Table Of ContentLecture Notes in Mathematics 1613
:srotidE
.A Dold, Heidelberg
E Takens, Groningen
:seiresbuS Institut de MatMmatiques, Universit6 de Strasbourg
Advisor: J.-L. Loday
regnirpS
nilreB
grebledieH
New kroY
anolecraB
Budapest
Hong Kong
nodnoL
Milan
siraP
Santa aralC
eropagniS
oykoT
J. Az4ma M. Emery
R A. Meyer M. Yor (Eds.)
S4minaire de
Probabilit4s XXIX
regnirpS
Editors
Jacques AzEma
Marc Yor
Laboratoire de Probabilitrs
Universit6 Pierre et Marie Curie
Tour 56, 3 ~me 6tage
4, Place Jussieu, F-75252 Paris Cedex, France
Michel Emery
Paul Andr6 Meyer
Institut de Recherche Mathrmatique Avanc6e
Universit6 Louis Pasteur
7, rue Ren6 Descartes, F-67084 Strasbourg, France
Cataloging-in-Publication Data.
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
S~mlnalre de probabllit~s ... - Berlin ; Heidelberg ; New York
; London ; Paris ; Tokyo ; Hong Kong ; Barcelona ; Budapest :
Springer.
ISSN 6678-0270
29 (1995)
(Lecture notes in mathematics (cid:12)9 Vol. )3161
SBN 3-540-60219-4 (Berlin ...)
NE: GT
Mathematics Subject Classification (1991): 60GXX, 60HXX, 60JXX
ISBN 3-540-60219-4 Springer-Verlag Berlin Heidelberg New York
This work is subject to copyright. All rights are reserved, whether the whole or part
of the material is concerned, specifically the rights of translation, reprinting, re-use
of illustrations, recitation, broadcasting, reproduction on microfilms or in any other
way, and storage in data banks. Duplication of this publication or parts thereof is
permitted only under the provisions of the German Copyright Law of September 9,
1965, in its current version, and permission for use must always be obtained from
Springer-Verlag. Violations are liable for prosecution under the German Copyright
Law.
(cid:14)9 Springer-Verlag Berlin Heidelberg 1995
Printed in Germany
Typesetting: Camera-ready TEX output by the authors
SPIN: 10479528 46/3142-543210 - Printed on acid-free paper
SEMINAIRE DE PROBABILITES XXIX
TABLE SED SEREITAM
A. M. Chebotarev, F. Fagnola : On Quantum Extensions of the Az~ma
Martingale Semi-Group.
F. Delbaen, W. Schachermayer : An Inequality for the Predictable
Projection of an Adapted Process. 17
N. V. Krylov : A Martingale Proof of the Khintchin Iterated Logarithm
Law for Wiener Processes. 25
Ph. Biane : Intertwining of Markov Semi-Groups, some Examples. 30
W. Werner : Some Remarks on Perturbed Reflecting Brownian Motion. 37
M. Chaleyat-Maurel, D. Nualart : Onsager-Machlup Functionals for
Solutions of Stochastic Boundary-Value Problems. 44
S. Attal, K. Burdzy, M. ~mery, Y. Y. Hu : Sur quetques filtrations
et transformations browniennes. 65
M./krnaudon : Barycentres convexes et approximations des martingales
continues dans les vari~t~s. 70
E. Cdpa : ~quations diff~rentielles stochastiques multivoques. 86
L. Overbeck : On the Predictable Representation Property for Super-
processes. 108
A. Dermoune : Chaoticity on a Stochastic Interval 0, T. 117
J. Bertoin, M. E. Caballero : On the Rate of Growth of Subordinators
with Slowly Varying Laplace Exponent. 125
S. Fourati : Une propri~td de Markov pour les processus indexes par .R 133
D. Williams : Non-Linear Wiener-Hopf Theory, :1 an Appetizer. 155
Y. Chiu : From an Example of L~vy's. 162
D. Applebaum : A Horizontal LSvy Process on the Bundle of Ortho-
normal Frames over a Complete Riemannian Manifold. 661
IV
S. Cohen : Some Markov Properties of Stochastic Difi'erential Equations
with Jumps. 181
J. Franchi : Chaos multiplicatif : un traitement simple et complet de la
fonction de partition. 194
Z. M. Qian, S. W. He : On the Hypercontractivity of Ornstein-
Uhlenbeck Semigroups with Drift. 202
Y. Z. Hu : On the Differentiability of Functions of an Operator. 218
D. Khoshnevisan : The Gap between the Past Supremum and the
Future Infimum of a Transient Bessel Process. 220
K. Burdzy, D. Khoshnevisan : The Level Sets of Iterated Brownian
