thematical Analysis of
Tandom Phenomena Proceedings of the International Conference
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Mathematical Analysis of
Random
Phenomena 0
12 - 17 September 2085
Hammamet, Tunisia
Editors
Ana Bela Cruzeiro Brupo de Fisica-Matemiitica &. UniversidadeTknica de bisboa, Portugal
Habib
Ouerdiane
Universityof Tunis El Manar, Tunisia
Nobuaki Obata Tohoku University, Japan
N E W JERSEY
LONDON
*
SINGAPORE
*
BElJlNG
*
SHANGHAI
*
HONG KOMG
*
TAIPEI
-
CHEMNAI
Published by
World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224
USA ofice: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 U K office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.
MATHEMATICAL ANALYSIS OF RANDOM PHENOMENA Proceedings of the International Conference Copyright 0 2007 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts tliereoj may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any inforniation storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ISBN-I3 978-981-270-603-4 ISBN-10 981-270-603-8
Printed in Singapore by World scientific Printers (S)Pte Ltd
Preface This volume contains research articles resulting of the “International Conference on Mathematical Analysis of Random Phenomena” that took place in Hammamet, Tunisia, from 12 to 17 September 2005 and was Coorganized by the Portuguese Mathematical Society (SPM) and the Tunisian Mathematical Society (SMT). This meeting was devoted to the exposition of recent developments in the mathematical analysis of random phenomena: stochastic analysis and its applications, mathematical physics, infinite dimensional analysis, probability theory and their interactions. One can read in this volume eighteen articles on the following topics: stochastic analysis and infinite dimensional analysis, white noise analysis, Malliavin calculus and applications, mathematical finance, Poisson analysis, hydrodynamics, statistical mechanics, and probability in quantum physics. We are grateful to all Tunisian and Portuguese institutions which have brought to the scientific organizing committee their moral and financial supports. These are, in particular,
. Ministhre Tunisien de 1’Enseignement Suphrieur, . Ministbe de la Recherche Scientifique, de la Technologie et du Dhveloppement des Comphtences, Tunisia GRICES,
. UniversitQ de Tunis El Manar, . University of Lisbon, Grupo de Fisica MatemLtica, . University of Madeira, Centro de Cihcias Matemgticas,
. Socihth Mathhmatique de Tunisie, . Sociedade Portuguesa de MatemLtica, . Faculth des Sciences de Tunis. We wish also to thank the authors for their contribution to a book of high quality, accessible to a large scientific public, as well as the colleagues who helped us with their anonymous and careful referee work. The Editors, ANABELACRUZEIRO, NOBUAKI OBATA,HABIBOUERDIANE Lisbon
/
Sendai
V
/ Tunis, July
12, 2006
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Contents Preface HBLENEAIRAULT Geometry and integration by parts on H \ Diff(S') PAULMALLIAVIN HBLENEAIRAULT, Invariant measures for Ornstein-Uhlenbeck operators ABDULRAHMAN AL-HUSSEIN Backward stochastic differential equations with respect to martingales WIDEDAYED,NOBUAKI OBATA,HABIBOUERDIANE Partial unitarity arising from quadratic quantum white noise SONIACHAARI,SOUMAYA GHERYANI, HABIBOUERDIANE Schilder's theorem for Gaussian white noise distributions F . CIPRIANO, H. OUERDIANE, J . L. SILVA,R. VILELAMENDES A nonlinear stochastic equation of convolution type FERNANDA CIPRIANO, ANABELACRUZEIRO Variational principle for diffusions on the diffeomorphism group with the H 2metric DIOGOAGUIARGOMES On a variational principle for the Navier-Stokes equation HANNOGOTTSCHALK, HABIBOUERDIANE, BOUBAKER SMII Convolution calculus on white noise spaces and Feynman graph representation of generalized renormalization flows TAKEYUKI HIDA,SI SI Characterizations of standard noises and applications YUH-JIALEE, HSIN-HUNG SHIH Analysis of stable white noise functionals PAULLESCOT Unitarizing measures for a representation of the Virasoro algebra, according to Kirillov and Malliavin: state of the problem YUTAOMA, NICOLASPRIVAULT FKG inequality on the Wiener space via predictable representation R. VILELAMENDES Path-integral estimates of ground-state functionals GIULIADI NUNNO,BERNTDKSENDAL A representation theorem and a sensitivity result for functionals of jump diffusions VON WALDENFELS WILHELM Creation and annihilation operators on locally compact spaces JEAN-CLAUDE ZAMBRINI From the geometry of parabolic PDE to the geometry of SDE List of participants
vii
V
1 23 31 45 57 73
85 93
101 111 121
141 155 167
177 191 213 231
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GEOMETRY AND INTEGRATION BY PARTS ON H \ Diff (S1) HELENEAIRAULT (INSSET, Saint-Quentin
/ L A M F A , Amiens)
We study various tensor fields on the Lie algebra diff (S') and we give their expressions in the trigonometrical basis. We define a bounded operator on diff(S1) modulo su(1,l). With Q, we obtain an integration by parts formula on H\Diff(S1).
0 . Introduction
In [l],the Levi-Civita connection on H \ Diff(S1), the quotient space Diff (Sl) modulo the homographic transformations has been explicited in real and complex coordinates and the existence of the parallel transport on H \ Diff(S1) has been established. With an extension to the infinite dimensional case of the Fang-Malliavin structure equations (see [12]), integration by parts formulae have been obtained on H \ Diff (S1). In the first part, we deepen the study of the geometry of tensor fields on diff(S1) as started in [6], [7], [8], [15], [16], [4], [l],[14]. Both manifolds H \ Diff(S1) and Diff(S1)/Rot(S1) have a structure of Kahlerian manifold. In [6], [7], [8],[15], [16], [4], [I],[14], the Ricci tensor has been proved to be a diagonal operator in the trigonometrical basis. It is a multiple of the metric tensor and these manifolds are Einstein manifolds. The bracket on the Lie algebra diff(S1) is defined by [ u , ~=u2r'-u'v ]
for uEdiff(S1),
Y
€diff(S1)
(1)
and the Hilbert transform is linear and it is given on trigonometric functions bY
J cosk0 = sink0
and
J sink0 = -cosk0
for k
2 1,
J1 = 0. (2)
In [l],it has been proved in particular that with the Levi-Civita tensor field r, the operators r(cospO)2 r(sinp0)2 are diagonal operators in the trigonometrical basis. In this work, in relation with the Levi-Civita tensor field on H \ Diff(S1), we study tensor fields r(u): diff(S1) + diff(S1) which (i) commute with J , (ii) are torsionless, (iii) for p 2 1, the operators I?(cosp0)2 f I'(sinpO)2 are diagonal operators in the trigonometrical basis
+
1
H ~ L E NAIRAULT E
2
(e,)p21 = {cosk~,sinke}k>l; - we resume these three conditions as
v] (torsionless condition) (i) I'(u)v - r ( v ) u = [u, (ii) r ( u ) J v = J r ( u ) v
+ [r(cospe)2+ I ' ( ~ i n p e )cos ~ ] ke = &,k
Let
a ( k ) = ak3 (rn
(3)
[r(cospe)2 ~ ' ( s i n p ~sin ) ~k e] = A p , k sin k e ,
(iii)
then
forzL,v E diff(S1),
+ j)a(rn
(2j
-
j)
+ bk
where a
(4)
cos Ice.
> 0;
(5)
+ ( j - 2 r n ) a ( j )+ ( 2 j - rn)a(rn)= 0,
+ rn)a(rn)= (rn - j)a(rn+ j ) + (2rn + j ) a ( j ) .
(6)
+
+
We shall take a ( k ) = k3 - k. It satisfies (k 2 ) a ( k ) = (k - l ) a ( k 1). Conversely, it is remarked in section 3 of the present work that the condition (6) on a ( k ) implies that a ( k ) is of the form (5) up to a constant term. On diff(S1), let ( I ) be a pseudo metric defined by the conditions, for k 2 1, (coskt9 I cosk13) = (sink0 I sinke) = a ( k ) , and {coske,sinke}k>~ are orthogonal vectors on diff(S1). The interest of this metric is due first to its close relationship and well adaptedness to the trigonometrical basis, secondly to its remarkable properties. In particular the second fundamental two-form is closed. In [3], it has been proved that the metric a ( k ) = k3 - k is the unique one (up to the multiplication by a constant) such that Ad(h) is unitary for any homographic transformation h. In the following, we extend the construction of diagonal operators associated to the Levi-Civita tensor field, (see [l],section 2) to more general tensor fields I?. We study the tensor fields I' satisfying (4) and such that I'(cosp8) sin Ice
+ k)O + yp(k) l p > k + l
C O S ( ~-
Uk
6; cos Ice
= &(k) C O S ( ~
+ pp(lc)ik2p+lcos(lc - p ) e +
k)B (7)
where 15: is the Kronecker symbol. We obtain that, for p 2 1, the operator I'(cospO)2 I'(sinpO)2 is diagonal in the trigonometric basis {cos kf3, sin ke}k>l - if and only if y,(k) l p l k + l = 0. In the same way, the operator
+
a,
: 2~ + [I'(cospO)u,cospB]
+ [I'(sinpO)u,sinpe]
(8)
is diagonal in the trigonometric basis {coskB,sinke}~>l if and only if y p ( k )lP2k+l = 0. In that case, Z, defined by Z, = aP r(cospe)2
+
+
Geometry and integration by parts on H
\ Diff(S1)
3
I'(sinpQ)2 is diagonal and Z, sin kQ = X t ( k ) sin k0, Z, cos k0 = $(k) cos k0 with
z
A, (k) = -llctp+i ( P
1
-(2p - k) + P p ( k - p ) + k) [:
- ka&.
(9)
(21$$iF , this ) value of Pp(k) gives the Levi-
If we assume that Pp ( k ) = Civita connection, then the trace
is finite. In particular, when operator 9 defined by
r is the Levi-Civita tensor field, we study the
where for h, u E diff(S'),
4(fj)h = [ E j , hl.
(12)
The operator 9 is bounded on diff (S1).For more general F satisfying (3), we obtain more bounded operators on diff(S1) with finite trace as (10). The second part is stochastic analysis. With 9 and an adaptation of Fang's integration by parts on loop groups [ll],we establish the following integration by parts formula for the Levi-Civita connection on H\ Diff ( S ' ) . Consider the canonical Brownian motion z ( t ) on diff(S1), see [17], [2], [13]. Let R(*) be the stochastic parallel transport of the frames above H \ Diff(S1). See [l]for the existence of a(.).We have the SDE
dz(t) =
C n,(t) * d Z a ( t ) ;
&(t)
+ r(a,(t),* d z ( t ) ,E k ) = o
va, vk .
01
(13) y ( t ) from [0, +oo[ t o 1-I \ Diff ( S ' ) is denoted 740). We call X the Wiener space of such continuous maps. For h E V and p E R, (exp(ph))(0) = Q + p h ( O ) + . . . (14) We have
A continuous map t
4
exP(-PWt)h) od exp(pQ(t)h) = ( I - pO(t)h) o p d R ( t ) h = p d R ( t )h
+ terms in p j ,
+ terms in p 3 , j
2 2.
j
22 (15)
HELENEAIRAULT
4
In the same way,
+
d exp(pa(t)h) o exp(-pR(t)h) = pdO(t)h terms in p 3 , j 2 2.
(16)
We consider the process
Denoting ( I ) the metric on V , then
Part I. Geometry on trigonometric functions 1. Bracket, metric and structure constants
Let 6; be the Kronecker symbol, with the bracket (l),we have 2 [COS Ice, cos pel = 2[sin Ice, sinpel =
cq2[(~c p)b,k+p + (~c+ p)6,k-p - (IC + p )64"-k]sin qe, cq>l [ ( p- 1 c ) 6 , ~ + +~( ~ c+ p)b,k-p - (IC+ p)64"-k]sin@, -
2[cos1c0,sinp0]= C q > l [ ( p - ~ ) 6 , (kk ++p~ ) b+ ,k-p+ ( k + p ) 6 , ~ - ~ ] c o s q e 2Ic6,p. (20) In complex coordinates, for m, n E 2 , [eime,cine] = i ( n - m)ei(m+n)e. From (20),
+
\
{
(2)
(ii)
Ice, sinpe] - [cosIce, cospO] = ( p - Ic) sin(lc + p)B, [COS Ice, sinpel + [sin Ice, cospd] = ( p - Ic) cos(Ic + p ) e . [sin
We take a ( k ) = k3 - Ic. For j
2 2, we put
(21)
Geometry and integration by parts on H
\ Diff(S1)
5
For j 2 2, k 2 2,
where KTk, STk are antisymmetric in ( j , k ) ,
2. Tensor fields on diff(S1), their expressions in the
trigonometrical basis The Hilbert transform J possesses the Nijenhuis property with respect t o the Lie bracket (1).For u,w E V ,
[ J u ,J v ] - [u,W] = J ( [u, JW] + [ J u ,4).
(23)
We define the following tensor fields, for u,w E diff ( S l ) ,see [ 4 ] ,
i
E ( u ,W ) = [ J u , J w ]- [u,w], J v ] [ J u W], , F ( u ,v) = [u,
+
i
+
G(u,V) = [ J u , J v ] [u,w ] , H ( u , W) = [u,J v ] - [Ju, w].
(24)
We remark that E , F , G are antisymmetric in u,u , whereas H is symmetric in u,W. On the other hand, (23) is the same as
E ( u ,W) - J F ( u , W) = 0.
(25)
HEL@NE AIRAULT
6
We obtain
H ( u , v) = G(u,J v ) = -G(Ju, v ) ,
“U,
E ( J u , Jv) = -E(u, v) and
1 = --(E(u, 2
v) - G(u,v)),
(26)
1 J [ J u ,v] = - ( E ( u ,?J)- J H ( u , v)). 2 We put
, - [u, Jv]. A(u,v) = J [ u ,v] - [ J u ,v] and B ( u ,v) = J [ u w] Since J 2 = -1, we have when u
(27)
# v,
A(u,v) = J A ( J u ,v) and B ( u ,v) = J B ( u ,Jv). A(u,Jv) A( J u , v) = 0
+
(28)
A and B are neither symmetric nor antisymmetric in u,w , the decomposition in symmetric and antisymmetric part is given by
We express these different tensor fields in the trigonometrical basis, we have ~ ( c okso , cospe) = ( p - IC) cos(p
+ k)e,
F(cos k 0 , sinp0) = ( p - k ) sin(p
+ k)0,
and
F ( J u , Jw) = - F ( u , v), J F ( u ,v) = F ( u , Jv). (30)
For j 2 1, k
2 1,
+ j ) sin(j - k ) 8 , H(cosje, cos Ice) = ( k + j ) cos(j - k)e, H(sinj0, cos k 0 ) = ( k
and
H ( J u , Jv) = H ( u ,v). (31)
The symmetric tensor
has been found by [14]. For j 2 1, k
2 1,
+ j ) cos(k - j ) e - b j ) ( k + j ) sin(j - k ) e
2Q(sinj0, cos k0) = -(lj2k- & ) ( k ~ Q ( c o scos ~ k~0 ,) = ( l j l k
and Q ( J u ,Jv) = Q ( u ,w).
(33)
Geometry and integration by parts on H
\ Diff(S')
7
In the same way,
Ice) = - ( k + j ) cos(k - j ) e G(cosj0,cos Ice) = (Ic + j ) sin(j Ic)e
G(sinj6, cos
and
G(Ju, Jv) = G(u, v)
-
(34) and
1 1 P ( u , v )= -JG(u,v) = Q ( J u , w )= - J ( [ J u , J w ] + [ u , w ] )
(35)
Ice) = (ljlk - lklj)(lc + j ) sin(Ic - j ) e , 2p(cOsje, cos Ice) = -(ijlk - iklj)(lc + j ) cos(j - k)e,
(36)
2
2
satisfies 2P(sinje, cos
P ( J U , Jw) = P ( U , w). For A, B ,
+ j ) sin(Ic - j ) e qcosje,cos Ice) = iklj(lc+ j ) c o s ( ~- jp
A(sinj0, cos Ice) = -lklj(k
and A(Ju,Jv) = A(u,v),
(37)
Ice) = - i j l k ( I c + j ) sin(j - k)B B(cosje,cos Ice) = + j ) cOs(lc - j ) e B(sinj0, cos
and B ( J u ,Jv) = B ( u ,v).
(38) Let
1 D(u ,U) = - ( [ J u , J w ] [u,711) 2
+
then
Q(u,W)
=
1 -G(u, w); 2
+ D ( u ,W) = [u,W ] + J [ u , J v ] = - J B ( u , w),
Q ( u ,W) - D(u,V) = -[u, U]- J [ J u ,V ] = JA(u, v). On the other hand. let
then 2{cOs Ice, cospe)
= ( p - ~ c cos(p )
+qe,
Ice, sinpe} = (Ic - p ) cos(p + k)B, ~ { C OIce, S sinpe} = ( p - Ic) sin@ + k ) e . 2{sin
(39)
(40)
HELENEAIRAULT
8
3. The fundamental two-form and the metric
The identities (6) give the closure condition of the symplectic form on diff (Sl).It is the same as
( m - n ) a ( p ) + ( p - n ) a ( m ) + ( m - p ) a ( n=) 0
with
m+n+p=O (43)
or equivalently
det
:; ) =o.
X
(i
y
(44)
a(-(z+y)>
1 -(z+t)
We look for solutions of (44) and assume that a has no singularity at zero. The function a ( z ) = Ax p is a solution of (44). We may assume that a(0) = 0 and a’(0) = 0. With y = 0 in (44), we obtain that a is an odd function. In (44), we take the derivative with respect to y, and we put y = 0 and a’(0)= 0; we obtain -3a(z) za’(z)= 0, thus a(.) = bx3.
+
+
With (6), we obtain (see [4]),
“%4 I
J w ) + (Iw14 I J v )
and for the fundamental two-form @
@(u,.)
=
(u
it gives
4. The Levi-Civita connection on H
\ Diff (S1)
Let u E V , 2) E diff(S1), w E V, we define the Levi-Civita tensor field (see [I]) rl(zt)u with
2(rl(v)u I w)= “w774 I u)+ ([WI’LLI and we put
I
I
- ([WI w ) )
(48)
Geometry and integration by parts on H
\ Diff(S')
9
We have r l ( v ) J u = J r l ( v ) u and
( r l ( v ) u I W) = -(u I r1(v)W).
(50)
Both rl and A1 are torsionless. The expression of I'l in the trigonometrical basis has been given in [l].If v = 1, w = cos0, v = sine, then 2(rl(v)u I w) = ( [ w ,I u) ~ ]- ( [ u , v ]I w), thus 2(Fl(l)u I W ) = -(w' I U ) (u' I W ) = 2(u' I w). = ul, (51)
+
rl(i)u
i
2rl(cosB) coske = - l k > 3 x ( k
+ 1)sin(k
-
l ) e - (k - 1)sin(k
+ l)e,
2rl(cose)sinke= lk23 x ( k + l ) c o s ( k - l ) O + ( k -
2rl(sin8) sinke
=
-1k23 x ( k
+ 1)sin(k - l)e +
1)~0~(k+1)8, (52) ( k - 1)sin(k l)e,
2rl(sin6)cosk9 = - 1 k 2 3 x (k+1)cos(k--l)B+(kand, for p 2 2,
Thus we have
+
1)cos(k+1)8,
HELENEAIRAULT
10
We obtain A ~ ( w ) J= uJ~~(v)u and both
A2
and
r2
(59)
are torsionless. For p 2 2,
Proof of (60)-(61). As for (54)-(55), we have 4(Az(cospO)sink0 I cosme) = 2(-[~0sme,sinpe]I -
2( [sin Ice, C
C O S ~ ~ )~
( [ c o s ~ ~ , c sin o sp ~e )~ ]
I
O S ~ ~cos ] me)
= ( m- p ) a ( k ) b r + P - ( p
+ m )a(k)S,"+k
+ P)a (k )Sh+p m )a(p)S,m+k - ( k + m )a(p)Sh+P - (m
+ ( k + m)cY(p)S?+P + (k + ( k + p ) a(m)6;2"+P+ ( k + p ) a(m)s,m+k+ (k - p ) a(m)S$+P. -
5. Commuting with the Hilbert transform, torsionless and antisymmetry
We prove that rl of last section is characterized by torsionless condition, commutation with J and antisymmetry condition. W e put ourselves o n E = diff ( S ' ) . Then hawing discussed the properties of r o n diff(S1), we take the orthogonal projection 7r : diff(S1) + V and we define, for w E diff(S1), the operator
Geometry and integration by parts on H
\ Diff(S1)
11
where I'(u)lv denotes the restriction of r ( u ) to the subspace V . In fact, the metric on the linear subspace V of diff ( S ' ) will determine the curvature of the quotient space. Torsionless condition and commutation with J on diff (5''). In the following lemma, we characterize I? when (i) r(u)w - r ( u ) u= [u, u] (torsionless condition) (ii) I'(u)Ju = Jr(u)w
for w E diff(S1).
(63) Notice that when u = 1 and u is in the subspace of B generated by {cos kB, sinkB}k>l, then with r ( l ) u = u',r ( v ) ( l ) = 0 and r ( l ) J v = (Jw)'= Ju', the conditions (63) are satisfied. We consider the case where u and u are in the subspace generated by {cos kB, sin k B } k > l . Since [cospB,sin kB] is expressed in terms of cos, we put for p , k 2 1,
+ k)B + yp(k)1p2k+lcos(p k)B + p p ( q i k r p f l - p)e + ak6; cos ke.
I'(cospB) sin kB = & ( k ) cos(p
-
C O S ~
Lemma. Assume that
r
satisfies (63)-(64); then
+ k)B + yp(k)lplk+l + / l p ( k ) l k > p + l cos(k - P)B + a k 6gj r(cosp8) coskB = -&(k) sin(p + k)B - T p ( k ) l p / k + l sin@ I'(cosp6) sink0 = & ( k ) cos(p
-Pp(k)bp+l
I n particular,
COS(P -
sin(k
-
P)Q,
k)B
-
k)B
(64)
H ~ L - ~AIRAULT NE
12
Proof. From (63), and since cos Ice = -J sin Ice, we obtain r(cosp8) cos Ice. With (63) (i) (torsionless condition), F(cosp0) cos Ice = I'(cos cospe [cospe,cos Ice]. This gives the condition (66). With the torsionless condition, and with (64), we calculate
Ice)
r(sinp0) cos Ice = r(cos Ice) sinpe
+
+ [sin@,cos Ice].
With (63) (ii) (commutation with J ) , we find r(sinp0) sin Ice. The torsion0 less condition on r(sinp6) sin yields again the conditions (66).
Ice
Lemma. Assume that r satisfies the conditions of the previous lemma, i.e., it is given by (65)-(66); then, for p 2 1, the operator r(cospO)2 I'(sinpe)2 is diagonal in the trigonometric basis {cos Ice, sin kf3}k2l i f and only if y,(k) Ip>k+l = 0. I n that case, we have, for Ic 2 1,
+
= I n particular, i f diagonal operator.
Proof.
Thus
and
rl
or
r
= A2, then [I'(cospe)2
+ r ( ~ i n p O ) ~is] a
Geometry and integration by parts on H
\ Diff(S')
13
Adding, we find
We proceed in the same way with
( [r(cosPe)2+ ~ ' ( s i n p ~cos ) ~k]e I C O S ~ ~ ) = ( [qcOspq2 + ~ ( s i n p e )sink6 ~ ] 1 sinje). Corollary 1. For I' = rl, p 2 0, k 2 2, ~ I ' ( c o s sin ~ ~k e) + ~ ~ ( s i n psin ~ k)e ~
+
Moreover Cp12 I'(cospe/&($2 r(sinpO/&($2 ator. The coefficients o n the diagonal are given by
is a diagonal oper-
where the series converge.
+
[r(cose)2 I'(sin q2]sin ke = - ( k 2 1'(1)2sin k0 = -k2 sin k€J
-
2) sin k e
f o r k 2 2.
(71)
In the same order of idea, looking for diagonal operators, we have
Lemma. With the assumptions (65)-(66) on I?, the operator
aP : u -, [I'(cospO)u,cospe] + [r(sinpe)u,sinpel
(72)
is diagonal in the trigonometric basis { c o s k ~ , s i n k ~-} k if > ~and only i f yp(k) l p 2 k + l = 0. I n that case we have, f o r k 2 1, @.,(sin k 6 ) = $(k) sin Ice and @,(cos k e ) = $(k) coske, where @ 1 A, (k) = - Z ( P
+ k ) ( 2 p - k) b P + l + ( 2 p + k)P (k) -
P
-
ka&.
(73)
HBLENEAIRAULT
14
Proof. [r(cospQ)sin kQ,C O S ~ I ~ ] 1 =-
c [pp(k)s;+k +
T>l,j>l
r,(k)6,p-k
+p p ( k ) S 3
x [(r- p)6jrfP + ( r + p)6jr-P - ( r + p)6jPPT]sinjo
+ ak 6; 11, Cospe], [r(sinpQ)sin kQ,sinpQ] 1 =-
c [P,(W,P+lc + rp(w;-k +
T>l,j>l
x [ ( p- r),jr+P
(pup(k)- ( P + k))6,k-"I
+ ( r + p1~jr-P
-
( r + p)~jP-'] sinjo.
Adding, we see that the terms corresponding to j = k - 2p and j = 2 p - k vanish if and only if p p ( k ) = $ ( p k ) . This gives the condition on a, t o be diagonal. In that case, we calculate X z ( k ) . 0
+
Theorem. We keep the assumptions (65)-(66) o n operator Z, defined by Z, = a,
r.
For p 2 1 , the
+ r(cOspq2 + r(sinp8)2
(74)
is diagonal and Z, sin kQ = $ ( k ) sin kQ, Z, cos kQ = $ ( k ) cos k Q with x Zp ( k ) =-1k>p+l(P+k) -
"
2(2P-k)+pp(k-P)
I
-kak6;.
Corollary 1. With (75), f o r p 2 0 , we assume that pp(k) = (notice that this value of P,(k) is in r l ) . T h e n
Moreover, 2
c
13 2k(ak - 1 ) X,Z(k) = -- 6 4k) . P W
Proof. For p 2 1, p p ( k - p ) remark that
c
l
=
(2:zEituk(f)
(77)
(p+is$-p). To calculate the sum, we
( P + k ) ( 2 k - P ) = ' 3 k ( k 2 - 1 ) - 2 k ( k - 1)
4k)
(75)
6
4k) '
Geometry and integration by parts on H
\ Diff(S1)
15
then, we add the two terms corresponding to p = k and p = 0. We see that the corresponding operator is bounded and that if
ak =
13 1, then 2 c x f ( k ) = --. 6
P20
+
We have [I'(cos e)u,cos 01 [I'(sin e)u,sin 01 and [I?( l ) ~ 11 , r(1 ) 2 = ~ 0.
+
+ r(cos q 2 u + I'(sin 6q2u = 0 0
Theorem (of unicity). Let I' be given by (64),
+ k)e + y p ( k )ip2k+l COS(P + p p ( k ) I k ? p + l cos(k - P ) e + a k h i .
r(cosp0)sin k0 = pp(k)C O S ( P
Assume that I'(u)w condition
-
r(v)u
(r(v)U
implies that
E =
k)e
[u,v] and r ( u ) J v = J r ( u ) v . T h e n the
I W ) = E(r(v)wI U )
with e2 = 1
(78)
-1 and
(2P+ 2a(p
W k =)
=
-
+ k)
and for k 2 p
+k + 1, y k ( p ) = 0 , p p ( k ) = P2 . (79)
Proof. We define H ( u ) w with ( H ( u ) w I v) = (I'(v)u H ( u ) w in the trigonometric basis,
The relation H(sin ke) cospe = eH(cosp0)sin k0 gives
I w).
We calculate
HELENEAIRAULT
16
In a similar way, we have
The condition
It implies that y = 0. The second equation in (ii) can be written as
Replacing in P k ( p ) a ( k
+ p ) = ~ ( p k ( +k p )
Then the condition & ( p ) - & ( k )
=
-
(2k
+ P ) ) a ( p ) , we find
$ ( p - k) determines
E =
-1.
0
Part 11. Stochastics on H \ Diff (Sl) and integration by parts formula We identify the quotient space diff(S1)/su(l, 1) with the subspace V of The following is valid when r diff(S1) generated by {cosk0,sink0}k>2. is the Levi-Civita connection. Let ( ~ j ) j > 2be the orthonormal basis of V defined by (22). Let ( z j ( t ) ) j > obe independent Brownian motions. We put
Geometry and integration by parts on H
\ Diff(S’)
We denote ‘*d’ the Stratonovitch differential and maps. On 7-1 \ Diff (S1),let y ( t ) be the solution of
dy(t) =
‘0’the
17
composition of
c(tj
oy(t)) * d z j ( t ) with y(0) = Id.
(81)
C(tjo y ( t ) )* d z j ( t ) .
(82)
j22
We denote
* d z ( t )o y ( t )=
j
For fixed j , then t j z j ( t ) is a process in diff(S1). More generally, consider random vectors ( y j ( t ) ) j with y j ( t ) E diff(S1) for any j. Let
We put
*dY ( t )0 y(t) =
c
*dYj ( t )0 y(t).
(84)
j
For the Levi-Civita connection transport given by (see [l])
r, for
any h in V, consider the parallel
d R ( t ) h = r(*dz(t))R(t)h = C r ( e j ) R ( t ) h
* dzj(t).
(85)
j
With the notations of (14)-(17), we consider the process YLL(t) = exP(Pfl(t)h)oy(t).
(86)
Theorem (Integration by parts). Let R(*)h be a solution of (85). For F : X --+ R, we define D L F ( y ( 0 ) )as in (18), t h e n (19) holds.
Proof. It is an adaptation of Fang’s proof [ll]. Since the adaptation is not straightforward, we give the details and we divide the proof into three steps. Step 1. Consider the process y p ( t ) = exp(pR(t)h)oy(t) as in (86). We construct a tangent process TLL(t) such that (i), (ii) and (iii) are fulfilled,
(9 T o b t ) = y ( t ) ,
+
(ii) dTLL(t)= ( d y p ( t ) p z ( t ) d t ) o T p ( t ) with yp(0) = Id,
(87)
HELENEAIRAULT
18
Ji
where y ” ( t ) = exp(pr(R(s)h))dz(s). Since the operator r(w(s)h) is antisymmetric from V to V, then yp(t) is a Brownian motion on diff(S1). Construction of the tangent process “J, and the expressions of z ( s ) and dyp(s) in (ii).
Proposition 1. Let
then M ( t ) satisfies dM(t) = *dy(t) o y ( t )
+ *dz(t) o M ( t )
(89)
and the differential of y ( t ) in It6’s f o r m is d y ( t ) = r ( R ( t ) h ) dz(t)
+ QR(t)hdt.
(90)
Proof. Notice that dz(t) E diff(S1) and I’(R(t)h) is an antisymmetric operator on diff(S1). Taking the stochastic derivative of y p ( t ) in (17), we obtain d,Yp(t) = *d(exP(@(t)h)) 0 exP(-PWt)h) o y p ( t )
+ ( e x P ( P w ) h ) ) ’ o * d-dt) = ( p * d(R(t)h) + terms in pj + (exP(PWt)h))‘0 * dy(t).
with j 2 2) 07” (91)
On the other hand, because of (81),
We differentiate this last identity with respect to p and take p
= x [ R ( t ) h ,~
j o ]y ( t )
* dzj(t) + ~j o M ( t ) * dzj(t),
= 0,
it gives
Geometry and integration by parts on H
\ Diff(S’)
19
€4
The bracket [R(t)h,cj] is given by [R(t)h,cj] = (R(t)h))’ocj - oR(t)h. Notice that in our case the Lie bracket is different from the one in [ll].We deduce that
d ~ ( t=) C ( r ( c j ) n ( t ) h + [ n ( t ) h , E j ]oy(t) ) * d z j ( t ) + C e oj ~ ( t* d)z j ( t ) . j
j
(94) We put dyj(t) =
+
(r(Ej)R(t)h [ a ( t ) h , c j ]*) d z j ( t )
*
= r(o(t)h)Ej d z j ( t ) ,
(95)
where the last equality is a consequence of
+[qt)h,
r(+qt)h
= r(cqt)h)Ej.
(96)
Compare with (49). The equation (95) is a Stratonovitch equation. Let
Y(t)
=
c
Yj(t>.
j
We have in Stratonovitch differential
dy(t) = C r ( n ( t ) h ) c *j d z j ( t ) j
and we have
d M ( t ) = *dy(t)oy(t)
+ *dz(t)o M ( t ) .
From (95), we calculate It6’s stochastic differential of y(t). tractions are given by
Ijdt
and
+ [R(t)h, d z j ( t ) ) = [r(cj)r(cj)R(t)h+ 4(Ej)r(cj)a(t)h) dt
=
(d(r(cj)R(t)h
C Ij dt
~j]),
= PR(t)h d t ,
j
where P and qh are given by (11)-(12). We see that Itb’s differential of y ( t ) is given by (90). This proves Proposition 1. 0 Definition. For t
2 0, let Qf be the solution of
dQf = r(R(t)h)Qf dp with Q: = IddiE(Si).
(102)
HELENEAIRAULT
20
We have Qf
= r(R(t)h).
(103)
Since r ( R ( t ) h )is antisymmetric on diff(S1), then Qf is orthogonal with the scalar product. We define
then y o ( t ) = z ( t ) and y”(t) is a Brownian motion on diff(S1) from the orthogonality of Qs: the It6 contraction
( d y p ( t ) ,d y p ( t ) ) = C l l Q t ~ j l l Z d = t dt.
(105)
j
Proposition 2. Let
yp be the solution
of the Stratonovitch equation
d;J, = ( * d y p ( t ) + p @ R ( t ) h d t )0 7 ~
we have ;);o(t)= y ( t ) since yo@)= z ( t ) . Moreover,
M
(106)
=M
Proof.
We have t o verify that
or equivalently
(110)
To verify (110), we differentiate d y p ( t ) = Q f d z ( t ) with respect to p. It gives
($1
p=O
y,>
=
($lp=o
Q f ) dx(t).
Since M ( t ) and g(t)satisfy the same stochastic differential equation, we conclude that =M .
Geometry and integration by parts on H
\ Diff (S’)
21
Step 2. The Girsanov formula is written for the tangent process 7,. Let
T, : 7 and the density
7,
K,
with +(s) = @,R(s)hand where O ( s ) is the parallel transport along y. From Girsanov theorem, for F : X 4 R,
E [ ( F o T , ) x K,] Step 3.
= E[F].
We differentiate with respect to p the previous formula,
& I,,=o~,
Since formula.
= - J t ( +(s)
I d s ( s ) ) , we obtain the integration by parts 0
References 1. H. Airault, P. Malliavin, “Quasi-invariance of Brownian measures on the group of circle homeomorphisms and infinite-dimensional Riemannian geometry”, J . Funct. Anal. (2006). 2. H. Airault, J. Ren, “Modulus of continuity of the canonic Brownian motion “on the group of diffeomorphisms of the circle” ”, J . Funct. Anal. 196,395426 (2002). 3. H. Airault, P. Malliavin, Anton. Thalmaier, “Canonical Brownian motion on the space of univalent functions and resolution of Beltrami equations by a continuity method along stochastic flows”, J. Math. Pure Appl. 83,955-1018 (2004). 4. H.Airault, “Riemannian connections and curvatures on the universal Teichmuller space”, C. R. Acad. Sci. Paris, Aout 2005. 5. J. M. Bismut, Large deviations and Malliavin Calculus, Birkhauser, Base1 (1984). 6. M. J. Bowick, S. G. Ftajeev, “String theory as the Kahler geometry of loop space”, Phys. Rev. Letter 58, no. 6 (1987). 7. M. J. Bowick, S. G. Rajeev, “The holomorphic geometry of closed bosonic string theory and Diff(S’)/S’”, Nuclear Physics B 293,348-384 (1987). 8. M.J. Bowick, A. Lahiri, “The Ricci curvature of Diff(S1)/SL(2, R)”,J . Math. Phys. 29,no. 9, 1979-1980 (1988).
22
H ~ L ~ AIRAULT N E
9. A. B. Cruzeiro, P. Malliavin, “Renormalized differential geometry on path space: structural equation, curvature”, J . Funct. Anal. 139,no. 1, 119-180 (1996). 10. B. Driver, “Integration by parts and quasi-invariance for heat kernel measures on loop groups”, J . Funct. Anal. 149,470-547 (1997). 11. S. Fang, “Integration by parts for heat measures over loop groups”, J . Math. Pures Appl. 7 8 , 877-894 (1999). 12. S. Fang, P. Malliavin, ‘Stochastic analysis on the path space of a Riemannian manifold”, J . Funct. Anal. 118,no. 1, 249-274 (1993). 13. S. Fang, “Canonical brownian motion on the diffeomorphism group of the circle”, J. Funct. Anal. (2002). 14. M. Gordina, P. Lescot, “Riemannian geometry on Diff(S1)/S’”, J . Funct. Anal. (2006). 15. D. K. Hong, S. G. Rajeev, “Universal Teichmuller space and Diff S1/S1, Commun. Math. Phys. 135,401-411 (1991). 16. A. A. Kirillov, D. V. Yurev, “Kahler geometry of the infinite-dimensional homogeneous space M = Diff+(S’)/ Rot(S1)”, Translated from Funlct. Anal. i Ego Priloz. 20, no 4, 79-80 (1986). 17. P. Malliavin, “The canonic diffusion above the diffeomorphism group of the circle”, C. R. Acad. Sci. Paris, Serie 1, 325-329 (1999). HELENEAIRAULT INSSET, 48, rue Raspail, 02100 Saint-Quentin, Aisne
and LAMFA, UMR 6140 CNRS 33, rue Saint-Leu, 80039 Amiens, France hairau1tQinsset.u-picardie.fr
INVARIANT MEASURES FOR ORNSTEIN-UHLENBECK OPERATORS HELENEAIRAULT(INSSET, Saint-Quentin), PAULMALLIAVIN (Paris) We produce a proof susceptible of generalization of the following result: the classical Ornstein-Uhlenbeck operator has for invariant measure the law v at time 1 of the Brownian motion starting from 0 at time 0. Let M be a Riemannian manifold. On M , let 2 be a vector field and v be the law of a diffusion with 1 = X Z y A where X and y are infinitesimal generator A. The condition [A,2 two constants permits t o obtain that the semi-group Pt = exp(t(A c Z ) ) has for invariant measure the transition probability associated to the semi-group exp(tA).
+
+
1. The Ornstein-Uhlenbeck operator on a Berezinian space Given a Kahlerian manifold M , of dimension n, let w be its symplectic form, assume the existence of a globally defined Kahler potential K , that is K is a globally defined C2 function such that i%’K = i w . Define a Berezinian measure as a probability measure p of the form p, := y exp(-cK) ( w ) * ~ ,
where c is a positive constant and where y is a normalizing constant. Berezinian measures appear in the theory of representations of finite dimensional Lie groups; in infinite dimension, it has been recently discovered. See [3, Theorem (4.2.4)] that unitary representations of Virasoro algebra can been described in a suitably defined Berezinian context: the right hand side of (1) becomes in infinite dimension meaningless and the measure p is then defined as the reversible invariant probability measure associated to the elliptic operator
A - C2V K * V
(2)
where A is the infinitesimal generator of the Brownian motion on M (that is the Laplace-Beltrami operator of classical differential geometry). See [7]. The pole will be the point mo where K reaches its minimum. We denote by 7r,(mo,dm) the law of the Brownian motion at time s, conditioned t o start from mo at time t = 0. The question object of this work is when p, = T, for a suitable choice of s? As the law of Brownian motion
23
HELENEAIRAULT,PAULMALLIAVIN
24
perturbed by a drift in infinite dimension is often constructible this identity will furnish an alternative route for construction of Berezinian measures in infinite dimension. This procedure was started by P. Malliavin in [7]. In the present, we discuss the finite dimensional setting. In [l],they consider real valued processes. Taking the law of a Levy Brownian motion at time 1, they obtain an invariant measure of the process and study the associated Dirichlet forms. In [9], the Ornstein-Uhlenbeck process was constructed in a Riemannian context. Here, we develop some properties of the OrnsteinUhlenbeck operator on a finite dimensional complex Kahlerian manifold. In particular, for the Poincar6 disk (see for example [4]), we obtain that in the case of the Ornstein-Uhlenbeck process, we have t o consider the radial part of the process and the drift a t the point m is given by the Poincar6 distance from m to the origin, this in some sense ties with [9]. The radial part can be studied with the projection method of [2]. 2. The classical one-dimensional Ornstein-Uhlenbeck process On the real line R, consider the measure pt(z, d y ) defined by
and denote We have
Thus d v ( z ) is an invariant measure for the semi-group pt. By a change of variables, -Y2/2 d y
and the density
7rt(z,y)
satisfies
Z T t ( z , y ) = --.rrt(x,y) d
at
with
Invariant measures for Ornstein-Uhlenbeck operators
25
On the other hand, let Pt be the semi-group associated to the one dimensional Brownian motion on the real line,
then the invariance condition (5) is the same as (PI(&f))(O) = 0.
The fact that (5) implies (6) is immediate since (5) implies
-
To prove that (6) implies (5) is more delicate. We assume that (8) is true for f = P,g. Then we use the semi-group property for Ft t o deduce that the condition Fttf(y) v(dy) = 0 for f = F,g implies that it holds
It=,
s
$1 s t=O
Ftg(y)v(dy) = 0. This is true for any s, thus (5) holds. The previous proof likely extends to more general situations. In the following, our aim is to provide the algebraic framework for such a generalization. Therefore, t o stay more simple, we leave aside the difficulties that arise from the discussions of the functions spaces on which the operators are well defined, We call test function any function for which the formula has a meaning. We leave t o the reader to take care of the details. Moreover the method of path spaces as presented in [8],may provide a better framework for possible extensions than a rigorous development in functions spaces. 3. The Ornstein-Uhlenbeck process on an Euclidean space E
Denote
A the infinitesimal generator
and denote by process. Then
d the
of the Brownian motion on E :
infinitesimal generator of the Ornstein-Uhlenbeck
d=
1
-2
Xkak. k
H - ~ L ~ AIRAULT, NE PAULMALLIAVIN
26
As d2xk = 2dk
+ x&,
we have the commutation property
[&A] = A ,
(9)
commutation which implies the following commutations, for any integer m>l,
[A,nrn] = mAm
and finally
Introducing the semi-group Pt = exp(tA), we want to prove that for every test function A := ( P I ( &f ) ) ( 0 ) = 0. (11) With (lo), we have
then using the fact that the localization a t the point 0 of the two operators A, A coincide, we get
relation which implies (11). From (ll),it results that v , the law at time 1 of the Euclidean Brownian motion starting from 0 is an invariant measure for the process generated by b. 4. Invariant measures under a commutator hypothesis The identities (9), (lo), (11) can be generalized as follows
Theorem 4.1. Let A be the infinitesimal generator of a diffusion o n a = A - CZ where Z is a vector field and c is a manifold M . Consider constant. W e assume that
[A,Z]=XA+yZ
(12)
where X and y are constants. W e define L = X A +yZ. Assume c > 0 , c y > 0 and denote p = (c y), consider the semi-group associated to L, then f o r every test function f
+
+
Invariant measures for Ornstein-Uhlenbeck operators
27
Proof.
