SINGULAR PERTURBATION AND STATIONARY SOLUTIONS OF PARABOLIC EQUATIONS IN GAUSS-SOBOLEV SPACES

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1 Communications on Stochastic Analysis Vol. 2, No. 2 (28) Serials Publications SINGULAR PERTURBATION AND STATIONARY SOLUTIONS OF PARABOLIC EQUATIONS IN GAUSS-SOBOLEV SPACES PAO-LIU CHOW* Abstract. The paper is concerne with a class of parabolic equations in a Gauss-Sobolev space containing a small parameter ε >. They egenerate into elliptic equations as ε tens to zero. It is proven that, uner appropriate conitions, the solution to the Cauchy problem for such a parabolic equation converges, as ε, to a it that satisfies the reuce elliptic equation. This singular perturbation problem is shown to be closely relate to the stationary solution of the parabolic problem as t. An application of this result to the asymptotic evaluation of a certain functional integral is given. 1. Introuction The subject of parabolic equations in infinite imensions has been stuie by many authors, (see e.g., the papers [6, 1, 13] an the books [3, 7, 8]). In finite imensions, the singular perturbation problems for partial ifferential equations have been stuie extensively [9]. It is natural to initiate the investigation of such problems in infinite imensions. In a recent paper [4], we treate the singular perturbation problem for the Kolmogorov equation in a Gauss-Sobolev space when the iffusion coefficient approaches zero. In contrast the current paper is concerne with a class of parabolic equations containing a small positive parameter ε such that, as ε, the equations become elliptic. In particular we are intereste in the iting behavior of solutions to the associate Cauchy (initial-value) problems as the parameter tens to zero. As it turns out, such problems are closely relate to the existence question of the stationary solutions to the parabolic Cauchy problems as the time t. In both of the linear an the semilinear cases, the existence of strong solutions to the associate Cauchy problems was prove in a recent paper [2]. Here the main technical problem is to show that such strong solutions will converge in a proper sense to the reuce elliptic equations. Moreover we will show the connection of the singular perturbation problems to that of asymptotic solutions of the parabolic equations as t. To be specific, the paper is organize as follows. In Section 2, we recall some basic results in the Gauss-Sobolev spaces to be neee in the subsequent sections. Section 3 pertains to the strong solutions of parabolic equations in Gauss-Sobolev spaces. Some a priori estimates an inequalities for the linear equations are given 2 Mathematics Subject Classification. Primary 6H; Seconary 6G, 35K55, 35K99, 93E. Key wors an phrases. Singular perturbation, stationary solution, parabolic equation in infinite imensions, Gauss-Sobolev space, mil an strong solutions, Feynman-Kac formula. *This research was supporte in part by a grant from the Acaemy of Applie Sciences. 289

2 29 PAO-LIU CHOW* by Lemmas 3.1 to 3.3, an the existence of strong solutions to the semilinear parabolic equations is ensure by Theorem 3.4. Then a singular perturbation problem an the relate stationary solutions for linear parabolic equation are consiere in Section 4. The results on the convergence of solutions are presente in Theorem 4.1 an Theorem 4.2, respectively. In Section 5 the corresponing results for the semilinear parabolic equations are given in Theorem 5.4 an Theorem 5.5. As an application, Theorem 5.5 is applie to the asymptotic evaluation of a certain functional integral in Section Preinaries Let H be a real separable Hilbert space with inner prouct (, ) an norm an let V H be a Hilbert subspace with norm. Denote the ual space of V by V an their uality pairing by,. Assume that the inclusions V H = H V are ense an continuous [11]. Suppose that A : V V is a continuous close linear operator with omain D(A) ense in H, an W t is a H-value Wiener process with the covariance operator R. Consier the linear stochastic equation in a istributional sense: u t = Au t t + W t, t, u = h H. (2.1) Assume that the following conitions (A.1) (A.3) hol: (A.1) Let A : V V be a self-ajoint, coercive operator such that Av, v β v 2, for some β >, an ( A) has positive eigenvalues < γ 1 γ 2 γ n, counting the finite multiplicity, with γ n as n. The corresponing orthonormal set of eigenfunctions {e n } is complete. (A.2) The resolvent operator R λ (A) an covariance operator R commute, so that R λ (A)R = R R λ (A), where R λ (A) = (λi A) 1, λ, with I being the ientity operator in H. (A.3) The covariance operator R : H H is a self-ajoint operator with a finite trace such that T r R <. Then, by a irect calculation or applying Theorem 4.1 in [5] for invariant measures, we have the following theorem. Theorem 2.1. Uner conitions (A.1) (A.3), the stochastic equation (2.1) has a unique invariant measure µ on H, which is a centere Gaussian measure supporte in V with covariance operator Γ = 1 2 A 1 R. Remark 2.2. Let e ta enote the semigroup of operators generate by A. Without conition (A.2), the covariance operator of the invariant measure µ is given by R = e ta Re ta t, which cannot be evaluate in a close form. Though a L 2 (µ) theory can be evelope in the subsequent analysis, one nees to impose some other conitions which are not easily verifiable.

