EXISTENCE OF S 2 -ALMOST PERIODIC SOLUTIONS TO A CLASS OF NONAUTONOMOUS STOCHASTIC EVOLUTION EQUATIONS

Elecronic Journal of Qualiaive Theory of Differenial Equaions 8, No. 35, 1-19; hp://www.mah.u-szeged.hu/ejqde/ EXISTENCE OF S -ALMOST PERIODIC SOLUTIONS TO A CLASS OF NONAUTONOMOUS STOCHASTIC EVOLUTION EQUATIONS PAUL H. BEZANDRY AND TOKA DIAGANA Absrac. The paper sudies he noion of Sepanov almos periodiciy (or S -almos periodiciy) for sochasic processes, which is weaker han he noion of quadraic-mean almos periodiciy. Nex, we make exensive use of he so-called Acquisapace and Terreni condiions o prove he exisence and uniqueness of a Sepanov (quadraic-mean) almos periodic soluion o a class of nonauonomous sochasic evoluion equaions on a separable real Hilber space. Our absrac resuls will hen be applied o sudy Sepanov (quadraicmean) almos periodic soluions o a class of n-dimensional sochasic parabolic parial differenial equaions. 1. Inroducion Le (H,,, ) be a separable real Hilber space and le (Ω, F,P) be a complee probabiliy space equipped wih a normal filraion {F : R, ha is, a righconinuous, increasing family of sub σ-algebras of F. The impeus of his paper comes from wo main sources. The firs source is a paper by Bezandry and Diagana [], in which he concep of quadraic-mean almos periodiciy was inroduced and sudied. In paricular, such a concep was, subsequenly, uilized o sudy he exisence and uniqueness of a quadraic-mean almos periodic soluion o he class of sochasic differenial equaions (1.1) dx() = AX()d + F(, X())d + G(, X())dW(), R, where A : D(A) L (P; H) L (P; H) is a densely defined closed linear operaor, and F : R L (P; H) L (P; H), G : R L (P; H) L (P; L ) are joinly coninuous funcions saisfying some addiional condiions. The second sources is a paper Bezandry and Diagana [3], in which he auhors made exensive use of he almos periodiciy o sudy he exisence and uniqueness of a quadraic-mean almos periodic soluion o he class of nonauonomous semilinear sochasic evoluion equaions (1.) dx() = A()X() d + F(, X()) d + G(, X()) dw(), R, where A() for R is a family of densely defined closed linear operaors saisfying he so-called Acquisapace and Terreni condiions [1], F : R L (P; H) L (P; H), G : R L (P; H) L (P; L ) are joinly coninuous saisfying some addiional condiions, and W() is a Wiener process. Mahemaics Subjec Classificaion. 34K14, 6H1, 35B15, 34F5. Key words and phrases. Sochasic differenial equaion, sochasic processes, quadraic-mean almos periodiciy, Sepanov almos periodiciy, S -almos periodiciy, Wiener process, evoluion family, Acquisapace and Terreni, sochasic parabolic parial differenial equaion. EJQTDE, 8 No. 35, p. 1

The presen paper is definiely inspired by [, 3] and [7, 8] and consiss of sudying he exisence of Sepanov almos periodic (respecively, quadraic-mean almos periodic) soluions o he Eq. (1.) when he forcing erms F and G are boh S -almos periodic. I is worh menioning ha he exisence resuls of his paper generalize hose obained in Bezandry and Diagana [3], as S -almos periodiciy is weaker han he concep of quadraic-mean almos periodiciy. The exisence of almos periodic (respecively, periodic) soluions o auonomous sochasic differenial equaions has been sudied by many auhors, see, e.g., [1], [], [9], and [17] and he references herein. In paricular, Da Prao and Tudor [5], have sudied he exisence of almos periodic soluions o Eq. (1.) in he case when A() is periodic. In his paper, i goes back o sudying he exisence and uniqueness of a S -almos periodic (respecively, quadraic-mean almos periodic) soluion o Eq. (1.) when he operaors A() saisfy he so-called Acquisapace and Terreni condiions and he forcing erms F, G are S -almos periodic. Nex, we make exensive use of our absrac resuls o esablish he exisence of Sepanov (quadraic mean) almos periodic soluions o an n-dimensional sysem of sochasic parabolic parial differenial equaions. The organizaion of his work is as follows: in Secion, we recall some preliminary resuls ha we will use in he sequel. In Secion 3, we inroduce and sudy he noion of Sepanov almos periodiciy for sochasic processes. In Secion 4, we give some sufficien condiions for he exisence and uniqueness of a Sepanov almos periodic (respecively, quadraic-mean almos periodic) soluion o Eq. (1.). Finally, an example is given o illusrae our main resuls.. Preliminaries For deails of his secion, we refer he reader o [, 4] and he references herein. Throughou he res of his paper, we assume ha (K, K ) and (H, ) are separable real Hilber spaces and ha (Ω, F, P) sands for a probabiliy space. The space L (K, H) denoes he collecion of all Hilber-Schmid operaors acing from K ino H, equipped wih he classical Hilber-Schmid norm, which we denoe. For a symmeric nonnegaive operaor Q L (K, H) wih finie race we assume ha {W() : R is a Q-Wiener process defined on (Ω, F,P) wih values in K. I is worh menioning ha he Wiener process W can obained as follows: le {W i () : R, i = 1,, be independen K-valued Q-Wiener processes, hen W 1 () if, W() = W ( ) if, is Q-Wiener process wih he real number line as ime parameer. We hen le F = σ{w(s), s. The collecion of all srongly measurable, square-inegrable H-valued random variables, denoed by L (P; H), is a Banach space when i is equipped wih norm X L (P;H) = E X EJQTDE, 8 No. 35, p.

