Chaos expansions and local times

Publicacions Ma emá iques, Vol 36 (1992), 827-836 . A bs ac CHAOS EXPANSIONS AND LOCAL TIMES DAVID NUALAII'I' AND JOSEP VIVES In his no e we p o e ha he Local Time a ze o o a mul ipa a- ne ic Wiene p ocess belongs o he Sobole space Dk - z -,,2 o any e > 0 . We do his compu ing i s Wiene chaos expansion . We see also ha his expansion con e ges almos su ely . Finally, us- ing he same ecl nique we p o e simila esul s o a eno malized Local Time o he au o¡ n e sec ions o a plana B ownian mo ion . 0 . In oduc ion and no a ions In his no e we i s ob ai,n he Wiene chaos decomposi ion o he local ime a ze o o a mul ipa ame e Wiene p ocess . We also show ha he Wiene chaos se ies con e ges almos su ely, and he local ime belongs o he Sobole space Dk-!/2- ,2, o any e > 0, whe e k ; is he numbe o pa ame e s o he Wiene p ocess . The las pa o he pape is de o ed o show he exis en e o a eno malized local ime o he au oin e sec ions o a, plana B ownian mo ion (Va adhan eno maliza- ion), by ineans o he Wiene chaos expansion . Le (T, 5, p) be a a- ini o a omless measu e space . We will deno e by H he Hilbe space L 2 (T, B, p) which is assumed o he sepa able . Le W = h E H} be a ze o-mean Gaussian p ocess wi h co a iance unc ion E [W ( ) W (g)] = ( , g) H de ined en so ne p obabili y space (S2, F, P) . We will suppose ha .'F is he Q- ield gene a ed by {W (h), h E H} . I is well-known ha any squa e-in eg able unc ional on 52 has an o hogonal decomposi ion o he o m 00 F=E[F]+E I, ( , ), n=l whe e ~, E L2(T') (symme ic squa e in eg able ke nel), and I deno es he mul iple Wiene -I ó s ochas ic in eg al . 828  D . NUA AW , J . VIVES In his amewo k we can conside he de i a i e ope a o D which ac s on mul iple s ochas ic in eg als in he ollowing o m, o n >_ 1, E T . NVe can in oduce he Sobole spaces Dn,2 o a E R, as i is done in [11] . A unc ional F E L 2(Q) wi h he de elopmen (1) belongs o Da,2 i and only i n>1 Se D-,2 = naERDa , 2 and D«-,2 = n7<,D 7,2 o all a E R . and E[H,,(Y)] = 0 i n is odd . D . In ( a( i 1. . .; n)) = n II,,- 1 ( n( l, . . . . ,L-1, )) n!(1 + n)~ 11 ,L II2 < 00 . 1 . P elimina ies Le us i s ecall he S oock o mula (c . [8]) ha gi es he Wiene chaos decomposi ion o a unc ional F belonging o D°°,2 : L J  1! In (E[Dn F]) .  .-o n . We will also make use o he He mi e polynomials . Fo each n >_ 0, we will deno e by H, (x), he n h He mi e polynomial de ined by (3)  H,, (x)  en2/2 d 7  e-~ 2/2 )  n> 0 . ,! dx 7 Le p,(x) be he cen e ed Gaussian ke nel wi h a ian e E > 0 . The ol- lowing equali y, which ollows immedia ely om (3), ela es he de i a- i es p ( , n) (x) wi h he He mi e polynomials : p (n)( x )  ñ ! E-n/2 pe(x) H,, (-) '  n> 1 . Lemma 1 .1 . Le Y be a andom a iable wi h dis ibu ion N(0, o-2) . Then 2 ! (0-2 - 1) . . 