3 PARABOLIC EQUATIONS IN GAUSS-SOBELEV SPACES 291 Let H = L 2 (H, µ) with norm efine by Φ = { Φ(v) 2 µ(v)} 1/2, an the inner prouct [, ] given by [Θ, Φ] = Θ(v)Φ(v)µ(v), for Θ, Φ H. H H Let n = (n 1, n 2,, n k, ), where n k Z +, the set of nonnegative integers, an let Z = {n : n = n = k=1 n k < }, so that n k = except for a finite number of n k s. Let h m(r) be the stanar one-imensional Hermite polynomial of egree m. For v H, efine a Hermite (polynomial) functional of egree n by H n (v) = h nk [l k (v)], k=1 where we set l k (v) = (v, Γ 1/2 e k ) an Γ 1/2 enotes a pseuo-inverse, by restricting it to the range of Γ 1/2. For a smooth functional Φ on H, let DΦ an D 2 Φ enote the Fréchet erivatives of the first an secon orers, respectively. The ifferential operator AΦ(v) = 1 2 T r[rd2 Φ(v)] + Av, DΦ(v) (2.2) is well efine for a polynomial functional Φ with DΦ(v) lies in the omain D(A) of A. It was shown in [1] that the following hols. Proposition 2.3. The set of all Hermite functionals {H n : n Z} forms a complete orthonormal system in H. Moreover we have where λ n = n γ = k=1 n kγ k. AH n (v) = λ n H n (v), n Z, We now introuce the Gauss-Sobolev spaces. For Φ H, by Proposition 2.2, it can be expresse as Φ = n Z Φ n H n, where Φ n = [Φ, H n ] an Φ 2 = n Φ n 2 <. Let H m enote the Gauss- Sobolev space of orer m efine as for any integer m, where the norm H m = {Φ H : Φ m < } Φ m = (I A) m/2 Φ = { n (1 + λ n ) m Φ n 2 } 1/2, (2.3) with I being the ientity operator in H = H. For m 1, the ual space H m of H m is given by H m, an the uality pairing between them will be enote by, m with, 1 =,. Clearly, the sequence of norms { Φ m } is increasing, that is, Φ m < Φ m+1,

4 292 PAO-LIU CHOW* for any integer m, an, by ientify H with its ual H, we have H m H m 1 H 1 H H 1 H m+1 H m, for m 1, an the inclusions are ense an continuous. Of course the spaces H m can be efine for any real number m, but they are not neee in this paper. Owing to the use of the invariant measure µ, it is possible to evelop a L 2 -theory of infinite-imensional parabolic an elliptic equations connecte to stochastic PDEs similar to the finite-imensional ones. In particular the following properties of A are crucial in the subsequent analysis. So far the ifferential operator A given by (2.2) is efine only in the set of Hermite polynomial functionals. In fact it can be extene to a self-ajoint linear operator in H. To this en, let P N be a projection operator in H onto its subspace spanne by the Hermite polynomial functionals of egree N an efine A N = P N A. Then the following theorem hols (Theorem 3.1, [1]). Theorem 2.4. The sequence {A N } converges strongly to a linear symmetric operator A : H 2 H, so that, for Φ, Ψ H 2, the following ientity hols: (AΦ, Ψ) µ = µ = H H(AΨ)Φ 1 (RDΦ, DΨ) µ. (2.4) 2 H Moreover, A has a self-ajoint extension, still enote by A, with omain ense in H. For Φ C 2 b (H) being a boune C2 -continuous functional on H, let P t enote the transition operator efine by [P t Φ](v) = E{Φ(u t ) u = v} = Ψ t (v). Then, for v D(A), Ψ t (v) satisfies the Kolmogorov equation in the classical sense: t Ψ t(v) = AΨ t (v), t >, (2.5) Ψ (v) = Φ(v). In fact the transition operator P t can be extene to be a boune linear operator on H an it is possible to efine the equation (2.5) for µ-a.e. v H. Theorem 2.5. Uner conitions (A.1) (A.3), the transition operator P t is efine on H for all t an {P t : t } forms a strongly continuous semigroup of linear contraction operators on H with the infinitesimal generator à = A in H Basic Estimates an Existence Theorem Consier the Cauchy problem for the linear parabolic equation: t Ψ t(v) = A α Ψ t (v) + Q t (v), µ a.e. v H, t >, Ψ (v) = Φ(v), (3.1)

