RANDOM SINGULAR INTEGRAL OF RANDOM PROCESS WITH SECOND ORDER MOMENT

2005,25B(2):376-384.MalhemslelaHtientia数学物理学报RANDOM SINGULAR INTEGRAL OF RA NDOMPROCESS WITH SECOND ORDER MOMENT 1Wang Chruanrong (王传荣)Department of Mathematics, Fuzhou University, Fuzhou 350002, ChinaAbstract This paper discussses the random singular integral of random process withsecond order moment, establishes the concepts of the random singular integral and provesthat it's a linear bounded operator of space H^ (L)(m, s). Then Plemnelj formula and someother properties for random singular integral are proved.Key words Random singular intergral, second order moment, Plemelj formula2000 MR Subject Classification 30E25, 47G101 IntroductionThe purpose of the present paper is to discuss the random singular integral of randomprocess with second order moment, we establish the concepts and some statistic propertiesof random singular integral, and we prove that random singular integral operator is a linearbounded operator in space Ha(L)(m, s) and Plemelj formula hold. It is well known that singularintegral equations and boundary value problems of analytic function and random process areclosely connected with many physical and engineering problems such as elastic mechanics, crackmechanics and aero-dynamics, etc. Therefore it is expected that the results of present paperwill be applied.2 Random Singular IntegralLet L be a simple, smooth and closed curve. It devides the complex plane into inter domainD+ and outer domain D-. (S,F,P) is a probability space, g(w,S) is a random process withcomplex value on (M, F, P), which depends on parametersξ on L. From now on, we will assumethat it is a second-order moment process. Namely, for any ξ on L, the means and variance ofg(w,S) always exist.Definition 1 Let g(w,S) and 9n(w,S)(n= 1,2,中国煤化Ilent process. Iflim E{|9n(w,S) - g(w,S)MHCNMHG1Received May 22, 2003. The project is supported by the NSF of China (10271098) and the EducationFoundation of Fujian (JB02083) and Science & Technical Development Foundation of Fuzhou University (2003xy-11).No.2Wang: RANDOM SINGULAR INTEGRAL OF RANDOM PROCESS377we can say that gn(w, 5) coriverges to g(w, 5) on the point 6 in the sense of mean square matricand denotelim 9n(w,C)=g(w,5) (m.s)orgn(w,5) : g(w,5)If we replace n by a continous variables considering 9x(w,S), then lim gx(w,5) can bedefined similarly. Furthermore, the meaning of the derivative and integral of random processesin the sense of mean square metric may refer to Ch4 of [1].Definition2 Letζo∈ L,l8= {ζ∈L:|5-6o|<δ},Lε= L-ls. g(w, S) is a second-ordermoment process. If there is a random process f(w, 5), such thatg(w,S)limE6- Sodζ - f(w,5o)0，(2.1)then we say that there exists random singular integral with Cauchy kernelg(w,S),-ζdζ= f(w,So).(2.1)'In order to find the conditions, such that there exists random singular integral, first, we recallspace Ha(L) and H(ar,a2)(L1 x L2)(a,Q1,Q2∈(0,1)H°(L)=f∈C():HfH8(L)=. sup[f(1)- f(62)|<∞ζ1,62∈L|S1- 62|aIfHa(L) = ll|l() + Ifl|Hg() = sup|f(S)| +supf(1)- f(62)I.