where he expecaion E is defined by E[X] = Ω X(ω)dP(ω). Le K = Q 1 (K) and le L = L (K ; H) wih respec o he norm Φ L = Φ Q 1 = Trace(Φ Q Φ ). Throughou, we assume ha A() : D(A()) L (P; H) L (P; H) is a family of densely defined closed linear operaors, and F : R L (P; H) L (P; H), G : R L (P; H) L (P; L ) are joinly coninuous funcions. In addiion o he above-menioned assumpions, we suppose ha A() for each R saisfies he so-called Acquisapace and Terreni condiions given as follows: There exis consans λ, θ ( π, π), L, K, and α, β (, 1] wih α + β > 1 such ha (.1) Σ θ { ρ(a() λ ), R(λ, A() λ ) K 1 + λ and (A() λ )R(λ, A() λ )[R(λ, A()) R(λ, A(s))] L s α λ β for all, s R, λ Σ θ := {λ C { : argλ θ. Noe ha he above-menioned Acquisapace and Terreni condiions do guaranee he exisence of an evoluion family associaed wih A(). Throughou he res of his paper, we denoe by {U(, s) : s wih, s R, he evoluion family of operaors associaed wih he family of operaors A() for each R. For addiional deails on evoluion families, we refer he reader o he landmark book by Lunardi [11]. Le (B, ) be a Banach space. This seing requires he following preliminary definiions. Definiion.1. A sochasic process X : R L (P; B) is said o be coninuous whenever lim s E X() X(s) =. Definiion.. A coninuous sochasic process X : R L (P; B) is said o be quadraic-mean almos periodic if for each > here exiss l() > such ha any inerval of lengh l() conains a leas a number τ for which supe X( + τ) X() <. R The collecion of all quadraic-mean almos periodic sochasic processes X : R L (P; B) will be denoed by AP(R; L (P; B)). 3. S -Almos Periodiciy Definiion 3.1. The Bochner ransform X b (, s), R, s [, 1], of a sochasic process X : R L (P; B) is defined by X b (, s) := X( + s). EJQTDE, 8 No. 35, p. 3

Remark 3.. A sochasic process Z(, s), R, s [, 1], is he Bochner ransform of a cerain sochasic process X(), if and only if for all R, s [, 1] and τ [s 1, s]. Z(, s) = X b (, s), Z( + τ, s τ) = Z(s, ) Definiion 3.3. The space BS (L (P; B)) of all Sepanov bounded sochasic processes consiss of all sochasic processes X on R wih values in L (P; B) such ha X b L ( R; L ((, 1); L (P; B)) ). This is a Banach space wih he norm ( X S = X b L (R;L ) = sup R = sup R ( +1 ) 1/ E X( + s) ds E X(τ) dτ) 1/. Definiion 3.4. A sochasic process X BS (L (P; B)) is called Sepanov almos periodic (or S -almos periodic) if X b AP ( R; L ((, 1); L (P; B)) ), ha is, for each > here exiss l() > such ha any inerval of lengh l() conains a leas a number τ for which sup R +1 E X(s + τ) X(s) ds <. The collecion of such funcions will be denoed by S AP(R; L (P; B)). The proof of he nex heorem is sraighforward and hence omied. Theorem 3.5. If X : R L (P; B) is a quadraic-mean almos periodic sochasic process, hen X is S -almos periodic, ha is, AP(R; L (P; B)) S AP(R; L (P; B)). Lemma 3.6. Le (X n ()) n N be a sequence of S -almos periodic sochasic processes such ha sup R +1 Then X S AP(R; L (P; B)). E X n (s) X(s) ds, as n. Proof. For each >, here exiss N() such ha +1 X n (s) X(s) ds, R, n N(). 3 From he S -almos periodiciy of X N (), here exiss l() > such ha every inerval of lengh l() conains a number τ wih he following propery Now +1 E X N (s + τ) X N (s) ds <, R. 3 EJQTDE, 8 No. 35, p. 4