[&,,A] = (cX+y)A-yA
= cL,
L := XA+yZ,
[A,L] = (c+y)L.
(14)
Introducing the semi-group Pt = exp(tL) the commutation (16) implies that
Using the fact that the localization at the point mo of the two operators L coincide, we get
A,
relation which implies (13).
Main Theorem. With the assumptions of Theorem 4.1, let Pt = exp(tL) and rt(mo,dx) be the transition probability associated to this semi-group, z.e.,
then r~ (mo,dx) is an invariant measure for the process generated by A, P
/(An(Y,
7.r;
(mo,dY) =
/f
(Y)
7r;
(mo,dY) v t 2 0.
For the proof, see the equivalence of (6) and (5).
HELBNEAIRAULT.PAUL MALLIAVIN
28
5.
The following one dimensional elementary example has been considered in [ 2 ] .It is straightforward that
and the previous theorems apply. Consider the image by a map g of a one dimensional Brownian motion. Assume that g is differentiable invertible from R to an interval (a,b) in R. For example, g(r) = tanh(r). We take on the interval ( a ,b) the metric ds2 = dx2/a(x)with a ( x ) = [g’(g-1(x))]2 (see [2]p. 379). The Laplace-Beltrami operator associated to ds2 is
and A,(f)
=
A(fog)of-’. The associated semi-group Pf is given by
the probability of transition is
We put Z = a(x)d/dx,assume that [A,, Z ] = X A, +yZ where X and y are constants. Then we must have y = 0 and
-
where c is a constant. Moreover, let A, = A,
-
c Z , then the process
A,
starting from x and generated by has nf(z,dz)for invariant measure with t = l / p . For X = 2 and x = g(r) = tanh(r), we have and
d a(x)dx
d dr
= r-
+ c-.drd
With 1x1 = x, the coefficient r of d/dr is the hyperbolic distance from 0 to Z.
Invariant measures for Ornstein-Uhlenbeck operators
29
References 1. L. Accardi, V. Bogachev, “The Ornstein-Uhlenbeck process associated with the Levy Laplacian and its Dirichlet form”, Probab. Mat. Statist. 17 no. 1, Acta Univ. Wratislav, n 1928, 95-114 (1997). 2. H. Airault, “Projection of the infinitesimal generator of a diffusion”, J. Funct. Anal. 85 no. 2 (1989). 3. H. Airault, P. Malliavin, “Unitarizing probability measures for representation of Virasoro algebra”, J. Math. Pures A&. (9) 80, no. 6, 627-667 (2001). 4. H. Airault, “Stochastic analysis on finite dimensional Siegel disks, approach to the infinite dimensional Siegel disk and upper half-plane”, Bull. Sc. Math. 128,605-659 (2004). 5. A. B. Cruzeiro, P. Malliavin, “Non perturbative construction of invariant measure through confinement by curvature”, J . Math. Pures AppZ. (9) 77, no. 6, 527-537 (1998). 6. B. Gaveau, J. Vauthier, “Annulations et calculs infinitesimaux de laplaciens pour un fibre non integrable”, BuZ1. Sci. Math 100 no. 4, 353-368 (1976). 7. P. Malliavin, “The canonic diffusion above the diffeomorphism group of the circle”, C. R. Acad. Sc. Paris Ser. 1 Math. 329 no. 4, 325-329 (1999). 8. P. Malliavin, “It6 atlas”, to appear in Proceedings of Abel conference, Oslo (2005), Springer. 9. D. Stroock, “The Ornstein-Uhlenbeck process on a Riemannian manifold”, in First International Congress of Chinese Mathematicians (Beijing, 1998), Amer. Math. SOC.,Providence, RI, 2001, pp. 11-23. HELENEAIRAULT INSSET, Universitk de Picardie, 48, rue Raspail, 02100 Saint-Quentin, France hairault8insset.u-picardie.fr
PAUL MALLIAVIN 10, rue Saint-Louis en l’ile, 75004, Paris, France
[email protected]
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BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH RESPECT TO MARTINGALES* ABDULRAHMAN AL-HUSSEIN (Al-Qassim University)
This paper is devoted to proving the existence and uniqueness of the solutions of backward stochastic differential equations driven by infinite dimensional martingales.
1. Introduction
Backward stochastic differential equations (BSDEs in short) have been widely studied over the last decade. These equations take usually the form (1) below. The appearance of such equations was first in the work of Bismut in [5] and later in the joint work of Pardoux and Peng in [12]. The main reason of studying such type of equations is t o involve them in some mathematical problems; for instance in theory of PDEs, stochastic control and in finance. In [2] we studied those BSDEs driven by a Wiener process on a Hilbert space H . The solutions of such equations were required t o be adapted to the filtration generated by this driving Wiener process, i.e., the Wiener filtration. The question arising now is whether we are able t o deal with such sort of BSDEs with a given arbitrary filtration, not necessary the Wiener filtration; for example the filtration & ( M ) = a { M ( s ) , 0 5 s 5 t } , t 2 0 , where M is a square integrable cadlag martingale in H . In this work we will be concerned with giving answers t o this question. Another example also could be the filtration generated by two independent cylindrical Wiener processes Wl and W2 on H . Note that if the terminal value E of the concerned BSDE is measurable with respect to F T ( W ~and ) is independent of F T ( W ~the ) , solution (Y,2 )of the following equation can only be adapted t o the filtration (Ft(W1)VFt(W2),0 5 t 5 T } . One advantage of working with a more general filtration than just the Wiener filtration is to enable us to study more equations than those focussed on just the Wiener filtration, e.g. as in [2] and [3]; cf. [4].We shall study here on the space H the following backward stochastic differential equation *Research supported by Al-Qassim University, project no. SR-D-006-003.
31
ABDULRAHMAN AL-HUSSEIN
32
(BSDE):
i
-
d Y ( t ) = f ( t , Y ( t ) Z, ( t ) )d t
-
Z ( t )d M ( t ) - d N ( t ) , 0 5 t 5 T,
Y(T) = E ,
where M is a given square integrable martingale in H . For this we shall look for a triple (Y,2,N) of adapted processes, square integrable and satisfy, for each t E [0,TI, the integral form of this equation. Here N is a martingale required to be very strongly orthogonal (V.S.O.) t o M , a notion will be given in Section 2. These equations are in fact backward stochastic differential equations driven by martingales. The main purpose of this paper is to prove the existence and uniqueness of the solutions of this type of equations; see Theorem 3.1 in Section 3. These results generalize the work of El Karoui et al. [7] in finite dimensions, and also generalize the study of the usual BSDEs (see the equation (1) below), which was considered by Pardoux and Peng in [12]. For this issue, see the discussion following the proof of Theorem 3.1. Moreover, these BSDEs can be considered somehow as a generalization of some reflected BSDEs; see e.g. [8]. In fact, since the solutions of the reflected equations take usually values in R,to make use of our result here, one should first reformulate the results in Theorem 3.1 for the case when the martingale M lies in the space H and Y lies in R. This is however straightforward. Such type of BSDEs can also be applied in finance to construct the Follmer-Schweiser strategy. These BSDEs were applied also in [6] to study the approximation of the usual BSDEs as follows. Consider as in Pardoux and Peng [12] setting the following BSDE: - d Y ( t ) = f ( t ,Y ( t ) Z , ( t ) )d t - Z ( t )d W ( t ) , 0
Y(T) = E ,
5 t 5 T,
(1)
with W being a genuine Wiener process (e.g. a Brownian motion in R). By taking a martingale approximation of this W (see [6] for the definition), one obtains a sequence of equations, all of which are of the type (3). It was shown in [6] that this sequence of solutions actually converges t o the solution started with. This result was done in fact for the finite dimensional case. This paper is organized as follows. In Section 2 we recall some information on Hilbert space valued martingales and stochastic integration with respect t o them. Section 3 contains the main results.
Backward stochastic differential equations with respect t o martingales
33
2. Basic elements of an infinite dimensional martingale and stochastic integration
A. Let (R,.F, {.Ft}t>O- ,P)be a complete filtered probability space, such - is right continuous. Fix 0 < T < 03. Let R be the algebra that {Ft}t>o generated by elements of R x (0, T ]of the form F x ( t ,s],where F E 3 t and t , s E [0,TI. Define P to be the a-algebra generated by R.The family of sets P is called the predictable a-algebra. Let H be a separable Hilbert space. An H-valued process is said to be predictable if it is P / B ( H ) measurable. We shall write H @ H for the tensor product of H with itself, denoting by z 63 y the tensor product of x E H and y E H . Let Mf0,,](H) denote the vector space of right continuous square integrable martingales { M ( t ) ,0 5 t 5 T } , taking values in H , that is S U P ~ ~ [ , , ~ I[lM(t)l&] IE < 03. It is a separable Hilbert space with respect to the inner product ( M ,N ) H E [ ( M ( T )N(T)),q], , if we agree to identify P-equivalence classes. We say that the two elements M and N of Mf0,,](H) are very strong orthogonal (V.S.0.)if IE ( M ( u )@ N ( u ) )= E ( M ( 0 )@ N(O)),for all [0, TI-valued stopping times u. For example, if moreover N ( 0 ) = 0, then A4 and N are V.S.O. if and only if E ( M ( u )@ N ( u ) ) = 0, for all such stopping times. Note that ( M ( t ) N , ( ~ ) )= Htr(M(t) @ N ( t ) ) . This notion of orthogonality is stronger than the usual definition of strong and weak orthogonality. For further details see [lo] and [9]. In [l]we considered notions of orthogonality (strong and very strong) slightly stronger than these used in this paper. In fact we found that our definitions here are more suitable; for instance, they are invariant under the shift by a constant. Let us now recall the definition of Dole'ans measure associated with [MI&. Define d l M I ~on elements A = F x ( t , s ] of R by dlM1c(A) := iE [lF(IM(s)l&- IM(t)l$)]. This function can be extended uniquely to a measure ( Y M on F. This measure is called the DolQans measure associated with \MI& (see [lo] or [ 9 ] ) . Analogously, we associate on P the H & H valued a-additive DolQans measure p~ of M @ M . Here the space H & H is the completed nuclear tensor product, that is the completion of H 63 H for the nuclear norm. Recall that the linear form trace, denoted here by tr, is defined as the unique continuous extension to H & H of the mapping z @Y (&?AH. For a square integrable martingale M we write ( M ,M ) (or shortly ( M ) ) for the increasing Meyer process associated with the DolQansmeasure of the submartingale /MI&,that is the unique (up to IF-equivalence) predictable, right continuous, increasing, real valued process, vanishing at zero such that
34
ABDULRAHMAN AL-HUSSEIN
\ M I L - (111)is a martingale. It exists since IM($ is a submartingale. We recall the following proposition from [lo, Theorem 14.3.1, p. 1671. Proposition 2.1. (1) There is one predictable H&H-valued process Q M , defined up to aM-equivalence such that for every G E P
Moreover, Q M takes its value in the set of positive symmetric elements of H & H and t r & M ( w , t ) = 1, ( Y M a.e. (2) The HI& H-valued process
has finite variation, is predictable, admits p~ as its Dole'ans measure, and is such that M @ M - ( ( M ) )is a martingale. From this we conclude that M and N are V.S.O. if and only if ( ( M ,N ) ) = 0, where ( ( M ,N ) ) is the unique (up to P-equivalence) predictable H & H valued process with paths of finite variation vanishing a t zero such that M @ N - ((MIN ) ) is an H&H-valued martingale. To illustrate the above notions, let us for example consider the case of a 2-dimensional Brownian motion B = (B1,B 2 ) , where B1 and B2 are two independent Brownian motions in R. It is obvious that ( ( B ) ) t= (h '1) =: t I 2 , and so ( B ) , = 2 t and Q B = !j I 2 . Moreover, p~gis the product measure ( 1 @ B)12 and ag = (2 1 @ P),where I is the Lebesgue measure on
([O,TIl~([O,TI)). Denote by L 1 ( H ) the space of nuclear operators on H . It is known that elements of H 6 1 H can be identified with elements of L1 ( H) . So we can let OM be the identification of &M in L 1 ( H ) . Denote also by & ( H ) the Hilbert space of all Hilbert-Schmidt operators from H t o itself. We shall write that G E L$"(H)if G & Z E L2(H).
B. Now we are ready to set the definition of stochastic integration with respect to elements of Mf0,,](H). First, let L * ( H ; ' P , M ) be the space of processes a, the values of which are (possibly non-continuous) linear operators from H into itself with the following properties: (i) the domain of @(w,t ) contains
a z ( w , t ) ( H )for every ( w ,t ) ,
Backward stochastic differential equations with respect t o martingales
35
(ii) for every h E H , the H-valued process @ o a z ( h ) is predictable, (iii) for every (w, t ) E R x (0, TI, @(w,t )o t ) is a Hilbert-Schmidt operator and
eT(w,
s
This space is complete with respect to the scalar product ( X , Y ) H t r ( X o O M o ~ * ) d a M cf. ; [g, Proposition 22.2, p. 1421. See also
G.(o'T1 Denote by E ( L ( H ) ) the space of R-simple processes and A2(H;P , M )
the closure of E ( L ( H ) )in L * ( H ;P , M ) . It is therefore a Hilbert subspace of L * ( H ;P , M ) . For a simple @ of the form
c n
@ =
1 F t X ( T Z , S J Uil
Ui E
L ( H ; K ) ,pi
E FT, ,
i= 1
we define
This gives an isometric linear mapping from E ( L ( H ) )into M f o , T l ( Hgiven ), by @ H @ d M . Extend this mapping to A 2 ( H ;P , M ) . The image @ dM of @ in Mf0,,](H) by this mapping is called the stochastic integral of @ with respect to M . For such @ E A 2 ( H ; P , M )the stochastic integral N = dM can easily be seen to satisfy the following two properties:
s
s
s
(1) ( N ) t =
1
ti-(@0
OM
0
@*) d ( M ) ,
(O>!l
for every t 2 0. The following representation property is due to [ l l ] see ; also [9, E. 8, p. 1601. Theorem 2.1. Let M E M f o , T I ( Hand ) 3-11 := {
J X d M :X E A 2 ( H ; P , M ) }c M i , , ] ( H ) .
Let 'Fl2 be the orthogonal complement of in Mf0,.](H). Then every element of 3-12 is V.S.O. to every element of in 3-11. I n particular, every
ABDULRAHMAN AL-HUSSEIN
36
L
E
Mf0,,](H) can be written uniquely as L=
s
XdM+N,
Note that since M E
X E A 2 ( H ; P , M ) ,NE‘FI2.
(2)
XI,the martingales M and N are V.S.O.
3. Main results A. This section contains the proof of the existence and uniqueness of the solution of the following type of BSDEs.
i
- d Y ( t ) = f ( t , Y ( t )Z , ( t ) )d t
- Z ( t )d M ( t ) - d N ( t ) , 0 5 t 5 T ,
Y ( T )= E .
(3)
<
The mappings f , and M are required to satisfy the following conditions. First, let y > 0 be fixed.
f is P @ B ( H ) @ B ( L f M(H))/B(H)-measurable. 3 k > 0 such that ‘dy, y’ E H , ‘d z , z‘ E L f M( H )
If(t, Y,2 ) - f(t7 Y’,z%f I k (IY
-
Y’I&
1’
+ lz 42 L p (H) -
-
uniformly in ( t ,w ) . M E Mr0,,](H), cadlag and ( ( M ) ) t = b(s)b(s)*dc,, for some adapted continuous and increasing R+-valued process { c s , s 2 0) such that co = 0, and an Lz(H)-valued predictable process b. Here b(s)* is the adjoint of b(s). In other words ( ( M ) )is absolutely continuous with respect to c.
IE [s,’eycs If(s,0,0)1&dc,] < 00. IE [eY c~ 1<1$] < 00. From (H3) it follows that, for all t ,
and
Thus the condition (H2) becomes
If(t,Y , ).
-
f(t,
Y’l
z’)l?z
L k (IY
- Y’lC
+ I(.
- 2’)
b(t)li2(H)).
(4)
Backward stochastic differential equations with respect to martingales
37
Let us now introduce the following spaces.
Lg(0, T ;H ) := { 4 : [0,TI x R such that
-+
IE
A 2 ( H ;P , M ) := { 4 : [0,TI x R
H , progressively measurable and
[6eYct14(t)l&dct] <
4
CQ}.
L 2 ( H ) , predictable and such that
E [s,'eYCtI4(t) w l ; 2 ( H ) dctl < CQ>. S 2 ( H ):= {$ : [O,T]x R -+ H , cadlag, adapted and such that
IE [ SUP (eYCtI4(t)l&)]< > .. OltlT
MfO,T](H)
:= { N E
MfO,T](H)? IE
[JTeYctd ( N ) , ]<
O0}'
B 2 ( H ) := Lg(O,T;H ) x A 2 ( H ;P , M ) . Then B 2 ( H )is a separable Hilbert space with the norm
A solution of (3) is a triple (Y,2,N ) E B 2 ( H )x hfO,T1(H) such that for all t E [O,T],we have a s .
N ( 0 ) = 0 and M and N are V.S.O. The main result is the following theorem. Theorem 3.1. Suppose that (Hl)-(H5) hold with a parameter y being large enough. Then there exists a unique solution (Y,2,N ) E B 2 ( H )xMf0,,,(H) of (3). Moreover, Y E S 2 ( H ) .
B. In preparation t o proving this theorem we give the following proposition which gives more properties to the integrand X and to the martingale N in the above representation identity (2). Recall the information in Theorem 2.1.
ABDULRAHMAN ALHUSSEIN
38
Proposition 3.1. Under the conditions in Theorem 2.1, i f moreover the martingale L belongs to the space Mfo,Tl(H), then the processes X and N in the representation identity (2) lie in the spaces A 2 ( H ;P , M ) and MfO,T1 ( H I , respectively. Proof. Recall first that
Thus
[I
T
= IE
e y c 3d(L),]
< 00. This shows that ( X ,N ) E A 2 ( H ;P , M ) x &ffo,T,(H).
0
Before we can establish the proof of Theorem 3.1 we need t o give some lemmas. Lemma 3.1. Assume that (Hl)-(H5) hold. Let (y,z) E B 2 ( H ) . The process Y defined by
belongs to the space i % ( O , T ;H ) , and moreover Y E S 2 ( H ) .
Proof. Note that it holds
1
< 1 ePpct
-P
T
epc8If ( s ,y ( s ) ,z ( s ) ) l $ dc,
,
(7)
Backward stochastic differential equations with respect t o martingales
39
a s . for all t E [O,T]and for all ,L? > 0. Therefore, putting ,L? := y/2 in this inequality and using Fubini's theorem give
1' (4' eyct
2
If(s,!/(s), z ( s ) ) I H dcs)
dct
This together with (4) yields
Now by applying Cauchy-Schwartz inequality and this inequality, we find that
In particular, the conditions (H4) and (H5) and this result show that
ABDULRAHMAN AL-HUSSEIN
40
It remains to prove that Y E S 2 ( H ) . Applying (6), (7) (with p = Y), Doob’s inequality (see e.g. [13]) and the assumptions (4), (H4) and (H5) shows that
which completes the proof.
0
Lemma 3.2. Under the same conditions in Lemma 3.1 the process
lies in
~ 2 (f H ~) . , ~ ~
Proof. Let ( 2 , N ) be the unique processes in A 2 ( H ; P , M )x M f o , T I ( H ) given by applying Theorem 2.1 through the formula:
K ( t ) = K(O)
+
l
Z ( s )d M ( s )
+N(t),
0 5 t 5 TI
(13)
such that N ( 0 ) = 0 and M and N are V.S.O. Recall that K ( 0 ) = Y ( 0 ) and K ( t ) = Y ( t ) f ( s ,y(s), z ( s ) ) dc, for each t , which comes from the definition of Y in (6). We can then apply the integration by parts and use
+ s,”
Backward stochastic differential equations with respect to martingales
41
this fact to find that
Thus
IE [ I ’ e y c s d ( K ) . ]
Note that
by using (11). Now substituting (16) in (15) and applying (7) (with ,O = r), (4),(10) and ( 9 ) give the following result.
I 36IE [eycTI<[&]
126k +IE Y
[I
eyc8Iy(s)1& dc,]
+- Y 126 Y Thus by applying the conditions in the lemma we complete the proof.
0
ABDULRAHMAN ALHUSSEIN
42
C . We are now ready to establish the proof of Theorem 3.1. Proof of Theorem 3.1. Define the mapping @ on B 2 ( H ) by @(y,z) = (Y,Z ) , such that Y is the right continuous version of the semimartingale given by (6) and 2 (and also the martingale N ) are the unique processes given by the decomposition (13) as discussed earlier at the beginning of the proof of the preceding lemma. Recall that N is V.S.O. t o M and N ( 0 ) = 0. According to Lemma 3.2 and Proposition 3.1 we conclude that 2 E A 2 ( H ;P , M ) and N E M f o , T l ( H ) .Therefore @ maps B 2 ( H )into itself, and (Y, 2,N ) is the solution of the following BSDE:
Y ( t )= t
+
l f(s, T
Y(S), 4 s ) ) dcs -
1
T
1
T
Z ( s >d M ( s ) -
d N ( s ) , (18)
where t E [O,T].The rest of the proof is standard, but we give it here for completeness. We shall show in the following that @ is a contraction mapping on B 2 ( H ) . Take two elements (yl, z2) and (y2, z2) of B 2 ( H ) and let ( Y I 21) , and (Y2,Z2) denote respectively their images in B 2 ( H ) under @. Thus Zi, Ni) is the solution of the BSDE (18) with generator f ( t , yi(t), z i ( t ) ) and the terminal value t, for i = 1,2. Denote b y = Y1 - Yz, 6 2 = 21 - 2 2 and 6N = N1 -N2. It is clear that (SY, 6 2 ) E B 2 ( H )and 6N E &fo,Tl(H). We have a s . for all t E [O,T],
(x,
Since this equation is of the sort of the BSDE (18), we can obtain two estimates similar to those in (10) and (17). In particular, we must have
(19) for some positive constant C. Hence, if we choose y > C, we find that @ is a contraction mapping on B 2 ( H ) . Consequently, @ has a unique fixed point in B 2 ( H ) ,call it (Y,2 ) . Now from the definition of @ it can be seen
Backward stochastic differential equations with respect to martingales
43
that (Y,2,N ) is the unique solution of the BSDE (3), where the martingale N E Mf0,,,(H) is given with the help of Theorem 2.1 by
t
Z(s) d M ( s )
+N(t),
0 5 t 5 T.
Finally, this solution Y lies in S 2 ( H ) as deduced from (11).
0
D. Note that the process Y which solves the BSDE (3) is only known to have a right continuous version, so it may develop a jump. In [4] we give a condition on the filtration {Fi}t2~ to guarantee the continuity of the martingale N and hence Y . We now close the paper by the following remark. Assume for simplicity that the space H is the real space R.Assume also that M is the martingale given by the formula M ( t ) = J , ” f ( s ) d B ( s )t, 2 0, where f E L 2 ( [ 0 , T ] ; (or even random) and B is a Brownian motion taking its values in R. If f(s) > 0 for each s 2 0, then we find that F t ( M ) = Ft(B) for each t 2 0. Therefore by making use of the unique representation of martingales in Theorem 2.1 and the Brownian martingale representation theorem (see [13, Theorem 3.4, p. 200]), one concludes that the martingale N in Theorem 2.1 vanishes almost surely. This tells in particular that the BSDE (3) becomes similar to the BSDEs studied by Pardoux and Peng in [12] but with the variable Zf replacing Z there; see the BSDE (1). References 1. A. Al-Hussein, Backward stochastic evolution equations in infinite dimensions, Ph.D. thesis, Warwick University, UK, 2002. 2. A. Al-Hussein, “Backward stochastic differential equations in infinite dimensions and applications”, Arab J . Math. Sc. 10, no. 2, 1-42 (2004). 3 . A. Al-Hussein, “Backward stochastic evolution equations”, preprint (submitted). 4. A. Al-Hussein, “Backward stochastic partial differential equations in infinite dimensions”, Random Oper. and Stoch. Equ. 14,no. 1, 1-22 (2006). 5. Jean-Michel Bismut, “ThBorie probabiliste du contrBle des diffusions”. Mem. 4, no. 167 (1976). Amer. Math. SOC. 6. Philippe Briand, Bernard Delyon, Jean MBmin, “On the robustness of backward stochastic differential equations”, Stochastic Process. Appl. 97,no. 2, 229-253 (2002).
44
ABDULRAHMAN ALHUSSEIN
7. N. El Karoui, S.-J. Huang, “A general result of existence and uniqueness of backward stochastic differential equations”, Backward stochastic differential equations (Paris, 1995-1996), 27-36, Pitman Res. Notes Math. Ser., 364, Longman, Harlow, 1997. 8. N. El Karoui, E. Pardoux, M. C. Quenez, “Reflected backward SDEs and American options”, Numerical methods in finance, 215-231, Publ. Newton Inst., Cambridge Univ. Press, Cambridge, 1997. 9. Michel Metivier, Semimartingales. A course on stochastic processes. de Gruyter Studies in Mathematics, 2. Walter de Gruyter & Co., Berlin-New York, 1982. 10. Michel Metivier, Jean Pellaumail, Stochastic Integration. Probability and Mathematical Statistics, Academic Press, Harcourt Brace Jovanovich, Publishers, New York-London-Toronto, Ont., 1980. 11. Jean-Yves Ouvrard, “Reprbsentation de martingales vectorielles de carre integrable B valeurs dans des espaces de Hilbert reels separables” (French), Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 33,no. 3, 195-208 (1975/76). 12. 8. Pardoux, S. G. Peng, “Adapted solution of a backward stochastic differential equation”, Systems Control Lett. 14,no. 1, 55-61 (1990). 13. Daniel Revuz, Marc Yor, Continuous martingales and Brownian motion, third edition, Grundlehren der Mathematischen Wissenschafien [Fundamental Principles of Mathematical Sciences], 293, Springer-Verlag, Berlin, 1999. 14. B. L. RozovskiY, Stochastic evolution systems. Linear theory and applications to nonlinear filtering, translated from the Russian by A. Yarkho, Mathematics and its Applications (Soviet Series), 35, Kluwer Academic Publishers Group, Dordrecht, 1990. ABDULRAHMAN AL-HUSSEIN Department of Mathematics, College of Science, Al-Qassim University, P. 0. Box 237, Buraidah 81999, Saudi Arabia
[email protected],
[email protected]
PARTIAL UNITARITY ARISING FROM QUADRATIC QUANTUM WHITE NOISE WIDEDAYED(Inst. Prkp. aux Etudes d'IngEnieures, El Merezka), NOBUAKI OBATA( Tohoku University, Sendai), HABIBOUERDIANE (Universite' de Tunis El Manar) In general, the solution to a normal-ordered white noise differential equation involving quadratic quantum white noise is a white noise operator and is not an operator acting in the L2-space over the original Gaussian space where the quantum white noise is defined. The solution happens to be a unitary operator on a certain subspaxe of the L2-space over a Gaussian space with different variance. This regularity property is referred to as partial unitarity.
1. Introduction Given a quantum stochastic process { L t } , we consider a normal-ordered white noise differential equation (1)
where o is the Wick product (or normal-ordered product). Roughly speaking, the unique solution is always found in a space of white noise operators, suitably chosen according to the coefficient { L t } and the initial value SO, see e.g., Chung-Ji-Obata [4] and Ji-Obata [6]. Let { a t ,a;} be the quantum white noise. If Lt is a linear combination of {afat,at, a;, l}, the equation (1) is reduced essentially to a usual quantum stochastic differential equation for which the quantum It6 theory works well, see Parthasarathy [15]. As is well known, the higher powers of quantum white noise have rather singular nature but are well formulated in quantum white noise theory. The case when { L t } involves a quadratic quantum white noise {a:, a;'} is a non-trivial step going beyond the traditional quantum It6 theory and the regularity properties of the solution are of great interest. Recall also that the quadratic quantum white noise is related to the L6vy Laplacian, see Ji-Obata-Ouerdiane [9] and Obata [14]. This paper is devoted to one of the simplest cases. We consider
45
46
WIDEDAYED,NOBUAKI OBATA,HABIBOUERDIANE
where y,a , b E C are constant numbers and 7-ta,b a Fourier-Gauss transform. In general, the solution is merely a white noise operator. We shall prove that the solution happens to be unitary on a certain subspace of L2-space over a Gaussian space whose variance is different from the one of the original space where the quantum white noise is defined. This property is called partial unitarity. Our result is relevant to unitarity of a (generalized) FourierGauss transform investigated by Ji-Obata [7,8]. The main results will be stated in Section 5. There are different approaches to the quadratic quantum white noise, see e.g., Accardi-Amosov-Franz [l],Accardi-Franz-Skeide [2], Lytvynov [12], and references cited therein. 2. Generalized Fourier-Gauss transforms We adopt mostly the same notations as in [7]. Let us start with a real Gelfand triple
N
= S(R) c H = L2(R,dt)c
N* = S’(R),
(3)
where H = L2(R) is the Hilbert space of R-valued square-integrable functions on the real line R with respect to the Lebesgue measure d t , S(R) the space of rapidly decreasing functions and S’(R) the space of tempered distributions. The canonical bilinear form on N* x N is denoted by (., .), which is compatible with the inner product of H . By the same symbol we denote the canonical C-bilinear form on N: x NQ:, where the suffix means the complexification. With a E C and E E NQ:we associate a continuous function ~ $ ~ on > e N’ defined by 4a,E(X)
= e( X L F a ( C ? C ) / 2 ,
x E N’,
(4)
which we call a coherent vector or an exponential vector. Let & be the linear space spanned by {&,E ; E E A&}. Due to the obvious relation
the space & does not depend on the choice of a E C. In general, two locally convex spaces X,y we denote by L ( X ,y ) the space of continuous operators equipped with the bounded convergence topology. In the next, we will use such space for X and Y are equal to Nc or W or their dual spaces.
Partial unitarity arising from quadratic quantum white noise
47
With a pair A E L(Nc,N@*) and B E L(Nc,Nc) we associate an operator G(A,B ) on E defined by
G(A,B ) 4 1 , = ~ e(AE7E)/241,BE,
EENC.
The above formula is sufficient to define a linear operator on E since the exponential vectors {q5,,~ ; E &} are linearly independent. The operator G(A,B ) is called a generalized Fourier-Gauss transform. Our definition is due to Chung-Ji [3], while an equivalent definition is given by Lee-Liu [ll] in terms of an integral formula.
Lemma 2.1.
+
(1) G ( A i ,Bi) G(A2,B2) = G(BZAiB2 A2, B1B2). (2) G(A,B ) = 1 (the identity operator o n E ) i f and only if A = 0 and B = 1 (the identity operator on Nc). ( 3 ) G ( A , B ) is invertible i f and only i f so is B , i.e., B E GL(&). I n
1.
that case, G(A,B)-' = G(-(B-l)*AB-' 7 B-'
The proof is immediate from definition. In particular,
becomes a group of linear automorphisms of E. If both A , B are scalar operators, say, A = a1 and B = Pl, we write simply G(a,p) and is called a Fourier-Gauss transform. We have
G(a,P ) 4 1 , = ~ ea(EiE)/241,pE,
E E NC.
(6)
We naturally come to a subgroup of 8 : 8 0 =
(G(a,P); Q
E
c, p E ex} = c x ex,
where ex is the multiplicative group of non-zero complex numbers. For later use we define one-parameter subgroups of 8 0 . First, for a E C we define
T, = G(a, 1) :
41,~ H ea(EyE)/2q51,E,
< E Nc.
It follows immediately that
T,T,l = T,+,i
,
TL'
= T-,
.
48
WIDEDAYED,NOBUAKI OBATA,HABIBOUERDIANE
-
Moreover, by a straightforward computation we obtain Ta-l+2b
: &,€
Next, for a , b E C let
E E Nc , a, b E @.
,
eb(E3E)41,E
(7)
be a linear operator on & by x a > b $a,E
++
E E Nc.
4a,bE,
(8)
Obviously,
xi,:= x U , b - l
x u , b x a , b ' = 'Fta,bb',
for b
# 0.
On the other hand, by straightforward computation we obtain xa,b4l,E
= e(.-w-b2)(E>E)/2
($l,bE
,
which reads
3. Unitarity The Gaussian measure with variance a
> 0 is a
probability measure p a on
N* uniquely specified by
Then we have (($a,E, 4 a , v ) ) p L a
J*
4 a , c ( x ) 4 a , q ( x )p u ( d x )
= ea(~lq), E ,
v E ~zlc. (10)
Lemma 3.1. For a > 0 and Ibl = 1, the linear automorphism extends uniquely to a unitary operator f i a , b on L2(N*,p a ) .
xa,b
of &
Proof. Note that the inner product of L2(N*,pa) is defined by ((f, 9)). Since & c L2(N *,p a ) is a dense subspace, it is sufficient t o show that
,
((xa,bf xu,bg))pa
=
((7,g))pa
I
f
7
E 8.
Verification of the above identity is straightforward from (10).
0
Let I c R be a closed (finite or infinite) interval. We denote by &I the subspace of & spanned by { 4 a ,; ~ E Nc, supp< c I } . By (5), &I does not depend on the choice of a E @ either. In view of the action ( 6 ) , we are ready to claim the following
Partial unitarity arising from quadratic quantum white noise
Lemma 3.2. Each G(cx,p) E 80 induces a linear automorphism of particular, so as Na,b f o r any pair a, b E @ with b # 0.
49
&I.
In
For an interval I let 11 denote the indicator function. The associated multiplication operator is denoted by the same symbol. For a > 0 we define a linear map E," from & into L2(P , pa) by
E E J%. It is shown that E," extends to a projection on L2(N+, pa), which is denoted E," : 4a,e
H
4a,lrc,
by the same symbol. The image of this projection will be denoted by L2(pal I ) . It is noted that &I is a dense subspace of L 2 ( p a (I ) . Now we may state a generalization of Lemma 3.1, the proof of which is similar. We only need t o note that f i a , b commutes with the projection EF.
Lemma 3.3. Let I c E% be a closed interval and a , b E @ a pair of complex numbers with a > 0 and Ibl = 1. T h e n the linear automorphism N a , b t &I extends uniquely to a unitary operator o n L 2 ( p a l I ) , which coincides with %,b t L2(paII). 4. White Noise Operators
We take a white noise triple
w c r(Hc)= L ~ ( N * ,c~W* ~)
(11)
constructed in the standard manner [5,6,10,13]. Recall that I?(&) is the Boson Fock space over He which is canonically identified with L2(P, p1) through the Wiener-It6-Segal isomorphism. For instance, we may take the Hida-Kubo-Takenaka space for (11). The canonical @-bilinear form on W * x W is denoted by ((., .)). In general, a continuous operator from W into W* is called a white noise operator. Since the canonical injection W -+ W* is continuous, we have a natural inclusion L ( W ,W ) c L ( W ,W * ) . By simple application of the famous characterization of operator symbols [6,13]we see that every generalized Fourier-Gauss transform G(A,B ) extends uniquely t o a white noise operator in L ( W ,W ) . In fact, the symbol is given by
so the check is straightforward. The continuous extension is also called a generalized Fourier-Gauss transform and is denoted by the same symbol. Moreover, we note the following
50
WIDEDAYED,NOBUAKI OBATA,HABIBOUERDIANE
Proposition 4.1. Every B(A,B ) E 6 is a topological linear automorphism of W . I n this sense 6 is a subgroup of G L ( W ) . We say that {Lt ; t E R} is a quantum stochastic process if t H Lt E C ( W ,W * )is continuous. Let at and a; be the annihilation and creation operators at a time point t E R,respectively. It is known that both
t ++ at
E
q w ,W ) ,
t ++ a;
E
qw*,W * ) ,
are Coo-maps [6]. The pair { a t , a ; ; t E R} is called the quantum white noise process. We then see that higher powers of quantum white noise (in normal-order) are well defined white noise operators. As is mentioned in Introduction, we focus on the normal-ordered white noise differential equation:
where y,a , b E C are constant numbers and X a , b is defined in (8). Recall that Xa,b is a Fourier-Gauss transform and hence, is a white noise operator. By the general theory [4,6] there exists a unique solution to (12) in a space of white noise operators suitably chosen and is given by
Here the Wick product o is replaced with the usual product (composition) of operators since the integral contains only annihilation operators. 5. The Main Results
Theorem 5.1. Let a , b, y E C satisfy the following conditions:
Let {Et} be the solution to (12), i.e., given as in (13). Then, for any t > 0 , the white noise operator Et possesses the following properties: (1) Et 1 &[o,t~extends uniquely to a unitary operator on L2(parI [O, t ] ) . and EL 1 & I ~ , + ~extend ) uniquely (2) I f a > 0 in addition, Et &(-,,o] to unitary operators on L2(paI ( - 0 0 , O l ) and L2(paI [t,+00)), respectively.
Partial unitarity arising from quadratic quantum white noise
51
As a matter of fact, it will be seen that
-
=t =
i
xa',b
on &[o,t]
%,b
on &(-m,o] u &[,,+a).
Taking into account the canonical factorizations:
L 2 ( N ' , p a ) = L~(cL~I(--oo,oI) @L2(paI [o,tl)g ~ ~ ( p It,+m)>>, a1 L2(",Pa4
= L2(cLa,I(-m,0]) '8L2(pa4[ O , t ] ) ' 8 L 2 ( p a 4[t,+m)),
we see that &(-,q ' 8 & [ 0C3&[t,+m) ,~] becomes their common dense subspace. Then Theorem 5.1 says that, according to this factorization, we have
Zt = x a , b '8 x a ' , b '8 X a , b and each factor in the right hand side extends t o a unitary operator on the corresponding subspace of L2(N',p a ) or L2(N',p a l ) . We call this property of Et the partial unitarity. In fact, we prove the following more general result.
Theorem 5.2. Given a , b , y E C, let Et be defined as in (13). Assume Ibl = 1, b # f l and choose a', b' E C in such a way that 1 2
- ( a - a'
(1)
+ b')(l
If a' > 0, the restriction
-
b2)
TFIEtTbi
+ y = 0.
(14)
1 E [ O , ~extends ] uniquely to a
unitary operator o n L2(pa,l[O, t ] ) . (2)
If a''
a'
-
2y > o , 1 - b2
(15)
then the restrictions T ; ' E t T b ! 1 &(-,,ol and T F I E t T b , 1 &it,+,) extend uniquely to unitary operators on L2(pa,,I (-m,O]) and L 2 ( p a !I/ [t,+m)), respectively. Theorem 5.1 follows immediately from Theorem 5.2 by setting b' = 0. The proof of Theorem 5.2 will be divided into a few steps. The Gross Laplacian process is defined by t
Gt In fact, t
H
=
a:ds,
Gt E C(W,W ) is a Coo-map.
t 2 0.
WIDEDAYED,NOBUAKI OBATA,HABIBOUERDIANE
52
Lemma 5.1. For any y E C we have
Proof. Since at$q,c
= J ( t ) $ l , ~ ,we
have
which implies (17).
0
Lemma 5.2. Given a , b , y E @, let Et be defined as in (13). For any a’, b‘ E C and E E Nc we have
Proof. Combining (9) and (17), we obtain the solution (13) written in terms of generalized Fourier-Gauss transforms: Et Then, using we have
= ‘Fla,b 0 expyGt = G ( ( a - 1)(1- b 2 ) ,b ) G(2yl[o,t],1). Tbl
= G(b’, 1) and applying the composition rule (Lemma 2.1),
t o obtain We take the action on &,,E = e(l-a‘)(E>c)/zc#q,E
from which (18) follows immediately.
Partial unitarity arising from quadratic quantum white noise
53
Lemma 5.3. Given a , b,y E @, let Et be defined as in (13). Assume b # f l and choose a', b' E CC in such a way that 1
-2( a
- a'
+ b')(l
- b2)
+ y = 0.
(19)
Then, f o r t > 0 we have
where
Proof. Let
5 E Nc with s u p p t c [0,t]. Then, by (18) and (19) we see that
TF1%Tb/4a',E=eXP{ z1 ( " - a ' + b ' ) ( l - b 2 ) ( ~ , E ) + ' Y ( E . F ) } Q o l , b ( = d)a',bE
.
Hence TF1E:tTb,4a,,E = x a / , b d ) a / , Eand the first part of (20) is proved. We next take [ E Nc with s u p p t c (-oo,O]U [t,+m). Again, in view of (18) and (19) we see that
TclE;tTb!c$a!,E = e-r(E'E'4ai,bE,
namely,
Therefore we have
= G ((a' -
27
- 1) (1 - b2)l b ) + 1 , ~ 1 - b2
Taking (9) and (21) into account, we conclude that
which proves the second half of (20).
54
WIDEDAYED,NOBUAKI OBATA,HABIBOUERDIANE
Remark 5.1. Lemma 5.3 becomes uninteresting when b = 61. In fact, in t h a t case y = 0 so t h a t Et is reduced t o a constant independent of t , see (13). Proof of Theorem 5.2. (1) We already know from Lemma 5.3 t h a t
TG1%Tbl t
&[o,t]
= xd,b t
&[o,t].
(22)
Noting by assumption t h a t Ibl = 1 and a’ > 0, we see from Lemma 3.3 t h a t (22) extends t o a unitary operator o n L2(patl [ O , t ] ) . T h e proof of (2) is similar. 0
References 1. L. Accardi, G. Amosov, U. Franz, “Second quantized automorphisms of the renormalized square of white noise (RSWN) algebra”, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7,183-194 (2004). 2. L. Accardi, U. Franz, M. Skeide, “Renormalized squares of white noise and other non-Gaussian noises as Levy processes on real Lie algebras”, Comm. Math. Phys. 228, 123-150 (2002). 3. D. M. Chung, U. C. Ji, “Transforms on white noise functionals with their applications to Cauchy problems”, Nagoya Math. J. 147,1-23 (1997). 4. D. M. Chung, U. C. Ji, N. Obata, “Quantum stochastic analysis via white noise operators in weighted Fock space”, Rev. Math. Phys. 14, 241-272 (2002). 5. R. Gannoun, R. Hachaichi, H. Ouerdiane, A. Rezgui, “Un thBoritme de dualit6 entre espaces de fonctions holomorphes B croissance exponentielle” , J . Funct. Anal. 171,1-14 (2000). 6. U. C. Ji, N. Obata, “Quantum white noise calculus”, in Non-Commutativity, Infinite-Dimensionality and Probability at the Crossroads (N. Obata, T. Matsui and A. Hora, eds.), World Scientific, 2002, pp. 143-191. 7. U. C. Ji, N. Obata, “Unitarity of Kuo’s Fourier-Mehler transform”, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7,147-154 (2004). 8. U. C. Ji, N. Obata, “Unitarity of generalized Fourier-Gauss transforms”, to appear in Stoch. Anal. Appl. (2006). 9. U. C. Ji, N. Obata, H. Ouerdiane, “Quantum LBvy Laplacian and associated heat equation”, preprint, 2005. 10. H.-H. Kuo, White Noise Distribution Theory, CRC Press, 1996. 11. Y.-J. Lee, C.-F. Liu, “A generalization of Mehler transform”, in International Mathematics Conference ’94, World Scientific, 1996, pp. 107-116. 12. E. Lytvynov, “The square of white noise as a Jacobi field”, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7,619-629 (2004). 13. N. Obata, White Noise Calculus and Fock Space, Lect. Notes in Math. vol. 1577, Springer-Verlag, 1994. 14. N. Obata, “Quadratic quantum white noises and LBvy Laplacian” , Nonlinear Analysis 47,2437-2448 (2001).
