LIMIT THEOREMS FOR SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATION PROBLEMS

In this paper linear differential equations with random processes as coefficients and as inhomogeneous term are regarded. Limit theorems are proved for the solutions of these equations if the random processes are weakly correlated processes.


I. IRTROIEUTION.
At the research of physical and engineering problems it is of great ortance the approach to the dierentia1 equations th ochaic process as eiciemts reectively th ochaic boda or Int cdiions.ere s a sees of papas wch deal th ch a probl.e fir ments of the slution are oen calcated fr the fir moments of the occ pcess ovi the pb- I.For the applications te is inteest an Implant problem (see 4]). e calcation of the die.buttons of the sulutlon pro- eess fr the dibutions of the involvi pcess is o more dft.s pblem contains already diiculties for ve le pbs (e.g. for the tial value probl of a linear d- na derental equation of he flr order th ochaIc coei- ents).If one ecializ the ocic pcess olvi that probl then one succeeds In a f cas n obtai atents on he dibion of the luon.A ele of ch a re,It we rer t the paper of G.E. beck L.S. Ornen 8].ey d such ochastlc inputs wch do not poss a "dsant eect", .e. the valuea of the pcess d not sss a lation if the ce between the obatlon points is .As a result they c sh tt the solution of a special nitial value problem wth ch a process without "distant effect" as the gh side is approxtvly a -called Oein-lenbeck-pcess, .e. a process for wch the fir diibutlon ction s a Gaussian dstrlbuton ction.
e pcess without "distt effect" were defined exactly by the uthors in the paper 6 toh the pcess class of the "weakly correlated processes" and they were applied in this paper at the con- sideraton of ocstic eigenvalue problems d bodary value prob- lems.A limit theorem is obtained for the eigenvalues, eigenfunctions of stochastic eigenvalue problems respectively for the solutions of stochastic boundary problems, with weakly correlated coefficients.Ths 1mlt theorem shos the appro:mate Gaussan dstributon of the first distribution function of the solutions of eigenvalue problems and boundary value problems.
In the present paper the conception of the weakly correlated pro- ces s defined more gvnerally than in the paper K6S (.e. the sta- tonarty of the process falls out the suppositionS.The correlation Yenggh enotes the ,tin,mum distance between bservaton points of a wly CO:Telate process that the walues of the process do not af- fect in observation poimts which possess a dstance larger than la section 2 a few theorems wil be proved about functonals of weevily correlated rocesse which re mportant for the apllcaon a eigenalue problems, boundary value poblem$ and nita value problems in the following sections: Section , deals wit sochasic elgenvalue problems for odinary dilerentisl equations wth deterministic boundary conditions where the eoefficient of the difTerential opeato are independent, weaEly eo--elated processes of he eorrelstlon length .We prove hat the egenvalues and the eigenfunctions, as 550 possess a Gausslan distri- bution.For instance the eigenfunctions of the tochasic elgenvalue problem converge in the distribution as 0 to Gausslan processes.
Methods of the perturbation theory are essentially used.In a general example it is eferred to a few remarkable appearances.
In section 4 we deal with stochastic boundary problems and we tan similar results as fo stochastic eigenvalue problems.The case of a Sturm-Liouville-operator with a stochastic inhomogeneous tem i16 J.V. -SCHEIDT-a'qI W. PURKERT (y corelated which was dealt with by .E. Boyce n [ ] is ncluded n the result of thie section.Soae limltatons relative to the Iness f the toehastic coefficlents of the operator but not of the' nhoageneoue term ae assumed a at the eigenvalue probleae, too.
e ehow a calculation f the correlation fYmctlon by the Ritz-aethod to eiminate the Green function f the average problea from the cor- relation fYhnction of the limi pocesa of the sluton of the bouary value problem.
At last, sectoa 5 deals" with stochastic nitial value problems of ordinary differential equations.The inhomogeneous temas are weakly correlated processes.The result n this theory of the weakly cor- related procesees as 0 esemble the results of the It-theory if the nhomogeneeus temms are replaced by Gaussan white noae accord- m to the I-theor and the formed ItS-equatlon s solved according to the It-theory.The practice by the help of the weakly correlate procese iffers princpally from the It-theory.One obtains the lmit theorems by use of the weakly correlated processes f at first a formula for the a.s.continuous differentiable sample functions of the solutions is derived (%0) and then we go to the limit (0).
ith it we get an approximation of the sIutfon of the initial value problem with weakly correlated processes (<< ,0).In the It--theory one goes, to the limit in the differential equation and this equation is solved by a well worked out mathematical theory.We gt different results at this different practice in problems of differ- emtial equations in which the coefficients are weakly correlated procesees and not the inhomogeneous term.We do not deal with such prob- lems in this: paper.