5 PARABOLIC EQUATIONS IN GAUSS-SOBELEV SPACES 293 for Q L 2 ((, ); H) an Φ H, where α is a positive parameter an A α = (A αi). (3.2) It was shown in [1] that the solution of (3.1) has the regularity: Ψ C([, T ]; H) L 2 ((, T ); H 1 ). Let G t enote the Green s operator associate with equation (3.1) given by G t = e αt P t. In view of Theorem 2.4, the solution of (3.1) can be expresse as Ψ t (v) = [G t Φ](v) + [G t s Q s ](v)s. (3.3) In what follows, we assume that conitions (A.1) (A.3) are satisfie. Then we have the following technical lemmas. Lemma 3.1. The Green s operator G t : H H is linear an boune such that, for Φ H an Q L 2 ((, ); H), we have an G t s Q s s 2 1 α G t Φ < e αt Φ, (3.4) e α(t s) Q s 2 s, t. (3.5) Proof. For Φ H, in terms of the Hermite functionals, we can write It follows that G t Φ = n e β nt [Φ, H n ]H n, with β n = α + λ n, (3.6) G t Φ 2 = n e 2βnt [Φ, H n ] 2 e 2αt n [Φ, H n ] 2 = e 2αt Φ 2, which verifies (3.4). Similar to (3.6), for Q L 2 ((, T ); H), we have G t s Q s = n e β n(t s) [Q s, H n ]H n, with β n = α + λ n, so that G t s Q s s 2 = n where we let Q n s = [Q s, H n ]. Since { e β n(t s) Q n s s} 2 1 { β n t 1 α e β n(t s) Q n s s} 2, (3.7) e β n(t s) Q n s 2 s, e α(t s) Q n s 2 s,

6 294 PAO-LIU CHOW* the inequality (3.5) can be obtaine from (3.7) as follows G t s Q s s 2 1 α = 1 α n e α(t s) Q n s 2 s e α(t s) Q s 2 s, where the interchange of the summation an integration can be justifie ue to the monotone convergence. Lemma 3.2. For Q L 2 ((, T ); H m 1 ), the following inequality hols: G t s Q s s 2 m 1 e α(t s) Q s 2 m 1 s, (3.8) α 1 where α 1 = min{α, 1} an =. Proof. By efinition (2.3) an an expansion in terms of the Hermite functionals, similar to (3.7), we can obtain G t s Q s s 2 m = n where we let Q n s = [Q s, H n ]. Since { the equation (3.9) yiels (1 + α n ) m { e β n(t s) Q n s s} 2 1 β n e βn(t s) Q n s s} 2, (3.9) e β n(t s) Q n s 2 s, G t s Q s s 2 m (1 + λ n ) m e βn(t s) Q n s 2 s β n n 1 e βn(t s) Q s 2 α m 1s, 1 (3.1) after changing the orer of the summation an integration. The inequality (3.8) now follows from (3.9). Lemma 3.3. The linear operator A α : H 2 H, as efine by (3.2), has a boune inverse A 1 α such that an A 1 α Θ 2 1 α 2 Θ 2, for Θ H, (3.11) A 1 α Φ 2 1 αα 1 Φ 2 1, for α 1 = min{α, 1}, Φ H 1. (3.12) Further, for any Θ H, the following equation hols G s Θ s = A 1 α Θ. (3.13)

7 PARABOLIC EQUATIONS IN GAUSS-SOBELEV SPACES 295 Proof. The linear operator is clearly invertible since, by equation (2.4), A α Φ, Φ α 1 Φ 2 1 for Φ H 2. The inequalities (3.11) an (3.12) can be verifie similarly by expaning Φ in terms of the Hermite functionals. So, for brevity, we will only show the latter one. We can get or A 1 α Φ 2 = (α + λ n ) 2 Φ 2 n n 1 (1 + λ n ) 1 Φ 2 αα n, 1 n A 1 α Φ 2 1 Φ 2 αα 1. 1 Since A α generates a contraction semigroup G t in H, the equation (3.13) is known to be true in the semigroup theory [12]. Now consier the Cauchy problem for the nonlinear evolution equation in H in a istributional sense: t Ψ t = A α Ψ t + B(Ψ t ) + Q t, t >, Ψ = Θ, (3.14) where B : H 1 H is boune an continuous. Then it follows from Theorem 4.2 [2] that the following theorem hols. Theorem 3.4. Suppose that A α is given as before, an B satisfies the following conitions: (1) B(Φ) 2 C 1 {1 + Φ 2 + R 1/2 DΦ 2 }, (2) B(Φ) B(Φ ) 2 C 2 { Φ Φ 2 + R 1/2 D(Φ Φ ) 2 }, for some constants C 1, C 2 > an for any Φ, Φ H 1. Then, for Θ H an Q L 2 ((, ); H), the Cauchy problem (3.14) has a unique strong solution Ψ C([, T ]; H) L 2 ((, T ); H 1 ), for any T >, so that the following equation hols for every t [, T ], Φ H 1 : [Ψ t, Φ] = [Θ, Φ] + A α Ψ s, Φ s + [B(Ψ s ), Φ] s + [Q s, Φ] s. Lemma 3.5. Let F t be a boune continuous H-value function on [, ). Suppose there is F H such that Then, for any α >, we have F (t) F =. e α(t s) F (s) F 2 s =. (3.15)