Partial unitarity arising from quadratic quantum white noise
55
15. K. R. Parthasarathy, A n Introduction to Quantum Stochastic Calculus, Birkhauser, 1992. WIDEDAYED Dbpartement de Mathbmatiques, Institut Preparatoire aux Etudes d’hgbnieures, El Merezka, Nabeul, 8000, Tunisia NOBUAKI OBATA Graduate School of Information Sciences, Tohoku University, Sendai, 980-8579, Japan HABIBOUERDIANE Ddpartement de Mathbmatiques, Facult6 des Sciences, Universit6 de Tunis El Manar, Campus Universitaire, Tunis 1060, Tunisia
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SCHILDER’S THEOREM FOR GAUSSIAN WHITE NOISE DISTRIBUTIONS SONIACHAARI, SOUMAYA GHERYANI, (University of Tunis El Manar) HABIBOUERDIANE In the theory of large deviations, one of the main results is Schilder’s theorem. It gives the large deviation estimates for a family {pCLe,e > 0) of measures on some Polish space X, which tends weakly to the Dirac measure 6p at the point p E X . In this paper, we investigate analogous problems for a family {-ye,E > 0) of white noise Gaussian measures with mean 0 and variance E on the Schwartz distributions space S’(R). Applications to stochastic differential equations are given.
1. Introduction
In this paper we give an example of large deviation result for a certain family of measures on an infinite dimensional space. For this purpose, let X be a reel nuclear F’r6chet space. A function I : X [0,+a] is said to be a good rate function, if it is lower semi-continuous and {x E X , I ( x ) 5 L } are compact for all L 2 0. We say that a family {ye,E > 0) of Bore1 probability measures on the space X satisfies large deviation principle (LDP) with good rate function I if the following conditions are satisfied
-
1. (UPPERBOUND)for all closed subsets F in X limsupElog(y,(F)) I - inf I(y), UEF
E’O
2. (LOWERBOUND)for all open sets G in X
liminf Elog(y,(G)) 2
-
inf I(y). YEG
E’O
Let A, be the Logarithmic moment generating function, i.e.,
We denote by A; the Legendre transform of A,, i.e., for every cp E S’(R),
57
58
SONIACHAARI,SOUMAYA GHERYANI, HABIBOUERDIANE
Then, the rate function is the Legendre transform of the corresponding Logarithmic moment generating function which is also introduced in both Cramer's [3] and Schilder's theorems [16], see also [17]. We remark that, in the one dimensional case where X = R, the rate function I is given by: 22
I(z) = Ry(x)= R;(z) = -.
2
The present paper is organized as follows. In section 2 we recall the structure and concepts of white noise distributions. In section 3 we prove that a family { Y ~ , E> 0) of white noise Gaussian measures with mean 0 and variance E on the Schwartz distributions space S'(R) satisfies the large deviation principle with rate function "1; given in (9), see lemma 3.1. Section 4 is devoted to applying this large deviation results to the distributions measures associated to the solution of some stochastic differential equations. 2. Notation and preliminaries
+
Let N = X ZX the complexification of the reel nuclear FrCchet space X and suppose that its topology is defined by a family { I . l p l p E R?} of increasing Hilbertian norms. We have the representation
N =
nN, P>O
= proj lim
N,
P-+W
where N, is the completion of N with respect to the norm .1, Denote by N-, the topological dual space of the space N,,then the dual N' of N can be written as N' = N-, = ind lim N-,.
u
P20
P-+m
Let 8 : R+ + R+ be a Young function, i.e., 8 is continuous, convex, strictly increasing and satisfies O(0) = 0 and limx--tm = +cm. Denote 8* the Legendre transform of 8: 8*(z) = sup{ts - 8 ( t ) ;t > 0) for all z 2 0, which also a Young function. Given a complex Banach space ( B ,11. 1 1) , let H ( B ) be the space of entire functions on B , i.e., the space of continuous functions from B to @, whose restriction to all affine lines of B are entire on C. Let Exp(B, 8, m) denote the space of all entire functions on B with exponential growth of order 8, and of finite type m > 0: Exp(B, 8, m) =
{ f E H ( B ) ; Ilflle,m = sup If(~)Ie-'(~''~ll)< +w}. xEB
Schilder’s theorem for Gaussian white noise distributions
Let also
I l f l l ~ , ~ =, ~
sup If(x)le-’(mlxlp) for
f
59
E Exp(Np,8, m). The inter-
uENp
section
n
.Fe(N’) =
EXP(~-,A~),
p>O,m>O
equipped with the projective limit topology, is called the space of entire functions on N’ of 9-growth and minimal type. The union
equipped with the inductive limit topology, is called the space of entire functions on N of 9-growth and (arbitrarily) finite type. Denote by .Fo(N’)* the strong dual of the test function space .Fo(N’). In the sequel we take N = X i X , the complexification of a nuclear FrBchet space X. Let .Fo(N’)+ denote the cone of positive test functions] i.e., f E .Fo(N’)+ if f(x 20) 2 0 for all y in the topological dual X’of X.
+
+
Definition 2.1. The space .Fo(N’); of positive distributions is defined as the space of 4 E .Fo(N’)* such that (4,f ) 2 0 ; f E Fo(N’)+. We recall the following results on the representation of positive distributions; see [14]:
Theorem 2.1. Let 4 E .Fo(N’);, then there exists a unique Radon measure p+ o n X‘,such that
4(f)=
/
X‘
f(Y
+ i 0 ) d P d Y ) ; f E &d”>.
Conversely, let p be a finite, positive Bore1 measure o n X‘. Then p represent a positive distribution in .Fo(N’); if and only i f p is supported b y some X-,,p E N*,and there exists some m > 0 such that: eo(mlyl-p)d p ( y ) < 00.
(4)
We recall also the following estimates given in [15]. For a given 6 E X and x E R,let
Ac,x = {Y E X’: ( Y I O > denote the half-plane in
X‘ associated to [ and x.
(5)
60
SONIACHAARI,SOUMAYAGHERYANI, HABIBOUERDIANE
Theorem 2.2. Let $ E .Fo(N’)$ such that $ defines a positive Radon measure p+ on X’. Then f o r all E E X and x > 0 , there exists m > 0 and p E N such that:
where
6is the Laplace transform of $.
3. Large deviation for Gaussian measures on S’(R)
In this section, we take X = S(R)the Schwartz space of real-valued rapidly decreasing functions on R, and X‘ the corresponding dual space, i.e., X’ = S’(R) the Schwartz distributions space. For every integer n let H n ( x ) = (-l)nez2 (&)ne-z2 be the Hermite polynomial of degree n and
be the corresponding Hermite function. Then the set {e,;n 2 0} is an orthonormal basis for the Hilbert space L2(R). Now for each p 2 0 , define
where (., .) is the inner product of L2(R). Let S, = { f E L2(R);If , 1 < cm}. Then we have S(R) = np20S,(R) endowed with the projective topology. By the general theory of duality, S’(R) the dual space of S(R) can be written as S’(R) = Up20S--p(R) endowed with the inductive topology, where S-,(R) denotes the topological dual space of S,(R). Then we have the Gelfand triple
- -
S(R)
LZ(R,dX)
S’(R).
(7)
Using the Bochner-Minlos theorem, see [6] and [8], there exists a unique measure y on S’(R) such that
L R ,
<
ei(Ytc)d y ( y ) = e - ~ l ~ l ~ E, S(R),
where (., .) denotes the dual paring between S’(R) and S(R) which is realized as an extension of inner product (., .) on L 2 ( R , d z ) and 1.10 the corresponding norm.
Schilder’s theorem for Gaussian white noise distributions
61
We begin by introducing the logarithmic moment generating function:
The function A, can be extended t o the space L2(R). In fact, if E E L2(R) there exists a sequence ( J n ) n E ~in S(R)converging to in L2(R, dx). Then the sequence { ( . , J n ) } n E N of random variables is Cauchy in L2(S’(R),y).
<
Define ( . , J ) := lim 71-00
(.7tn), in L~(s’(R),~).
This limit is independent of the choice of the sequence (&),EN, see for example [12]. So for E L2(R) the logarithmic moment generating function A, is defined bv
Note that A, is lower semi-continuous and convex function. It is not hard t o see that the Legendre transform Al; of A, given in (3) is a lower semicontinuous and convex function, see [17] and [Ill.
Lemma 3.1. The Legendre transform of A, has the following expression: q c p ) =
;lcpl;
if cp E L2(R,dx),
= Ay((P),
if cp E S’(R)\L2(R7dx).
{+m.
(9)
Proof. Let cp E L 2 ( R , d x ) then , we have:
A,(E); E E S W } 1 = SUP{ (PiE ) - T ( € ,E ) ; E E S(W}
q c p ) = SUP{ (cp,€)
-
1
1
= #J,‘p)
=
ZIcpIi.
So we obtain
A;(’p)
= A,(p),
for all cp E L2(R,dz).
To prove that A;(cp) = oc7 for cp E S’(R)\L2(R,dz), suppose that Al;(cp) < 00 for ‘p E S’(R). Since the space L2(R,dx)is densely embedded in S’(R) with respect t o the weak topology, see [5],so for all cp E S’(R) there exists (cp), E L2(R)such that cp = limn,,cpn weakly. By the lower semicontinuity property of the function A;, we have:
X,(’p)= lim A * ( n-00
7
lim vn) = n-00
1 2
--l‘pnIg =
1
--Icplg < 00.
Thus, we conclude that cp E L2(R7 dx) and therefore lemma (3.1) holds.
0
SONIACHAARI, SOUMAYA GHERYANI, HABIBOUERDIANE
62
For each
E
> 0, let ye denote the image measure of y under the map
i.e., the Laplace transform is given by
<
Given E S-,(R) c S’(R), r > 0 and p > 0, we denote by B P ( ( , r )the open ball of radius r around a point 6 , and z P ( ( , r )the corresponding closed ball: BP(E7.) = {Y E S’(R); IY - E l - p I .}.
Lemma 3.2. Let r > 0 such that
for all
E
E
E S’(R) be given. Then for each S
> 0. In particular, if K is a compact subset lim sup E log(y,(K)) E-+O
> 0 there exists
of S’(R), then
I - inf A; K
Proof. First, note that
For all y E S’(R), and
’p,E
E S(R), there exists p E N such that
So for all y E B P ( [ , r we ) have (y, ):
2 ( E , $) - $ - I ‘ p l p . Hence
Schilder’s theorem for Gaussian white noise distributions
- $ and If A;(<) = 00, choose cp E S(R) such that (J,cp) - A,(cp) 2 2(l+l~l,). and r = 1 So we have
If A;(() r=-
1
+
63
< 00, we choose cp E S(R) such that (J,cp) -A,(cp) 2 A;(J) U+IVlP).
To prove the relation (12), let C = infK A;. Since K is a compact of S’(R) (and by definition of a compact set in dual space of nuclear space see [5]),we can choose G , [ z , . . . ,<, E K , r l , r z , . . . , r , E R; and p l , p 2 , . . . ,p, E N*, so that K c U;=, B p k ( J k , r k ) , and for k E {1,2,.. . , n } , B p k ( J k , r k ) satisfies the inequality (11), then
and therefore limsupElog(-y,(K)) 5 -min EO ’
Finally, let b \ 0, then we obtain lim sup E log(y E(K)) 5
-
E-0
inf A; K
.
Together with lemma 3.2 we need the following lemma to prove the inequality (1).
Lemma 3.3. There exists some m > 0, p E N* so that
Moreover, the set KL = {y S’(R) f o r each L > 0 and
E
k} is a compact subset of
S’(IR);?jlyl’?, I
limsupElog(y,(Kf)) 5 -L, E’O
where KE is the complement of K in S’(R).
(14)
64
SONIACHAARI,SOUMAYA GHERYANI, HABIBOUERDIANE
Proof. The relation (13) is an immediate consequence of the integrability condition (4) for the particular case where p is the Gaussian measure y. For all E > 0
0
Together with (13), this surely leads to (14).
Proposition 3.1. Let F a closed set of S'(R); we have limsupElog(y€(F))I: -inf(A1;). F €-+0
Proof. Let e = infF A;, and for L > 0 set FL= F compact set produced in the lemma 3.3. Then:
YdF)
(15)
nK L , where K L is the
I Y€(FL)+ r@E)
and so by lemma 3.2 and lemma 3.3, we have -
After letting L
-
lim clog(y,(F)) 5 - min(1, L ) .
E-0 00,
we obtain the desire results.
0
To prove the inequality (2) for a sequence of Gaussian measures, we need the following result concerning the quasi invariance of the Gaussian measure y on S'(R); see for example [6], [9] and [12].
-
<
Lemma 3.4. Given E L2(R), let yc denote the image measure of y under the map x x <. So
+
i.e., the Gaussian measure y is quasi-invariant under the translation by any in the Radon-Nikodym derivative.
< E L2(R) and
Schilder’s theorem for Gaussian white noise distributions
65
Proposition 3.2. For every open G c S’(R), lim inf E log(yE(G)) 2 - inf A; G
E+O
.
(16)
Proof. For every open G c S’(R), there exist r > 0 , p > 0 and E E S-,(R) so that I?,([, r ) C G. Since the space L2(R) is densely imbedded in S’(R) for the weak topology, there exist ( & ) n E L2(R) so that t = limn+m& weakly. By the lower semi-continuity property of the function A;, we have: lim At(&) = A;([).
n-cc
Hence, we need only to prove (16) for
Since, for [ E L2(R), A,([)
= A;(J),
t E L2(R).
we see that the relation (16) holds.
Propositions 3.1 and 3.2 prove in the following theorem that the family
{re,E > 0 ) satisfies the full large deviation principle with rate function A;. Theorem 3.1. For every measurable I? in S’(R), we have -inf(At) ro
I: liminf clog(y,(r)) 5 limsupElog(y,(I‘)) 5 -igf(A;). r
E’O
E’O
(17)
In particular, for a given 5 E S(R) and z E R, consider the half-plan in S’(R) associated t o [ and z,given by (5). Then we obtain the following result:
Corollary 3.1. For a given that - inf
sup
YEA€..: X E S ( I )
and z > 0 , there exist p
(u) 5 2
1 2
5 E S(R)
liminf clog(y,(AC,.))
140
EO ’
1
-2
> 0 such
66
SONIACHAARI, SOUMAYA GHERYANI, HABIBOUERDIANE
Proof. Using the definition of A;, the relation (9) and the fact that the topology on S(R) is defined by a family { I.lP,p E N} of increasing Hilbertian norms, we note that for all q E N
hl;(Y) 1
1 &
for all Y
E
S’(R),
(19)
and there exists p E N such that
we prove Combining the equations (19), (20) and the definition of the right inequality of the equation (18). To prove the left inequality of equation (18), we observe that for all X E S(R), y E S’(R):
finally we obtain the desired results.
Remark 3.1. 1. Theorem 2.2 obtained in [15] gives for the Gaussian measure yEthe following tail estimate:
(
Z;),
3 p E N : yE(Ac,,) 5 Cexp -- which implies that
1 x2 liminf Elog(y,(AE,z)) 5 limsupElog(yE(Ac,z))5 -- -. €-0 E+O 2 El;
(22)
So the inequality (22) is only the right hand inequality of (18). Therefore corollary 3.1 generalizes and precises the result obtained in [15]. In fact, the image measure of ya by the map (10) is given by yaE=yet. So if E 4 0 then E‘ 4 0 and
Schilder's theorem for Gaussian white noise distributions
67
2. Analogously we recover the same results if we replace the measure y by the Gaussian measure ya defined on S'(R) with mean 0 and variance a
> 0, i.e.,
3. In the particular case where yEis the Gaussian measure with mean 0 and variance E on R, the large deviation principle given in (18) becomes the following equality: 'X
4. Application to stochastic differential equations
4 . 1 . Generalized Gross heat equation It is well known that in infinite dimensional complex analysis the convolution operator on a general function space is defined as a continuous operator which commutes with the translation operator. Let us define the convolution (a * cp of a distribution (a E Fo(N')* and a test function cp E Fo(N') to be the function
where txcp is the translation operator, i.e.,
Note that (a * cp E Fo(N') for any cp E Fo(N') and the convolution product is given in terms of the dual pairing as ((a * cp)(O) = (((a,cp)) for any (a E Fe(N')* and cp E Fo(N'). We can generalize the above convolution product for generalized functions as follows. Let (a, Q E Fe(N')* be given, then (a** is defined by
) cp at z E N' is given by The Gross Laplacian A ~ c p ( z of AGq(2) =
x(n+
n>O
2)(n
+ l)(z@",
(7,(P'"'')),
SONIACHAARI,SOUMAYA GHERYANI, HABIBOUERDIANE
68
for cp E Fe(N') represented by cp(z) = CnlO(z@'", cp'")) and T is the trace q ) , 5, q E N . For more information on operator defined by ( T , @ q ) = the Gross Laplacian, see [7],[8],[10]and [12]. In fact, the Gross Laplacian AG is a convolution operator given by
c
(c,
where I is the distribution in Fe(N')* such that its Laplace transform is given by ?(z) = ( T , z@').
Theorem 4.1. (11 Let 6 be a Young function satisfying lim,,+, 6 ( r ) / r 2< 00 and F E Fo(N')*. Then the following generalized Gross heat equation perturbed by the white noise Wt
au,
---
at
1 2
-AcUt
+ aWt,
t 2 0,
U ( 0 ) = F, a E R,
(26)
has a unique solution in Fo(N')* given by
Ut = F
* 5+
I"
rt-,
* Wsds ;
is a positive distribution in Fe(N')* given by ((rt, 9))=
where $(<)
L,
= eit(c>c),
cp(z) d r t ( z ) , f o r all cp E Fe(N'),
EX.
From theorems 4.1 and 3.1 we obtain the following result:
Corollary 4.1. Let put be the associated measure with the solution of Cauchy problem (26) for the particular case where a = 0 , the Young function 6 is given by 6(x) = x2/2 and F is a the standard Gaussian distribution o n S'(R). Then the family of measures {pst,E > 0 } of image measure of the measure put under the map (10) satisfies the full large deviation principle (1 7) with rate function A;,, . Proof. If a = 0, the Young function 8 is given by O(x) = x2/2 and F is the standard Gaussian distribution on S'(R), then the solution of (26) is a positive generalized function and given by the explicit formula
Schilder’s theorem for Gaussian white noise distributions
69
So theorem 2.1 guarantees the existence and uniqueness of a Radon measure put on S’(R) associated with Ut such that
Therefore, the desired result is a consequence from theorem 3.1.
0
4 . 2 . Langevin Equation We can apply Schilder’s theorem for the measure associated with the solution of Langevin equation. In fact, the following Langevin equation d& = -aVtdt
+ udWt,
a
> 0, u > 0,
V ( 0 )= vo, where Wt is the Wiener process, has a unique solution given by:
& = VoeCat+ c
t
e-a(tCs)dW, .
The process & is called the Ornstein-Uhlenbeck process, and if VOhas a Gaussian distribution and is independent of W , Vt is a Gaussian distribution with parameters:
E(K) = E(&) e-at , var(&) = Var(&) e-2at
U2 + -(I 2a
- eCZat).
In particular, if VOhas a Gaussian distribution independent of W with mean 0 and variance the solution of (27) has a Gaussian distribution with mean 0 and variance $. And the family {pb,} of image measures of p~ under the map (10) satisfies the full large deviation principle (17).
g,
4.3. Ventcel and Freidlin’s estimate Our third application of Schilder’s theorem will be t o Ventcel and Freidlin’s estimate on the large deviations of randomly perturbed dynamical systems (see [IS]). The theory of Ventcel and Freidlin deals with families of measures {Pe : E > 0 ) on S’(R) of which the following is a typical example. For a given bounded, uniformly Lipschitz continuous function b : R 4 R, define the map X : S’(R) H S(R) by
SONIACHAARI,SOUMAYA GHERYANI, HABIBOUERDIANE
70
This integration equation is equivalent t o the following stochastic differential equation:
i
dXt(E) = dE(t)
+ f(&(E))ds
1
X(016) = E ( 0 ) .
> 0,
let PE= y Eo X-’ be t h e image measure of ye under the map H X(6). It is easy to see that the map E E S’(R)H X(<) E S(R) is a continuous one, see [13]. Thus, if G C S(R)is open, then Schilder’s theorem says t h a t
For c
(
liminf cln(P,(G)) = liminf Eln y€(X-’(G))) O’€
E’O
L
X(E) E G} = - inf(hl; o X-’) = - inf(R;), -inf{K,(E), G
G’
where G’ = X-l(G) c S’(R) which is also an open set. F 2 S(R) is closed, then limsup clog(y,(F)) €-+0
where F’ = X-’(F)
Similarly, if
I - inf(R;), F’
c S’(R).
References 1. A. Barhoumi, H. H. Kuo, H. Ouerdiane, “Generalized Gross Heat equation with noises”, Soochow Journal of Mathematics 32, no. 1, 113-125 (2006). 2. M. Ben Chrouda, M. El Oued, H. Ouerdiane, “Convolution calculus and applications to stochastic differential equations”, Soochow Journal of Mathematics 28, no. 4, 345-388 (2002). 3. H. Cramer, “Sur un nouveau th6or8me-limite de la th6orie des probabilit6s”, Actualit& Scientifiques et Industrielles, 736, 5-23 (1938), Colloque consacre‘ ci la the‘orie des probabilite‘s, vol. 3, Hermann, Paris. 4. R. Gannoun, R. Hachaichi, H. Ouerdiane, A. Rezgui, “Un th6orhme de dualit6 entre espaces de fonctions holomorphes B croissance exponentielle” , J. Func. Anal. 171(1), 1-14 (2000). 5. I. M. Gelfand, N. Ya. Vilenkin, Generalized Functions, vol. IV, Academic Press, New York and London, 1968. 6. N. Obata, White Noise Calculus and Fock Space, vol. 1577, L.N.M SpringerVerlag, Berlin, Heidelberg, and New York, 1994. 7. L. Gross, P. Maliavin, “Hall’s transformation and the Segal-Bargmann map”, in It6’s Stochastic Calculus and Probability, Springer-Verlag, Tokyo, 1996, pp. 73-116.
Schilder’stheorem for Gaussian white noise distributions
71
8. T. Hida, Brownian Motion, Springer-Verlag, New York, 1980. 9. T. Hida, H. H. Kuo, J. Potthoff, L. Streit, White Noise And Infinite Dimensional Calculus, Kluwer, Dordrecht, 1993. 10. H. Holden, B. Bksendal, J. Uboe, T. Zhang, Stochastic Partial Dafferential Equations: A Modeling White Noise Approach Birkhauser, Boston, Basel, Berlin, 1996. 11. Jean-Dominique Deuschel, Daniel W. Stroock, Large Deviations, AMS Chelsea Publishing, American Mathematical Society, . Providence, Rhode Island, 2001. 12. H. H. Kuo, White Noise Distribution Theory, CRC Press, Boca Raton, 1996. 13. H. H. Kuo, Introduction to Stochastic Integration, Universitex, Springer, 2006. 14. H. Ouerdiane, A. Rezgui, “Un th6orhme de Bochner-Minlos avec une condition d’int6grabiliW , Infinite Dimensional Analysis, Quantum Probability and Related Topics 3,no. 2, 297-302 (2000). 15. H. Ouerdiane, N. Prilvaut, “Asymptotics estimates for white noise distributions”, C. R. Acad. Sci. Paris, Ser. I 3 3 8 , 799-804 (2004). 16. M. Schilder, “Some asymptotics formulae for Wiener integrals”, Trans. Amer. Math. SOC.125,63-85 (1966). 17. D. W. Stroock, An Introduction to the Theory of Large Deviation, Universitex, Springer-Verlag, 1984. 18. S. R. S. Varadhan, “Asymptotic probabilities and differential equations”, Comm. Pure Appl. Math. 19,261-286 (1966). SONIA CHAARI University of Tunis El Manar, Faculty of Sciences of Tunis, Department of Mathematics, Campus Universitaire, 1060 Tunis, Tunisia Sonia.Chaari0fsb.rnu.tn
SOUMAYA GHERYANI University of Tunis El Manar, Faculty of Sciences of Tunis, Department of Mathematics, Campus Universitaire, 1060 Tunis, Tunisia
[email protected]
HABIBOUERDIANE University of Tunis El Manar, Faculty of Sciences of Tunis, Department of Mathematics, Campus Universitaire, 1060 Tunis, Tunisia
[email protected]
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A NONLINEAR STOCHASTIC EQUATION OF CONVOLUTION TYPE F. CIPRIANO ( G F M / FCT-UNL, Lisboa), H. OUERDIANE (Faculte' des Sciences de Tunis), J. L. SILVA(University of Madeira), R. VILELAMENDES( C M A F , Lisboa) A nonlinear equation, similar to the Burgers equation with the usual product replaced by a convolution product, is studied. The initial condition is a generalized function. By using the Laplace transform in a general white noise analysis setting, a general solution is found in (1 n) dimensions.
+
1. Introduction
The aim of this paper is t o study the following nonlinear stochastic equation of convolution type -
+ ( 3 * Q3= v A 3 + f'* 3,
Z(0,z) = 3o(z), -,
where u' is a Rn-generalized vector field, f is a n-dimensional generalized function, v > 0 a real constant, t E [ O , o o ) the time parameter, z = ( 5 1 , . . . ,z,) E Rn the spatial variable, A the Laplacian operator in R", V the gradient and * the convolution product for generalized functions (see [l],[ll]and subsection 2.2 for more details), f'* u' is a driving term, u' * 9 denotes the differential operator
2%* azj d
j=1
and the initial condition u'o = ( u o , .~. ,. ,210,") is a n-dimensional generalized function, see section 3 for more details. Problem (1) with * replaced by the usual product coincides with the classical Burgers equation well known in the literature ([3], [5] and references therein). However the physical interpretation of (1) is quite different. This is easily seen by comparing the j-component, for any j, 1 5 j 5 n,
73
74
F. CIPRIANO, H. OUERDIANE, J. L. SILVA,R. VILELA MENDES
the Fourier transform of the nonlinear kinetic term (Z.V)ii in the Burgers equation, namely n
with the j-component of the Fourier transform of the nonlinear term in ( l ) ,
(ii * V)G
C.z(t,k)kifiij(t,k),
(3)
i=l
denoting the Fourier transform of ui. In the Burgers case the expression ( 2 ) implies that the Fourier modes at length scale and control the eddies at scale f , consistent with the phenomenological description of the inertial range in the turbulence cascade. However, in the convolution case the nonlocal nonlinearity corresponds to a self-interaction of the modes at each length scale. Nevertheless, nonlocal nonlinearities are also important in models of transport in magnetized plasmas, see [4], and also in the modeling of convection driven by density gradients as it arises in geophysical fluid flows, see [12], [13], [14] and [15]. By restricting oneself to solutions of gradient type, the Burgers equation may be linearized by the Cole-Hopf transformation. This provides the most general solution for (1 1) dimensions but not for (1 n ) dimensions. For our equation (l),using the Laplace transform in a general setting, we obtain a general solution for (1 n) dimensions. The paper is organized as follows: In Section 2 we provide the mathematical background needed to solve the Cauchy problem stated above, namely spaces of test and generalized functions, the characterization theorem of generalized functions and the convolution product as well as some of its properties. In Section 3 we combine the convolution calculus and the characterization theorem in order to find an explicit solution of the problem
fii
+
+
+
(1).
2. Preliminaries
2.1. Test and generalized functions spaces In this section we introduce the framework needed later on. The starting point is the real Hilbert space 'H = L2(B,Rd) x R', d , r E N with scalar
A nonlinear stochastic equation of convolution type
product (.,
a)
and norm
I . I.
75
More precisely, if
then
Let us consider the real nuclear triplet
M'
= S'(R, Rd)x
R' II 1-I II S(R,Rd)x R' = M .
(4)
The pairing (., .) between M' and M is given in terms of the scalar product in 1-I, i.e., ( ( w , x),( t , ~:=) ()w , E ) L ~ (Z,P)RP, ( w , x) E M' and ( 6 , ~E) M . Since M is a Frkchet nuclear space, it can be represented as
+
n Sn(R,Rd) 00
M
=
nM n , 00
x R' =
n=O
n=O
+
where Sn(R,Rd)x R' is a Hilbert space with norm squared given by I .I: 1 . I&, see e.g. [8] or [2] and references therein. We will consider the complexification of the triple (4) and denote it by
N'
3
2 II N ,
(5)
+
+
where N = M iM and 2 = 1-I i7-i. On M' we have the standard Gaussian measure y given by Minlos's theorem via its characteristic functional, namely for every ( 6 , ~ E) M
In order to solve the (1+n)-dimensional equation of convolution type (1) we need to introduce an appropriate space of vectorial generalized functions. We borrow this construction from [9]. Let 8 = (81,8:!) : Rw"++ R, (tl, t 2 ) H el(t1) 8 2 ( t 2 ) where el,82 are two Young functions, i.e., Oi : R+ -+ R+ continuous convex strictly increasing function and
+
&(t) t
lim -= 00, &(O)
t-+m
= 0,
i
=
1,2.
F. CIPRIANO, H. OUERDIANE, J . L. SILVA,R. VILELAMENDES
76
where
Ilfllo,m,n
:= SUP { I f ( z ) l exp(-e(mlZI-n). ZEN-n
+
Here, for each z = ( w , z ) we have e(mlzI-n) := B l ( m l l W I - n ) 6%(m2Izl). Now we consider as test function space the space of entire functions on N’ of (61, &)-exponential growth and minimal type
endowed with the projective limit topology. We would like to use Fo(N’) t o construct a triple centered in the complex Hilbert space L 2 ( M ’ , y ) . To this end we need another condition on the pair of Young functions (Ql,ez). Namely, lim ei(t) < co, i = 1,2.
t-+m
t2
This is enough to obtain the following Gelfand triple
where FL(N‘)is the topological dual of Fo(N’) with respect t o L 2 ( M ’ , y ) endowed with the inductive limit topology. In applications it is very important t o have the characterization of generalized functions from .FL(N’). First we define the Laplace transform of an element in FL(N’). For every fixed element ( < , p ) E N the exponential function exp((<,p)) is a well defined element in Fe(N’), see [7]. The Laplace transform C of a generalized function @ E FL(N’) is defined by
q<, P ) := (C@)(C,P):= ((@,eXP((<,P)))).
(8)
We are ready to state the characterization theorem (see, e.g., [7] and [l]for the proof) which is the main tool in our further consideration.
Theorem 2.1. 1. The Laplace transform is a topological isomorphism between FL(N‘)
and the space
Gp(N),where Gp(N)is defined by
A nonlinear stochastic equation of convolution type
77
and G o * , ~ ( is N the ~ ) space of entire functions o n N, with the following 8-exponential growth condition
Here Q* = ( O T , 8,*), where O f ( % ) = sup,,,(tz - 8,(t)) is the Legendre transform associated to the function 6, i = 1 , 2 . 2. I n the particular case 8 ( x ) = ( 8 l ( x ) , 8 2 ( x ) )= ( x , ~ )we , denote the space 3L(N’) by F;(N’). Then the Laplace transform realizes a topological isomorphism between the distributions space 3;(N‘) and the space Holo(N) of holomorphic functions o n a neighborhood of zero of N. 2.2. The Convolution Product
*
It is well known that in infinite dimensional complex analysis the convolution operator on a general function space F is defined as a continuous operator which commutes with the translation operator. Let us define the convolution between a generalized and a test function. Let Q, E FL(N‘) and cp E Fo(N’) be given, then the convolution CP * cp is defined by
It is not hard the see that Q,*pE .FO(N’), cf. [7]. The convolution product is given in terms of the dual pairing as (CP * cp)(O,O) = ((CP,cp)) for any Q, E FL(N’) and cp E Fo(N‘). We can generalize the above convolution product for generalized functions as follows. Let CP, Q E .FA(”) be given, then Q, * Q is defined by
This definition of convolution product for generalized functions will be used later for the solution of the equation (1). We have the following equality, (see [ll,Proposition 31):
78
F. CIPRIANO, H. OUERDIANE, J . L. SILVA,R. VILELAMENDES
As a consequence of the above equality and definition (9) we obtain
L(@* 9)= L @ L 9 , @ ] 9 E Fh(N’)
(10)
which says that the Laplace transform maps the convolution product on Fh(N’)into the usual pointwise product in the algebra of functions Q p ( N ) . Therefore we may use Theorem 2.1 to define the convolution product between two generalized functions as @ * Q = Lc-l(L@L9).
This allows us to introduce the convolution exponential of a generalized function. In fact, for every @ E FL(N’)we may easily check that exp(L@)E Gee* ( N ) .Using the inverse Laplace transform and the fact that any Young function 0 verifies the property (0.). = 0 we obtain that L-’(ge.*(N)) = Ftee.)* (n/‘). Now we give the definition of the convolution exponential of @ E F;(N‘), denoted by exp* @ exp*
:= L-’ (exp(L@)).
Notice that exp* @ is a well defined element in F;eo* ) * (N’)and therefore the distribution exp* @ is given in terms of a convergent series exp*
= 60
+ C n!1 n= 1
where is the convolution of @ with itself n times, @*’ := 60 by convention with 60 denoting the Dirac distribution at 0. We refer t o [l]for more details concerning convolution product on Fi(MI). A one parameter generalized stochastic process with values in .?h(N’) is a family of generalized functions { @ ( t )t, 2 0) c FL(N’). The process @ ( t )is said to be continuous if the map t ++ @ ( t )is continuous. For a given continuous generalized stochastic process ( X ( t ) ) t l owe define the generalized stochastic process
Y ( t 1 w , 2 )=
I”
X ( s , w , z ) d s E Fh(”)
The process Y ( t , w , z )is differentiable and we have & Y ( t , w , x ) = X ( t , w , z ) . The details of the proof can be seen in [lo, Proposition 111.
A nonlinear stochastic equation of convolution type
79
2 3 . Convolution inverse of distributions Let @ be a fixed element on the distribution space FL(N’)and consider the following convolution equation
Applying the Laplace transform t o the convolution equation (13) we obtain
6.G = 1 If 6(E,q)# 0 for every ( J , q ) E N , then using the division result in the space Be*(N)(see [ 6 ] ) we obtain
Moreover, by the Laplace transform isomorphism (see Theorem 2.1), we prove the existence and uniqueness of the solution 6 E Fh(N’) in the equation (13). If we denote this solution XQ by a*-’, we have
This division result is also true in the limit case O(z) = (z,z); i.e.,
6
E
Be* ( N )= Halo ( N ) . 3. Solution of the n-dimensional convolution e q u a t i o n
We are now ready t o solve the Cauchy problem stated in (1) which we recall for the reader convenience, namely
- + (Z* ?)C
= vAG
+ f * Z,
Z(0,5) = i i o ( 5 ) . The different terms in (14) are as follows: GO(.) = (uo,l(z),. . . , u ~ 5 , ~) )( is a generalized function; uo,j(x) E F;(N’), v > O a real constant, II: = + (21,. . . ,zn)E Rn, ii = ( ~ 1 , .. . ,un), f = ( f l , . . . , fn), with fj = f j ( t , 5 ) 9
80
F. CIPRIANO, H. OUERDIANE, J . L. SILVA,R. VILELAMENDES
n
j=1
n .j=1
We are now ready t o prove the main result of this paper, namely we obtain the explicit solution of the equation (14), using the tools from Section 2.
Theorem 3.1. Let fio(z)= (uo,l(z),. . . ,uo,n(z))and f = ( f l , . . . , f n ) be such that U O , k , f k E FL(N'), k = 0 , . . . , n . Then the solution G(t,w,x) of the nonlinear equation of convolution type (14) is given explicitly by the following system: 'LLk(t, z) = u O , k ( z )
* e* ri fk(S)dS * ')'2vt n
* (60 +
dju0,j
*
t
e*r:
fj(s)ds
* ~ 2 , d, r~
j=1
)
*-l
(15)
with k = 1,.. . , n and 60 is the Dirac measure at point zero and 79,,t is the Gaussian measure with variance 2ut on Rn.
Proof. We denote be written as
[
at = dtUk
such that the n-dimensional equation (14) may
+I
n U j
* a j U k = V A U k + fk * U k , (16)
j=1 uk(o,w,z)
= UO,k(W,z),
where k = 1,.. . , n . w e denote by v k = ? ' k ( t , < , q ) , gk = g k ( t , t , q ) and v0,k = v O , k ( < , q ) , E S(R,Rd), q E Rn, the Laplace transforms of the generalized functions uk = u k ( t , w, z), f k = fk(t, w, x) and initial condition UO,k = UO,k(W, z), respectively, for k = 1,.. . ,n. Applying the Laplace transform to the system
<
A nonlinear stochastic equation of convolution type
81
(16) we obtain n
f
atvk
2 qjvjvk = vq v k
-
+ gkvk (17)
j=1 Vk(0,q) = VO,k(Q).
Changing the variables, to the following:
Sk =
&, k = 1,. . . , n,the system (17) is equivalent
which can be integrated. The solution is
sk= SO k s1e S J ( g 1 ( ~ ) - g k ( ~ ) ) d ~ ,k = 1,.. . ,n, so, 1 where for simplification of notation gj(s) we have the relation
Introducing the expression of system of equations
= gj(s,J, q), j =
(22)
1,.. . , n. Then
2 in (18) we deduce the following linear
a2
F. CIPRIANO, H. OUERDIANE, J . L. SILVA,R. VILELAMENDES
For any fixed Ic, the solution of the homogeneous equation is given by
where X is a constant. Then the solution of (24) is given by the method of variation of constants as
where the constant X is determined by the initial conditions; S k ( O , J , q ) = SO,^ = A. Then (26) may be written as
1
Since Sk(t,<,q) =
V k ( t ,€1
4)
, we obtain
In fact, it is easy to show that for every t 2 0, the function
belongs to the space Holo(h/) and satisfies Y ( t ,0,O) = 1 # 0. Then there exists U a neighborhood of (0,O) of N, such that Y ( t ,q , I )# 0 for every ( J , q ) E U.Therefore 1 Y ( t , q , J ) Holo(N)
which implies that
A nonlinear stochastic equation of convolution type
83
Finally, to obtain t h e solution of the equation (14) we use the following equalities:
u k ( t , z), k = 1,.. . , n is given by the Laplace inverse transform according t o the theorem 2.1, as in (15). 0
and then
Corollary 3.1. If the potential f i n the equation (14) does n o t depend o n f(x), t h e n the solution is given by the t i m e variable t , i e . ,
f=
% ( t ,z) = U O , k ( Z ) * e*tfk * Y2vt
with Ic = 1,.. . , n. In particular, i f f = 0 the solution has the f r o m Uk(t1.)
= UO,k(%)
* Y2vt *
(.
*-I
+v
.Uo
* I’YbTdi)
with Ic = 1 , . . , , n and V. represents the divergence operator.
Acknowledgment We thank Martin Grothaus for useful discussions. Financial support by GRICES, Portugal/Tunisia, 2004 and FCT, POCTI - Programa Operacional CiGncia, Tecnologia e Inova@io, FEDER are gratefully acknowledged.
References 1. M. Ben Chrouda, M. El Oued, H. Ouerdiane, “Convolution calculus and
applications to stochastic differential equations”, Soochow J . Math. 28(4), 375-388 (2002). 2. Yu. M. Berezansky, Yu. G. Kondratiev, Spectral Methods in Infinite-Dimensional Analysis, volume 1, Kluwer Academic Publishers, Dordrecht, 1995. 3. J. M. Burgers, The Nonlinear Diffusion Equation. D. Reidel Publishing Company, Dordrecht, Holland, 1974. 4. J. D. Callen, C. C. Hegana, E. D. Held, T. A. Gianakon, S. E. Kruger, C. R. Sovinec, “Nonlocal closures for plasma fluid simulations”, Phys. Plasmas 11(5), 2419-2426, (2004).
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F. CIPRIANO, H. OUERDIANE, J . L. SILVA, R. VILELAMENDES
5. Weinan E,K. Khanin, A. Mazel, Ya. Sinai, Invariant measures for Burgers equation with stochastic forcing, Ann. of Math. (2) 151(3), 877-960 (2000). 6. R. Gannoun, R. Hachaichi, P. Kree, H. Ouerdiane, “Division de fonctions holomorphes a croissance 8-exponentielle” , Technical Report E 00-01-04,BiBoS University of Bielefeld, 2000. 7. R. Gannoun, R. Hachaichi, H. Ouerdiane, A. Rezgui, “Un theorbme de dualit6 entre espaces de fonctions holomorphes B croissance exponentielle” , J. Funct. Anal. 171(1), 1-14 (2000). 8. T. Hida, H. H. Kuo, J. Potthoff, L. Streit, White Noise. A n Infinite Dimensional Calculus. Kluwer Academic Publishers, Dordrecht, 1993. 9. H. Ouerdiane, “Infinite dimensional entire functions and applications to stochastic differential equations”, Not. S. Afr. Math. SOC. 35(1), 23-45 (2004). 10. H. Ouerdiane, J. Silva, “On the heat equation with positive generalized stochastic process potential”, Methods Funct. Anal. Topology 10(3), 54-63 (2004). 11. H. Ouerdiane, J. L. Silva, “Generalized Feymann-Kac formula with stochastic potential”, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5(2), 1-13 (2002). 12. A. J. Roberts, “Planform evolution in convection - an embedded centre manifold”, J. Austral. Math. SOC.Ser. B 34(2), 174-198 (1992). 13. A. J. Roberts, “The swift-Hohenberg equation requires non-local modifications to model spatial pattern evolution of physical problems”, technical report, arXiv:patt-s01/94i2002(2002). 14. P. VBn, “Weakly nonlocal continuum theories of granular media: restrictions from the second law”, International Journal of Solids and Structures 41(21), 5921-5927 (2004). 15. P. VBn, T.Fulop, “Weakly nonlocal fluid mechanics - the Schrodinger equation”, technical report, arXiv :quant-ph/0304062 (2004). F. CIPRIANO
GFM e Departamento de Matemcitica FCT-UNL, Av. Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal ciprianoQgfm.cii.fc.ul.pt
H. OUERDIANE
D6partement de M a t h h a t i q u e s , Facult6 des Sciences de Tunis, 1060 Tunis, Tunisia habib.ouerdianeQfst.rnu.tn
J. L. SILVA University of Madeira, CCM, 9000-390 Funchal, Portugal luismuma.pt
R. VILELAMENDES
CMAF, Complexo Interdisciplinar, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal
[email protected]
VARIATIONAL PRINCIPLE FOR DIFFUSIONS ON THE DIFFEOMORPHISM GROUP WITH THE H 2METRIC FERNANDA CIPRIANO ( G F M - UL / FCT- UNL, Lisboa), ANA BELACRUZEIRO ( G F M - U L / IST, Lisboa)
We describe a stochastic variational principle associated to the H 2 metric on the group of diffeomorphisms of the torus as well as the corresponding dynamics. This variational principle corresponds to a minimal entropy principle.