E X( + τ) X() = E X( + τ) X N ( + τ) + X N ( + τ) X N () + X N () X() and hence sup R +1 which complees he proof. Similarly, E X( + τ) X N ( + τ) + E X N ( + τ) X N () + E X N () X() E X(s + τ) X(s) ds < 3 + 3 + 3 =, Lemma 3.7. Le (X n ()) n N be a sequence of quadraic-mean almos periodic sochasic processes such ha Then X AP(R; L (P; B)). supe X n (s) X(s), s R as n Using he inclusion S AP(R; L (P; B)) BS (R; L (P; B)) and he fac ha (BS (R; L (P; B)), S ) is a Banach space, one can easily see ha he nex heorem is a sraighforward consequence of Lemma 3.6. Theorem 3.8. The space S AP(R; L (P; B)) equipped wih he norm is a Banach space. X S = sup R ( +1 E X(s) ds )1/ Le (B 1, B1 ) and (B, B ) be Banach spaces and le L (P; B 1 ) and L (P; B ) be heir corresponding L -spaces, respecively. Definiion 3.9. A funcion F : R L (P; B 1 ) L (P; B )), (, Y ) F(, Y ) is said o be S -almos periodic in R uniformly in Y K where K L (P; B 1 ) is a compac if for any >, here exiss l(, K) > such ha any inerval of lengh l(, K) conains a leas a number τ for which sup R +1 for each sochasic process Y : R K. E F(s + τ, Y ) F(s, Y ) B ds < Theorem 3.1. Le F : R L (P; B 1 ) L (P; B ), (, Y ) F(, Y ) be a S - almos periodic process in R uniformly in Y K, where K L (P; B 1 ) is compac. Suppose ha F is Lipschiz in he following sense: E F(, Y ) F(, Z) B M E Y Z B 1 for all Y, Z L (P; B 1 ) and for each R, where M >. Then for any S - almos periodic process Φ : R L (P; B 1 ), he sochasic process F(, Φ()) is S -almos periodic. EJQTDE, 8 No. 35, p. 5

4. S -Almos Periodic Soluions Le C(R, L (P; H)) (respecively, C(R, L (P; L )) denoe he class of coninuous sochasic processes from R ino L (P; H)) (respecively, he class of coninuous sochasic processes from R ino L (P; L )). To sudy he exisence of S -almos periodic soluions o Eq. (1.), we firs sudy he exisence of S -almos periodic soluions o he sochasic non-auonomous differenial equaions (4.1) dx() = A()X()d + f()d + g()dw(), R, where he linear operaors A() for R, saisfy he above-menioned assumpions and he forcing erms f S AP(R, L (P; H)) C(R, L (P; H)) and g S AP(R, L (P; L )) C(R, L (P; L )). Our seing requires he following assumpion: (H.) The operaors A(), U(r, s) commue and ha he evoluion family U(, s) is asympoically sable. Namely, here exis some consans M, δ > such ha U(, s) Me δ( s) for every s. In addiion, R(λ, A( )) S AP(R; L(L (P; H))) where λ is as in Eq. (.1). Theorem 4.1. Under previous assumpions, we assume ha (H.) holds. Then Eq. (4.1) has a unique soluion X S AP(R; L (P; H)). We need he following lemmas. For he proofs of Lemma 4. and Lemma 4.3, one can easily follow along he same lines as in he proof of Theorem 4.6. Lemma 4.. Under assumpions of Theorem 4.1, hen he inegral defined by X n () = U(, ξ)f( ξ)dξ belongs o S AP(R; L (P; H)) for each for n = 1,,... Lemma 4.3. Under assumpions of Theorem 4.1, hen he inegral defined by Y n () = U(, ξ)g( ξ)dw(ξ). belongs o S AP(R; L (P; L )) for each for n = 1,,... Proof. (Theorem 4.1) By assumpion here exis some consans M, δ > such ha U(, s) Me δ( s) for every s. Le us firs prove uniqueness. Assume ha X : R L (P; H) is bounded sochasic process ha saisfies he homogeneous equaion (4.) dx() = A()X()d, R. Then X() = U(, s)x(s) for any s. Hence X() MDe δ( s) wih X(s) D for s R almos surely. Take a sequence of real numbers (s n ) n N such ha s n as n. For any R fixed, one can find a subsequence EJQTDE, 8 No. 35, p. 6

