Flow Decomposition and Large Deviations

Transcription

1 ounal of funconal analy (1995) acle no. 97 Flow Decompoon and Lage Devaon Ge ad Ben Aou and Fabenne Caell Laboaoe de Mode laon ochaque e aque Unvee Pa-Sud (Ba^. 425) Oay Cedex Fance Receved July We udy lage devaon popee elaed o he behavo a = goe o of dffuon pocee geneaed by = 2 L 1 +L 2 whee L 1 and L 2 ae wo econd-ode dffeenal opeao exendng ecen eul of Do and Soock and Rabehemanana. The man ool he decompoon heoem fo flow of ochac dffeenal equaon poved by Bmu and Kuna. We gve anohe applcaon of flow decompoon n a nonlnea fleng poblem Academc Pe Inc. 1. INTRODUCTION The pupoe of h wok o how how he decompoon heoem fo flow of ochac dffeenal equaon poved by Bmu [4] and Kuna [9] can be ued o oban new lage devaon pncple fo he dffuon geneaed by = 2 L 1 +L 2 when L 1 and L 2 ae wo econd-ode dffeenal opeao and when =. Th poblem now clacal when L 2 f ode (ee Fedln and Wenzell [8] Azenco [1]). I ha alo been eaed when L 2 he Laplacan (Bezudenhou [3]) and when L 1 can be wen a a um of quae of veco feld L 1 = 1 X 2 2 whee he Le algeba geneaed by he X abelan (Do and Soock [7]) o nlpoen (Rabehemanana [11]). Thee auho gve a lage devaon pncple fo he law of he andom vaable R = pacula veon of he condonal law of X = elave o (=B) whee X = a oluon o he Saonovch ochac dffeenal equaon X = =x+= : =1 l _ (X = ) db + : _~ (X = ) db + _~ (X = ) d: (1.1) =1 v x # R n ; v #[1]; v Fo all # [1... ] and all # [... l] _ and _~ ae uffcenly mooh veco feld on R n Copygh 1996 by Academc Pe Inc. All gh of epoducon n any fom eeved.

2 24 BEN AROUS AND CASTELL v B and B ae wo ndependen Bownan moon wh value n R and R l epecvely defned on he Wene pace W=C ([ 1] R ) and W =C ([ 1] R l ). We wll denoe by P (epecvely P ) he Wene meaue on W (epecvely W ). One could wonde whehe a lage devaon pncple fo he law of X = aanable. A a mae of fac no a Do and Soock have poned ou. Indeed he uppo of he law of he nonpeubed dffuon X no compac n geneal. Snce he ae funcon * aocaed o he lage devaon of X = vanhe on he uppo of P he level e [*L] canno be compac. Howeve X = can be condeed a a andom vaable X = : (W P) (L p (W C x ([ 1] R n )) (=B) (B X. = (x)). In h cae hee no obvou conadcon o have a lage devaon pncple fo he law P = of X =. The uppo of P now a pon of L p (W C x ([ 1] R n )) whch obvouly compac. And we do oban a lage devaon pncple fo he law of X =. Ou eul conan he eul of [7 11] and exend hem o he geneal cae wh no hypohe a all on he Le algeba. The key obevaon he fac ha a conacon pncple can be ued f one ha wo ngeden 1. he decompoon pncple: we ecall h eul n Secon a lage devaon pncple fo flow of ochac dffeenal equaon. Such a lage devaon pncple fo flow ha been obaned by Mlle Nuala and Sanz-Sole [1] and Bald and Sanz-Sole [2]. We need a lgh exenon of o be able o conol devave of he flow. We gve he poof of h lage devaon pncple n Secon 3 and we ge n Secon 4 o he man heoem ha we now ae. Theoem 7. Le P = be he law of he andom vaable X = (P = a pobably meaue on L p (W C x ([ 1] R n ))). Then P = afe a lage devaon pncple wh ae funcon 4 defned fo all z # L p (W E x ) by 4(z)=nf { 1 2 &h&2 H h # H uch ha P a.e. z =x+ : =1 _ (z ) h4 d+ : l =1 _~ (z ) db + _~ (z ) d =.

3 FLOW DECOMPOSITION 25 Th conan a lage devaon pncple fo he condonal law R = of X = elave o (=B) (a n Do and Soock [7] and n Rabehemanana [11]) a a val conacon pncple how. Fnally n Secon 5 we how how h mehod can be ued fo a poblem n nonlnea fleng exendng eale eul of Do [6] and Rabehemanana [11]. 2. FLOW DECOMPOSITION FOR STOCHASTIC DIFFERENTIAL EQUATIONS We menon hee he eul of [4 9] fo lae ue. Fo # [... k] le X and Y be C m b veco feld on Rn (ha dffeenable up o ode m bounded wh bounded devave). Le u conde he Saonovch dffeenal equaon k dx = : =1 x =x. X (x ) db +X (x ) d Then hee a veon of ( x) [ x (x) whch a flow of C m -dffeomophm n R n ha an elemen of D n whee [1]_R D #{: n R n (x)[ (x) uch ha \ #[1] n a C m -dffeomophm of R n \l # N n l m l (x) l ( ) &1 x l x l = (x) ae connuou n ( x). Le (x) denoe h eenally unque veon of x (x). Almo uely fo all # [ 1] we can hen defne he ochac veco feld &1 &1 V Y ( y)= \ x (y) Y + ( ( y)). Le u conde hen he Saonovch dffeenal equaon k dy = : =1 y = y. &1 V Y ( y ) db +&1 V Y ( y ) d (2.2)

4 26 BEN AROUS AND CASTELL Theoem Thee a ong oluon o (2.2) defned on [ 1]. 2. Le z # ( y ( y)). Then z oluon o he ochac dffeenal equaon k dz = : (X +Y )(z ) db +(X +Y )(z ) d =1 (2.3) z = y. Poof. Le! be he ong oluon of Eq. (2.3). Le u conde he poce defned fo all #[1] by y~ = &1 (! ). Then by he genealzed Io^ fomula (ee Theoem 4.1 n [4]) y~ oluon o Eq. (2.2). Theefoe 1 and 2 ae poved. K Theoem 1 ha Io^ counepa. Theoem 2. Le u defne v v v (X +Y )* (x)#x (x)+y (x)+ 1 2 : k Y *(x)=y (x)+ 1 2 : k =1 Y (x)=y *(x)+ 1 2 : k & 1 2 : k =1 =1 =1 Y $(x) Y (x). (X +Y )$ (x)(x +Y )(x). &1 V [X Y ](x) 2 x 2 (&1 V Y (x) &1 V Y (x)) whee [X Y] he Le backe of he veco feld X and Y. Le u conde he Io^ ochac dffeenal equaon Then k dy = : =1 y = y. &1 V Y ( y ) $B + &1 V Y ( y ) d (2.4) 1. Thee a ong oluon o (2.4) defned on [ 1]. 2. Le z = ( y ). Then z oluon o he Io^ ochac dffeenal equaon k $z = : (X +Y )(z ) $B +(X +Y )* (z ) d. =1 Poof. The ame a Theoem 1. K

5 FLOW DECOMPOSITION 27 We wll ue hee Theoem 2 n he followng conex: v k=l+; v B=(B 1... B B 1... B l ); v \ # [1... ] Y =; v \ # [+1... l+] X = o ha Eq. (1.1) pl n wo ochac dffeenal equaon one dven by he Bownan B he ohe by he Bownan B. 3. LARGE DEVIATIONS FOR STOCHASTIC FLOWS 3.1. Noaon and Reul In h econ we wll conde he Saonovch dffeenal equaon d! = == : _ (=! = ) db =1 (3.5)! = =x whee v _ (= } ) ae C m+2 b veco feld on R n fo ome mn+1. We wll aume ha _ (= } ) convege n C m b unfomly on compac ube of Rn o ome veco feld _ when = goe o. v B a andad Bownan moon defned on he Wene pace (W P) whee W he pace C ([ 1] R n ) wh he opology of unfom convegence and P he Wene meaue. Le D n be defned a n Secon 2. D n wll be endowed wh he C k o k C -opology defned fo all km by v n w C k ff \K compac ube of R n v n w C k up x#k; #[1]" : x : : k (x)& : n x : ff \K compac ube of R n up x#k;#[1]" x : : k : (x)& : n x : (x) " (x) " ww n " : ( + ) &1 (x)& : ( n )&1 (x) " x : x ww. : n

6 28 BEN AROUS AND CASTELL Le 8 = (x) be he veon of!=(x) whch an elemen of Dn. Ou pupoe o how a lage devaon pncple fo he law of 8 = (pobably on D n wh he C k -opology). Mlle Nuala and Sanz-Sole [1] Bald and Sanz-Sole [2] have aleady hown h eul fo he C opology. Befoe ang he eul we wll noduce fuhe noaon. Le H be he CameonMan pace ove R [1]R H ={h: h()= h aboluely connuou wh epec o= Lebegue meaue uch ha 1 &h4 & 2 d<. H a Hlbe pace fo he nne poduc (h g) H = 1 h4 g* d. Gven h # H we aocae o (3.5) he odnay dffeenal equaon dx (h)= : =1 x (h)=x. _ (x (h) h4 d (3.6) Unde he aumpon made on he veco feld x(h) an elemen of D n. Thu we defne a map F: H D n (3.7) h [ ( x [ x (h)(x)). Ung he eul of Bmu [4] F can be exended n a meauable way o W: P-a.e. F ( )(x) wll be a oluon o he ochac dffeenal equaon d! = : =1 $ =x _ (! ) db Th exenon wll ll be denoed by F. We defne now he ae funcon aocaed o he lage devaon of he ochac flow 8 =. Le be n D n. I()#nf[ 1 2 &h&2 H h # H F(h)=]. (3.8) When A a ube of D n we wll denoe by I(A)=nf[I() # A]. We have hen he followng eul. Theoem 3. D n povded wh he C k -opology fo ome km&1& [n2]. Then 1. I lowe em-connuou and fo all L> [IL] a compac ube of D n.

