Carleman Estimate for Stochastic Parabolic Equations and Inverse Stochastic Parabolic Problems

Carleman Estimate for Sto c hastic P arab olic Equations and In v erse Sto c hastic P arab olic Problems ∗ Qi L ¨ u † Abstract In this pap er, w e establish a global Carleman estimate for sto c hastic p a rab olic equa- tions. Based on this estimate, w e study t w o inv erse p r oblems for stochasti c parab olic equations. One is concerned with a determination pr ob lem of the history of a sto c has- tic heat pr ocess through the observ ation a t the final time T , for whic h w e obtai n a conditional stabilit y estimate. The other is an in v erse sour ce problem with observ ation on the lateral b oundary . W e deriv e the u n iqueness of the source. 2010 Ma thematics Sub ject Classification . Primary 65N21, 60 H15. Key W ords . Sto c hastic par abolic equations, Carleman estimate, conditional stability , in- v erse source problem 1 In tro ductio n In this pap er, w e study t w o differen t in v erse problems for sto c hastic para bolic equations by establishing a global Carleman estimate. W e first in tro duce some no tations. Let T > 0, G ⊂ R n ( n ∈ N ) b e a giv en b ounded doma in with a C 2 b oundary Γ. Put Q △ = (0 , T ) × G, Σ △ = (0 , T ) × Γ . Let (Ω , F , {F t } t ≥ 0 , P ) b e a complete filtered probability space on whic h a one dimensional standard Bro wnian motion { B ( t ) } t ≥ 0 is defined. Let H be a Banac h space. Denote by L 2 F (0 , T ; H ) the Banac h space consisting of all H -v alued {F t } t ≥ 0 -adapted pro cess es X ( · ) suc h that E ( | X ( · ) | 2 L 2 (0 ,T ; H ) ) < ∞ , with the canonical no r m; b y L ∞ F (0 , T ; H ) the Banach space consisting of all H -v alued {F t } t ≥ 0 -adapted b ounded pro ces ses; b y L 2 F (Ω; C ([0 , T ]; H )) ∗ This work was partially suppo rted b y the NSF o f China under gr an ts 1110 1070, and Grant MTM2008 - 03541 of the MICINN, Spain, Pro ject PI2 010-04 of the Bas q ue Gov ernmen t, the ER C Adv anced Grant FP7-24 6775 NU MERIW A VES and the E SF Research Net w orking Pr ogramme OPTPDE. † Basque Center for Applied Mathematics (BCAM), Mazarredo 14, 48 0 09, Bilbao, Basque Country , Spain; and Sch o ol of Mathematical Sciences, Universit y o f Electro nic Science and T echnology of China , C he ng du 61005 4, China . E-mail: l uqi59@163.com . 1 the Banac h space consisting of all H -v alued {F t } t ≥ 0 -adapted pro cesses X ( · ) satisfying tha t E ( | X ( · ) | 2 C (0 ,T ; H ) ) < ∞ , w ith the canonical norm(similarly , one can define L 2 F (Ω; C k ([0 , T ]; H )) for any p ositiv e k ). Throughout this pap er, w e mak e the follow ing assumptions on the co efficien ts b ij : Ω × Q → l R , ( i, j = 1 , 2 , · · · , n ) : (H1) b ij ∈ L 2 F (Ω; C 1 ([0 , T ]; W 2 , ∞ ( G ))) and b ij = b j i ; (H2) T h er e is a c onstant σ > 0 such that n X i,j =1 b ij ( ω , t, x ) ξ i ξ j ≥ σ | ξ | 2 , ( ω , t, x, ξ ) ≡ ( ω , t, x, ξ 1 , · · · , ξ n ) ∈ Ω × Q × l R n . (1.1) Let a 1 ∈ L ∞ F (0 , T ; L ∞ ( G ; l R n )) , a 2 ∈ L ∞ F (0 , T ; L ∞ ( G )) a 3 ∈ L ∞ F (0 , T ; W 1 , ∞ ( G )) , f ∈ L 2 F (0 , T ; L 2 ( G )) and g ∈ L 2 F (0 , T ; H 1 ( G )) . Consider the follow ing sto c hastic para bolic equation:              dy − n X i,j =1 ( b ij y x i ) x j dt =  ( a 1 , ∇ y ) + a 2 y + f  dt + ( a 3 y + g ) dB in Q, y = 0 on Σ , y (0) = y 0 , in Ω , (1.2) where y 0 ∈ L 2 (Ω , F 0 , P ; L 2 ( G )) and y x i = ∂ y ∂ x i . W e first recall the definition o f the weak and strong solution of equation (1.2) and giv e some w ell-p osedne ss results. Definition 1.1 We c al l a sto chas tic pr o c ess y ∈ L 2 F (Ω; C ([0 , T ]; L 2 ( G ))) ∩ L 2 F (0 , T ; H 1 0 ( G )) ∩ L 2 F (Ω; C (( 0 , T ]; H 1 0 ( G ))) a we ak solution of e quation (1.2) if for any t ∈ [0 , T ] and any p ∈ H 1 0 ( G ) , it ho l d s that Z G y ( t, x ) p ( x ) dx − Z G y 0 ( x ) p ( x ) dx = Z t 0 Z G n − n X i,j =1 b ij ( s, x ) y x i ( s, x ) p x j ( x ) +  a 1 ( s, x ) , ∇ y ( s, x )  + a 2 ( s, x ) y ( s, x ) + f ( s, x )  p ( x ) o dxds + Z t 0 Z G  a 3 ( s, x ) y ( s, x ) + g ( s, x )  p ( x ) dxdB , P -a.s. (1.3) 2 Definition 1.2 A pr o c es s y ∈ L 2 F (Ω; C ([0 , T ]; H 2 ( G ) ∩ H 1 0 ( G ))) is said to b e a str ong solution of e quation (1 .2) if for any t ∈ [0 , T ] , it hol d s that y ( t ) = y 0 + Z t 0 n − n X i,j =1  b i,j ( s ) y i ( s )  j +  ( a 1 ( s ) , ∇ y ( s )) + a 2 ( s ) y ( s ) + f ( s )  o ds + Z t 0 Z G  a 3 ( s ) y ( s ) + g ( s )  dB , P -a.s. (1.4) Ob viously , strong solution of equation ( 1 .2) is also its w eak solution. W e ha v e the following w ell-p osedness results for equation (1.2), whose pro of can b e found in [11, Chapter 6]. Lemma 1.1 The r e exists a unique we ak solution of e quation (1.2) . F urthermor e, it holds that | y | L 2 F (Ω; C ([0 ,T ]; L 2 ( G ))) + | y | L 2 F (0 ,T ; H 1 0 ( G )) ≤ C r 1 | y 0 | L 2 (Ω , F 0 ,P ; L 2 ( G )) . (1.5) Here and in the sequel, r 1 △ = | a 1 | 2 L ∞ F (0 ,T ; L ∞ ( G ;l R n )) + | a 2 | 2 L ∞ F (0 ,T ; L ∞ ( G )) + | a 3 | 2 L ∞ F (0 ,T ; W 1 , ∞ ( G )) + 1 . Lemma 1.2 L et y 0 ∈ L 2 (Ω , F 0 , P ; H 2 ( G ) ∩ H 1 0 ( G )) , b 1 ∈ L ∞ F (0 , T ; W 1 , ∞ ( G ; l R n )) , b 2 ∈ L ∞ F (0 , T ; W 1 , ∞ ( G )) , and b 3 ∈ L ∞ F (0 , T ; W 2 , ∞ ( G )) . The n ther e exists a unique str ong s o lut ion of e quation (1 .2) . Next, we recall the following Itˆ o’s formula, whic h plays a ke y role in the sequel. Lemma 1.3 [ It ˆ o’s form ula ] L et X ( · ) ∈ L 2 F (0 , T ; H 1 0 ( G )) b e a c ontinuous pr o c es s with val- ues in H − 1 ( G ) . Supp ose that ther e exist X 0 ∈ L 2 (Ω , F 0 , P ; L 2 ( G )) , Φ( · ) ∈ L 2 F (0 , T ; H − 1 ( G )) and Ψ( · ) ∈ L 2 F (0 , T ; L 2 ( G )) such that for any t ∈ [0 , T ] , it holds that X ( t ) = X 0 + Z t 0 Φ( s ) ds + Z t 0 Ψ( s ) dB , P -a.s. (1.6) in H − 1 ( G ) . Then we hav e that | X ( t ) | 2 L 2 ( G ) = | X (0) | 2 L 2 ( G ) + 2 Z t 0  X ( s ) , Φ( s )  H 1 0 ( G ) ,H − 1 ( G ) ds +2 Z t 0  X ( s ) , Ψ( s )  L 2 ( G ) dB + Z t 0 | Ψ( s ) | 2 L 2 ( G ) ds (1.7) for arbitr ary t ∈ [0 , T ] . Remark 1.1 Her e we o n ly pr esent a sp e cia l c ase for the Itˆ o’s formula. It is e n ough for the pr o of in our p a p er. The gener al form c an b e fo und in [2 5, Chapter 1]. 