(s nk ) k N (s n ) n N such ha s nk < for all k = 1,,... By leing k, we ge X() = almos surely. Now, if X 1, X : R L (P; H) are bounded soluions o Eq. (4.1), hen X = X 1 X is a bounded soluion o Eq. (4.). In view of he above, X = X 1 X = almos surely, ha is, X 1 = X almos surely. Now le us invesigae he exisence. Consider for each n = 1,,..., he inegrals and X n () = Y n () = U(, ξ)f( ξ)dξ U(, ξ)g( ξ)dw(ξ). Firs of all, we know by Lemma 4. ha he sequence X n belongs o S AP(R; L (P; H)). Moreover, noe ha +1 +1 E X n (s) ds E U(s, s ξ) f(s ξ) dξ ds { +1 M e δξ E f(s ξ) ds dξ { M f S e δξ dξ Since he series M δ f S e δn (e δ + 1). M δ (eδ + 1) n= e δn is convergen, i follows from he Weirsrass es ha he sequence of parial sums defined by n L n () := X k () k=1 converges in he sense of he norm S uniformly on R. Now le for each R. Observe ha l() := X n () n=1 l() = and hence l C(R; L (P; H)). U(, ξ)f(ξ)dξ, R, EJQTDE, 8 No. 35, p. 7

Similarly, he sequence Y n belongs o S AP(R; L (P; L )). Moreover, noe ha +1 E Y n (s) ds = TrQ +1 M TrQ E U(s, s ξ) g(s ξ) d(ξ) ds e δξ { +1 M δ TrQ g S e δn (e δ + 1). E g(s ξ) ds dξ Proceeding as before we can show easily ha he sequence of parial sums defined by n M n () := Y k () k=1 converges in sense of he norm S uniformly on R. Now le for each R. Observe ha m() := Y n () n=1 m() = U(, ξ)g(ξ)dw(ξ), R, and hence m C(R, L (P; L )). Seing X() = U(, ξ)f(ξ) dξ + U(, ξ)g(ξ) dw(ξ), one can easily see ha X is a bounded soluion o Eq. (4.1). Moreover, +1 E X(s) (L n (s) + M n (s)) ds as n uniformly in R, and hence using Lemma 3.6, i follows ha X is a S -almos periodic soluion. In view of he above, i follows ha X is he only bounded S -almos periodic soluion o Eq. (4.1). Throughou he res of his secion, we require he following assumpions: (H.1) The funcion F : R L (P; H) L (P; H), (, X) F(, X) is S -almos periodic in R uniformly in X O (O L (P; H) being a compac). Moreover, F is Lipschiz in he following sense: here exiss K > for which E F(, X) F(, Y ) K E X Y for all sochasic processes X, Y L (P; H) and R; EJQTDE, 8 No. 35, p. 8

(H.) The funcion G : R L (P; H) L (P; L ), (, X) G(, X) be a S - almos periodic in R uniformly in X O (O L (P; H) being a compac). Moreover, G is Lipschiz in he following sense: here exiss K > for which E G(, X) G(, Y ) L K E X Y for all sochasic processes X, Y L (P; H) and R. In order o sudy (1.) we need he following lemma which can be seen as an immediae consequence of ([16], Proposiion 4.4). Lemma 4.4. Suppose A() saisfies he Acquisapace and Terreni condiions, U(, s) is exponenially sable and R(λ, A( )) S AP(R; L(L (P; H))). Le h >. Then, for any >, here exiss l() > such ha every inerval of lengh l conains a leas a number τ wih he propery ha for all s h. U( + τ, s + τ) U(, s) e δ ( s) Definiion 4.5. A F -progressively process {X() R is called a mild soluion of (1.) on R if (4.3) X() = U(, s)x(s) + + for all s for each s R. s s U(, σ)f(σ, X(σ)) dσ U(, σ)g(σ, X(σ)) dw(σ) Now, we are ready o presen our firs main resul. Theorem 4.6. Under assumpions (H.)-(H.1)-(H.), hen Eq. (1.) has a unique S -almos period, which is also a mild soluion and can be explicily expressed as follows: X() = U(, σ)f(σ, X(σ))dσ+ U(, σ)g(σ, X(σ))dW(σ) for each R whenever ( Θ := M K ) δ + K Tr(Q) < 1. δ Proof. Consider for each n = 1,,..., he inegral R n () = U(, ξ)f( ξ) dξ + where f(σ) = F(σ, X(σ)) and g(σ) = G(σ, X(σ)). Se X n () = U(, ξ)f( ξ) dξ and, U(, ξ)g( ξ) dw(ξ). EJQTDE, 8 No. 35, p. 9