7 FLOW DECOMPOSITION Fo all # D n uch ha I()< hee ex a unque h # H uch ha =F(h) and I()= 1 2 &h&2. Moeove f _(x)=(_ H 1(x)}}}_ (x)) and f V# x # R n Ke _(x) hen d-a.e. h4 # V =. 3. \A/D n &I(A1 )lm nf = = 2 log P(8 = # A)lm up = 2 log P(8 = # A)&I(A ). = Thee eul eman ue when D n povded wh he C k -opology Poof of Theoem 3 Lage Devaon fo he C k -opology F of all le u noe ha he map F defned by (3.7) connuou fom H #[h # a H &h& H a] endowed wh he unfom convegence o (D n C k ). Le ndeed f and g be wo funcon n H a and le x=f( f ) and y=f( g). Then &x (x)&y (x)& 2 2 " 2Ka 2 (_(x )&_(y )) f4 d " 2 +2 " &x & y & 2 d+2 " And an negaon by pa hen yeld _(y )( f4 & g* ) d " 2 _(y )( f4 & g* ) d " 2. &x & y &K &_( y )( f & g )&+ &_$( y ) y* ( f & g )& d +2Ka 2 &x &y & 2 d K & f & g& \ 1+ +2Ka 2 &x &y & 2 d. By Gonwall' lemma we have hen: &_$( y )&&_(y )g*&d + &x &y &K & f & g& (1+Ka). The ame agumen hold fo he devave of x& y. Theefoe Lemma 1.3 [1 p. 69] enue ha I a ``good'' ae funcon. Moeove when I()< he nfmum n he defnon of I eached.

8 3 BEN AROUS AND CASTELL We ae now gong o pove ha he pon whee he nfnum eached unque. So le # D n be uch ha I()<. Le h # H be uch ha I()= 1 2 &h & 2 H and =F(h ). Le u defne he map 6: H H (3.9) h [ 6h: 6h = P V =h4 d whee P V = he ohogonal poecon on V =. Then eay o check ha v &6h& H &h& H ; v &6h& H =&h& H d-a.e. h4 # V = ; v F(h)=F(6h). Thu = F(h )=F(6h ) and &h & H = &6h & H. Theefoe d-a.e. h4 ()#V =. Le u aume now ha =F(h )=F(h 1 ) I()=12 &h & 2 H =12 &h 1& 2 H. The equaon afed by F(h 1 ) and F(h ) yeld \ \x _( (x)) h4 ()=_( (x)) h4 1(). Ung he dffeomophm popey of we deve ha \ \x _(x)(h4 ()&h4 1())=. Theefoe h4 ()&h4 1()#V. Bu we aleady know ha h4 ()&h4 1()#V =. Theefoe h =h 1 and 2 poved. Followng Azenco [1] we begn wh he ``quaconnuy'' of he map F n ode o oban 3. Lemma 4. \K compac ube of R n \a> \L> \R> hee ex b and = > uch ha \bb \== \h # H &h& H a P[&8 = &F(h)& C k ([1]_K) R; &=B&h&b]exp(&L=2 ). Poof of Lemma 4 Thoughou he pape C a conan whch can dffe fom one expeon o he ohe. Snce mk+1+[n2] Sobolev' embeddng heoem gve &}& C (K)C &}& k W m2 (K) whee W m2 he pace of funcon dffeenable up o ode m whoe devave ae quae negable wh epec o Lebegue meaue. W m2 (K) a Hlbe pace fo he nom & f & W m2 (K)= K : : m" : f x :" 2 dx.

9 FLOW DECOMPOSITION 31 Theefoe P[&8 = &F(h)& C k ([1]_K) R; &=B&h&b] P _ up &8 = &F (h)& W m2 (K) R #[1] C ;&=B&h&b &. So we have o how ha \K compac e of R n \a R L> hee ex b = > uch ha bb == &h& H a mply 6 1 #P[ up &8 = &F (h)& W m2 (K)R; &=B&h&b]exp(&L= 2 ). #[1] In he followng we wll denoe { = R =nf[ uch ha &8= &F (h)& W m2 (K)R]. Then 6 1 =P[up #[1] &8 = 7{ = &F R 7{ = (h)& W R m2 (K)R; &=B&h&b]. Snce he veco feld _ ae C m b one can ealy check ha hee a conan M (dependng on a and K) uch ha up #[1] up &F (h)& W m2 (K)M. h; &h& H a Fom he defnon of { = R eul hen ha up { = R &8= & W m2 (K)R+M. Theefoe nce mn+1 poved n Appendx 1 ha hee a conan C uch ha \ # [1... ] \= \ #[1] &_ (8 = 7{ = R )&_ (F 7{ = R (h))& W m2 (K)C &8 = 7{ = R &F 7{ = R (h)& W m2 (K) &_ (= 8 = 7{ = R )&_ (8 = 7{ = R )& W m2 (K)C&_ (=})&_ (})& C m (B R+M ) (whee B R+M he ball of adu R+M n R n ). Thu \ #[1] &8 = 7{ = R &F 7{ = R (h)& W m2 (K) ": =1 7{ = R + : 7{ =1 + : _ (= 8 = ) d(=b &h ) "W m2 (K) = R = R =1 7{ &_ (= 8 = )&_ (8 = )& W m2 (K) h4 d &_ (8 = )&_ (F (h))& W m2 (K) h4 d

10 32 BEN AROUS AND CASTELL " : =1 7{ = R +C _a : + 7{ = R =1 _ (= 8 = ) d(=b &h ) "W m2 (K) &_ (=})&_ (})& C m (B R+M ) &8 = &F (h)& W m2 (K) h4 d &. I follow fom Gonwall' lemma ha \ #[1] up &8 = 7{ = &F R 7{ = (h)& W R m2 (K) #[1] Theefoe 6 1 P _up " : +P _ : C \ up #[1]" : =1 + : =1 =1 7{ = R =1 7{ &_ (=})&_ (})& C m (B R+M )+. = R _ (= 8 = ) d(=b &h ) "W m2 (K) _ (= 8 = ) d(=b &h ) R "W m2 (K) 2C ;&=B&h&b & &_ (=})&_ (})& C m (B R+M ) R 2C ;&=B&h&b &. Fom he unfom convegence of _ (= }) o_ ( } ) we can chooe = uch ha == O &_ (= })&_ (})& <R2C. C m (B R+M ) Thu we ae led o how ha \K compac of R n \R R$ L a> hee ex b and = uch ha bb == &h& H a mply 6 2 #P _up " : exp \&L = 2+ = R =1 7{ _ (= 8 = ) d(=b &h ) R$; &=B&h&b & "W m2 (K) An negaon by pa yeld 6 2 P 1 +P 2 +P 3 +P 4 whee P 1 =P _ up { = R : =1 =B &h &_ (= 8 = )& W m2 (K)> R$ 4 ;&=B&h&b & P 2 =P _up " : (_ (= 8 = ) =B ) = 7 { > R$ =1 R"W m2 (K) 4 ;&=B&h&b &

11 FLOW DECOMPOSITION 33 P 3 =P _up " : = R =1 7{ > R$ 4 ;&=B&h&b & whee $ denoe he Io^ dffeenal P 4 =P up _=2 " : =1 7{ > 2R$ 4 ;&=B&h&b &. _ $(= 8 = ) _ (= 8 = )(=B &h ) =$B "W m2 (K) = R (=B &h ) Tace(_*_ "_)(= 8 = ) d "W m2 (K) Teamen of P 1. Snce up &8 = 7 { = & W R m2 (K)R+M follow fom Appendx 1 ha hee a conan C uch ha \=1 \ # [1... ] up &_ (= 8 = )& W m2 (K)C. Theefoe P 1 P[Cb>R$4]= fo uffcenly mall b. Teamen of P 2. : =1 (_ (= 8 = ) =B ) = 7 { ==2 : R = R =1 7{ _ $(= 8 = ) _ (= 8 = ) d. Appendx 1 yeld hen a conan C uch ha P 2 P[= 2 C>R$4].e. P 2 = fo = 2 R$4C. Teamen of P 4. mall. P 4 P[C= 2 b>2r$4]= fo = and b uffcenly Teamen of P 3. The conol of P 3 gven by an exponenal nequaly fo mangale wh value n ome Hlbe pace poved n Appendx 2. Le (e n ) n be an ohonomal ba n W m2 (K). Le u denoe v M = (x)= _$ (= 8 = ) _ (= 8 = )(=B &h ) =$B. v T = =nf[ uch ha &=B b &h &b] v S = =nf[ uch ha &M = & R$ W m2 (K)R$] v {={ = R 7T= b 7S=. R$ We have hen o how ha \K compac of R n \L R R$ a> hee ex b and = uch ha \== \bb \h &h& H a 6 3 =P[ up &M = & 7{ W m2 (K)R$]exp(&L= 2 ). #[1]