2- m! P oo . . I ollows easily om he explici o mula o He mi e polynomials : [n/21  k n-2k n  k ! (n - 2k) ! 2k ' k=o and he momen s o a Gaussian andonl a iable, E[Y2,] _í 2 2 ~ . Lemma 1 .2 . Le , {FE}E>o be a a,mily o squa e in eg able andoni a iables wi h he expansions Assum,e ha i) ., con e ges in L 2 (T`), when E 10, o so ne lLnc ion , E L2(T'L) . ,~=o E 11 ~s Slip {n!  112} < W . Tl~,e1 hc allLily F E con e ges in L 2 (Q) o F = °° o I'n ( z) . P o0 . I is animmedia e consequence o he Lebesgue do nina ed con e gen e heo e n . Le So be he Di ac del a unc ion a ze o . NVe can conside 6o(W(h)) as a, dis ibu ion on he Wiene space in he sense o Wa anabe (c . [11]) . Using he in eg a ion by pa s o mula on he Wiene space one can show ha p,(W(h)) con e ges in B-1,2 o bo(6V(h,)) (see [5]) . We will i s compu e he Wiene chaos expansion o p (W(h)), and om i we will deduce he expansion o 6o(W(h)) . By o mulas (2) and (3) we ha e p, ( 1 +'(h)) = CI-IAOS GNPANSIONS AND LOCAL TIMES  829 2 . Chaos expansion o b o (W(h)) W (n!1E  . 2E ~pE(IV(h))Hn . ( w ) )] I~ .(h®,~) n=0 00 _l E p('n)(W(h))] I,l(h®'y) 1L! ,L=o The expec a ion appea ing in he abo e o mula anislies i n is odd because pE and H,, a e e en unc ions . On he o he hand, using Lemma 1 .1 o n= 2-ni we ob ain x H2, L (-) PEMPIJhll2(x)dx E _ ( 21 (11h11 2 +E))-'/2 J x E ) p_-111111 2 1 (E+1111 .11 1 )(x)dx _ (2~(11h~~ 2 +E)) 1 ~2 V2711,! 2 , 11 ni! (~I ,Ia -E h (1 2 + E) 830  D . NUALART,J . VIVES Finally, om (6) and (7), we ge he ollowing expansion (10) e>0 . ( -1)m I2m(h ®2m ) pe(W(h)) _ 1 :  -+1/2 ?n-o  27 2m m ! (11 h 112 + E) Le ing E end o ze o we deduce he Wiene chaos expansion o bo(W(h)) : 00 (9) bo(W(h)) =o ( -1 ) m I 2m (h®2 n) 27 2 m m,! II h II 2-+1 This se ies does no con e ge in L 2 (9), because 11 bo(W(h)) 112 = m=0 (2m) !  _ 00 . 22m (m!)2 27 il hij 2 -  ; by he S iling o mula . Obse e ha om (9) and (10) we ob ain i) bo(W(h))E D-1/2-,2 ii) bo(W(h))0 D -1 / 2 ' 2 and he se ies (9) con e ges in he no m o he space D-1/2-6,2, o any Rema k . Mo e gene ally we can ob ain he chaos expansion o b, ; (W (h» when x 7~ 0 6 . (W (h))  =  -p 11 lZ II ()  x  I n (h ®n ) 2x H ~h~~~ ~~h~~ z n! . n=o 3 . Wiene chaos expansion o he local ime o a mul ipa ame ic Wiene p ocess In his sec ion we will assume ha T is [0, 1] k , wi h k >_ 1 .  Then W ={W ( ), E T} will be he s anda d Wiene p ocess on T . We will deno e by [0, ] he ec angle [0, 1] x . . . x [0, k], whe e = ( ,, .. . , k) . We will also se 1 1 = 1 -...