8 296 PAO-LIU CHOW* Proof. Let f t = F (t) F 2. Then f t is a boune continuous real-value function on [, ) with f t an f t as t. Thus the equation can be verifie easily. e α(t s) f s s =, 4. Linear Parabolic Equations Consier the singularly perturbe form for the linear parabolic equation (3.1), which will be regare as an evolution equation in H (in a istributional sense): ε t Ψε t = A α Ψ ε t + Q ε t, t >, Ψ ε = Θ, (4.1) where ε (, 1] is a small parameter an Q ε t is a given H-value function epening on ε. Therefore the solution, enote by Ψ ε t, also epens on ε. Suppose that Q ε t is boune an continuous in t [, ) for each ε (, 1], an there exists Q H such that ε sup Q ε t Q =, (4.2) t [δ,t ] for any finite interval [δ, T ] (, ). For a fixe t > we are intereste in the asymptotic behavior of the solution as ε. Suppose that, as ε, the solution Ψ ε t converges in H 2 to a it Ψ for t >. Then formally, in view of conition (4.2), the equation (4.1) reuces to the elliptic equation for Ψ: A α Ψ = Q,. (4.3) This result can be mae precise by the following theorem. Theorem 4.1. Let Θ H an let Q ε t be a H value function on [, ) such that it is boune an continuous for all ε (, 1]. Suppose that conitions (A.1) (A.3) an conition (4.2) hol. Then, for any t >, there exists Ψ H 2 such that ε Ψε t Ψ =, (4.4) an the it is uniform in any finite interval [a, b] [δ, ). Moreover the it Ψ satisfies the elliptic equation (4.3). Proof. In view of equation (3.3), the solution of equation (4.1) can be written as Ψ ε t = [G ε t Θ] + 1 ε where we set G ε t = G t/ε. By Lemma (3.1), we have, for Θ H, so that G ε t Θ e αt/ε Θ [G ε t sq ε s]s, (4.5) ε Gε t Θ =. (4.6)

9 PARABOLIC EQUATIONS IN GAUSS-SOBELEV SPACES 297 Let an V ε t = 1 ε By (3.13) in Lemma 3.3, we get U ε t = 1 ε G ε t sq s = G ε t sq ε s s, (4.7) /ε G s Q s. (4.8) V t ε = G s Q s = A 1 α Q, Q H. (4.9) ε By invoking Lemma (3.2), we can euce from (4.7) an (4.8) that = 1 ε = Ut ε Vt ε 1 ε εt T T e α(t s)/ε Q ε s Q s e α(t s)/ε Q ε s Q s + 1 ε e ατ Q ε t ετ Q τ + /ε e ατ Q ε t ετ Q τ + M α e αt, T εt e α(t s)/ε Q ε s Q s e α(t s) Q ε εs Q s where M = sup ( Q ε t + Q ). t,<ε 1 For any η > an t [a, b], choose T > b such that M α e αt < η/2. For this fixe T, we let ε be so small that (a εt ) δ. Then we have sup a t b T e ατ Q ε t ετ Q τ 1 α sup Q ε t Q, δ t T (4.1) which, in view of conition (4.2), can be mae less than η/2 for a sufficiently small ε. Hence it follows from (4.1) an the above estimates that, for any η >, there exists ε > such that sup Ut ε Vt ε < η, a t b for < ε < ε. That is, by noting (4.9), U t ε = Vt ε = A 1 α Q, (4.11) ε ε for any t [a, b]. By taking (4.6), (4.7) an (4.11) into account, we euce from (4.5) that, for t >, the it (4.2) hols true an Ψ ε t converges to Ψ = A 1 α Q H 2 as ε so that it satisfies equation (4.3) as asserte. Now let us consier the stationary solution of the following Cauchy problem as t : t Φ t = A α Φ t + Q t, Φ = Θ. (4.12) As it turns out, this can be reformulate as a singular perturbation problem so that the above theorem is applicable. In fact we have the following theorem.

