110 Chapter 5. Linear, Degenerate BSPDEs Theorem 2.3. Let m > 1 and (H)m hold for {A, B, a, b, c}. Let (1.6) and (2.2) hold. Let A > 0 such that { e-~(')f 9 L2~(O,T;Hm(]Rn)), (2.14) e -~ (")g 9 L~r (ft; gm(Rn)). Then BSPDE (1.1) admits a unique adapted weak solution (u, q), such that the following estimate holds: T EHe-~()u(t,.)l]2H m +El He-~()q(t,.)ll2H.~-,dt max tE[0,T] Jo + E E fr/{ ((A- BBr)D[O~(e-~(')u)],D[O~(e-~(')u)]) (2.15) [al_<m -Io JR ~" + BT{D[OC~(e-A(')u)]} + OC~(e-X(')q) 2}dxdt- _< + where the constant C > 0 only depends on m, T and Kin. Furthermore, if m > 2, the weak solution (u, q) becomes the unique adapted strong solution of (1.1); and if m > 2 + n/2, then (u, q) is the unique adapted classical solution of (1.1). In the case that (2.2) is replaced by (2.6) or (2.7), the above conclusion rema/ns true and the estimate (2.15) can be improved to the following: T ElJe-~(')u(t,.)[[~m + E[ [le-~(')q(t,.)[[2H.~dt max t~[O,T] Jo /o'/o (2.16) + ~ E . (AD[O~(e-~(')u)l,D[O~(e-~(')u)])dxdt [aI<m Finally, if in addition, (1.9) holds for some 5 > O, then (2.16) can further be improved to the following: max Elle-~ <" >u(t,.)ll~m te[O,Tl (2.17) +E { lie-:' <" >u(t, ")115m+1 +lle-a(')q(t,')llhm}dt Clearly, (2.14) means that f and g can have an exponential growth as Ix I -+ ce. This is good enough for many applications. w Uniqueness of adapted solutions 111 We note that {A,B,a,b.c} satisfies (H),~ if and only if {A,B,~d,b,c} satisfies (H),~, where ~ and b are given by (1.4). Thus, we have the exact statements as Theorems 2.1, 2.2 and 2.3 for BSPDE (1.5) with a and b replaced by ~ and b. w Uniqueness of Adapted Solutions In this section, we are going to establish the uniqueness of adapted weak, strong and classical solutions to our BSPDEs. From the discussion right before Proposition 1.3, we see that it suffices for us to prove the uniqueness of adapted weak solutions. w Uniqueness of adapted weak solutions For convenience, we denote (3.1) s ~ V.[ADu] + ( a, Du ) +cu, 3dq ~ V.[Bq] + ( b, q ) . Then, equation (1.1) is the same as the following: f du=-{s f}dt+(q, dW(t)), (t,x) 9215 '~, (3.2) U[t=T = g. In this section, we are going to prove the following result. Theorem 3.1. Let (2.3) hold and the following hold: (3.3) A 9 L~(O,T;L~176 BE L~/Oj:~, T', LO~/R,~.~ , ~n • d)), a 9 Lcff(O,T;L~176 b 9 L~176176 c 9 L~(0, T; n~176 Then, the adapted weak solution (u, q) of (3.2) is unique in the class (3.4) u e Cf([0, T]; L2(a; H l(]Rn))), q 9 L~=(0, T; L2(Rn; Nd)). To prove the above uniqueness theorem, we need some preliminaries. First of all, let us recall the Gelfand triple H 1 (~n) ~+ L 2 (~n) ~_+ H-1 (IRa). Here, H-I(IR ~) is the dual space of Hl(~n), and the embeddings are dense and continuous. We denote the duality paring between H 1 (]R n) and H-I(~ '~) by (-,-)0, and the inner product and