1. Introduction
In his celebrated paper [l],Arnold has shown that the Euler equation for incompressible fluids can be characterized as the geodesic equation in the group of volume preserving diffeomorphisms with respect to the weak L2 invariant metric. His work, developed by Ebin and Marsden [6], introduced Lagrangian formalism in hydrodynamics and showed the relevance of infinite dimensional geometry in this area. Considering the H1 right invariant metric, the geometry of the volume preserving diffeomorphism group and the corresponding equations of motion were studied in [13], inspired by a hydrodynamical model which had been previously introduced by Holm, Marsden and Ratiu [9]. In [3], and in the two dimensional periodic case, we proposed a generalization of Arnold’s result that consists in obtaining Navier-Stokes equations as critical points of a stochastic variational action defined in the space of measure preserving homeomorphisms with respect to the L2 metric. It is an extension of the deterministic variational principle in the sense that the Laplacian term of Navier-Stokes is associated to a random perturbation of the deterministic Euler system. This approach found its inspiration in the work [12]. On the other hand it relies on the possibility of solving some stochastic differential equations on the space of homeomorphisms of the torus and for this uses the recent developments on the subject ([ll],[7]). Other variational characterizations of Navier-Stokes equations can be found in the literature and follow different points of view. Without trying to be exhaustive, we refer t o [lo] and to [8]. Here we adopt a similar point of view of [3]but consider a H 2 metric. We describe the stochastic variational approach of the associated dynamics.
85
86
FERNANDA CIPRIANO, ANABELACRUZEIRO
The higher regularity of the metric allows to reinterpret the variational principle for the corresponding energy functional as a minimal entropy principle. The same methods can actually be applied to H l f e with any E > 0; then all the corresponding results hold true, except the partial differential equation coming from the variational principle (Theorem 2), which has t o be modified accordingly. 2. Diffusions on the homeomorphism group of the torus We denote by T the two-dimensional torus, dB its volume element measure.
Gs= { g : T -+ T bijection, 9,g-l
E
H"}
is the group of homeomorphisms of the torus belonging t o the Sobolev class H" and GG the subgroup of those which preserve the volume. For g E H " , G" is locally diffeomorphic to the Hilbert space
H i = { 2 vector field on T : 7roZ = g } , where 7r is the natural projection from the tangent space of T to T.More precisely, a local chart is defined by wezp : H l 4 G", wew ( 2 )= e x p o 2 , where exp is the Riemannian exponential map of the metric on T. For the case of volume preserving diffeomorphisms, the Lie algebra of GG, denoted here by GG, is the space of H" Sobolev vector fields on the torus with zero divergence. The Euler equation corresponds t o a flow on GO, which is critical for the action functional defined by the square of the right invariant metric L2 and defined on GO,, namely,
We consider the complete system of nomials,
G c given by the trigonometric poly-
where k = ( k l , k z ) E Z2, k # 0. Notice that we are considering vector fields with mean zero, an assumption that is not restrictive from the point of view of the dynamics we study.
Variational principle for diffusions on the diffeomorphism group with . . .
87
We follow the methodology of [ll]and [7] in order to construct a Brownian motion that takes values in the group of homeomorphisms (for more details about this construction in the two dimensional case, see [3]). We consider a sequence of R2-valued independent Brownian motions z k = (z;, z;) which are adapted t o the usual increasing filtration (generated by events before time t ) and define
This sum can be shown t o converge uniformly in [O,T]x T and t o have quadratic variation
Consider the stochastic differential equation
which is a condensed form of writing
d g 2 ( t )=
c k#O
1
3[A2k(g(t))O d d ( t ) + B,2(g(t))O d 4 ( t ) ] . lkl
Then the process g ( t ) is well defined and lives in GO,. An important observation (cf. [3]) is that the It6 contraction vanishes and therefore Stratonovich stochastic differentiation in the definition of g ( t ) coincides with It6 differentiation. This can be checked by direct computation. Define the generator of the process g ( t ) by
w ) ( g ) ( e )= ";o +(F(s(t))(e) - W
4).
Then, using It6 calculus, we can prove the following Theorem 1. ([S'])
W h e n considered o n functions o n the torus, namely when F ( g ) ( Q = ) f (g(0)) f o r f smooth, the generator of the process g ( t ) applied t o F coincides with ky/lk16, the usual Laplacian of f multiplied by the constant c = 2 w g ) ( e ) = cAf M e ) ) .
xkZo
88
FERNANDA CIPRIANO, ANA BELACRUZEIRO
Let ut(g)be a vector field on G$, continuous in t and satisfying
We can solve, proceeding as in [7], the following stochastic differential equation
dg"(t) = ( f i d ~ ( t )+ w)(g"(t)),
gu(0) = id,
(3)
weakly via a Girsanov transformation. The law of this process at each time t has for density with respect to the law of g ( t ) the martingale (with respect t o the future filtration)
where we have used the notation:
uAk= ( u , A k ) p ,uBk= (u, B k ) L z , if kl
> 0 and I c Z
arbitrary;
uAk= uBk= 0, otherwise. 3. The variational principle and the associated dynamics
We denote by S the set of continuous semimartingales c(t),t E [0,TI with initial condition g(0) = id and taking values in GO,. On S we consider the action functional
where D is the mean derivation operator
D F ( < ( ~= ) )a.s.,6-0 lim 1 [ F ( t+ S , <(t+ 6)) - F ( t ,[ ( t ) ) ] . 6E F ~ The variations will be defined as follows. For w E C'([O,T];Q,") with w(0) = 0 we consider the GF-valued deterministic path et(w) which is the solution of the ordinary differential equation
d -et(v) = ir(t,e t ( v ) ) ,
dt
eo = i d .
Variational principle for diffusions on the diffeomorphism group with . . .
We compute the left derivative of the action functional S at a process this left derivative being defined as
89
E,
For E(t) = g"(t) with u as in (2), we have
Using Taylor expansion, the perturbation of the process g"(t) in the direction e t ( u ) can be written as
+
e t ( 6 v ) 0 g U ( t )= g U ( t ) bvt(g"(t))
+ o(6).
We have
S[et(Sv)0 g"(t)l =
l T p/ 11u(g"(t)) + C " ( v ) ) ( g " ( t ) )+ o(s)11;2
dtl
0
where C" denotes the infinitesimal generator of the process g", namely avi
+
C " ( V )= ~ - u.Vvi + vAvi,
at
i = 1,2.
Therefore
// T
=E
O
(A2u,
T
dV at + U.VV+ V A V()g U ( t ) d) d d t .
Using the fact that g" preserves the measure dd and integrating by parts,
// T
DL,,S[g"] =
0
T
a
(--Azut
at
+ uA3ut
-
(ut.V)A2ut)vt dt.
We have deduced the following Theorem 2. T h e process g " ( t ) , with ut E H 2 as in (2), is a critical value of the action functional S in (4) i f and only i f u satisfies the partial d i e r e n t i a l equation
a
+
-A2u = vA3u - (u.V)A2u V p , dt for some function p .
UT
E H2,
divu = 0,
90
FERNANDA CIPRIANO, ANABELACRUZEIRO
4. The minimal entropy principle On a countably generated measurable space (E,B), and for u and p two probability measures on El the relative entropy of u with respect to p is defined as
if u is absolutely continuous with respect to p and +oo otherwise. On R = C([O,11;GO,,Fr) we consider the canonical realization of the Brownian motion g(t) defined in paragraph 2. Its law on R will be denoted by P. For a probability measure Q on R denote by Qt the marginal law of g(t) under Q. Assume Qo = 6, is the Dirac mass at the identity and QT = p ~ If. Q is absolutely continuous with respect t o PI let Mt be the
Then the coordinate process satisfies the equation
M t ) = ( d Z ( t )+ 4 M t ) ) I where Z ( t ) is a Q-Brownian motion and 1 u(tlg(t)) = lim -EQ((g(t 6 ) - g(t))lF:) 6-0 6
+
= Dg(t).
It can be proved (cf. [4]) that the regularized velocity term of the action functional, namely
1
T
S(Q) = ;EQ
IIDg(t)llHz dt
coincides with the relative entropy, i.e., S(Q) = hF; (QI Pa,).
An application of Csiszar’s minimization theorem ([5])provides an unique probability measure Q that attains the infimum of S(Q) for Q probability measures on R with QO = 6, and QT = p ~ if,this infimum is not equal to infinity] which is true by Theorem 1. Moreover the corresponding coordinate process is Markov. Other properties of this process could be studied, notably the multiplicative nature of its law (cf. [4]).
Variational principle for diffusions on the diffeomorphism group with . . .
91
Acknowledgements Research of both authors has been carried out in the framework of the
Project POCI/MAT/55977/2004 (co-financed by FCT/OE and FEDER through POCI2010).
References 1. V. Arnold, “Sur la gkomktrie diffkrentielle des groupes de Lie de dimension infinie et ses applications a I’hydrodynamique des fluides parfaits”, Ann. Inst. Fourier 16, 319-361 (1966). 2. P. Cattiaux, C. Leonard, “Minimization of the Kullback information for some Markov processes”, in Seminaire d e Probabilites X X X , Lect. Notes i n Mathematics 1626, Springer, 1996, pp. 288-311. 3. F. Cipriano, A. B. Cruzeiro, “Navier-Stokes equation and diffusions on the group of homeomorphisms of the torus”, preprint. 4. A. B. Cruzeiro, W. Liming, J. C. Zambrini, “Bernstein processe associated with a Markov process”, in Stochastic Analysis and Mathematical Physics, Trends i n Mathematics, ed. R. Rebolledo, Birkhauser, 2000. 5. I. Csiszar, “I-divergence geometry of probability distributions and minimization problems”, A n n . Probab. 3(1), 146 (1975). 6. D. Ebin, J. Marsden, “Groups of diffeomorphisms and the motion of an incompressible fluid”, Ann. of Math. 92, 22-46 (1970). 7. S. Fang, “Solving stochastic differential equations on Homo ( S l ) ” , J. Funct. Anal. 216, 22-46 (2004). 8. D. A. Gomes, “A variational formulation for the Navier-Stokes equation”, Comm. Mathem. Physics 257(1), 227-234 (2005). 9. D. D. Holm, J. Marsden, T. S. Ratiu, “Euler-Poincar6 models of ideal fluids with nonlinear dispersion”, Phys. Rev. Letters 80, 4273-4277 (1998). 10. A. Inoue, T. Funaki, “A new derivation of the Navier-Stokes equation”, Comm. Mathem. Physics 65, 83-90 (1979). 11. P. Malliavin, “The canonic diffusion above the diffeomorphism group of the circle”, C. R. Acad. Sci. Paris 329, 325-329 (1999). 12. T. Nakagomi, K. Yasue, J. C. Zambrini, “Stochastic variational derivations of the Navier-Stokes equation”, Letters in Math. Physics 160, 337-365 (1981). 13. S. Shkoller, “Geometry and curvature of diffeomorphism groups with H 1 metric and mean hydrodynamics”, J . Funct. Anal. 5, 545-552 (1981). FERNANDA CIPRIANO Grupo de Fisica-Matemitica UL and Dep. de Matemitica FCT-UNL, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal cipriano(0cii.fc.ul.pt
ANA BELACRUZEIRO Grupo de Fisica-Matemitica UL and Dep. Matemhtica IST, Av. Rovisco Pais, 1049-001 Lisboa, Portugal abcruzhath.ist.ut1.pt
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ON A VARIATIONAL PRINCIPLE FOR THE NAVIER-STOKES EQUATION DIOGOAGUIARGOMES( I S T , Lisboa) In this paper we study the variational principle for the Navier-Stokes equation described in [14],and clarify the role of boundary conditions. We show that in certain special cases this variational principle gives rise to new models for fluid equations.
1. Introduction Several authors, for instance [19], [MI, [3], [2], [S], have studied representation formulas for solutions of the Navier-Stokes equation using probabilistic methods, and, in fact, the idea of using random maps instead of deterministic ones can be traced back to Chorin [6] and Peskin [17]. In a completely different setting, Arnold [l],and Ebin and Marsden [12], proved a variational principle for the solutions of Euler equation. It is of course natural to ask whether there is a probabilistic analog for the Navier-Stokes equation. In fact, there is at least one, non-probabilistic variational principle for the Navier-Stokes equation[l3]. However, recently stochastic variational principles for the Navier-Stokes equation have drawn attention and at least three different ones have been proposed by several authors: [15], [5], and [14]. This paper builds upon the last variational principle and is a contribution to its study. This variational principle can be stated as follows:
Theorem 1.1. Let Bt be a Brownian motion defined on a probability space R, and let E denote the expected value o n R. Let O(X,x) : Rn x Rn 4 Rn be a smooth function with suitable growth conditions. Suppose that for each w E R, (u, q5w) is a critical point, u smooth in space-time, and 4 smooth in space and C' in time, of
under the constraint divu = 0 , and
a&
93
DIOGOAGUIARGOMES
94
and 4"(x, 0 ) = x . Assume further that the function II" which satisfies the differential equation
any + n p u j
-
at
0
Bt 0 4u = 0,
with the terminal condition
n u ( x , T )= o x @ ( 4 w ( x , T ) , z ) , is, at t = 0 , non-random. Then u is a solution to the Navier-Stokes equation ut+u.Vu+Vp=
1 2
-Au,
where p is the scalar pressure field. One can relate some of the objects that arise in this last theorem t o the works by P. Constantin in [9], [lo], [ l l ] . In fact, consider a critical point (u, 4"') as in the previous theorem, and define
A
=E
[(4")-l o BL1].
The next proposition shows that the vector A satisfies exactly the advection-diffusion equation as in [lo]
Proposition 1.1. The vector A satisfies dA
+
1
- (u V ) A- -2A A at
=0,
with A(x,O) = x . Note that this does not depend on u being a solution to the Navier-Stokes equation. In this paper we study a variation of theorem 1.1,with a special terminal cost which enable to remove the probability from the variational principle. However, the vector will not be in general progressively measurable, hence the critical points may not be solutions to the Navier-Stokes equation. This gives rise to a new model for incompressible viscous fluids and we study some of its properties.
n
2. A non-probabilistic variational principle Consider the case of a terminal cost which depends only on the active transport vector A, and given deterministic data. For instance
On a variational principle for the Navier-Stokes equation
Then, we may replace the variable consider the problem
4'" by
95
the variable A , and so we may
under the constraints div u = 0,
&A
+ (u. V ) A- -21 A A = 0.
(4)
Theorem 2.1. Let ( u ,A ) be a critical point of the functional (3) under the constraints (4). Let be the solution of
<
with the terminal condition
Then u = Pm, where mi = <j&Aj satisfies the modified Navier-Stokes equation in magnetization form:
atmi
1 + ujDjmi + mjDiuj = ZAmi - d k ((d&)(&Aj)).
(6)
Remark. (6) should be compared with the corresponding identity for the Navier-Stokes equation, see (8) in the appendix. Remark. The functional (3) has as domain the velocity fields u E L 2 , for which the corresponding weak solution of the active transport equation satisfies A ( z ,T ) - $(z) E L2(Rn).However, the theorem, as stated, is only valid for smooth critical points. Remark. The Leray projection Pm is defined as the L2 projection of the vector field m into the divergence free vector fields in Rn.
<
Proof. We will use the variable as a Lagrange multiplier to make the computation of the Euler-Lagrange equation easier. Thus consider the equivalent problem of minimizing
DIOGOAGUIAR GOMES
96
under the same constraints. By making variations A obtain rT
+ d A and u + ESU we
r
l o lin S U ) + (C,(SU (21,
*
V ) A )= 0,
which implies u = Pm with mi = CjjaiAj, and
1 / (C, T
0
at6A
1 + (u . V ) 6 A- 5A6A)
in
( A ( z , T )- $ ( z ) , S A ( z , T )= ) 0, which shows that
C(z,T ) = A ( z ,T)- $(z), after integration by parts (in space and time) and using the definition of Finally, we must show that m satisfies (6). To this end, observe that
<.
+ Cjai (Uk&Aj + 5AAj l 1) = &(CjUk)diAj + (UkdkAj) + 5 (CjaiAAj - ACj&Aj) = ukdkmi + mkaiuk + -21 (CjaiAAj ACjaiAj).
&(<jjaiAj)= dtCjjdiAj
<j&
-
To finish the proof, we just have to observe that
By applying the Leray projection t o (6) we obtain that u solves
Finally, we should observe that in the case of no-viscosity, the proof of this theorem is a very short and elementary proof of the well known result by Arnold [l],and Ebin and Marsden [12], for the solutions of Euler's equation, as using the active vector A which is the inverse of the flow map instead of the flow map simplifies a lot the computations.
On a variational principle for the Navier-Stokes equation
97
3. Conservation of Energy Whereas the original variational problem in [14] does involve some explicit time dependence through the Brownian motion, the modified variational principle we study in this problem does not involve time explicitly. Therefore, by Noether’s theorem, there should be a conserved quantity, the energy, corresponding t o this symmetry. This is exactly the content of the next theorem. There should be also further symmetries, for instance associated with translation or rotation invariance, although we do not investigate this issue here, also note that this issue was not addressed yet for the original variational problem, although it is not clear that it will produce new identities.
Theorem 3.1 (Energy conservation). The energy
is a constant of motion for the modified Navier-Stokes equation ( 6 ) .
Proof. Although this conservation law can be derived directly, it is quite instructive to do the proof following the steps of Noether’s theorem. We have
1
-
-AA(s, t - h ) ) 2
is independent of h. Therefore, if we differentiate in h at h = 0 we obtain
+ ( C , a ~ t A + b ) t u . V A + u . V a t A -ZAatA). 1 Now observe that
DIOGOAGUIARGOMES
98
Furthermore, integrating by parts, we have
Since &A = -u
V A - $ A A we conclude that
is a constant of motion. Taking into account equation (5) we finish the proof. 0
Appendix. Navier-Stokes equation in magnetization variables In this appendix we review the magnetization form for the Navier-Stokes equation in R” for the velocity field u(z,t ) of an incompressible fluid:
ut
1 + (u . V ) u+ V p = ,Au,
divu
= 0,
(7)
with initial condition ult=O = uo. The variable p(z,t) is the pressure and is necessary t o impose the incompressibility condition div u = 0. For our purposes in this paper, it is convenient to rewrite (7) in new variables, the magnetization variables. These have been used t o study the Euler equation by several authors, namely Buttke [4],Oseledets [16], Russo and Smereka [20], among others. We will follow Chorin [7] in the summary of results we present next. The magnetization variable m is obtained by adding t o the velocity field u a gradient u=m+Vk. The scalar function k ( z , t ) is arbitrary at t = 0 and its evolution is chosen conveniently. This transformation is a change of gauge, of which there are several possible choices, as discussed in [20]. Clearly, from m one can compute u by using the Leray projection on the divergence free vector fields:
u = Pm.
On a variational principle for the Navier-Stokes equation
99
With an appropriate choice for k, the equation for the evolution of m is 1 & m i + u j D j m i + m 3. D a. u3 .-- 2 -Am.2 ‘
(8)
A main difference from (7) is that equation (8) does not involve pressure, nor d i v m = 0. Furthermore, to any solution of (8) with u = Pm, corresponds a solution u to (7). In t h e other direction, to any solution of (7) and initial value of k there exists a solution of (8) such that u = Pm for all times. Acknowledgments Supported in part by FCT/POCTI/FEDER and POCI/MAT/55745/2004.
References 1. V. Arnold, “Sur la gkomktrie diffkrentielle des groupes de Lie de dimension infinie et ses applications ? l’hydrodynamique i des fluides parfaits”, Ann. Inst. Fourier (Grenoble) 16(fasc. l ) , 319-361 (1966). 2. Barbara Busnello, Franc0 Flandoli,Marco Romito, “A probabilistic representation for the vorticity of a 3-dimensional viscous fluid and for general systems of parabolic equations”. 3. Barbara Busnello, “A probabilistic approach to the two-dimensional NavierStokes equations”, Ann. Probab. 27(4), 1750-1780 (1999). 4. Tomas F. Buttke, “Velocity methods: Lagrangian numerical methods which preserve the Hamiltonian structure of incompressible fluid flow”, in Vortex flows and related numerical methods (Grenoble, 1992), volume 395 of N A T O Adw. Sci. Inst. Ser. C M a t h . Phys. Sci.,Kluwer Acad. Publ., Dordrecht, 1993, pp. 39-57. 5. Ana Bela Cruzeiro, Fernanda Cipriano, preprint, 2005. 6. Alexandre Joel Chorin, “Numerical study of slightly viscous flow”, J . Fluid Mech. 57(4), 785-796 (1973). 7. Alexandre J. Chorin, Vorticity and Turbulence, volume 103 of Applied Mathematical Sciences. Springer-Verlag, New York, 1994. 8. P. Constantin, G. Iyer, “A stochastic Lagrangian representation of 3dimensional incompressible Navier-Stokes equations” (2005). 9. Peter Constantin, “An Eulerian-Lagrangian approach for incompressible fluids: local theory”, J. Amer. Math. SOC.14(2), 263-278 (electronic) (2001). 10. Peter Constantin, “An Eulerian-Lagrangian approach to the Navier-Stokes equations”, Comm. Math. Phys. 216(3), 663-686 (2001). 11. Peter Constantin, “Near identity transformations for the Navier-Stokes equations”, In Handbook of mathematical fluid dynamics vol. 11, North-Holland, Amsterdam, 2003, pp. 117-141. 12. David G. Ebin, Jerrold Marsden, “Groups of diffeomorphisms and the notion of an incompressible fluid”, Ann. of Math. (2) 92, 102-163 (1970).
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DIOGOAGUIARGOMES
13. Nassif Ghoussoub, “Anti-selfdual hamiltonians: Variational resolutions for navier-stokes and other nonlinear evolutions”, preprint. 14. Diogo Aguiar Gomes, “A variational formulation for the Navier-Stokes equation” Comm. Math. Phys. 257(1), 227-234 (2005). 15. Atsushi Inoue, Tadahisa Funaki, “On a new derivation of the Navier-Stokes equation”, Comm. Math. Phys. 65(1), 83-90 (1979). 16. V. I. Oseledets, “A new form of writing out the Navier-Stokes equation. Hamiltonian formalism” Uspekhi Mat. Nauk 44(3(267)), 169-170 (1989). 17. Charles S. Peskin, “A random-walk interpretation of the incompressible Navier-Stokes equations”, Gomm. Pure Appl. Math. 38(6), 845-852 (1985). 18. D. L. Rapoport, ‘Stochastic differential geometry and the random integration of the Navier-Stokes equations and the kinematic dynamo problem on smooth compact manifolds and Euclidean space”, Hadronic J . 23(6), 637-675 (2000). 19. D. L. Rapoport, “Random representations of viscous fluids and the passive magnetic fields transported on them”, Discrete Contin. Dynam. Systems (added volume), 327-336 (2001). Dynamical systems and differential equations (Kennesaw, GA, 2000). 20. Giovanni Russo, Peter Smereka, “Impulse formulation of the Euler equations: general properties and numerical methods”, J . Fluid Mech. 391, 189-209 (1999). DIOGOAGUIAR GOMES
Departamento de MatemAtica, Instituto Superior TBcnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal dgomes(9math.ist.utl.pt
CONVOLUTION CALCULUS ON WHITE NOISE SPACES AND FEYNMAN GRAPH REPRESENTATION OF GENERALIZED RENORMALIZATION FLOWS HANNOGOTTSCHALK ( Universitat Bonn), HABIBOUERDIANE ( Universite' d e Tunas El Manar), BOUBAKER SMII(Universite' de Tunis El Manar / Universitat Bonn) In this note we outline some novel connections between the following fields: 1) convolution calculus on white noise spaces, 2) pseudo-differential operators and LBvy processes on infinite dimensional spaces, 3) Feynman graph representations of convolution semigroups, 4) generalized renormalization group flows and 5) the thermodynamic limit of particle systems.
Convolution semigroups on infinite dimensional spaces are the mathematical backbone of the Wilson-Polchinski formulation of renormalization group flows [6]. There, a "time" (in this context one should say: scale) dependent infinite dimensional heat equation
vff
governs the flow of the effective action between two scales T, TO,where T is the scale on which the system is observed and TOis the cut-off scale. The "time" dependent infinite dimensional Laplacian A t , ~ o = g A t , ~ is o
with Gt,To= Gt - G T ~the covariance function of the Gaussian random field that has t o be integrated out t o intermediate between the random field with fundamental cut-off scale TOand covariance G T ~ and the random field with cut-off t and covariance Gt describing Gaussian fluctuations at some other scale t # TO(t > TOfor ultra violet and t < TOfor infra red problems). The functional derivative acts on a function F(q5) = $63") via
c,"=o(Fnl
&
b
-F($) M x )
00
=
Cn(D,F,,$@n-')l
D,F,(x2,..
.,x,) = Fn(X1221.. . ,%).
n=O
Here the kernel functions F, are assumed to be symmetric.
101
102
H. Gottschalk, H. Ouerdiane, B. Smii
The usefulness and mathematical beauty of the Wilson-Polchinski approach to renormalization is given by the following key features 1. The renormalization equation is the generating equation of a infinite dimensional diffusion process; 2. The perturbative solution of 1) can be represented as a sum over Feynman graphs. 3. Renormalization conditions at a scale T can be imposed by a change of the initial condition Vinitialat scale TOleading to a finite theory if the cut-off TOis removed.
In this note we outline the generalization of (1)replacing the infinite dimensional Laplacian by a pseudo-differential operator in infinite dimensions. We show that under this generalization the key features of the renormalization group approach all prevail: Instead of infinite dimensional diffusion processes one obtains jump-diffusion type LBvy processes and instead of classical Feynman graphs generalized Feynman graphs studied in [2]. The idea of renormalization as given in item 3 above then carries over unchanged. We will illustrate this for the special case of particle systems in the continuum. White noise analysis is a natural framework to rigorously formulate equations like (1) and their generalizations: In fact, following [ 3 ] ,we let S = S(Rd,@) be the space of rapidly decreasing test functions equipped with the Schwartz topology which is generated by an increasing sequence of Hilbert seminorms {l.l,},G~ and let S’ the dual of S. By S, we denote the closure of S w.r.t. ] . I p and by S-, the topological dual of S,. For 8 ( t ) , t 5 0, a Young function (nonnegative, continuous, convex and strictly increasing s.t. limt,me(t)/t = 00) we set e*(t)= supz>o(zt - e ( x ) ) ,the Legendre transform of 0, which is another Young function. Given a complex Banach space (B,II . [I), let H ( B ) denote the space of entire function on B , i.e., the space of continuous functions from B t o @, whose restriction to all affine lines of B are entire on @. Let Exp(B,0, rn) denote the space of all entire functions on B with exponential growth of order 0, and of finite type m > 0
In the following we consider the white noise test functions space
F~(s’) =
n
p r o , m>O
E ~ ~ ( s - ,e,, m).
(4)
White noise convolution calculus and Feynman graphs
103
Let Fo(S’)*, the space of white noise distributions, be the strong dual of the space Fo(S’) equipped with the projective limit topology. For any f E S and 0, the exponential ef : S’ -+ @, ef(q5) = e(4if) is in .Fo(S’). Thus, the Laplace transform C : Fo(S’)* x S -+ R, C ( Q , ) ( f )= (a,e f ) , is well defined. Recalling the definition of the space
GdS) =
u
-%4N,,O,m)
(5)
p 2 0 , m>O
which is equipped with the topology of the inductive limit, we get [3] that L : Fo(S’)*4 & * ( S ) is a topological isomorphism. Using the property of Young functions limt-.03 O*(t)/t = co it is easy t o see that Go* is an algebra under multiplication. Thus, for Q, Q, E Fo(S’)* one can define the convolution Q * Q, = L-l(C(Q)L(Q,)) as an element of Fo(S’)*. Let us assume for a moment that limt,mO(t)/t2 exists and is finite. Under this condition we have that Fo(S’) -+ L$(S&,dvo) -+ Fo(S’)* is a Gelfand triplet, where vo is the white noise measure, cf. [3]. Suppose that Q E .Fo(S’) has a Taylor series Q(q5) = xf=o(Qn7q56n) with Q n E Sgn. One can then show that Q * Q, E Fo(S’) and Q * a(+) = Q(-&)Q,(q5) with
Here the rule for the evaluation of the pseudo-differential operator a(&) is that first the n-th order differential operators are applied to Q, and then the result is summed up over all n. Hence we see that the equation
is the correct generalization of the renormalization flow equation (1). If now Vinitial Q,,T, : R+ -+ Fo(S’)* is continuous and eE Fo(S’)*, i t has been proven in [5] that (7) has an unique solution in F((,V-~).(S‘)*, namely
Again, the above solution is of particular interest if a probabilistic interpretation can be given. This is the case when C ( q t , T o ) for every t is a conditionally positive function, i.e. C ( Q t , T o ) ( 0 = ) 0, n
n
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H. Gottschalk, H. Ouerdiane, B. Smii
and the opposite inequality holds for t > TO.Under these conditions, by the Bochner-Minlos theorem, the transition kernel C-1( estTo m % T o ) d s ) is a family of probability measures on S' that fulfills the Chapman-Kolmogorov equations and thus defines a stochastic process with state space S'. In general, this process will be of jump-diffusion type, as it follows from the L6vy-It6 decomposition of conditionally positive definite functions. Let us now come t o the Feynman graph expansion. We take an initial condition of the type
p=o
with kernel (vertex) functions X(P) E S6P, for simplicity (this condition can clearly be relaxed). It is also assumed that Vinitia1(4)2 -C for some Vinitial C > 0 and all 4 E S'. Then, eE L2(S',dvo).If now e(t) fulfills limt-.oo e ( t ) / t 2= c < co,then by the theorem cited above, the solution of (7) exists. The next step is to expand (7) in a formal power series in Vinitial.At least in the case where, for t fixed, C-' (eJ;o L ( " ~ T o ) d s ) is a measure on S&, this expansion is an asymptotic series, cf. [2, Lemma 2.21. We note that the Laplace transform of a white noise distribution is an analytic function [3]. One can thus consider the Taylor series in f of eos; L ( \ i ! 3 * T o ) ( f ) d S at zero given by
for f E S with mn,t,To E (S')63n the n-th moment. The connected moment functions, mk,t,Toby definition are the Taylor coefficients of the logarithm of the generating functional of the moment functions, i.e.,
Jlb
The well-known linked cluster theorem, cf. e.g. [2, Appendix A], then gives the combinatorial relation between moments and connected moments, namely k
I€P(l,...,n ) I = l 1={11 ,...J k }
White noise convolution calculus and Feynman graphs
105
where P(1,.. . , n) is the set of all partitions of (1,. . . , n } into disjoint non empty subsets 11,. . . ,Ik, Ic E N arbitrary, 11 = { j t , .. . , j r ' } . After these preparations one obtains by straight forward calculation
K C n (s,q)€K
Here we used the following notation: R ( l , m ) ,. . . , ( m ,11,.. . , ( m , p r n ) )and for A
=
R(p1,. . . ,pm) = ((1,l),. . . ,
C R, A = {(sl,ql),. . . , ( s T , q T ) ) , .. . P(R\K) is the set of partitions of R \ K . mt,To(A) (c) = mT,t,To(z8:, (c) We also made use of the fact that (Vinitia1)rn(4), being a polynomial with test functions as coefficients, is a white noise test function and that the convolution between a white noise distribution @ and a white noise test function 0 is @ * 0(4) = @ ( 0 4 ) where O#~(cp) = 0 ( p - 4)is a shift. Generalized Feynman graphs can now be used to order the combinatorial sum on the r.h.s. of (14). A generalized (amputated) Feynman graph is a graph with three types of vertices, called inner full e, inner empty o and outer empty @ vertices, respectively. By definition full vertices are distinguishable and have distinguishable legs whereas empty vertices are non distinguishable and have non distinguishable legs. Outer empty vertices are met by one edge only. Edges are non directed and connect full and empty (inner and outer) vertices, but never connect two full or two empty vertices. The set of generalized Feynman graphs with m inner full vertices with p l , . . . ,pm the number of legs of the inner full vertices such that p j 5 p and X(PJ) # 0, j = 1,.. . ,m, is denoted by F(m).
106
H. Gottschalk, H. Ouerdiane, B. Smii
Figure 1. Construction of a generalized Feynman graph from the set K and the partition I = {zI,I z , 13)
To obtain the connection with (14) we consider an example where m = 3, p l = 4, p2 = 3 and p 3 = 4. For each element in f2 = R(4,3,4) we draw a point s.t. points belonging t o the same pi are drawn close together. Then
we choose a subset K and a partition I , cf. figure 1 (top). The generalized Feynman graph can now be obtained by representing each collection of points by a full inner vertex with pl legs, for each set Il in the partition we draw a inner empty vertex connected to the legs of the inner full vertices corresponding to the points in 11.Finally we draw an outer empty vertex connected t o the leg of an inner full vertex corresponding t o each point in K . We then obtain the generalized Feynman graph figure 1 (bottom). In this way, for fixed m, one obtains a one t o one correspondence between the index set of the sum in (14) and F(m). We want to calculate the contribution to (14) directly from the graph G E F(m)without the detour through the above one to one correspondence. This is accomplished by the following Feynman rules: Attribute a vector
White noise convolution calculus and Feynman graphs
x
107
E Rd to each leg of a inner full vertex. For each inner full vertex with p
legs multiply with X(P) evaluated a t the vectors attributed to the legs of that vertex. For any inner empty vertex with 1 legs, multiply with a connected moment function m&Toevaluated with the 1 arguments corresponding to the legs that this inner empty vertex is connected with. For each outer empty vertex multiply with -$(x) where x is the argument of the leg of the inner full vertex that the outer empty vertex is connected with. Finally integrate . . . dx over all the arguments z that have been used to label the legs of the inner full vertices. In this way one obtains the analytic value 1/"G](t,2'0, $). The perturbative solution of (7) then takes the form
SRd
where the identity is in the sense of formal power series. The linked cluster theorem for generalized Feynman graphs proven in [2,4] then implies that can be calculated as a sum over connected Feynman graphs
vff
-Vff(4)=
c7c O0
(-1y
VGI(t,570,4).
(16)
GEFc(m)
m=O
Let us now apply the above renormalization group scheme t o the problem of taking the thermodynamic limit of a particle system. To this aim let
where T is a probability measure on [-c,c], 0 < c < co, at(.) = a ( x / t ) where a is a continuously differentiable function with support in the unit ball and Va(0) = 0 such that a (0 ) = z > 0. ,PI&) at the same time is the Laplace transform of the Poisson measure pt representing a system of noninteracting, charged particles in the grand canonical ensemble with intensity measure (local density) at, see e.g. [l]. and a white noise distribution pt E .F@(S')* for any O (due t o the linear exponential growth of (17) in f we have that the r.h.s. is in G p ( S ) for any O ) , cf. [3]. Both objects can thus be identified. Furthermore we assume that ax) is monotonically decreasing in cy for cy > 0. Then d-t(x) = dat(x)/dt 2 0 for t > 0 , x E Rd. It is then standard to show that C ( @ , ) ( f )= -
// Wd
[-c,c]
(esf(l)- 1) d r ( s )bt(x)dx
(18)
108
H. Gottschalk, H. Ouerdiane, B. Smii
in fact fulfills (9) for all t < To. Thus the pseudo differential operator $(S/Sq5) is the generator of a jump-diffusion process with state space S’ with backward time direction. Let X(P) = X(P)(x1,.. . ,xp)be a set of C” functions that are of rapid decrease in the difference variables xi - xj, a # j . For a distribution q5 of compact support we define Vinitia1(q5) as in (10). We assume that Vinitia1(q5) > -C for each such 4. Let q5 E S’ have compact support. The non normalized correlation functional p t ( 4 ) of the particle system with infra-red cut-off t is defined as
Note that p~~ has support on the distributions supported on a ball of diameter TO,B T ~ .Therefore, for q5 fixed and t < TOone can replace XP(x1, . . . ,2,) with X(P)(x1, . . . ,x , > ~ ~ . , x ( xwith ~ ) x being a test function that is constantly one on supp q5 u B T ~without changing (19). Under this replacement, Vinitialmeets the conditions from above that X(P) E S6P. Vinitial E L2(S’,dvg)and Furthermore, under this replacement in Vinitial,evinitial E .Fo(S’)* if limt,, O(t)/t2 finite. Furthermore, one can hence eargue as above to see that 9t is in .Fo(S’)* for O arbitrary. Thereby, the requirements of white noise convolution calculus are met. We shall neglect the inessential distinction between Vinitialand its replaced version in the following. R o m (17)-( 19) we see that the non normalized correlation functional pt fulfills the renormalization group equation (7). The thermodynamic limit, which is achieved as TO4 00, is thus governed by this equation (and thus by a Lkvy process with infinitely dimensional state space). Let us now come to the issue of the normalization of p~ at a time T = 0, for simplicity. A normalized correlation functional should fulfill p~ (0) = 1. But -VGff(0) = logpT(0) N T t in our case where the divergent (as To 4 co) parts originate from the so-called vacuum to vacuum diagrams, i.e., such Feynman graphs in F,(rn)that do not have outer empty vertices [2, Thm. 6.61. All other contributions remain finite in the limit TO -+ 00 [2, Section 71. The normalization -V;ff(0) = 0 can now be achieved perturbatively by re-defining Vinitialby a counter term
and replacing A(’) by A(’) - bX,(0) since this removes properly the vacuum to vacuum diagrams at any order m of perturbation theory of Vffand other
White noise convolution calculus and Feynman graphs
109
diagrams give vanishing contribution for 4 = 0. The (perturbative) thermodynamic limit TO-+ 00 of p!j?‘($) can now be taken achieving at once the finiteness of the perturbation expansion and the normalization of ~‘“”(4)since V[G](T= O,To,4 ) converges as TO-+ 00 for G E F,(rn) not a vacuum t o vacuum diagram, cf. [2,Section 71. Of course, the above renormalization problem is rather trivial as the particle system only has short range p b o d y forces for p 5 p. But having put the problem of TD limits of particles in the continuum in the language of the (generalized) renormalization group now paves the way t o the use of typical renormalization techniques, as e.g. differential inequalities and inductive proofs of renormalizability order by order in perturbation theory [6],t o tackle less trivial problems in the thermodynamics of particle systems with long range forces.
Acknowledgements
H. Gottschalk has been supported through the German Research Council (DFG) project “Stochastic methods in Q F T ” , while B. Smii has been supported by the German Academic Exchange Service (DAAD); the authors gratefully acknowledge this. H. 0. would like t o thank S. Albeverio for for his warm hospitality and DAAD and DFG financial support when being in Bonn. The authors also thank Sergio Albeverio for his encouragement and interesting discussions.
References 1. S. Albeverio, H. Gottschalk, M. W. Yoshida, “Systems of classical particles in
2.
3. 4.
5. 6.
the grand canonical ensemble, scaling limits and quantum field theory”, Rev. Math. Phys. 17,no. 2, 356-369 (2005), arXiv:math-ph/0601021. S. H. Djah, H. Gottschalk, H. Ouerdiane, “Feynman graph representation of the perturbation series for general functional measures”, J. Funct. Anal. 227, 153-187 (2005), arXiv:math-ph/0408031. R. Gannoun, R. Hachaichi, H. Ouerdiane, A. Rezgui, “Un theBorkme de dualit6 entre espaces de fonctions holomorphes B croissance exponentielle”, Journ. Funct. Anal. 171,1-14 (2000). H. Gottschalk, B. Smii, H. Thaler, “The Feynman graph representation of convolution semigroups and applications to LBvy statistics”, Bonn preprint 2005, arXiv:math.PR/0601278. H. Ouerdiane, N. Privault, “Asymptotic estimates for white noise distributions”, C. R . Acad. Sci. Paris, Ser. I, 338,799-804 (2004). M. Salmhofer, Renormalization, Springer Verlag, Berlin/Heidelberg, 1999.
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HANNO GOTTSCHALK Institut fur angewandte Mathematik, Wegelerstr. 6, D-51373 Bonn, Germany gottscha0wiener.iam.uni-bortn.de HABIBOUERDIANE Dgpartement de Mathkmatique, Universite de Tunis El Manar, Campus Universitaire, TN-1006, Tunis
[email protected] BOUBAKER SMII Institut fur angewandte Mathematik, Wegelerstr. 6 , D-51373 Bonn, Germany boubaker0wiener.iam.uni-b011n.de
CHARACTERIZATIONS OF STANDARD NOISES AND APPLICATIONS TAKEYUKI HIDA(Meijo University, Nagoya), SI SI (Aichi Prefectural University, Aichi-ken)
1. Introduction
As a background of white noise analysis, we take two basic standard noises; one is Gaussian noise (white noise) and the other is Poisson noise. The second is immediately generalized to compound Poisson noise. Those noises are idealized elemental stochastic processes. The reason why we take those noises as basic object comes from our idea of stochastic analysis. Suppose we are given a complex random system to be analyzed, we shall proceed as follows. First form an elemental system of random variables that contain the same information as the given random system, i.e., Reduction. We are therefore given a system of independent random variables which can be taken to be the variables and the given random system is to be represented as a functional of those variables, i.e., Synthesis. Finally follows Analysis of the function in order to investigate the random system in question. With this plan, we first discuss elemental random systems, to fix the idea we take elemental noises. The basic noises are Gaussian or Poisson, as we can understand by the LBvy decomposition (or LBvy-It6 decomposition) of a L6vy process. It leads us to find out characteristic properties of them for detailed investigation of the random system expressed as functions of noises. 2. Invariance of the noises
Following the idea of reduction, we take two standard systems of idealized elemental random variables. Such a system should consist of i.i.d. (independent identically distributed) and be parameterized by the time variable t . As is suggested by the book by Gel’fand-Vilenkin [5],it is natural to consider a (Gaussian) white noise and Poisson noise as elemental noises. They are stationary, generalized stochastic process. They have flat spectrum, so that they are called white noise. Customary, Gaussian white noise is called
111
TAKEYUKI HIDA.SI SI
112
simply white noise and Poisson type elemental system is called Poisson
noise. 2.1. Two noises The two noises may intuitively be defined as time derivatives of a Brownian motion B ( t ) and a Poisson process P ( t ) , respectively. Sample functions of their time derivatives are no more ordinary functions, but generalized functions. Their probability distributions are therefore introduced in the space of generalized functions, denoted by E * , which is a dual space of some nuclear space E ( C L 2 ( R ' ) ) . Given B ( t ) and P ( t ) ,their characteristic functionals are easily computed as follows: CG(E)= exp
and
Cp(<) = exp
[-; [x
llCIl2]
, E E E, - 1)dt] .