Y n () = U(, ξ)g( ξ) dw(ξ). Le us firs show ha X n ( ) is S -almos periodic whenever X is. Indeed, assuming ha X is S -almos periodic and using (H.1), Theorem 3.1, and Lemma 4.4, given >, one can find l() > such ha any inerval of lengh l() conains a leas τ wih he propery ha for all s, and U( + τ, s + τ) U(, s) e δ ( s) +1 for each R, where η() as. E f(s + τ) f(s) ds < η() For he S -almos periodiciy of X n ( ), we need o consider wo cases. Case 1: n. +1 +1 = + E X n (s + τ) X n (s) ds E U(s+τ, s+τ ξ)f(s+τ ξ)dξ U(s, s ξ)f(s ξ)dξ ds +1 +1 M +1 + +1 M + Case : n = 1. We have +1 +1 = 3 U(s + τ, s + τ ξ) E f(s + τ ξ) f(s ξ) dξ ds U(s + τ, s + τ ξ) U(s, s ξ) E f(s ξ) dξ ds e δξ E f(s + τ ξ) f(s ξ) dξ ds e δξ E f(s ξ) dξ ds e δξ { +1 e δξ { +1 E X 1 (s + τ) X 1 (s) ds E +1 E f(s + τ ξ) f(s ξ) ds dξ E f(s ξ) ds dξ U(s+τ, s+τ ξ)f(s+τ ξ)dξ U(s, s ξ)f(s ξ)dξ ds U(s + τ, s + τ ξ) E f(s + τ ξ) f(s ξ) dξ ds EJQTDE, 8 No. 35, p. 1

+3 +3 +1 +1 3 M +1 +3 +1 U(s + τ, s + τ ξ) U(s, s ξ) E f(s ξ) dξ ds U(s + τ, s + τ ξ) U(s, s ξ) E f(s ξ) dξ ds e δξ E f(s + τ ξ) f(s ξ) dξ ds e δξ E f(s ξ) dξ ds+6m +1 { +1 3 M e δξ E f(s + τ ξ) f(s ξ) ds dξ { +1 { +1 +3 e δξ E f(s ξ) ds dξ+6m e δξ e δξ E f(s ξ) dξ ds E f(s ξ) ds dξ which implies ha X n ( ) is S -almos periodic. Similarly, assuming ha X is S -almos periodic and using (H.), Theorem 3.1, and Lemma 4.4, given >, one can find l() > such ha any inerval of lengh l() conains a leas τ wih he propery ha for all s, and U( + τ, s + τ) U(, s) e δ ( s) +1 E g(s + τ) g(s) L ds < η() for each R, where η() as. The nex sep consiss in proving he S -almos periodiciy of Y n ( ). Here again, we need o consider wo cases. Case 1: n +1 +1 = TrQ + TrQ E Y n (s + τ) Y n (s) ds E U(s + τ, s + τ ξ) g(s + τ ξ) dw(ξ) U(s, s ξ) g(s ξ) dw(ξ) ds +1 +1 TrQ M +1 U(s + τ, s + τ ξ) E g(s + τ ξ) g(s ξ) L dξ ds U(s + τ, s + τ ξ) U(s, s ξ) E g(s ξ) L dξ ds e δξ E g(s + τ ξ) g(s ξ) L dξ ds EJQTDE, 8 No. 35, p. 11

+ TrQ +1 TrQ M + TrQ Case : n = 1 +1 +1 = +1 n 3 TrQ +3 TrQ +3 TrQ e δξ E g(s ξ) L dξ ds { +1 e δξ E g(s + τ ξ) g(s ξ) L { +1 e δξ E g(s ξ) L E Y 1 (s + τ) Y 1 (s) ds E ds dξ U(s + τ, s + τ ξ) g(s + τ ξ) dw(ξ) U(s, s ξ) g(s ξ) dw(ξ) ds +1 +1 +1 3 TrQ M +1 +3 TrQ +1 +6 TrQ M +1 ds dξ U(s + τ, s + τ ξ) E g(s + τ ξ) g(s ξ) L dξ ds U(s + τ, s + τ ξ) U(s, s ξ) E g(s ξ) L dξ ds U(s + τ, s + τ ξ) U(s, s ξ) E g(s ξ) L dξ ds e δξ E g(s + τ ξ) g(s ξ) L dξ ds e δξ E g(s ξ) L dξ ds e δξ E g(s ξ) L dξ ds { +1 3 TrQ M e δξ E g(s + τ ξ) g(s ξ) L { +1 +3 TrQ e δξ E g(s ξ) L ds dξ { +1 +6 TrQ M e δξ E g(s ξ) L ds dξ, which implies ha Y n ( ) is S -almos periodic. Seing ds dξ X() := U(, σ)f(σ, X(σ)) dσ + U(, σ)g(σ, X(σ)) dw(σ) EJQTDE, 8 No. 35, p. 1