the norm in L2(IR ~) by 112 Chapter 5. Linear, Degenerate BSPDEs (',')o and ]. [0, respectively. Then, by identifying L2(]R ~) with its dual L 2 (R~)* (using Riesz representation theorem), we have the following: ( r ~ )o = (r ~)o (3.5) JR f" ~b(x)~o(x)dx, Vr E L2(~Ln), ~ E H'(]Rn), and (3.6) n i~lOir E H-I(]Rn), "~n ( Z o~r ~/o = - i:1 ]R~ Vr E L2(IRn), 1 < i < n, r V99 E Hi(IRa). Next, let (u, q) be an adapted weak solution of (3.2) satisfying (3.4). Note that in (3.4), the integrability of (u, q) in x is required to be global. By (3.5)-(3.6), we see that (3.7) s + A4q E L~(0,T; H-l(~n)). In the present ease, from (1.15), for any ~ E HI(IR n) (not just C~(Rn)), we have {d(u,~o)o=-(s f,~)o+((q,~)o,dW(t)), tE[0, T], (3.8) (u, ~)0 It=T = (g, ~)o. Here, (q, ~O)o ~((ql, ~)o,'", (qd, ~)0) and q = (ql,'", qd). Sometimes, we say that (3.2) holds in H-I(IR '~) if (3.8) holds for all ~ E HI(Rn). In proving the uniqueness of the adapted weak solutions, the following special type of It6's formula is very crucial. Lemma 3.2. Let ~ E L2(O, T; H-I(~)) and (u, q) satisfy (3.4), such that du=~dt+(q, dW(t)>, tE[0, T]. (3.9) Then (3.10) 0 t lu(t)l~ = lu(O)l~ + {2 (r + Iq(s)l~}ds .// + 2 ((q(s),u(s))o,dW(s)), t E [0, T]. Although the above seems to be a very special form of general It6's formula, it is enough for our purpose. We note that the processes u, q and take values in different sp aces H 1 (IR~), L 2 (iRa) and H - 1 (]R~), respectively. This makes the proof of (3.10) a little nontrivial. We postpone the proof of Lemma 3.2 to the next subsection. w Uniqueness of adapted solutions 113 Proof of Theorem 3.1. Let (u, q) be any adapted weak solution of (3.2) with f and g being zero, such that (3.4) holds. We need to show that (u, q) 0, which gives the uniqueness of adapted weak solution. Applying Lemma 3.2, we have (note (3.7)) P T Elu(t)l 2 = E.L {2 (s + .A4q(s), u(s) )0 - Iq(s)12} ds = E , { - ( ADu, Du ) + ( a, D(u 2) ) +2eu 2 2 (q, BTDu) +2 (bu, q) -]ql2}dxds - Iq + BTDu - bul 2 + [b 2 + 2c - V-(a + Bb)]u2}ds /* T < C/ Elu(s)l~ds , t 9 [0, T]. Jt By Gronwall's inequality, we obtain El(t)l ~ = O, t 9 [0, T]. Hence, u = 0. By (3.11) again, we must also have q = 0. This proves the uniqueness of adapted weak solutions to (3.2). [] w An It5 formula In this subsection, we are going to present a special type of It6's formula in abstract spaces for which Lemma 3.2 is a special case. Let V and H be two separable Hilbert spaces such that the embedding V ~-+ H is dense and continuous. We identify H with its dual H' (by Riesz representation theorem). The dual of V is denoted by V'. Then we have the Gelfand triple V r H = H' ~ + V'. We denote the inner product and the induced norm of H by (-, ")0 and [-10, respectively. The duality paring between V and V' is denoted by (.,.)0, and the norms of V and V' are denoted by 1]" I[ and I[" []*, respectively. We know that the following holds: (3.12) (u,v)0=(u,v)o, VueH, v9 Due to this reason, H is usually called the pivot space. It is also known (see [Lions]) that in the present setting, there exists a symmetric linear operator A e s V'), such that (3.13) (Av, v)o <-Hvll 2, VveV. Now, let us state the following result which is more general than Lemma 3.2. 