Existence and Uniqueness of Solutions for Non-Linear Stochastic Partial Differential Equations

Exis ence and uniqueness o solu ions o Non–Linea S ochas ic Pa ial Di e en ial Equa ions TOM´ AS CARABALLO GARRIDO Depa amen o de An´alisis Ma em´a ico, Uni e sidad de Se illa, Apa ado de Co eos 1.160. 41080-SEVILLA, Spain. ABSTRACT: We s a e some esul s on exis ence and uniqueness o he solu ion o non linea s ochas ic PDEs wi h de ia ing a gumen s. In ac , we conside he equa ion dx( )+(A( , x( )) + B( , x(τ( ))) + ( )) d = (C( , x(ρ( ))) + g( )) dw , whe e A( , .), B( , .)and C( , .)a e sui able amilies o non linea ope a o s in Hilbe spaces, w is a Hilbe alued Wiene p ocess, and τ , ρ a e unc ions o delay. I Asa is ies a coe ci i y condi ion and a mono onici y hypo hesis, and i B , C a e Lipschi z con inuous, we p o e ha he e exis s a unique solu ion o an ini ial alue p oblem o he p eceden equa ion. Some examples o in e es o he applica ions a e gi en o illus a e he esul s. 1. In oduc ion The s udy o s ochas ic PDE’s has g ea ly de eloped o e he las yea s. S ochas ic PDE’s a e used in modelling physical phenomena [5], popula ion biology [7], il e ing [13], e c. The main aim o his pape is o s udy his ype o equa ion wi h delay e ms. In ac , we p o e exis ence and uniqueness o solu ion (in I ˆo’s sense) o a a he gene al ype o s ochas ic PDEs wi h non linea mono one ope a o s and wi h delays. We deal wi h he ollowing s ochas ic pa abolic equa ion: (1) ½dx( )+[A( , x( )) + B( , x(τ( ))) + ( )] d = [C( , x( ho( ))) + g( )] dw , > 0 x(0) = x0, whe e A( , .), B( , .), C( , .) a e amilies o ope a o s in Hilbe spaces, non linea e en ually, and sa is ying a mono onici y condi ion; w is a Hilbe alued Wiene p ocess, and τ , ρ a e delay unc ions. 1 When he e a e no delays ( τ( ) = ρ( ) = ), he equa ion (1) has been s udied: in he case B=C= 0, o Anon linea , in Bensoussan [2] and Cu ain [6], and o some ype o non linea ope a o s A, in Bensoussan–Temam [3,4] and Ma cus [10]; in he case C6= 0 , B = 0, o linea Aand C, in Balak ishann [1], o linea Aand non linea Cin Dawson [6], and o non linea mono one Aand Lipschi z con inuous Cin Pa doux [12]. In he case wi h de ia ing a gumen s, Real [14,15] s udies a a he gene al case when all o he ope a o s a e linea and he e exis s a e m which is a non con inuous ma ingale. Howe e , we ha e no ound in he li e a u e he case we a e going o analyze he e. We will adap o ou p oblem one o he mos impo an me