and proceeding as in he proof of Theorem 4.1, one can easily see ha +1 E X(s) (X n (s) + Y n (s)) ds as n uniformly in R, and hence using Lemma 3.6, i follows ha X is a S -almos periodic soluion. Define he nonlinear operaor Γ by ΓX() := U(, σ)f(σ, X(σ)) dσ + U(, σ)g(σ, X(σ)) dw(σ). In view of he above, i is clear ha Γ maps S AP(R; L (P; B)) ino iself. Consequenly, using he Banach fixed-poin principle i follows ha Γ has a unique fixed-poin {X (), R whenever Θ < 1, which in fac is he only S -almos periodic soluion o Eq. (1.). Our second main resul is weaker han Theorem 4.6 alhough we require ha G be bounded in some sense. Theorem 4.7. Under assumpions (H.)-(H.1)-(H.), if we assume ha here exiss L > such ha E G(, Y ) L for all R and Y L (P; H), hen Eq. (1.) L has a unique quadraic-mean almos period mild soluion, which can be explicily expressed as follows: X() = whenever U(, σ)f(σ, X(σ))dσ+ U(, σ)g(σ, X(σ))dW(σ) for each R ( Θ := M K ) δ + K Tr(Q) < 1. δ Proof. We use he same noaions as in he proof of Theorem 4.6. Le us firs show ha X n ( ) is quadraic mean almos periodic upon he S -almos periodiciy of f = F(, X( )). Indeed, assuming ha X is S -almos periodic and using (H.1), Theorem 3.1, and Lemma 4.4, given >, one can find l() > such ha any inerval of lengh l() conains a leas τ wih he propery ha for all s, and U( + τ, s + τ) U(, s) e δ ( s) +1 E f(s + τ) f(s) ds < η() for each R, where η() as. The nex sep consiss in proving he quadraic-mean almos periodiciy of X n ( ). Here again, we need o consider wo cases. Case 1: n. E X n ( + τ) X n () = E U( + τ, + τ ξ) f( + τ ξ) dξ EJQTDE, 8 No. 35, p. 13

+ M + U(, ξ) f(s ξ) dξ U( + τ, + τ ξ) E f( + τ ξ) f( ξ) dξ U( + τ, + τ ξ) U(, ξ) E f( ξ) dξ M + M n Case : n = 1. e δξ E f( + τ ξ) f( ξ) dξ e δξ E f( ξ) dξ e δξ E f( + τ ξ) f( ξ) dξ e δξ E f( ξ) dξ n+1 E X 1 ( + τ) X 1 () = E E f(r + τ) f(r) dr + n U( + τ, + τ ξ) f( + τ ξ) dξ n+1 [ 3 E U( + τ, + τ ξ) f( + τ ξ) f( ξ) dξ [ +3 E U( + τ, + τ ξ) U(, ξ) f( ξ) dξ [ +3 E U( + τ, + τ ξ) U(, ξ) f( ξ) dξ [ 3 M E e δξ f( + τ ξ) f( ξ) dξ [ +3 E e δ ξ f( ξ) dξ] + 1 M E ] [ E f(r) dr U(, ξ) f( ξ) dξ ] ] ] ] e δξ f( ξ) dξ Now, using Cauchy-Schwarz inequaliy, we have ( ) ( ) 3 M e δξ dξ e δξ E f( + τ ξ) f( ξ) dξ ( ) ( ) +3 e δ ξ dξ e δ ξ E f( ξ) dξ EJQTDE, 8 No. 35, p. 14