I% is principally no distinction in the proof of limit theorems for the nitial value problems and boundary value respectively eigen- value problems with weakly correlated processes.A widening of the l-theory on boundary value respectively eigenvalue problems with white noises for the coefficients seems to contain a few fundamental difficulties.
Every finite set (x ,...,x n) split unique in disjoint maximum -adjoining subset s.DEFINITION  x ,...,x n) lits n th mam -adjoini subsets (pi=n).= If the pces f(x,) is wey correlated with the corlation leth , then we get for its corlaton ction f(x}ry)p = (x,y) for y Kt(x) 0 for y K(x) where K Cx)-yg R |x-y|a.
The existence of especially aatonary weakly correlated processes has been proved in the paper 6.In this paper it is also proved that weakly correlated processes with smooth sample functions exist.
In the following a few theorems will be proved about weaEly cot- related processes.These theorems will be used essentially in the applications at equations of the mathematical physics.
Before we denote the more important theorem 3, we prove a simple theorem.
PROOF.By n(bl,b2,...,bn) it is g(1 '" ,Xn)1 Cnl I dx2"'" and wth it the tatement of the theorem2 THEOEEM 3. Le% f (x,) be a sequence of weakly correlated pro- e O.The absolute moments <If(x,)l J>=cj(x) are to exist (:esees as and cj.(x) Cj.Let gi(x,y), i=1,2,...,n, be in [a-,bi+ ] differenti- able functions relativ to x (q > O,b i> a) and i,x,y Then we have for ri (x,)= if (Y')gi (y,x) dy The eends for all splittis of (1,...,n) in pairs (I'i2)' "'',(n-l,in th ir 4 1 fer every par (ir,il).Splttis re equal wch differ toh the sequence of the pairs.e pof of this theorem subts silar to the proof of theorem 7 in 6.S by the help of the above given theorem and it also bmi%s the proof of the followi theorem 4 from the proof of the theorem s n [].
THEOREM 4. Let flz (x,),"',fn(X,9) be sequences of independent weakly correlated processes as aO.Let gij(x,y), =1,...,k,j=1,...,n, be in [a-,bi+],i=1 ,...,k, differentiable fUnctions relatv to x and f r; p) r( ) lira<tOP ) rCP \ p I... c-n 2 p e in the te for A p eends for all splitters of i p p CI,...,i, ) in pairs like in teorem P As an application of the theorems to 4 we will prove the theorem 5. THEORE 5. Let fi (x,w),...,fn(X,W) be sequences of independent, weakly correlated processes with eontlnuous eample functions as the correlation length 0. Let gj(x,y) be differeniable functions in [a-,b+]x[a,b relative to x with the condition8 of he limited like in theorem 4. Then the stochastic vector process r(x,) = (ri (x,))T ) r(x,) = j=1 a gij(y'x)fj ,(y )dy (I)   with converges in the distribution as 0 to a Gausslan vector process I) Matrices are denoted by underlining.
Let () be the eigenvalues of Eq.C4).Further on let 1 the egenvalues and wl(x) the eigenfUnctlons of the averaged problem.
(A Cn a n laa laka [ 1 ) ()I_ li () d_ show that ts men wth aq ,...,s_, converts to the adequate k-th moment of AI fir we calculate the order of a te of the fo as 0 if ...r tiies the condition ..rp Because of the independence of the pcess i(x')' we mu deal wth tes of the fo , =onsidmrstons as in the pof of theorem we obtain the theorem We will show in the followi considtions that the theorem 7 m s-)T inead of 'IiI s-)T.For this we denote the conve nce ifoy in .We will deal with the idea of ts proof at the bounda value problems because we use for the proof dental estimations from the perturbation theol.
3.2.The results of this section can also be used of eigenvalue problems of the form Lu %h(x,)u, Ui[ul O, i=1,2,...,2m, Ox&1. ( The operator L is a deterministic differential operator of the order where the coefficient fr(X) is continuous differentiable of r-th order.It is for the stochastic process h(x,) the equation h(x,)= h o(x)+g(x,).Let ho(X] be positive d let the process g(x,) be a weakly correlated process of the correlation eth with g(x,) ( sufficient small).e deterministic bounda conditions Ui[u]=O,i=1,2,...,2m, are constituted in that maer that the problem (11) is sefadjoint and positive definite.
)ho w k(x)wl(x)hO(x)dx 1" We define as n section 3.1.
