Chaos expansions and local times

Publicacions Matemátiques, Vol 36 (1992), 827-836 . A bstrac t CHAOS EXPANSIONS AND LOCAL TIMES DAVID NUALAII'I' AND JOSEP VIVES In this note we prove that the Local Time at zero for a multipararnetric Wiener process belongs to the Sobolev space Dk - z -,,2 for any e > 0 . We do this computing its Wiener chaos expansion . We see also that this expansion converges almost surely . Finally, using the same teclrnique we prove similar results for a renormalized Local Time for the auto¡ ntersections of a planar Brownian motion . 0 . Introduction and notations In this note we first obtai,n the Wiener chaos decomposition of the local time at zero for a multiparameter Wiener process . We also show that the Wiener chaos series converges almost surely, and the local time belongs to the Sobolev space Dk-!/2-f,2, for any e > 0, where k ; is the number of parameters of the Wiener process . The last part of the paper is devoted to show the existente of a renormalized local time for the autointersections of a, plana Brownian motion (Varadhan renormalization), by ineans of the Wiener chaos expansion . Let (T, 5, p) be a a-finito atomless measure space . We will denote by H the Hilbert space L 2 (T, B, p) which is assumed to the separable . Let W = h E H} be a zero-mean Gaussian process with covariance function E [W (f) W (g)] = (f, g) H defined en sorne probability space (S2, F, P) . We will suppose that .'F is the Q-field generated by {W (h), h E H} . It is well-known that any square-integrable functional on 52 has an orthogonal decomposition of the form 00 F=E[F]+E I,r(f,r), n=l where f~, E L2(T') (symmetric square integrable kernel), and I denotes the multiple Wiener-Itó stochastic integral . 828  D . NUAr AWr, J . VIVES In this framework we can consider the derivative operator D which acts on multiple stochastic integrals in the following form, for n >_ 1, t E T . NVe can introduce the Sobolev spaces Dn,2 for a E R, as it is done in [11] . A functional F E L 2(Q) with the development (1) belongs to Da,2 if and only if n>1 Set D-,2 = naERDa , 2 and D«-,2 = n7<,D 7,2 for all a E R . and E[H,,(Y)] = 0 if n is odd . Dt . In (fra(ti 1. . .; tn)) = n II,,- 1 (fn(tl, . . . . t,L-1, t)) n!(1 + n)~ 11 f,L II2 < 00 . 1 . Preliminaries Let us first recall the Stroock formula (cf . [8]) that gives the Wiener chaos decomposition of a functional F belonging to D°°,2 : L J  1! In (E[Dn F]) .  .-o n . We will also make use of the Hermite polynomials . For each n >_ 0, we will denote by H, t (x), the nth Hermite polynomial defined by (3)  H,, (x)  en2/2 d 7t  \ e-~ 2/2 )  n> 0 . rr,! dx 7 t Let p,(x) be the centered Gaussian kernel with variante E > 0 . The following equality, which follows immediately from (3), relates the derivatives p ( , n) (x) with the Hermite polynomials : p (n)( x )  ñ ! E-n/2 pe(x) H,, (-) '  n> 1 . Lemma 1 .1 . Let Y be a random variable with distribution N(0, o-2) . Then 2rrt ! (0-2 - 1) . . 2m! Proof . . It follows easily from the explicit formula for Hermite polynomials : [n/21  k n-2k n  k ! (n - 2k) ! 2k ' k=o and the moments of a Gaussian randonl variable, E[Y2,] _í 2 2 ~ . Lemma 1 .2 . Let, {FE}E>o be a fa,mily of square integrable randoni variables with the expansions Assum,e that i) f ., converges in L 2 (T`), when E 10, to sorne flLnction f, E L2(T'L) . ,~=o E 11 f~s Slip {n!  112} < W . Tl~,e1t thc fallLily F E converges in L 2 (Q) to F = r°° o I'n (fz) . Pro0f . It is animmediate consequence of the Lebesgue dorninated convergente theorern . Let So be the Dirac delta function at zero . NVe can consider 6o(W(h)) as a, distribution on the Wiener space in the sense of Watanabe (cf . [11]) . Using the integration by parts formula on the Wiener space one can show that p,(W(h)) converges in B-1,2 to bo(6V(h,)) (see [5]) . We will first compute the Wiener chaos expansion of p f (W(h)), and from it we will deduce the expansion of 6o(W(h)) . By formulas (2) and (3) we have p, ( 1 +'(h)) = CI-IAOS GNPANSIONS AND LOCAL TIMES  829 2 . Chaos expansion of b o (W(h)) W (n!1E  .t2E ~pE(IV(h))Hn . ( w ) )] I~ .