hod o sol ing non linea PDEs (see Lions [9]): he mono onici y me hod. Pa doux [12] also used an adap a ion o ha me hod o ano he ype o non linea mono one equa ions: when B= 0 and wi hou delays. In Sec ion 2 we shall s a e he p oblem and he no a ion we a e going o use. Uniqueness o solu ion will be p o ed, in Sec ion 3, using I ˆo’s o mula. In Sec ion 4, we s a e he exis ence esul s. Some ex ensions o he esul s a e gi en in Sec ion 5. Finally, we illus a e ou heo y wi h se e al impo an examples appea ing in he applica ions. 2. S a emen o he p oblem The heo y o s ochas ic in eg als in Hilbe spaces is well de eloped (see [8], [12], o example). The a ia ional me hod we a e going o use, o ces us o wo k wi h he classical pai o eal sepa able Hilbe spaces V , H sa is ying V ,→H(injec ion con inuous and dense). We iden i y Hwi h i s dual space, and deno e by V0 he dual o V. Then, we ha e V ,→H≡H0,→V0. We will deno e by k.k,|.|and k.k∗ he no ms in V,Hand V0 espec i ely; by h., .i he duali y p oduc be ween V0, V , and by (.,.) he scala p oduc in H. Le us ix T > 0 and, le w be a Wiene p ocess de ined on he comple e p obabili y space (Ω,F, P) and aking alues in he sepa able Hilbe space K, wi h inc emen al co a iance ope a o W. Le (F ) ≥0be he σ-algeb a gene a ed by {ws,0≤s≤ }, hen w is a ma ingale ela i e o (F ) ≥0and we ha e he ollowing ep esen a ion o w : w = ∞ X i=1 βi ei, whe e (ei) is an o hono mal se o eigen ec o s o W,βi a e mu ually independen eal Wiene p ocesses wi h inc emen al co a iance λi>0, W ei=λieiand W=P∞ i=1 λi( deno es he ace o an ope a o , see [8], [12], [13]). 2 As an abuse o no a ion, we also use |.| o he no m in he linea con inuous ope a o space L(K, H). We deno e by Ip(0, T;V), o p > 1, he space o V– alued p ocesses (x( )) ∈[0,T ](we will w i e x( ) o sho ) measu able ( om [0, T ]×Ω in V), and sa is ying: i) x( ) is F −measu able a.e. in (in he sequel, we will w i e a.e. .) ii) ERT 0|x |pd < +∞. I is easy o check ha Ip(0, T;V) is a closed subspace o Lp(Ω×[0, T],F ⊗B([0, T]), dP ⊗d ;V), whe e by B([0, T]) we deno e he Bo el σ–algeb a. Fo sho , we shall w i e L2(Ω; C(−h, T;H)) ins ead o L2(Ω,F, dP ;C(−h, T;H)) ,whe e C(−h, T;H) deno es he space o con inuous unc ions om [−h, T] o H. Le A( , .) : V→V0be a amily o non linea ope a o s de ined a.e. ., and le p > 1.We make he ollowing hypo heses: (a.1) Coe ci i y: ∃α > 0, λ ∈Rsuch ha : 2hA( , x), xi+λ|x|2≥αkxkp,∀x∈V , a.e. . (a.2) Mono onici y: 2hA( , x)−A( , y), x −yi+λ|x−y|2≥0,∀x, y ∈V , a.e. . (a.3) Boundedness: ∃β > 0 : kA( , x)k∗≤βkxkp−1,∀x∈V , a.e. . (a.4) Hemicon inui y: θ∈R→ hA( , x +θy), zi ∈ Ris con inuous ∀x, y, z ∈V , a.e. . (a.5) Measu abili y: ∈(0, T)→A( , x)∈V0is Lebesgue −measu able ∀x∈V , a.e. . Le B( , .) : H→Hbe a amily o ope a o s de ined a.e. ., and sa is ying: (b.1) B( , 0) = 0 (b.2) Lipschi z condi ion: ∃k1such ha |B( , x)−B( , y)| ≤ k1|x−y|,∀x, y ∈H , a.e. . (b.3) Measu abili y: ∈(0, T)→B( , x)∈His Lebesgue–measu able, ∀x∈V . And le C( , .) : H→ L(K, H) be ano he amily de ined a.e. . and e i ying: (c.1) C( , 0) = 0 (c.2) Lipschi z condi ion: ∃k2such ha |C( , x)−C( , y)| ≤ k2|x−y|,∀x, y ∈H , a.e. . (c.3) Measu abili y: ∈(0, T)→C( , x)∈ L(K, H) is Lebesgue–measu able ∀x∈H . We also conside wo measu able unc ions (o delay) ρ, τ : [0, T]→[0, T] , such ha (ρ.τ) 0 ≤ρ( ), τ( )≤ , ∀ ∈[0, T]. Fo , g we suppose ha 3 ( .g) ∈I2(0, T;H), g ∈I2(0, T;L(K, H)). And inally, we a e gi en an ini ial alue x0∈L2(Ω,F0, P;H). Now, we s a e he ollowing p oblem: (PC)   To ind a p ocess x∈Ip(0, T ;V)∩L2(Ω; C(0, T;H)) such ha : x( ) + R 0[A(s, x(s)) + B(s, x(τ(s))) + (s)] ds =x0+R 0[C(s, x(ρ(s))) + g(s)] dws, P −a.s., ∀ ∈[0, T ]. Rema k 2.1. We obse e ha , i x∈L2(0, T ;H) hen, by (b.1)–(b.3), B(x)∈L2(0, T;H) , whe e B(x)( ) = B( , x( )) .Mo eo e , x∈L2(0, T;H)→B(x)∈L2(0, T ;H) is con inuous, and so, measu able. Since x∈H→B( , x)∈His con inuous a.e. ., i ollows ha , i x( ) is an H– alued s ochas ic p ocess and F –adap ed, hen B( , x( )) also is. In addi ion, i x∈ L2(Ω ×(0, T); H), hen B(x)∈L2(Ω ×(0, T ); H) oo. Finally, i xnis a bounded sequence in L2(Ω ×(0, T); H), B(xn) also is bounded. Simila obse a ions a e deduced om (c.1)–(c.3) o C:L2(0, T ;H)→L2(0, T;L(K, H)) de ined by C(x)( ) = C( , x( )) . These ema ks and he measu abili y o ρand τimply ha he in eg als appea ing in (P C) a e well de ined. 3. Uniqueness o solu ion In his Sec ion we will p o e ha he e exis s a mos one solu ion o (PC). We will ob ain his esul om (a.2) and I ˆo’s o mula (see [8], [13] o ha o mula). Theo em 3.1 Assume he hypo heses in Sec ion 2. Then, he e exis s a mos one solu ion o (P C)in Ip(0, T;V)∩L2(Ω; C(0, T ;H)) . P oo . Suppose ha x, y ∈Ip(0, T;V)∩L2(Ω; C(0, T ;H)) a e solu ions o (PC). Then, applying I ˆo’s o mula, we ob ain E|x( )−y( )|2 (3.1) =−2EZ 0 hA(s, x(s)) −A(s, y(s)), x(s)−y(s)ids −2EZ 0 (B(s, x(τ(s))) −B(s, y(τ(s))), x(s)−y(s)) ds +EZ 0 £(C(s, x(ρ(s))) −C(s, y(ρ(s)))) W(C(s, x(ρ(s))) −C(s, y(ρ(s))))∗¤ds. 4 Now, by pu ing z( ) = x( )−y( ) and using condi ions (a.2), (b.2), (c.2), i ollows E|z( )|2≤λE Z 0 |z(s)|2ds + 2k1EZ 0 |z(τ(s))||z(s)|ds(3.2) +k2 2 (W)EZ 0 |z(ρ(s))|2ds . We a e going o es ima e he e ms on he igh -hand side o (3.2). Fi s , λE Z 0 |z(s)|2ds ≤ |λ|Z 0 sup ∈[0,s] E|z( )|2ds.