- k . The local ime o W can be o mally de ined as (11)  L( ; x) =  &, (14 7 s ) d .s ;  E T,  x E R . !o~l Al hough o any ixed s, b~(61 s) is no an o dina y andom a iable bu a dis ibu ion on he Wiene space, i ums ou ha he in eg al in (11) has a smoo hing e ec , and L( ,x) is a well-de ined andom a iable o any ixed poin _, , no on he axes . We will es ic ou analysis o he case x = 0, and we will se L( ) = L( , 0) . We know ha L( ) = o .a] So(14~ s ) ds can be ob ained as he L 2 -limi o (12)  LE( ) =  pe(ws ) d, .s o, l when e ends Lo 0 (see, o ins an e, [2]) . In he nex heo em we will compu e he Wiene chaos expansion o L( ) . Theo em 3 .1 . We ha e ha L( ) belongs o he space ® k--,2 , o any poin no 2 o a he a es, and i holds ha L( ) = P oo .. ( - 1 . ), n 2k m ó 27 2m a!(1-712) CHAOS EXPANSIONE AND LOCAL TIMES  831 Mo eo e , L( ) does no belong o ®~`-2 ,2 . 00 (13)  L E ( ) = M=0 z  1,i  2, z,i i=1 We will i s compu e he Wiene chaos expansion o L E ( ) applying he esul s o he p e ious sec ion . F om (8) and (11) we ob ain I2z z 1®2m ( - 1) n  o s] 2~ 2  m . ío  ds . ,al  (I s I + . ),,+1/2 Then he se ies Z :,°0 1 X n con e ges a .s . (14) CHAOS EXPANSIONS ANll LOCAL TIMES As a consequence o his heo em, i F is a squa e in eg able andom a iable wi h he de elopmen (I .), and L = n=o hen he Wiene chaos expansion (1) con e ges a .s . In pa icula he condi ion (14) is sa is ied i F belongs o he Sobole space ®e,2 o any e > 0 . Consequen ly, applying Theo em 3 .1, and he abo e c i e ion (14), we deduce he almos su e con e gen e o he Wiene chaos expansion o he local ime o he mul ipa a ne e Wiene p ocess . 4 . Reno malized local ime o he au oin e sec ions o a plana B ownian mo ion Conside now W = { (W¿, W, 2 ), E [0,1] } a s anda d plana B ow- nian mo ion . Le us w i e [X] = X - .E (X) o any in eg able andom a iable X . I is known om [6] ha (15)  L E _  [pE (W - Ws ) pE (W¿- 1V .,2)] ds d o <4< <1 con e ges in L2(Q) ; as E ends o ze o . The pu pose o his sec ion is o gi e a new p oo o his ac by means o he esul s ob ained on Sec ion 2 . Theo em 4 .1 . The amily o andom a iables L E con e ges as E ends o ze o, in ®1/2-6, 2 , o any 6 > 0 . In pa icula , his implies he con e gen e in L2(Q) . P oo .- Se ¿I = (s, ] . Applying he esul s o Sec ion 2, we ha e (16) I2e  1ó2e  1 2p ( 1021, , `  `'  ds d , 27 2 , 1  P, ! p !  Jo<s<e<1  (~ 0  + E)n+1 2+P= . n!(logn) Z 11 , 2 112 < oe, 833 whe e I2g and I2 P deno e, espec i ely, he mul iple s ochas ic in eg als wi h espec o he B ownian mo ions GV I and W 2 . When n a ios he 834  D . NUALART, J . VIVES e ms appea ing in he abo e sum a e o hogonal . The squa e o he L 2 -no m o he n h e co is gi en by (17) < L Obse e ha (2,,)2 22n C+p n  (e !)2 (p !) 2 E[I2e(lo)I2e(lo'), ~[ 1 Z p( l o) I2 p()] ds d du d ; [(,A~+E) (IA*I+e)]n+l whe e 0* = (u ; ] . We can es ima e his e m by (2n) !  :( n ! )2  (2e) !(2p) _  !(lo ; 1,J .) 2n (2~)222n(n!)2 P+p=n  e!p!  < (2n)! (I  I 0* I)n+I -_  (2n) ! (2 7 )222n ( n !)2 On he o he hand we claim ha ¡ (18)  //  1 (s ; ] 1 (u, ] 1 2n  ds d du d < J '< ( - S )n+I ( V - U )n+1  , 1 . 