(h®,~) n=0 00 _l E rp('n)(W(h))] I,l(h®'y) 1L! ,L=o The expectation appearing in the above formula vanislies if n is odd because pE and H,, are even functions . On the other hand, using Lemma 1 .1 for n= 2-ni we obtain x f H2,rL (-) fPEMPIJhll2(x)dx E _ ( 21 r(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, from (6) and (7), we get the following expansion (10) e>0 . ( -1)m I2m(h ®2m ) pe(W(h)) _ 1 :  -+1/2 ?n-o  27r 2m m ! (11 h 112 + E) Letting E tend to zero we deduce the Wiener chaos expansion of bo(W(h)) : 00 (9) bo(W(h)) =o ( -1 ) m I 2m (h®2rn) 27r 2 m m,! II h II 2-+1 This series does not converge in L 2 (9), because 11 bo(W(h)) 112 = m=0 (2m) !  _ 00 . 22m (m!)2 27ril hij 2 -  ; by the Striling formula . Observe that from (9) and (10) we obtain i) bo(W(h))E D-1/2-,2 ii) bo(W(h))0 D -1 / 2 ' 2 and the series (9) converges in the norm of the space D-1/2-6,2, for any Remark . More generally we can obtain the chaos expansion of b, ; (W (h» when x 7~ 0 6 . (W (h))  =  -p 11 lZ II ()  x  I n (h ®n ) 2x H ~h~~~ ~~h~~rz n! . n=o 3 . Wiener chaos expansion for the local time of a multiparametric Wiener process In this section we will assume that T is [0, 1] k , with k >_ 1 .  Then W ={W (t ), t E T} will be the standard Wiener process on T . We will denote by [0,t] the rectangle [0, t1] x . . . x [0, tk], where t = (t,, .. . , tk) . We will also set 1 1 = t 1 -...- tk . The local time of W can be formally defined as (11)  L(t ; x) =  &, (14 7 s ) d .s ;  t E T,  x E R . !o~l Although for any fixed s, b~(61 s) is not an ordinary random variable but a distribution on the Wiener space, it tums out that the integral in (11) has a smoothing effect, and L(t ,x) is a well-defined random variable for any fixed point t_, , not on the axes . We will restrict our analysis to the case x = 0, and we will set L(t) = L(t, 0) . We know that L(t) = fo .a] So(14~ s ) ds can be obtained as the L 2 -limit of (12)  LE(t) = f  pe(ws ) d, .s fo,tl when e tends Lo 0 (see, for instante, [2]) . In the next theorem we will compute the Wiener chaos expansion of L(t ) . Theorem 3 .1 . We have that L(t) belongs to the space ® k--,2 , for any point t not 2 ora the ares, and it holds that L(t) = Proof .. ( - 1 . ), n 2k m ó 27r 2mrra!(1-712) CHAOS EXPANSIONE AND LOCAL TIMES  831 Moreover, L(t) does not belong to ®~`-2 ,2 . 00 (13)  L E (t) = M=0 z  1,i  2,rz,i i=1 We will first compute the Wiener chaos expansion of L E (t) applying the results of the previous section . From (8) and (11) we obtain I2zrz 1®2m ( - 1)rn  fo s] 2~ 2  m . fío  ds . ,al  (I s I + . ),,+1/2 Then the series Z :,°0 1 X n converges a .s . (14) CHAOS EXPANSIONS ANll LOCAL TIMES As a consequence of this theorem, if F is a square integrable random variable with the development (I .), and L f = n=o then the Wiener chaos expansion (1) converges a .s . In particular the condition (14) is satisfied if F belongs to the Sobolev space ®e,2 for any e > 0 . Consequently, applying Theorem 3 .1, and the above criterion (14), we deduce the almost sure convergente of the Wiener chaos expansion of the local time of the multipararneter Wiener process . 4 . Renormalized local time for the autointersections of a planar Brownian motion Consider now W = { (W¿, W, 2 ), t E [0,1] } a standard planar Brownian motion . Let us write [X] = X - .E (X) for any integrable random variable X . It is known from [6] that (15)  L E _  [pE (WtWs ) pE (W¿- 1V .,2)] ds dt o <4<t<1 converges in L2(Q) ; as E tends to zero . The purpose of this section is to give a new proof of this fact by means of the results obtained on Section 2 . Theorem 4 .1 . The family of random variables L E converges as E tends to zero, in ®1/2-6, 2 , for any 6 > 0 . In particular, this implies the convergente in L2(Q) . Proof .