114 Lemma 3.3. Let (3.14) satisfying (3.15) Then Chapter 5. Linear, Degenerate BSPDEs [ ~ 9 c~([0, T]; V), 9 L~(0, T; Y'), du=~dt+(q, dW(t)), t9 ~0 t lu(t)[o ~ = lu(O)lo ~ + {2 (~(s),u(s))o + [q(S)lo~}ds (3.16) // + 2 ((q(s),u(s))o,aW(s)), t 9 [0,T]. In the above, q 9 L}(O, T; H) d means that q = (ql,'", qa) with qi 9 L~-(0, T; H). In what follows, we will see the expression q 9 L}(O, T; V) a whose meaning is similar. Before giving a rigorous proof of the above result, let us try to prove it in an obvious (naive) way. From (3.16), we see that the trouble mainly comes from ( since it takes values in V'. Thus, it is pretty natural that we should find a sequence (k 9 L~(O, T; H), such that (3.17) (k -+ (, in L~(0, T; V'), (k ~ oe), and let Uk be defined by // /o (3.18) u~(t) = u(0) + (k (s)ds + (q(s), dW(s) ), t 9 [0, r]. Since the processes Uk, (k and q are all taking values in H, we have // luk(t)[~ = lu(0)10 ~ + {2 (Sk(S),Uk(S) )0 + [q(s)12}ds (3.19) +2 ((q(s),uk(s))o,dW(s)), t9 This can be proved by projecting (3.18) to finite dimensional spaces, using usual ItS's formula, then pass to the limit. Having (3.19), one then hopes to pass to the limit to obtain (3.16). This can be done provided one has the following convergence: Uk + U, in L~:(0, T; V). However, (3.17)-(3.18) only guarantees Uk -+ u, in L~:(0, T; Y'). Thus, the convergence of Uk to u is not strong enough and such an approach does not work! In what follows, we will see that to prove (3.16), much more has to be involved. w Uniqueness of adapted solutions 115 Let us now state two standard lemmas for deterministic evolution equa- tions whose proofs are omitted here (see Lions [1]). Lemma 3.4. Let v : [0, T] -+ V ~ be absolutely continuous, such that (3.20) [ v 9 52(0, T; V), ( b 9 L2(0, T; V'). Then v 9 C([O,T];H) and (3.21) d[v(t)to 2 = 2(9(t),v(t))o, a.e.t 9 [0,T]. Let A 9 s V') be symmetric satisYying (3.13). Then for (3.25) Let (3.26) M(t) = fot Then, M 9 Cj:([O,T]; V) and v 9 L~(0, T;V), ~3 9 L2(0, T; V'), q 9 L2~(O, T; V) d. (q(s),dW(s) ), t 9 [0, T]. (3.27) fot fo' [M(t)l~ = 2 ((M(s),q(s))o,dW(s) ) + [q(s)l~ds, t 9 [0, T], a.s. Lemma 3.5. any vo E H and f E L2(0,T; V'), the following problem (3.22) ~7)=Av+f, tE[0,T], I v(0) = vo, admits a unique solution v satisfying (3.20) and /o' /o' (3.23) Iv(t)lo 2 + IIv(s)ll2ds <_ Ivolg + I]f(s)ll2.ds, t e [0,T]. Moreover, it holds /o' (3.24) Iv(t)lo 2 = Ivolo 2 + 2 (Av(s) + f(s), v(s) )ods, t 9 [0, r]. Now, we consider stochastic evolution equations. We first have the following result. Lemma 3.6. Let v be an {Ft}t>_o-adapted V'-valued processes which is absolutely continuous almost surely and q be an { ~t }t>_o-adapted H-valued process such that the following holds: 116 (3.28) Chapter 5. Linear, Degenerate BSPDEs d(v(t), M(t))o = (O(t), M(t) )odt + ((v(t), q(t))o, dW(t) ), a.e. t E [0, T], a.s. Proof. First of all, it is clear that M E C~=([0, T]; V) and (3.27) holds since we may regard both M and q as H-valued processes. We now prove (3.28). Take a sequence of absolutely continuous processes Vk with the following properties: (3.29) vk E L~(0, T;V), Ok E L~-(0, T; H), Vk + V, in L~:(0, T; V), Ok + 0, in L2(0, T; V'). Now, in H, we have (note (3.12)) (3.30) d(vk (t), M(t))o = (Ok (t), M(t))odt + ((Vk (t), q(t))o, dW(t) ) = (0k (t), M(t) )odt + ((Vk (t), q(t))o, dW(t) ). Pass to the limit in the above, using (3.29), we obtain (3.28). [] Lemma 3.7. Let A E s V') be symmetric satisfying (3.13). Then, for any f, q, u0 satisfying (3.31) f E L~(0, T;V'), q E L2(0, T; H) d, u0 E H, the following problem S du = (Au + f)dt + (q, dW(t) }, (3.32) [ = t e [0, T], admits a unique solution u E L2(0, T; V) M C7([0, T]; H), such that (3.33) ~0 t lu(t)J~ = luol~ + {2 (Au(s) + f(s), u(s))o + Jq(s)l~}ds // + 2 ((q(s), u(s))o, dW(s) >, Vt E [O,T], a.s. Proof. We first let q E L~(0, T; V) d and define M(t) by (3.26). Con- sider the following problem: (3.34) O=Av+f+AM, t ~ [0,T], v(0) = uo. w Uniqueness of adapted solutions 117 By Lemma 3.5, for almost all w E ~, (3.34) admits a unique solution v. Obviously (by the variation of constants formula, if necessary), v is {~t}t>o- adapted. Thus, we have e L~-(0, T, V'), which implies (by Lemma 3.4) v C Cj:([0, T]; H) and (by (3.24)) (3.35) Iv(t)t~ = lu~ + 2 (.Av(s) + f(s) + .AM(s), v(s) )ods, Vt ~ [0, T], a.s. Set u(t) = v(t) + M(t). Then, we see that u E g~(0, T; V) N C~-([0, T]; H) is a solution of (3.32). We now combining (3.27)-(3.28) and (3.34) (3.35) to obtain the following: lu(t)]~ = Iv(t)l~ + IM(t)]~ + 2(v(t), M(t))o = luolo2 +2 (.Av(s)+f(s)+AM(s),v(s))ods /0 /o +2 ((M(s),q(s))o,dW(s) ) + Iq(s)lgds + 2 (i~(s), M(s) )ods + 2 ((v(s), q(s))o, dW(s) ) (3.36) = 1~o1~+2 (Au(s)+S(s),v(s))odS /o /o + 2 ((u(s), q(s))o, dW(s) ) + Iq(s) lgds + 2 (Au(s) + f(s), M(s) )ods = luolg + {2 (A~(s) + S(s),~(s))o + Iq(s)lg}ds +2 ((~(s),q(s))o,aW(s)). Next, we claim that solution to (3.32) is unique (for any f, q and uo sat- isfying (3.31)). As a matter of fact, if ~ is another solution to (3.32), then u - ~ is a solution of (3.32) with f, q and uo all being zero. Applying (3.36) to u - ~, we obtain (see (3.13)) I~(t) - ~(t)lo ~ = 2 (A[~(s) - ~(s)], ~(s) - ~(s) )oas ___ 0, which results in u = ~. Thus, we have proved our lemma for the case q E L~(0, T; V) d. Now, for