(3.3) Using (ρ.τ) we ge 2EZ 0 |z(τ(s))||z(s)|ds ≤EZ 0 |z(τ(s))|2ds +EZ 0 |z(s)|2ds(3.4) ≤2Z 0 sup ∈[0,s] E|z( )|2ds , EZ 0 |z(ρ(s))|2ds ≤Z 0 sup ∈[0,s] E|z( )|2ds .(3.5) Consequen ly, (3.2) −(3.5) yield (3.6) sup ∈[0, ] E|z( )|2≤£|λ|+ 2k1+ 2k2 2 (W)¤Z 0 sup ∈[0,s] E|z( )|2ds . Finally, G onwall’s Lemma implies (3.7) sup ∈[0, ] E|z( )|2= 0 ,∀ ∈[0, T]. Ob iously, uniqueness ollows om (3.7). 4. Exis ence o solu ion Fi s , we s a e a heo em on exis ence and uniqueness o solu ion o a s ochas ic e olu ion equa ion, and an ene gy equali y. Nex , we will p o e he exis ence o solu ion o (PC) using his esul . Theo em 4.1 Assume he hypo heses in Sec ion 2, wi h λ= 0 . Then, he e exis s a unique p ocess x∈Ip(0, T;V)∩L2(Ω; C(0, T ;H)) such ha x( ) + Z 0 [A(s, x(s)) + (s)] ds =x0+M , P −a.s. , ∀ ∈[0, T], 5 whe e M is a H– alued con inuous, squa e in eg able F –ma ingale. The solu ion also e i ies he ollowing ene gy equali y: |x( )|2+ 2 Z 0 hA(s, x(s)), x(s)ids + 2 Z 0 ( (s), x(s)) ds(4.1) =|x0|2+ 2 Z 0 (x(s), dMs) + hhMii , P −a.s. , ∀ ∈[0, T], whe e hhMii deno es he quad a ic a ia ion o M (see M´e i ie and Pellaumail [11]). P oo . See [11], [15] and he e e ences gi en he e. Rema k 4.1. We obse e ha , in ou si ua ion (see he p oo o heo em 4.2), he ma ingale M will be R 0g(s)dwsand hence, he ene gy equali y yields E|x( )|2+ 2EZ 0 hA(s, x(s)), x(s)ids + 2EZ 0 ( (s), x(s)) ds(4.2) =E|x0|2+EZ 0 (g(s)Wg(s)∗)ds , P −a.s. , ∀ ∈[0, T]. In Pa doux [12,13] and Ichikawa [8] we can ind a a he gene al I ˆo’s o mula. Now, using a Pica d’s scheme, we can p o e he exis ence o solu ion o he p oblem (PC). Theo em 4.2 Assume he condi ions in Sec ion 2. Then, he e exis s a unique solu ion o (PC)in Ip(0, T;V)∩L2(Ω; C(0, T ;H)) . P oo . Uniqueness holds om heo em 3.1. Fo he exis ence, we conside he equa ions x1( ) + Z 0·A(s, x1(s)) + λ 2x1(s)¸ds +Z 0 (s)ds(4.3) =x0+Z 0 g(s)dws xn+1( ) + Z 0·A(s, xn+1(s)) + λ 2xn+1(s)¸ds(4.4) +Z 0 B(s, xn(τ(s))) ds +Z 0 (s)ds =x0+Z 0 λ 2xn(s)ds +Z 0 C(s, xn(ρ(s))) dws +Z 0 g(s)dws,∀n= 1,2,3, ... By (a.1)–(a.5), he amily A1( , .) : V→V0de ined by A1( , x) = A( , x) + (λ/2)x , sa is ies he assump ions in heo em 