10 298 PAO-LIU CHOW* Theorem 4.2. Let Q t H be boune an continuous on [, ) such that the following conition hols in H Q t = Q. (4.13) Then uner conitions (A.1) (A.3), for any Θ H, there exists the it an Ψ satisfies the elliptic equation (4.3). Φ t = Ψ, (4.14). Proof. Let us re-scale the time t by letting τ = εt. We set Φ t = Φ τ/ε = Ψ ε. τ an Q t = Q τ/ε = Q ε τ. Then, after the change of time, the equation (4.12) yiels equation (4.1). Notice that, for a fixe τ, the it ε implies that t. Therefore it is enough to show that, for a fixe τ >, ε Ψε τ = Ψ. By efinition an conition (4.13), we have ε sup Q ε τ Q = a τ b ε sup a τ b Q τ/ε Q = Q t Q =. This shows that the conition (4.2) for Theorem 4.1 is fulfille. So we can apply this theorem to conclue that the it (4.14) exists an it satisfies equation (4.3). Remark 4.3. Two special cases for the function Q ε t are of special interest. For Q ε t = Q t/ε, the conition (4.2) yiels while, for Q ε t = Q εt, Q = ε Q ε t Q = ε Q ε t = Q t = Q, = t Q t = Q. 5. Semilinear Parabolic Equations Now consier the Cauchy problem for the nonlinear parabolic equation: ε t Ψε t (v) = A α Ψ ε t (v) + F(v, Ψ ε t, DΨ ε t ) + Q ε t (v), < t < T, Ψ (v) = Φ(v), (5.1) where, uner suitable conitions, the nonlinear term F : V R H H can be efine µ-a.e.. Similar to the linear case (4.1), we regar (5.1) as the following evolution equation in H: ε t Ψε t = A α Ψ ε t + B(Ψ ε t ) + Q ε t, t >, Ψ ε = Θ, (5.2) where we let B(φ) = F(, φ, Dφ). As before we are intereste in proving the convergence of the solution Ψ ε t to a it as ε an the relate stationary

11 PARABOLIC EQUATIONS IN GAUSS-SOBELEV SPACES 299 solution. Unlike the linear case, uner certain conitions on B an Q ε t, we can only show that the solution Ψ ε t converges to a it Ψ in a weaker sense an it is a mil solution of the stationary equation: A α Ψ + B(Ψ) + Q =. (5.3) To procee we rewrite the equation (5.3) in the integral form: Ψ t = G t Θ + G t s B(Ψ ε s) s + In particular we impose the following conitions: G t s Q s s. (5.4) (B.1) Let B( ) : H 1 H a continuous mapping with B() =. Suppose there exist positive constants b 1, b 2, such that b 2 < b 1 < α an B(φ) B(ψ) 2 b 1 φ ψ 2 + b 2 R 1 2 D(φ ψ), for any φ, ψ H 1. (B.2) The map B can be extene to be a continuous operator from H into H 1 such that, for some constant κ >, the following hols B(φ) B(ψ) 1 κ φ ψ, for any φ, ψ H. (B.3) Let Q ε t = Q t/ε such that Q t is a boune continuous H-value function on [, ) with the it Q t = Q. Similar to the linear case, by a simple change of the time-scale, we efine Φ t = Ψ ε εt. Then it follows from equation (5.2) an conition (B.3), that Φ t satisfies the equation: t Φ t = A α Φ t + B(Φ t ) + Q t, t >, Φ = Θ,. (5.5) To show Ψ ε t Ψ as ε, it suffices to prove that Φ t = Ψ. To this en we shall first present some lemmas. In what follows, the conitions (A.1) (A.3) are always assume to be true without further mentioning. Lemma 5.1. Suppose the conitions (B.1) to (B.3) hol. Then, for Θ H, the Cauchy problem (5.5) has a strong solution Φ C([, T ]; H) L 2 ((, T ); H 1 ). Moreover the following energy equation hols t Φ t 2 = R 1/2 DΦ t 2 2α Φ t 2 +2[B(Φ t ), Φ t ] + 2 [Q t, Φ t ], (5.6) Φ 2 = Θ 2.