2.2. (Gaussian) white noise
We start with the characteristic functional CG(E).So there is a probability measure p on a measurable space ( E * ,B), where E* is a space of generalized functions and where B is a sigma-field generated by cylinder subsets of E*, in such a way that
To describe invariance of p , we must introduce the infinite dimensional rotation group. Take the nuclear space E as before and define a rotation g of E t o be i) a linear isomorphism of E and ii) orthogonal : 11ge11 = IIEll, where 1) . 11 is the L2(R')-norm. The collection of rotations forms a group which is called the infinite dimensional rotation group and is denoted by O ( E ) . We introduce the compact-open topology to O ( E )to let it be a topological group. The collection of adjoint transformations g* of g E O ( E ) forms a group denoted by O * ( E * ) .The correspondence 9
+
(g*)-l
Characterizations of standard noises and applications
113
is bijective, so that we can topologize O*(E*)to be a topological group isomorphic t o O ( E ) . The transformation g* is B-measurable, so that the product g*p is defined. The following assertions (Proposition 2.1 and Proposition 2.2) are known. however we list to remind them. Proposition 2.1. The white noise measure p is O*(E*)-invariant. Namely, for any g* in O * ( E * ) it holds that
This is the reason why the infinite dimensional rotation group plays an important role in white noise analysis. In fact, by the use of the rotation group we can carry on an infinite dimensional harmonic analysis. This is one of the big advantages of our analysis. A converse direction of Proposition 2.1 is significant. To claim the assertion, we provide a notation. The probability measure determined by the characteristic functional e~p[-gll[11~]is denoted by p, and is called a Gaussian measure with variance c2. Proposition 2.2. The probability measure v determined by a characteristic functional C(€J which is O(E)-invariant, i.e., C(g€J = C(<)f o r every g in O(E), then u is expressed in the form u = a60
+
(2)
where a 2 0 and dm is a bounded Bore1 measure on (0,co). If we further assume that u is non-trivial and atomic (or O*(E*)-ergodic), we are given a single Gaussian measure p,, that is to be an elemental noise. Our infinite dimensional rotation group O ( E ) involves two important families of subgroups. To define the first family, we use a complete orthonormal system, say {&}. We require that each &, is a member of E l otherwise quite arbitrary. With the help of the tn,we can define, in an obvious manner, a subgroup Gn isomorphic to SO(n). Also, their inductive limit G, can be defined. It is noted that this family contains much bigger subgroups like the LBvy group which involves essentially infinite dimensional rotations (see e.g. [lo, chapter 51). Another family comes from a group of diffeomorphisms acting on the parameter space R' . This class consists of one-parameter subgroups, called
TAKEYUKI HIDA,SI SI
114
whiskers which describe profound probabilistic properties by using geometric structure of the parameter space. There is, among others, an interesting subgroup, denoted by C(1), which is generated by whiskers and is isomorphic t o the conformal group acting on R1. It illustrates latent traits of white noise; e.g. it shows the projective invariance of Brownian motion. In the higher dimensional parameter case, say Rd, the family is much larger, in particular, we can define the conformal group C ( d ) which is locally isomorphic to S O ( d 1 , l ) . (For detail see [lo].) With this review, given as a background, we come t o a generalization.
+
2.5’. Two-dimensional valued white noise.
Having been suggested by S. Albeverio and A. Sengupta [3], we discuss two-dimensional valued white noise and applications to path integral. Assume that the collection of (z, y)’s is the product space Ej x Ey* of the spaces E,” and E; consisting of generalized functions, respectively. The product measure v ( d z ) = p x ( d z ) x p y ( d y ) , with d z = d x A d y is introduced to E,” x E t . It is obvious that
Proposition 2.3. T h e probability measure v ( d x d y ) is invariant under the rotations acting o n (x,y)-space. In particular, under the transformation
z
= (z, y)
--$
.(
+ y , z - y ) / f i = (z’, y’)
the measure v ( d z ) is kept invariant. We may understand that the renormalized inner product (z,y), z E E,”,y E Ey* is defined below:
+
JJ’:
where IIxII = ~ ( t: dt. ) ~We can see that (x g) and (z - y) are independent under the Gaussian measure v. Hence, there is no difficulty to deal with generalized functionals of (z, y ) . As an example, the functional eic(xJ’)can be reduced to the product of independent Gauss kernels of the form ’p1(.’)
= Nexp[c/
and the result is of the form
: z’(t)2: dt] ,
’pl(z’)‘p2(y’).
Characterizations of standard noises and applications
115
Thus, we have the following theorem. Although the assertion seems obvious, some background is necessary, as we have seen above. Theorem 2.1. The integral
is well defined if 9 is a test functional of z . There are various publications in this direction by many authors; for instance, S. Albeverio, A. Hahn and A. Sengupta [2] and others. It is our hope that interesting developments will be obtained in line with white noise analysis. It is easy t o generalize the results to higher dimensional valued white noise. Take a tensor product El @ . . . @ En of spaces E3,each of which is a copy of a nuclear space E . A product Gaussian measure v is introduced on thespace ET@...@E;. Wedefinethedirect product O * ( E T ) x . . . x O * ( E Z ) under which v is invariant. Theorem 2.2. The direct product group O*(ET)x ... x O*(EZ) and the rotation group o n E; @ . . . @ E; give invariances of the measure v.
2.4. Gauge transformations The infinite dimensional rotation group O ( E ) can be complexified t o have the infinite dimensional unitary group U(E,), where E, is the complexification of the nuclear space E. Many interesting subgroups have been discovered so far (see [7], [lo]). This note extends the Heisenberg group and shows an invariance of complex white noise. The Heisenberg group that we have introduced in [7] involves three oneparameter groups: Let be in E,.
<
a. The gauge transformation It is defined by
It : <(u)--+ (It<)(.)
= eit<(u).
b. The shift St is st :
c. Multiplication
7rt
<(u)
+
(St<)(.)
= <(u- t ) .
is such a transformation that 7rt
: <(u)4 (7rt<)(u) = eiut<(u).
TAKEYUKI HIDA,SI SI
116
We now define Iat by
where a is a member of the (real) Schwartz space S. d. The collection {Ia,,,a E S } forms a subgroup of U(E,) and is called S-gauge transformation group. (It is abelian in Physics.) We now list the infinitesimal generators of the whiskers in a, b, c, and d.
ill
d du'
s = --
7r
= iu.,
ia E
s.
The following theorem can be proved without any difficulty.
Theorem 2.3.
i) T h e generators listed above f o r m a Lie algebra. ii) T h e S-gauge transformations act o n z = (x,y)-space as rotations, and Y measure is invariant under those rotations. The assertion ii) is a generalization of what we have discussed in section 2.3.
3. Poisson noise We shall discuss characteristics of Poisson noise, although they are latent, by comparing with the characteristics of Gaussian noise (white noise). The Gaussian case has been discussed in [ 8 ] .The Poisson case can be discussed in rather similar manner, but somewhat different to be interesting. A general setup is this. We follow [15]. The characteristic functional of Poisson noise is
C p ( J )= exp
[.k ( e i c ( t )
- l)dt],
J E E.
With this in mind we wish t o discover latent characteristics of Poisson noise. Let a Poisson process P ( t , w ) , w E R(P), be given. We restrict our attention t o the unit time interval [ 0 , 1 ] .Setting
A,
=
{ w , P ( l , w ) = n},
we are given a measurable partition { A n ,n 2 0, } of R ( P )
UA,
= 0,
A, n A ,
=
0, n # m a.s.
Characterizations of standard noises and applications
117
We have
Let T ~ ( w be ) the j - t h jump point of P ( t , w ) , t E [0,1]. If w is restricted to A,, there are exactly n jump points, 7 1 , . . . ,T,. Those jump points are distributed independently on the unit time interval. The joint probability density p ( z 1 , . . . ,z,) is invariant under the permutation of the variables z~j. Obvious assertion is given in other words. Proposition 3.1. Let P, be the conditional probability measure o n A,. The conditional joint probability density p ( z 1 , . . . ,x,) o f j u m p points i s invariant under the symmetric group S ( n ) that acts as permutation group of variables.
We can further show another characteristic of Poisson noise by restricting the sample space t o be A,. If we may consider n discrete random points that are distributed over the unit time interval, in general, t o form a point process, the case of Poisson noise is the optimal case where the system of those n points enjoys maximum entropy. In this sense Poisson noise has a latent optimality in the sense of information theory. The converse direction of the study is as follows. Indeed, this is more interesting part. Start with a partition { A n , n 2 0) of IR. The probability P(A,), which is non zero, will be determined later. Introduce random variables on A, as many as n, denoted by 7 1 , . . . ,7,. Let them be distributed over the unit time interval. Now consider the case where 1) the joint probability density q ( z 1 , . . . , 2), of the random vector (71,. . . ,7,) is invariant under the symmetric group S ( n ) acting as permutation of the component of the vector, and 2) the random vector has maximum entropy. Proposition 3.2. Under the assumptions 1) and 2), the joint distribution q is in agreement with the p introduced above.
We are now given a characterization of Poisson noise provided that the probability P(A,) is determined as in (3). There are many reasonable ways to determine these probabilities like in [15], however we do not go into details.
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TAKEYUKI HIDA,SI SI
Finally we come to the conclusion to observe that the conditional characteristic functional C:([) is expressed in the form
and the weighted sum, by using the values of the P(A,)'s, is
The sum (4) is the characteristic functional of Poisson noise with parameter space [0, I]. In this way, a characterization, using latent properties, of Poisson noise has been given.
4. Application to information sociology
There are various applications of the characterization of Gaussian and Poisson noises discussed in this paper. Among others, we consider an application to information sociology, where stable distribution appears. Then, we may think that the data may be a distribution of a random variable of a stable process evaluated at a fixed instant. Namely, one should check that the random phenomena in question can be embedded in a stable stochastic process of some exponent a. Suppose this is confirmed in some way. It is a superposition of Poisson processes with various magnitudes of jump. Anyhow elemental process is a Poisson process. According to our discussion, one can speak of latent characteristics such as optimality and symmetry. It is interesting t o find that those characteristic properties correspond to the properties of the actual phenomena Suppose we are given observed data, the cumulative distribution function will be close to a stable distribution. At the same time one can think of the weight to have a superposition of Poisson noises, i.e., the LBvy measure, in order t o find some relation to the structure of random phenomena. The LBvy measure for a stable process with exponent Q is proportional to lul-("+')which is the weight associated to the Poisson noise with magnitude u of jump. This may tell us some meaning of the phenomena.
Characterizations of standard noises and applications
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References 1. L. Accardi et al., eds., Selected papers of Takeyuki Hida, World Sci. Pub. Co., 2001. 2. S. Albeverio, A. Hahn, A. Sengupta, “Chern-Simons theory, Hida distributions, and state models”, Infinite Dim. Analysis, Quantum Probab. and Related Topics 6,65-81 (2003). 3. S. Albeverio, A. Sengupta, “The Chern-Simon functional integral as an infinite distribution”, Nonlinear Analysis, Theory, Methods and Applications 30, 329-335 (1997). 4. A. Borodin, G. Olshanski, “Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes”, Ann. Math. 161, 13291422 (2005). 5. I. M. Gel’fand, N. Ya. Vilenkin, Generalized Functions, vol. 4, Applications of Harmonic Analysis, Academic Analysis, Academic Press, 1964. Russian original, Moscow, 1961. 6. T. Hida, Stationary Stochastic Processes, Princeton Univ. Press, 1970. 7. T. Hida, Brownian motion, Springer-Verlag. 1980 (original Japanese edition, 1975). 8. T. Hida, H. Nomoto, “Gaussian measure on the projective limit of spheres”, Proc. Japan Acad. 40, 301-304 (1964). 9. T. Hida, Si Si, Innovation Approach to Random Fields: A n Application of White Noise Theory, World Sci., 2004. 10. T. Hida, Si Si, Lectures on White Noise Functionals, World Sci. Pub. Co., 2006. 11. P. Lbvy, Processus stochastiques et mouvement brownien, Gauthier-Villars, 1948. 2lime ed., 1965. 12. N. Obata, “Certain unitary representations of the symmetric group I ” , “II”, Nagoya Math. J. 105,109-119 (1987); ibid. 106,143-162 (1987). 13. I. Ojima, “Macro-Micro duality in quantum physics”, Stochastic Analysis: Classical and Quantum. Perspectives of White Noise Theory, ed. T. Hida, World Sci. Pub. Co., 2005, pp. 143-161. 14. S. Rohde, 0. Schramm, “Basic properties of SLE”, Ann. Math. 161,883-924 (2005). 15. Si Si, “Effective determination of Poisson noise”, Infinite Dim. Analysis and Quantum Probab. and Related Topics 6,609-617 (2003). 16. Si Si, “Note on Poisson noise”, Quantum Information and Complexity, eds. Hida, SaitB and Si Si, World Sci., 2004, pp. 411-425. 17. Si Si, A . Tsoi, Win Win Htay, “Invariance of Poisson noise”, Stochastic Analysis: Classical and Quantum, ed. T.Hida, World Sci. Pub Co., 2005, pp. 199210. TAKEYUKI HIDA Meijo University, Nagoya, Japan SI SI
Aichi Prefectural University, Nagakute, Aichi-ken, Japan
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ANALYSIS OF STABLE WHITE NOISE FUNCTIONALS YUH-JIA LEE* (National University of Kaohsiung), HSIN-HUNG S H I H(~K u n Shan University, Tainan) The white noise analysis for L6vy processes developed in our previous paper [7] is extended to stable processes of order a: E ( 0 , 2 ) . In the course of our investigation, the Segal-Bargmann transform for stable processes is introduced.
1. Introduction
Let X = { X ( t ) : t E IK} be a LBvy process, which is an additive process with stationary increments on a probability space ( R , F , P ) with X ( 0 ) = 0 almost surely. By the well-known LBvy-Khintchine formula,
for r E IK and -co < s < t < +co, where f x is called the Lkvy function of X which is uniquely expressed by
where p is a real constant, and P is a positive finite measure on IK with uz = P ( ( 0 ) ) . Conversely, for a given triple ( p , CT, P), where p,CT E R and ,8 is a positive finite measure on R, there exists a LBvy process X such that the equality (1) holds. X is called a LBvy process with the generating triple ( p , a , P ) and the measure dPo(u) = (1 u 2 ) / u 2 d P ( u )on R \ (0) is called the L6vy measure of X . In our previous paper [7], under the assumption that P has finite absolute second moments, we have shown that the Lkvy white noise measure A exists and defined as a Bore1 measure on the space S' of tempered distributions. Then the analysis of LBvy white noise functional were studied based on the following representation of a Lkvy process X ( t ) :
+
*Research supported by the National Science Council of Taiwan +Research supported by the National Science Council of Taiwan
121
YUH-JIALEE, HSIN-HUNGSHIH
122
Under the above assumption on the moments of p, Gaussian, Poisson, Gamma processes and Meixner are included in our investigation, but the stable processes with characteristic exponent a E (0,2) are excluded since, in this case, the corresponding measure p has no second moments. Thus we wish to extend the theory developed in [7] to all a-stable processes. Let 0 < a < 2. A LQvyprocess X = {X(t) : t E R} with X(0) = 0 8.5. is called an a-stable process on a probability space (R, 3,P ) if its LBvy function has the following representation: There are real constants p , c1, c2 with c1, c2 2 0 and c1 c2 > 0 such that for any r E R,
+
f x ( r )=
)
i p r + S (eiur - 1 - iur 1 +u2
(CI
+
l ( - w , ~ ) ( u )cz l ( ~ , + ~ ) ( u ) ) I u Idu. -~-~
I4>0
In other words, a stable process is a LBvy process with given by Po(u) =
(C11(-m,0)(4
CT
=0
and
0 being
+ c2 l ( 0 , + m ) ( 4 ) 1 4 - 1 - a .
(3)
Lemma 1.1 ( [ l l ,Theorem 14.151). Let 0 < a < 2 and X be an a-stable process o n (R, 3,P ) . Then there are constants c, p with c > 0 and p E [-1,1] so that
r
fx(r)=ipr-c\rla
(4)
1
where w ( r ,a ) = tan ?f for a # 1 and w(r, a) = In Irl for a = 1. Here, p , c , and p are uniquely determined by P o X(l)-'. Conversely, for any c > 0 and p E [-1, 11 and p E R,there is an a-stable process X with f x satisfying (4). 2. a-stable white noise functionals
By Lemma 1.1, there is a constant
Ifx(r)I I
i
K:
> 0 such that for each T E R,
if a # 1, (Irl + Irla), ~ ( I r l ~ r l n l r ~ ~if) a, = 1. K:
+
Let 0 be the class given by 0 = L1 fl L a @ , m ) if a
# 1, and
(5)
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123
if a = 1, where m is Lebesgue measure on W. L1 n L"(W, m ) ,0 regarded as a metric space with the metric d, defined by
< a < 2, is
I4 - 4 E t t W , m ) } . By (5) fx(q5) is Lebesgue integrable for each 4 E 0. Lemma 2.1. Let &, n E N,and 4 be elements in the class 0 which satisfy d"(47+)
= m={I4
-
?h1(W,rn),
(i) {&} converges in measure t o q5 (with respect to m); and (22)
(222)
J?:
JLE (I4n(t)I + 14n(t)Ia) dt = J-'," ( I + ( ~ )+ I 14(t>1") dt for a # 1; n-cc lim J-'," I4n(t)I(1+ IlnI4n(t)II)dt=J?: 14(t)1(1+ 11n14(t)11>dt for a = 1.
Proof. (Sketch) -
-
For any continuous function g on W, it is not hard t o prove that {g(&)} converges in measure to g ( 4 ) . Applying the preceding result to g(t) = Itl" for 0 < a < 2, a # 1 and g(t) = Itlnltll for a = 1, the estimation (5) and the Lebesgue dominated convergence theorem, we obtain immediately +cc
S_,
+cc
fx(4n(t))dt
-+
S_,
fx(4(t))dt.
Let (., .) stand for the S'-S pairing and observe that S the complex-valued functional CX on S by
C X ( ~= ) exp
(1:
fx(v(t)) dt)
7
77 E S.
c 0. Consider (6)
Clearly C(0) = 1 and CX is a continuous positive-definite functional on the space S (by [l, p. 2781). It follows from Minlos's theorem, there exists a unique probability measure A on (S',B ( S ' ) ) ,such that CX
(77) =
L,
ei (297)A (dIz:) I
rl E Sl
where B(S') is the Bore1 field which is exactly the a-field generated by all cylinder sets. A will be called the a-stable white noise measure.
YUH-JIALEE, HSIN-HUNG SHIH
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Theorem 2.1. [9] Let 4 be an element in the class 0 . Then there is a of elements of the space S so that ( i ) 4n converges to 4 sequence (4,) in L1 n L a ( R , m ) ,and (ii) {(.,&)} converges in probability to a random variable, denoted by (., 4), with lE[eiT('>@) 1 = exp(S?E f x ( r 4 ( t ) ) d t ) ,r E
R. Remark 2.1. 1. In Theorem 2.1, 4,s' can be chosen so that 4, E Ic, the space of infinitely differentiable functions with compact supports. 2. It 4 E S , then (.,4) = (.,$) [A]-a.e..
Corollary 2.1. For each 4 E 0 with # 0 [m]-a.e., the random variable (., 4) has an a-stable distribution, where the corresponding L6vy function has the f o r m as given in (4) associated with the triple ( p , c , p ) of constants by
or
according to a
# 1 or a = 1.
Proof. Applying (4) to ward.
s_'," f x ( r $ ( t ) )d t , r E R, the proof is straightfor0
Note that, for any bounded interval A , the indicator function 1~ of A belongs t o the class 0. By Corollary 2.1, the a-stable process X on (S',B(S'),A) can be represented by:
X ( t ; x) =
i
(x,l [ O , t ] ) , -(x, l I t , O ] ) ,
ift if t
2 0, < 0, IC E S'.
P r o p o s i t i o n 2.1. Let {a,},{b,} be two sequences of real numbers with Then f o r each 4 E S , a, \ -ca and b, /" +a.
.>-
J," 4(t) d X ( t ;
(x,4)
for [A]-almost all x E S' and the above integral is understood to be a
Riemann-Stieltjes integral.
Analysis of stable white noise functionals
125
It follows from Proposition 2.1 that we may formally interpret the integral 4 4 4(t) dt by
La
+a
From the above interpretation we regard elements of S’ as the sample path so that X ( t ;z) = z(t). Members of L2(S’,A) are referred as squareintegrable a-stable white noise functionals. R e m a r k 2.2. If
4
E 0
\ S,
we define
s_’,” 4 ( t ) d X ( t )as the limit of
convergence in probability of s_’,” &(t)d X ( t ) , n E N,where {&} c K is a sequence satisfying the conditions (i) and (ii) stated in Theorem 2.1.
3. Chaos decomposition of a-stable white noise functionals Let Bb(R:) be the class of all bounded Bore1 subsets E of
\ { ( t ,0) :
lR:
= R2
t E R}, away from the t-axis. For E E: Bb(R:), let N ( E ;.) be a random variable on (S’,B(S’)) defined by N ( E ; z )= I{(t,u)E E : X ( t ; z )- X ( t - ; z ) = u}I, z E S‘. Then N ( E ;z), E E Bb(R:) and z E S’, is a Poisson random measure with the intensity measure v , where d y ( t , u ) = (c1 l(-m,O)(u)
+ c2 l ( O , + a ) ( ~ ) ) l ~ I - l - a
dudt.
Let
where dNo(t,u) = d N ( t , u ) - dv(t,u). Then M ( E ) , E E independent random measure with zero mean and
a(@),is
an
lE[M(E)M ( F ) ]= X(E fl F ) for any E , F E a(@),where d X ( t , u ) = u2dv(t,u). Let In(gn), gn E Lz((R:)n, be the nth order multiple integral of the kernel function gn with respect to M , where L2((lR:)n,XBn) denotes the space of symmetric h
I
YUH-JIALEE, HSIN-HUNG SHIH
126
complex-valued L2-functions on (Rq)n with respect t o XBn. Then we have the isometry
The following chaos decomposition theorem is in fact a reformulation of [3, Theorem 21, due to K. It& on the probability space (S’,B(S’),h)for stable processes.
Theorem 3.1. For any cp E L2(S’,A), there exists a sequence of kernel functions q$, E LZ((R:)”, X B n ) , n E W u { 0 } , such that cp can be eqressed uniquely as an orthogonal direct sum h
cc n=O
of In(&), n = 1 , 2 , .. . . I n notation, we write cp
N
(&).
The well-known LBvy-It6 decomposition theorem for L6vy processes (see [11, Theorem 19.21) asserts that there is a set A E f?(S’)with R ( A ) = 1 such that for any x E A and b > a,
X ( b ;X ) - X ( a ;X) = p(b-a)
+ n-cc lim
U
1 +u2
where the limit is uniformly convergent in a, b on any bounded interval. In the case of stable processes, we have
X ( b ) - X ( U )=
Denote by 7,
=p
+
(c2 - c1)
x‘ lz
-d u , n 2 2,
and
T
= lim n’cc
7,.
(8)
We note that T exists if and only if 1 < Q < 2. Unlike the cases in [7], X ( b ) - X ( a ) - T ( b - a ) is not a member of L2(S’,A) even though T exists. It will be defined as a generalized stable white noise functional (see Section 7).
Analysis of stable white noise functionals
127
4. The Segal-Bargmann transform of square-integrable a-stable
white noise functionals For an arbitrarily given ‘p E L 2 ( S ’ , A ) ,the Segal-Bargmann (or the S-) transform S’p of ‘p is a complex-valued functional on L:(Rz, A) defined by
where &x(g) = C:=, sociated with g. If ‘p
-$
In(gBn), called the coherent state functionals as(&), then
In [6],we derived a closed form of the Segal-Bargmann transform S’p(g) for ‘p E L 2 ( S ’ , A ) ,and g belonging to some dense subset of Lz(Rz,A). There measure O, in [6] was assumed to satisfy the moment condition. If one carefully goes through the proof one would found that the moment condition is superfluous except some minor change. The major difference is that some regularity properties, such as the analyticity of characteristic functionals and square-integrability of cylinder functions are no longer hold. For the sake of clarity we sketch the proof by showing the key steps as follows. Let 4 be the class of all bounded B(Rq)-measurable complex-valued functions which are supported on a compact set of Rq. Then, for any g E 4, we have
where g*(t,u)= ug(t,u)for (t,u)E Rq. For any complex-valued B(Rq)-measurable function g, let
Then, by making use of the integral formula ( l o ) , we have
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128
Theorem 4.1. [lo] Let g E 4 and gn E G((RB)n,A@n). Then
n
h
Corollary 4.1. [lo] Let gn E L:((RB)", A@'"). T h e n , for a n y q5 E 0 ,
n n
x
@ i ~ ~ l e , ( t j , u j ) d X ( t l , .u. l. )d ~ ( t n , u n ) .
j=1
Corollary 4.2. [lo] T h e class of all functions subset of Lz(RB, A).
@i,,g,lR*,
7 E S , is a total
Combining Corollary 4.1 with (9) we immediately have
Proposition 4.1. For a n y cp E L2(S',A) and 4 E 0,
cp(x)e i ( z i 4 )A(&)
= IE[ei('94)] . S'cp(@i4@lR*).
S,l
Let Q be the closed subspace {g E Lz(Rq, A);g*E LA(@, v)} of Lz(Ra, A). Let g E Q be fixed. By [6, Proposition 2.11 there is a set A, E B ( S ' ) with A(A,) = 1 such that Ig*(t,jx(t;x))l< +cc
for any z E A,,
tER
w h e r e j x ( t ; z ) = X ( t ; z ) - X ( t - ; x ) , t E R. It implies that l + g * ( t , j x ( t ; x ) ) # 0 except only for finitely many t E R and the infinite product (1 t g*(t,jx (t;x)) is absolutely convergent. Define a functional Tx (9) associated with g by
n,,,
Analysis of stable white noise functionals
129
Then 'Y'x(g), g E 9, is an entire function belonging to L2(S',A). Moreover, for any x E S',
and
g E 9.
5 . Test and generalized functionals
From now on, the parameter cl,c2 in (3) will be always assumed t o be C1,CZ > 0.
A Gel'fand triple on L2(R~,X) Let A = -d2/dt2 (1 t 2 ) be a densely defined self-adjoint operator on L2(R,dt) and {hn;n E NO} be eigenfunctions of A with corresponding eigenvalues 2n 2, n E No(= N U {0}), where h,'s are Hermite functions on R. Then {hn;n E No} forms a complete orthonormal basis (CONS, in abbreviation) of L2(R,m). For any p E R and 7 E L2(R,m ) ,define (71, := I A P q l ~ z ( a and , ~ ) let S, be the completion of the class (7 E L2(R,m ) ; 171, < +m} with respect to I . Ip-norm. Then Sp is a real separable Hilbert space and we have the continuous inclusions:
+ +
+
s= l@S,
c s,c s,c L2(R,m) c s-,c s-,c S'
= 12s-,, P>O
P>O
p>q>O, and
s c P ( R , m )c S' forms a Gel'fand triple. Next, consider the real Hilbert space L2(R*,y), where dy(u) = w ( u )du with
w(u) =
(C11(-00,0)(4
+ c2 ~ ( o , + o o ) ( ~ ) ) 1 4 1 - - c r ,
21
E R*.
YUH-JIALEE, HSIN-HUNGSHIH
130
en(.)
Let = h n ( u ) / m ,u E R,, n E No.Then { e n ; n E NO}is a CONS of L2(R,, 7 ) . Define a linear operator A , densely defined on L2(R,, y) by Aacn = (271 2)cn for n E No and for each p 2 0, let Ep be the set of all 4 E L2(R,,y) with l A z c / L 2 ( R * , 7 )< +m, which is a real separable Hilbert space with the inner product (., .), given by
+
We use the notation I ., , , I to denote the induced norm by (., .),. Denote by E-, the dual of E,. Then E-, is isometrically isomorphic to the completion of L2(R,,y) with the inner product (., .)-,, and 1 . I-,,,-norm by
Let E = l@p,o &., and
Then & is a nuclear space with the dual E' =
E
E-,
c L2(R,,y) c E'
also forms a Gel'fand triple. There is a connection between EP and S,, p 2 0, as follows.
Proposition 5.1. Let 4 E L2(R,,y). Then for each p 2 0, 4 E Ep if and only if there is an element qb of S, such that fi'4 = qb, [m]-a.e.o n R,. Moreover, f o r a sequence {&} in E,, 4n -+ 4 in &, if and only if q b n + qb in S,. Corollary 5.1. Let f E &-, such that for every 4 E E ,
p
> 0. Then there is an element qf of S-,
(f,4 ) E ' , E
=
(Vf
7
774).
Moreover, i f f is a regular distribution defined b y a B(R,)-measurable function, still denoted by f , that is, (f,$ ) E ~ , E = f ( u )4(u)d y ( u ) , then qf = J?J. f , [m]-a.e. o n R,.
sw,
Remark 5.1. A crucial difference between the above argument and [7] is that in [7] we can apply the Gram-Schmidt orthogonalization method to { 1 , u , u 2 , .. .} to obtain a CONS of L 2 ( R , , y ) as in [7]. However in the case of a stable process, none of any polynomial functions p ( u ) , u E R,, are square-integrable with respect t o y without the moment condition so that we can not apply the Gram-Schmidt orthogonalization method anymore.
Analysis of stable white noise functionals
131
Example 5.1.
1. All polynomial functions p ( u ) , u E R,, are regular elements in E’. In fact, let {&} c & such that 4, -+ q5 in &. Then, by Proposition 5.1, ~ b , ,-+ q b in S and
I
Vbn (u) P(’u.)I =(da(-cO,0)(4
+ Jc21(O,+cO,(U))lrlbn(4P(u)llUI+-$
(fi+ 6) Irl+,(u)p(u)l Iul+-%
.{ <(
if 0
< IuI < 1,
( ~ + ~ ) 1 U 2 r l b n ( U ) P ( u ) I I u I - q - q ,if14
11,
ul < 1, c . lul+-?, if 0 < l c . l u l 3- ~ - ~if, lul 2 I,
where c > 0 is a constant not depending on n. By the dominated convergence theorem,
and hence p ( u ) E &’. 2. S, E E’ for any u E
R,.
Next, for p E R, denote by Np the Hilbert space tensor product S, @ &, with 1. Ip-norm defined by I h, @ 1, = lhnlp and let N = S @ E with the dual N’ = S’ @ E’. Then Af c L2(R:, A) c N’ forms a Gel’fand triple and we have a continuous chain N c N, c N, c L2(R:,A) c N-, C N-, c N’, p > q 2 0. In the following, we denote
cm
lcmlp,a,
where denotes the symmetric tensor product. Also, we relabel the CONS { h , @ Cm; n,m E NO} by { f o , f i , . . .}, fo = ho @ 6, and let X j = I ( A @ A,) f j lofor any j E NO. Applying Proposition 5.1 and Corollary 5.1 we have the following Proposition 5.2.
( i ) N, is the space consisting of all functions g on R: of the form given by d t , u )= T g ( t , u, ( t ,u)E R?,
a’
132
YUH-JIALEE, HSIN-HUNGSHIH
where Tg E S, 8 S,. Moreover, f o r a sequence {gn} and g in N,, gn + g in N,if and only if Tgn Tg in S, 8 S,. (ii) Let F E N,'. There is an element TF in S L 8 S; such that for every 9 E Nc, ( F , g)N;,Nc = ( T F , T g ) s ~ @ z , s ~ z . --f
Moreover, i f F is regular, then TF(t,U)
=m . F ( t , u ) ,
[mB2]-a.e.( t , u )E Wq.
(iii) Let 2) be the class consisting of all complex-valued functions g ( t , u ) , ( t ,u ) E IRq, in S, 8 S,, which have compact supports away from the t-axis. Then D is dense in N,. Remark 5.2. 1. For each g E N,, the function J w , T g ( t , u ) a d u , t E IR, is an element in S,. 2. Each element f in SL can be identified with an element in N,' by regarding f as f 8 lw,. More precisely, for each g E N,,
Construction of test and generalized a-stable white noise functionals Spaces of test and generalized a-stable white noise functionals will be constructed by using the second quantization operator r ( ( A8 A,),) of ( A 8 A,),, p E R, and applying the same procedure as in white noise analysis (see [ 5 ] ) .For p E IR and 'p E L2(S',A), define
ll'pll;
=
IlU(A 8
'pll;z(s~,*,.
and let C, be the completion of the class {'p E L 2 ( S ' , R ) ; llpll, < +co} with respect to 11 . Il,-norrn. Then C,, p E IR, is a Hilbert space with the inner product ((., .)), induced by (1 . Ilp-norm and we naturally come to the following facts. For the details, we refer the reader to [7].
Fact 5.1.
4,C, c C, and the embedding C,
1. For q - p > Schmidt type.
-
C, is of Hilbert-
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133
2. Let C =
C,. Then C is a nuclear space with its dual C' and we have a continuous inclusion: C c C, c C, c CO= L 2 ( S ' , h )c C-, c C-, c C' = lim C-,, p > q > 0. +p>o 3. For 'p (&) E C, with p > 0, & E N$: for each n E No and
-
IIPII~= C,"=on!I+nI,.
2
C-,, p > 0. Then there is a sequence {Fn} with Fn N?Lc, n E No,such that for every 'p (&) E C,,
4. Let- F
E
-
E
00
M
where ((., .)) denotes the C'-C pairing, denoted by F = C,"==, In(Fn) or in short, F (Fn).
-
C will serve as the space of test functions and the dual space C' of C the space of generalized functions, and members of C' are called generalized a-stable white noise functionals. The S-transform on L2(S',A) can be extended to all generalized a-stable white noise functionals as in [7] by S F ( g ) = ((F,E x ( g ) ) ) , F E C',
E Nc.
(11)
The following properties of S F can be derived directly from (11). See also [7,8] for details. Proposition 5.3. Let F
-
(F,) be in L Pp,E R. Then
( i ) S F is an entire function in NP+.satisfying ISF(g)I I ~ ~ F I l - , e ~ for ~ g each ~ ~ g E Np,c. ( i i ) DnSF(0)(hl,... , h n ) = ((F,In(h16$...%hn)))for a n y h l , . . . , h, E NP,C.
(iii) The S-transform is an unitary operator from C-, onto the Bargmann-Segal-Dwyer space F 1(Np,c)over Np,c.In fact,
where D" is the nth Fre'chet derivative of SF and (1. IIHS(n)(H) as the Hilbert-Schmidt operator norm of a n-linear functional on a Halbert space H .
YUH-JIALEE, HSIN-HUNGSHIH
134
( i v ) [8] (Characterization theorem) Suppose that a complex-valued function G defined o n N, is analytic and satisfies the following growth condition:
Then there exists a unique F E Cp-4 such that S F = G with IIFllp-+ 5 K c , where K is a constant independent of the choice of F .
As an application of Proposition 5.3(iii), we have Corollary 5.2. For a n y p E R, the class { € x ( g ) ;g E N,} is a total subset of
c-,.
Proposition 5.4. Let 2) be the class consisting of all functions g ( t , u ) , ( t ,u)E Wq,in S, @ S,, which have compact supports away from the t-axis.
( i ) For each g E D, Qg E N, and e''(g) E C. Moreover, for each F E C', ((F,eJ1(g))) = ~ [ e I l ( 1g.)SF(@.,). (ii) The set {eI1(g);g E D} is a total subset of C. Proof. (i) For a fixed g E D,it is obvious by Proposition 5.2 that Q g lies in N,. Now, for any F E C', we take a sequence {cpn} c L 2 ( S ' , A ) and cpn + F in C'. By Theorem 4.1,
This implies that eJ1(g)E C and ((F,ell(g)))= E[e''(g)] . SF(@,). (ii) It suffices t o show that if F E L' such that S F ( Q g )= 0 for all g E D, F = 0. Since the function r g ( t , u )= log(1 + g * ( t , u ) ) / u , ( t , u )E Rq, is in D. Then for g E D with Re(1 + g * ) > 0, S F ( g ) = S F ( Q T g = ) 0. By Proposition 5.3, S F 0 on D and then by Proposition 5.2(iii), SF = 0 on N,. Thus F = 0.
=
Analysis of stable white noise functionals
135
6. Annihilation, creation, and conservation operators
Annihilation and creation operators Let F E C, and E E N--p,c, p E R. The GGteaux derivative
in the direction E is an analytic function on Cauchy integral formula, one can show that (dldz)l,=o S F ( .
N-,,+. In fact, by using the
+ 2 6 ) E cp-1.
Define
8,F=S-’((d/dz)lz=0 S F ( . + z J ) ) . Then, by Proposition 5.3(iv), we have 8, F in C p - s . Since 8, is continuous from C into itself, its adjoint operator 8; is then defined by
((8; F, cp)) := ((F,8,cp)) for F E C’ and cp E L. 8, is called the annihilation operator and 8; is called the creation operator. Conservation operator For every p 2 0, denote by M , the class consisting of all functions h in n/, so that the associated multiplication operator Mh* , which is defined by Mh. ( 9 ) = h* g for g E Np,c,acts continuously from Np,cinto L:(R2, A), where h*(t,u)= u h(t,u),( t ,u)E R2. For h E M,, let a h be the differential second quantization of Mh* , and let 8; be the linear operator on the linear space ‘H spanned by In(gl%.. .% g n ) , 91,. . . ,gn E Np,cand n E N, defined by 8; In(gl% ‘ ’ ‘ % gn) = In ( a h ( g l % ’ ’ ‘ 6g n ) ) . Then, for cp E ‘If with cp
-
(&)I
where “sym” means “the symmetrization of”. Moreover,
where llMh*11 is the operator norm of Mh*.
YUH-JIALEE, HSIN-HUNG SHIH
136
According the above estimation, we can extend the domain of 8; to all cp E C p . 8; is called the conservation operator indexed by h.
Let h E M , with p 2 0 and g E N,. The LBvy product formula (see [7, Theorem 4.11) still holds for a-stable processes as follows:
Il(h) = m( h,g)Im-l(g@m-l ) Im+l(hG g B m )
+
+ mIm(Mh*( g ) Gg@'"-')
+ a; Im(g@'m) + a; Im(gBrn).
= ah I m ( g @ y
In fact, by the same argument as in [7], we have
-
Theorem 6.1. Let h E M , with p 2 0 and cp (&) E C, with q - p >_ 1. Then 8h9, aicp, and a;cp are in L:(S',A). Moreover, for [A]-almostall 2
E K',
11(h)(z)cp(z) = a h cp(z)
+ ai cp(z) + % Cp(z).
(12)
7. A quantum decomposition of stable processes Regarding the Lkvy process X ( t ;z) as a multiplicative operator acting on test Lkvy white noise functionals, then X ( t ;z) has a quantum decomposition provided that T = p + udp(u)exists. If T does not exist, we have the quantum decomposition for the renormalized LBvy process X ( t )- rt. The former includes the cases such as Gaussian processes, Poisson processes, Gamma processes and the processes in the Meixner class; while the latter includes stable processes as special cases. This establishes a connection between Lkvy white noise analysis and quantum probability theory. Now, let X ( t ; z ) ,t E R and z E S', be a fixed a-stable process with 0 < a < 2. We need the following lemma for the further discussion.
s-",
Lemma 7.1. There is a fixed po E N such that for any 77 E K , the space of infinitely differentiable functions on R with compact supports, the mapping from N,,,,, into Lz(R:, A) by
g(t1u)
Url(t)g(tlu)I
(tl'LL)E
@I
is continuous. Proof. For any g E N,, let T g ( tu), , ( t ,u)E R:, be defined as in Proposition 5.2(i). Then Tg E S, 8 S,. Let p > 0 such that 1 C3 u2 E S-, 8 S-, and let q > p so that Ihlcl, 5 Const. Ihlq Ikl, for any h, k E S, 8 S,. Then, for 77 E K l
Analysis of stable white noise functionals
Let po = q and then we complete the proof.
137
0
By Lemma 7.1, the definition of the conservation operator 8; can be for 7 E K. extended to h E K @ 1. For convenience, we write 8; as Moreover, as in [7, Theorem 6.31 the conservation operator can also be written as follows: for 7 E K and cp E C,
where 6(t,,)
= bt @ 6,
and the integral exists in the sense of Bochner.
Proposition 7.1. For cp E C and 4 E 0 , let {&} c K be a sequence converging to 4 in L1(R,m). Then for any increasing sequence {A,} of compact subsets of R, with n-icc lim An = R,,
exists. W e denote such a limit by I l ( 4 )cp Proof. (Sketch) -
-
{8$n8iAn cp} and cp} are convergent t o 84 cp and 8; cp in C’ respectively. By Lemma 7.1, it is not hard t o prove that for q - po 2 1 and 7 E K,
Then, by the preceding result, we obtain that
YUH-JIALEE. HSIN-HUNGSHIH
138
-+O -
asn-+m.
Apply the above estimation. Then, by (13) and Theorem 6.1 for 11(&), the proposition follows immediately. 0
For any real-valued random variable Y on (S',B(S')),let M y denote the multiplication operator by Y , i.e., Mycp = Y (p, (p E L2(S',A). Proposition 7.2. Let 4 E 0. Then there exist sequences (6,) with properties as in Proposition 7.1 such that
(
and { A , }
)
exp i t r ~ S, _ + _ , 4 , ( t ) d t + i t M i , ( m n ~ l ~ , ) e x P ( i t M ( . , d in the sense of strong convergence for any t E R, where JA,
T A ~=
p
+
udp(u)'
+
Proof. Let Y,,t = Il((sgn(t) l [ o A t , o v t ] ) 8 l~,) r, t, n E N,t E R, where B, = {u;1/n 5 I u I 5 n}. By the LBvy-It6 decomposition theorem, there is a set A E B(S') with A(A) = 1 such that for any 77 E K and x E A,
For 4, we take a sequence {&} c K: with the property stated in Theorem 2.1. Then (., 4m) 4 (., 4) in probability as m 4 00. Since $m(t)dY,,, -+ (., &) in probability as n 00, we can choose a subsequence {k,} of {n} such that s_'," $m(t)dYk,,t + (., 4) in probability as rn + 00, and
JTz
Analysis of stable white noise functionals
{en}
139
thus there is a subsequence of { m } such that Zn =: -+ (., 4 ) [A]-almost surely. Therefore,
It implies that eit M Z n -+ ei
s_’,” q5tn( t)dYktn
,t
strongly for any t E R. Since
M ( , j + )
r+m
0
the proof is completed.
Apply the above propositions for 4 = 1 p t ] ,t 2 0; or 4 = 1 p 0 1 , t < 0. It is natural t o regard “ X ( t )- rt” as a generalized stable white noise functional by
( ( X ( t )- r t , V)) = 71-00 lim
((Il($n @ l A n ) , V)),
E L.
By Theorem 6.1 and Proposition 7.1, we can obtain the quantum decomposition of stable processes as follows.
Theorem 7.1. The “renormalized”L h y process X ( t )- r t is a continuous operator from L into Lf and we have
1
+00
+ c2
cp d u } dt.
u2-
(14)
If r is finite, we obtain the quantum decomposition for X ( t ) from the identity (14). If we formally take the derivative in both sides of the identity (14) with respect to t we obtain f m
s_,
( x ( t ;X ) - T ) + ) ) m d t
=
/
+00
q ( t ) (a,+a,*+a,o)cp(X)dt,
E
K,
-00
at
a;
a;t,
a;
where = as,, = and = a&. Symbolically we may write : X ( t ) : = X ( t ) - t which is called a renormalization of X ( t ) . Then the quantum decomposition of the generalized functional : X ( t ) :is given by
: X ( t ) := a, +a;
+a;.
140
YUH-JIALEE, HSIN-HUNG SHIH
Acknowledgement T h e final version of t h e present paper was completed while t h e first author was visiting CCM (Centro de Ci6ncias MatemAticas, Universidade da Madeira) in March, 2006. T h e first author would like t o t h a n k the Universidade da Madeira for financial support while visiting CCM.