3 Remark 1.2 Ob viously, b oth the we ak and str ong solution of e quation (1.2) satisfy the as- sumptions for L emma 1.3. I n this p ap er, we sometimes use the differ ential form the the ab ove Itˆ o’s fo rmula, that is, d ( X 2 ) = 2 X dX + ( dX ) 2 , for the s implicity of notations. In this pap er, we establish a Carleman estimate for equation (1.2). The so-called Carle- man estimate is a class of w eigh ted energy estimates whic h is in connection with (sto c hastic) differen tial op erators. As far as w e kno w, t he first example of suc h kind of estimate app eared in Carleman’s pioneer w o rk for the uniqueness of the solution of first order elliptic system with tw o v ariables(see [8]). The idea w as generalized to g et the uniqueness of the solutions for general Cauc hy problems in [6]. Now it is a useful to ol for studying the uniqueness and unique con tinuation prop ert y for partial differen tial equations(see [15 ] for example). Suc h kind of estimate has b een in tro duced to solving inv erse problems in [4], and w ere comprehen- siv ely studied in [18, 22]. No w it is a helpful metho dology for solving in verse problems ( e.g. [18, 21, 22, 29 , 30]). Altho ug h the form of Carleman estimate seems to b e very complex, the idea b ehind them is v ery simple. One can understand it b y the follow ing example. Let    dx dt = a ( t ) x in (0 , T ] , x (0) = x 0 . (1.8) Here x 0 ∈ l R a nd a ( · ) ∈ L ∞ (0 , T ). W e pro ve that there exists a constan t C > 0 suc h that for an y x 0 ∈ l R, | x ( T ) | ≤ C | x 0 | by Carleman estimate. This result is almost trivial. And one can pro v e it without utilizing Carleman estimate. Ho w ev er, the pro of employ ed here show s all the ideas of Carleman estimate. Let ˜ x ( t ) = e − ς t x ( t ) with ς ≥ 0. Then w e hav e d ˜ x 2 dt = 2 a ( t ) ˜ x 2 − 2 ς ˜ x 2 = 2  a ( t ) − ς  ˜ x 2 . If w e c ho ose ς ≥ | a | L ∞ (0 ,T ) , then we know that d ˜ x 2 dt ≤ 0, whic h implies that ˜ x 2 ( T ) ≤ ˜ x 2 (0). Hence, w e get e − 2 ς T x 2 ( T ) ≤ x 2 (0) . (1.9) F rom t his, w e o btain | x ( T ) | ≤ e T | a | L ∞ (0 ,T ) | x 0 | immediately . Th us, we prov e the desired result and we kno w C can b e c hosen to b e e T | a | L ∞ (0 ,T ) . Inequalit y (1 .9) is a kind of Carleman es tiam t e. The function e − ς t is called weigh t function and ς is a para me ter whic h can b e c hosen for our purp ose. By means of the c hoice of ς , w e con tro l the lo wer order term a ( t ) x and o btain inequality (1.9). F or (sto c hastic) part ia l differen tial equations, b oth the c ho ice o f the w eight function and the computation a re m uch more complex. Ho wev er, they enjoy the same idea. No w w e in tro duce the Carleman estimate to b e established in this pap er. T o start with, w e giv e some f unc tions. Let s ∈ (0 , + ∞ ) , t ∈ (0 , + ∞ ), and ψ ∈ C ∞ (l R) with | ψ t | ≥ 1, whic h is indep enden t of the x -v ariable. Put ϕ = e λψ and θ = e sϕ . (1.10) W e hav e the fo llo wing r es ult. 4 Theorem 1.1 L et δ ∈ [0 , T ) . L et ϕ and θ b e given in (1.10) . Ther e exists a λ 1 > 0 such that for al l λ ≥ λ 1 , ther e exists an s 0 ( λ 1 ) > 0 so that fo r al l s ≥ s 0 ( λ 1 ) , it holds that λ E Z T δ Z G θ 2 |∇ y | 2 dxdt + sλ 2 E Z T δ Z G ϕθ 2 y 2 dxdt ≤ C E h θ 2 ( T ) |∇ y ( T ) | 2 L 2 ( G ) + θ 2 ( δ ) |∇ y ( δ ) | 2 L 2 ( G ) + sλϕ ( T ) θ 2 ( T ) | y ( T ) | 2 L 2 ( G ) + sλϕ ( δ ) θ 2 ( δ ) | y ( δ ) | 2 L 2 ( G ) + Z T δ Z G (1 + ϕ ) θ 2  f 2 + g 2 + |∇ g | 2  dxdt i , (1.11) Her e y is arbi tr ary we ak so l ution of e quation (1 .2) . Here and in the sequel, the constant C dep ends o nly on G , ( b ij ) n × n , T , δ and ψ , which ma y c ha nge from line to line. Although there are n umerous results for the global Carleman estimate for deterministic parab olic equations(see [13, 29] for example), people kno w very little ab out the sto c hastic coun terpar t . In fa ct, a s f a r as w e kno w, [2, 28] are the only tw o published pa pers addressing the global Carleman estimate fo r sto c hastic para bolic equations. In [2, 28], some Carleman- t ype inequalities w ere established, for deriving the nu ll controllabilit y o f sto c hastic parab olic equations. Note f urther that the w eigh t f unction θ used in this pap er (whic h play s a k ey role in the sequel) is quite differen t f rom that in [2, 28]. It seems that the Carleman estimate in [2, 28] cannot b e a pplie d to studying the in v erse problems in tro duced in ths sequel. Indee d, the w eigh t f unction θ in [2, 28] is supp osed to v anish at 0 and T , a nd therefore it do es not serv e the purp ose of proving Theorem 1.2 and Theorem 1 .4. As applicatio ns of Theorem 1.1, w e study tw o in v erse