12 3 PAO-LIU CHOW* Proof. Uner the conitions (B.1) (B.3), we can apply Theorem 3.4 to assert the existence of a unique strong solution with the inicate regularity property. So we have Φ t H 1 an t Φ t H 1. The equation (5.6) can be erive easily from (5.5). We have t Φ t, Φ t = A α Φ t, Φ t + [B(Φ t ), Φ t ] + [Φ t, Q t ], which yiels equation (5.6) with the ai of the integration by parts formula given in Theorem 2.3. Lemma 5.2. As in Lemma 5.1, uner conitions (B.1) (B.3), the the following inequality hols where γ = (α b 1 ). Φ t 2 Θ 2 e γt + 1 γ e γ(t s) Q s 2 s, (5.7) Proof. For any β, γ >, we can euce from equation (5.6) that t Φ t 2 R 1/2 DΦ t 2 2α Φ t 2 + (β Φ t β B(Φ t) 2 ) +( γ Φ t γ Q t 2 ), which, by invoking conition (B.1), yiels the following t Φ t 2 (1 b 2 β ) R1/2 DΦ t 2 (2α β b 1 β γ) Φ t γ Q t 2. (5.8) Now we set β = b 1 an γ = α b 1 in (5.8) to get t Φ t 2 (1 b 2 b1 ) R 1/2 DΦ t 2 (α b 1 ) Φ t γ Q t 2 γ Φ t γ Q t 2, which implies the esire result (5.7). To show the convergence of Φ t as t, we shall aopt a backwar asymptotic approach which was use effectively in the stochastic case (Lemma , [8]). To this en, by extening the initial time backwar in (5.5), consier the Cauchy problem: t Φ t(s) = A α Φ t (s) + B(Φ t (s)) + Q t, t > s, (5.9) Φ s (s) = Θ, for s, where the extene function Q t is efine by Q t = Q t for t, an Q t = Q t for t <. Then we have Φ t = Φ (t). Hence Φ (t) Ψ implies Φ t Ψ as t.

13 PARABOLIC EQUATIONS IN GAUSS-SOBELEV SPACES 31 Theorem 5.3. Suppose the conitions (B.1) (B.3) are satisfie. For Θ H, the solution Φ t of the Cauchy problem (5.9) converges to a it Ψ H inepenent of Θ as t. Proof. Consier the Cauchy problem (5.9). In view of (5.7) in Lemma 5.2, we shift the initial time to s to get Φ t (s) 2 Θ 2 e γ(t+s) + 1 α Θ 2 + M γ e γ(t τ) Q τ 2 τ s e γ(t τ) τ Θ 2 + M (5.1) s αγ, where M = sup Q t 2. It follows from (5.1) that, for any t, t sup Φ t (s) 2 <. (5.11) s> Now, for r > s, let Φ t (r) be the solution of (5.9) with s replace by r an efine Then it satisfies the equation Φ t (s, r) = Φ t (s) Φ t (r). t Φ t(s, r) = A α Φ t (s, r) + [B(Φ t (s)) B(Φ t (r))] t > s, Φ s (s, r) = Θ Φ s (r), for s. (5.12) As in Lemma 5.1, we can obtain from (5.12) the following equation t Φ t(s, r) 2 = R 1/2 DΦ t (s, r) 2 2α Φ t (s, r) 2 +2[B(Φ t (s)) B(Φ t (r)), Φ t (s) Φ t (r)], (5.13) Φ s (s, r) 2 = Θ Φ s (r) 2. In view of conition (B.1), similar to (5.8), equation (5.13) yiels the following inequality t Φ t(s, r) 2 (1 b 2 b1 ) R 1/2 DΦ t (s, r) 2 (α b 1 ) Φ t (s, r) 2 which implies γ Φ t 2, Φ t (s, r) 2 Θ Φ s (r) 2 e γ(t+s) 2 ( Θ 2 + Φ s (r) 2 )e γ(t+s). Therefore, in view of (5.11), we have, for any t, r > s, Φ t(s, r) 2 = s Φ t(s) Φ t (r) 2 =. (5.14) r,s

14 32 PAO-LIU CHOW* In particular we set t = in (5.11) an (5.14) to conclue that the family {Φ s = Φ (s) : t > } yiels a Cauchy sequence in H so that Φ t = Ψ. (5.15) To show the it Ψ is inepenent of the initial state Θ, suppose that Φ = Θ Θ an enote the corresponing solution of the Cauchy problem (5.5) by Φ t. Then, similar to the estimate in Lemma 5.2, we can show that the ifference (Φ t Φ t) has the following boun: (Φ t Φ t) 2 C Θ Θ 2 e γt for some C >, which implies that Ψ = Ψ an the it is inepenent of the initial state as claime. Returning to the singular perturbation problem (5.1) or (5.2), the following theorem assert that, as ε, the solution Ψ ε t converges to the mil solution of the stationary equation. Theorem 5.4. Suppose that conitions (B.1) (B.3) hol. Then, for any t > an Θ H, the solution Ψ ε t of the Cauchy problem (5.1) converges to the it: ε Ψε t = Ψ, (5.16) an Ψ is the mil solution of (5.3) which satisfies the following equation:. Ψ = A 1 α [B(Ψ) + Q], (5.17) where A 1 α B( ) = A 1 α B( ) is a boune linear operator on H. Moreover the solution of equation (5.17) is unique if, in conition (B.2), κ < αα 1 with α 1 = min{α, 1}. Proof. It follows from Theorem 5.3 that, by re-scaling the time, we can conclue that the it given by (5.16) exists. To verify the equation (5.17), again we work with the re-scale equation (5.5) an show that the it (5.15) satisfies (5.17). To this en we rewrite (5.5) as the integral equation (5.6). We claim that equation (5.17) follows from equation (5.6) by taking the it term-wise as t. Clearly we have Φ t Ψ an G t Θ by Lemma 3.1. Similar to the proof of equation (4.11), we can show that G t s Q s s = A 1 α Q. So it remains to prove that A 1 B is boune an α G t s B(Φ s ) s = A 1 α B(Ψ), (5.18) The bouneness follows from Lemma 3.3 an conition (B.2) as follows A 1 α B(Ψ) 1 αα1 B(Φ) 1 κ αα1 Φ.