References 1. I. M. Gel’fand, N . Y . Vilenkin, Generalized Functions, vol. IV, Academic Press, New York, 1964. 2. T. Hida, Analysis of Brownian Functionals, Carleton Mathematical Lecture Notes no. 13, 2nd ed., 1978. 3. K. It6, Spectral type of shift transformations of differential process with stationary increments, ‘Trans. Amer. Math. SOC.81,253-263 (1956). 4. Y. Ito, Generalized Poisson functionals, Probab. Theory Relat. Fields 77, 1-28 (1988). 5. H.-H. Kuo, White Noise Distribution Theory, CRC Press, 1996. 6. Y.-J. Lee, H.-H. Shih, “The Segal-Bargmann transform for LBvy functionals” , J . Funct. Anal. 168,46-83 (1999). 7. Y.-J. Lee, H.-H. Shih, “Analysis of generalized L6vy white noise functionals”, J . Funct. Anal. 211, 1-70 (2004). 8. Y.-J. Lee, H.-H. Shih, “A characterization of generalized LBvy white noise functionals” , Quantum Information and Complexity, World scientific, 2004, pp. 321-339. 9. Y.-J. Lee, H.-H. Shih, “LBvy white noise measure on infinite dimensional spaces: existence and characterization of measurable support”, J . Funct. Anal. (2006), in press. 10. Y.-J. Lee, H.-H. Shih, 11. K.-I. Sato, Lkvy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1999. YUH-JIALEE Department of Applied Mathematics, National University of Kaohsiung,
Kaohsiung, TAIWAN 811 HSIN-HUNG SHIH
Department of Accounting Information, Kun Shan University, Tainan, TAIWAN 710
UNITARIZING MEASURES FOR A REPRESENTATION OF THE VIRASORO ALGEBRA, ACCORDING TO KIRILLOV AND MALLIAVIN: STATE OF THE PROBLEM PAULLESCOT (INSSET, Saint-Quentin)
0. Introduction
In this paper, we review results due to Airault-Bogachev ([l]),AiraultMalliavin ([2]) and Kirillov ([7],[8],[9]).In the first paragraph, we define the Virasoro algebra Vir,,h (depending upon two real parameters c > 0 and h 2 0) as a natural central extension of the (complexified algebra of) the Lie algebra diff (S1)of Cm-diffeomorphisms of the unit circle S1, and introduce a complex structure on its subspace diffo(S1). In the second paragraph we explain the identification of the (infinite-dimensional) homogeneous space Diff(S1)/S1 with a certain space M of univalent functions on the (open) unit disk; this is due to Kirillov ([7], [8], [9]), as well as the definition of a doubly infinite sequence ( L k ) k E ~of differential operators on M , and its link to the Neretin polynomials; our exposition of these matters in $2 and the first half of $3 follows [2] and [9]. In the second half of $3 the previous ingredients are combined in order to define a representation p of the Virasoro algebra; this is due t o Airault-Malliavin ([2]), and, in a more general case, to Airault-Malliavin-Thalmaier ([3]). To these authors is also due the notion of a unitarizing measure on M for p, that we define in $4; we conclude by giving a proof of the result from [l]that such a measure, should it exist, would admit the above-mentioned Neretin polynomials as an orthogonal family. This entails as a consequence the non-existence of a unitarizing measure in the case h = 0, obtained in [l]in an almost purely algebraic way, and previously in [2] by a geometrical argument. The content of this paper is based upon a talk given at Bielefeld University on July 27th, 2004; I have tried to uniformize notation, sign conventions, etc., and I have slightly amended the definition of p given in [l]. A preliminary version appeared as a preprint in the Bielefeld BIBOS series in 2004; I am indebted to Professor Michael Roeckner for the invitation to lecture in Bielefeld upon these topics, and t o Professors Ana Bela Cruzeiro and Jean-Claude Zambrini, resp. Carlos Florentino, for invitations to give
141
PAULLESCOT
142
talks in Lisbon upon this material, in April, resp. June, 2005. To Professors HQlBneAirault, Philippe Blanchard and Francesco RUSSO,as well as t o the anonymous referee, I address my thanks for numerous remarks on the first version of the paper. My recent joint work ( [ 5 ] )with Professor M. Gordina is an attempt a t a new approach t o some of the questions described here. Of related interest is the paper [4]. 1. Preliminaries Let Diff (Sl) denote the group of C", orientation-preserving diffeomorphisms of the unit circle S1. Its Lie algebra diff(S1) can be naturally identified with the set of C" vector fields on S1,i.e.: diff(S')
=
{ 4(Q)-4 4 : R
-+
I
R, Cw,2.1r-periodic .
We shall often identify, without further warning, the function 4 and the vector field 4(Q)&,. A topological basis (for the obvious Frkhet space topology) of diff(S1) is given by the ( f k ) k l O and the ( g k ) k > l , where:
and def gk
d dQ
= sin(k0)-
ef
Let diff@(S1) diff(S1) @R CC denote the complexified Lie algebra of diff (S1);it is now clear that a topological basis of diff@(S1)is given by the { e k } k E Z where
One has the commutation relations
The Lie algebra diff@(S1)contains
as a Lie subalgebra, dense in the natural FrQchet space topology.
Unitarizing measures for a representation of the Virasoro algebra . . .
143
Setting Lk = - i e k , one finds that: [ L k r Lk’]=
(k’
-
k)Lk+k’,
whence
A 21 Der@(C[t,t-’1). Here Lk corresponds, through this isomorphism, t o tk+’-$, which is equivalent t o setting t = eie. The algebra Virc,h is defined by: def
Virc,h = diff@(S’)@ Cn as a vector space, with the following Lie bracket for any ( f , g ) E diff@(S1)2,
[.,fl
= 0,
and [f,glViTc,h = [ f , g I
+ Wed,g).
7
where
The so-called Gelfand-Fuks cocycle is W O , ~ . It turns out that Virc,h is the unique nontrivial central extension of diff@(S1)(see [ S ] ) .An easy computation yields Proposition 1.1. For any ( m , n )E Z2
It is easy to deduce from [6, chapter 7, exercises 7.1 and 7.131, that the wc,h are exactly the continuous cocycles a! on diffc(S1) such that for any f in diff@(S’) a(eo,f ) = 0 . We shall denote by V i e h the natural “real” Lie subalgebra of Virc,h,i.e.
Vi&
efdiff (5’’) @ Rn.
From now on, we shall assume c > 0 and h
2 0.
I I’“
Let
PAULLESCOT
144
On diffo(S1), one defines a complex structure as follows: for
the sequences { a k } k y o and m 2 0 one has
{bk}k>l -
are rapidly decreasing, i.e., for any
+
nnm(lanI Ibnl) n+<w
0.
We now set
k=l and we have J+ E Cm(S1). Lemma 1.1 ( [ 2 ,p. 6301). For any f in diff(S1)
whence (by Proposition 1.l):
k2l
Taking into account the relations: for any k bk = i(ck - c-k), the result follows.
2
1,
Uk
=
Ck
+ C-k
and 0
Unitarizing measures for a representation of the Virasoro algebra . . .
145
2. Kirillov's construction of an action of Diff(S1) on a space of univalent functions Let D = D(0,l) denote the open unit disk in C, and let M denote the set of C" functions f : D + @, injective, holomorphic on D , with f(0) = 0, f'(0) = 1, and f'(z) # 0 for any z E 0. Each f E M can be written as I
f ( z ) = z(1
+"
\
+Xwn), z E D, n= 1
whence the imbedding
M
L)
f H
C"'
(Cl,C2,...).
In fact, by De Branges's solution of Bieberbach's conjecture, one has IcnI 5 n 1, thus one may identify M with an open subset of ITn21&(O, n 2); one therefore obtains a structure of (contractible) manifold on M . For f E M , r = f(S') = f ( a D ) is a Jordan curve, therefore one has a decomposition into connected components:
+
+
( @ u { ~ } )= \ rr +urE r-. By a combination of Riemann's Representation
with 0 E ?I and 03 Theorem and Caratheodory's Theorem, there exists a holomorphic mapping
4f (ax}..{ such that
4f(co) = 00.
\ D ) + F =r- ur
Let us then define gf by gf :
s1-+ s1 eis H f - ' ( 4 f ( e i e ) )
Then gf E Diff (Sl),and gf is well-defined up to multiplication on the right by an holomorphic automorphism of C \ D stabilizing 03, i.e., a rotation, whence a mapping
IC : M
-+
Diff(S1)/S1.
Theorem 2.1 (Kirillov, [9, p. 7361). IC is a bijection. Therefore, by transport of structure, Diff (S1)/S1acquires a structure of contractible complex manifold. Using J and &,h, this manifold can be equipped with a Kahlerian structure(see [2], [ 7 ] ) .
PAULLESCOT
146
Definition 2.1 (Kirillov action). For w = $(O)& E diff(S1) and f E M let us write w(eie) = $(O), and define K,(f): D 4 C by the following equality, for any z E D (we set t = eie)
Definition 2.2. For n E Z,let def
L n = -ZKe,, , a differential operator in the variables k. For nonnegative n, it is very easy t o compute L,.
Proposition 2.1.
Proof. (1) In this case, the expression for K, becomes, with t = eie,
(by Cauchy's formula)
= iZ"+lf'(Z) +m k=l
therefore +m
Ln(f)(z) = Zn+'
+ x ( k + 1)ckzk+"+', k=l
Unitarizing measures for a representation of the Virasoro algebra . . .
147
whence the result. (2) The computation is similar, taking into account the pole at 0, and yields
Lo(f)(z)= z f b ) - f ( z )
7
whence the result.
Lemma 2.1. One has the commutation relations, for any ( m , n ) E Z2, [Lm,Lnl
=
( m - n)LTn+n
Proof. [ 2 , p. 6551 for m 2 0 and n this relation using Proposition 2.1.
(*)
’
2 0, it is actually very easy t o check 0
3. The Neretin polynomials and the representation p def c Let Tk = -(k3 12
-
k), and Pk
=
o for k < 0.
Theorem 3.1 (Kirillov-Neretin). There exists a unique sequence (P,),?O of polynomials in the (&)i>l such that: ( 1 ) Pk depends only upon c1,. . . , C k ;
(2) Po = h; (3)
v k 2 1vn 2 1 Lk(pn) = ( n + k)pn-k
(4)
Vn 2 1 Pn(0)= 0 .
+ Tkbk,n ; +
Proof. Given Po,. . . , P, ( n 2 0 ) , the relations (3) (with n 1 in place of n ) are trivially satisfied for any polynomial P,+1 in c1,. . . , cn+l and any k > n 1; for 1 5 k 5 n + 1, the relations determine, by descending induction on k, the apn+l in a unique way, therefore they determine Pn+l 0 up to a constant; (4)for n 1 now determines a unique P,+1.
+
+
The first few terms of the sequence are easily computed:
Po = h , Pi = 2hc1 , P2
=
(4h + :)c2
-
( h + :)c:.
If each ck is given the weight k, it is easily seen that Pk is homogeneous of weight k.
PAULLESCOT
148
Let us remind the reader of the definition of the Schwarzian derivative of a holomorphic function f: def f”’(z) 3 (.f”(z)). S(f)(z) = -- - - . f (). 2 f”z)
The following result could have been used as definition of the polynomials
Pk Proposition 3.1 ([9, p. 742, Theorem]). For any f E M
Proposition 3.2. For any k 2 0, p 2 0 ,
L - k ( P p ) - L - p ( p k ) = (P - k ) P p + k ; in particular, the formula of Theorem 3.1 (3) remains valid for k Proof. [2, p. 6631.
= 0.
0
Let def
Qk
=
{
pk 0
for k # 0, for k = O .
Theorem 3.2. Let us set, for each k E Z,
and p ( ~= ) iId.
Then p defines a representation of the Lie algebra Virc,h into the Lie algebra of differential operators o n M . Proof. As, obviously, [ p ( e k ) , ~ ( I E= ) ]0 is enough to prove that [p(em),p(en)l = P([em,e n ] ) . Taking Proposition 1.1 into account, this is easily reduced t o checking the relation:
Unitarizing measures for a representation of the Virasoro algebra . . .
149
But, for m 2 0 and n 2 0, that relation is trivially satisfied; for m = 0 and n < 0, as well as for n = 0 and m < 0, it follows from the relation for any n21 Lo(Pn) = nPn ; in the case m < 0 and n < 0, setting p prove that for any p 2 1 and k 2 1
L-k(pp)
-
L-p(Pk)
=
-m and k = -n, it is enough to
=
(P - k)pp+k
,
but both these facts follow from Proposition 3.2. There remains the case m 5 -1 and n 2 1 (or the other way round); in this case, we need to prove, setting k = -m 2 1, that:
i.e., for any k 2 1 and n
2 1,
+
( n k)P,-n if n # k , if n = k . 2hk + yk As PO= h, this follows from Theorem 3.1(3).
0
4. Definition of an unitarizing measure and a non-existence result Definition 4.1. A Bore1 probability measure p on M is said to be unitarizing for the representation p if and only if for any u E V i e , P ( u ) * = -P on the space 'HLE(M)of p-square integrable holomorphic functions on M .
Lemma 4.1 ([l,Theorem 1, p. 4331). If p exists, then, setting zk = Lk -=(k 2 0), one has for any F E cm(M)
z k ( F ) d p = - JM Fpkdp 7
where
(1)
PAULLESCOT
150
Proof. From the definition follows that for any w E Virc,h p(v)* = -p(v)
.
By a density argument, one may assume that F = holomorphic; then
=
lM
(P(Qk - Q-k)FdP
'p$,
with
'p
and $
7
by the hypothesis on p and since 'p and $ are holomorphic. Whence we have the result with Pk
= Q-k - Q k =
-
i-
2 1,
-Pk
for k
0
for k = O .
Theorem 4.1 ([I, Theorem 3 and Corollary 4,p. 2341). (1) If p exists then the sequence 1,PI, P2, dots is a sequence of orthogonal polynomials in L ~ ( Mp ), ; more precisely:
(Pm,Pk)L2(p) =
4- 2hk
ifm#k, if m = k >_ 1, if m = k = 0 .
Unitarizing measures for a representation of the Virasoro algebra . . .
151
( 2 ) If h = 0 then there is no unitarizing measure on M for p. Proof.
+
(1) Let us set, for each k 2. 0, and Hk = 2: PkZk; it follows from Lemma 4.1 applied to z k ( F ) that, for each k 2. 0, one has: for any F E C"(M) and k 2 0
But it follows from the definition of the Neretin polynomials (Theorem 3.1(3)) and from the last remark in Proposition 3.2 that for any k 2. 0 and n 2 1
+ k)Pn-k + Yk6k,n) + Pk((n+ k)Pn-k + Ykbk,n) = (n + k)nPn-2k + ( n + k)Ykbk,n-k + ( n + k)PkPn-k + PkYkdk,n.
Hk(Pn) = Lk((n
(3) By (2) one has for any k 2. 0 and n 2. 1
Applying ( 3 ) for k = 0 and n 2 1, one finds that for any n 2. 1
H o ( P n ) = n2Pn 7 whence (4) yields that for any n 2. 1 JM
(5)
Pndp =0.
From Lemma 4.1 applied to F = 1 follows that for any k 2 0
Taking now k 2 1, m 2. 1 and n
= m+ k ,
( 3 ) and (4) together yield:
PAULLESCOT
152
from t h e fact t h a t
O
for n
2 I(5)
0
for n
< 0 (by definition)
and from (6), we get:
Recall that p k = - P k for k 2 1 and therefore the result holds. (2) Let us remind the reader that PI = 2hcl. Clearly,
whence
which is impossible for h = 0. A more geometrical proof of this nonexistence result had previously been given in [3, Theorem 2.2, p. 6251.
References 1. H. Airault, V. Bogachev, “Realization of Virasoro unitarizing measures on the set of Jordan curves”, C.R.Acad.Sci.Paris, Ser. I 336, 429-434 (2003). 2. H. Airault, P. Malliavin, “Unitarizing probability measures for representations of Virasoro algebra”, J . Math. Pures Appl. 80(6), 627-667 (2001). 3. H. Airault, P. Malliavin, A. Thalmaier, “Support of Virasoro unitarizing measures”, C. R. Acad. Sci. Paris, Ser. I 3 3 5 , 621-626 (2002). 4. H. Airault, P. Malliavin, A. Thalmaier, “Canonical Brownian motion on the
space of univalent functions and resolution of Beltrami equations by a continuity method along stochastic flows”, J . Math. Pures Appl. 8 3 , 955-1018 (2004). 5 . M. Gordina, P. Lescot, “Riemannian Geometry of Diff(S’)/S’”, Journal of Functional Analysis (2006).
Unitarizing measures for a representation of the Virasoro algebra
...
153
6. V. G. Kac, Infinite Dimensional Lie Algebras. 7. A. A. Kirillov, “Kahler structures on K-orbits of the group of diffeomorphisms of a circle”. 8. A . A . Kirillov, D. V. Yuriev, “Representations of the Virasoro algebra by the orbit method”, J . Geom. Phys. 5, no. 3, 351-363 (1988). 9. A. A. Kirillov, “Geometric approach to discrete series of unirreps for VIR”, J . Math. Pures Appl. 77, 735-746 (1998). PAULLESCOT
INSSET
-
Universiti: de Picardie,
48 Rue Raspail, 02100 Saint-Quentin, France paul.lescotQu-picardie.fr
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FKG INEQUALITY ON THE WIENER SPACE VIA PREDICTABLE REPRESENTATION YUTAOMA ( Wuhan University, Hubei / Universite' de La Rochelle), NICOLAS PRIVAULT (Universite' de La Rochelle) Using the Clark predictable representation formula, we give a proof of the FKG inequality on the Wiener space. Solutions of stochastic differential equations are treated as applications and we recover by a simple argument the covariance inequalities obtained for diffusions processes by several authors.
1. Introduction
Let (R, 3,P,5 ) be a probability space equipped with a partial order relation 5 on R. An (everywhere defined) real-valued random variable F on (R, 3,P,5 ) is said to be non-decreasing if
F ( w i ) 5 F(u2) for any w1, w2 E R satisfying w1 5 w2. The FKG inequality [4] states that if F and G are two square-integrable random functionals which are nondecreasing for the order 5 , then F and G are non-negatively correlated: Cov(F, G) 2 0.
It is well known that the FKG inequality holds for the standard ordering on R = R,since given X, Y : R 4 R two non-decreasing functions on R we have:
155
156
YUTAOMA, NICOLAS PRIVAULT
On the Wiener space ( R , 3 ,P) with Brownian motion (Wt)tGw+, Barbato [3] introduced a weak ordering on continuous functions and proved an FKG inequality for Wiener functionals, with application t o diffusion processes. In this paper we recover the results of [3] under weaker hypotheses via a simple argument. Our approach is inspired by Remark 1.5 stated on the Poisson space in Wu [14],page 432, which can be carried over to the Wiener space by saying that the predictable representation of a random variable F as an It6 integral, obtained via the Clark formula 1
F = %?I+
WtFl3tI d W t ,
yields the covariance identity
where D is the Malliavin gradient expressed as
From (2) we deduce that D F is non-negative when F is non-decreasing, which implies Cov(F, G) 2 0 from (1). Applications are given to diffusion processes and in Theorem 3.2 we recover, under weaker hypotheses, the covariance inequality obtained in Theorem 3.2 of [7] and in Theorem 7 of [31. We proceed as follows. Elements of analysis on the Wiener space and applications to covariance identities are recalled in Section 2. The FKG inequality and covariance inequalities for diffusions are proved in Section 3. We also show that our method allows us t o deal with the discrete case, cf. Section 4.
2. Analysis on the Wiener space In this section we recall some elements of stochastic analysis on the classical Wiener space ( R , 3 , P ) on R = Co([O,l];R), with canonical Brownian motion (Wt)tG[o,l] generating the filtration (.Tt)tE[o,l]. Our results extend without difficulty t o the Wiener space on Co(R+;IR).Let H denote the Cameron-Martin space, i.e., the space of absolutely continuous functions with square-integrable derivative:
h : [0,1]
---f
R : / ' \ h ( ~ ) \ ~ d<sm}. 0
FKG inequality on the Wiener space via predictable representation
157
Let I n ( f n ) ,n 2 1, denote the iterated stochastic integral of fn in the space I,:([0, 11,) of symmetric square-integrable functions in n variables on [0,l],, defined as
with the isometry formula %(fn)
Im(gm)l = n!l { n = m }(fnrgm)L2([o,l]n).
Every F E L2(s2) admits a unique Wiener chaos expansion
n=l with f, E L:([O,lIn), n L 1. Let (ek)k?l denote the dyadic basis of L 2 ( [ 011) , given by
ek
= 2n'2
lIk-2"
k+;_zn],
T >
2,
Ik I 2,+l
- 1,
12
E
N.
Recall the following two equivalent definitions of the Malliavin gradient D and its domain Dom(D), cf. Lemma 1.2 of [8] and [lo]: a) Finite dimensional approximations. Given F E L2(s2), let for all n E N G, = 4 l l ( t . ~ n ) ,... , I 1 ( e ~ + 1 - 1 ) ) , and F, = EIFIGn], and consider f, a square-integrable function with respect to the standard Gaussian measure on lR2n, such that
F,
.
= f i z ( ~ 1 ( e 2 n.) ,. ,Il(e2n+1-1)).
Then F E Dom(D) if and only if f, belongs for all n 2 1 t o the Sobolev space W2.1(W2n) with respect to the standard Gaussian measure on W2n, and the sequence
converges in L2(R x [0,1]). In this case we let
D F := lim DF,. n-+w
YUTAOMA, NICOLASPRIVAULT
158
b) Chaos expansions. Let G
E
L2(R) be given by
n=l Then G belongs to Dom(D) if and only if the series 00
n=l
converges and, in this case,
In case (a) above the gradient ( D F , , A ) L Z ( [ ~ ,h~ ~E) ,H , coincides with the directional derivative
P F n , &2([0,1])
d
= -&(h(ezn)
. . .,
h)L2([0,1]),
++an,
E=O
d = -Fn(w dc
+~h)l
&=O
,
where the limit exists in L2(R). Similarly, the Ornstein-Uhlenbeck semi-group (Pt)tE~+ admits the following equivalent definitions, cf. e.g. [9], [12], [13]: a) Integral representation. For any F
E
L2(R) and t
E
R+, let
b) Chaos representation. For any F E L2(R) with the chaos expansion 00
n=1
we have
c 00
PtF = E[F]
+
e-ntIn(fn),
n=1
t
E
R+.
(4)
FKG inequality on the Wiener space via predictable representation
159
The operator D satisfies the Clark formula, i.e., PI
cf. e.g. [12]. By continuity of the operator mapping F E L2(R) to the adapted and square-integrable process ( u t ) t E R + appearing in predictable represent ation rl
+
F = IE[F] /o
ut dWt,
the Clark formula can be extended to any F E L2(R) as in the following proposition.
Proposition 2.1. The operator F tinuous operator on L2(R).
H
(IEIDtFIFt])tEp,l~ extends as a con-
Proof. We use the bound
for F E Dom(D).
0
Moreover, by uniqueness of the predictable representation of F E L2(R), an expression of the form 1
F=c+I
utdWt,
where c E R and ( u t ) t E R + is adapted and square-integrable, implies ut = IEIDtFIFt],dt x dP-a.e. The Clark formula and the It6 isometry yield the following covariance identity, cf. Proposition 2.1 of [6].
Proposition 2.2. For any F , G E L2(R) we have r -1
1
This identity can be written as
[I
1
Cov(F,G) = IE
IEIDtFIFt]Dt G d t ] ,
(9)
provided G E Dom(D). The following lemma is an immediate consequence of (8).
160
YUTAOMA, NICOLAS PRIVAULT
Lemma 2.1. Let F I G E L2(R) such that E[DtFIFt] . EIDtGl.Ft] 2 0,
dt x dP-a.e.
Then F and G are non-negatively correlated: Cov(F,G) 2 0. If G E Dom(D), resp. F, G E Dom(D), the above condition can be replaced by IEIDtFI.Ft] 2 0 and DtG 2 0, dt x dP-a.e., resp.
D t F 2 0 and DtG 2 0,
dt x dP-a.e..
As recalled in the introduction, if X is a real random variable and f,g are C1(R) functions with non-negative derivatives f’,g’, then f ( X ) and g ( X ) are non-negatively correlated. Lemma 2.1 provides an analog of this result on the Wiener space, replacing the ordinary derivative with the adapted process W t l ) t E [ O ,I]. 3. FKG inequality on the Wiener space We consider the order relation introduced in [3].
Definition 3.1. Given w1, w2 E R, we say that w1 5 w2 if and only if we have 0 I tl I t2 I 1. Wl(t2) - W l ( t 1 ) I w z ( t 2 ) - W 2 ( t l ) , The class of non-decreasing functionals with respect to 5 is larger than that of non-decreasing functionals with respect to the pointwise order on R defined by W l ( t ) I w2(t), t E [O, 11, w1,w2 E Q.
Definition 3.2. A random variable F : R 4 R is said to be non-decreasing if w1 5 w2 F ( w l ) 5 F ( w ~ ) , P(dw1) @P(dw2)-a.s. Note that unlike in [3], the above definition allows for almost-surely defined functionals. The next result is the FKG inequality on the Wiener space. It recovers Theorem 4 of [3] under weaker (i.e., almost-sure) hypotheses.
FKG inequality on the Wiener space via predictable representation
161
Theorem 3.1. For a n y non-decreasing functionals F, G E L2(s1) we have Cov(F,G) 2 0. The proof of this result is a direct consequence of Lemma 2.1 and Proposition 3.1 below.
Lemma 3.1. For every non-decreasing F E Dom(D) we have d t x dP-a.e..
DtF 2 0,
Proof. For n E N, let .rr, denote the orthogonal projection from L 2 ( [ 01, 1) onto the linear space generated by (ek)2nIk<2n+~. Consider h in the Cameron-Martin space H and let
Let A, denote the square-integrable and &-measurable random variable
From the Cameron-Martin theorem, for all n E N and G',-measurable bounded random variable G, we have, letting F, = E[F 1 &]:
hence F,(w
+ h,)
= E[F(.
+ h,)lG,](~),
P(dw)-a.s.
If h is non-negative, then 7r,h is non-negative by construction hence w 5 w h,, w E R, and we have
+
F(w) 5 F(w
+ h,),
P(dw)-a.s.,
since, from the Cameron-Martin theorem, P({w+h,
:w E
R})
= 1.
Hence
YUTAOMA, NICOLASPRIVAULT
162
with the notation of Section 2,
F,(W
+ h) = fn(ll(ean)+ (e2n,h)Lz(p,1]), .. ., 11 - 1 ) + (ezn+1-11 h ) L 2 ( [ 0 , l ] ) ) = f n ( I l ( e 2 4 + (e2n,7hh)L2([0,1]), .. 1 1 (e2"+'--1)+ (e2nf1-1, ([OJ])) . I
= F,(w =
%&)L2
+ h,)
W ( +. hn)lGnl(W)
2 lqFlGnl(w) = F,(w),
i.e., for any
~1
P(dw)-as,
5 e2 and h E H such that h is non-negative we have F,(w
+ ~ i hI ) Fn(w + &ah), F,(w + eh) is non-decreasing in E
and the smooth function E I+ P(dw)-a.s. As a consequence,
on R,
for all h E H such that h 1 0, hence DF, 2 0. Taking the limit of ( D F , ) n E ~as n goes to infinity shows that DF 2 0. 0 Next, we extend Lemma 3.1 to F E L 2 ( R ) .
Proposition 3.1. For any non-decreasing functional F E L2(R) we have
IEIDtF I Ft] 2 0,
d t x dP-a.e.
Proof. Assume that F E L2(s2) is non-decreasing. Then Pl/,F, n 2 1, is non-decreasing from (3), and belongs to Dom(D) from (4). From Lemma 3.1 we have
DtPl/,F 2 0 ,
d t x D-a.e.,
hence
IE[DtPl/,F I Ft]2 0,
dt x 0 - a . e .
Taking the limit as n goes t o infinity yields IE[DtF I Ft] 1 0 , d t x 0 - a . e . from (7) and the fact that Pl/,F converges to F in L2(R) as n goes to infinity. 0
FKG inequality on the Wiener space via predictable representation
163
Conversely it is not difficult to show that if u E L2([0,11) is a non1 negative deterministic function, then the Wiener integral utdWt is a non-decreasing functional. Note however that the stochastic integral of a non-negative square-integrable process may not necessarily be a nondecreasing functional. For example, consider ut = Gl[a,ll,t E [0,1], where G E L2(R,Fa)is non-negative and decreasing, then
so
6'
ut dWt = G(B1- B,)
is not non-decreasing.
Example: maximum of Brownian motion By Proposition 2.1.3 of [9] the maximum M = up^<^<^ W ( t )of Brownian ~ ]0, where 7 motion on [0,1] belongs to Dom(D) and satisfies D k 1 [ 0 ,2 is the a.s. unique point where M attains its maximum. Here, M is clearly an increasing functional. Example: diffusion processes Consider the stochastic differential equations
and
where b, b, el a are functions on R+ x R satisfying the following global Lipschitz and boundedness conditions, cf. [9], page 99:
6)
let(.)
(ii) t
H
-
Ct(Y)I
+ Ibt(z)
ct(0) and t
H
- bt(Y)l I K l z - YI, Z,Y E R, t E [O, 11, bt(0) are bounded on [0,1],
for some K > 0. Lemma 8 of [3] shows that the solutions (Xt)tE[0,1], (Xt)tElo,ll of (10) and (11) are increasing functionals when a(.), a(z) are differentiable with Lipschitz derivative in one variable and satisfy uniform bounds of the form
O < ~ < e ( z ) < M < m and O < E < 6 ( z ) I u < o o ,
z€R
Thus from Proposition 3.1 it satisfies the FKG inequality as in Theorem 7 of [ 3 ] . Here the same covariance inequality can be obtained without using the FKG inequality, and under weaker hypotheses.
YUTAOMA, NICOLAS PRIVAULT
164
Theorem 3.2. Let s , t E [0,1] and assume that a , 6 satisfy the condition a,(z)6?.(y)2 0,
z, y E
IR, 0 I r I s A t.
Then we have C o v ( f ( X , ) , g ( ~ t ) )2 0, for all non-decreasing Lipschitz functions f , g.
Proof. From Proposition 1.2.3 and Theorem 2.2.1 of [ 9 ] ,we have f (X,) Dom(D), s E [0,1],and
D,~(x,) = I [ ~ , + ] ( T a,(x,.) ) f'(x,)el:
a u d ~ u + l : ( ~ ~ - ; a ; ) ~7u
E
(13)
r , s E [0,1],where ( ( u , ) , ~ [ ~ , ~and I (bu)uE[~,ll are uniformly bounded adapted processes. Hence we have
T, s
E [0,1]. Similarly we show that E[D,.g(Xt) I FYIhas the form
~ [ D ? . s ( XI tF )TI
= l [ O , t ] ( T ) S?.(X?.)lE [
g'm
14,
~- u d W U + . r ~ ( P U - ~ 6 ~ ) d u
r, t E [0,1],and we conclude the proof from Lemma 2.1.
I7
Note that (12) has also been obtained for s = t and X = X in [7], Theorem 3.2, by semigroup methods. In this case it also follows by applying Corollary 1.4 of [5] in dimension one. The argument of [7]can in fact be extended t o recover Theorem 3.2 as above. Also, (12) may also hold under local Lipschitz hypotheses on a and 6 , for example as a consequence of Corollary 4.2 of [l]. 4. The discrete case
Let R = {-1,l)" and consider the family ( X k ) k >-l of independent Bernoulli { -1,l)-valued random variables constructed as the canonical projections on R, under a measure IF' such that
p, Let .?-I
= P(X, = 1)
and q, = P(X,
= -1),
= { 0 ,R} and
Fn = a(X0,. . . ,X,),
nE
N.
n E N.
FKG inequality on the Wiener space via predictable representation
165
Consider the linear gradient operator D defined as
D k F ( w ) = Jprc4k ( F ( ( w i l { i f k }+ l{i=k})iEW)- F(wil{i#k}- l{i=k})iEN), (14) k E N. Recall the discrete Clark Formula, cf. Proposition 7 of [ l l ] :
defines a normalized i.i.d. sequence of centered random variables with unit variance. The Clark formula entails the following covariance identity, cf. Theorem 2 of [ 1 11: m
Cov(F, G ) = E x E [ D k F 1 [k=O
Fk-11
E[DkG I Fk-11
which yields a discrete time analog of Lemma 2.1.
1
L e m m a 4.1. Let F , G E L2(R) such that
E[DkF 1
Fk-11
*
E[DkG I Fk-11 2 0 ,
k E
N.
T h e n F and G are non-negatively correlated:
Cov(F,G) 2 0. According to the next definition, a non-decreasing functional F satisfies D k F 2 0 for all k E N. Definition 4.1. A random variable F : R if for all w1, w2 E R we have
Wl(k)I WZ(k), Vk E N,
+R
3
is said to be non-decreasing
F(w1) e F(w2).
The following result is then immediate from (14) and Lemma 4.1, and shows that the FKG inequality holds on R. P r o p o s i t i o n 4.1. If F , G E L2(R) are non-decreasing then F and G are non-negatively correlated:
Cov(F,G) 2 0 . Note however that the assumptions of Lemma 4.1 are actually weaker as they do not require F and G to be non-decreasing.
166
YUTAOMA, NICOLASPRIVAULT
References 1. E. A16s, C. 0. Ewald, “A note on the Malliavin differentiability of the Heston volatility”, Economics and Business Working Papers Series, no. 880, Universitat Pompeu Fabra, 2005. 2. D. Bakry, D. Michel, ‘(Sur les inkgalitks FKG” in Skminaire d e Probabilitks, X X V I , volume 1526 of Lecture Notes i n Math., Springer, Berlin, 1992, pp. 170-188. 3. D. Barbato, “FKG inequality for Brownian motion and stochastic differential equations”, Electron. Comm. Probab. 10, 7-16 (electronic) (2005). 4. C. M. Fortuin, P. W. Kasteleyn, J. Ginibre, “Correlation inequalities on some partially orderd sets”, Comm. Math. Phys. 2 2 , 89-103 (1971). 5 . I. Herbst, L. Pitt, “Diffusion equation techniques in stochastic monotonicity and positive correlations”, Probab. Theory Related Fields 87, 275-312 (1991). 6. C. Houdrk, N. Privault, “Concentration and deviation inequalities in infinite dimensions via covariance representations”, Bernoulli 8(6), 697-720 (2002). 7. Y . Z. Hu, “It6-Wiener chaos expansion with exact residual and correlation, variance inequalities”, J. Theoret. Probab. 10(4), 835-848 (1997). 8. D. Nualart, “Markov fields and transformations of the Wiener measure”, in T. Lindstr~m,B. Bksendal, A. S. Ustunel, editors, Proceedings of the Fourth Oslo-Siliuri Workshop on Stochastic Analysis, volume 8 of Stochastics Monographs, Oslo, 1993. Gordon and Breach. 9. D. Nualart, The Malliavin Calculus and Related Topics. Probability and its Applications, Springer-Verlag, 1995. 10. D. Ocone. “A guide to the stochastic calculus of variations”, in H. Korezlioglu, A. S. Ustunel, editors, Stochastic Analysis and Related Topics, volume 1316 of Lecture Notes in Mathematics, Silivri, 1988, Springer-Verlag. 11. N. Privault, W. Schoutens, “Discrete chaotic calculus and covariance identities”, Stochastics and Stochastics Reports, 72,289-315 (2002). 12. A. S. Ustunel, A n introduction to analysis on Wiener space, volume 1610 of Lecture Notes in Mathematics, Springer-Verlag, 1995. 13. S. Watanabe, Lectures on Stochastic Differential Equations and Malliavin Calculus, Tata Institute of Fundamental Research, 1984. 14. L. Wu, “A new modified logarithmic Sobolev inequality for Poisson point processes and several applications”, Probab. Theory Related Fields 118(3), 427-438 (2000). YUTAOMA
Department of Mathematics, Wuhan University, 430072 Hubei, China and Laboratoire de Mathkmatiques et Applications, Universitk de La Rochelle, 17042 La Rochelle, France
[email protected]
NICOLASPRIVAULT Laboratoire de Mathematiques et Applications, Universitk de La Rochelle, 17042 La Rochelle, France
[email protected]
PATH-INTEGRAL ESTIMATES OF GROUND-STATE FUNCTIONALS R. VILELAMENDES(CMAF- UL, Lisboa) Based on a rigorous version of the path-integral ground-state method (currently used in molecular and condensed matter physics), a closed form representation of the ground-state functional is obtained. This provides a ground-state approximation scheme displaying non-perturbative features already at the lowest order.
1. Introduction
The path integral ground state (PIGS) method [l,21 is currently used for practical calculations in molecules as well as in infinite systems like liquid and solid 4He [3,4]. It is based in an asymptotic estimate of the ground state wave function, namely
&, (x)0: lim 7 ’ 0 0
s
dz’K (z, z’, T ) &- (d) ,
(1)
where x refers to the set of configuration space variables of the quantum many-body problem, $T (d)is a trial wave function and
K (z, z’,
T)
= (z, ePTHz’)
(2)
is the imaginary time propagator for the Hamiltonian H , computed by dividing the time in small intervals AT and summing over polygonal paths E statistical sampled with a probability density
p (E)
0: @T
(x)d‘T
(z’)
i”
1
K ( z j ,zj+l,AT) .
j=O
(3)
Starting from a rigorous version of (l),the purpose of this paper is to obtain a ground state representation providing an approximation scheme with convergence properties better than a perturbative expansion. The approximation scheme uses a Gaussian approximation for paths centered at each point of the wave function, thus corresponding to a different expansion for each individual configuration point. In Section 2, using conditions on the Hilbert spaces and Hamiltonians well established for finite-dimensional systems, several ground state representation formulas are established. They are rigorous versions of the formulae
167
R. VILELAMENDES
168
established in [5] by heuristic means and used in [6] to study nongeneric strata. In Section 3 a finite dimensional system is studied t o exhibit the non-trivial nature of the leading order term in the ground state representation. Finally, in the conclusions, conditions are established which would allow for the extension of the results t o infinite dimensional systems, in particular to the Yang-Mills theory in the Hamiltonian formulation. 2. The ground state path integral representation Consider an Hilbert space 7-l represented as L2 ( X ,dv) for some measure space X with measure dv. In this space let H be an Hamiltonian 1 2
H = --A+V(z),
zEX,
(4)
with V ( z ) continuous and bounded from below and H essentially selfadjoint. Then the kernel Kt (z, z’) of e-tH is strictly positive. This implies [7] the existence of a unique ground state R and for any E L2 ( X ,dv) with 11+11 # 0 and $ 2 0
+
Eo being the lowest eigenvalue of H . From the spectral representation of H
eCtH =
/
e-tEdP ( E )
one obtains
Under the stated conditions on H , one also has a Feynman-Kac representation for the kernel of e-tH,
dWi,,, being the Wiener measure for continuous paths on X going from t o z in time t. Then 1
R (z) = -lim etEo (Q, Ict) t-+m
/
dz’exp
(- J_”, V (rs)
ds) dW:,,,+
(z’)
2‘
, (9)
Path-integral estimates of ground-state functionals
169
where ys is a Brownian path pinned down to x at time zero and t o x' a t time -t. Equation (9) is a rigorous version of the path integral ground state method referred to in (1)-(3). Practical calculations in quantum theory are made by some kind of approximation method, usually based on the calculation of the first few terms in a power series expansion. In the Feynman-Kac formula for the kernel (after the integrable part of the Hamiltonian is included in the measure), the perturbation expansion would be obtained by expanding the remaining exponential
Usefulness of this expansion depends on the norm of V, , that is on the strength of the coupling constant. On the other hand, the semiclassical approximation would be obtained by reintroducing the Planck constant h and expanding on it. Here, instead, one uses an expansion of the argument of the exponential. For a strictly positive kernel, the ground state R may be chosen to be a strictly positive function. Therefore R2 (x) contains the same information as R (x).By a change s -+ --s in the time variable one converts the integral in (9) from ( - t , O ) t o ( 0 , t ) . Then
a2(z)
0;
lim
P
o
t-i"
s
dx'b ( 7 0 - x) exp
(- ltv
(7s)
d s ) dW2,,/?(, (z') >
(11) the Brownian paths being pinned to x a t time zero. Noticing that ?(, (z') is an arbitrary positive function, we may compute the integral (11) using a sequence of functions that are constant in successive larger domains (and zero outside). Choosing as normalization constant the same stochastic integral without the S (70- x) one obtains
R2 (x)=
J 6 ( 7 0 - x) exp (- J-", v (7s) ds) dW" J exp (- J-", v (7s) d s ) dW"
7
(12)
dW" denoting the Wiener measure for Brownian paths for t E (-00,00). To derive now from (12) a closed form expression, from which useful approximations may be obtained, one changes variables
7s = x + z ,
(13)
170
R. VILELAMENDES
and adds a source term to the potential
with z, E S' and J , E S , S' and S being respectively the Schwartz distribution and test function spaces. Then, assuming analyticity of the potential and denoting by G (2, 2,) the higher order terms,
S/SJ, denoting the Gateaux derivative. 6 ( 7 0 - x) = S (20) may be represented by
Substituting (15) and (16) in (12), one is left with Gaussian integrals over the Wiener measure. Computing these integrals one obtains
R2 (x)=
where
{.. .}+ meaning symmetrization. In the J
-+0
limit, L (x)reduces to
Notice that the Gaussian integrations that were performed assume the positivity of the matrix S (x).
Path-integral estimates of ground-state functionals
171
By computing successive terms in exp(J ds G(x, S / S J s ) ) one obtains successive approximations for the ground state. Notice the highly non-trivial nature of the leading term
3. A finite-dimensional example
Consider the Hamiltonian
H
=
1 d +V(y) 2 dy2
with
1 (22) 2 In a neighborhood of the origin this potential is qualitatively similar to the quartic anharmonic potential. However t o illustrate the approximation scheme developed in the previous section it is a more convenient potential because its ground state is exactly known
V(y) = -y2 - p y t a n h p y .
Making in (22) the change of variables y=x+z
(24)
and computing the linear and quadratic terms in an expansion of the potential around x , one obtains
r (x) = p
- p2x (1 - tanh2 (px)) - p tanh (px) 1 S (x) = - - p3x (tanh3 (px) - tanh (px)) - p2 (1 - tanh2 (px)) 2
with a leading term approximation for the ground state
R. VILELA MENDES
172
Figure 1. Exact ground state and leading order approximation
In the figure (for a coupling constant value p = 2 ) we compare the exact ground (23) (continuous line) with the approximation (25) (dotted plot). The region suppressed, near z = 0, corresponds t o the region where S (z) < 0, outside the validity of the Gaussian integrations leading to (25). Notice the good qualitative agreement of R(o) with the exact solution, whereas from an expansion in the coupling constant
H
= - -l- d
2 dy2
(i
+ --p
2 ) y2
+-y$
4
- - p2 y 6 15
6
f...
(26)
one concludes that, for this value of the coupling constant, a perturbative treatment would be hopeless. 4. Conclusions
1 - The reason why one has obtained results beyond the perturbation expansion is because, for the ground state wave function, the expansion is performed around each x. Therefore one is effectively dealing with a different expansion at each point. The improved convergence properties of expansions when the expansion center changes at each step of the calculations is a familiar feature for example in the Newton method or in KAM calculations.