4.1. Consequen ly, (4.3) has a unique solu ion x1∈Ip(0, T;V)∩ L2(Ω; C(0, T;H)) . We no e ha , om (b.2),(c.2) and he measu abili y o he unc ions ρ , τ , i ollows: 6 i) The map ( , ω)∈(0, T )×Ω→B( , x1(τ( )) ∈Hbelongs o I2(0, T;H). ii) The map ( , ω)∈(0, T)×Ω→C( , x1(ρ( )) ∈Hbelongs o I2(0, T;L(K, H)) ,and so, R· 0C(s, x1(ρ(s))) dwsis a con inuous and squa e in eg able F –ma ingale. Again, by hese ema ks, we can apply heo em 4.1 and we ge ha he e exis s a unique p ocess x2∈Ip(0, T;V)∩L2(Ω; C(0, T;H)) , which is solu ion o (4.4) o n= 1 . By ecu ence, we ob ain a sequence o solu ions o (4.3) −(4.4) , {xn}n≥1⊂Ip(0, T;V)∩L2(Ω; C(0, T;H)) . In he sequel, we shall p o e ha he sequence {xn}is con e gen o a p ocess in Ip(0, T;V)∩ L2(Ω; C(0, T;H)) , and his p ocess is he solu ion o (PC). We shall spli his p oo in ou s eps. STEP 1.– {xn}is a Cauchy sequence in L2(Ω; C(0, T;H)). Indeed, o n > 1, i ollows om I ˆo’s o mula o he p ocess xn+1( )−xn( ), |xn+1( )−xn( )|2+ 2 Z 0 hA(xn+1)−A(xn), xn+1 −xnids(4.5) +λZ 0 |xn+1 −xn|2ds + 2 Z 0¡B(xn τ)−B(xn−1 τ), xn+1 −xn¢ds =λZ 0¡xn+1 −xn, xn−xn−1¢ds + 2 Z 0¡xn+1 −xn,(C(xn ρ)−C(xn−1 ρ)) dws¢ +Z 0 h¡C(xn ρ)−C(xn−1 ρ)¢W¡C(xn ρ)−C(xn−1 ρ)¢∗ids , whe e, by de ini ion: xn:= xn(s), A(xn) := A(s, xn(s)) , B(xn τ) := B(s, xn(τ(s))) and C(xn ρ) := C(s, xn(ρ(s))) . ¿F om (a.2), |xn+1( )−xn( )|2≤|λ|Z 0 |xn+1 −xn||xn−xn−1|ds(4.6) + 2 ¯¯¯¯Z 0¡xn+1 −xn,(C(xn ρ)−C(xn−1 ρ)) dws¢¯¯¯¯ +Z 0¯¯¯ h¡C(xn ρ)−C(xn−1 ρ)¢W¡C(xn ρ)−C(xn−1 ρ)¢∗i¯¯¯ds + 2 Z 0 |B(xn τ)−B(xn−1 τ)||xn+1 −xn|ds . Consequen ly, (4.6) yields E·sup 0≤θ≤ |xn+1(θ)−xn(θ)|2¸≤ |λ|EZ 0 |xn+1 −xn||xn−xn−1|ds(4.7) + (W)EZ 0 |C(xn ρ)−C(xn−1 ρ)|2ds + 2EZ 0 |B(xn τ)−B(xn−1 τ)||xn+1 −xn|ds + 2E·sup 0≤θ≤ ¯¯¯¯Z 0¡xn+1 −xn,(C(xn ρ)−C(xn−1 ρ)) dws¢¯¯¯¯¸. Now, we es ima e he e ms on he igh -hand side o (4.7), and we apply he inequali y 2ab ≤a2 l2+l2b2, a, b ∈R, l > 0, 7 o sui able l. |λ|EZ 0 |xn+1 −xn||xn−xn−1|ds(4.8) ≤1 4E·sup 0≤θ≤ |xn+1(θ)−xn(θ)|2¸+λ2TZ 0 E·sup 0≤θ≤s |xn(θ)−xn−1(θ)|2¸ds . (W)EZ 0 |C(xn ρ)−C(xn−1 ρ)|2ds(4.9) ≤ (W)k2 2Z 0 E·sup 0≤θ≤s |xn(θ)−xn−1(θ)|2¸ds . 2EZ 0 |B(xn τ)−B(xn−1 τ)||xn+1 −xn|ds(4.10) ≤1 4TEZ 0 |xn+1 −xn|2ds + 4k2 1TE Z 0 |xn(τ(s)) −xn−1(τ(s))|2ds ≤1 4E·sup 0≤θ≤ |xn+1(θ)−xn(θ)|2¸+ 4k2 1TZ 0 E·sup 0≤θ≤s |xn(θ)−xn−1(θ)|2¸ds . Bu kholde –Da is–Gundy’s inequali y implies 2E·sup 0≤θ≤ ¯¯¯¯Z 0¡xn+1 −xn,(C(xn ρ)−C(xn−1 ρ)) dws¢¯¯¯¯¸ (4.11) ≤6 (W)E·µsup 0≤θ≤ |xn+1(θ)−xn(θ)|2¶Z 0 |C(xn ρ)−C(xn−1 ρ)|2ds¸1/2 ≤1 4E·sup 0≤θ≤ |xn+1(θ)−xn(θ)|2¸+ 36k2 