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Content: The SemiMartingale Property of the Square of White Noise Integrators Luigi Accardi and Andreas Boukas SPDEs Leading to Local, Relativistic Quantum Vector Fields with Indefinite Metric and Nontrivial SMatrix Sergio Albeverio, Hanno Gottschalk, and JiangLun Wu Considerations on the Controllability of Stochastic Linear Heat Equations Viorel Barbu and Gianmario Tessitore Stochastic Differential Equations for TraceClass Operators and Quantum Continual Measurements Alberto Barchielli and Anna Maria Paganoni Invariant Measures of Diffusion Processes: Regularity, Existence, and Uniqueness Problems Vladimir I. Bogachev and Michael Roeckner On the Theory of Random Attractors and Some Open Problems Tomas Caraballo and Jose Antonio Langa Invariant Densities for Stochastic Semilinear Evolution Equations and Related Properties of Transition Semigroups Anna ChojnowskaMichalik On Some Generalized Solutions of Stochastic PDEs PaoLiu Chow Riemannian Geometry on the Path Space B. Cruzeiro and P. Malliavin A Note on Regularizing Properties of OrnsteinUhlenbeck Semigroups in Infinite Dimensions Giuseppe Da Prato, Marco Fuhrman, and Jerzy Zabczyk White Noise Approach to Stochastic Partial Differential Equations T. Deck, S. Kruse, J. Potthoff, and H. Watanabe Some Results on Invariant States for Quantum Markov Semigroups Franco Fagnola and Rolando Rebolledo Stochastic Problems in Fluid Dynamics Franco Flandoli Limit Theorems for Random Interface Models of GinzburgLandau "j Type Giambattista Giacomin Second Order HamiltonJacobi Equations in Hilbert Spaces and Stochastic Optimal Control Fausto Gozzi Approximations of Stochastic Partial Differential Equations Istvan Gyoengy Regularity and Continuity of Solutions to Stochastic Evolution Equations Anna Karczewska Some New Results in the Theory of SPDEs in Sobolev Spaces N. V. Krylov Lyapunov Function Approaches and Asymptotic Stability of Stochastic Evolution Equations in Hilbert SpacesA Survey of Recent Developments Kai Liu and Aubrey Truman Strong Feller InfiniteDimensional Diffusions Bohdan Maslowski and Jan Seidler Optimal Stopping Time and Impulse Control Problems for the Stochastic NavierStokes Equations J. L. Menaldi and S. S. Sritharan On Martingale Problem Solutions for Stochastic NavierStokes Equation R. Mikulevicius and B. Rozovskii SPDEs Driven by a Homogeneous Wiener Process Szymon Peszat Applications of Malliavin Calculus to SPDEs Marta SanzSole Stochastic Curvature Driven Flows Nung Kwan Yip
stochastic partial differential equations and applications
edited by Giuseppe Da Pnato Scuola Normale Super!ore di Pisa Pisa, Italy
Luciano Tubaro Universita degli Studi di Trento Trento, Italy
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PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes
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Preface In recent years the theory of stochastic partial differential equations has had an intensive development and many important contributions have been obtained. The first results on stochastic evolution equations started to appear in the early 1960s and were motivated by physics, filtering, and control theory. An important development, concerning potential theory on infinite dimensional spaces, has been initiated by L. Gross and Yu. Daleckij. Basic results on existence and uniqueness of solutions of SPDEs were obtained in the 1970s by A. Bensoussan, R. Temam, E. Pardoux, N. V. Krylov, B. Rozovski, M. Viot and D. Dawson, and many others. Since then the literature on the subject has been constantly growing. As an influential reference we cite the Saint Flour course (1984) by Walsh devoted to stochastic evolution equations driven by the brownian sheet, which also surveys previous work. SPDEs are also deeply connected with the theory of parabolic and elliptic equations with an infinite number of variables (Kolmogorov equations) and the theory of Dirichlet forms. Specific aspects of the theory of evolution equations and applications have been major topics of numerous conferences and the development of the subject is becoming more and more rapid. The purpose of the Trento meetings (the fifth of the series being the one which gave rise to the present publication) was to provoke reflections on the "state of the art" of the subject, in order to identify those results that are most promising for future developments. The aim of these lecture notes is to present the topics treated in the meeting, essentially in a review form, to provide a quick overview of current research on the subject. The contributions include the following topics:
1
Stochastic partial differential equations: general theory
2
Specific stochastic partial differential equations
3
Diffusion processes (finite and infinite dimensional)
4
Stochastic calculus
5
Theory of interacting particles
6
Quantum probability
7
Stochastic control
Giuseppe Da Prato Luciano Tubaro
Copyright © 2002 Marcel Dekker, Inc.
Contents Preface Contributors 1.
2.
3.
4.
5.
Hi vii
The SemiMartingale Property of the Square of White Noise Integrators Luigi Accardi and Andreas Boukas
1
SPDEs Leading to Local, Relativistic Quantum Vector Fields with Indefinite Metric and Nontrivial SMatrix Sergio Albeverio, Hanno Gottschalk, and JiangLun Wu
21
Considerations on the Controllability of Stochastic Linear Heat Equations Viorel Barbu and Gianmario Tessitore
39
Stochastic Differential Equations for TraceClass Operators and Quantum Continual Measurements Alberto Barchielli and Anna Maria Paganoni
53
Invariant Measures of Diffusion Processes: Regularity, Existence, and Uniqueness Problems Vladimir I. Bogachev and Michael Rockner
69
6.