+1 M ( 3 M 1 ) ( ) e δξ dξ e δξ E f( ξ) dξ E f(r + τ) f(r) dr +3 E f(r) dr + 1 M 1 E f(r) dr, which implies ha X n ( ) quadraic mean almos periodic. Similarly, using (H.), Theorem 3.1, and Lemma 4.4, given >, one can find l() > such ha any inerval of lengh l() conains a leas τ wih he propery ha U( + τ, s + τ) U(, s) e δ ( s) for all s, and +1 E g(s + τ) g(s) L ds < η for each R, where η() as. Moreover, here exiss a posiive consan L > such ha sup σ RE g(σ) L L. The nex sep consiss in proving he quadraic mean almos periodiciy of Y n ( ). Case 1: n E Y n ( + τ) Y n () = E U( + τ, + τ ξ) g(s + τ ξ) dw(ξ) U(, ξ) g( ξ) dw(ξ) TrQ + TrQ U( + τ, + τ ξ) E g( + τ ξ) g( ξ) L dξ TrQ M + TrQ U( + τ, + τ ξ) U(, ξ) E g( ξ) L dξ e δξ E g( + τ ξ) g( ξ) L dξ e δξ E g( ξ) L dξ n+1 n+1 TrQ M E g(r + τ) g(r) L dr + TrQ E g(r) L dr. Case : n = 1 n E Y 1 ( + τ) Y 1 () = E U( + τ, + τ ξ) g(s + τ ξ) dw(ξ) n EJQTDE, 8 No. 35, p. 15

3 TrQ +3 TrQ U(, ξ) g( ξ) dw(ξ) +1 U( + τ, + τ ξ) E g( + τ ξ) g( ξ) L dξ ( + ) U( + τ, + τ ξ) U(, ξ) E g( ξ) L dξ 3 TrQ M e δξ E g( + τ ξ) g( ξ) L dξ +3 TrQ e δξ E g( ξ) L dξ + 6 TrQ M e δξ E g( ξ) L dξ 3 TrQ M E g( + τ ξ) g( ξ) L dξ +3 TrQ E g( ξ) L dξ + 6 TrQ M E g( ξ) L dξ 3 TrQ M 1 E g(r + τ) g(r) L dr +3 TrQ E g(r) L dr + 6 TrQ M E g( ξ) L dξ 1 3TrQM E g(r+τ) g(r) L dr+3trq E g(r) L dr+6trqm L, 1 1 which implies ha Y n ( ) is quadraic-mean almos almos periodic. Moreover, seing X() = U(, σ)f(σ, X(σ)) dσ + U(, σ)g(σ, X(σ)) dw(σ) for each R and proceeding as in he proofs of Theorem 4.1 and Theorem 4.6, one can easily see ha supe X(s) (X n (s) + Y n (s)) as n s R and hence using Lemma 3.7, i follows ha X is a quadraic mean almos periodic soluion o Eq. (1.). In view of he above, he nonlinear operaor Γ as in he proof of Theorem 4.6 maps AP(R; L (P; B)) ino iself. Consequenly, using he Banach fixed-poin principle i follows ha Γ has a unique fixed-poin {X 1 (), R whenever Θ < 1, which in fac is he only quadraic mean almos periodic soluion o Eq. (1.). EJQTDE, 8 No. 35, p. 16

5. Example Le O R n be a bounded subse whose boundary O is boh of class C and locally on one side of O. Of ineres is he following sochasic parabolic parial differenial equaion (5.1) (5.) d X(, x) = A(, x)x(, x)d + F(, X(, x))d + G(, X(, x)) dw(), n n i (x)a ij (, x)d i X(, x) =, R, x O, i,j=1 where d = d d, d i = d, n(x) = (n 1 (x), n (x),..., n n (x)) is he ouer uni normal dx i vecor, he family of operaors A(, x) are formally given by n A(, x) = i,j=1 x i ( a ij (, x) x j ) + c(, x), R, x O, W is a real valued Brownian moion, and a ij, c (i, j = 1,,..., n) saisfy he following condiions: We require he following assumpions: (H.3) The coefficiens (a ij ) i,j=1,...,n are symmeric, ha is, a ij = a ji for all i, j = 1,..., n. Moreover, a ij C µ b (R; L (P; C(O))) C b (R; L (P; C 1 (O))) S AP(R; L (P; L (O))) for all i, j = 1,...n, and c C µ b (R; L (P; L (O))) C b (R; L (P; C(O))) S AP(R; L (P; L 1 (O))) for some µ (1/, 1]. (H.4) There exiss δ > such ha n a ij (, x)η i η j δ η, i,j=1 for all (, x) R O and η R n. Under previous assumpions, he exisence of an evoluion family U(, s) saisfying (H.) is guaraneed, see, eg., [16]. Now le H = L (O) and le H (O) be he Sobolev space of order on O. For each R, define an operaor A() on L (P; H) by n D(A()) = {X L (P, H (O)) : n i ( )a ij (, )d i X(, ) = on O and, i,j=1 A()X = A(, x)x(x), for all X D(A()). Le us menion ha Corollary 5.1 and Corollary 5. are immediae consequences of Theorem 4.6 and Theorem 4.7, respecively. EJQTDE, 8 No. 35, p. 17