2 (W)Z 0 E·sup 0≤θ≤s |xn(θ)−xn−1(θ)|2¸ds . I we se (4.12) ϕn( ) = E·sup 0≤θ≤ |xn+1(θ)−xn(θ)|2¸, hen, (4.7) −(4.11) yield (4.13) ϕn( )≤3 4ϕn( ) + (λ2T+k2 2 (W) + 4k2 1T+ 36k2 2)Z 0 ϕn−1(s)ds , and so, he e exis s k > 0 such ha (4.14) ϕn( )≤kZ 0 ϕn−1(s)ds . By i e a ion om (4.14), we ge (4.15) ϕn( )≤kn−1Tn−1 (n−1)! ϕ1(T),∀n > 1,∀ ∈[0, T]. The e o e, (4.16) E·sup 0≤θ≤T |xn+1(θ)−xn(θ)|2¸≤kn−1Tn−1 (n−1)! ϕ1(T),∀n > 1. 8 Ob iously, (4.16) implies ha {xn}is a Cauchy sequence in L2(Ω; C(0, T ;H)) . STEP 2.– The sequence {xn}is bounded in Ip(0, T;V). Indeed, I ˆo’s o mula o |xn|2,wi h n≥2, yields E|xn(T)|2+ 2EZT 0 hA(xn), xnids +λE ZT 0 |xn|2ds(4.17) =E|x0|2−2EZT 0 (B(xn−1 τ), xn)ds −2EZT 0 ( , xn)ds +λE ZT 0 (xn, xn−1)ds +EZT 0 h¡C(xn−1 ρ) + g¢W¡C(xn−1 ρ) + g¢∗ids . The e o e, 2EZT 0 hA(xn), xnids +λE ZT 0 |xn|2ds(4.18) ≤E|x0|2+ 2EZT 0 |B(xn−1 τ)||xn|ds + 2EZT 0 | ||xn|ds +|λ|EZT 0 |xn||xn−1|ds + (W)EZT 0 |C(xn−1 ρ) + g|2ds . Since {xn}is con e gen in L2(Ω; C(0, T;H),i will be bounded in his space. Now, i is no di icul o check ha he e exis s a posi i e cons an , k0,such ha he igh -hand side o (4.18) is bounded by his cons an . As an example, we will es ima e one o hose e ms: 2EZT 0 |B(xn−1 τ)||xn|ds ≤2k1EZT 0 |xn−1(τ(s))||xn(s)|ds ≤k1EZT 0£|xn−1(τ(s))|2+|xn(s)|2¤ds ≤k1EZT 0·sup 0≤θ≤T |xn−1(θ)|2+ sup 0≤θ≤T |xn(θ)|2¸ds ≤Tk1·Eµsup 0≤θ≤T |xn−1(θ)|2¶+Eµsup 0≤θ≤T |xn(θ)|2¶¸ ≤Tk1³kxnk2 L2(Ω;C(0,T ;H)) +kxn−1k2 L2(Ω;C(0,T ;H))´. This ac , (4.18) and (a.1) lead o he ollowing inequali ies: (4.19) αZT 0 Ekxn(s)kpds ≤2EZT 0 hA(xn), xnids +λE ZT 0 |xn|2ds ≤k0, and S ep 2 is p o ed. STEP 3.– We can ake limi s in (4.4). Indeed, om S ep 1, xn→x , o some xin L2(Ω; C(0, T ;H)) .Since (b.2) and (c.2) hold, we also ha e B(xn τ)→B(xτ) (in L2(Ω; L∞(0, T;H)) ), and C(xn ρ)→C(xρ) (in L2(Ω; L∞(0, T;L(K, H))) ). On he o he hand, by S ep 2, {xn}has a subsequence which is weakly con e gen in Ip(0, T;V) . Bu , since xn→xin L2(Ω; C(0, T;H)) ,we can assu e ha xn→xweakly in 9 such ha du( , x)−∆u( , x)d +u3( , x)d +ϕ1(u(τ( ), x)) d =ϕ2(u(ρ( ), x)) dw ,in (0, T)× O , u( , x) = ψ( , x) in (−h, 0) × O , u( , x) = 0 in (−h, T)×∂O. Example 6.3.