- Set ¿I = (s, t] . Applying the results of Section 2, we have (16) I2e  1ó2e  1 2p ( 1021, , `  `'  ds dt, 27r 2 , 1  P, ! p !  Jo<s<e<1  (~ 0  + E)n+1 2+P=rr . n!(logn) Z 11f, 2 112 < oe, 833 where I2g and I2 P denote, respectively, the multiple stochastic integrals with respect to the Brownian motions GV I and W 2 . When n varios the 834  D . NUALART, J . VIVES terms appearing in the above sum are orthogonal . The square of the L 2 -norm of the nth terco is given by (17) f< L Observe that (2,,)2 22n C+p n  (e !)2 (p !) 2 E[I2e(lo)I2e(lo'), ~[ 1 Z p( l o) I2 p()] ds dt du dv ; [(,A~+E) (IA*I+e)]n+l where 0* = (u ; v] . We can estimate this term by (2n) !  \ :( n ! )2f  (2e) !(2p) _  !(lo ; 1,J .) 2n (2~)222n(n!)2 P+p=n  e!p!  <v (2n)! (I  I 0* I)n+I -_  (2n) ! (2 7r )222n ( n !)2 On the other hand we claim that ¡ (18)  //  1 (s ; t] f1 (u, v] 1 2n  ds dt du dv < J '<t (t - S )n+I ( V - U )n+1  , 1 . t 2 (19) 2 fu<s<V<t (7L) (TL p ~ /// ) p e (2n) Jus<L e+p=n 2e <v `ei  < (n + 1)  max (2ze n) O<Q<n In order to show (18) we will decompose the integral by considering the different positions of s, t, u and v . We have that the left hand side of (18) is equal to (v - (t _ s)n+1 S )2n ds dt du dv (v - u)TL+1 +2 J eL<s<t<v Q n Q * I2n (I 0 11 0* I) n+l (e) 2 (2e) <n+1 . (v - u)n+1 The second summand in (19) can be estimated as follows 2  (v _  ~  - s)n  du ds dv =  1  < 1 . n  <s<v  (v - u)n+ i n (n + 1)  n2 d .s dt du dv d .s dt du dv . ds dt du dv . For the first terco, we have < CHAOS EXPANSIONE AND LOCAL TIMES  835 2 1  (v - s)' 2 1  (v - s)'  ds dv dt n  (t  +I ds dt, di) -  ~ + I  ~ <v<t  - S)  n  <v<t (t -8 )  v - 1  2 ~  (v - $)2"  ds dv +  2  ~  (v - s )n ds dv n (n + 1)  n 2  S<v  (1 - .s) ,, vn  n 2 , s<v  vn _  1  2  (v V - n  [l_ (, v - s)n n (n + 1) + n ] 2  s <v  v n  (1 - s)n 1 1 2 n (n + 1) + ?t2 <_ n2' which completes the proof of (18) . Therefore the square of the L 2 nornl of each terco of (16) can be estimated by (2n) !  3(?z, + 1) (27r )222n ( n !)2 Referentes n 2 d .s dv which is equivalent to a constant times i7, -3 /2 . Then Lernma 1 .2 allows to complete the proof of the theorem . 1 .  N . BOULEAU ANDF . HIRSCI-i, "Dir-ichlet forras and analysis on Wiener space," Walter de Gruyter, 1991 . 2 . D . GEMAN AND 3 . HOROWITZ, Occupation densities, Annals of Probability 8 (1980), 1-67 . 3 .  J . F . LE GALL, "Sur le temps local d'intersection du mouvement brownien plan et la méthode de renormalisation de Varadhan," Sein . Prob . XIX, Lecture Notes in Math . 1123, 1984, pp . 314-331 . 4 .  D . NUALART AND E . PARDOUX, Stochastic calculus with anticipating integrands, Prob . Theory and Rel . Fields 78 (1988), 535-581 . 5 .  D . NUALART AND J . VIVES, Srnoothness of Brownian local times and related functionals, Preprint . 6 .  J . ROSEN, "A renormalized local time for multiple intersections of planar Brownian motion," Seco Prob . XX, Lecture Notes in Math . 1204, 1985, pp . 515-531 . 7 .  W . STOUT, "Almost sure convergente," Academic Press ; 1984 . 8 .  D . W . STR .OOCK, "Homogencous Chaos revisited," Sem . Prob . XXI ; Lecture Notes in Math . 1247, 1987, pp . 1-7 . 836  D . NUALART, J . VIVES 9 .  H . SuciTA, Sobolev spaces of Wiener functionals and Malliavin's calculus, J . Math . Kyoto Univ . 25, 1 (1985), 31-48 . 10 .  J . B, WALSH, The local time of the brownian sheet, Astérisque 52, 53 (1978), 47-61 . 11 . S . WATANABE, "Lectures on stochastic differential equations and Malliavin Calculus," Springer, 1984 . David Nualart :  Josep Vives : Facultat de Matemátiques  Departament de Matemátiques Universitat de Barcelona  Universitat Autónorna de Barcelona Gran Via, 585  08193 Bellaterra (Barcelona) 08007 Barcelona  SPAIN SPAIN Rebut el 2 de Mari de 1992