problems for sto c hastic parab olic equations. There are abundant w orks addressing the in v erse problems for PD Es . And it is ev en imp ossible to list the related pap ers owing to the big amoun t . Ho w ev er, there exist a v ery few w orks addressing in v erse problems for sto c hastic PDEs (see [3, 9, 1 6] for ex am- ple). Although there are some p eople considering the in v erse source pro blem for para bolic equations with random noise in the measuremen t (see [20] for example), to the b est o f o ur kno wledge, there is no pap er considering the in v erse problem for sto c hastic par a bolic equa- tions. No w w e intro duc e the inv erse problems studied in this pap er. Consider the following sto c hastic parab olic equation:            dy − n X i,j =1 ( b ij y x i ) x j dt = [( a 1 , ∇ y ) + a 2 y ] dt + a 3 y dB in Q, y = 0 on Σ , y (0) = y 0 in Ω . (1.12) Here y 0 ∈ L 2 (Ω , F 0 , P ). The first in v erse problem is concerned with the f o llo wing pr o blem : Sto c hastic parab olic equation bac kward in t ime: L e t 0 ≤ t 0 < T . Determine y ( · , t 0 ) , P -a. s . fr o m y ( · , T ) . 5 F or deterministic parab olic equations, suc h kind of problem has lots of applications in mathematical ph ysics (e.g. [1]) and is studied extensiv ely (see [30] for a nice surv ey). Gener- ally sp eaking, the problem of (sto c hastic) parab olic equation backw ard in time is ill-p osed. Small errors in the measuring of the terminal dat a ma y cause h uge deviations in final results, that is, there is no stabilit y in this problem. F ortunately , if w e a ss ume a priori bound for y (0) (suc h assumption is reasonable from a practical viewp oin t), then w e can regain the stability in some sense. The concept o f conditional stabilit y is used to describ e suc h kind of stability . In general framew ork, the conditional stabilit y problem can b e form ulated as follows: L et t 0 ∈ [0 , T ) , α 1 ≥ 0 , α 2 ≥ 0 and M > 0 . Put U M ,α 1 △ = { f ∈ L 2 (Ω , P , F 0 ; H α 1 ( G )) : | f | L 2 (Ω ,P , F 0 ; H α 1 ( G )) ≤ M } . If y 0 ∈ U M ,α 1 , then c an we cho ose a function β ∈ C [0 , + ∞ ) satisfying the fol lowing pr op erties:        1 . β ≥ 0 and β is strictly incr e asing ; 2 . lim η → 0 β ( η ) = 0; 3 . | y ( t 0 ) | L 2 (Ω , F t 0 ,P ; H α 1 ( G )) ≤ β  | y ( T ) | L 2 (Ω , F T ,P ; H α 2 ( G ))  . Remark 1.3 Her e we exp e ct the e x i s tenc e of β w i th the assumptions that y 0 b elongs to a sp e cia l set U M ,α 1 , w hich me an s that y 0 enjoys a priori b o und in s o m e sense. Gener al ly sp e aking, β dep ends on M and α 1 . On c e we cho ose M and α 1 , we a dd s o me c onditions to the initial data of e quation (1.12) . Henc e, that the stability r esult impli e d by β dep ends on our choic e of the initial da t a. This is why we c al l i t “c ondition a l stability”. Remark 1.4 Th e first pr op erty for β me a ns that we only ch o ose β in a sp e cial cl a ss of functions, that is, the strictly incr e as i n g functions. The se c ond and the thir d pr op erty g uar- ante e the c onditional stability. Without assuming pr op erty 2, we c an alw ays c onstruct β as β ( x ) = C + x with a c onstant C which is lar g e enough. However, such kind of functions do not make any sen s e for c on ditional stability. Remark 1.5 On c e β exists, it is not unique. F or example , ˜ b ( x ) = β ( x ) + x is another function satisfying the thr e e pr op erties. In this pap er, w e obtain the follo wing in terp olation inequalit y fo r the w eak solution of equation (1.12 ), whic h implies a conditional stabilit y result for equation (1.12 ) bac kw ard in time. Theorem 1.2 L et t 0 ∈ [0 , T ] . Then ther e ex i s t a c onstant θ ∈ (0 , 1) a n d a c onstant C > 0 such that | y ( t 0 ) | L 2 (Ω , F t 0 ,P ; L 2 ( G )) ≤ C | y | 1 − θ L 2 F (0 ,T ; L 2 ( G )) | y ( T ) | θ L 2 (Ω , F T ,P ; H 1 ( G )) , (1.13) for any y solving e quation (1 .12) in the sense of w e ak solution. 6 As a consequence, we obtain the following result. Theorem 1.3 L et y 0 ∈ U M , 0 , α 2 = 1 and β ( x ) = C M 1 − θ x θ with a c onstant C inde p endent of y (0) . The n we have | y ( t 0 ) | L 2 (Ω , F t 0 ,P ; L 2 ( G )) ≤ β  | y ( T ) | L 2 (Ω , F T ,P ; H 1 ( G ))  . The pro of of Theorem 1.3 follows Lemma 1.1 and Theorem 1.2 immediately . W e omit it here. In deterministic setting, a result which is stronger than Theorem 1.2 w as o btained in [26], where the autho r s study the following equation: ( y t − ∆ y = by in Q, y = 0 on Σ . (1.14) Here b is a suitable function. With the assumption tha t G is conv ex, they get | y (0) | 2 L 2 ( G ) ≤ C exp  | y (0) | L 2 ( G ) | y (0) | H − 1 ( G )  | y ( T ) | 2 L 2 ( G 0 ) . (1.15) Here G 0 is an y op en subset of G . Compared with Theorem 1 .2, only | y ( T ) | 2 L 2 ( G 0 ) is in v olv ed in the righ t hand side of the inequalit y . They prov e this result by emplo ying some special frequency functions, whic h w ere first constructe d for proving the doubling prop erty of the solution o f heat equations. Ho w ev er, since the solution of equation (1.12) is no n- differen tiable with resp ect to t , it seems that their method cannot be easily a do pted to solv e our problem. As another consequence of The orem 1.2, w e g et a backw ard uniquene ss for equ ation (1.12). Corollary 1.1 Assume that y is a we ak solution of e quation (1.12). If y ( T ) = 0 in G , P -a.s., then y ( t ) = 0 in G , P -a.s. for al l t ∈ [0 , T ] . The uniquenes s problems