J.V. SCHEIDT AND .PURKERT u lx i,) for with x i 0,.Thn a dvlopmnt xists li() for which we asse the ,onvergence of llil) It is I for i:0 and Ali= w l(xi) for i#O li= G l(x,y)g(y,W)wl(y)dy for i$0 where Gl(X,y) denotes the generalized Green nction to ho d the boundary conditions Ui, i=1,2,...,2m, d b (w) (w ())2g(y,)dy.By IIio G(x i,y)wl(y) for i@O we can foulate a to theorem 7 adequate theorem.TOS 8. Let x 1,x2,... be from 0,I and let g(,) be a quence of weakly correlted process with the correlation leith and for y K(x)' limo a ,x+y)dy=a(x (a(x)O) uniformly in x.Then the random vector '41 oI sis i i ,'" 11 11 s-)T formed from the stochastic eigenvalue problem (I with g(x,) converges in the distribution to a Gaussian random vector )T with 11ii"''' Isis pi = a(y)IG (y)lq (y)dy.
q pi p 0, plqi q o p Giq RE,AEK.We can also prove such a theorem for the more general stochastic eigenvalue problem Lu + L ()u A(hoU + M (W)u), Ui[uI = O, i=I ,2,...,2m, O gx I.
Here the operator L is a deterministic differential operator like in Eq. ( 12) and m-1 with L ()B = (M () =0 where the coefficients ak(x,) bk(X,) are independent, weakly correlated processes.Perturbation serieses relativ to L (), M () form the basic.These serieses were deduced in 5S.
--,--__ # -# / Fig. 2 # i By b-zO the variation of the limit distribution does not dependlon the number of the egenvalues.In this case the corre]ation loko= for k#l is independent of k and I.This effect is expli- cable from the fact that the operator L(@) determines the eigen- values respectively the eigenvalues () for a o determines about the random operator L(@) the other eigenvalues () for this o" (b) The random vector (u l(x)-w l(x),uk(x)_wk(x )T converges in the dstributioh as I0, to the Gaussian vector process (I(x)'k(x))T wh <,[p(x)> : O, o P P for p, q Ik,.
We can also calculate the correlation values of ths Gaussian vector process from <p(X)(x)> = lm <Ulp(X)U (y)> Zo q if Ulp(X,) denotes the first term in the development of u (x W) p Wp(X) relativ to a(x,) and b(x,w).One obtains PP o after a few c Iculations with (x,)=a(x,w)-pb(x,) and then from (14) the correlation function of the limit process of ],(Ul-W I) x) )w. (x)w The figure 3 shows this variation for the parameters =I,2,3,4.e ea ske very good statements about the behaviour of the limit pro(C)esm of the egsnfunctions because of the limit process l(x) is a auealan process.f right for all permissible functions u,v (i.e.functions, which pseeas 2m continuous derivatives end fulfil the boundary condi- tions), i.e. the boundary terms of the integration by parts mue be zero.Then the stochastic operator L() is symmetric reltiv to aI1 permssdble functions.
We make for the solution u(x,) of (I 5) the statement uCx, = k=o when uk(x,) denotes the homogeneous part of k-th order of u(x,) in the (C)oefficents r and .S ubstituting this statement in (15) eads to the boundary value problem for Uk(X,) <L>u k -L (W)Uk_ U link] = 0 for k=2,3,...
Let G(x,y) be the Green function corresponding to <L> and to the boundary conditions U.[.=O then we obtain u o () = ( ) is a deterministic funetiono ih (16) fr u() obtained.
The theom 8 is ped.
Now we deal th questions of the convergence of he delopment.
].This theorem 9 proves the convergence n the diri- bution of the processes (u(,)-uo(x)) whCeh has been constituted by the solution u(x,) of (]5) and the solntion of the avered problem to (]5) u o(x) to the Gaussian process (x,) th We put (x,)KUk(X,9).
we see with similar eons PROOF.
Then we obtain with theorem )-u o(x)) converges in the Hence we obtain in this case that (u(x, distribution to a Gaussian process (x,) wh (x) = O, (x)(y) I ag(z)G(,z)G(y,z)dz.
0 .E. Boyce deals ih this case of a sochasic bounda value problem n [] and he shoed hat (u(x,)-Uo(X)) is in the limit a Gauss random vaable (x) for any x [0,I.
3. Let wi(x) be the eigenfctions of the avered operator KL> th w i(x)wj(x)dij d the eigenvalues of o en we can calculate the correlation ction of the limi process of (u(x')-u(x)) toh Specially we have for a wie-sense stationary, weakly correlated process -r pW (X)Wp(y)+ nard----Cpqw (X)Wq(y) Cpq o p ese statements can be proved easly wth G(x,y)= (see also [7]).