– A s ochas ic non–linea mono one pa abolic equa ion. Le p > 2 . Now, we conside V=W1,p 0(O),and we de ine A:V→V0by hA(u), i= N X i=1 ZO¯¯¯¯ ∂u ∂xi¯¯¯¯ p−2∂u ∂xi ∂ ∂xi dx +ZO |u|p−2u dx , ∀u, ∈V , and le B , C , ρ , τ , w as in example 6.2. I is easy o check ha (a.1)–(a.5) hold, wi h α= 1/2, λ = 0 and β=N+ 1 . Consequen ly, gi en ψ∈Ip(−h, 0; V)∩L2(Ω; C(−h, 0; H)) and , g ∈I2(0, T;H) , he e exis s a unique p ocess uin Ip(−h, T;V)∩L2(Ω; C(−h, T;H)) solu ion o (PC)0. In o he wo ds, du( , x) = ÃN X i=1 ∂ ∂xiﯯ¯ ∂u( , x) ∂xi¯¯¯¯ p−2∂u( , x) ∂xi!d +|u( , x)|p−2u( , x)!d + (ϕ1(u(τ( ), x)) + ( , x)) d + (ϕ2(u(ρ( ), x)) + g( , x)) dw ,in (0, T)× O , u( , x) = ψ( , x),in (−h, 0) × O , u( , x) = 0 ,in (−h, T)×∂O. Re e ences [1] A. Balak ishnan, S ochas ic bilinea pa ial di e en ial equa ions, U.S.–I aly Con e ence on Va iable S uc u e Sys ems, O egon (1974). [2] A. Bensoussan, Fil age op imal des sys emes lin´eai es, Dunod. [3] A. Bensoussan and R. Temam, Equa ions aux d´e i ´ees pa ielles s ochas iques non lin´eai es, Is ael J. Ma h.,11 (1972), 95–129. [4] A. Bensoussan and R. Temam, Equa ions s ochas iques du ype Na ie –S okes, J. Func ional Analysis, 13,2 (1973), 195–222. [5] P.L. Chow, S ochas ic Pa ial Di e en ial Equa ions: Tu bulence and Rela ed P oblems, P ob. Analysis and Rela ed Topics, 1, A.T. Bha ucha-Reid, Academic P ess, New Yo k (1978). 16 [6] R. Cu ain, S ochas ic di e en ial equa ions in Hilbe spaces, Ph. D. Thesis, B own Uni e si y (1969). [6] D. Dawson, S ochas ic e olu ion equa ion, Ma h. Biosc.,15 (1972) [7] W.H. Fleming, Dis ibu ed Pa ame e S ochas ic Sys ems in Popula ion Biology, Lec u e No es in Economics and Ma hema ical Sys ems, ol. 107, Sp inge , Be lin-New Yo k (1975). [8] A. Ichikawa, S abili y o Semilinea S ochas ic E olu ion Equa ions, J.Ma h.Anal.Appl.90 (1982), 12-44. [9] J.L. Lions, Quelque m´e hodes de ´esolu ion des p oblemes aux limi es non lineai es, Dunod Gau hie –Villa s, Pa is (1969). [10] R. Ma cus, Pa abolic I o equa ions, T ans. Am. Ma h. Soc.,198 (1974), 177–190. [11] M. M´e i ie and J. Pellaumail, S ochas ic In eg a ion, Academic P ess, New Yo k, (1980). [12] E. Pa doux, ´ Equa ions aux D´e i ´ees Pa ielles S ochas iques non Lin´eai es Mono ones, Thesis, Uni e si y o Pa is XI (1975). [13] E. Pa doux, S ochas ic Pa ial Di e en ial Equa ions and Fil e ing o Di usion P ocesses, S ochas ics 3, (1979) 127-167. [14] J. Real, Con ibuci´on al es udio de una clase de Ecuaciones en De i adas Pa ciales Es- oc´as icas con Re a do, Thesis, Uni e si y o Se illa (1980). [15] J. Real, S ochas ic Pa ial Di e en ial Equa ions wi h Delays, S ochas ics 8, 2 (1982-83), 81-102. [16] M. Vio , Solu ions aibles d’´equa ions aux d´e i ´ees pa ielles s ochas iques non lin´eai es, The- sis, Uni e si y o Pa is VI (1976). 17