general case, i.e., q e L~(O, T; H) e, we take a 118 Chapter 5. Linear, Degenerate BSPDEs sequence qk E L2(0, T; V) d with qk + q, in L~(0, T; H) d. Let uk be the solution of (3.32) with q being replaced by qk. Then applying (3.36) to Uk ue, we have (note (3.13)) Eluk(t) - u,(t)l~ + 2El t Iluk(s) - ut(s)ll2ds (3.37) dO // _< E Iqk(S) q~(s)12 ds ~ O, k,e + oc. This means that the sequence {Uk} is Cauchy in L2(0, T; V)AC~:([0, T]; H). Hence, there exists a limit u of {u}k in this space. Clearly, u is a solution of (3.32). Also, we have a similar equality (3.33) for each uk. Pass to the limit, we obtain the equality (3.33) for u (with general q E L~(0, T; H)d). [] Now, we are ready to prove Lemma 3.3. Proof of Lemma 3.3. Set no = u(O) C H, f ~= ~ - Au 9 L~(0, T; V'). Then u is a solution of (3.32) with (3.31) holds. Hence, (3.33) holds, which yields (3.16). [] Now, by taking V = HI(R~), H = L2(R ~) and V' = H-I(IR~), we see that Lemma 3.2 follows immediately from Lemma 3.3. w Existence of Adapted Solutions The proofs of existence of adapted solutions is based on the following fun- damental lemma. Lemma 4.1. Let the parabolicity condition (1.6) and the symmetry con- dition (2.2) hold. Let (H)m hold for some m >_ 1. Then there exists a " constant C > O, such that for any u G C~(]R n) and q E C~(~n; ]l~d), it holds (4.1) { ((A - BBT)D(O%),D(0%)) I~l<m +lBTD(Oau)+OC~q]2} + E IOaql2} dx I~1_<,~-1 I,~1<~ a.e. t ~ [O,T], a.s. w Existence of adapted solutions 119 If (2.6) or (2.7) holds instead of (2.2), following: (4.2) the above can be replaced by the / {E I~l_<m <c/ E ( AD(Oau)'D(Oau))+ E la<'ql2} dx I~l_<m { - 2(a%)a"(s + Mq)+ la'ql 2 + la"ul2}d=, a.e. t E [0, T], a.s. Furthermore, if (2.6) or (2.7) holds and A( t, x) is uniformly positive definite, then (4.2) can be improved to the following: S { ) ]- la<'ul'+ ~ iraqi'}a,, Io<l<m+l Io, l<m (4.3) < C mR. E {- 2(oc~u)oc~(s + Mq)+ 1O"ql2 + tO=ul2}dz, I~l<m a.e. t e [0, T], a.s. We note that the square root of the left hand side of (4.1) is a norm in the space C~(R n) • C~~ ]Rd). Thus, if we denote the completion of the space C~~ n) • C~(~n; ~d) under this norm by 7-lm(tl w) (note that it depends on (t, w) E [0, T] • fl), then we have the following inclusions: cF(~ n) • cF(n~; n ~) c u~(t,w) c_ H~(n n) • H~-I(~; ~). It is clear that estimate (4.1) also holds for any (u,q) E 7/m(t,w). A similar argument holds for (4.2) and (4.3). Since the proof of the above lemma is rather technical and lengthy, we postpone its proof to the next section. Before going further, let us recall the following fact concerning the dif- ferentiability of stochastic integrals with respect to the parameter. Let h E L~(O, T; C~(Rn; ~d)). Then it can be shown that the stochastic inte- gral with parameter: f~ ( h(s, x, .), dW(s) ) has a modification that belongs to n~:(0, T; c~-l(~n; lRm)) and it satisfies f ' x, f[ On .In ( h(s, .), dW(s) ) = ( O~h(s, x, .), dW(s) ), (4.4) for [a 1=l,2, ,m-1. Consequently, if h C L2(0, T; C~), then L " ( h(s, .), dW(s) ) e L~(O, T; C~), 0 ~ and (4.4) holds for all multi-index a. [...]