for the solutions of bo th deterministic and sto c hastic par t ia l differen tial equations ha v e b een studie d for a long time. T here are a great man y p ositiv e results and some negativ e results. In cas e of time rev ersible systems, the bac kward uniqueness is equiv alen t to the classical (forw ard) uniqueness. If one considers time irrev ersible systems, suc h as para bolic equations, the situation is quite differen t. The back w ard uniqueness implies the classic al (forw ard) uniquenes s, how ev er, generally sp eaking, the con v erse conclusion is un tr ue . On account of the plen tiful applications, suc h as studying the long time b eha vior o f solutions and establishing the approximate controllabilit y from the null con t r o llabilit y , the bac kward uniqueness fo r parab olic equations draw s lots of atten tion(see [12, 14, 23, 24, 27 ] and the reference s cited therein). It is w ell understo o d no w. On the contrast, as far as w e kno w, [5] is the only pap er concerned with back w ard uniqueness f or sto c hastic parab olic equations in the literature. In [5], the authors obtained t he back w ard uniqueness for semilin- ear sto c hastic para bolic equations with deterministic co efficien ts. They employ ed some deep 7 to ols in Sto chas tic Analysis to establish the result. Ho w ev er, it seems that their metho d dep ends on the very fact t ha t the co efficien ts are deterministic and one cannot simply mimic their metho d to obtain Corollary 1 .1, since the co efficien ts are random. The other in v erse problem studied in this pap er is a bout the g lobal uniqueness of a n in verse source problem for sto c hastic parab olic equations. W e first giv e a precise formulation of the problem. Let x = ( x 1 , x ′ ) ∈ l R n and x ′ = ( x 2 , · · · , x n ) ∈ l R n − 1 . Consider a special G as G = (0 , l ) × G ′ , wh ere G ′ ⊂ l R n − 1 b e a b ounded do main with a C 2 b oundary . W e cons ider the follo wing sto c ha stic para bolic equation:        dy − ∆ y = [( b 1 , ∇ y ) + b 2 y + h ( t, x ′ ) R ( t, x )] dt + b 3 y dB ( t ) in Q, y = 0 on Σ , y (0) = 0 in G. (1.16) Here b 1 ∈ L ∞ F (0 , T ; W 1 , ∞ ( G ; l R n )) , b 2 ∈ L ∞ F (0 , T ; W 1 , ∞ ( G )) , b 3 ∈ L ∞ F (0 , T ; W 2 , ∞ ( G )) , and R ∈ C 2 ([0 , T ] × G ) , h ∈ L 2 F (0 , T ; H 1 ( G ′ )) . The inv erse source problem studied here is as fo llo ws: L et R b e given an d 0 < t 0 < T . De termi ne the sour c e function h ( t, x ′ ) , ( t, x ′ ) ∈ (0 , t 0 ) × G ′ , by me ans of the obse rv ation of ∂ y ∂ ν    [0 ,t 0 ] × ∂ G . Here ν = ( ν 1 , · · · , ν n ) ∈ l R n is the outer nor mal v ector of Γ. W e hav e the fo llo wing uniqueness r esult ab out the a bov e problem. Theorem 1.4 L et | R ( t, x ) | 6 = 0 for al l ( t, x ) ∈ [0 , t 0 ] × G. (1.17) If ∂ y ∂ ν = 0 on [0 , t 0 ] × ∂ G, P -a.s. , then h ( t, x ′ ) = 0 for al l ( t, x ′ ) ∈ [0 , t 0 ] × G ′ , P -a.s. Remark 1.6 On e c an fol low the pr o of of The or em 1.4 to show that T h e or em 1.4 also holds when ∆ y is substitute d by n X i,j =1  b ij y i  j . Her e we c onsider e quation (1 .16) for the sake of pr esenting the key ide a i n a si m ple way. 8 In practical problems, it is imp ortan t to sp ec ify some prop er data so that the parameter to b e reconstructed is uniquely iden tifiable. In our mo del, the data utilized is the b oundary normal deriv ativ e of the solution. This type of in v erse problem is imp ortan t in man y branc hes of engineering sciences. F or example s, an accurate estimation of a p ollution source in a river, a determination of mag nitude of groundw ater p ollution sources. In the literature, determining a spacewis e dep end en t source function for parab olic equa- tions has b een considered comprehensiv ely(see [7, 10, 18, 19, 30] and t he references cited therein). A classical result fo r the deterministic setting is as follows. Consider the follow ing para bolic equation: ( y t − ∆ y = c 1 ∇ y + c 2 y + Rf in Q, y = 0 on Σ . (1.18) Here c 1 and c 2 are suitable functions on Q . R ∈ L ∞ ( Q ), R t ∈ L ∞ ( Q ) and R ( t 0 , x ) 6 = 0 in G for some t 0 ∈ (0 , T ]. f ∈ L 2 ( G ) is indep enden t of t . The aut ho rs in [17] prov ed the following result: Assume that y ∈ H 2 , 1 ( Q ) and y t ∈ H 2 , 1 ( Q ) , then ther e exi s ts a c onstant C > 0 such that | f | L 2 ( G ) ≤ C  | y ( t 0 ) | H 2 ( G ) +    ∂ y t ∂ ν    L 2 (0 ,T ; L 2 (Γ 0 ))  , (1.19) wher e Γ 0 is any op en subse t of Γ . Compared with Theorem 1.4, inequality (1 .19) giv es an explicit estimate fo r the source term by | y ( t 0 ) | H 2 ( G ) and    ∂ y t ∂ ν    L 2 (0 ,T ; L 2 (Γ 0 )) . A k ey step in the pro of of equality (1.19) is to differen tiate the solution of (1.18 ) with res p ect to t . Unfortunately , the solution of ( 1 .16) do es not enjo y differen tiability with resp ec t to t since the effect of the sto c hastic noise. How ev er, w e can b orro w some idea from the pro of of inequalit y (1 .1 9). Although it is imp ossible for us to assume that the solutio n of equation (1 .16) is differen tia ble with r esp ect t o t , w e can sho w that it is differen tiable with respect to x with some a ssumptions(the a ssumptions in this pap er is enough). In this case, we can show the uniqueness of h if h is indep enden t of some x i ( i = 1 , · · · , n ). Here w e supp ose that h is indep enden t of x 1 . Ob viously , b oth equation (1.12) and equation (1.16) are sp ecial examples of equation (1.2). Remark 1.7 As w e