Motion. 132
J.-C. Gruet, Z. Shi : On the Spitzer and Chung Laws of the Iterated
Logarithm for Brownian Motion. 237
L. E. Dubins, K. Prikry : On the Existence of Disintegrations. 248
N. Eisenbaum, H. Kaspl : A Counterexample for the Markov Property
so Local Time for Diffusions on Graphs. 260
N. Eisenbaum : Une version sans conditionnement du th6or6me d'iso-
morphisme de Dynkin. 266
Y. Y. Hu: Sur la rep%sentation des (S-t- = a{B;-, s <~ t}) martingales. 290
S. Song : C-Semigroups on Banach Spaces and Functional Inequalities. 297
On quantum extensions
of the Az6ma martingale semigroup
by A.M. Chebotarev and F. Fagnola
1. Introduction
In this note we study some quantum extensions of classical Markovian semigroups
related to the Az~ma martingales with parameter ~ (fl # 0, flr -1) (see 1, 5,
6). The formal infinitesimal generator given by
(,Cof)(x) = (fiX) -2 (f(cx) - f(x) - flxf'(x))
on bounded smooth functions f can be written formally as follows (see 9)
s =Gm I + L'rolL + rnyG*
where m I denotes the multiplication operator by f, the operator G is the infinites-
imal generator of a strongly continuous contraction semigroup on L2(~;r (see
Section 2) and L is related to G by the formal condition G + G* + L*L = .O The
associated minimal quantum dynamical semigroup, can be easily constructed, for
example as in 2, 3, 4, 8. We show that this semigroup is conservative if/3 < ~.
and it is not if fl >/3. where/3, is the unique solution of the equation
exp(/3) +/3 + 1 = .O
Therefore it is a natural conjecture that the minimal quantum dynamical semigroup
is a ultraweakly continuous extension to B(h) of the Az@ma martingale semigroup
when fl < fl.. However we can not prove this fact because the characterisation of
the classical infinitesimal generator is not known. The above quantum dynamical
semigroup is not such an extension when/3 >/3. because the corresponding classical
Markovian semigroup is identity preserving.
We were not able to study the critical case/3 =/3. although it seems reasonable
that conservativity holds also in this case. In fact, as shown by Emery in 5, the
Az6ma martingale with pm'ameter 3/ starting from x # 0 can hit 0 in finite time
only if/3 > /3.. The operators G and L we consider are singular at the point 0,
hence, in this case, boundary conditions on G at 0 should appear to describe the
behaviour of the process.
The cases when fl </3. and fl >/3. are studied in Section 3 by checking a necessary
and sufficient condition obtained in 2. In Section 5 we apply a sufficient condition
for conservativity obtained in 3. This condition yields the previous result when
3/ < -1.5; since /3. = -1.278..., it is quite "close" to the necessary and sufficient
one.
2. Notation and preliminary results
Let fl be afixed real number with fl ~ -1, fl ~ 0 and let c = fl+l. Let h -- L2(Z~;~)
and denote by B(h) the *-algebra of all bounded operators on h. Let us consider
the strongly continuous contraction semigroup P on h defined by
(p(t)u) (x) = 1 - -a-~ ~, x 1 - -a~ if 1 - ~ > o
0 if 1 - ~-2~ t <0
The dual semigroup P* can be easily computed
0 ifl+~-~ <0
Let Do be the linear manifold of infinitely differentiable functions with compact
support vanishing in a neighbourhood of 0. The infinitesimal generators G and G*
of the semigroups P and P* satisfy
2-#~z u( x )
for all u E Do. In fact P has been obtained integrating the first order partial
differential equation
oOw(t, x) 10w(t, x) fl - 1
o~ - ~ ~ o~ + 2-fifi w ( t ' ~ )
by the characteristics method. Consider the operator M on h defined by
D(M) = { u (cid:12)9 h x-lu(x) (cid:12)9 h }, Mu(x) = (fl%)--lu(x).
and let S be the unitary operator on h
Su(x) = Icl-1/2u(c-lx).
The form E on h given by
(%/:(f)u) = (G'v, fu> + (SMv, fSMu) + (v, fa*u)
for all u,v (cid:12)9 Do, transforms f (cid:12)9 D o into the multiplication operator on h by
(s = (fix) -2 (f(cx) - f(x) - flxf'(x)).
