From the discussion right before Proposition 1.3, we see that it suffices for us to prove the uniqueness of adapted weak solutions.. In this section, we are going to prove the following
Trang 1110 Chapter 5 Linear, Degenerate BSPDEs
T h e o r e m 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; g m ( R n ) )
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 + E l He-~()q(t,.)ll2H.~-,dt max
+ E E f r / { ( ( A - BBr)D[Õ(e-~(')u)],D[Õ(e-~(')u)])
(2.15) [al_<m -Io JR ~"
+ BT{D[OC~(e-Ắ)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
m a x
/ó/o
(2.16) + ~ E (AD[Õ(e-~(')u)l,D[Õ(e-~(')u)])dxdt
[aI<m
Finally, if in ađition, (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-ắ)q(t,')llhm}dt
Clearly, (2.14) m e a n s that f a n d g can have an exponential g r o w t h as
Ix I -+ cẹ This is g o o d e n o u g h for m a n y applications
Trang 2w 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 B S P D E (1.5) with a and b replaced by ~ and b
w U n i q u e n e s s 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 U n i q u e n e s s o f a d a p t e d w e a k solutions
For convenience, we denote
3dq ~ V.[Bq] + ( b, q )
Then, equation (1.1) is the same as the following:
(3.2)
U[t=T = g
In this section, we are going to prove the following result
T h e o r e m 3.1 Let (2.3) hold and the following hold:
(3.3)
A 9 L~(O,T;L~176
B E L~/Oj:~, T', LO~/R,~.~ , ~ n • d)),
a 9 Lcff(O,T;L~176
b 9 L~176176
c 9 L ~ ( 0 , T; n ~ 1 7 6
(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 ( I R ~) is the dual space of H l ( ~ 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
Trang 3112 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),
a n d
(3.6)
n
" ~ n
( Z o~r ~ / o = -
Vr E L2(IRn), 1 < i < n,
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 , d W ( t ) ) , t E [ 0 , T], (3.8) (u, ~)0 It=T = (g, ~)o
Here, (q, ~O)o ~((ql, ~)o,'", (qd, ~)0) and q = ( q l , ' " , qd) Sometimes, we say that (3.2) holds in H - I ( I R '~) if (3.8) holds for all ~ E H I ( R n )
In proving the uniqueness of the adapted weak solutions, the following special type of It6's formula is very crucial
L e m m a 3.2 Let ~ E L2(O, T; H - I ( ~ ) ) and (u, q) satisfy (3.4), such that
(3.9)
Then
(3.10)
0 t
lu(t)l~ = lu(O)l~ + {2 (r + Iq(s)l~}ds
.//
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
Trang 4w Uniqueness of adapted solutions 113
with f and g being zero, such that (3.4) holds We need to show t h a t (u, q) 0, which gives the uniqueness of adapted weak solution Applying Lemma 3.2, we have (note (3.7))
P T
- Iq + B T D u - bul 2
+ [b 2 + 2c - V-(a + Bb)]u2}ds
/ * 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
w A n 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 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 , V u e H , v 9
Due to this reason, H is usually called the pivot space It is also known (see [Lions]) t h a t in the present setting, there exists a symmetric linear operator
Now, let us state the following result which is more general than Lemma 3.2
Trang 5114
L e m m a 3.3 Let
(3.14)
satisfying
(3.15)
Then
Chapter 5 Linear, Degenerate BSPDEs
[ ~ 9 c~([0, T]; V),
9 L~(0, T; Y'),
d u = ~ d t + ( 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 = ( q l , ' " , 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
and let Uk be defined by
(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)
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 t h a t to prove (3.16), much more has to be involved
Trang 6w 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])
L e m m a 3.4 Let v : [0, T] -+ V ~ be absolutely continuous, such that
( b 9 L2(0, T; V')
Then v 9 C([O,T];H) and
(3.21) d [ v ( t ) t o 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
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) [M(t)l~ = 2 fot ((M(s),q(s))o,dW(s) ) + fo' [q(s)l~ds,
t 9 [0, T], a.s
L e m m a 3.5
any vo E H and f E L2(0,T; V'), the following problem
(3.22) ~ 7 ) = A v + f , t E [ 0 , T ] ,
I v(0) = vo,
admits a unique solution v satisfying (3.20) and
(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 ( A v ( s ) + f(s), v(s) )ods, t 9 [0, r ]
Now, we consider stochastic evolution equations We first have the following result
L e m m a 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:
Trang 7116
(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) []
L e m m a 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 = A v + f + A M , t ~ [0,T],
v ( 0 ) = u o
Trang 8w Uniqueness of adapted solutions 117
By L e m m a 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 L e m m a 3.4) v C Cj:([0, T]; H ) and (by (3.24))
Vt ~ [0, T], a.s
is a solution of (3.32) We now combining (3.27)-(3.28) and (3.34) (3.35)
to obtain the following:
= luolo2 + 2 ( A v ( s ) + f ( s ) + A M ( s ) , v ( s ) ) o d s
= luolg + {2 ( A ~ ( s ) + S ( s ) , ~ ( s ) ) o + Iq(s)lg}ds
+ 2 ( ( ~ ( s ) , q ( s ) ) o , a W ( s ) )
Next, we claim t h a t solution to (3.32) is unique (for any f , q and uo sat- isfying (3.31)) As a m a t t e r 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
Trang 9118 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))
//
_< 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 ~= ~ - A u 9 L~(0, T; V')
Then u is a solution of (3.32) with (3.31) holds Hence, (3.33) holds, which
Now, by taking V = H I ( R ~ ) , H = L2(R ~) and V' = H-I(IR~), we see
t h a t Lemma 3.2 follows immediately from Lemma 3.3
w E x i s t e n c e o f A d a p t e d S o l u t i o n s
The proofs of existence of adapted solutions is based on the following fun- damental lemma
L e m m a 4.1 Let the parabolicity condition (1.6) and the s y m m e t r y 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 - B B T ) 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
Trang 10w Existence of adapted solutions 119
following:
(4.2)
the above can be replaced by the
/ {E
I~l_<m
<c/ E
I~l_<m
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,,
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:
c F ( ~ n) • c F ( 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
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
(4.4)
for [a 1 = l , 2 , , m - 1 Consequently, if h C L2(0, T; C ~ ) , then
L " ( h(s, 0 ~ .), dW(s) ) e L~(O, T; C~),
and (4.4) holds for all multi-index a