have p ointe d out, the non-diff er entiability wi th r esp e ct to the variable with noise (s a y, the time v a riable c onsider e d in this p a p er) of the solution of a sto chastic PDE usual ly le ads to substantial ly new difficulties in the s t udy of inverse pr oblems f or sto ch a stic PDEs. Another tr ouble for studying the inverse pr oblem of sto chastic PDEs is that the usual c omp actness emb e dding r e sult do es not r e main true for the solution sp a c es r elate d to sto chastic PDEs. Due to these new difficulties, so m e useful metho ds for solving inverse pr oblems for deterministic PDEs (se e [18, 22] for example) c annot b e use d to solve the c orr esp onding inverse pr oblem s in the sto chastic setting. The rest pap er is o rganized as follows. In Section 2, we prov e Theorem 1.1. Section 3 is addressed the pro of of Theorem 1.2. A t last, in Section 4 , w e giv e a pro of for Theorem 1.4. 9 2 Carleman esti mate for st o c hastic p arab o lic equations In this section, we prov e Theorem 1.1. W e first giv e a weigh ted identit y , whic h pla ys an impo rtan t role in the pro of of Theorem 1.1. Prop osition 2.1 Assume that u is an H 2 ( R n ) -value d c on tinuous semi- m artingale. Put v = θ u (r e c al l (1.10) for the de fi nition of θ ). Then we have the fol lowing e quality: − θ h n X i,j =1 ( b ij v x i ) x j + sλϕψ t v ih du − n X i,j =1 ( b ij u x i ) x j dt i + 1 4 λθ v h du − n X i,j =1 ( b ij u x i ) x j dt i = − n X i,j =1  b ij v x i dv + 1 4 b ij v x i v dt  x j + 1 2 d  n X i,j =1 b ij v x i v x j − sλϕψ t v 2 + 1 8 λv 2  −  1 2 n X i,j =1 b ij dv x i dv x j + 1 2 n X i,j =1 b ij t v x i v x j dt − 1 4 λ n X i,j =1 b ij v x i v x j dt  + 1 2 sλ 2 ϕψ 2 t v 2 dt + 1 2 sλϕψ tt v 2 dt − 1 4 sλ 2 ψ t ϕv 2 dt + 1 2 sλϕψ t ( dv ) 2 − 1 8 λ ( dv ) 2 + h n X i,j =1 ( b ij v x i ) x j + sλϕψ t v i 2 dt. (2.1) Pr o of : The pro of is based on some direct computation b y Itˆ o’s sto c hastic calculus. The first term in the left hand side of equalit y (2 .1) reads as − θ h n X i,j =1 ( b ij v x i ) x j + sλϕψ t v ih du − n X i,j =1 ( b ij u x i ) x j dt i = − h n X i,j =1 ( b ij v x i ) x j + sλϕψ t v ih dv − n X i,j =1 ( b ij v x i ) x j dt − sλϕψ t v dt i = − n X i,j =1 ( b ij v x i ) x j dv − sλϕψ t v dv + h n X i,j =1 ( b ij v x i ) x j + sλϕψ t v i 2 dt = − n X i,j =1 ( b ij v x i dv ) x j + 1 2 n X i,j =1 d ( b ij v x i v x j ) − 1 2 n X i,j =1 b ij dv x i dv x j − 1 2 n X i,j =1 b ij t v x i v x j dt − 1 2 d ( sλϕψ t v 2 ) + 1 2 sλ 2 ψ 2 t ϕv 2 dt + 1 2 sλϕψ tt v 2 dt + 1 2 sλψ t ϕ ( dv ) 2 + h n X i,j =1 ( b ij v x i ) x j + sλϕψ t v i 2 dt. 10 The second term in the left hand side of equalit y (2.1 ) satisfies 1 4 λθ v h du − n X i,j =1 ( b ij u x i ) x j dt i = 1 4 λv h dv − n X i,j =1 ( b ij v x i ) x j dt − sλϕψ t v dt i = 1 4 λv dv − 1 4 λv n X i,j =1 ( b ij v x i ) x j dt − 1 4 sλ 2 ϕψ t v 2 dt = 1 8 λdv 2 − 1 8 λ ( dv ) 2 − 1 4 λ n X i,j =1 ( b ij v x i v ) x j dt + 1 4 λ n X i,j =1 b ij v x i v x j dt − 1 4 sλ 2 ψ t ϕv 2 dt. This, tog ethe r with equalit y (2 .2), implies equalit y (2.1). No w w e a re in a p osition to prov e Theorem 1 .1 . Pr o of of The or em 1.1 : Applying Prop osition 2 .1 to eq uation (1.2) with u = y , in tegrating equalit y (2.1) on [ δ, T ] × G for some δ ∈ [0 , T ), and taking mathematical exp ectation, w e get that − E Z T δ Z G θ h n X i,j =1 ( b ij v x i ) x j + sλϕψ t v ih du − n X i,j =1 ( b ij u x i ) x j dt i dx + 1 4 λ E Z T δ Z G θ v h dy − n X i,j =1 ( b ij y x i ) x j dt i dx = − E Z T δ Z G n X i,j =1  b ij v x i dv + 1 4 λb ij v x i v dt  x j dx + 1 2 E Z T δ Z G d  n X i,j =1 b ij v x i v x j − sλϕψ t v 2 + 1 8 λv 2  dx − E Z T δ Z G  1 4 λ n X i,j =1 b ij v x i v x j dt + 1 2 n X i,j =1 b ij t v x i v x j dt − 1 2 n X i,j =1 b ij dv x i dv x j  dx + E Z T δ Z G h 1 2 sλ 2 ϕψ 2 t v 2 dt + 1 2 sλϕψ tt v 2 dt − 1 4 sλ 2 ϕψ t v 2 dt + 1 2 sλϕψ t ( dv ) 2 − 1 8 λ ( dv ) 2 i dx + E Z T δ Z G h n X i,j =1 ( b ij v x i ) x j + sλϕψ t v i 2 dxdt. (2.2) No w w e estimate the terms in the rig ht hand side of equalit y (2.2) one by one. F or t he first one, since y | Σ = 0, w e ha v e that v | Σ = 0. The refore, it ho lds that − E Z T δ Z G n X i,j  b ij v x i dv + 1 4 λb ij v x i v dt  x j dx = − E Z T δ Z Γ n X i,j =1 b ij  v x i dv + 1 4 λv x i v dt  ν j d Γ = 0 . (2.3) 11 F or t he second one, w e hav e 1 2 E Z T δ Z G d  n X i,j =1 b ij v x i v x j − sλϕψ t v 2 + 1 8 λv 2  dx ≥ − C E  |∇ v ( T ) | 2 L 2 ( G ) + |∇ v ( δ ) | 2 L 2 ( G ) + sλϕ ( T ) | v ( T ) | 2 L 2 ( G ) + sλϕ ( δ ) | v ( δ ) | 2 L 2 ( G )  . (2.4) Since E Z T δ Z G 1 2 n X i,j =1 b ij dv x i dv x j dx = 1 2 E Z T δ Z G n X i,j =1 b ij θ 2  a 3 y + g  x i  a 3 y + g  x j dxdt, the third one r eads as E Z T δ Z G  1 4 λ n X i,j =1 b ij v x i v x j dt + 1 2 n X i,j =1 b ij t v x i v x j dt − 1 2 n X i,j =1 b ij dv x i dv x j  dx ≥ E Z T δ Z G h 1 4 λσ |∇ v | 2 − C | ∇ v | 2 − C  a 2 3 |∇ v | 2 + |∇ a 3 | 2 v 2 + θ 2 |∇ g | 2 + θ 2 | g | 2  i dxdt ≥ 1 4 λ E Z T δ Z G σ |∇ v | 2 dxdt − C ( | a 3 | 2 L ∞ F (0 ,T ; W 1 , ∞ ( G )) + 1) E Z T δ Z G ( |∇ v | 2 + v 2 ) dxdt − C E Z T δ Z G θ 2 ( |∇ g | 2 + g 2 ) dxdt. (2.5) F or t he forth one, r ec alling that | ψ t | ≥ 1 and utilizing that E Z T δ Z G h 1 2 sλϕψ t ( dv ) 2 − 1 8 λ ( dv ) 2 i dx = E Z T δ Z G θ 2 h 1 2 sλϕψ t ( a 3 y + g ) 2 − 1 8 λ ( a 3 y + g ) 2 i dxdt, w e see E Z T δ Z G h 1 2 sλ 2 ϕψ 2 t v 2 dt + 1 2 sλϕψ tt v 2 dt − 1 4 sλ 2 ϕψ t v 2 dt + 1 2 sλϕψ t ( dv ) 2 − 1 8 λ ( dv ) 2 i dx ≥ 1 4 sλ 2 E Z T δ Z G ϕv 2 dxdt + sO ( λ ) E Z T δ Z G ϕv 2 dxdt − C sλ E Z T δ Z G (1 + ϕ ) θ 2 g 2 dxdt. (2.6) Th us, w e kno w that there exists a λ 0 > 0 suc h t ha t for all λ ≥ λ 0 , it holds that E Z T δ Z G h 1 2 sλ 2 ϕψ 2 t v 2 dt + 1 2 sλϕψ tt v 2 dt − 1 4 sλ 2 ϕψ t v 2 dt + 1 2 sλϕψ t ( dv ) 2 − 1 8 λ ( dv ) 2 i dx ≥ 1 8 sλ 2 E Z T δ Z G ϕv 2 dxdt − C sλ E Z T δ Z G (1 + ϕ ) θ 2 g 2 dxdt. (2.7) 12 No w, we estimate the terms in the left hand side one by one. By equation ( 1.2) and noting that − E Z T δ Z G θ h n X i,j =1 ( b ij v x i ) x j + sλϕψ t v i ( a 3 y + g ) dB dx = 0 , w e know that − E Z T δ Z G θ h n X i,j =1 ( b ij v x i ) x j + sλϕψ t v ih dy − n X i,j =1 ( b ij y x i ) x j dt i dx = − E Z T δ Z G θ h n X i,j =1 ( b ij v x i ) x j + sλϕψ t v ih ( a 1 , ∇ y ) + a 2 y + f i dtdx ≤ E Z T δ Z G h n X i,j =1 ( b ij v x i ) x j + sλϕψ t v i 2 dxdt + E Z T δ Z G θ 2 h ( a 1 , ∇ y ) + a 2 y + f i 2 dtdx ≤ E Z T δ Z G h n X i,j =1 ( b ij v x i ) x j + sλϕψ t v i 2 dxdt + 3 E Z T δ Z G θ 2  | a 1 | 2 |∇ u | 2 + a 2 2 u 2 + f 2  dxdt ≤ E Z T δ Z G h n X i,j =1 ( b ij v x i ) x j + sλϕψ t v i 2 dxdt + 3 | a 1 | 2 L ∞ F (0 ,T ; L ∞ ( G ;l R n )) E Z T δ Z G |∇ v | 2 dxdt +3 | a 2 | 2 L ∞ F (0 ,T ; L ∞ ( G )) E Z T δ Z G v 2 dxdt + 3 E Z T δ Z G θ 2 f 2 dxdt, (2.8) and that 1 4 λ E Z T δ Z G θ v h du − n X i,j =1 ( b ij u x i ) x j dt i dx = 1 4 λ E Z T δ Z G θ v h ( a 1 , ∇ y ) + a 2 y + f i dtdx ≤ 1 64 λ E Z T δ Z G v 2 dxdt + E Z T δ Z G θ 2 h ( a 1 , ∇ y ) + a 2 y + f i 2 dtdx ≤ 1 64 λ 2 E Z T δ Z G v 2 dxdt + 3 E Z T δ Z G θ 2  | a 1 | 2 |∇ u | 2 + a 2 2 u 2 + f 2  dxdt ≤ 1 64 λ 2 E Z T δ Z G v 2 dxdt + 3 | a 1 | 2 L ∞ F (0 ,T ; L ∞ ( G ;l R n )) E Z T δ Z G |∇ v | 2 dxdt +3 | a 2 | 2 L ∞ F (0 ,T ; L ∞ ( G )) E Z T δ Z G v 2 dxdt + 3 E Z T δ Z G θ 2 f 2 dxdt. (2.9) 13 F rom (2 .2)–(2.9), w e find 1 4 λ E Z T δ Z G |∇ v | 2 dxdt − C  | a 1 | 2 L ∞ F (0 ,T ; L ∞ ( G ;l R n )) + | a 3 | 2 L ∞ F (0 ,T ; W 1 , ∞ ( G )) + 1  E Z T δ Z G |∇ v | 2 dxdt − C ( | a 2 | 2 L ∞ F (0 ,T ; L ∞ ( G )) + | a 3 | 2 L ∞ F (0 ,T ; W 1 , ∞ ( G )) + 1) E Z T δ Z G |∇ v | 2 dxdt +  1 8 sλ 2 − 1 64 λ 2  E Z T δ Z G ϕv 2 dxdt ≤ C E h |∇ v ( T ) | 2 L 2 ( G ) + |∇ v ( δ ) | 2 L 2 ( G ) + sλϕ ( T ) | v ( T ) | 2 L 2 ( G ) + sλϕ ( δ ) | v ( δ ) | 2 L 2 ( G ) + sλ Z T δ Z G (1 + ϕ ) θ 2  f 2 + g 2 + |∇ g | 2  dxdt i . (2.10) Recalling that r 1 = | a 1 | 2 L ∞ F (0 ,T ; L ∞ ( G ;l R n )) + | a 2 | 2 L ∞ F (0 ,T ; L ∞ ( G )) + | a 3 | 2 L ∞ F (0 ,T ; W 1 , ∞ ( G )) + 1 , from inequ alit y (2.1 0 ) , w e kno w that there ex ists a λ 1 ≥ max  C r 1 , λ 0  suc h that for all λ ≥ λ 1 , there exists a s 0 ( λ 1 ) > 0 so tha t for all s ≥ s 0 ( λ 1 ), it holds that λ E Z T δ Z G |∇ v | 2 dxdt + sλ 2 E Z T δ Z G ϕv 2 dxdt ≤ C E h |∇ v ( T ) | 2 L 2 ( G ) + |∇ v ( δ ) | 2 L 2 ( G ) + sλϕ ( T ) | v ( T ) | 2 L 2 ( G ) + sλϕ ( δ ) | v ( δ ) | 2 L 2 ( G ) + sλ Z T δ Z G (1 + ϕ ) θ 2  f 2 + g 2 + |∇ g | 2  dxdt i , (2.11) whic h implies inequalit y (1.11) immediately . 3 Pro of for Theo r e m 1 .2 This section is dev oted to the pro of of Theorem 1.2. W e b orro w some ideas from [30]. Pr o of of The or e m 1 . 2 : Cho ose t 1 and t 2 suc h that 0 < t 1 < t 2 < t 0 . Set α k = e λt k ( k = 0 , 1 , 2). Let ρ ∈ C ∞ ( R ) suc h that 0 ≤ ρ ≤ 1 a nd tha t ρ = ( 1 , t ≥ t 2 , 0 , t ≤ t 1 . (3.1) Let z = ρy , b y means o f y solves equation (1.1 2 ), w e kno w that z solv es              dz − n X i,j =1 ( b ij z x i ) x j dt =  ( a 1 , ∇ z ) + a 2 z + ρ t ( t ) y  dt + a 3 z dB ( t ) in Q, z = 0 on Σ , z (0) = 0 in G. (3.2) 14 Applying Theorem 1 .1 with ψ = t and δ = 0 to equation (3.2), f or λ ≥ λ 1 and s ≥ s 0 ( λ 1 ), w e hav e λ E Z Q θ 2 |∇ z | 2 dxdt + sλ 2 E Z Q θ 2 ϕ | z | 2 dxdt ≤ C E h θ 2 ( T )   ∇ z ( T )   2 L 2 ( G ) + sλϕ ( T ) θ 2 ( T )   z ( T )   2 L 2 ( G ) + Z Q θ 2 | ρ t ( t ) y | 2 dxdt i . (3.3) F rom the c hoice of ρ , we see that E Z Q θ 2 | ρ t ( t ) | 2 y 2 dxdt ≤ C Z t 2 t 1 Z G θ 2 y 2 dxdt ≤ C θ 2 ( t 1 ) | y | 2 L 2 F (0 ,T ; L 2 ( G )) . (3.4) This, tog ethe r with inequalit y (3 .3 ) , implies that λθ 2 ( t 0 ) E Z T t 0 Z G |∇ y | 2 dxdt + sλ 2 θ 2 ( t 0 ) E Z T t 0 Z G ϕ | y | 2 dxdt ≤ λ E Z Q θ 2 |∇ z | 2 dxdt + sλ 2 E Z Q θ 2 ϕ | z | 2 dxdt ≤ C θ 2 ( t 1 ) | y | 2 L 2 F (0 ,T ; L 2 ( G )) + C E  θ 2 ( T )   ∇ y ( T )   2 L 2 ( G ) + sλϕ ( T ) θ 2 ( T )   y ( T )   2 L 2 ( G )  . (3.5) Here w e utilize the fact that θ ( t ) ≤ θ ( s ) for t ≤ s . F rom inequality (3.5), we see λ E Z T t 0 Z G |∇ y | 2 dxdt + sλ 2 E Z T t 0 Z G ϕ | y | 2 dxdt ≤ C θ 2 ( t 1 ) θ − 2 ( t 0 ) | y | 2 L 2 F (0 ,T ; L 2 ( G )) + C E  θ 2 ( T )   ∇ y ( T )   2 L 2 ( G ) + sλϕ ( T ) θ 2 ( T )   y ( T )   2 L 2 ( G )  . (3.6) By means of d ( y 2 ) = 2 y d y + ( dy ) 2 , we obtain that E Z G | y ( t 0 ) | 2 dx = E Z G | y ( T ) | 2 dx − E Z T t 0 Z G  2 y dy + ( dy ) 2  dx = E Z G | y ( T ) | 2 dx − E Z T t 0 Z G n 2 y h n X i,j =1 ( b ij y x i ) x j + ( a 1 , ∇ y ) + a 2 y i + ( a 3 y ) 2 o dxdt ≤ E Z G | y ( T ) | 2 dx + C E Z T t 0 Z G |∇ y | 2 dxdt +  | a 1 | 2 L ∞ F (0 ,T ; L ∞ ( G ;l R n )) + | a 2 | L ∞ F (0 ,T ; L ∞ ( G )) + | a 2 | 2 L ∞ F (0 ,T ; L ∞ ( G ))  E Z T t 0 Z G y 2 dxdt ≤ E Z G | y ( T ) | 2 dx + C E Z T t 0 Z G |∇ y | 2 dxdt + C r 1 E Z T t 0 Z G y 2 dxdt, (3.7) 15 Recalling ϕ ≥ 1, f rom inequalit y (3.7), w e kno w that there exists a λ 2 > 0 suc h that for all λ ≥ λ 2 , it holds that E Z G | y ( t 0 ) | 2 dx ≤ E Z G | y ( T ) | 2 dx + C  λ E Z T t 0 Z G |∇ y | 2 dxdt + sλ 2 E Z T t 0 Z G ϕy 2 dxdt  . (3.8) Com bing inequalit y (3.6) and inequality (3.8), for an y λ ≥ max { λ 1 , λ 2 } and s ≥ s 0 ( λ 1 ), w e ha v e E Z G | y ( t 0 ) | 2 dx ≤ C θ 2 ( t 1 ) θ − 2 ( t 0 )   y   2 L 2 F (0 ,T ; L 2 ( G )) + C E  θ 2 ( T )   ∇ y ( T )   2 L 2 ( G ) + sλϕ ( T ) θ 2 ( T )   y ( T )   2 L 2 ( G )  . (3.9) No w w e fix λ 3 = max { λ 1 , λ 2 } , fr o m inequality (3.9), we get E Z G | y ( t 0 ) | 2 dx ≤ C θ 2 ( t 1 ) θ − 2 ( t 0 )   y   2 L 2 F (0 ,T ; L 2 ( G )) + C θ 2 ( T ) E   y ( T )   2 H 1 ( G ) . (3.10) Replacing C b y C e s 0 e λ 3 T , fro m inequality (3.10), fo r a n y s > 0, it holds that E Z G | y ( t 0 ) | 2 dx ≤ C e − 2 s ( e λ 3 t 1 − e λ 3 t 0 )   y   2 L 2 F (0 ,T ; L 2 ( G )) + C e C s E   y ( T )   2 H 1 ( G ) . (3.11) Cho osing s ≥ 0 whic h minimize the rig h t-hand side o f inequalit y (3.11), w e obtain that E   y ( t 0 )   2 L 2 ( G ) ≤ C | y | 1 − θ L 2 F (0 ,T ; L 2 ( G )) E   y ( T )   θ H 1 ( G ) , (3.12) with θ = 2( e λ 3 t 0 − e λ 3 t 1 ) C + 2( e λ 3 t 0 − e λ 3 t 1 ) . 4 Pro of of Theor e m 1 .4 Thie section is dev oted to proving Theorem 1 .4. W e b orro w some ideas in [3 0 ] a gain . Pr o of of The or em 1.4 : F rom the assumptions on b 1 , b 2 , b 3 , R and h , and by Lemma 1 .2, w e know equation ( 1 .16) admits a unique stro ng solution. F or arbitrary small ε > 0, w e c ho ose t 1 and t 2 suc h that 0 < t 0 − ε < t 1 < t 2 < t 0 . Let χ ∈ C ∞ (l R) b e a cut-off function suc h that 0 ≤ χ ≤ 1 a nd that χ = ( 1 , t ≤ t 1 , 0 , t ≥ t 2 . (4.1) 16 Put y = R z (recall (1.17) fo r R ) in [0 , t 2 ] × G . Since y is a strong solutio n of equation (1.16), we kno w that z solv es                      dz − ∆ z dt = h ( b 1 , ∇ z ) +  2 ∇ R R , ∇ z  +  b 2 + ∆ R R − 2( ∇ R, ∇ R ) R 2 − R t R +  ∇ R R , b 1  z i dt + hdt + b 3 z dB ( t ) in [0 , t 0 ] × G, z = ∂ z ∂ ν = 0 on [0 , t 0 ] × Γ , z (0) = 0 in G. (4.2) Setting u = z x 1 , noting z is the strong solution o f equation (4.2) and z x 1 = ∂ z ∂ ν = 0 on  { 0 } × G ′  ∪  { l } × G ′  , w e know that u is the w eak solution of t he following equation:                                  du − ∆ udt = h (( b 1 ) x 1 , ∇ z ) + ( b 1 , ∇ u ) +  2 ∇ R R  x 1 , ∇ z  +  2 ∇ R R , ∇ u  +  b 2 + ∆ R R − 2( ∇ R, ∇ R ) R 2 − R t R +  ∇ R R , b 1  x 1 z +  b 2 + ∆ R R − 2( ∇ R, ∇ R ) R 2 − R t R +  ∇ R R , b 1  u i dt +( b 3 ) x 1 z dt + b 3 udB ( t ) in [0 , t 0 ] × G, u = 0 on [0 , t 0 ] × Γ u (0) = 0 in G. (4.3) Set w = χu . Then we kno w that w is a we ak solution of the fo llo wing equation:                                  dw − ∆ w dt = h (( b 1 ) x 1 , χ ∇ z ) + ( b 1 , ∇ w ) +  2 ∇ R R  x 1 , χ ∇ z  +  2 ∇ R R , ∇ w  +  b 2 + ∆ R R − 2( ∇ R, ∇ R ) R 2 − R t R +  ∇ R R , b 1  x 1 χz +  b 2 + ∆ R R − 2( ∇ R, ∇ R ) R 2 − R t R +  ∇ R R , b 1  w i dt +( b 3 ) x 1 χz dB ( t ) + b 3 w dB ( t ) − χ ′ udt in [0 , t 0 ] × G, w = 0 on [0 , t 0 ] × Γ , w = 0 in G. (4.4) By means of u = z x i and z ( t, 0 , x ′ ) = y ( t, 0 , x ′ ) = 0 for ( t, x ′ ) ∈ (0 , t 0 ) × G ′ , we see χz = χ Z x 1 0 u ( t, η , x ′ ) dη = Z x 1 0 w ( t, η , x ′ ) dη . (4.5) This, together with equation (4.4), implies that w is the weak solution of the follo wing 17 equation:                                                dw − ∆ w dt = h ( b 1 , ∇ w ) +  2 ∇ R R , ∇ w  +  ( b 1 ) x 1 , ∇ Z x 1 0 w ( t, η , x ′ ) dη  +  2 ∇ R R  x 1 , ∇ Z x 1 0 w ( t, η , x ′ ) dη  +  b 2 + ∆ R R − 2( ∇ R, ∇ R ) R 2 − R t R +  ∇ R R , b 1  w +  b 2 + ∆ R R − R t R +  ∇ R R , b 1   x 1 Z x 1 0 w ( t, η , x ′ ) dη i dt +( b 3 ) x 1 χ Z x 1 0 u ( t, η , x ′ ) dη dB ( t ) + b 3 w dB ( t ) − χ ′ udt in [0 , t 0 ] × G, w = 0 on [0 , t 0 ] × Γ , w = 0 in G. (4.6) Applying Theorem 1.1 to equation ( 4 .6) with ψ ( t ) = − t , noting that w (0) = 0, and that w ( t 0 ) = χ ( t 0 ) u ( t 0 ) = 0, we get E Z t 0 0 Z G θ 2  λ |∇ w | 2 + sλ 2 w 2  dxdt ≤ C E Z t 0 0 Z G θ 2 | χ ′ u | 2 dxdt + C r 1 E Z t 0 0 Z G θ 2     Z x 1 0 w ( t, η , x ′ ) dη    2 +    Z x 1 0 |∇ w ( t, η , x ′ ) dη |    2  dxdt. (4.7) Since    Z x 1 0 w ( t, η , x ′ ) dη    2 ≤ l Z l 0 | w ( t, η , x ′ ) | 2 dη , w e know Z t 0 0 Z G θ 2    Z x 1 0 w ( t, η , x ′ ) dη    2 dxdt ≤ l Z l 0 dx 1 Z t 0 0 Z G ′ Z l 0 θ 2 | w ( t, η , x ′ ) | 2 dη dx ′ dt ≤ l 2 Z t 0 0 Z G θ 2 | w ( t, η , x ′ ) | 2 dη dx ′ dt. (4.8) By virtue of ∇ Z x 1 0 w ( t, η , x ′ ) dη = Z x 1 0 ∇ w ( t, η , x ′ ) dη + w ( t, 0 , x ′ ) = Z x 1 0 ∇ w ( t, η , x ′ ) dη , w e get that Z t 0 0 Z G θ 2    ∇ Z x 1 0 w ( t, η , x ′ ) dη    2 dxdt = Z t 0 0 Z G θ 2    ∇ Z x 1 0 w ( t, η , x ′ ) dη    2 dxdt ≤ l Z l 0 dx 1 Z t 0 0 Z G ′ Z l 0 θ 2 |∇ w ( t, η , x ′ ) | 2 dη dx ′ dt ≤ l 2 Z t 0 0 Z G θ 2 |∇ w ( t, η , x ′ ) | 2 dη dx ′ dt. (4.9) 18 F rom inequality (4.7) – (4.9), we obtain that E Z t 0 0 Z G θ 2  λ |∇ w | 2 + sλ 2 w 2  dxdt ≤ C E Z t 0 0 Z G θ 2 | χ ′ u | 2 dxdt + C l 2 r 1 E Z t 0 0 Z G θ 2  |∇ w | 2 + | w | 2  dxdt. (4.10) Th us, w e kno w that there is a λ 4 = max { C r 1 , λ 1 } suc h that for all λ ≥ λ 4 , there exists an s 1 ( λ 4 ) > 0 so that fo r all s ≥ s 1 ( λ 4 ), it holds that E Z t 0 0 Z G θ 2  λ |∇ w | 2 + sλ 2 w 2  dxdt ≤ C E Z t 0 0 Z G θ 2 | χ ′ u | 2 dxdt. (4.11) Fix λ = λ 4 , by the prop ert y of χ (see (4.1 ) ), w e find E Z t 0 0 Z G θ 2 | χ ′ u | 2 dxdt ≤ e 2 se − λ 4 t 1 E Z Q | u | 2 dxdt ≤ e 2 se − λ 4 t 1 | y x 1 | 2 L 2 F (0 ,T ; L 2 ( G )) . (4.12) This, tog ethe r with inequalit y (4 .1 1) , implies that fo r a ll s ≥ s 1 , it holds that e 2 se − λ 4 ( t 0 − ε ) E Z t 0 − ε 0 Z G  |∇ w | 2 + sw 2  dxdt ≤ E Z t 0 − ε 0 Z G θ 2  |∇ w | 2 + sw 2  dxdt ≤ E Z t 0 0 Z G θ 2  |∇ w | 2 + sw 2  dxdt ≤ C e 2 se − λ 4 t 1 | y x 1 | 2 L 2 F (0 ,T ; L 2 ( G )) . (4.13) F rom inequality (4.13), we ha ve | w | 2 L 2 F (0 ,T ; H 1 ( G )) ≤ C e 2 s ( e − λ 4 t 1 − e − λ 4 ( t 0 − ε ) ) | y x 1 | 2 L 2 F (0 ,T ; L 2 ( G )) . (4.14) Recalling that t 0 − ε < t 1 , w e kno w e − λ 4 t 1 − e − λ 4 ( t 0 − ε ) < 0. Letting s → + ∞ , w e obtain that w = 0 in (0 , t 0 − ε ) × G, P -a.s. This, tog ethe r with equalit y (4.5), implies that z = 0 in (0 , t 0 − ε ) × G, P -a.s. , whic h means h = 0 in (0 , t 0 − ε ) × G ′ , P -a.s. Since ε > 0 is arbitrary , the pr o of of Theorem 1.4 is completed. Ac knowledgmen ts . The autho r would lik e to thank the a non ymous referees for helpful commen ts. 19 References [1] W. F. Ames and B. Straughan, Non-Standard and Impro perly Posed P roblems, Academic Press , San Diego, 1 997. [2] V. Barbu, A. R˘ ascanu a nd G. T essitore, Carleman estimate and c ontr ol lability of line ar sto chastic he at e quatons, Appl. Math. Optim., 47 (200 3 ), 97–120. [3] G. B a o, S.-N. Chow, P . Li and H. Zhou, N u meric al solution of an inverse me dium sc attering pr oblem with a sto chastic s ou rc e, Inv ers e Problems, 26 (2010), 0 74014 (23pp). [4] A. L. B uhgeim and M. V. Klibanov, Glob al uniqueness of a class of multidimentional inverse pr oblems, Soviet Mathematics Dokla dy , 24 (19 81), 244–24 7. [5] Z. Brzezniak a nd M. Neklyudov, Backwar d uniqueness and the existenc e of the sp e ctr al limit for some p ar ab olic SPDEs, h ttp://arxiv.o r g/abs/080 6 .0616. [6] A. P . Calde r ´ on, Uniqueness in the Cauchy pr oblem for p artial differ ential e quations, Amer. J. Math., 80 (1958 ), 1 6–36. [7] J. R. Cannon and S. P´ erez Estev a , Uniqueness and stabili ty of 3D he at sour c es, In verse P r oblems, 7 (1991), 5 7–62. [8] T. Ca rleman, Sur un pr obl` eme d’unicit´ e p our les syst` emes d’ ´ equ ations aux d´ eriv´ ees p artiel les ` a deux variables r´ eel les, Ark. Ma t. Astr. Fys., 17 (1 9 39), 1–9. [9] L. Cav alier and A. Tsybako v, Sharp adaptation for inverse pr oblems with r andom noise, Probab. Theor y Relat. Fields, 123 (2002 ), 323–354 . [10] M. Choulli and M. Y ama moto, Conditional stability in determining a he at sour c e, J. Inv erse Ill-Posed Probl., 12 (2 004), 233–24 3. [11] G. Da Pra to and J . Zab czyk, Sto c hastic Equations in Infinite Dimensions, Cambridge Universit y P ress, Cambridge, 1992. [12] L. Escaur iaza, G. Seregin and V. ˇ Sver´ ak, Ba c kward uniquenes s for pa rabolic equa tions, Ar ch. Ration. Mech. Anal., 169 (2 003), 147–15 7. [13] A. V. F ursik ov and O. Y u. Imanuvilo v, Controllabilit y of Evolution Eq uations, Lectures Notes Series, vol. 34, Seoul National Universit y , Seoul, 1996 . [14] J. M. Ghidaglia, Some Backwar d Un iq ueness Re sults, Nonlinea r Analys is . Theor y , Methods and Appli- cations, 1 0 (1986), 777 –790. [15] L. H¨ ormander, The Analysis of Linea r Partial Differen tial Op erators, vol I II, Springer -V erlag , 19 8 5. [16] I.A. Ibragimov and R.Z. Khas ’mins k ii, Estimation pr oblems for c o efficients of st o chastic p artial differ- ential e quations. Part I, Theory Probab. Appl., 4 3 (1999), 370 –387. [17] O. Y u. Imanuvilov and M. Y amamo to, Lipschitz stability in inverse p ar ab olic pr oblems by the Carleman estimate, Inv erse Proble ms , 14 (1998), 12 2 9–1245. [18] V. Isa k ov, In v erse Pro blems for Partial Differential Equations, Springer, New Y ork, 20 06. [19] B. T. Joha nsson and D. Lesnic, A variatio nal metho d for identifying a sp ac ewise dep endent he at sou rc e, IMA J. Appl. Math., 7 2 (2007), 748–7 60. [20] B. T. Johansson and M. P ricop, A metho d for identifying a sp ac ewise dep endent he at sour c e under sto chastic noise interfer enc e, Inv er se probl. Sci. En., 18 (201 0), 51–63. [21] M. V. Klibanov, Inverse pr oblems and Carleman estimates, In verse Problems, 8 (1 992), 575–59 6. [22] M. V. Klibanov and A. Timonov, Carleman Es timates for Co efficien t Inv erse Problems and Numerical Applications, Inv erse and Ill-Posed Pro ble ms Se r ies, VSP , Utrech t, 200 4 . 20 [23] I. K uk a vica, Backwar d uniqueness for solutions of line ar p ar ab olic e quations, P . Am. Math. So c., 132 (2004 ), 17 55–1760. [24] M. Lees and M. H. Protter, Unique c ontinu atio n for p ar ab olic differ ential e quations and ine qualities, Duk e Ma th. J., 28 (1 961), 369–38 2. [25] E. P ardoux, ´ Equations aux d´ eriv´ ees par tielles sto c ha stiques non lin ´ eaires mo no tones, Ph D Thesis, Univ ersit´ e Paris XI, 19 7 5. [26] K.-D. Ph ung a nd G. W ang, Quantitative unique c ontinu atio n for the semiline ar he at e quation in a c onvex domain, J. F unct. Anal., 259 (201 0 ), 12 30–1247. [27] D. D. Santo and M. Prizzi, Backwar d uniqueness fo r p ar ab olic op er ators whose c o efficients ar e non- Lipschitz c ontinuous in time, J. Math. P ures Appl., 84 (2005), 471–4 91. [28] S. T ang and X. Zhang, Nu l l c ontr ol lability for forwar d and b ackwar d sto chastic p ar ab olic e quations, SIAM J. Control Optim., 48 (200 9), 2191–221 6. [29] M. Y a mamoto, Uniqueness and st ability in multidimensional hyp erb olic inverse pr oblems, J. Ma th. Pures Appl., 78 (1 999), 65–98. [30] M. Y amamoto, Carleman estimates for p ar ab olic e quations and applic ations, Inv er se Pro blems, 25 (2009 ), 1 23013. 21