Thus the restriction of E to D O coincides with the restriction to Do of the infinites-
imal generator of an Azfima-Emery martingale with parameter fl (see 5).
The domain Do, however, may be too small for the operators G and G*. In fact
it can be shown that Do is a core for G if and only if fl > -1/2 and is a core for G*
if and only if fl < 1/2. The domains of G and G* however can be described as the
range of the resolvent operators R(1; G) = (1 - G) -1 and R(1; G*) -- (1 - G*) -1.
Proposition 2.1. For all u E h des a function oU by
sgn(z)oo
f exp(-flyZ/2)y-(#+)/2#yu(y)dy if 31 > 0
- o exp(-#y212)lYl-(Z+nl2nyu(y)dy if # < 0
Then the operator R(1; G*) is given by
(R(I; G*)u) (x) =/3Ix (#+1)/2# exp(flx2/2)Uo(X). (2.1)
Proof. For all u, v C h we have
(
By the change of variables y = xx/1 + 2tlflx 2 in the integral with respect to t we
obtain the representation formula (2.1).
Proposition 2.2. The domain of the operator M contains the domain of the op-
erator G*. Moreover the necessary condition for 7- to be conservative,
(G'v, )u + (SMv, SMu) + (v, G'u) = 0 (2.2)
for all vectors u, v in the domain of G*, is fullilled.
Proof. Clearly, to establish the inclusion D(G*) C D(M), it suffices to show that,
for all u E h, the integral
/x -2 I(R(1; c*)~)(x)l 2 xd (2.3)
is convergent. To this end, let us first fix r C (0, 1) and denote by I(r) the set
(-r -1, -r)U (r, r-l). Integrating by parts we have
f(
,)(#x) -2 I(R(1; a*)u)(x)l ~ dx = f(,)(#~)-'l~l '/~ ./3sgn(x) xd21)X(oUl)~x#(pxe
- m-1/z exp(#~ -~) (1~o(,--')1 ~ - I~o(-~-')1 ~)
+ #~'/# exp(#, ~ ) (I,~o(-,-)1 ~ - i,.,o(,.)1 ~)
- 2 f(~) #2 ixl,+l/# xdZi)X(oul)2x#(pxe
_
+ 2~ f~(,.)#lxl ('+#)/2z exp(#:~2/2),~(X)Uo(z)dx
The first two terms vanish as r tends to zero. In fact consider, for example, the case
fl > 0, then, using the Schwartz inequality, we can write the estimate
/?
exp(flr~)~'/~
r 1/# exp(flr2)uo(r) 2 < llbi ~ exp(-fly2)yl-UZdg. (2.4)
Clearly, when fl > 1/2, the right-hand side integral is bounded, therefore (2.4)
vanishes as r tends to 0. On the other hand, when fl C (0, 1/2, by the De L'Htpital
rule, we have
lim r 1/~ exp(-~y2)yl-1/~dy = lira ~rl+l/Bexp(~r2)rl-1/~ = .O
0*---r + r~0+
In a similar way one can compute the other limits and show that the first two terms
vanish. The third and fourth term clearly converge to
-2 IIR(1; C*)uH ~ , 2~ (u, R(1; G*)u)
respectively. Therefore the integral (2.3) is convergent. Moreover, by the identity
R(1; G*) - I = G'R(1; G*), we have
IMR(1; V*)u)) e = -2 IR(1; G*)ull e + 2~e (u, R(1; G*)u)
= (R(1; G'R(1; C*)u)
Therefore we proved also the identity (2.2), with v -- u, because the operator S is
unitary. The proof for arbitrary v,u is the same.
Having found a Lindblad form for the infinitesimal generator of the classical pro-
cess we investigate whether the corresponding minimal quantum dynamical semi-
group (abbreviated to m.q.d.s, in the sequel) on B(h) is identity preserving i.e.
conservative. Recall that the m.q.d.s. 7- is the ultraweakly continuous semigroup
on B(h) defined as follows (see 2, 3, 4, 8). For all positive element X of B(h),
let us consider the increasing sequence
=
The bounded operator ~(X) is given by
= sup
n>0
Proposition 2.3. The abelian subalgebra L~176162 of B(h) is invariant for the
m.q.d.s. T.
Proof. In fact, for every X C L~176 a straightforward computation shows
that
e
for all t >_ 0 and all integer n > 0.
Let/~, be the unique solution of the equation
exp(fl)+~+l =0.
It is easy to check the inequality
-1.2785 < ,3~ < -1.2784.
In the following sections we shall prove the