2 (19) 2 u<s<V< (7L) (TL p ~ /// ) p e (2n) Jus<L e+p=n 2e < `ei  < (n + 1)  max (2ze n) O<Q<n In o de o show (18) we will decompose he in eg al by conside ing he di e en posi ions o s, , u and . We ha e ha he le hand side o (18) is equal o ( - ( _ s)n+1 S )2n ds d du d ( - u)TL+1 +2 J eL<s< < Q n Q * I2n (I 0 11 0* I) n+l (e) 2 (2e) <n+1 . ( - u)n+1 The second summand in (19) can be es ima ed as ollows 2  ( _  ~  - s)n  du ds d =  1  < 1 . n  <s<  ( - u)n+ i n (n + 1)  n2 d .s d du d d .s d du d . ds d du d . Fo he i s e co, we ha e < CHAOS EXPANSIONE AND LOCAL TIMES  835 2 1  ( - s)' 2 1  ( - s)'  ds d d n  (  +I ds d , di) -  ~ + I  ~ < <  - S)  n  < < ( -8 )  - 1  2 ~  ( - $)2"  ds d +  2  ~  ( - s )n ds d n (n + 1)  n 2  S<  (1 - .s) ,, n  n 2 , s<  n _  1  2  ( V - n  [l_ (, - s)n n (n + 1) + n ] 2  s <  n  (1 - s)n 1 1 2 n (n + 1) + ? 2 <_ n2' which comple es he p oo o (18) . The e o e he squa e o he L 2 no nl o each e co o (16) can be es ima ed by (2n) !  3(?z, + 1) (27 )222n ( n !)2 Re e en es n 2 d .s d which is equi alen o a cons an imes i7, -3 /2 . Then Le nma 1 .2 allows o comple e he p oo o he heo em . 1 .  N . BOULEAU ANDF . HIRSCI-i, "Di -ichle o as and analysis on Wiene space," Wal e de G uy e , 1991 . 2 . D . GEMAN AND 3 . HOROWITZ, Occupa ion densi ies, Annals o P obabili y 8 (1980), 1-67 . 3 .  J . F . LE GALL, "Su le emps local d'in e sec ion du mou emen b ownien plan e la mé hode de eno malisa ion de Va adhan," Sein . P ob . XIX, Lec u e No es in Ma h . 1123, 1984, pp . 314-331 . 4 .  D . NUALART AND E . PARDOUX, S ochas ic calculus wi h an icipa - ing in eg ands, P ob . Theo y and Rel . Fields 78 (1988), 535-581 . 5 .  D . NUALART AND J . VIVES, S noo hness o B ownian local imes and ela ed unc ionals, P ep in . 6 .  J . ROSEN, "A eno malized local ime o mul iple in e sec ions o plana B ownian mo ion," Seco P ob . XX, Lec u e No es in Ma h . 1204, 1985, pp . 515-531 . 7 .  W . STOUT, "Almos su e con e gen e," Academic P ess ; 1984 . 8 .  D . W . STR .OOCK, "Homogencous Chaos e isi ed," Sem . P ob . XXI ; Lec u e No es in Ma h . 1247, 1987, pp . 1-7 . 836  D . NUALART, J . VIVES 9 .  H . SuciTA, Sobole spaces o Wiene unc ionals and Mallia in's calculus, J . Ma h . Kyo o Uni . 25, 1 (1985), 31-48 . 10 .  J . B, WALSH, The local ime o he b ownian shee , As é isque 52, 53 (1978), 47-61 . 11 . S . WATANABE, "Lec u es on s ochas ic di e en ial equa ions and Mallia in Calculus," Sp inge , 1984 . Da id Nuala :  Josep Vi es : Facul a de Ma emá iques  Depa amen de Ma emá iques Uni e si a de Ba celona  Uni e si a Au óno na de Ba celona G an Via, 585  08193 Bella e a (Ba celona) 08007 Ba celona  SPAIN SPAIN Rebu el 2 de Ma i de 1992