Now we regard the bodary value problem (;5) with the conditions above.It denotes i a system of fctions of the given energetic space H<L>. is system be complete in HL>.The solution u(x,) of the equation L()u=g() is the minimum of the energetic ctional (Lu,u)-2(g,u) n the energetic space.The Ritz-method with the co-ordinate fctions i conducts to the foowi stem an approximation of the correlation function how it has been rep- resented in the remark 3. The formula is used by an approximation of the correlation function <q(x)(y)> and ( 24) is suitable for a explicit calculation of ((x)(y) if the _Green function is ealculable difficult.with f(x) = sh2((I-x)) (sh(2x) x). a is tie pareter wNch helots to the wide-sense stationa, wetly correlated process g(x,).I iafor bO and th f(x):-sin((-x)) in(2x)-x).
with the initial condition _X(to)o.A(t) is a nxn deterministic matrix, A(t)=(aij(t))lli,jan, x s the vector x(t)=(x i(t))T olin' o(i)'iAn^a nd f(t,)=(fi(t,))T is a stochastic vector pro- I, in cess.Let fi(t,) be processes of which a.s. the trajectories are continuous and Q(t,t o) be the principal matrix associated with A_(t) (.e. _Q(to,to)=_l (I is the dentity matrix) and (t,o)= A(t)Q_(,t o )), then the unique solution a.s.In the folowing we consider the solution (26) if f(t,)= E(t,) denotes a vector process with independent, weakly correl- ated processes gi(t') as components.This leads to the following theorem.
THEOREM I0.Let (t,) a sequence of weakly correlated vector processes for $ 0 with independent components gi(t') and gi(t)gi(s)} : Rit(t's) for s K(t) Ri 0 for sK(t)' lmo E. (t t+s)dsa i(t) PROOF.The proof is following from theorem.5 then we obtain (27) o ;s) I_ i ,jn de"   with the dfinition of the n t qi x(t,) = [(t) + _ t (qik(t,to;S)gk$(S,))iAindS o and from this for the limit process (t,)=(i(t,))1i n with the notations of theorem 5 n i (tl)j(t2) = _ <ik(tl )jk(t2 ) k__nmnlt1't2)ak(s)qik(tl,to;S)qjk(t2,to;s)ds. o In generally (t,) does not denote a wide-sense stationary vector process while (tl)T(t2) is not a function which only depends on t2-tI.We assume that the processes giz(t,) are wide-sense sta- tionary and weakly correlated and A(t)=A is a constant matrix, the egenvalues of which have negative real parts.Then we obtain for t T > the formula Q(t,t)o=e(-tO)o--0 and in the same t eA (t_s)   way the solutions (h_(s) (s) )s and eA(-)((s)+(.))s of the system of differential equations differ by a solution of the homogeneous sy'tem of (2"/) which is near by zero for t Z Tt o.Hence the solution of ( 27) is described by _Z(t,w) = h(s)ds + (s,w)ds.
rk(X;Y, ,k) dyl -..dY k for the tes Uk(X,) of the development of the solution u(x,) of the boundary value problem (I 5) with uo(r r r(X;y) = o.r 2 Hr(X;y) = G(x,y) p and a are constants.By properties of the Green" ction C,C 0 u(2m)(x,&; = -,o(x)I o = (v)fr Uo o o v=o for aost alI U then lu ) =,= =(c*)c o d throh induction for almost all for Op2m and =E[0,1].(x,u)I converges almost surely and ifoNy in Hence o 0&x I when tt ((C+a))-1 for almost all , ) =_ r(y,)uo(r) it follows as in the proof of the theorem 7 lm u )...u (Xap)B.= [ (X)br(Y) = o arZ) G(x"Z)0z r G(y.Z)r (u r) (z)) 2dz ily the statement of the theorem for ((x )-u o(x ) complete pof of ts theorem follows as In theorem 7 from the ifo convergence relativ to lira

. 2 .
YAFLE.Let g(x,) be a wide-sense stationary, weakly correlated process then the simple boundary value problem -u" + bu = g(x,), u(O) = u(1) = 0 with b=const, possesses the averaged problem-w" + bw = O, w(O) = w(1) = O.This problem has only the trivial solution when b$-(kl) 2 is and the Green fUnction Gb(X,y) is sh(,x.).sh({(1.-y)for 0 x < y G b Cx,y) = sh(y) ah((1-x) the correlation function for the limit process b(X,) from theorem 9 o d the variance 2(x,z)dz = a (f(x) + f(1-x)) Eecialy we have bo b-( b In the following Fig.4 the variance of the limit process b(X,9) is potted for ,ome values of the parameter b.Since <b 2(x) =<b2(1-x)>, we have plotted ths function for O x1/2 only.The following values submi for b= by contrast to it: INITIAL VALUE PROBLEMSWe consider a system of ordinary differential equations of the first order dx --A_Ct )x_ + _C ,) of the nitial value problem (26) may be written in the form t x(t,) = Q(t,to)(o + _Q-1(s,t )f(s,)ds)The ntegral is defined by the integral of sample functions.In generally we cannot calculate the distribution of the solutfon x_(t,) f we know the distribution of f(t,).Let _f(t,;) be a Gausslan vector process, then is Q_-I (s,t)_f(s,)ds also a Gaussian to o vector process and in the same way the solution x(t,) of this system of linear differential equations (see /2/).