Corollary 5.1. Under assumpions (H.1)-(H.)-(H.3)-(H.4), hen Eqns.(5.1)-(5.) has a unique mild soluion, which obviously is S -almos periodic, whenever M is small enough. Similarly, Corollary 5.. Under assumpions (H.1)-(H.)-(H.3)-(H.4), if we suppose ha here exiss L > such ha E G(, Y ) L L for all R and Y L (P; L (O)). Then he sysem Eqns. (5.1)-(5.) has a unique quadraic mean almos periodic, whenever M is small enough. References 1. P. Acquisapace and B. Terreni, A Unified Approach o Absrac Linear Parabolic Equaions, Tend. Sem. Ma. Univ. Padova 78 (1987), pp. 47-17.. P. Bezandry and T. Diagana, Exisence of Almos Periodic Soluions o Some Sochasic Differenial Equaions. Applicable Anal.. 86 (7), no. 7, pp. 819-87. 3. P. Bezandry and T. Diagana, Square-mean almos periodic soluions nonauonomous sochasic differenial equaions. Elecron. J. Diff. Eqns. Vol. 7(7), no. 117, pp. 1-1. 4. C. Corduneanu, Almos Periodic Funcions, nd Ediion. Chelsea-New York, 1989. 5. G. Da Prao and C. Tudor, Periodic and Almos Periodic Soluions for Semilinear Sochasic Evoluion Equaions, Soch. Anal. Appl. 13(1) (1995), pp. 13-33. 6. T. Diagana, Pseudo Almos Periodic Funcions in Banach Spaces. Nova Scieence Publishers, New York, 7. 7. T. Diagana, Sepanov-like Pseudo Almos Periodiciy and Is Applicaions o Some Nonauonmous Differenial Equaions. Nonlinear Anal. (in press). 8. T. Diagana, Sepanov-like Pseudo Almos Periodic Funcions and Their Applicaions o Differenial Equaions, Commun. Mah. Anal. 3(7), no. 1, pp. 9-18. 9. A. Ya. Dorogovsev and O. A. Orega, On he Exisence of Periodic Soluions of a Sochasic Equaion in a Hilber Space. Visnik Kiiv. Univ. Ser. Ma. Mekh. 115 (1988), no. 3, pp. 1-3. 1. A. Ichikawa, Sabiliy of Semilinear Sochasic Evoluion Equaions. J. Mah. Anal. Appl. 9 (198), no. 1, pp. 1-44. 11. A. Lunardi, Analyic Semigroups and Opimal Regulariy in Parabolic Problems, PNLDE Vol. 16, Birkhäauser Verlag, Basel, 1995. 1. D. Kannan and A.T. Bharucha-Reid, On a Sochasic Inegro-differenial Evoluion of Volerra Type. J. Inegral Equaions 1 (1985), pp. 351-379. 13. T. Kawaa, Almos Periodic Weakly Saionary Processes. Saisics and probabiliy: essays in honor of C. R. Rao, pp. 383 396, Norh-Holland, Amserdam-New York, 198. 14. D. Keck and M. McKibben, Funcional Inegro-differenial Sochasic Evoluion Equaions in Hilber Space. J. Appl. Mah. Sochasic Anal. 16, no. (3), pp. 141-161. 15. D. Keck and M. McKibben, Absrac Sochasic Inegro-differenial Delay Equaions. J. Appl. Mah. Soch. Anal. 5, no. 3, 75-35. 16. L. Maniar and R. Schnaubel, Almos Periodiciy of Inhomogeneous Parabolic Evoluion Equaions, Lecure Noes in Pure and Appl. Mah. Vol. 34, Dekker, New York, 3, pp. 99-318. 17. C. Tudor, Almos Periodic Soluions of Affine Sochasic Evoluions Equaions, Sochasics and Sochasics Repors 38 (199), pp. 51-66. EJQTDE, 8 No. 35, p. 18

(Received Augus 1, 8) Deparmen of Mahemaics, Howard Universiy, Washingon, DC 59, USA E-mail address: pbezandry@howard.edu Deparmen of Mahemaics, Howard Universiy, Washingon, DC 59, USA E-mail address: diagana@howard.edu EJQTDE, 8 No. 35, p. 19