12 34 BEN AROUS AND CASTELL Fo { M = # W m2 = n K and f we denoe M #(M = e n) can be checked ha M = n =: (_ $(= 8 = (}))_ (=8 = (}))e n)(=b &h ) =$B by wng he ochac negal a L 2 -lm of Remann um. Theefoe (M =n M =m ) = : l = 2 (_ $(= 8 = ) _ (= 8 = ) e n)(=b &h ) _(_ $(= l 8 = ) _ (= 8 = ) e m)(=b l &hl ) d. Th allow u o conol he quane appeang n Appendx 2. N : 7 { n m Moeove N : k=1 M = n M =m : d(m = n M = m ) =: = 2 7{ 2 ( $(= 8 = ) _ (= 8 = ) P N(M = ))(=B &h ) d & (whee P N he ohogonal poecon on Span[e N]) C= 2 b 2 : 7 { : &_ $(= 8 = ) _ (= 8 = )&2 &P W m2 (K) N(M = )&2 d W m2 (K) C= 2 b 2 R$ 2 by Appendx 1. (M =k ) 7{ =: = 2 7{ k _ : 2 (_ $(= 8 = ) _ (= 8 = ) e k)(=b &h ) d & C= 2 b 2 : 7 { : &P N (_ $(= 8 = ) _ (= 8 = ))&2 d W m2 (K) C= 2 b 2 by Appendx 1. Choong = 2 R$ 2 Cb 2 we oban by Appendx 2 &= 2 b 2 C) exp _&(R$2 8C= 2 b 2 R$ & exp \ 2+ &L 2 = fo b and = uffcenly mall. The poof of Lemma 4 hen complee. Fom Lemma 4 and fom he connuy of F fom H a o (Dn C k ) nequale of lage devaon ae now clacal. We efe he eade fo nance o Azenco [1]. K

13 FLOW DECOMPOSITION 35 Lage Devaon fo he C k -Topology. In he followng he & ndex wll concen he C k -opology. Ung he dffeenal equaon afed by he nvee flow eay o ee ha he funcon F defned by (3.7) connuou fom H o a (Dn C k ). Theefoe we deve a pevouly ha I a good ae funcon n C k opology. Now \A/D n A /A and A1 /A1 o ha I(A )I(A ) and I(A1 )I(A1 ). Bu h doe no allow u o conclude. The man pon ha when no a C m -dffeomophm I()=. Theefoe I(A )=nf[i() # A and C m -dffeomophm]. Aume hen ha I(A )< (he cae I(A )= obvou). Le # A C m -dffeomophm be uch ha I()=I(A ). Le n be a equence n A uch ha n w Ck. Snce a C m -dffeomophm we deduce fom he fac ha [ &1 an open mappng ha n w C k. Theefoe # A and I(A )I()=I(A ). A mla agumen hold fo he open e. K 4. LARGE DEVIATIONS FOR PERTURBED STOCHASTIC DIFFERENTIAL EQUATIONS We wll be neeed n h econ n he peubed ochac dffeenal equaon (1.1); (1.1) wll be wen n Io^ fom dx = ==2 _ *(X = ) d+= : X = =x =1 _ (X = ) $B +_~ *(X = ) d+ : l =1 _~ (X = ) $B (4.1) whee _ *( y)= 1 2 : =1\ _ x (y) _ ( y) + _~ *( y)=_~ (y)+ 1 2 : l =1\ _~ x (y) _~ (y) +: v B and B ae wo ndependen andad Bownan moon epecvely defned on he Wene pace W=C ([ 1] R ) and W = C ([ 1] R l ). W and W ae endowed wh he opology of unfom convegence and he Boelan _-feld. We wll denoe by P (epecvely P ) he Wene meaue on W (epecvely W ) and by P he meaue PP on W_W.SoE(epecvely E E) wll be he expecaon unde P (epecvely P P).

14 36 BEN AROUS AND CASTELL v E x wll be he pace C x ([ 1] R n ) of all connuou pah ang fom x wh value n R n endowed wh he unfom convegence. v _ and _~ ae aumed o be n C k b wh kmax(n+1 4+[n2]) ``Peudo'' Lage Devaon fo X = We deve fom Theoem 3 ome exponenal lowe and uppe bound fo X = exendng he eul of [7 11]. Thoughou we wll denoe by H l he CameonMan pace aocaed o he Wene pace W. When h # H and h # H l G(h h ) wll be he oluon o he odnay dffeenal equaon x =x+ : =1 _ (x ) h4 d+ l : =1 Popoon Le A be an open ube of E x : _~ (x ) h4 d+ _~ (x ) d. lm nf = 2 log P(X = # A) = &nf[ 1 2 &h&2 h # H H uch ha _h # H l G(h h )#A] 2. Le A be a cloed ube of E x : lm up = 2 log P(X = # A) = &nf { 1 2 &h&2 H h # $> [g#h _h #H l G(gh )#A $ ] =. The cloue aken wh epec o he unfom convegence and A $ =[y#e x _z#a&z&y&<$]. The eade efeed o Secon 4.3 fo he poof of Popoon Lage Devaon fo a Peubed Sochac Dffeenal Equaon Ung he Bu kholdedavegundy nequaly eay o check ha X = n all L p (W_W E x ). Bu ha aleady been poned ou ha one can no expec a lage devaon pncple fo he law of X =. Howeve Fubn' heoem allow u o conde he andom vaable X = defned fo p2 by X = : WL p (W E x ) (4.12) [ X = ( }).

15 FLOW DECOMPOSITION 37 Le P = be he law of X = (pobably meaue on L p (W E x )). Ou pupoe o how a lage devaon pncple fo he famly (P = ) =. Th wll be done by wng he oluon o Eq. (1.1) n em of he ochac flow defned by he ochac dffeenal equaon (3.5) and by applyng he conacon pncple. Befoe ang he eul we defne he ae funcon. Le be n D n. We aocae o he veco feld n R n ~ ~ ( y )=&1 V _~ ( y) \ # [1... l] ( y )=&1 V _~ *( y)& 1 l 2 =1\ : &1 x (y) + We conde hen he Io^ ochac dffeenal equaon _ 2 x ( y)(~ ( y ) ~ ( y )) 2 &. (4.13) dz~ z~ l =~ ( z~ ) d+ : = x =1 ~ ( z~ ) $B (4.14) Whou aumpon on he exence of a ong oluon o (4.14) no enued. So we wll ec ouelve o flow of dffeomophm n D n b whee ={ # D n up D n y # R n #[1]" m ( ) &1 ( y) m=1 2 = x m "< b (4.15) up y # R n #[1]" m x (y) " < m=123. m D n an open e of b Dn wh he C 3 opology. We defne he opology on D n a he nduced opology. When n b Dn he veco feld ~ b ae bounded and Lpchz n y o ha (4.14) ha a ong oluon. Moeove eay o ee ha E (up #[1] & (z~ ) & p )<. Fo p2 we can hen conde he map D D: D n b L p (W E x ) [ ( ~ [.(z~. ( ~ ))). (4.16) D allow u o anfe on L p (W E x ) he ae funcon I defned by (3.8). \z # L p (W E x ) 4(z)=Inf[I() # D n b D()=z] (4.17) (whee Inf(<)=+).

16 38 BEN AROUS AND CASTELL When A a ube of L p (W E x ) we wll denoe by 4(A)= Inf[4(z) z # A]. Ung he expeon of he ae funcon I we have he followng expeon fo 4. Popoon 6. Fo all z # L p (W E x ) 4(z)=nf { 1 2 &h&2 H h # H uch ha P a.e. \ z =x+ : _ =1 (z ) h4 d l + : =1 _~ (z ) db + _~ (z ) d =. (4.18) Poof. I ealy een fom he expeon (3.8) of I ha 4(z)=nf[ 1 2 &h&2 H h # H F(h)#D n b DbF(h)=z] whee F he map defned by (3.7). Now ung he odnay dffeenal equaon afed by he devave of F(h) and F(h) &1 one can deduce fom Gonwall' lemma ha F(H ) ncluded n D n b. Fuhemoe Theoem 2 how ha D b F(h) a oluon o he ochac dffeenal equaon z =x+ : =1 l _ (z ) h4 d+ : =1 _~ (z ) db + _~ (z ) d. K Once he ae funcon 4 defned we can ae he lage devaon pncple fo he famly (P = ). Theoem 7. v 4 a good ae funcon. v \A/L p (W E x ) &4(A1 )lm nf = = 2 log P = (A)lm up = 2 log P=(A)&4(A ). (4.19) = Befoe povng Theoem 7 we would lke o undelne ha exend o he nonnlpoen cae he eul of Do and Soock [7] when he veco feld _ commue and hoe of Rabehemanana [11] when he Le algeba geneaed by he _ nlpoen. In hee wo pape a pacula veon R =} of he condonal law of he poce X = elave o (=B) condeed and a lage devaon pncple obaned fo he law Q = of he andom vaable # W [ R = =B( ) # M 1 (E x ) (Q = an elemen of M 1 (M 1 (E x ))).

