PERIODIC IN DISTRIBUTION SOLUTION FOR A TELEGRAPH EQUATION

In this paper we study an abstract stochastic equation of second order and stochastic boundary problem for the telegraph equation in a strip. We prove the existence of solutions, which are d-periodic (periodic in distribution) random processes.


Introduction
Let H be a separable Hilbert space, A E C(R,(H)) be a periodic function, and a E R. Let w {w(t)" t R} be an H-valued Wiener process.Consider the following abstract stochastic boundary problem for the telegraph equation u" tt aui(t )-uxz tt, x) + x (t, x) A(t)u(t, x) + g(x)w'(t), (t, x) e Q; u(t, O) u(t, r) O  (1 where 0 denotes the zero element in H and Q: Rx[0,r],g:[0,r]C.We are interested in d-periodic time variable t and a w-adapted solution u for problem (1) in the sense defined below.The existence of periodic solutions for deterministic partial differential equations are intensively studied, see for example, the well-known book [19].The problem of the existence of stationary and d-periodic solutions to stochastic ordinary differential equations is also well-known, see the books [7,13], and the survey [8] for more references.During the past years, it has become apparent that it is natural and more adequate in many applications to consider an input source for partial differential equations as random source or a random disturbance.Thus the problem of investigating stochastic partial differential equations is of importance, see [3, 9, 18], where the problems of this kind were studied.We prove the existence of d-periodic solutions for stochastic boundary problem (1).
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Periodic Solution for Ordinary Stochastic Equation
First, we recall some standard notations and terminology.Let (H, (.,.),I I II) be a complex separable Hilbert space, 0 the zero element in H, and (H) the Banach space of bounded linear operators on H, with the operator norm denoted also by the symbol I1" II.The adjoint of D E (H) will be denoted by D*.For an H-valued or (H)-valued function, the continuity and differentiability means correspondingly the continuity and differentiability in the norm.Let I be the identity operator.
In what follows, we shall consider all random elements on the same complete probability space (,,P).The uniqueness of a random process, satisfying some equation, means its uniqueness up to stochastic equivalence.We consider only H- valued random functions which are continuous with probability one.All equalities for random elements are assumed to hold with probability one.
Definition 1: Let v > 0 be fixed.A random H-valued process {x(t)'t R} is called a d-periodic (in distribution) with period r if where (H) is the Borel a-algebra in H.
To prove the existence of periodic solutions for problem (1), we first consider the following stochastic differential equation x'(t) A(t)x(t) + w'(t), R, (3) where A C(R,(H)), and Vt e R, A(t + r)-A(t).Definition 2: An H-valued continuous zbt-adapted random process {x(t): t R} is called a nonanticipating solution of equation (3) if for every t E R, the random ele- ment x(t) is bt-measurable E I I x(t)II 2 < + c, and for every s < t with probability 1, x(t)-x(s) / A(u)x(u)du + w(t)-w(s).