Path-integral estimates of ground-state functionals
173
2 - When X is a finite dimensional space (X c Rd) and under the stated conditions on the Hilbert space N ,namely representatibility as L2 ( X ,dv), a real potential bounded from below and an essentially self-adjoint Hamiltonian, the ground state representation formulae (11) or (12) are valid. An important question for applications to Yang-Mills, Chern-Simons or string theories is whether the results can be extended to those infinite dimensional cases. Of particular relevance would be to extend the rigorous formulation to the Yang-Mills case, in particular to establish a rigorous foundation for the mass gap results obtained in [5]. A classical Yang-Mills theory consists of four basic objects: (i) A principal fiber bundle P ( M ,G) with structural group G and projection T : P -+ M , the base space M being an oriented Riemannian manifold. (ii) An affine space C of connections w on P , modelled by a vector space A of 1-forms on M with values on the Lie algebra 9 of G. (iii) The space of differentiable sections of P , called the gauge group W . (iv) A W-invariant functional (the Lagrangian) C : A 4 R. Choosing a reference connection, the a%ne space of connections on P may be modelled by a vector space of 6-valued 1-forms (C" (A' @ 6 ) ) . Likewise the curvature F is identified with an element of (C" (A2 @ 6 ) ) . The configuration space of the gauge theory is the quotient space d / W . For an Hamiltonian formulation of Yang-Mills, with AP = Agta ( { t a }a basis for the Lie algebra), one takes
as canonical variables. To extend the ground state discussion to the YangMills case one writes the Yang-Mills first-order action as
'I
I=g2
1 d42n o ~ A . E + - ( E ~ + B A ~' ) (V - E 2
{
with B, = - $ e i j k F i k . Then the Hamiltonian is
H = / d 3 2 x (E,"
+ B:)
a
and A' being a Lagrange multiplier, the constraint is
+ [A,E l ) }
(28)
174
R. VILELAMENDES
and the canonical brackets
The constraint (Gauss’s law) simply means t h a t the allowed physical states should be gauge invariant. For path integral calculations a measure in d / W would be desirable, but no such measure has been found for Sobolev connections. Therefore it is more convenient t o work in a space of generalized connections 2,defining parallel transports on piecewise smooth paths as simple homomorphisms from the paths on M t o the group G, without a smoothness assumption[8]. Then, there is in The same applies t o the generalized gauge group _ _ d / W an induced Haar measure, the Ashtekar-Lewandowski measure[9][lo]. Sobolev connections are a dense zero measure subset of the generalized connections [ 111. The extension of ground state representation, derived in the Section 2, t o this case (in the continuum), depends on: (i) the possibility of representing the Hilbert space 3-1 of states as L~ (2, dpA) for some measure P A , (ii) the existence of a Wiener measure on paths on 2, (iii) a Feynman-Kac formula in this infinite-dimensional setting. A natural setting for these purposes will be the construction of a Gelfand triplet (sd)c L 2 ( d , B , p A ) c (sd)* 1
w.
p~ being the Ashtekar-Lewandowski measure.
References 1. D. M. Ceperley, Rev. Mod. Phys. 67,279 (1995). 2. A. Sarsa, K. E. Schmidt, W. R. Magro, “A path integral ground state method”, J . Chem. Phys. 113, 1366 (2000). 3. J. E. Cuervo, P.-N. Roy, M. Boninsegni, “Path integral ground state with a fourth-order propagator: Application to condensed helium”, J . Chem. Phys. 122,114504 (2005). 4. W. Purwanto, S. Zhang, “Quantum Monte Carlo for the ground state of many-boson systems”, Phys. Rev. E 70,056702 (2004). 5. R. Vilela Mendes, “Stochastic processes and the non-perturbative structure of the QCD vacuum”, Z. Phys. C - Particles and Fields 54, 273-281 (1992). 6. R. Vilela Mendes, “Stratification of the orbit space in gauge theories. The role of nongeneric strata”, J . Phys. A : Math. Gem 37,11485-11498 (2004). 7. L. Gross, “Existence and uniqueness of physical ground states”, J . Funct. Anal. 10, 52-109 (1972).
Path-integral estimates of ground-state functionals
175
8. A. Ashtekar, C. J. Isham, “Representations of the holonomy algebras of gravity and nonabelian gauge theories”, Class. Quant. Grav. 9, 1433-1468 (1992). 9. A. Ashtekar, J. Lewandowski, “Differential geometry on the space of connections via graphs and projective limits”, J . Geom. Phys. 17,191-230 (1995). 10. A. Ashtekar, J. Lewandowski, “Projective techniques and functional integration for gauge theories”, J.Math. Phys. 36,2170-2191 (1995). 11. D. Marolf, J. M. MourBo, “On the support of the Ashtekar-Lewandowski measure”, Commun. Math. Phys. 170,583-606 (1995). R. VILELAMENDES CMAF, Complexo Interdisciplinar, Universidade de Lisboa,
Av. Gama Pinto, 2, 1649-003 Lisboa, Portugal
[email protected]
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A REPRESENTATION THEOREM AND A SENSITIVITY RESULT FOR FUNCTIONALS OF J U M P DIFFUSIONS GIULIADI NUNNO(University of Oslo), BERNTQKSENDAL (University of Oslo / NHH,Bergen) We use white noise calculus for LBvy processes t o obtain a representation formula for the functionals of a jump diffusion. Then we use this to find an explicit formula for the Donsker delta function of a jump diffusion and we suggest its application to sensitivity analysis in mathematical finance for the computation of the Greeks.
1. Introduction
A difficult, but crucial, task in the analysis of option prices is the prediction of their variation. To this aim it is important to locate which are the factors contributing to the fluctuation of prices and their effect. The sensitivity analysis is carried over the parameters appearing in the models for the price dynamics and the so-called Greeks represent a form of measure for the price sensitivity to some factors. For example, the “delta” is related t o the initial price of the option, the “theta” is related to the time until maturity, the “rho” t o the interest rate, the “vega” is the sensitivity t o the volatility, etc. Efficient techniques for the computation of the Greeks rely on numerical finite difference methods and simulation. See [16], [14], for example and references therein. However, too often some restriction on the regularity of the price processes has t o be imposed. In the recent years high attention was dedicated to finding more efficient and more general methods t o apply numerics and simulation for the computation of the Greeks. The papers [13] and [12] proved that, with a preliminary application of sophisticated tools of stochastic analysis, soiae better formulae could be derived which would ease a direct application of Monte Carlo simulation. Their method, based on Malliavin calculus, applies to price dynamics driven by Brownian motion only. See also [5], [15], [24], [26],for example, and references therein. Several forms of generalization or extension to include dynamics driven by Poisson processes or combinations of independent Brownian motions and Poisson processes have been suggested. We can refer to [4], [8], [ll],[32], for example. In this paper we present a representation formula for functionals of jump
177
GIULIADI NUNNO,BERNTDKSENDAL
178
diffusions (see Theorem 3.1) which, if applied to the sensitivity analysis context, gives a computational efficient formula for the Greek "delta". We frame our method in the setting of white noise analysis. A short introduction to this framework with the preliminary results is given in Section 2. Section 3 presents the representation formula for functionals of a jump diffusion. Moreover, we apply this result to give an explicit representation of the Donsker delta function. Our approach is in the same line as [27]. This results gain importance in view of the applications of the Donsker delta function for the computation of hedging portfolios in mathematical finance. See [2] for the Brownian motion setting and [9] for the pure jump LQvyprocesses case. Section 4 is dedicated to the sensitivity analysis. 2. Framework Let (R, 3,P ) be a complete probability space and &(P) the standard (complex) L2-space of the random variables with finite norm ll
<
where Q E R, a2 > 0 are constants and v(dz), z E Ro, is a a-finite Bore1 measure on Iwo := R \ (0). Note that
namely we assume that r](t)E L2(P) for all t 2 0. Throughout this paper we will always consider the chdlbg modification r](t),t 2 0, of the stochastic process above. We can refer to [6], [7], [33], for example, for general and detailed information about LQvyprocesses. In particular we recall that, for every t , the random variable r](t)admits a representation in the form
where the standard Brownian motion B ( t ) ,t 2 0, and the compensated Poisson random measure
N ( d t ,d z ) := N ( d t ,d z ) - v ( d z ) d t , t >_ 0, z E R,
A representation theorem and a sensitivity result for functionals o f . . .
179
are independent - cf. [20]. Inspired by the stochastic integral representation (1) it is natural to consider stochastic processes [ ( t ) , t 0 , of the form
>
where a ( t ) ,P ( t ) and y ( t , z ) , t 2 0 , z E Ro, are deterministic functions satisfying
On the other side, in line with the approach suggested in [3], we could consider a representation of type (2) embedded in a multidimensional framework as follows. Let us consider the probability space (0,3,P) as a product of two complete probability spaces, i.e., R=01
X02,
3 = 3 1 @ 3 2
P=Pi@Ppz.
(4)
In such a framework we could consider stochastic processes [ ( t ) , t 2 0 , on (0,3,P) such that
I ( t ,w1, w2) = Y +
for y E R constant and a(t),P(t)and Y ( t , z ) , t 2 0 , z E Ro, deterministic functions satisfying ( 3 ) . We equip the probability space (01,31,PI) with the filtration F:, t 0 , (3& = 3 1 ) generated by B ( t ) ,t 2 0 , augmented of all PI-null sets and the space ( 0 2 , 3 2 , P 2 ) with the filtration I-:, t 2 0 (32= 3 2 ) generated by the values of f i ( d t , d z ) , t 2 0, z E Ro, augmented of all P2-null sets. Then on the product (0,3,P) we fix the filtration
>
t 3 t := 3;@3?,
> 0.
In the sequel we apply white noise analysis and techniques. Thus we choose t o set (01,3 1 , PI) to be a Gaussian white noise probability space and ( 0 2 , 3 2 , P 2 ) a Poissonian white noise probability space. General references to white noise theory for Gaussian processes are e.g. [17], [18], [19], [25], [28]. As for a white noise theory to non-Gaussian
GIULIADI NUNNO,BERNT0 K S E N D A L
180
analysis we can refer to e.g. [l],[lo], [22], [23], [29], [31]. In order to keep this presentation moderate in size we recall here only the Poisson white noise framework in the approach and notation of [lo] and [29]. To ease the notation we drop the index of (02,F2,P2) and we write ( 0 2 , 3 2 , P2) = (0,F, P ) from now up to the end of this section. From now on we assume that for every E > 0 there exists p > 0 such that
This condition implies that the polynomials are dense in L2(p) where p(dz) = z2v(dz). It also guarantees that the measure Y integrates all polynomials of degree greater than or equal to 2. Let A denote the set of all multi-indices (Y = ((YO, a l l .. .) which have only finitely many non-zero values ai E N \ (0). In the space L2(0,F,P ) = L2(02,32,P2) we construct the orthogonal basis K,, a E A, as follows. First of all we consider the orthonormal basis (pi, i E N, in & ( A ) constituted by the Laguerre functions (order 1/2). Here and in the sequel A(&) = d t denotes the Lebesgue measure on the real line. Moreover we take an orthonormal basis $ j , j E N,in L ~ ( vof) polynomial type. See e.g. [29] for further details. Then we can consider the products Ck(t,z ) =
cpi(t)$ j ( Z )
(7)
for Ic = k ( i , j ) as a bijective mapping k : N x N --+ N (e.g. the diagonal counting of the Cartesian product N x N). For any a E A with max{i : ai # 0) = j and la1 := a{ = m, we can define
xi
< y ( t I , Zl),. . . , ( tm,z m ) ) :=
and = 1. Moreover, we denote the corresponding symmetrized tensor product by (6,. We can now construct an orthogonal basis K,, a E A, in L , ( P ) as follows: K , := I I , ~ ( < ~ ~ )E, A, (8) where
A representation theorem and a sensitivity result for functionals of . . .
181
is the It6 iterated integral with respect to the centered Poisson stochastic measure. See [20]. Here f E &(A x v ) is~ symmetric in the n pairs (tl,zl), . . . , (tn,zn).Note that
- q L z ( g ).
Mf)] = 0,
m # n,
and for all symmetric g E L2(A x v ) and ~ f E & ( A x v)* (m,n E Hence every E E & ( P ) admits the chaos expansion
N).
CYEA
and CY€A
CYEA
where a! := a l ! a 2 ! .. . for a = (a1,a2,.. .) E A. Thanks to these chaos expansions we can characterize the following spaces and chain of embeddings. By ( S ) , (0 5 p 5 1) we denote the space of all random variables f = C a E A c C Y KEa L , ( P ) such that
I I ~ I I=~c, (~a ! ) l + p c ; ( 2 ~ )<~03~
for all k E N,
CY€A
where (2N)kCY := ( 2 . 1 ) k C Y l ( 2 . 2 ) " 2 . . . ( 2 . j ) ~ C iYf~j = m u { i : aj # 0). And by (S)-, we denote the space of all random variables F = ~ a E A ~ , K E , L2(P) such that
~
~
C
F = ~ (~~ ! )~' - P ~ c ; ( ~, N )- -< ~ ~~ cm
for some IC E N.
CY€A
The subspaces (S), and ( S ) - , are respectively equipped with the projective topology and the inductive topology induced by the above seminorms. Note that for any F = CaEA uaKa E (S)-, and f = CaEA b,K, E ( S ) , the action uCY baa!
( F , f ):= CYEA
is well-defined and thus the space (S)-, is the dual of (S),, i.e., ( S ) - , (S);. We remark that, for p = 0, the spaces (S):= (S)o and (S)* = (S):
= =
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GIULIADI NUNNO.BERNTD K S E N D A L
(S)-Oappear respectively as a LQvyversion for the Hida test function space and Hida distribution space for pure jump Levy processes. See e.g. [18], [19], [25], [28]. For p = 1, the spaces (S)1and (S)-l are the Levy version of the Kondratiev test function space and the Kondratiev distribution space respectively. See [21] and also [19], for example. The following relationships hold true
(S)l c ( S ) , c ( S )c L 2 ( P ) c (S)*c ( S ) - , c (S)-l. The relevance of these spaces will be clarified in the sequel. For instance ( S ) * is rich enough to contain the white noise of the centered Poisson stochastic measure and of the pure jump L6vy process as its elements. In fact, let us consider the random variable = N ( t , B ) E L,(P), for any Bore1 set B E B(R\{O}), then the following chaos expansion can be written:
<
via the application of the orthogonal basis in L 2 ( X ) arid Lz(v) and the diagonal counting - cf. (7). Moreover, for any k E N,the multi-index € k = (el, k c2, k . . .) is defined by 6;
:=
1, i = k
0, otherwise. Here we refer to [29] for all the details. Then we can define the white noise f o r the centered Poisson stochastic measure as the element
in ( S ) * ,for almost all t Nikodym derivative
2 0, z
E R. Naturally it appears as the Radon-
The Livy- Wick product F o G of two elements F CPEA bpKp in (S)-l is defined by
=
CaEd aaKa
and G =
A representation theorem and a sensitivity result for functionals of . . .
183
It can be shown that the spaces (S)1, ( S ) ,(S)* and (S)-l are closed under Wick products. One of the useful features of the Wick product is the following relationship within It6 stochastic integration and Bochner integration:
J,”J,
Y ( S , z ) fi(ds,d z ) =
itl
Y ( s ,z ) o
k(s, z ) v(dz)ds.
We also mention that for all F E (S)-l one can define the expo F E (S)-l by M
expo F :=
C ~~
n=O
(13)
Wick exponential
-
I - F~~ n!
and the following property E(expo F) = exp{EF} holds true. 3. A representation theorem for functionals a, a class o
diffusions Let J ( t ) = Jy(t), t E [O,T],be a stochastic process on ( R , F , P )- cf. (4), of the form
i
dJ(t) = a ( t ) d t
J(0)
+ D(t) d q t ) + J, y(t, z ) m t , d z ) ,
t E [O, TI,
(16)
=Y E
where a(t),D(t) and y(t, z ) , t E [O,T],z E R, are deterministic functions satisfying (3) - cf. (5). For X E R define Y(t) = exp(XJ(t)), t E [O,TI. Then by the It6 formula (see e.g. [30, Chapter 11) we have
(17)
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GIULIADI NUNNO,BERNTDKSENDAL
Using white noise notation and Wick calculus this can be written
+ @ 2 p 2 ( t ) + X@(t)B(t)
d Y ( t ) = Y ( t - )0 Xa(t) dt
{exY(s,z) - 1 - Xy(s, z ) } v ( d z )
(20)
Comparing (20) with (17) we get the following formula for the Wick exponential. Lemma 3.1. With [ ( t ) as in (16) and X E R we have
Using this we obtain the following result:
Theorem 3.1 (Representation theorem for functionals of a jump diffusion). Let g : R 4 R be a function with Fourier transform
and satisfying the Fourier inversion property
A representation theorem and a sensitivity result for functionals of . . .
185
where
Proof. Applying (21) with i X instead of X we get
Corollary 3.1. Let g be a real function as in Theorem 3.1. Then we have (24) where
GA(t)=
it{
i X a( s )- i X 2 p 2 ( s )
+
s,
(ei~7(59~) - 1(25)
Proof. This follows from Theorem 3.1 plus the fact that (see (15)) E[expOXi@)]= exp(E[Xi(t)]). In the last part of this section we obtain an explicit formula for the Donsker delta function of <(t)= < Y ( t ) , t 2 0. This is derived as an application of Theorem 3.1. The Donsker delta function is a generalized white noise functional, we can refer e.g. [17], [18], [25], for general information. Here we give its definition
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GIULIADI NUNNO,BERNT0KSENDAL
within the white noise framework we have introduced, in the line of [3] and [27]. Note that the Donsker delta function has been used for giving explicit representation formulae for the hedging portfolio in some market models driven by Brownian motion or pure jump L6vy processes, see [3] and [9].
Definition 3.1. For a given random variable X E (S)-1the Donsker delta function of X is a continuous function b.(X) : R 4 (S)-1 such that
for all Bore1 real functions h on R for which the integral is well-defined in (+I. Following [27] we consider a measure P2 in P = PI @I P2 (see (4)) which satisfies the condition: there exists E E ( 0 , l ) such that lim Iul-('+')Re 14--+~
1 - iuz) v ( d z ) } = 0 0 .
(26)
Using Theorem 3.1 we can obtain an explicit formula for the Donsker delta function of [ ( t ) :
Theorem 3.2. Assume that (26) holds. Then the Donsker delta function b,([Y(t)), u E R, ofEY(t), t E [O,T],exists in (S)* and is given by
I
+ i X a ( s ) - !jX2p2(s) ds + iXy
-
1
iXu dX.
(27)
Proof (sketch). Formally this follows from (22) by using the Fubini theorem in (S)*,as follows. By (22) we have
For justification and more details we refer t o the proof in [27, Theorem 3.1.41. 0
A representation theorem and a sensitivity result for functionals of
., ,
187
4. Application to sensitivity with respect to the starting point Let X ( t ) = X " ( t ) , t E [O, TI, be a jump diffusion of the form
+a ( t ) d B ( t )+
p(t)dt
s,
O(t,z)a(dt,dz)] (28)
X(0)=x > 0 where p ( t ) ,a ( t )and e(t,z ) , t E [0, TI, t E R, are deterministic, e(t,z ) for a.a. t , z and
> -1
- cf. (3). By the It6 formula for LBvy processes (see e.g. [30, Theorem
1.141, the solution of this equation is
+
( I n ( l + e(t,z)) - e(t,z ) ) v ( d t ) ,
a ( t )= p ( t ) - + g 2 ( t )
IR
p(t) = a ( t ) , y(t, t) = ln(1 + - cf. (16). Therefore, if h : R -+
(30)
e(t,z ) ) and y = l n x
R then
JW(X"(T))l= E[h(exp(EY(T)))l= -wEY(T))l, where g ( u ) := h(exp(u)),
u E R.
If this g satisfies the conditions of Theorem 3.1 then
/
d d i(X)exp(iXlnx - -E[h(X "(T ))] = -[-Eg(J1""(T))] = dx dx dx w =Lij(X)
iX 31 exp(iX1nx + G x ( T ) )d X ,
where G x ( T )is given by (25). We have proved
+ G x ( T ) )dX
GIULIADI NUNNO,BERNT0 K S E N D A L
188
Theorem 4.1. Suppose h : R -+ R is such that g ( u ) := h(exp(u)), u E R, satisfies the conditions of Theorem 3.1 and that
Then d
--E[h(X"(T))] = dx
/
w
iX
@(A) 1exp(iX1nx
+ G x ( T ) )d X .
(31)
Example 4.1. Choose h ( u ) = X [ H , K ] ( u ) , u E R ( H , K > 0 ) . Then h ( X " ( T ) ) may be regarded as the payoff of a digital option on a stock with price X z ( T ) . In this case
4.u.) = X[H,K](e"), u E R, and
Therefore
d -dx -E[X[H,K](Xz(T))]
/
= w
H-ix - K-ix X
exp(iX1nx
provided that the integral converges. A sufficient condition for this is that, for some 6
X2
lT{
I
+ G x ( T ) )dX ,
(32)
> 0,
,B2(s)-+ L ( l - cos(Xy(s, z ) ) ) v ( d z ) ds 2 bX2 for all X
E
R,
which is a weak form of non-degeneracy of the equation (28). Thus, in spite of the fact that h is not even continuous, (31) is a computationally efficient formula for $ E " [ h ( X " ( T ) ) ] .
References 1. S. Albeverio, Y. G. Kondratiev, L. Streit, "HOWto generalize white noise analysis to non-Gaussian spaces", in Ph. Blanchard et al. (eds), Dynamics of Complex and Irregular Systems, World Scientific, 1993. 2. K. Aase, B. Bksendal, J. Uboe, "Using the Donsker delta function to compute hedging strategies", Potential Analysis 14, 351-374 (2001).
A representation theorem and a sensitivity result for functionals of . . .
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3. K. Aase, B. Bksendal, N. Privault, J. Ubme, “White noise generalizations of the Clark-Hausmann-Ocone theorem with application to mathematical finance”, Finance Stoch. 4,465-496 (2000). 4. M.-P. Bavouzet, M. Massaoud, “Computation of Greeks using Malliavin’s calculus in jump-type market models”, Report 5482, INRIA, Rocquencourt, France, 2005. 5. E. Benhamou, “Optimal Malliavin weighting function for the computation of the Greeks”, Conference on Applications of Malliavin Calculus in Finance (Rocquencourt, 2001). Math. Finance 13,37-53 (2003). 6. Yu. M. Berezansky, Yu. G. Kontratiev, Spectral Methods in Infinite-Dimensional Analysis, Kluwer Academic Publishers, 1995. 7. J. Bertoin, Ldwy processes, Cambridge University Press, 1996. 8. M. H. A. Davis, M. P. Johansson, “Malliavin Monte Carlo Greeks for jump diffusions”, Stochastic Process. Appl. 116, 101-129 (2006). 9. G. Di Nunno, B. Bksendal, “The Donsker delta function, a representation formula for functionals of a LBvy process and application to hedging in incomplete markets”, Preprint Series in Pure Math. 11, Dept. of Mathematics, University of Oslo (2004). To appear in Sem. Congres. Ac. Sci. 10. G. Di Nunno, B. Bksendal, F. Proske, “White noise analysis for LBvy processes”, Journal of Functional Analysis 206, 109-148 (2004). 11. Y . El-Khatib, N. Privault, “Computations of Greeks in a market with jumps via the Malliavin calculus”, Finance Stoch. 8, 161-179 (2004). 12. E. FourniB, J.-M. Lasry, J . Lebuchoux, P.-L. Lions, “Applications of Malliavin calculus to Monte-Carlo methods in finance. 11”,Finance Stoch. 5, 201-236 (2001). 13. E. FourniB, J.-M. Lasry, J. Lebuchoux, P.-L. Lions, N. Touzi, “Applications of Malliavin calculus to Monte-Carlo methods in finance.” Finance Stoch. 3, 391-412 (1999). 14. P. W. Glynn, “Optimization of stochastic systems via simulation”, in Proceedings of the 1989 Winter simulation Conference, San Diego: Society for Computer Simulation, 1989, pp. 90-105. 15. E. Gobet, R. Munos, “Sensitivity analysis using It6-Malliavin calculus and martingales, and application to stochastic optimal control”, S I A M J. Control Optim. 43,1676-1713 (2005). 16. P. Glasserman, D. D. Ym, “Some guidelines and guarantees for common random numbers”, Manag. Sci. 38,884-908 (1992). 17. T. Hida, “White noise analysis and its applications”, in L.H.Y. Chen (ed.), Proc. Int. Mathematical Conf., North-Holland, Amsterdam, 1982, pp. 43-48. 18. T. Hida, H.-H. Kuo, J. Potthoff, L. Streit, White Noise, Kluwer, Dordrecht, 1993. 19. H. Holden, B. Bksendal, J. Uboe, T.-S. Zhang, Stochastic Partial Differential Equations - A Modeling, White Noise Functional Approach. Birkhauser, Boston, 1996. 20. K. It6, “Spectral type of the shift transformation of differential processes with stationary increments”, Trans. A m . Math. SOC.81,253-263 (1956).
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21. Yu. G. Kondratiev, Generalized Functions in Problems in Infinite Dimensional Analysis, Ph.D. Thesis, University of Kiev, 1978. 22. Y. Kondratiev, J. L. Da Silva, L. Streit, “Generalized Appell systems”, Methods Funct. Anal. Topology 3,28-61 (1997). 23. Y . Kondratiev, J. L. Da Silva, L. Streit, G. Us, “Analysis on Poisson and gamma spaces”, Inf. Dim. Anal. Quant. Prob. Rel. Topics 1(1), 91-117 (1998). 24. A. Kohatsu-Higa, M. Montero, “Malliavin calculus in finance”, Handbook of computational and numerical methods in finance, Birkhauser, 2004, pp. 111174. 25. H. H. Kuo, White Noise Distribution Theory, Prob. and Stoch. Series, Boca Raton, FL, CRC Press, 1996. 26. P. Malliavin, A. Thalmaier, Stochastic calculus of variations in mathematical finance, Springer Finance, 2006. 27. S. Mataramvura, B. Bksendal, F. Proske, “The Donsker delta functin of a Lkvy process with application to chaos expansion of local time”, Ann. Inst. H. Poincare‘ Probab. Statist. 40, 553-567 (2004). 28. N. Obata, White Noise Calculus and Fock Space, LNM, 1577, Springer-Verlag, Berlin, 1994. 29. B. Bksendal, F. Proske, “White noise of Poisson random measures”, Potential Analysis 21,375-403 (2004). 30. B. Bksendal, A. Sulem, Applied Stochastic Control of Jump Diffusions, second edition, Springer, 2006. 31. N. Privault, “Splitting of Poisson noise and LBvy processes on real Lie algebras”, Infin. Dimen. Anal. Quantum Probab. Relat. Top. 5,21-40 (2002). 32. N. Privault, X. Wei, “A Malliavin calculus approach to sensitivity analysis in insurance”, Insurance Math. Econom. 35,679-690 (2004). 33. K. Sato, Le‘vy Processes and Infinitely Divisible Distributions, Cambridge University Press, 1999. GIULIA DI NUNNO Centre of Mathematics for Applications (CMA), Dept. of Mathematics, University of Oslo, P.O. Box 1053 Blindern, N-0316 Oslo, Norway
[email protected]
BERNT0 K S E N D A L Centre of Mathematics for Applications (CMA), Dept. of Mathematics, University of Oslo, P.O. Box 1053 Blindern, N-0316 Oslo, Norway and Norwegian School of Economics and Business Administration, Helleveien 30, N-5045 Bergen, Norway oksendal(0math.uio.no
CREATION AND A N N I H I L A T I O N OPERATORS ON L O C A L L Y COMPACT SPACES WILHELM VON WALDENFELS ( Universitat Heidelberg) Denote by cs the point measure in the point z of a locally compact space X . Define the annihilation operator a ( € %and ) the creation operator a f ( & . ) ( d x ) .We establish the usual algebraic relations and prove a generalization of Wick's theorem. If a privileged measure on X is introduced, a duality theory can be established similar to the habitual one. We derive a generalization of a convolution formula due t o Meyer.
1. Introduction
A basic formula of quantum probability is [as,
41 = 6(s - t),
the commutator of the annihilation operator in the point s E R with the creation operator in the point t E R. As the appearance of Dirac's 6function indicates, at and a; are highly singular. Hudson and Parasarathy [ll]choose a way out in working with the differentials t+dt
a,ds,
dA,f =
4
t+dt
a , f d s , dAt =
4
t+dt
a$as ds.
Another solution is to use white noise analysis as proposed by Obata [lo]. As the physicists use the calculus based on the commutation relation naively and with big success, Accardi, Lu and Volovich [l]attempted to put the physicists' calculus on a solid base using the theory of Schwartz distributions. An old tool of physicists is to represent operators in a form of a power series
c 00
T ( f )=
l,m,n=O
. a ;tl,., . t,; u11. ' . a: . . . aGall; ' . ' a + t m a t ,. ' .atma,, ... au, dsl . . . dsl dtl.. . dt, dul . . . dun .
S f ( S 1I
. ,
1
1
I'LL,)
The idea introduced by Maassen [8] into quantum probability was to establish a formula T ( f ) T ( S )= T ( f * g ) ,
191
192
WILHELM VON WALDENFELS
and t o work with the convolution f ,g H f * g instead with the operators a t , a t . Maassen did so with operators not including the number operator] i.e., where the term a,f, ’ . ’ a:mat, . . atm does not arise. Meyer [9] succeeded to give a formula for the convolution including the number operator. The analysis using convolution becomes a lot easier, but the computations get very difficult, especially in the case with the number operator included, if one has t o do with the complicated formula due to Meyer [9, p. 921. In a previous paper the author generalized Maassen’s approach t o locally compact spaces in order t o use measures in the place of test functions [15]. But it seemed t o be too cumbersome to include Meyer’s theory. So we turn t o the ideas of Accardi, Lu, Volovich, and try to imitate in this paper the physicists’ approach by directly dealing with the operators at, a:. We do not use Schwartz distributions] but deal with measure theory in the sense of Bourbaki’s integration theory, especially his chapter about integration of measures [4,5]. Instead of R and the usual Fock space we work with a locally compact space X and the space X of finite sequences of elements of X . We have no privileged measure on X like the Lebesgue measure in the usual theory. If E, is the point measure in the point z E X , then at and a: are replaced by
In section 2 we introduce the basic notations. In section 3 we study admissible monomials, these are monomials in a(&,), a + ( c . ) ( d z ) ,where any variable may arise basically once with the exception that monomials of the form
* . . . * a+(&,)(&)* . . . * a(&,) * . . . * are allowed. In the contrary monomials of the form
are forbidden. In section 4 we prove a generalization of Wick’s theorem. In section 5 we introduce a distinguished measure replacing Lebesgue measure in the usual theory and treat duality. In order t o check our theory we derive in section 6 the generalization of Meyer’s convolution formula.
Creation and annihilation operators on locally compact spaces
.193
2. Preliminaries. Let X be a set. We denote by D ( X ) the set of all finite sequences of elements of X or words formed by elements of X and write for short X = D ( X ) and we have
X
=
0 + x + x 2+ ...
We use the plus sign for denoting disjoint union of sets. We denote by 6 ( X ) the set of finite subsets of X and by m(X)the set of finite multisets of X. A multiset is a pair rn = (S, T), where S is a set and T is a function T : S + N = {0,1,. . .}. The cardinality of m is #m = CsES ~ ( s ) . The multiset is finite if its cardinality is finite. We may write multisets in the form m = { a l , . . . , a n } , where not all the ai must be different. So, e.g., {2,1,1,2,3} is a multiset of the set {1,2,3,4} with r ( 1 ) = r(2) = 2, ) 0 and {1,1,2,2,3} denotes the same multiset. r(3) = 1, ~ ( 4 = If w = ( 2 1 , . . , s), is a word and u is a permutation, then
uw = (sg-l(l), . . . ,xg-l(")). The word w defines a multiset
rn, = { X I , . . . , s"}. If w' is another word, then m,, = rn, iff there exists a permutation changing w into w'. An n-chain is a totally ordered set of n elements a = ( a 1 , .. . , a n ) .
We denote by x a the word s g = (sal,...,xan).
If a is the underlying set of
a, we denote by
2 ,
=
{sal,...,san}
the corresponding multiset, as it does not depend on the order of a. A function f on X is called symmetric, if f(w) = f(uw) for all permutations of w. If a is an n-set, i.e., a set with n elements, then f(sCa) is well defined. Assume now that X is a locally compact space denumerable at infinity, provide X" with the product topology and X = D ( X ) with that topology
194
WILHELM VON WALDENFELS
where the X" are as well open and closed and where the restriction to X" coincides with the topology of X". Then X is locally compact as well, its compact sets are contained in a finite union of the X" and their intersections with the X n are compact. If S is a locally compact space, denote by K ( S ) the space of complex valued continuous functions on S with compact support and by M ( S ) the space of complex measures on S. If p is a complex measure on X,we write p = po
+ -l!1p 1 +
1 2!
-pz
+ .. .
where pn/n! is the restriction of p to X". We denote by 9 the measure given by
Q(f)= f(0). Then po is a multiple of 9. If
(1)
is an n-chain, we denote
~ ( d z , ) = pn(dza,,...7dzan).
If p is a symmetric measure, then p(dz,), where a is an n-set is well defined. One has
A hierarchy is a family of finite index sets
such that #an = n. We write
or simply
or
s,
P(a) f(a)
if the variable z is clear. Recall lemma 2.1 [15].
Creation and annihilation operators on locally compact spaces
195
Lemma 2.1 (Sum-integral lemma for measures). Let be given a bounded measure p(dW1,. . . dwk) 1
on
symmetric in each of the variables wi and write
where v is a bounded measure o n X and v=
c
1
-vn, n!
Here vn/n! is the restriction of v to X" and
where /?I . , . /?k are disjoint sets. Using hierarchies A l l .. . A k , B we may write
with
We denote by
K
= K,(X)
196
WILHELM VON WALDENFELS
the space of continuous, symmetric functions with compact support in X. We define creation and annihilation operators for symmetric functions and measures on X. Assume a function cp E I c ( X ) , a function f E Ic, a measure v E M ( X ) ,a symmetric measure p E M ( X ) . We define (a(v)f)(z1,...,4=
/
.(d~O)f(~O,~1,.~.,5n)
+
+
or, in another notation, where a c = a {c} means that the point c is added to the set a and similar using a \ c = a \ {c}
CECY
If @ is the function
@(0) = I, @ ( z a= ) O for a # 0, then
a(.)
= 0.
Similarly if 9 is the measure defined in (l),then .(cp)9 = 0.
One finds the commutation relations
and obtains
(7)
Creation and annihilation operators on locally compact spaces
197
We define the exponential measures and functions
If S is a locally compact space and p a measure on S and f a Bore1 function, we define the product f p by
/(flL)(W
4s) =
/
CL(ds) f (s) 9 4 s )
for cp E K ( S ) and write (fP)(dS) = (CLf)(ds) = f(s)cL(ds). Let S and T be locally compact spaces. We consider a function f : S M ( T ) . It can be considered as a function
4
f : s x K(T)-4 c and we write it
f X
= f(s,dt).
We extend the notion of the creation operator to functions + M ( X ) and define for g E K,(X) (a+(f)g)(%, dY) =
c
f@C,
dY) g(xa\c).
CE a
We consider the function E
: 3: E
/Ex(dY)
X
+-+ E ,
EM(X)
cp(Y) = cp(x),
so E, is the point measure in the point x, and
(a+(E)g)(xa,dY) = C E X C ( d Y )g(xa\c). CECY
f = f (x,d y ) :
198
WILHELM VON WALDENFELS
We may consider a+(&)as an operator valued measure and write
a+(&)= U + ( & ) ( d Y ) .
(16)
We obtain the commutation relations [4&z),
&/)I
[a+(W.), a + ( W y ) ] [4Ez),
.+(&)(dY)]
=0
(17)
=0
(18)
= &z(dY).
(19)
We extend this notion to a Bore1 function g : y E X write (a+(&)gy>(%,
dy) =
c
&z,(dY)
H
g y E K , ( X ) and
gy(zo\c).
CEol
Here appears the product of the measure sz,(dy) with the function Then A special case arises if gy =
gy.
So
is the operator analogous to the number operator
3. Admissible monomials Recall the definition of the space K of all continuous symmetric functions with compact support in X. Assume x and Q two disjoint index sets and define K X , @as the space of all functions
f :X XXe
+M(X")
with the following property:
(w, 'p, Y) E X x K ( X " ) x
xe
-+
1
f ( w , dz, Y) 'p(z)
X"
is symmetric in w, continuous and for fixed compact support in X x X @ . For c $ x Q define
'p
a continuous function with
+
ac
Kr,e
(acf)(GY, (dZp)p€n, (Zr)rEe+c)
+
G,e+c
= f(%+c,
(d%)pErrr ( z r ) r € e )
Creation and annihilation operators on locally compact spaces
and for
Here
If c @
e the right-hand
side yields
x E z t , ( d x c )f (xa\b,
(dzp)pE,rr
(zr)r€e)
bEa
and for c E p it yields &zb(dxc)f(xa\b,
(dxp)pE.rr,( % c , (x?-)r€@\c))
bEa
One calculates easily
Lemma 3.1. Assume c # c' and f E KT,@. If c,c' $! 7r
i f c, c' @
7r
then
+ a,+ a,tf
+ e then
- a,,a, + + f, -
a n d i f c $ ! r + e andc'@7r then
u , u $ ~= Ez,(dzci) f
+ a:ac f .
Definition 3.1. A sequence
w =( a 2 , . . . , a : ; ) with c l , . . . ,c, indices and
Bi =
f l and
u : = { a$ a,
for O = + 1 for 0
= -1
is called admissible if
i > j =+
{G # cj or {ci = cj and Oi = l , O j = -1)).
199
200
W I L H E L M VON WALDENFELS
So W is admissible if it contains only pairs (not necessarily being neighbors) of the form ( a9c ,a%’c , )with c # c’ or (a:, a,) and no pairs of the form ( a c , a c ) , ( a+, , a+ c or ( a c , a 3 . Definition 3.2. If W is an admissible sequence, define
Lemma 3.2. If W is admissible and Wl W is admissible, then
(a:, W ) is ( a c ,W ) is ( W , a z ) is (W,a,) is
If W
admissible admissible admissible admissible
c W , then Wl is admissible. I f c $i! w + ( W )
*
4
c w(W) c $i! w ( W ) c 4 w- ( W ) .
= (WZ, WI) is admissible, then
a:, W l ) is admissible (Wz,ac,W1) is admissible
(“2,
++
c $i! w ( W 2 ) U w + ( W l ) c 4 w - ( W 2 ) u w(W1).
Proposition 3.1. Assume
W = (a?, . . . , a::) to be an admissible sequence. Assume disjoint index sets
w + ( ~ ) n ~ = 0 w - ( w )n (T e) = 0.
+
Define for k = 1 , .. . , n
Set
TO
=T,
= e and
T
and
e and
Creation and annihilation operators on locally compact spaces
201
where f o r sets a,p ff
\P
= ff
\ ((.nP).
Then a$
"nk-1,ek-l
i 'rk,@k
and the iterated application
with
Definition 3.3. If
W = (a$,...,az;) is an admissible sequence we call
an admissible monomial.
Proof. We prove by induction. The case k etc.. Assume xk-1 ek-1
+
= @
= 1 is trivial.
Put W k = w ( w k ) ,
W+,k-lr
\ (W+,k-l \ w - , k - 1 )
+
w-,k-l
\ w+,k-l.
Assume 0 k = +l. In order that a& is defined, c k $ 7 r k - 1 . But assumption and c k $ W + , k - l , as w k is admissible. So
Now 7 r k - l - k C k = T + W + , l , = 7rk and it can be seen easily that Assume, now, that 0 k = -1. In order that ack is defined,
But
Ck
$ 7r + e by assumption and Ck $ W k - 1 , as w
k
Ck
@k-l\Ck
is admissible.
$?!
7r
by
= @k.
202
WILHELM VON WALDENFELS
But
and
Lemma 3.3. Assume W=(W2,W1) to be admissible, denote by M2 and Mi the corresponding monomials and let c # c’ be indices. If (W2,a!, a$, W1) is admissible, so is (W2,a $ , a!, W1) and
Mza:a>Ml = M2a>a:M1 M2acactM I = M2a;a: M I M2aCa;Ml = M2a>acM1
+E ~ , ( ~ X , J ) M ~ M ~ .
For the proof combine lemmata 3.1 and 3.2 4. Wick’s Theorem
If S is a finite chain, denote by (;Pz(S)the set of all pair partitions of S , i.e., the set of all
p = {(sirti),si > ti; {sirt i ) n { s j , t j } = 0 for i # j ;
U{
If # S is odd, (;P2(S)is empty. Define
writing for short
E(c,c’)= ~ ~ , ( d x ~ ~ ) .
Definition 4.1. Assume
w = (a;;, ...,a;:) t o be an admissible sequence, then define
c
(W)= (Wb PET2(Il,nl)
Creation and annihilation operators on locally compact spaces
203
If n is even and
P
=
{(i(l),j(l)),.* . 3 ( i ( n / 2 L j ( m ) l >
then
n
4 2
(W)P
=
("%(k) ' a ( k ) 7 ' c' J3 (( k ) )
.
k=l
If n is odd, ( W )= 0. As there is a 1-1 correspondence between [1,n] and the chain ((c1,01),
. . . , (cn,On)), we may write 92([1,n]) = tpz(W). We denote by
);p20(W)
the set of those pair partitions where only pairs of the form
( a c ,a $ ) = € ( C , c') occur. Then ( W ) ,= 0 for p @ f&o(W) and
We want t o investigate ( W ) , for p E !&O(W). We define in p a relation of nearest right neighborhood
+
(abl,abz) D ( a b s , a & )
* b2 =
b3.
As for a given pair there is at most one nearest neighbor, the set p splits into a a family of maximal chains of the form
c = ( a l ,U:)
+) D . . . D (ak-1,a t ) .
D (02, U3
(*I
To that chain corresponds the quantity
fc = €(1,2) € ( 2 , 3 ) .* * €(k - 1,k) fc(Zi,dZ2,. . . ,dzk)
€(Zi,dzz)€(2~,d23)...E(Zk_l,dZk)
/fc(21,dxz ,...,dZk)(P(Zz,...,Zk) =(P(~i,...,Zk). 2,...,k
Proposition 4.1. Any p E (;Pzo(W)is the union of maximal chains Ci with
respect to the relation
D.
If 1
then
WILHELM VON WALDENFELS
204
Example 4.1. If
then !&O(W) consists of only one element, namely
P = { ( w a;), ,
..
(a21a,'),
. 1
(ak-1,
a;,>
and that is the only maximal chain, namely the chain C from equation (*).
Example 4.2. If W is antinormal ordered
+ . . ,a 2+k ) , w = ( m , .. . , a k , Uk+1,. then !&o(W) consists of k ! elements, namely Po
= {(a17
&I,.
. , ( a a,: ( k ) ) > , *
where CT runs through all bijections [l,k ] -+ [ k + 1 , 2 k ] . The maximal chains are all of length 1.
Definition 4.2. An admissible sequence is called normal ordered if it is of the form ' ' ' 7+ I:'
7
ucl)'
7 ' ' ' 7
Example 4.3. If W is normal ordered, then ';p2o(W)is empty. By interchanging the indices any admissible sequence can be normal ordered. As any interchanging of the indices in the intervals [l,k ] and [k+l, n] . . . a$k+la,k. . ' a,, , we can define does not change the monomial
:w:= ac+, . . a:k+, U C k *
if
*
'
. a,, ,
. . . ,aZk+,,a c k ,.. . ,a,,)is any normal ordering of W . We may write
:w:= a:+(W)aLJ-(W) with w*(W) defined in definition 3.2. Define by y(S)the set of all partitions of a finite chain S into singletons and ordered pairs. If p E ??3(S),denote by p' c p the subset of singletons ordered by the order of S and by p" c p the subset of ordered pairs.