ζ∈L61 ,62∈L1ζ1-62|0andH(a1,2)(L1x L2)= { f(1,62)∈C(LIx L2): sup|f(61,52)- f(6,62)L51-S1EL11ζ1-ζ1a1+|ζ2- S2|(a252,2∈L2|f(51,52) - f(51,52)|lIfg(gr.2)(L,x2)= sups.SfeL Gr-(1a1 +1S2一2Is°ζ2,S2∈L2Ilfll(L1xLx)= . max ,。|f(1,652),S1,62∈L1XL2IlH{aa2(LuxL2) = Ifl(Lx[)中国煤化工We need the following lemma which is from Ch 1.TYHCNMHGLemma 12] Let I be a piceswise smooth curve, f(T,S) ∈H(a1,a2)(L1 x L2), namelythere exist constants A and B, such that Vrj∈L,tj∈L,(j= 1,2)|f(x,t1)- f(r2,t2)l≤A|πn-r2|°1 + B|t1 - t2|2.378ACTA MATHEMATICA SCIENTIAVol.25 Ser.BAnd let l be a subarc of L, then Vto∈l, there eixsts constant M independent of 6, such that[f(t,S)- f(o,S)dr|≤M|l|°',T- towhere |lI| is the length of arc l.Lemma 2[2 Let L be a smooth closed curve, T is a bounded closed set on the complexplane, and f(t,T) ∈H(a1,a2)(Lx T), thenF(to,r)=1|' f(t,T)dt∈Ha,a2-e)(LxT) (to∈L, τ∈T),2π Jt-to.in the meanwhile,|\F(to, r)H(a1.c2-e)([xT)≤K\f(t, r)lHa1.az-<)(LxT),where K only depends on curve L, but independent of function f(t,r).0<ε < a2.Let Rg(S, 5") be the self-correlation function of 9(w, (), namelywe haveTheorem 1 If Rg(6,5')∈H(L x L), then there exists the random singular integralSrg(w,Co)= -1 { g(w,S),dζ,πiJx ζ- Soin the sense of mean square metric.Proof It is enough to prove that there existsδ > 0 for any E > 0 such thatg(w,S)g(w,C). I=E{么ζ- Sodζ -JL82 ζ- So<ε(2.2)with conditions of δ1,δ2 < δ. Worthy of note is thatEg(w,2)aζ _g(w,S)aclζ-ζog(w,()g(w,S),g(w,5')=E1ζ'-Ea5 |-dζ'ζ-ζ°J ζ-ζo'ζ°-ζo(2.3)g(w,5)。g(w,S')-Eaζ |a5'| + EdζJra, ζ-6o .JLs。ζ-ζo“Js ζ-6o≌η-I-13+ I4.According to the properties of mean square integral (see [1]), we haveg(w, 0rmI1=EId61(5-中国煤化工.CHCNMHGd1ζJzs (ζ-5o)(ζ°-ζ)ds_ . 厂Rg(G.Idg.ζ-ζoJL81ζ'一6oNo.2Wang: RANDOM SINGULAR INTEGRAL OF RANDOM PROCESS379Similarly,I2= Eg(w,S)g(w,(),{L.ζ-5o°ζ-ζaac$dζRg(S',S)=ζ-5oJLsn ζ'-SodS',g(w,5g(w,5')I3= E{ζ-ζo2d =0.Namely有好数据olds.384ACTA MATHEMATICA SCIENTIAVol.25 Ser.BTheorem 4 If g(w,5) is a normal process about 5 then the random singular integralf(w,()=πJ ζ°-ζ1” g(w,S)ds'is also a random normal process about 5.Proof From [1],g(w,S'),lζ'JL。ζ-ζis a normal process, so( g(wv,S)dζ= limg(w,$'))a<'J ζ°-ζas= -。 ζ°-ζis also a normal random process.References1 Wu Baotin, Li Qinsheng, Yang Yuewu. Random Processes and Random Differential Equations. Chengdu:Electronic Science and Technology Publication, 1994 (Chinese)2 Lu Jianke. Boundary Value Problems for Analytic Functions. World Scientifc, 19933 Gakhov F D. Boundary value problems (Russian). Moscow, 19774 Wang Chuanrong. Random Riemann boundary value problems. Journal of Sichuan Normal University,1994, 17(2): 81-82 (Chinese)5 Wang Chuanrong. Random Singular Integral and its Application. In: Lu Jianke, Wen Guochun eds.Boundary Value Problems Integral Equations and Related Problems. World Scientific, 2000. 191-197中国煤化工MYHCNMHG