15 PARABOLIC EQUATIONS IN GAUSS-SOBELEV SPACES 33 To show (5.18), we rewrite it as G t s B(Ψ s ) s = G t s B(Ψ) + Similar to (4.9), by means of Lemma 3.3, we get G t s [B(Φ s ) B(Ψ)] s. (5.19) G t s B(Ψ) s = A 1 α B(Ψ). (5.2) Now, by making use Lemma 3.2 an conition (B.2), we can obtain 1 α κ α G t s [B(Φ s ) B(Ψ)] 2 s t e α(t s) B(Φ s ) B(Ψ) 2 1 s e α(t s) Φ s Ψ 2 s, (5.21) which, by noting Lemma 3.5, converges to zero as t. In view of equations (5.19) (5.21), the result (5.18) is verifie. Therefore the equations (5.16) an (5.17) hol true. To show the uniqueness of solution to equation (5.17), let Φ be another solution. Then we have (Φ Ψ) = A 1 α [ B(Φ) B(Ψ)], from which, with the ai of Lemma 3.3 an conition (B.2), we can euce that Φ Ψ = A 1 α [ B(Φ) B(Ψ)] 1 αα1 B(Φ) B(Ψ) 1 κ αα1 Φ Ψ. Since κ/ αα 1 < 1 by assumption, the above inequality implies Φ Ψ = an hence the uniqueness. As seen from the above theorem, ue to the possible epenence of the nonlinear function B(Φ) on the graient DΦ, we have shown that the it function Ψ is only a mil solution of the stationary equation. However, if it is inepenent of DΦ, the it Ψ may become a classical solution. To show this possibility, instea of conitions (B.1) an (B.2), we assume that (B.4) Let B( ) : H H be a continuous mapping with B() =. Suppose there exists a positive constant b 1 such that b 1 < α 2 an for any φ, ψ H. B(φ) B(ψ) 2 b 1 φ ψ 2, Then it follows from Theorem 5.4 that the following result hols.

16 34 PAO-LIU CHOW* Theorem 5.5. Suppose that conitions (B.3) an (B.4) hol. Then, for any t > an Θ H, the solution Ψ ε t of the Cauchy problem (5.1) converges to a it Ψ H 2 as ε an Ψ satisfies the following equation: A α Ψ + B(Ψ) = Q. (5.22) Furthermore the solution of the above equation is unique. Proof. Given conition (B.4), following a similar line of proof for Theorem 5.4, we can show that the solution Ψ ε t of the problem (5.1) converges to Ψ H as ε. Moreover it is a mil solution that satisfies (5.17). Since B(ψ) H, Q H an : H H 2, the equation (5.17) shows Ψ H 2. So we can apply the operator A α to (5.17) to obtain the esire equation (5.22). For the uniqueness question, suppose Φ is another solution of equation (5.22). Then the ifference Θ = Ψ Φ satisfies the equation: A 1 α It follows that A α Θ + [B(Θ + Φ) B(Φ)] =. A α Θ, Θ = [ B(Θ + Φ) B(Φ), Θ]. (5.23) Since A α Θ, Θ α Θ 2, we can euce from (5.23) an conition (B.4) that Θ 2 1 B(Θ + Φ) B(Φ) Θ α b1 α Θ 2 < Θ 2, which implies Θ = Ψ Φ = an hence the uniqueness. Consier the linear Cauchy problem: 6. Application t Φ t(v) = (A α) Φ t (v) + U(v)Φ t (v) + Q t (v), t >, Φ (v) = Θ(v), (6.1) where, as before, AΨ(v) = 1 2 T r[rd2 Ψ(v)] + Av, DΨ(v), an U : H R is a boune continuous function. This equation is a special case of equation (5.5) with B(Φ) = U Φ. Assume conition (B.3) an sup U(v) 2 = b 1 < α 2. (6.2) v H Then it is easy to check that conition (B.4) hols. By applying Theorem 5.5, we can conclue that there is Ψ H 2 such that Φ t(v) = Ψ(v), an Ψ is the unique strong solution of the stationary equation: A α Ψ(v) + U(v) Ψ(v) = Q(v). (6.3)