... 0, Z~, Z~ and Z~ are all absent We now treat Z~, Z~, Z~ and Z~, separately w A proof of the fundamental lemma 1 27 Since A and B are C ~ +1 in x, we see immediately that (5.6) ]Z~'t + IZ~] -< C(tu]~ + 2 )qle-z)- Now, let us look at Z~ and Z~' Using integration by parts, we have Z~ = fa" { (AD(O(~u)'D(O'~u))+2(O'~q'BTD(O~u))+lO'~ql2 - ( a, n[(O"u)2]) -2c(O"u) 2 - 2 (b(O'~u), O"q ) }dx (5 .7) 9/R" { (... u E C ~ ( ~ n) and q E C~~ IRa), by definition of s and AA, and differentiation, we have Z~A= Sn { - 2(OC~u)O'~(s + Mq) + [OC'q[2}dx + V'[Bql+(b,q)]+[O'~ql2}dx = / n " { - 2(O~u) [~ (5.1) V'[AD(Oau)] + (a'D(O~u))+c(OC~u) + V.[B(O"q)] + ( b, Oaq ) ] + 0 0 only depending on T, m and Km We are going to prove that (u, q) is a weak solution of (3.2) To this end, let us take p E H 1(0, T) such that (4.16) p(0)=0, p(T)= l, O 0 be fixed and k _ ~ For any ~ E H ~ C C~(]Rn), from (4.8) and the fact Pk~o = ~, we have (gk, (fl)m = ~0 T {P(t)(uk(t), ~O)m... any ~ E H ~ C C~(]Rn), from (4.8) and the fact Pk~o = ~, we have (gk, (fl)m = ~0 T {P(t)(uk(t), ~O)m - p(t)(s + Mqk(t) + fk(t),~o),~}dt T + fo p(t) ((qk (t), qo)m, dW(t)) (4. 17) By the definition of/~ and Ad, using integration by parts, we obtain (gk,~)m : ~0 T {P(t)(uk(t),~)m p(t)[ - - (4.18) + ((a(t), + l(A(t)Duk(t) + B(t)qk(t), D~o),~ Duk(t) ) +c(t)uk(t) + (b(t), qk(t) ) +fk(t), p(t) ((qk (t), ~)m,... Iql2_1) "t- Note that (5.15) IO'~ql2 < 21BTD(O~u) + OC'ql2 + 21BTD(a~u)l 2 Using the parabolicity condition (1.6) and the definition of Ot (see (5.14)), we have IqlLl ~ (5.16) C((]}/-1 -1-lull) Consequently, from (5.14), we obtain 2 ~s ~ C ( ~ s ~- (]}~-1 ~- IU It), (5. 17) 1 < ~ < m On the other hand, for g = 0 (i.e., a = 0), we have JR- { - 2u(s + Mq) + Iql2}dx : i, {-'~ + ~a, ~ u ~ +.- + V.[Bq] + (5.18)... holds for all ~ E S Next, by HSrmander [1, p.161], Fourier transformation ~ ~ ~ is an isomorphism of S onto itself Applying Parseval's formula to (4.30), we obtain (4.33) UR, [ F ( ~ ) - ( 0 ( ~ ) , ~ ) ] ( E 1~12) ~ ( ~ ) d ~ = 0 ' I~l . 3.1. Let (2.3) hold and the following hold: (3.3) A 9 L~(O,T;L~ 176 BE L~/Oj:~, T', LO~/R,~.~ , ~n • d)), a 9 Lcff(O,T;L~ 176 b 9 L~ 176 176 c 9 L~(0, T; n~ 176 Then, the adapted. 2.1, 2.2 and 2.3 for BSPDE (1.5) with a and b replaced by ~ and b. w Uniqueness of Adapted Solutions In this section, we are going to establish the uniqueness of adapted weak, strong and classical. Gelfand triple H 1 (~n) ~+ L 2 (~n) ~_+ H-1 (IRa). Here, H-I(IR ~) is the dual space of Hl(~n), and the embeddings are dense and continuous. We denote the duality paring between H 1 (]R n) and