17 FLOW DECOMPOSITION 39 Applyng he conacon pncple we deduce h lage devaon pncple fom Theoem 7 whou nlpoence aumpon. Coollay The map D can be exended o F(W) whee F defned by (3.7). 2. When # F(W) le N be he law of he poce D(). Le u defne R =B by N F(=B). Then R =B a veon of he condonal law of X = elave o (=B). 3. When + an elemen of M 1 (E x ) le u defne 4 (+)=nf[ 1 2 &h&2 h # H H uch ha + he law of he poce z oluon o (4.18)]. If _=(_ 1 }}}_ ) a conan max uch ha a= * nveble hen 4 2 (+)={1 1 + * &E + ( )&E + (_~ *( ))& 2 d f + # a &1 [Rh h # H ] ohewe. 4. Le Q = be he law of he andom vaable [ R =B( ) and le A be a ube of M 1 (E x ). Then &4 (A1 )lm nf = = 2 log Q = (A)lm up = 2 log Q = (A)&4 (A ). (4.2) = Poof of Coollay 8. 1 and 2 ae conequence of Theoem 2 whee poved ha when n F(W) Eq. (4.14) ha a ong oluon defned on [ 1] and ha P-a.e. \ #[1] X = ( ~ )=D(F(=B( ))) ( ~ ). Saemen 3 que obvou nce f +=R h hen akng he expecaon n (4.18) yeld * Snce &E + ( ) * E + ( )=x+ _h4 d+ E + _~ *( ) d. * &E + (_~ *( ))& 2 a &1=nf[&x&2 _x=e + ( ) &E + (_~ *( ))] 1 2 &h&2 H &E +( )&E + (_~ *( ))& 2 a &1 d. Th nequaly beng ue fo all h uch ha +=R h alo hold fo 4 (+). Moeove le P be he ohogonal poecon on G#(Ke(_)) = and defne 6h by 6h = Ph4 * * d; hen +=R 6h _6h=E + ( )&E + (_~ *( )). Bu nce * nveble _ G : G R n nveble and * 6h=_ &1 (E * G +( )&E + (_~ *( ))).

18 4 BEN AROUS AND CASTELL Thu &6h& 2 H = 1 * &E + ( )&E + (_~ *( ))& 2 (_ G _* G ) &1 = 1 * &E + ( )&E + (_~ *( ))& 2 a &1 and 4 (+) 1 2 &6h&2 1 H 2 1 &E * +( )&E + (_~ *( ))& 2 a &1 d. 4 deved fom he conacon pncple. Indeed he map L p (W E x ) P = M 1 (E x ) Z [ he law of Z unde P (4.21) connuou when M 1 (E x ) endowed wh he opology of weak convegence. Moeove anfom P = no Q =. K Remak. Le q + ( } ) be he conugaed quadac fom of E + (_( )) E + (_( ))* ha Then Popoon 6 of [7] ay ha 4 2 (+)={1 1 q + ( x)=nf[&w& 2 E + (_( ))w=x]. * q + ( E + ( )&E + (_~ *( ))) d f + # [R h h # H ] + ohewe. I eem ha h aeon fale. Le u conde he cae v n==l=1; v _~ #1; _(x)=x; _~ =; v x=. Then he law + of he OnenUhlenbeck poce dz =z d+$b can be expeed a R h wh h =. Fo all E(_(z ))=E(z )=. Thu q + ( x)= { + f x= ohewe 1 * q + ( E + ( )&E + (_~ *( ))) d=. Bu f 4 (+)= hen +=R ; ha + he law of he Bownan moon.

19 FLOW DECOMPOSITION Poof of Popoon 5 and Theoem 7 Lemma 9. When D n b povded wh he C 3 opology he map D defned by (4.16) connuou. Poof of Lemma 9. uch ha n w C 3 Le n and be flow of dffeomophm n D n b &D( n )&D()& L p (W E x )=E [ up & n(z~ n )& (z~ )& p ] 1p T 1 +T 2 #[1] whee T 1 =up y " n x ( y) " E [up &z~ n & z~ & p ] 1p T 2 =E [up & n (z~ )& (z~ )& p ] 1p. Teamen of T 1. The f devave of n convege unfomly on compac e of [ 1]_R n o he f devave of (whch ae bounded). Hence up n y &( n x)( y)&<+. We have now o how ha E [up &z~ n ] www n +. Fo # [ 1] le f n ()=E [up #[] &z~ ] 1p. The angle nequaly n L p and he convexy of x [ x p fo p1 yeld f n ()E _ &~ n ( z~ n )&~ n ( z~ )& p d & 1p +E _ &~ n ( z~ )&~ ( z~ )& p d & 1p l +E _ up #[]" =1 p& : 1p (~ n (u z~ n )&~ ( u z~ )) $B & u u u. Ung he Bu kholdedavegundy nequaly hee ex a conan C uch ha l #[]" =1 : E _ up CE _\ (~ l : =1 p& 1p n (u z~ n )&~ ( u z~ )) $B & u u u &~ n ( u z~ n )&~ ( u z~ )& 2 u u du + p2 1p &

20 42 BEN AROUS AND CASTELL l C : =1 E _\ &~ l +C : E _\ =1 l C : =1 u n y" ~ n ( y ) x n (u z~ n )&~ n u (u z~ )& 2 n du + p2 1p & &~ n (u z~ u )&~ ( u z~ u ) & 2 du + p2 1p & " E _ &z~ n & z~ & p u u du & 1p l +C : E _ &~ n (u z~ u )&~ ( u z~ u )& p du & 1p. =1 Bu E [ &z~ n & z~ & p u u du] 1p ( f n(u) p du) 1p 3. Fuhemoe he C convegence of n o mple he unfom convegence on compac e of [ 1]_R n of ~ n and f devave o ~ and f devave whch ae bounded nce an elemen of D n. Theefoe b up n y &~ n ( y )x& <+. Thu hee ex a conan C uch ha f p ()C n _ l f p () d+ : n Gonwall' lemma yeld hen = \ #[1] f p n ()C : l and we have o how ha = E \ 1 &~ n (u z~ )&~ ( u z~ )& p u u du +&. E \ 1 &~ n (u z~ )&~ ( u z~ )& p n u du + ec \ # [... l] E \ 1 &~ n (u z~ u )&~ ( u z~ u )& p du + www. n + Le R be a pove eal numbe and B( R) he ball of adu R n R n E \ 1 &~ n (u z~ )&~ ( u z~ )& p u u du + up #[1]y#B(R) &~ n p ( y )&~ ( y )& +2 p&1 E [1 up #[1] &z~ & R up (&~ n ( y )& p + &~ ( y )& p ]. y

21 FLOW DECOMPOSITION 43 The convegence of n o mple ha up n y (&~ n < ; hu hee a conan C uch ha E \ 1 &~ n (u z~ u )&~ ( u z~ u ) & p du + up #[1]y#B(R) ( y )& + &~ ( y )& ) &~ n ( y )&~ ( y )& p + CP ( up &z~ & R ). #[1] Le '>. Snce z~ oluon o a ochac dffeenal equaon wh bounded coeffcen we can fnd R uch ha CP (up #[1] &z~ & R )< '2. Le n # N be uch ha \nn up #[1]y#B(R) &~ n ( y )& ~ ( y )& p < '2. Then fo nn E ( 1 &~ n (u z~ )&~ ( u z~ ) & p u u du)<'. Teamen of T 2. Snce n and ae elemen of D n and b n w C 3 one can chooe conan K n and K uch ha v \ #[1]\y#R n v up n K n <+. Theefoe T p 2 up y # B(R) & n (y)&k n(1+&y&) & (y)&k(1+&y&); & n (y)& (y)& p +C (up K n +K) E [up (1+&z~ & p ) 1 up &z~ & p > R ] n up & n ( y)&( y)& p y # B(R) +CE (up (1+&z~ & 2p )) 12 P [up < &z~ & > R ] 12. Le '>. Le R be uch ha he econd em le han '2. Fom he unfom convegence of n o we can hen fnd n uch he f em le han '2. Lemma 9 follow. K We ae now able o pove Theoem 7 and Popoon 5. The key o we X = n em of he ochac flow defned by (3.5) ung he map D and hen o apply he conacon pncple. The only echncal pon ha he pobably fo he ochac flow o be n D n b cly le han

22 44 BEN AROUS AND CASTELL one. So we wll have o uncae he veco feld _. Fo all R> we appoxmae he veco feld _ by ome C k veco feld b _R uch ha _ R ( y)= f &y&2r _ R (y)=_ (y) f &y&r. Le X = R be oluon o he Saonovch ochac dffeenal equaon =x+= : =1 X = R l + : =1 _ R (X=R ) db _~ (X =R ) db + _~ (X = R ) d. A long a X = ay n B( R) X =R =X =. Moeove X = and X =R ae oluon o ochac dffeenal equaon wh bounded coeffcen. Theeby one can fnd conan C R > uch ha fo RR and =1 P(up &X = &R)C exp(&r 2 C ) E(up &X = R &X = & p )C exp(&r 2 C ). (4.22) The advanage n condeng X = R nead of X = ha f 8 = R he ochac flow aocaed o he ochac dffeenal equaon d! = R = ) db 8= R # D n nce b 8= R (x)=x wheneve x B( 2R). = =1 _R (!=R Poof of Popoon 5. The lowe bound ha aleady been poved n [7] and n [11] n he geneal cae. So we only gve he poof of he uppe bound. Le A be a cloed ube of E x. Le u fx L> and = =LR (whee R choen o ha (4.22) hold). Then fo == P[X = # A]P[X =L= # A]+P[up &X = &L=] P[D(8 = L= )(B )#A]+C exp(&l 2 C = 2 ) E[N 8 = L= (A)]+C exp(]&l 2 C = 2 ). We ecall ha N 8 he law of he poce D(8) o ha N 8 (A)1. Theefoe P[X = # A]P[N 8=L= (A)>]+C exp(&l 2 C = 2 ).

