemann-Stieltjes integrable with respect to α if

Moreover,if ?∈(X,τ)′then the map t→〈f(t),?〉is continuous and

De fi nition 2.2A semigroup(T(t))t≥0of linear‖·‖-continuous operators on X is said to be a bi-continuous C-semigroups(with respect to τ)if the following conditions hold:

(i)T(0)=C,T(t+s)C=T(t)T(s)for all t,s＞0;

(ii)The operator T(t)is exponentially bounded,i.e.,T(t)≤Meωtfor all t＞0and some constants M and ω∈R;

(iii) (T(t))t≥0is strongly τ-continuous,i.e.,the mapR+?t→T(t)x∈X is τcontinuous for each x∈X.

Theτ-generator of a bi-continuousC-semigroup(T(t))t≥0is de fi ned as

Clearly,every strongly continuous semigroup on a Banach space is a bi-continuousC-semigroup with respect toτ=‖·‖.

and the following conclusions hold:

(i)If B:(X,τ)→(X,τ)is a continuous linear operator from(X,τ)into(X,τ),then E(BT(t))=BE(T(t));

(ii)〈E(T(t),?)〉=E[〈T(t),?〉].

Lemma 2.2[7]Let X be a real-valued random variable.

Lemma 2.3Let(T(t))t≥0be an exponentially bounded bi-continuous C-semigroup,U and V be independent variables.E(etU)and E(etV)are limited at t=η.Then E[T(U)], E[T(U)]and E[T(U+V)]exist in B(X),and

Lemma 2.4Let(T(t))t≥0be an exponentially bounded bi-continuous C-semigroup,x∈D(Ar),and r≥1.Then for all s,t≥0,one has

Lemma 2.5[7]Let X be a non-negative real random variable with expectation E(X)=ξ and ΨX(δ)＜∞for some δ＞0.Let further{Xn|n∈N}be an i.i.d.sequence of random variables,identically distributed with X.Then we have

Proof.By Lemma 2.4 and the Hlder’s inequality,we have

Using Theorem 2.1 withr=1 orr=2 andr=3,we can get

Corollary 2.1Let(T(t))t≥0be an exponentially bounded bi-continuous C-semigroup,and

The following theorem gives the probabilistic approximation for a bi-continuousC-semigroup in terms of its generator.

Using the inequality:

Using Theorem 2.1 withr=1 andr=2,3,we can get

Theorem 2.3Let(T(t))t≥0be an exponentially bounded bi-continuous C-semigroup,N be a non-negative integer-valued random variable with E(N)=ζ,and Y be a real-valued random variable with E(Y)=0.Denote ξ=ζγ.Then

By Theorem 2.1 and Lemma 2.5,one has

Using Theorem 2.3 withr=1,we can get

Corollary 2.3Let(T(t))t≥0be an exponentially bounded bi-continuous C-semigroup,N be a non-negative integer-valued random variable with E(N)=ζ,Y be a real-valued random variable with E(Y)=(η)＜∞,and η＞0.Denote ξ=ζγ.Then

3 Further Discussion

De fi nition 3.1Let(T(t))t≥0be an exponentially bounded bi-continuous C-semigroup. For h＞0,we de fi ne

where ω2(x,h)is called the second C bi-continuous modi fi ed modulus of the bi-continuous C-semigroup(T(t))t≥0.

After some calculations,we can get the lemma as follows.

Lemma 3.1If ω2(x,h)is the second C-continuous modi fi ed modulus of the bi-continuous C-semigroup(T(t))t≥0,h＞0,then we can get

Lemma 3.2Let(T(t))t≥0be an exponentially bounded bi-continuous C-semigroup,N be a non-negative integer-valued random variable with E(N)=ζ,σ2(N)≥0,and Y be a realvalued random variable with E(Y)=γ and σ2(Y)＞00. Denote ξ=ζγ.Then

Using Lemma 2.6 and Corollary 2.1(2),fornsufficiently large,we can get

Theorem 3.1Under the condition of Lemma3.2,for λ＞0,we have

Proof.By Lemma 3.1,we have

Then,using Lemma 3.2,we get

Set

Then

Corollary 3.1Let Y=γ be a constant,N be a two point distribution with E(N)=ζ and σ2(N)=ζ(1?ζ).Then

Furthermore,Corollary 3.2Let Y=γ be a constant,N be a Polya distribution with E(N)=ζ and σ2(N)=ζ(1+αζ).Then

Furthermore,

Corollary 3.3Let Y=γ be a constant,N be a Possion distribution with E(N)=ζ and σ2(N)=ζ.Then

Furthermore,

Corollary 3.4Let Y=γ be a constant,N be an exponential distribution with E(N)=ζ and σ2(N)=ζ2.Then

Furthermore,

Corollary 3.5Let Y be an exponential distribution with E(Y)=γ,and N be a two point distribution with E(N)=ζ.Then

Furthermore,

Next,we discuss the case of continuous type.For eachε＞0,letN(ε)be a non-negative, integer-valued random variance withE(N(ε))=εζandσ2(N(ε))=εσ2(N(ε)).Assume that

Similar to Theorem 3.1,we can give the Theorem 3.2 as follows.

Corollary 3.6Let Y=γ be a constant and N be a Polya distribution with E(N)=ζ. Then

Furthermore,

Corollary 3.7Let Y be an exponential distribution with E(Y)=γ,N be a Polya distribution with E(N(ε))=ζε and σ2(N(ε))=ζ(1+αζ).Then

Furthermore,

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tion:47D05,47D09

A

1674-5647(2014)02-0106-11

10.13447/j.1674-5647.2014.02.02

Received date:Feb.6,2012.

Foundation item:The NSF(10671205)of China,Fundamental Research Funds(3142012022,3142013039 and 3142014039)for the Central Universities and the Key Discipline Construction Project(HKXJZD201402)of NCIST.

E-mail address:cdbjd@163.com(Cang D B).