17 PARABOLIC EQUATIONS IN GAUSS-SOBELEV SPACES 35 On the other han, the solution of the Cauchy problem (6.1) has a probabilistic representation. To this en let u t (s, v) be the solution of the stochastic equation: u t = Au t t + W t, t s, u s = v, (6.4) an enote its solution by u t (s, v). Assume that Θ C 2 b (H). Then, for Q, the corresponing homogeneous equation (6.1) can be represente by the Feynman- Kac formula [7] in terms of the solution of equation (6.4) as follows: Φ t (v) = E { Θ(u t (, v)) exp[ αt + an equation (6.3) becomes U(u t (s, v)) s ] } (6.5) A α Ψ(v) + U(v) Ψ(v) =. (6.6) Since Ψ is the unique solution of equation (6.6), it follows from (6.5) that E { Θ(u t(, v)) exp[ αt + U(u t (s, v)) ] } s =. (6.7) For Q not being ientically zero, as in the finite-imensional case, the solution of equation (6.1) can be represente as Φ t (v) = E { Θ(u t (, v)) exp[ αt + +E{ Q(u t (r, s), s) exp[ α(t s) + U(u t (s, v)) ] } s s U(u t (r, v)) r ] } s. Therefore we can euce from (6.7) an (6.8) that the following it exists, Ψ(v) = E{ Q(u t (r, s), s) exp[ α(t s) + an Ψ is the unique solution of the stationary equation (6.3). s (6.8) U(u t (r, v)) r ] } s, (6.9) Remark 6.1. The right-han sie of equation (6.9) can be regare as the asymptotic evaluation of a functional integral by means of the ifferential equation (6.3). This equation can be solve by the metho of iterations. To procee let it be rewritten as Ψ = K α Ψ + Ψ, where Ψ = A 1 α Q an K α Ψ = A 1 α UΨ. By setting Ψ n = K α Ψ n 1 + Ψ, for n = 1, 2,, we can obtain n Ψ n = (K α ) j Ψ, j= where (K α ) = I, (K α ) 1 = K α an (K α ) j = K α (K α ) j 1 for j 2. By (3.11) in Lemma 3.3 an conition (6.2), we can show that K α Ψ b 1 /α Ψ. Thus

18 36 PAO-LIU CHOW* the linear operator K α : H H is boune. Since b 1 /α < 1, the sequence Ψ n converges in H to the solution of (6.3) given by Ψ = (K α ) j Ψ. j= Acknowlegement. The author wishes to thank the referee for the careful review of the original manuscript an pointing out several typographical errors for corrections. References 1. Chow, P. L.: Infinte-imensional Kolmogorov equations in Gauss-Sobolev spaces; Stoch. Analy. an Applic. 14 (1996) Chow, P. L.: Infinte-imensional parabolic equations in Gauss-Sobolev spaces; Comm. Stoch. Analy. 1 (27) Chow, P. L.: Stochastic Partial Differential Equations. Chapman & Hall/CRC, Boca Raton- Lonon-New York, Chow, P. L.: Singular Perturbation of Kolmogorov equations in Gauss-Sobolev spaces; Stoch. Analy. an Applic. (To appear). 5. Chow, P. L., Khasminskii, R.Z.: Stationary solutions of nonlinear stochastic evolution equations; Stoch. Analy. an Applic. 15, (1997) Daleskii, Yu. L.: Infinite imensional elliptic operators an parabolic equations connecte with them; Russian Math. Rev. 22 (1967) Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambrige University Press, Cambrige, Da Prato, G., Zabczyk, J.: Secon Orer Partial Differential Equations in Hilbert Spaces. Cambrige University Press, Cambrige, Eckhaus, E.: Asymptotic Analysis an Singular Perturbations. North-Hollan, Amsteram, Kuo, H.-H., Piech, M. A.: Stochastic integrals an parabolic equations in abstract Wiener space; Bull. Amer. Math. Soc. 79 (1973) Lions, J. L., Mangenes, E.: Nonhomogeneous Bounary-value Problems an Applications. Springer-Verlag, New York, Pazy, A.: Semigroups of Linear Operators an Applications to Partial Differential Differential Equations. Spriger-Velag, New York, Piech, M. A.: The Ornstein-Uhleneck semigroup in an infinite imensional L 2 setting; J. Funct. Analy. 18 (1975) P.L. Chow: Department of Mathematics, Wayne State University, Detroit, Michigan 4822, USA aress: plchow@math.wayne.eu