23 FLOW DECOMPOSITION 45 Noaon ae he ame a n Coollay 8. The veco feld _ L= convege n C m unfomly on compac e o he veco feld _ b. Fom Theoem 3 eul hen ha lm up = 2 log P[N 8 =L= (A)>] = &nf[ 1 2 &h&2 F(h)#[ # H F(W)&Dn b N (A)>]]. Bu F(W)/F(H ) (cf. [4]) and he connuy of he map D yeld [ # F(W)&D n b N (A)>]/[ # F(H ) N (A)>] / $> / $> F[h # H N F(h) (A $ )>] F[h # H _h # H l G(h h )#A $ ]. The la ncluon gven by he uppo heoem fo dffuon nce N F(h) nohng bu he law of he dffuon defned by (4.18). Le B $ #[h # H _h # H l G(h h )#A $ ]. I eman o how ha I 1 #nf {12 &h&2 H h # $> B $= nf { 12 &h&2 H F(h)# F(B $ ) =#I 2. $> Th nequaly obvou when I 2 =+. Theefoe we aume ha I 2 <. Le h # H uch ha I 2 =12 &h& 2 H and uch ha #F(h)# $> F(B $ ). Fo all $> le (h $ ) be a equence n B n $ uch ha $ # n F(h $ n ). Le u defne he map M: F(H ) H [ he unque h uch ha {=F(h) I()=12 &h& 2 H. I poved n Appendx 3 ha M connuou when H endowed wh he unfom convegence. Theefoe M( $ ) M()=h. Bu n M($ n )=6h$ n (whee 6 defned by (3.9)). Moeove eay o check ha 6h $ n # B $. Thu h # $> B $ and I 1 12 &h& 2 =I H 2. K Poof of Theoem 7. F of all eul fom he connuy of D and he conacon pncple ha 4 a ``good'' ae funcon. Poof of he uppe bound. Le A be a cloed ube of L p (W E x ). Le u fx '> L> = =LR (whee R choen o ha (4.22) hold). A ' wll denoe he ube of L p (W E x ) defned by

24 46 BEN AROUS AND CASTELL A ' =[z # L p (W E x ) _y # A &y&z& L p (W E x )'] P = (A)=P(X = #A) P(X =L= # A ' )+P(&X =L= &X = & L p (W E x )>') P(D(8 = L= )#A ' )+P(E (up P(D(8 =L= )#A ' )+ 1 ' p E(up = L= &X &X = & p )>' p ) = L= &X &X = & p ). Fo == 7 1 L=R. We deve fom (4.22) ha he econd em bounded up by C exp(&l 2 C = 2 = L= ). Fom Theoem 3 he law of 8 afe a lage devaon pncple wh ae funcon I. The map D beng connuou we u have o apply he conacon pncple o deve Theefoe \L> \'> lm up = 2 log P(D(8 = L= )#A ' )&4(A ' ). = lm up = 2 log P = (A) &nf(4(a ' ) L 2 C ). = Leng L go o nfny we deve ha \'> lm up = = 2 log P = (A) &4(A ' ). 4 beng a good ae funcon 4(A ' ) ww 4(A). ' Poof of he lowe bound. Le A be an open ube of L p (W E x ). When 4(A)=+ he lowe bound val. So we aume ha 4(A)<+. Le g # A be uch ha 4( g)<+. Le L 1 be uch ha 4( g)<l 1. A beng open we can chooe '> uch ha B( g ')/A. Le u fx L>- C L 1 and = =LR (whee R C ae conan uch ha (4.22) hold). P = (A)P = (B(g ')) P(X =L= # B( g '2); &X =L= &X = & L p (W E x )<'2) P(X =L= # B( g '2))&P(&X =L= &X = & L p (W E x )'2) P(D(8 =L= )#B(g'2))& 2p ' E(up p By he conacon pncple = L= &X &X = & p ). lm nf = 2 log P(D(8 = L= )#B(g'2))&4(B(g '2))&4(g) =

25 FLOW DECOMPOSITION 47 Fo == = L= E(up &X &X = & p )C e &L2 C = 2 ' p 1=2 e&l p 2 fo uffcenly mall =. Fom 4(g)<L 1 follow ha lm nf = 2 log P = (A)&4(g) = Takng he upemum ove A we oban he eul. K 5. LARGE DEVIATIONS IN A NONLINEAR FILTERING PROBLEM Th econ deal wh anohe applcaon of flow decompoon. I concen a nonlnea fleng poblem ha ha been f uded by Do [6] and hen by Rabehemanana [11]. The poblem can be aed a follow. Le u conde he couple gnal-obevaon (X = Y = ) oluon o he yem of ochac dffeenal equaon dx = == : =1 dy = =1(X = ) d+db _ (X = ) db +=2 _~ (X = ) d+ : l =1 _~ (X = ) dy= (5.23) whee X = =x; Y = = v B B ~ afy he ame aumpon a n Secon 4. v 1 a uffcenly mooh funcon fom R n o R l. We wan o oban a lage devaon pncple fo he condonal law of he gnal X = elave o he obevaon Y =. Such a pncple ha been obaned n [6 11] unde ome nlpoence aumpon fo he veco feld. A n Secon 4 we would lke o fee ouelve of hee aumpon by ung flow decompoon and he lage devaon fo ochac flow. A done n [6 11] he f ep o oban uch a pncple o make a change of pobably n uch a way ha he new law of Y = he law of a Bownan moon ndependen of B. So we ae led o oban a lage devaon pncple fo he condonal law of he poce X = defned by (1.1) elave o he Bownan moon B.

26 48 BEN AROUS AND CASTELL 5.1. Lage Devaon fo he Condonal Law of X = Relave o B We begn by decompong he ochac dffeenal equaon (4.1) (o (1.1)). Le be n D n. We aocae o he veco feld ( y)=&1 V _ ( y) \ # [1... ] ( y)=&1 V _*( y)& 1 2 =1\ : &1 x ( y) + 2 x 2 ( y)( ( y) ( y)) &. (5.24) We conde hen he Io^ ochac dffeenal equaon dz = == 2 = ( z ) d+= : =1 z = =x. = ( z ) $B (5.25) ae connuou and locally Lpchz. Thu he aecoe of z = may explode. Nevehele (5.25) defne a map fom W o he pace of explove aecoe E x (R n ) (ee Azenco [1]) E x (R n )=[f:[1]r n _f()=x f connuou: f( )=O\ #[ 1]f()=]. When f # E x (R n ) we defne he exploon me of f a {( f )#nf[ f()=]. We wll ay ha a equence ( f n ) n n E x (R n ) convege o f # E x (R n ) f and only f ( f n ) n convege o f unfomly on compac ube of [ {( f )[. In pacula h mean ha {( f )lm nf {( f n ). Smlaly a n Secon 4 we defne hen he map D : D n L (W E x (R n )) [ ( [.(z. = )) (wh he convenon ()=). Le T = ( d ) denoe he law of he poce D ()(T = ( d ) a pobably meaue on E x (R n )). We wll how a lage devaon pncple fo he famly (T = ( d )) =. A n Secon 3 we begn wh he ``quaconnuy'' of he map = [ D ()(= ).

27 FLOW DECOMPOSITION 49 Popoon 1. Gven h # H and # D n we defne he poce x (h) a he oluon o he odnay dffeenal equaon x (h)=x+ : =1 ( x (h)) h4 d. (5.26) Then \ # D n \K compac e of R n \a> \L> \R> \T #]1] _b= uch ha \== \h # H uch ha &h& H a x (h)([ T])/K P[ up & (z = )& (x (h))&r; 2 &=B&h&b]e&L=. #[T] Poof of Popoon 1. \'> we wll denoe by K ' he e K ' =[y#r n _z#k&y&z&']. Le ' # ] 1[ be uch ha Then up & (y)& (z)&<r. #[1]yz#K 1 &y&z&' P[ up #[T] & (z = )& (x (h))&r; &=B&h&b] P[ up & (z = )& (x (h))&r; up #[T] #[T] +P[ up #[T] P[ up #[1] yz#k 1 & y&z&' +P[ up #[T] =P[ up #[T] &z = &x (h)&';&=b&h&b] & (y)& (z)&r] &z = &x (h)&';&=b&h&b] &z = &x (h)&';&=b&h&b]. &z = &x (h)&'] So we ae led o how ha \K compac ube of R n \T # ] 1] \a L> \R> ee ex = b> uch ha \== and \h # H &h& H a x (h)([ T])/K P[ up &z = &x 2 (h)&r;&=b&h&b]e&l=. #[T] Cae h=. We wll need he followng lemma whch ae he quaconnuy n he cae h=.

28 5 BEN AROUS AND CASTELL Lemma 11. Gven c: [1]_R n R n n C k we defne he pocee z = and x a he oluon (n E x (R n )) o he equaon z = = x + c( z x = =x+ c( x ) d. ) d+= 2 = ( z ) d+= : =1 ( z = ) $B Then \ # D n \L R> \T # ] 1] \K compac ube of R n uch ha x ([ T])/K _b = uch ha \== = P[ up &z & x & R; 2 &=B&b]e&L=. #[T] Poof of Lemma 11. Le % = R be he oppng me % = R=nf[ uch ha &z = & x & R]. When % = 7 T z = R # K R. Theefoe {(z = )>% = R7TP-a.e. Fuhemoe Fo all % = R 7 T &z P[ up &z #[T] =P[ = & x & &c( z = & x up % = R 7 T &z & R; &=B&b] = & x & R; &=B&b]. = )&c( x )& d+=2 += " =1 : ( z = ) $B " c up y # K R" x ( y) " &z + " = =1 : ( z = ) $B " & ( z = ) & d = & x & d+=2 up & ( y)& y # K R. By Gonwall' lemma we oban ha fo ome conan C (dependng on K and R) = up &z % = & x & C \ = 2 + up 7T" = : R 7T % = R =1 ( z = ) $B"+.

29 FLOW DECOMPOSITION 51 Theefoe P[ up % = R 7 T &z = & x & R; &=B&b] P C R 2& +P _ up T" = : ( z = _=2 % = R 7 =1 ) $B " R 2C ; &=B&b &. The f em vanhe when =<(R2C) 12. Thu we ae led o how ha \L R R$> _b = uch ha == O P _ up T" = : ( z = % = R 7 =1 ) $B " R$; &=B&b & 2 e&l=. Fo all nege n we defne v k =kn (k=... n). v n ( y)=( k y) \y # R n \ #[ k ; k+1 [. n = = v z = z k \ #[ k ; k+1 [. Le L be a compac e n R n uch ha [( y); #[1] y#k R ]/L: wh P _ up T" = : % = R =1 ( z = ) $B 7 " R$; &=B&b & P 1+P 2 +P 3 v P 1 =P[up = % &z R 7 T n = = & z & + & n & & >#] C 1 (L) v P 2 =P[up = % R 7 T&z n = = & z & + & n & & #; C 1 (L) up = % R 7 T&= =1 ( ( z = )& n n = ( z )) $B &R$2] v P 3 =P[up = % R 7 T &= =1 n = n ( z ) $B R$2; &=B&b]. Teamen of P 2. = =1 7%= R 7T ( ( z = )& n n = ( z _ )) $B a mangale wh quadac vaaon = 2 : =1 7% = R 7T & ( z = )& n ( z n = ) & 2 d C= 2 = n = ( up &z % = & z & + & n & & C 1 (L) )2 R 7T C= 2 # 2. Theefoe P 2 C 1 e &C 2R$ 2 = 2 # e&l= 2 fo # and = uffcenly mall.

30 52 BEN AROUS AND CASTELL Teamen of P 1. & n && C 1 (L) up $#[1] &$ <1n y # L : n\" :#N : 1 : " : &1 + ( y)& : &1 $ x : y)& : x ( $ ( y) " : x : x : ( y) "+. Thu he connuy n ( y) of he funcon ( y) &1 ( y) ( x)( y) (( x)( y)) &1 how ha & n && C 1 (L) n. Once # fxed poble o chooe n 1 uch ha fo nn 1 & n && C (L)<#2. Thu fo 1 nn 1 Now \=1 P 1 P _ up % = R 7 T &z n = = & z & > 2& #. n = = up &z % = & z & = up R 7 T % = R 7 T k; #[ k k+1 [" + : =1 ( z = ) =$B k " = c( z ) k d+ = 2 ( z = ) d k up y # K R (&c( y)&+& ( y)&) 1 n So P _ up % = R 7 T &z + up k; #[ k k+1 [" % = R 7 T n = = & z & > 2& # =1 : ( z = ) =$B k ". P _ up y # K R (&c( y)&+& ( y)&) 1 n >#4 & n&1 + : k= #[ k k+1 [" : k =1 % = R 7 T P _ up ( z = ) =$B" ># 4&.

31 FLOW DECOMPOSITION 53 The f em of he ummaon vanhe fo uffcenly lage n (nn 2 ). Fuhemoe he quadac vaaon of k = =1 ( z = ) $B bounded up by C= 2 n. Theefoe fo nup(n 1 n 2 ) P 1 C 1 e &C 2# 2 n= e&l= 2 fo lage n and mall =. Teamen of P 3. Fo #[ k k+1 [ n ( z yeld ha \% = 7 T R " = : =1 n&1 n n = ( z ) $B " = " = : k= : =1 ( k z n = )= ( k z = k ). Th = k )(B 7 k+1 &B 7 k ) " up & ( y)& : &=B k+1 &=B k & y # K R k Cbn. Theefoe P 3 P(CbnR$2)= fo b uffcenly mall. And he poof of Lemma 11 complee. K We eun now o he poof of Popoon 1 ha o he cae h{. Cae h{. Gven h # H &h& H a x (h)([ T])/K we defne v he poce W = =B &h = v he pobably P = on W by dp = =exp dp } _1 _(B ) = (h4 $B )& 1 2= &h4 & 2 d & 2 v A#[up T &z = (=B)&x (h)&r]&[&=b&h&b]. v B#[ 1 (h4 $B )<&*=]. Then P(A)P(B)+E = [(dpdp = ) 1 A 1 B c]. Snce 1 (h4 $B ) gauan wh mean and vaance &h& 2 H fo lage * and mall =: E = P(B) = &h& H * -2? exp \ & *2 + 2= 2 &h& 1 2 H 2 exp \ 2+ &L = _ dp dp 1 A1 = B c& exp \ * 2+ = exp \ a 2+ 2= _P = [up T &z = &x (h)&r; &=W = &b]

32 54 BEN AROUS AND CASTELL wh dz = == 2 = ( z ) d+ : =1 dx (h)= : =1 ( x (h)) h4 d. = ( z ) h4 d+= : =1 = ( z ) $W = Unde P = W = a andad Bownan moon. Applyng Lemma 11 wh c( y)= =1 ( y) h4 we oban an exponenal bound fo E = [(dpdp = ) 1 A 1 B c] and he poof of Popoon 1 follow. K Fom Popoon 1 we deve a uually a lage devaon pncple fo he famly (T = ( d )) =. Popoon 12. E x (R n ) by Fo fxed n D n we defne he ae funcon L on \z # E x (R n ) L (z)=nf[ 1 2 &h&2 H h # H uch ha z.=.(x. (h))] whee x (h) oluon o (5.26). Then we have v L a ``good'' ae funcon. v \A/E x (R n ) &L (A1 )lm nf = = 2 log T = ( A)= 2 lm up = 2 log T = ( A)&L (A ). = Moeove f {(z = )>1 P-a.e. and f \h # H {(x (h))>1 he eul eman ue when he opologcal pace E x (R n ) eplaced by he opologcal pace E x. Poof of Popoon 12. I eul a uual fom he quaconnuy and he connuy of he map h # H a [ (x (h)) # E x (R n ). K We deve fom Popoon 12 a lage devaon pncple fo a pacula veon of he condonal law of X = elave o B. A n Secon 4 we defne: F : H l D n h [ flow of dffeomophm aocaed o he odnay dffeenal equaon (5.27) l dx =_~ (x ) d+ : =1 _~ (x ) h 4. Accodng o he eul of Bmu [4] F can be exended n a meauable way o W. Th exenon wll ll be denoed by F. We defne hen fo P -almo all ~ :