8
The last integral is a Riemann integral of H-valued continuous function with probability one.
Let U: R(H) be the unique solution of the following problem" U'(t)-A(t)U(t), U(0)-.tER; It is well-known that for every t E R, the operator U(t) is invertible and Let u(t) u(t-,-)u(-)', t e n, , e z, Now we are prepared to prove the existence of d-periodic solution of equation ( 3).
Theorem 1" Stochastic equation ( 3) has a unique d-periodic with period " nonanticipating solution {x(t): t R}, with supo < < r I I x(t) I I 2 Wiener process w if and only if the following inequlit sup o I I gk()G( , )j I[ 2d < / o (4) j>l 0 < +oc, for every is satisfied for every orthonormal basis {ej'j >_ 1} in H.
In order to prove Theorem 1, the following lemmas will be needed.
Lemma 1: If the random process x is a nonanticipating solution of (3), then for every s < t with probability 1,   x(t) G(t,x)x(s) + / G(t, u)dw(u), () where the integral is a stochastic integral with respect to w.
Proof: The conclusion follows from a computation similar to the proof of Lemma 1.
Lemma 3: If the random process x is a d-periodic nonanticipating solution of (3), then the stationary process {x(nv)'n E N} in H is a stationary adapted to {(n)" n Z} solution of the following difference equation in H" x((n + 1)r) U(v)x(n) + c(n), n Z, where (n + 1)'r -r e(n):-/ G((n + 1)r,u)dw(u) / U(r)U-l(u)dw(nT" + u), n Z, n" 0 (7) is a sequence of Gaussian independent identically distributed random elements in H.
Proof: The proof follows from Lemma 2. It follows by a direct computation that the covariance operator S of the element e(0) is given by / U(7")U-I(u)WU-l(u)*U(7")*du.S 0 In addition, it can be proved as in [3, n. 4.1] that for the covariance operator S x of x(0) we have *.  7has stationary and adapted to {e(n):n Z} solution with E I I x(O) 2 < + c for every Wiener process w, then for every orthonormal basis {ej'j > 1} in H, the inequality holds 7" I I v (,la( < + j>l 0 Proof: This is a part of Theorem 1 [7, n. 4.1, p. 93] or Theorem 4.1 [5].Lemma 5: Let {x(n'):n Z} be a stationary and adapted to {e(n):n G Z} solution of equation (7)  Proof: The process x is continuous with probability 1, see [1] or [7, n.8.4.2] and [2, 3, 11, 17] for general results.The last property follows from computations.
Proof of Theorem 1: (1) Suppose that equation (3) has a unique d-periodic with period r and anticipating solution x such that sup0 < < r l] x(t)[I 2 < + oo for every Wiener process w.It follows from Lemma 3 that euaion (7) has an adapted to e stationary solution {x(nr):n E Z} with E I[ x(0)II 2 < + o.This solution of ( 7) is unique.Indeed, if the equation ( 7) had two different stationary and e-adapted solutions, then by Lemma 5, equation (3) has two different d-periodic solutions.By virtue of Lemma 4, we have ( 4).
The following studies will be concerned with the following stochastic equation: x"(t) + ax'(t) A(t)x(t) + w'(t), t G R. ( We first give the definition of a solution for equation (9).Definition 3: An H-valued random process {x(t):t R} is called a solution of equation ( 9) if for every t G R, the random element x(t) is fit-measurable, the processes x, are continuous with probability 1, and for every s < t, the equality x'(s) + a(x(t)-x(s)) / A(u)x(u)du + w(t)-w(s) (10) 8 holds with probability 1.The main result of this section is a theorem which establishes a criterion of the existence of d-periodic solutions for equation (9).Let H2: H x H and for Xl, x2, Yl, Y2 from H, ((Xl,Yl),(x2, Y2))2" (Xl,Yl)-[-(x2, Y2).
Then (H2,(.,.)2) is a Hilbert space.For G R, let l(t)" A(t)a where (R) is the zero operator in H and () W The following lemma is an immediate consequence of the above definitions.
Lemma 6: Let x be a nonanticipating solution of equation (9).Then the H 2- valued random process y: is a nonanticipating solution of the following equation in H2: y'(t) =/(t)y(t) + w', t e R. (11) Lemma 7: Let y be a nonanticipating solution of ( 11).Then Y2-Y'I with probability 1, and Yl is a nonanticipating solution of ( 9).
Proof: The proof is obvious.
Let us consider the function U, which is a unique solution of the following problem U'(t)-(t)U(t), t E R; where is the identity operator in H2.By a direct computation, we obtain the follow- ing lemma.
Lemma 8: We have vl where the functions V1, V 2 are the unique solutions of the equation V'(t) A(t)Vj(t) aVj(t), Now, we are prepared to prove the existence of d-periodic solutions of equation ( 9).
Let k E N and let Vlk V2k be unique solutions of the following equation in H: V'k(t (A(t) k2I)Vjk(t) aVjk(t), t E R; j G 1, 2, Uk--VlkV2k Now we prove the main result of this paper.
k=l By (ii), the series for u converges on Q with probability 1.
continuity of u, we have to show that the series To establish the for b < c, c-b _< 7 is convergent.The convergence of this series follows from the well-known submartingale type inequality for stochastic integrals, see [1, 10, 14] and from the existence of the moments for Gaussian elements [15, 16].The continuity of t! the random functions ut, u can be established by using similar arguments XX Differentiating u with respect to t and x and using (14) we can verify that u is a solution of problem (1).The proof of the uniqueness is the same as that in the proof of theorem 2.
Let us show now that (i)implies (ii).Let k N and the Wiener process w be given.Let u be a unique nonanticipating d-periodic solution of (1) of period v with respect to t such that supEllu(t,x) 112< +oo, supEIIu(t,x) 112< +oo Q Q for g(x)sin kx, x [0, r].Define vk(t)'-J u(t,x)sinkxdx, 0 tER.
It can be easily verified that v k is a continuous nonanticipating d-periodic of period r, H-valued process such that sup E I I v(t)II 2 < + c, sup E O<t<Tr O<t<-By virtue of Parseval's identity, see for example [7, p. 146], it follows that