Proposition 4.2. If W is an admissible sequence
w=(a:;,...,u:;)l
Creation and annihilation operators on locally compact spaces
205
then
with iEP'
{j,k}Ep",j>k
Proof. We proceed by induction. The case n = 1 is clear. We consider the mapping cp : !73([1,n])4 p ( [ l , n - 11) defined by erasing n. If p E p ( [ l , n ] ) is of the form p = { n > T k > ' * ' > T l } x{Sj,tj}
+
j
with s j > t j , then
If
then
Assume q E p([l,n - l]),
then with
for i = 1 , . . . , k. Assume
WILHELM VON WALDENFELS
h206
to be an admissible sequence and let L be the corresponding monomial. Then by induction hypothesis
c
L=
LVlq.
qE!P([l,n--ll)
Assume, now, that (u,, V) is admissible. So c $ {cn-l,. . . ,c1). Denote again by q’ and q” the subsets of singletons and pairs of q. Then
n (. n
u C ~ v l=qa , :
.
=
JJ
u::
(u2,afc:)
(j>k)Eq”
iEq’
u$u,+
iEq’
c
(uc,u2) :
n .:::) n
lc(q‘\i)
ZEq’
If q’ = { r k > . . . > T I } , we obtain ac1Vlq
=
LWlP,
+ * . . +LWlP,
=
(u$,u::).
(j>k)Eq”
c LWP. PElo-’(q)
Sum over q and obtain the formula for (ac,V). Assume (u:, V) to be admissible, then c $ w+ (V). As
Gq‘
iEq’
we obtain that aclVlq
=
LWlpLl=
c
LWlP,
PEv-’(q)
as for i = 1 , .. . ,k the quantity LWlP, contains the factor ( u z , u z ) = 0.
0
Recall the definitions of the function CJ and the measure 9 from equations (1) and (7). If M is a normal ordered monomial, then
BMCJ =
i
1 for M = 1, 0 otherwise.
One deduces from the last proposition
Theorem 4.1 (Wick’s theorem). If
w =( u k ,. . . ,ufc:) is un admissible sequence and M the corresponding monomial, then 9MCJ
= 9u;;.
. u f c p = (W). *
Creation and annihilation operators on locally compact spaces
207
So ( W ) depends only of the monomial M , defined by W , and we may write ( W )= ( M ) . From the last proposition we obtain furthermore Theorem 4.2. Assume
w =(a:;, ...,a:;), +
to be a n admissible sequence, and assume subsets I , J such that I J = [l,n], and let WI and WJ be the restrictions of W to I resp. J . Denote by M , M I ,M J the corresponding monomials. Then,
C
M =
:MI:(MJ).
I+ J = [ l , n ]
5. Duality Fix a positive measure X on X , denote by e ( A ) the corresponding measure on X and write for short =A .,
e(X)(dz,) = A(z,) We define a scalar product on
K
Using equation 10, we obtain for
= K,(X)
‘p
by
E K(X)
where CpX is the product of the function (p with the measure A. Extend the scalar product to measure valued functions.
Lemma 5.1. One has
or writing for short
208
WILHELM VON WALDENFELS
Proof. Using the sum-integral-lemma
Now as
Definition 5.1. Assume
w = ( a k ., . . ,a:;) t o be an admissible sequence, then define the formal adjoint sequence by
w+= (a,01,. . . ,a;n@"). If M is the monomial corresponding to W , we denote by M + the monomial corresponding to W+. Recall definition 3.2 and call w
= w ( W ) , etc.;
M K:
K"+ ,"-\"+
We may interpret M as a function Xu-\"+ set of linear operators K -+ K and write
M
4
then, by proposition 3.1,
'
M ( X " + )with values in the
= M((dzi)i€w+, (zJj€"-\"+).
Then MX,-\w+
= (ML\kJ+)(dzi)i€w = M((dxi)i€w+7 b j ) j e L \ " + )
X((dxk)k€w-\w+)
is a measure on X" with values in the linear operators way,
M+X"+\"-
=
K
(ML+\"- )(dzi)i€"
is a measure on X" too. By induction we prove, by the last lemma,
-+
K. In a similar
Creation and annihilation operators on locally compact spaces
209
Theorem 5.1. For f , g E K: we have
6. Meyer's formula
If W is normal ordered, then the corresponding monomial is of the form
for 0 , r , u in finite index sets. A normal ordered sequence is always admissible. As it has been pointed out in the last section, MA, is a measure on X'+T+u with values in the linear operators K 4 K. Assume a function
symmetric in every variable xu,x,, x, and consider
T ( f )=
/
f ( X U , xT, 2,)
a:++,aT+U(dzU,
dxT,
xV)
U,T>U
or, written in an explicit way,
. .a(&,,)
a(&yl) . . . a(&,,)
X(dz1). . . A(&,).
As f is of compact support, the sum contains only finitely many terms and the integral is well defined and yields an operator K 4 K. That is the generalization of the Maassen-Meyer kernel representation generalized to locally compact spaces. We write for short
We want to prove Meyers formula for the composition of two operators.
Theorem 6.1. If f , g are two functions in K ( X 3 ) symmetric in each variable, then
T(f 1 T ( g )= T ( h )
WILHELM VON WALDENFELS
210
with
where the sum runs through all indices a 1 +a2
a1,
+a3
. . . , 7 3 with
= 0,
+ P2 + P3 = 7, 7 1 +7 2 +7 3 =
P1
'u.
That is essentially Meyer's formula [9] p. 921. The difference is mainly that his formula is formulated for sets of coordinates, whereas our formula deals with sets of indices of coordinates. So in our formula CT stands for 2, = ( z s l , . . I z s L ) Meyer . indicates the formula for much more general functions. In order to generalize our formula t o more complicated functions we had to use the extension theorems of measure theory. Meyer's statement is formulated for X = R and X Lebesgue measure. It could be extended easily to locally compact X and a diffuse measure A.
Proof. We define vpEWa,P)c E a
where B(a,P) is the set of all bijections cp : a B(a,P)= 8 and &(a,/?) = 0. One shows easily that &(a1 + a 2 , P )
= P1
&(a, P1
+0 2 ) =
c c
-+
P.
If # a
# #P, then
E(Ql,Pl)E(a2,P2),
+Pz =13 & ( a l l P 1 ) &(a27 P 2 ) .
al+CXz=a
From there one concludes that &(a1 +a2,P1 +P2)
=~
(*I
~ ~ ~ 1 1 , P l l ~ E ~ a 1 2 , 8 2 1 ~ ~ ~ ~ 2 1 , 8 1 2 ~ E ~ ~ 2 2
where the sum runs through all indices all P11
+ +
all,.
. . , P 2 2 with
a 1 2 = a17
021
+a22 =a2,
=P1,
PZl
+ P22 = P 2 .
P12
Creation and annihilation operators on locally compact spaces
E(722
211
+ 2122, 0 1 2 + 7 1 2 ) =xE(7221,
0 1 2 1 ) & ( 7 2 2 2 , 7 1 2 1 ) E(V221, 0.122) E ( v 2 2 2 , 7 1 2 2 )
with
Using the sum-integral lemma,
+ 7221 -k 7222, 2121 + 21221 -k 21222) d 0 1 1 -k 0 1 2 1 + 0 1 2 2 , 7 1 1 + 7121 + 7122, 211) f(02,721
&(72211 0 1 2 1 ) 4 7 2 2 2 , 7121) E(U221, 0 1 2 2 ) &(21222,7122)
z:a
+Tz
1+Tzz 1+
n z z + q1 +TI
+vz 1 +rl 1 + T i 2
1 +TI22
+vl
7
where the integral runs over all indices. Put 02
= all
0 1 2 1 = 7221
= Q2,
c11 = Q3,
pi,
7222 = 7121
=p2,
711 = p31
7122 = 7 2 ,
211 = 7 3 ,
721 = 2121
= 711
V222
0 1 2 2 = V221 = K ,
where the equalities in the second column hold after integration. Define
WILHELM VON WALDENFELS
212
a1
+ a 2 + a3 = 0,
P1+
P2
+
P3
= 7,
y1+ yz
+
73
a n d obtain t h e theorem using the sum-integral lemma again.
= 21,
0
Acknowledgment T h e author wants t o thank Uwe F’ranz for carefully reading t h e manuscript and indicating some misprints and errors.
References 1. L. Accardi, Y.-G. Lu, I. V. Volovich, “White noise approach to classical and quantum stochastic calculus”, Preprint 375, Centro Vito Volterra, Universita Roma 2 (1999). 2. L. Accardi, Y.-G. Lu, I. V. Volovich, “Quantum theory and its stochastic limit”, Springer 2002. 3. S. Attal, “Problemes d’unicite dans les representations d’operateurs sur l’espace de Fock”, Seminaire d e Probabilites X X V I , LNM 1526, 1992, pp. 617-632. 4. N. Bourbaki, Integration, Paris, 1965, chap. 1-4. 5. N. Bourbaki, Integration, Paris, 1965, chap. 5. 6. J. M. Lindsay, “Quantum and non-causal stochastic calculus”, Prob. Theory Relat. Fields 97,pp. 65-80 (1993). 7. J. M. Lindsay, H. Maassen, A n integral kernel approach to noise, LNM 1303, 1988, pp. 192-208. 8. H. Maassen, “Quantum Markov processes on Fock space described by integral kernels”, in L N M 1136, 1985, pp. 361-374. 9. P. A. Meyer, Quantum Probability for Probabilists, Lecture Notes in Mathematics 1538, Springer, Berlin, Heidelberg, 1993. 10. N. Obata, White noise calculus and Fock space, LNM 1577, Springer, 1994. 11. K. R. Parthasarathy, A n Introduction to Quantum Stochastic Calculus, Birkhaeuser, Basel, Boston, Berlin, 1992. 12. L. Schwartz, Theorie des distributions I, Herrmann, Paris, 1951. 13. W. von Waldenfels, “Continous Maassen kernels and the inverse oscillator”, Seminaire des Probabilites X X X , LNM 1626, Springer, 1996. 14. W. von Waldenfels, “Continuous kernel processes in quantum probability”, Quantum Probability Communications vol. XII, World Scientific, 2003, pp. 237-260. 15. W. von Waldenfels, “Symmetric differentiatiation and Hamiltonian of a quantum stochastic process”, Infinite dimensional analysis, quantum probability and related topics 8 , 73-116 (2005). WILHELM VON WALDENFELS
Institut fur Angewandte Mathematik, Universitat Heidelberg, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany Wi1helm.WaldenfelsQT-Online.de
FROM THE GEOMETRY OF PARABOLIC PDE T O THE GEOMETRY OF SDE* JEAN-CLAUDE ZAMBRINI(GFM-UL, Lisboa) We consider various notions of integrability in classical, quantum and stochastic dynamics. In the first field there is a plethora of definitions, sometimes not equivalent. In the two last ones, there are no universally accepted definitions, although a number of recent works (in quantum chaos, for example) suggest that it would be useful t o have one. We show here that a deformation of the classical notion of integrability underlying contact geometry provides a new perspective on stochastic integrability, with potential consequences in quantum mechanics.
1. Introduction This is an essay on the notion of integrability in classical, quantum and stochastic dynamics. As a matter of fact, the status of this notion is quite distinct in these 3 contexts. We shall start with a short survey of the simplest version of integrability, the one due to J. Liouville in classical dynamics. The classical notion required for our “transversal” purpose is, in fact, more general than Liouville’s one, inspiring Symplectic Geometry, or even more general than the one involved in Poisson geometry. It is the contact geometrical one, expressed in terms of Cartan’s ideal of differential forms. In quantum mechanics, there are still discussions about what should be an integrable system. We shall explain why. Regarding the notion of dynamics provided by Stochastic Analysis, starting from It6’s theory of Stochastic Differential Equations (SDE), we will mention some of the difficulties one meets with when trying to define integrability. It seems, in particular, that despite the remarkable progress of Stochastic Analysis during the last 25 years (cf. [l]for example) a counterpart of the classical Frobenius Theorem for stochastic differential equations is not yet available. However, there is a dual version of Frobenius for differentiable forms and we shall see that a stochastic counterpart of this one could be within reach, a t least in some special, dynamical contexts. *This work is supported by the project POCI/MAT/55977/2004.
213
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JEAN-CLAUDE ZAMBRINI
Our vital lead, in this expository review, will be the idea that an appropriate deformation of classical integrability notions should allow to shed a new light on their partly missing quantum and stochastic counterparts. We shall conclude with a list of examples, open problems and prospects of the stochastic deformation strategy summarized here. 2. On classical integrability
Let us consider the following Hamiltonian system on the open set M c R2” where n is called the number of degrees of freedom:
The dot denotes the derivative with respect t o time, and h = h(q,p , t ) is the (scalar) Hamiltonian observable. Let us denote by
the Poisson bracket of the observable f and g (where we have used Einstein summation convention). A first integral of the systems is a function n = n ( q , p , t ) constant along the solutions of equation (l),i.e., satisfying
In particular, for conservative systems, namely those with a time independent Hamiltonian h = h(q,p ) , this observable is itself a first integral. Liouville’s Theorem [2] states that if the above mentioned Hamiltonian system admits n first integrals n i ( q , p ) , 1 i n, in involution (i.e., { n i , n j } = 0 , Vi,j ) and if the functions ni are functionally independent on compact level sets, then there is a canonical coordinates transformation (to “action-angle” variables) on a n-dimensional torus where the flow becomes linear in time. This “complete” integrability has been generalized in many ways. The first integrals can be time-dependent, the Lie algebra of the first integrals can be noncommutative (then the system is called “super integrable”), etc. . . . Cf. [2]. Liouville’s Theorem is a t the origin of the principle that completely integrable systems can be solved “by quadrature” in terms of their first integrals.
< <
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The non-degeneracy of the bracket (2) allows to endow M with a nondegenerate 2-form named after Liouville:
R
= dpi A
dqi
(4)
which is closed by the Jacobi identity: d R = 0. It gives a symplectic structure on M and defines the Poincar6 1-form
w = p i dqi such that R = dw.
(5)
Using it , we have
WX,IXf) = {f191 = X,(f)I defining the Hamiltonian vector fields
x
ag a
ag
a p i aq2
89%api .
------
a
It is often useful to extend the dimension of M by two, by introducing the extra pair of canonical variables time and (minus) energy: ( t ,- E ) . Then the starting 1-form becomes the one of PoincarBCartan: wpc = pidqi - E d t
(5’)
with the associated extended versions of Hamiltonian vector field and Poisson bracket. Conservative Hamiltonian systems with one degree of freedom (i.e., n = 2) are always integrable by quadrature. Their trajectories lie entirely on the level sets of the Hamiltonian and provide the simplest example of the reduction procedure. More generally, when the Hamiltonian system has 1 independent first integrals in involution it can be reduced to a system with ( n - 1) degrees of freedom. It is known at least since the seventies that generic Hamiltonians systems are not integrable. There is an obvious relation (but far from trivial to formulate exactly) between integrability and regularity of the trajectories of Hamiltonian flows. No one would expect a (deterministic) chaotic dynamics from an integrable Hamiltonian system, for example. The approach of integrability needed for our purpose is not as familiar as Liouville’s one. It is inspired by the geometric theory of partial differential equation, for us the Hamilton-Jacobi equation associated with the Hamiltonian system (1):
as + h(q,-vs, t ) .
--
at
(7)
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Let us recall that to each Hamiltonian system are, actually, associated two Hamilton-Jacobi equations, adjoint to each other with respect to the time parameter (cf. [3]). Our choice of (7) is not the traditional one but is natural for the stochastic deformation described afterwards (cf. [4]for the justification). We are going to specialize to one degree of freedom (and even, later on, to an elementary class of Hamiltonians h) without lack of generality for the method advocated. So let us consider ( 4 , t } , the coordinates of the base configuration manifold of independent variables for equation (7), and { S } the one of the dependent variable. The coordinates in the jet bundle of order 1, denoted J 1 , are defined by
but regarded as independent variables. It is, therefore, much safer to choose a new label for the 2 last variables,
In the 5 dimensional space J1,our PDE (7) becomes a 4 dimensional manifold, specified by a function f(41 t ,
s,
Pl
E ) = h(41Pl t ) - E = 0
(10)
called the characteristic of Hamilton- Jacobi equation. By definition, for a given S , the section of S lifts up the base configuration manifold into the jet space, namely
Notice that S = S ( q , t ) solves Hamilton-Jacobi equation (7) iff the 2dimensional “section” (11) lies entirely in the 4-dimensional manifold (10) specifying this equation. A symmetry of Hamilton-Jacobi equation will be a mapping of sections into other sections but staying in its defining manifold (10). Lie characteristic equations for any PDE defined by f as in (10) are given by
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217
‘
where the denotes the derivative with respect to a parameter, say u.Specializing equation (10) to the elementary class of Hamiltonians h(q,p , t ) = :p2 V ( q ,t ) , we obtain
+
where . denotes now $, since the parameter u can be identified with t. The last equation provides us with the definition of the Lagrangian L (the minus sign is due to our unusual choice of Hamilton-Jacobi). The vector field on J1corresponding to the characteristic (12), denoted by X f , is called a contact vector field ([5]):
Like an Hamiltonian vector field, Xf is determined uniquely by the (contact) Hamiltonian f . But contact geometry is distinct from the Symplectic one mentioned before. Instead of Liouville 2-form 0, its fundamental tool is the following contact 1-form, which takes into account the “extra” dimension S of J1: w = wPc dS, (15)
+
any contact Hamiltonian being defined from it by 4X.f) = f.
(16)
A bracket {., . } L called after Lagrange (some authors prefer “Jacobi bracket”) transfers the Lie algebra structure of contact vector fields t o the space C” ( J1)of contact Hamiltonians: W([XS,XfI) = { f , 9 } L .
(17)
Only when f and g are independent of the variable S , the two contact vector fields reduce to (extended) Hamiltonian vector fields and {., . } L to the acsociated Poisson bracket. In the contact geometric context, for a given f defining the PDE (7), like in ( l o ) , an infinitesimal symmetry n of Hamilton-Jacobi will be defined as
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a contact Hamiltonian n such that
{ f , n } r ,= 0.
(18)
For our elementary class of Hamiltonians involved in (13) it is a simple exercise t o verify that
4% t, s,P , E ) = X(q, t ) P - T ( t ) E -
+((?I
t)
(19)
is such an infinitesimal symmetry of the associated Hamilton-Jacobi equa-
tion when its coefficients X, T and 4 satisfy the following relations, known as determining equations in Lie group theory:
The most appropriate formulation of these ideas about symmetry and integrability of a PDE, in our stochastic deformation perspective, is E. Cartan’s one in terms of ideal of differential forms [6]. For Hamilton-Jacobi with the above elementary class of Hamiltonians, one writes w =pdq -E d t + d S
The first form is the contact 1-form (15), involving the extended PoincarB Cartan form (5’). The 2-form p (where the wedge product is understood) defines our Hamilton-Jacobi equation. The set A of all differential forms over any manifold form an algebra, whose multiplication is the wedge (or hook) product. An ideal I of the algebra is such that the result of the multiplication of elements of I by any element of the algebra is in I . When a PDE (or system of PDEs) is expressed as an ideal I , Cartan has shown that I must be closed with respect to the exterior derivative d . Then I is called a differential ideal. The reason is that we must be sure that the integrability conditions associated with the PDE are, indeed, satisfied. Our ideal (21) is not closed, but the remedy for that is simple: just add dw. Then ( w = p dq - E dt d S
I
=
1i R=
,B = E
+
=drdq -d E d t -
1 -p2
-
V(q, t )
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is a differential ideal. Its “sectioned” forms, traditionally denoted by a tilde, are annuled - by the base configuration submanifold of coordinates ( q ,t ) . For example, R = & = 0 implies the existence of S = S(q,t ) s.t. the relations (9) hold true. In our five variables space J 1 with coordinate ( q ,t , S, p , E ) the generator of a symmetry should be, a priori, defined as a vector field
a + xi-d + x,s-d + x:- + x,E d at
dS
dP
dE
(23)
with coefficients to be determined such that
.cx,(I)c I
(24)
where Lxn denotes the Lie derivative along X,. In fact, the appropriate coefficients of X , are precisely given by the characteristic associated with the contact Hamiltonian f in (14), for f = n as in (19), justifying our present notations.
3. Quantum integrability Even when we restrict ourselves t o the quantization of the elementary class of Hamiltonians of $2,
one faces a puzzling situation when trying t o define the notion of quantum integrability. The first idea is to try to define a quantum version of Liouville Theorem: For a given Hamiltonian observable H densely defined on IFI = L2(RWn), we could say that this system is integrable if there are n “independent” first integral observables Ni,1 6 i 6 n, i.e., such that [ H, N i ]= 0 and “i,Nj] = 0, 1 2 , j n. But J. Von Neumannn [7] has proved that the Hamiltonian H and the other first integrals Ni can be all expressed as functions of a single observable 6 of the system so that any naive version of functional independence of those first integrals is ruined. Interesting definitions of quantum integrability have been, nevertheless, put forward. A good description of the present state of knowledge on this issue is given in [S], together with a sufficient integrability criterion (lack of continuous spectrum for H ) . One feels, however, that many other quantum
<
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systems should also be integrable, like the free particle for example, i.e., V = 0 in (25). The present consensus is that, in strong contrast with classical systems, “most” quantum systems should be integrable (because of the central role of the Spectral Theorem) and that no future definition of quantum integrability could imply the integrability of the corresponding classical systems. This circle of ideas gained a new relevance with the hope that a quantum Hamiltonian whose spectrum is made of the zeros of the Riemann function should have a non integrable classical counterpart. And more generally with the study of the link between integrability of quantum Hamiltonian systems and the distribution of its energies. For more about this issue, cf. [9]. 4. On stochastic integrability
Consider the following system of stochastic differential equations (SDE) for a diffusion process zt = ( z i ) :
where wt
= (wl)are i-copies of one dimensional Wiener process, and : R” x I 4 Rn x I are given. Such SDEs have R” x I -+ R”, been introduced by K. It6, and regarded as stochastic perturbation of systems of first order differential equations like our Hamiltonian system (1). Conditions on and needed to assure existence and uniqueness of the solution of equation (26) have been known for a long time (cf. [lo], for example). The modern theory founded on SDE, Stochastic Analysis, has been developed (impressively) as a first order theory. Although for ODE, the equivalence between the system (1) and, say, the second order equation of Euler-Lagrange needed for Lagrangian mechanics is rather obvious, this is far from being the case of equation (26). But, classically, the dynamical content of the theory always appears in second order equations. This may explain, in part, why it is often so hard to relate results of Stochastic Analysis with problems inspired by theoretical and mathematical physics. Interestingly it seems that there is no general notion of integrability for systems of SDEs like (26). Of course, when-g (the “drift”) and are linear, equation (26) is explicitly integrable. For G independent of q = zt, our main interest here, the solution can, in fact, be explicitly expressed in terms of the solution of the corresponding deterministic linear equation ( E = 0 in
g
:
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(26)). This (rare) circumstance should certainly be a sufficient condition for a definition of integrability of SDE. Applying any diffeomorphism 6 : R" + R" t o such a solution together with It6's formula we could produce many "integrable" SDEs. Still no general ( a priori) criterion seems t o be known. Our thesis is that the difficulties we face in looking for such a definition of stochastic integrability are very similar to the ones experienced in trying to define quantum integrability. To illustrate this idea, we suggest to: (1) Restrict the class (26) of SDEs to those qualitatively closer t o Hamiltonian equations ( l ) ,in order to preserve as much as possible their dynamical content. (2) Use the probability measures associated with all the solutions of this class of SDE to deform the differential ideal (22) containing the symmetries and integrability conditions of the classical HamiltonianJacobi equation ( 7 ) ,for the elementary class of Hamiltonians of 52. (3) Choose, for (2), a quantum-like probabilistic deformation so that, after coming back in Hilbert space 'If, we could infer a reasonable definition of quantum integrability.
The rest of this paragraph will describe some recent progress along this line of stochastic deformation, resulting from joint works with P. Lescot [Ill. Our starting deformation of Hamilton-Jacobi equation (7) for the elementary class of Hamiltonians involved in (13) will be: -
as + 51 (%) as dt
-
tz a2s
V (q , t ) - - - = 0. 2
aq2
This (uniformly) parabolic equation goes under the name of HamiltonJacobi-Bellman equation, our deformation parameter being denoted by ti > 0. In analogy with (9), we define two variables by
B will become a drift later on. In equation (26) the drift is, of course, a given function of q and t , and this is why we have added a - a t the top of B . But in (28), B and E are two independent variables, like in J1. We shall keep such convention until the end of this exposition.
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The deformation of the ideal (22), relevant to Hamilton-Jacobi-Bellman equation is given by
I H J B=
{
+ E d t + d S wPc+ d S s2 = dw = d B d q + d E d t ,B = ( E + $ B 2- V )dqdt + $ d B d t . w = B dq
(29)
There are a few changes of signs in (29) with respect to the classical ideal (22). They are expected because from (22) to (29) the real time parameter of Hamiltonian mechanics became the “imaginary time” one needs t o make probabilistic sense of our h-deformation. Like their classical counterpart, the sectioned forms of (29) are annuled on the 2-submanifold of coordinates ( q ,t ) . Calling N a symmetry generator defined exactly like (23), with unknown coefficient denoted by N q , N t , N S ,N B ,N E , such that
L N ( I H J B5) I H J B ,
(30)
we characterize in this way a Lie algebra 9. It contains an infinite dimensional subalgebra originated in the observation that the change of variable
S ( 4 , t ) = -ti 1n v ( 4 ,t )
(31)
in equation (27) reduces this one to the linear heat equation
and so the superposition principle applies to it. The most interesting part of Q is the supplement N of the above subalgebra. When V = 0 in equation (27), for example, the dimension of N is 6, each element of its basis being associated with a symmetry of the system described by I HJ B (cf. [12]). Still, it is the symmetry underlying the change of variable (31) which is at the origin of our stochastic deformation method, since probabilistic interpretations of equation (32) have been known for a long time. Given our “deformed” differential ideal (29), the first step is purely algebrical, however, and does not appeal for any probability whatsoever: Theorem 4.1. Let N E
N
be a symmetry generator, with coeficients denoted as before, f o r the ideal I H J B of (29). W e shall call Lagrangian L the following function of 3 (independent) variables: 1
( 4 , B , t ) H L(q,B , t ) = 5
8 2
+ V ( 4 ,t ) .
(33)
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Then the following Lie dragging relations hold:
-DNS. where D is the differential operator defined by
The proof of this Theorem is not hard, but quite long. It is given in [ll].We shall concentrate here on its interpretation in our perspective: the integrability of Hamilton-Jacobi-Bellman equation (27). An interesting aspect of the proof of Theorem 4.1 is that it requires the introduction of the following counterpart of the symmetry contact Hamiltonian (19), associated here to N E n/:
where X N = N g , TN = N t , 4~ = -N" in terms of the above mentioned coefficients of our symmetry generator N . The corresponding deformation of the determining equation (20) is as follows:
Although Theorem 4.1 does not have, a priori, a probabilistic meaning, the presence of the operator (37) is showing us the way to remedy this: If B could be interpreted as the drift of a diffusion process like .zt in (26) (with n = 1, = ti) then the differential operator (37) would coincide with
where A is called infinitesimal generator of the process. Since (26) is a (SD) Equation, the drift should, in this case, not be a variable independent of q and t , like in Theorem 4.1, but a given function of those two variables. Now this is precisely what the section of the S variable, both for the classical Hamilton-Jacobi equation ( 7 ) (cf. (11)) and its deformation (27),
-
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was all about. Moreover, using (31) and (28), provided by the geometry of I H J B ,we know that the dependence should be of the form: &q,
a
-
a
t ) = fL-dq lnv(4, t ) , q q , t ) = fi-at In v(4,t ) .
(41)
To promote our variable q at time t to a diffusion process zt with drift and energy given by (41), we only need t o assume that the underlying solution q of (32) is positive. From now on, we shall denote by 6 the stochastic derivation operator (37) involving the above drift B , namely the infinitesimal generator of z t , built in term of such a positive q. To the geometrical operation of section, needed here to construct zt, corresponds an homomorphism of Lie algebra. Indeed, pick a symmetry generator as in Theorem 4.1. Then form the semigroup, for p E R,
e-ON : ( 4 , t , s,B , Jq+ ( q p , to, so, Bp,J%). Defining qp(qp,tp) by e-kso, in conformity with (31), we observe that qp solves (32) as well, in the variables (qp,tp). Then one can define a
“reduced” symmetry generator ?i so that
eo’
-
: q ++ q p
+ &
and check that N 4 N = N Q84Z N t - i N S is a group homomorphism. This means that as long as our probabilistic interpretation of the ideal of H J B equation is concerned, nothing will be lost by such a reduction to the (4, t ) submanifold. This submanifold is also the one on which we can randomize (Lie) characteristic equations (13) along the irregular paths t H zt = q , for our elementary class of Hamiltonians: Theorem 4.2. Let q a positive solution of (32), for t E I an interval of R, and ztl t E Il a solution of the SDE (26) for the drift ?i of (41) and = h i . Then the following stochastic characteristic equations hold almost everywhere:
We shall consider mainly the last equation of (42), crucial for our purpose. After time integration and using a relation often called Dynkin formula, we
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obtain the action
where Et denotes the conditional expectation given z t , and the integrand coincides with our Lagrangian (33). The results of Theorem 4.1, after sectioning to the ( q , t ) submanifold describe, in particular, all the symmetries of the action (43). Before elaborating on this aspect, let us observe that this action, which is a function of z t , can also be regarded as a (Boltz) functional J whose domain DJ contains all diffusions X . absolutely continuous w.r.t. Wiener's measure and solving a SDE of the form (26), with fixed = tr.3 but arbitrary drift E. Defining X E V J as an extremal of J if E [ V J [ X ] ( S X )=] 0 V S X : Qo 4 x, where Ro = { w E C(R+,R) s.t. w ( 0 ) = 0 ) is the Wiener space, x its Cameron-Martin subspace of absolutely continuous paths with square integrable derivative and V J [ X ] ( h X denotes ) the directional derivative of J a t X in the direction S X , one shows that the diffusion zt of Theorem 4.2 is indeed extremal of J and satisfies (cf. [13]) a.s.:
Now, by construction,
g solves the H J B equation
(27). Taking
$ of
this equation and introducing the definitions (28) of ? i one checks that the resulting relation coincides with (44), the boundary conditions being also satisfied. The other relations of (42) follow as easily. So the diffusion zt of Theorem 27 (or, equivalently, the H J B equation associated with it) satisfies the stochastic least action principle (cf. [4]for more on this) underlying the presence of the variable S in the ideal IHJ B . For the geometry of the action (43) we rely, in particular, on the last claim of Theorem 4.1. Let us consider (36) when tr. = 0. By construction, this should essentially correspond t o a geometrical statement of classical dynamics. Indeed it does: This is exactly the invariance property of the fundamental (action) integral under its symmetry group of transformations on the ( q , t ) manifold, which is the hypothesis of Noether's famous Theorem (cf. [14] p.204). The r.h.s. of (36) is, in general, not zero: one says that the action is invariant modulo the divergence term $ N S .
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Classically, i.e., for h = 0, the invariance condition (36) means explicitly,
where the coefficient NG = B N Q- qTd N t expresses the LLprolongation”to q of the transformation on the (9, t ) variable. Using the two first determining equations (20) (for X = N Q T , = N t , qh = - N S ) , NG can be rewritten as
N Q = dq5 -’
89
.dX d9
Q-.
In the stochastic case, i.e., along the random paths t H zt, (36) tells us that the same hold true, replacing all derivatives $ by their counterpart 6. The coefficient N B in (36) is of the form
for q h and ~ X N solving the deformed determining equations (39) and can be, indeed, rewritten as:
NR =EX,
-
i?i?~
in conformity with the classical coefficient N Q . The invariance condition (36) implies the stochastic version of Noether Theorem (cf. [ 4 ] )which won’t be needed here. 5. Examples, open problems and prospects
Let us consider the following (“free Boltz’s’’) action functional for [ t ,u]c 1 = [0,w-’], and w is a positive constant,
where
3u is of the form (31):
(47) The diffusion extremal for this action (46) solves the SDE WZT
dz, = -dr 1-wr
+h d w ,
(48)
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227
or, in dynamical terms, equation (44) for V = 0 with the boundary condition (47). This is checked by computing the drift (41) using the solution q(q,t) of the heat equation (32) for our given future boundary condition rlu, then substituting in the definition of 0. It can be easier to use, instead of qu, the following smooth initial condition for the heat equation (32):
qo(q)= e-%q2
(49)
wliosc solution coincides, for t E [O,w-'], with the one of the above final boundary problem. A Theorem of Rosencrans tells us how to perturb the free heat Hamiltonian H = HO underlying the action (46) by, say, a quadratic potential
V: H = H o + - qw2
2
(50)
2
in using only linear combinations of symmetry generators N of the free case (computed in [12]). Then a solution of the perturbed heat equation, for the same initial condition as before, can be written explicitly in terms of the unperturbed solution. Here the two free generators needed to do that, among the 6 spanning the algebra N of $4 are
g,
N1= NG= 2 q t8q
+ 2t 2 8 + (tit - q2)& + 2(q
-
tB)&
-
(2qB
+ 4tE + ti)a 8E
and the resulting explicit relation between any solution of the free and perturbed (by V = $ q 2 ) heat equations is given by
Using the definition (41), the corresponding relation between drift becomes
-
kv(q,t)=wqtanhwt+----B(-,-----). 1 q tanhwt cosh w t cosh w t w
(52)
This relation between stochastic derivatives can be integrated and provides the following one between the associated processes:
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By construction, any such diffusion z ” ( t ) solves the dynamical equation (44) for v(q)= g q 2 i.e., is extremal point of an action functional (43) with this potential. Coming back t o our special boundary condition (49) (or (47)) characterizing the solution (48) of our free dynamical equation (zT is called the “Brownian Bridge”) it is easy to check that the relation (50) reduces, then, to ‘ I V ( q , t )= e-%iEQ2+%t. (54) Equivalently, using (52) for the drift g(q,t ) of the Brownian Bridge given in the SDE (48), we obtain -V
( 4 , t ) = -wq
(55)
zv ( t )is the famous Ornstein-Uhlenbeck process, expressed here explicitly in terms of the Brownian Bridge ~ ( tvia ) the general relation (53), that we could call a “quadrature of z ( t ) ” (cf. [ll])by analogy with classical dynamics. Indeed, this type of explicit relations expresses manifestly a kind of “stochastic integrability” for our underlying dynamical system. It says that, somehow, the two equations (44) for V = 0 and V = $q2 are equivalent, or that there is a probabilistic version of symplectomorphism carrying one into the other. Since it follows from our construction that this relation is built on the one existing between two underlying H J B equations (27), we have here a simple example of integrability for this equation, resulting from our ti-deformation of the classical dynamical structure. We believe that this strategy should allow us t o construct a general Theorem of integrability for the n-dimensional Hamilton-Jacobi-Bellman, where all the ingredients of the classical mechanical proof are appropriately deformed. Such a proof would be useful for applications (given the versatility of (27)) but even more for its potential generalizations to other classes of processes, for example the one introduced in [15]. The explicit relations introduced between processes (like (53) for example) have a dynamical origin revealed only in second-order a s equations like (44). Their systematization should introduce a number of new structures hidden behind the traditional theory of stochastic processes and their first order SDE. Another interesting line of thought is to understand if it is possible (or not) to construct a probabilist interpretation not on the ( q , t ) submanifold as we did here but on the ( 4 , B , t ) submanifold of the jet bundle J 1 , i.e., on a stochastic counterpart of the classical phase space. Of course, after “projection” on ( q , t ) , the probabilistic interpretation looked for should
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reproduce the results given here. Such a probabilistic theory on phase space would be very interesting. In the non rigorous “probabilistic” deformation created by R. Feynman, and known as the method of path integrals [16], it is striking t o see that the status of the path integrals in phase space is much worse than the one in configuration space (which is supposed to have found its rigorous counterpart in Feynman-Kac formula. Cf. [4] for another viewpoint). This allusion to quantum theory is, of course, not accidental. As an illustration, let us observe that if a s second order equations like (44), resulting from characteristic equations (42), seem unusual in Stochastic Analysis, they are quite natural in a quantum perspective. Indeed, after expectation, namely d d %E[ZT]= E[B],--E[5] = E[VV(Z,)], dt they constitute probabilistic counterparts of Ehrenfest equations of motion for the position and momentum (Von Neumann [7]) observables of the system resulting from the quantization of Hamiltonian (25). For the specific relation between what we did and quantum mechanics, the probabilistic content presented here is only one half of the story of “Euclidean Quantum Mechanics”. The other half deals with the heat equation adjoint to (32) w.r.t. the time parameter (i.e., another filtration and its adapted backward SDE). These two adjoint PDE play the role of Schrodinger equation and its complex conjugate, and time reversible processes can be constructed from them, whose qualitative properties are very close to those of Feynman’s informal “processes”. Euclidean Quantum Mechanics is, in fact, our main source of explicit examples like the abovementioned one, even if the geometrical ideas summarized here are relevant far beyond this initial motivation. The analogy between our “imaginary time” framework and quantum mechanics is strong enough, for instance, to suggest new results in Hilbert space. This is what was found with the stochastic Noether Theorem alluded to at the end of $4 (cf. [17]). And this is also why we believe that the expected Theorem of integrability for H J B equation should help us to produce a reasonable definition of quantum integrability. Acknowledgments It is a pleasure to thank the local organizers for their new demonstration of the exceptional quality of Mediterranean hospitality.
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References 1. P. Malliavin, “Stochastic Analysis”, Grund der Math. Wiss. 313, Springer (1997). 2. B. A. Dubrovin, I. M. Krichever and S. P. Novikov, “Integrable Systems I” in Dynamical System IV (Symplectic Geometry and its Applications), eds. V. I. Arnol’d and S. P. Novikov, Springer-Verlag (1990). 3. J. L. Synge, “Classical Dynamics” in Encyclopedia of Physics, vol. III/l, ed. S. Fliigge, Springer-Verlag, 1960. 4. K. L. Chung, J. C. Zambrini, Introduction to Random Time and Quantum Randomness, New Edition, World Scientific, 2003. 5. M. Giaquinta, S. Hildebrandt, “Calculus of variations 11”, Grundl der Math. Wiss. 311,chapter 10, Springer, 1996. 6. B. K. Harrison, F. P. Estabrook, “Geometric approach to invariance groups and solution of partial differential systems”, J . Math. Phys. 12(4),653 (1971). 7. J. Von Neumann, Mathematical Foundation of Quantum Mechanics, Princeton U. Press, Princeton, 1955. 8. A. Enciso, D. Peralta-Salas, “On the classical and quantum integrability of Hamiltonians without scattering states”, to appear in Theor. Math. Physics. 9. P. Cartier, B. Julia, P. Moussa, P. Vanhove (eds.), Frontiers in Number Theory, Physics and Geometry I, Springer, 2006. 10. N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes North Holland, 1981. 11. P. Lescot, J. C. Zambrini, “Probabilistic deformation of contact geometry, diffusion processes and their quadrature”, to appear in Proceedings Ascona 2005, Progress i n Probability, eds. R. Dalang, M. Dozzi and F. Russo, Birkhauser. 12. P. Lescot, J. C. Zambrini, “Isovectors for the Hamilton-Jacobi-Bellman equation, formal stochastic differential and first integrals in Euclidean quantum mechanics”, in Progress in Probability, vol. 58, eds. R. Dalang, M. Dozzi and F. Russo, Birkhauser, 2004, p. 187. 13. A. B. Cruzeiro, J. C. Zambrini, “Malliavin calculus and Euclidean quantum mechanics. I Functional Calculus”, J . Funct. Anal. 96, 62 (1991). 14. D. Lovelock, H. Rund, “Tensors, Differential Forms and Variational Principles”, Dover Publications, New York, 1989. 15. N. Privault, J.C. Zambrini, “Markovian bridges and reversible diffusions processes with jumps”, Ann. Inst. H. Poincare‘ PR40, 599 (2004). 16. R. P. Feynman A. R. Hibbs, “Quantum Mechanics and Path Integrals”, McGraw-Hill, New York, 1965. 17. S. Albeverio. J. Rezende and J.C . Zambrini, “Probability and quantum symmetries I1 - The theorem of Noether in quantum mechanics”, to appear in J . Math. Physics (2006). JEAN-CLAUDE ZAMBRINI Grupo de Fisica-Matemiitica d a Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal zambriniQcii.fc.ul.pt
List of participants
EZEDDINE HAOUALA . Tunis TAKEYUKI HIDA. Meijo SAMEHHORRIGUE . Monastir MA’RTONISPA’NY. University of Debrecen, Hungary RAOUDHA JENANE-GANNOUN . Tunis RIADHKADDACHI . Tunis / Mannheim MOUNIRKHLIFI. Tunis PAUL KRBE. Paris 6 YUH-JIALEE . University of Kaohsiung PAUL LESCOT. Saint-Quentin, fiance PAULMALLIAVIN . Paris 6 PENKA MAYSTER. Tunis RUI VILELAMENDES. Lisbon GIULIADI NUNNO. Oslo NOBUAKI OBATA. Tohoku BERNTOKSENDALOslo MARIAJ. OLIVEIRA . Lisbon HABIBOUERDIANE . Tunis ANDREAPOSILICANO . Italy NICOLASPRIVAULT . La Rochelle HAFEDHREGUIGUI . Tunis ANIS RIAHI. Tunis BARBARA RUDIGER. Koblenz Luis SILVA. Madeira NEJIBSMAOUI . Kuwait University BOUBAKER SMII’ Tunis / B o n n FETHISOLTANI . Tunis LAKHDAR TANNECH RACHDI. Tunis ALEX UGLANOV . Yaroslavl, Russia WILHELM VON WALDENFELS . Heidelberg JEAN-CLAUDE ZAMBRINI. Lisbon
LUIGIACCARDI. Roma 2 AHMEDS . AL-RAWASHDEH . Jordan University of Science and Technology HELENEAIRAULT. Picardie ABDULRAHMAN AL-HUSSEIN . A1 Quassam, Saudi Arabia AHMEDAL SALAM. Yarmouk WIDEDAYED’ Tunis ABDESSATTAR BARHOUMI . Tunis SONIABELKADHI-CHAARI . Bizerte MOHAMED BEN CHROUDA . Monastir ANISBEN GHORBAL . Roma 2 / Tunis ZIED BEN SALAH. Tunis MADIBELCACEM . Guelma, Algerie PHILIPPE BLANCHARD . Bielefeld WISSEMBOUGHAMOURA . Tunis RADHOUAN BOUKHRIS . Tunis MOUNIRBEZZARGA . Tunis REFAATCHAABOUNI ENIT FERNANDA CIPRIANO . Lisbon ANABELACRUZEIRO . Lisbon HOUCINE CHEBLI. Tunis SIDI HAMIDOU DJHA. Tunis AHCENEDJOUDI. Annaba VITO CRISMALE . Bari MOHAMED EL OUED. Monastir NEJIBGHANMI . IPEIN SOUMAYA GHERYANI . Tunis DIOGOGOMES. Lisbon HANNOGOTTSCHALK . Bonn MARTINGROTHAUS . Kaiserslautern SKANDER HACHICHA . Tunis RACHEDHACHAICHI . Tunis
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