33 FLOW DECOMPOSITION 55 v he pobably N = ( ~ d )#T = (F ( ~ ) d ); v \h # H l! (h)( ~ )#F ( ~ )(x F ( ~ ) (h)); v he ae funcon l ~ =L F ( ~ ). We have hen he followng eul. Popoon \h # H! (h) a oluon o d! (h)=x+ : =1 _ (! (h)) h4 d+ _~ (! (h)) d l + : =1 _~ (! (h)) db and l ~ (z)=nf[12 &h& 2 h # H l H uch ha z=! (h)]. P -a.e. l ~ a good ae funcon. 2. N = ( ~ d ) a pobably meaue on E x whch a veon of he condonal law of X = elave o B. P -a.e.\a/e x : &l ~ (A1)lm nf = = 2 log N = ( ~ A)lm up = 2 log N = ( ~ A)&l ~ (A ). = Poof of Popoon 13. Pon 1 a conequence of Theoem 2 and of he defnon of l ~. We deve alo fom Theoem 2 ha P({(z F ( ~ ) = ( ))>1)=1 and P-a.e. X = ( ~ )=D (F ( ~ ))( ). Theefoe N = ( ~ d ) veon of he law of X = elave o B. Lage devaon nequale ae he ame a n Popoon 12. K 5.2. Applcaon o Nonlnea Fleng We conde now he ognal poblem ha lage devaon fo he condonal law of X = elave o Y = whee (X = Y = ) a oluon o (5.23). To begn wh we noduce ome noaon: v An elemen y of R n+1 wll be decompoed no ( y 1 y 2 ) whee y 1 # R n and y 2 # R. v We defne he followng veco feld n R n+1 : \y # R n+1 \ # [1... ] * ( y)# \_ ( y 1 ) + ; \ # [1... l] * (y 1 ) (y)# \_~ 1 (y 1 )+ ;

34 56 BEN AROUS AND CASTELL _~ ( y 1 ) * ( y)#\ + & 1 l 2 =1\ : 1 x ( y 1) _~ ( y 1 ) + * *( y)# 1 2 : =1\ * x ( y) * ( y) +. no v A flow of dffeomophm n R n+1 anpo hee veco feld &^ &^ ( y )=&1 V * ( y) =1... ( y )=&1 V **( y)& 1 2 =1\ : &1 x ( y) + 2 x 2 ( y)(^ ( y ) ^ ( y )) &. v Fo all # D n+1 we wll denoe by z^ = he poce n E x (R n+1 ) oluon o dz^ z^ = == 2 ^ ( z^ = ) d+= : = o =(x). =1 ^ ( z^ = v We map hen D n+1 o L (W E x (R n+1 )) by D : D n+1 L (W E x (R n+1 )) [ ( [.(z^. = ( ))). ) $B v The pocee C = (1) # E x(r n ) and C = (2) # E x(r 1 ) ae defned by D ()#(C = (1) C= (2) ). v Fnally we wll denoe by F he ``flow'' map F : W D n+1 ~ [ ochac flow aocaed o he ochac dffeenal l equaon d! =* (! ) d+ : We hen have hen he followng eul. =1 * (! ) db.

35 FLOW DECOMPOSITION 57 Theoem \A # B(E x (R n )) we defne M = ( ~ A)# E[1 = F ( ~ ) A(C (1) ) exp(c = F ( ~ ) E[exp(C = F ( ~ ) (2) 1 &1 2 1 F ( ~ ) &1(C= (2) 1 &1 2 1 (1) )& 2 d)] &1(C= F ( ~ ) (1) )& 2 d)] M = ( ~ v) a pobably meaue on E x whch a veon of he condonal law of X = elave o Y =. 2. P -a.e. \A/E x &l ~ (A1 )lm nf = = 2 log M = ( ~ A)lm up = 2 log M = ( ~ A)&l ~ (A ) = wh l ~ a n Popoon 13. Poof of Popoon 14. = \=> we defne a new pobably P on W W by = dp =exp dp} _(B _& 1(X = ) $B & 1 B ) 2 &1 (X = )&2 d & =exp _& 1(X = ) $Y = &1 (X = )& 2 d &. Unde P = Y = ha he law of a Bownan moon ndependen of B. Theefoe (X = Y = ) ha he ame law unde P = a (X = B ) unde P. Le G and H be wo meauable funcon epecvely defned on E x and E : E(G(X = ) H(Y = ))=E = We deduce hen ha P -a.e. \ G(X= ) H(Y = ) dp dp =+ =E _G(X = ) H(B ) exp \ 1 & &1(X = )&2 d +&. 1(X = ) $B E(G(X = ) Y = = ~ )= E[G(X= ) exp( 1 1(X = ) $B & &1(X = )&2 d) B = ~ ] E[exp( 1 1(X = ) $B & &1(X = )&2 d) B = ~ ] The poce S = =(X= 1(X= ) $B ) afe he ochac dffeenal equaon ds = == : =1 S = =(x). * (S = ) db + : l =1 * (S = ) db +* (S = ) d.

36 58 BEN AROUS AND CASTELL Thu he decompoon of ochac dffenal equaon yeld F ( ~ v P-a.e. {(z^ ) ( ))>1. v P-a.e. S = =D (F ( ~ ))( ). By ndependence of B and B eul ha P -a.e. \A # B(E x ) E(1 A (X = ) Y = = ~ )= E[1 = F ( ~ ) = F ( ~ ) A(C (1) ) exp(c (2) 1 &1 2 1 F ( ~ ) &1(C= (1) )& 2 d)] = F ( ~ ) E[exp(C (2) 1 & = F ( ~ ) &1(C(1) )& 2 d)] Th pove 1. =M = ( ~ A) Poof of he lowe bound. Le A be an open e of E x. When l ~ (A)=+ he lowe bound val le u uppoe ha l ~ (A)<+. Le h # H be uch ha z#f ( ~ )(x F ( ~ ) (h)) # A (whee x (h) defned by (5.26)) 1 2 &h&2 =l H ~ (z)=l ~ (A). le u noduce v he andom vaable U = #exp \C = F ( ~ ) (2) 1 &1 2 1 v he poce W = =B &h = v he pobably meaue P = on W = F ( ~ ) &1(C (1) )& 2 d + (5.28) dp} = =exp _1 _(B ) = (h4 $W = )+ 1 2= &h4 & 2 d &. 2 Then M = = F ( ~ ) ( ~ A)=E(1 A (C(1) ) U = )E(U = ). By Ganov' anfomaon = F ( ~ ) E(1 A (C (1) )U = ) =E = \ 1 = F ( ~ ) A(C (1) ) U dp =+ = dp = F ( ~ ) =E \1 A(C (1) ) U = exp \&1 = (h4 $B )& 1 2= &h4 & 2 d 2 ++

37 FLOW DECOMPOSITION 59 whee &C = ( )=D () \ +h =+ Theefoe &U = =exp \C = F ( ~ ) (2) 1 &1 2 1 = F ( ~ ) &1(C (1) ) " 2 d +. = F ( ~ ) E(1 A (C (1) ) U = = F ( ~ ) )E(1 A (C (1) ) U = 1 1 h4 $B K) ~ (A) _exp \&l = + exp \ &K =+. 2 Now ung well-known eul abou he aympoc behavo of peubed = dynamc yem P-a.e. C ww (z )ne = x (R n ) whee z a oluon o he odnay dffeenal equaon z = \ x + + =1 : ^ ( z ) h4 d. Thu P-a.e. U = ww = U wh U > and ndependen of B. Moeove ealy een fom defnon of F ha \ #[1] \(x 1 x 2 )#R n+1 F ( ~ )(x 1 x 2 )= \ F ( ~ )(x 1 ) x 2 +G (x 1 ~ )+ fo ome funcon G and F defned by (5.27). Th yeld ha v \ # [1... ] F ( ~ ) v \#[1] z F ( ~ ) ( x 1 ) F ( ~ ) ^ ( x 1 x 2 )=\ +. & G F ( ~ ) ( x 1 ) x 1 F ( ~ ) 1 =x = F ( ~ ) Theefoe C (1) F ( ~ )(x ha = F ( ~ ) lm nf E(1 A (C (1) ) U = F ( ~ ) (h). (h))=z. By Faou' lemma we deduce hen = 1 1 )E(1 h4 $B K A(z) U 1 1 ) h4 $B K =U P \ 1 h4 $B K + U 2

38 6 BEN AROUS AND CASTELL fo uffcenly lage K. I eul hen ha lm nf = = 2 = F ( ~ ) log E(1 A (C (1) ) U = )&l ~ (A). Now = 2 log M = ( ~ A)== 2 = F ( ~ ) log E(1 A (C (1) )U = )&= 2 log E(U = ). The lowe bound follow hen fom he fac ha P -a.e. \p1 up =1 E((U = ) p )< whch poved n Appendx 4. Poof of he uppe bound. Le A be a cloed ube of E x. Ung Ho lde' nequaly \p>1 = 2 log M = ( ~ A) 1 p =2 = F ( ~ ) log E(1 A (C (1) ))+ 1 q =2 log E((U = ) q ) &= 2 log E(U = ) (wh 1p+1q=1). = F ( ~ ) The pacula fom of F ( ~ ) mple ha C (1) =F ( ~ )(z F ( ~ ) = ) whee z = defned by (5.25). Snce P -a.e. \p1 up =1 E((U = ) p )< we have lm up = 2 log M = ( ~ A) 1 = p lm up = 2 log P(F ( ~ )(z F ( ~ ) = )#A) = & 1 p l ~ (A) by Popoon 13. Leng p deceae o 1 we oban he eul. K APPENDIX 1 Le F and G be wo funcon n C m b (Rn R). Le be a bounded doman n R n. Le : R n be n W m2 (). We aume ha hee a conan R> uch ha && W m2 ()R && W ()R. m2 Then f mn+1 1. F b # W m2 () and hee a conan C> uch ha up &Fb& W m2 ()C. && W m2 () R 2. When F # C m+1 b &Fb&Fb& W m2 ()C&&& W m2 (). 3. &F b &G b & W m2 ()C&F&G& Cb m.

39 FLOW DECOMPOSITION 61 Poof. \: # N n : m D : (Fb)(x)= : ;#N n ; : D ; F((x)) P ; (x) whee P ; a polynomal n he devave of whoe each monomal > l =1 D# (x) afe l =1 # =:. Theefoe &D : (F b )(x)& 2 dxc : ;#N n ; : up &D ; F(x)& 2 &P ; (x)& 2 dx. x Thu we have o conol em uch a > l =1 &D# (x)& 2 dx wh # =:. When # m&[n2]&1 he Sobolev embeddng heoem enue ha up &D # (x)&c && W ()CR. m2 x Bu hee a mo one em wh # >m&[n2]&1. Indeed f # m&[n2] nce mn+1. Theefoe : { # = : & # _n 2& m& _ n 2& &1 l ` &D # (x)& 2 dx ` up &D # (x)& 2 =1 { &D # (x)& 2 dx x CR 2l&2 && 2 W m2 () CR2l. Th pove 1. 2 and 3 ae obaned n a mla way: &D : (F b )(x)&d : (Fb)(x)& 2 dx C : ;#N n ; : +C : ;#N n ; : up &D ; F(x)& 2 &P ; (x)&q ; (x)& 2 dx x up &D ; F((x))&D ; F((x))& 2 &Q ; (x)& 2 dx. x

40 62 BEN AROUS AND CASTELL Q ; he ame polynomal a P ; fo he devave of. Snce F n C m+1 b up x &D ; F((x))&D ; F((x))& 2 C up &(x)&(x)& 2 x C &&& 2 W m2 (). We oban an uppe bound (dependng on R) o &Q ; (x)& 2 (x)dx a pevouly. To conol &P ; (x)&q ; (x)& 2 dx we have o bound up em uch a l " ` =1 l D # (x)& ` D # (x) " 2 dx =1 l C : D =1 "` # (x)(d # (x)&d # (x)) ` D # (x) " 2 dx. < > The ame agumen a fo 1 mple 2. Fnally &D : (F b )(x)&d : (Gb)(x)& 2 dx C : ;#N n ; : &(D ; F((x))&D ; G((x))) P ; (x)& 2 dx C &F&G& 2 m C b : &P ; (x)& 2 dx. K ;#N n ; : APPENDIX 2 Le H be a Hlbe pace and le (e n ) n be an ohonomal ba of H. Le ( A (F ) #[1] P) be a pobably pace. Le M be a F -adaped poce wh value n H. We aume ha fo all n M n #(M e n ) a eal F -mangale and ha hee ex conan K L> uch ha \N # N* \ #[1] N : k=1 (M k ) K P-a.e. N : k l=1 M k Ml d(mk M l ) L P-a.e.

41 FLOW DECOMPOSITION 63 Then \R uch ha R 2 K &K) 2 P[ up &M & H R]exp #[1] _&(R2 8L &. Poof. Le V N #Span(e 1... e N ) and le P N denoe he ohogonal poecon on V N : P[ up &P N M & H R]=P[ up &P N M & 2 HR 2 ] #[1] #[1] P _ N up #[1]\ : (M k )2 &(M k ) k=1 + R2 &K &; N k=1 (Mk )2 &(M k ) a eal mangale whoe quadaque vaaon A a eul: N 4 : k l=1 M k Ml d(mk M l ) 4L. &K) 2 P[ up &P N (M )& H R]exp #[1] _&(R2 8L &. Leng N go o nfny Beppo Lev' lemma yeld he eul. K APPENDIX 3 The map M: F(H ) H [ he unque h uch ha {=F(h) I()=12 &h& 2 H connuou when H endowed wh he unfom convegence. Poof. Le ( n ) and be uch ha n C k. Le h n =M( n ) and h=m(). By Theoem 3 d-a.e. h4 n and h4 # V =. Moeove n (x)& (x)= (_( n (x))&_( (x))) h4 d + _( n (x))(h4 n &h4 ) d.

42 64 BEN AROUS AND CASTELL Ung he popee of _ we deve ha fo all K compac ube of R n up x # K" _( n (x))(h4 n &h4 ) d " ww. (5.29) n Now le u noe ha V= x # R n Ke _(x)= x # I Ke _(x) whee I a fne e. Le K be a compac e of R n uch ha I/K. The convegence of n o yeld up x # K &( n )&1 (x)& up x # K &( ) &1 (x)&. Thu hee a compac e K of R n uch ha [( n )&1 (x) x # K #[1]n#N]/K. Now up x # I & _(x)(h4 n & h4 ) d& up x # K & o _(n(x))(h4 n &h4 ) d&. Ung (5.29) we oban ha up &_(x)(h n &h )&. x # I Le I=[x 1... x n ] _ #_(x ) V #Ke _ and P poecon on V = : be he ohogonal whee up &_ (h n &h )&=up &_ (P (h n &h ))& * mn ((_ V = )* _ V = ) 12 up &P (h n &h )& v fo all max S ymmec and pove * mn (S) he lowe egenvalue of S. v _ V = : V = Im(_ ) y [ _ ( y). By defnon _ V = nveble; hu * mn ((_ V = )* _ V = )>. Theefoe \ # [1... n] up &P (h n &h )&. Bu h n &h # V = =( V ) = =V = and we oban ha up &h n&h &. K APPENDIX 4 Le U = be defned n Secon 5 by (5.28). Then\p P -a.e. up =1 E[(U = ) p ]<. Poof. We wll n fac how ha E [up =1 E[(U = ) p ]]<:

43 FLOW DECOMPOSITION 65 Le E [up =1 a oluon o E[(U = ) p ]]E[up(U = ) p ] =1 E _up exp \ p 1 1(X = ) $B & p =1 2 1 &1 (X = )&2 d +& E _up exp \ p 1 1(X = ) $B =1 +&. + X = M = #p 1(X= )$B ; V = #\M = = dv = =* (V = ) db +* (V = ) db +* (V = ) d =\+ x V = = whee \m+=\ x =_ (x) x _~ (x) * + * \m+=\p1 (x)+ = = x _~ (x) * \m+=\12 1$ (x) 1 (x)+. = Thu all hee veco feld ae C m b on Rn _R + _[ 1]. Theefoe hee a veon of V = whch connuouly dffeenable n (x =). Moeove V = = a oluon o he vaaonal equaon d \V = = + =: *$ (V = ) V = = db +: +*$ (V = ) V = = d V = = =\+. 1 * $ (V = ) V = = db

44 66 BEN AROUS AND CASTELL Theefoe E[up exp(m = )]E =1 _ 1 exp(m = ) } M = = } d= & +E(exp(M )) E[exp(2M = )] 12 E d=+e(exp(m _\M= = + & )). v Snce (M = ) =p 2 &1(X= )&2 d bounded by ome conan K fo all =1 E[exp(2M = )]exp(2k). v E _\M= = + 2& =E _ M= = & =p 2 E _: \ C 1$ (X = ) X = 2 d = + & = 2 E _"X = " & d. Applyng Io^ fomula eay o check ha hee a conan C uch ha \=1. E _"X = = " 2& C+CE _ : " +CE _: C \1+ _ (X = )+=_$ (X = ) X = 2 d = " & " _~ $ ( = ) X = 2 d = " & = 2 E _"X = " & d +. Anohe applcaon of he Gonwall lemma complee he poof. K REFERENCES 1. R. Azenco Gande de vaon e applcaon. Ecole d'e e de Pobable de San- Flou VIII 1978 Lecue Noe n Mah. Vol. 774 pp. 176 Spnge-Velag BelnHedelbegNew Yok P. Bald and M. Sanz-Sole Modulu of connuy fo ochac flow Poge n Pobably Vol. 32 Bkha ue Bael 1993.

45 FLOW DECOMPOSITION C. Bezudenhou Sngula peubaon of degeneae dffuon Ann. Pobab. 15 No. 3 (1987) J. M. Bmu A genealzed fomula of Io and ome ohe popee of ochac flow Z. Wach. 55 (1981) H. Do Len ene e quaon dffe enelle ochaque e odnae Ann. In. H. Poncae Nouv. Se. Sec. B 13 (1977) H. Do Un nouveau pncpe de gande de vaon en he oe du flage non lne ae Ann. In. H. Poncae 27 No. 3 (1991) H. Do and D. W. Soock Nouveaux e ula concenan le pee peubaon de ye me dynamque J. Func. Anal. 11 No. 2 (1991) M. I. Fedln and A. D. Wenzell Small andom peubaon of dynamcal yem Ruan Mah. Suvey 25 (197) H. Kuna On he decompoon of oluon of ochac dffeenal equaon. Sochac Inegal n ``Poceedng LMS Duham Sympoum 198'' Lecue Noe n Mahemac Vol. 851 pp Spnge-Velag BelnHedelbegNew Yok A. Mlle D. Nuala and M. Sanz-Sole Lage devaon fo a cla of ancpang ochac dffeenal equaon Ann. Pobab. 2 No. 4 (1992) J. T. Rabehemanana ``Pee peubaon de ye me dynamque e alge be de Le nlpoene'' The e de l'unvee Pa