On the Theory of Random Attractors and Some Open Problems Tomas Caraballo and Jose Antonio Langa
7.
Invariant Densities for Stochastic Semilinear Evolution Equations and Related Properties of Transition Semigroups Anna ChojnowskaMichalik
89
105
8.
On Some Generalized Solutions of Stochastic PDEs PaoLiu Chow
121
9.
Riemannian Geometry on the Path Space A. B. Cruzeiro and P. Malliavin
133
10.
A Note on Regularizing Properties of OrnsteinUhlenbeck Semigroups in Infinite Dimensions Giuseppe Da Prato, Marco Fuhrman, and Jerzy Zabczyk
167
White Noise Approach to Stochastic Partial Differential Equations T. Deck, S. Kruse, J. Potthoff, and H. Watanabe
183
11.
Copyright © 2002 Marcel Dekker, Inc.
vi
12.
Contents
Some Results on Invariant States for Quantum Markov Semigroups Franco Fagnola and Rolando Rebolledo
197
13.
Stochastic Problems in Fluid Dynamics Franco Flandoli
14.
Limit Theorems for Random Interface Models of GinzburgLandau Vcp Type Giambattista Giacomin
235
Second Order HamiltonJacobi Equations in Hilbert Spaces and Stochastic Optimal Control Fausto Gozzi
255
15.
209
16.
Approximations of Stochastic Partial Differential Equations Istvan Gyongy
17.
Regularity and Continuity of Solutions to Stochastic Evolution Equations 309 Anna Karczewska
18.
287
Some New Results in the Theory of SPDEs in Sobolev Spaces
325
N. V. Krylov
19.
Lyapunov Function Approaches and Asymptotic Stability of Stochastic Evolution Equations in Hilbert Spaces—A Survey of Recent Developments Kai Liu and Aubrey Truman
337
20.
Strong Feller InfiniteDimensional Diffusions Bohdan Maslowski and Jan Seidler
21.
Optimal Stopping Time and Impulse Control Problems for the Stochastic NavierStokes Equations J. L. Menaldi and S. S. Sritharan
389
On Martingale Problem Solutions for Stochastic NavierStokes Equation R. Mikulevicius and B. Rozovskii
405
22.
23.
SPDEs Driven by a Homogeneous Wiener Process Szymon Peszat
24.
Applications of Malliavin Calculus to SPDEs Maria SanzSole
25.
Stochastic Curvature Driven Flows Nung Kwan Yip
Copyright © 2002 Marcel Dekker, Inc.
373
417 429 443
Contributors Luigi Accardi Universita degli Studi di Roma Tor Vergata, Rome, Italy Sergio Albeverio Rheinische FriedrichWilhelmsUniversitat Bonn, Bonn, Germany Viorel Barbu Romania
Institute of Mathematics of the Romanian Academy, Ia§i,
Alberto Barchielli Politecnico di Milano, Milan, Italy Vladimir I. Bogachev Moscow State University, Moscow, Russia Andreas Boukas American College of Greece, Athens, Greece Tomas Caraballo Universidad de Sevilla, Sevilla, Spain Anna ChojnowskaMichalik University of Lodz, Lodz, Poland PaoLiu Chow Wayne State University, Detroit, Michigan Ana Bela Cruzeiro Universidade de Lisboa, Lisbon, Portugal Giuseppe Da Prato Scuola Normale Superiore di Pisa, Pisa, Italy T. Deck Universitat Mannheim, Mannheim, Germany Franco Fagnola Universita degli Studi di Geneva, Genoa, Italy Franco Flandoli Universita di Pisa, Pisa, Italy Marco Fuhrman Politecnico di Milano, Milan, Italy Giambattista Giacomin Universite Paris 7 and Laboratoire de Probabilites et Modeles Aleatoires, Paris, France Hanno Gottschalk Rheinische FriedrichWilhelmsUniversitat Bonn, Bonn, Germany Fausto Gozzi Universita di Roma "La Sapienza," Rome, Italy Istvan Gyongy University of Edinburgh, Edinburgh, Scotland, United Kingdom Anna Karczewska Technical University of Zielona Gora, Zielona Gora, Poland
Copyright © 2002 Marcel Dekker, Inc.
viii
Contributors
S. Kruse Universitat Mannheim, Mannheim, Germany Nicolai V. Krylov University of Minnesota, Minneapolis, Minnesota Jose Antonio Langa Universidad de Sevilla, Sevilla, Spain
Kai Liu University of Wales Swansea, Swansea, Wales, United Kingdom Paul Malliavin Paris, France
Bohdan Maslowski Mathematical Institute of the Academy of Sciences, Prague, Czech Republic Jose Luis Menaldi Wayne State University, Detroit, Michigan
Remigius Mikulevicius University of Southern California, Los Angeles, California Anna Maria Paganoni Politecnico di Milano, Milan, Italy
Szymon Peszat Institute of Mathematics of the Polish Academy of Sciences, Krakow, Poland Jurgen Potthoff
Universitat Mannheim, Mannheim, Germany
Rolando Rebolledo Universidad Catolica de Chile, Santiago, Chile Michael Rockner Universitat Bielefeld, Bielefeld, Germany Boris Rozovskii University of Southern California, Los Angeles, California Marta SanzSole Universitat de Barcelona, Barcelona, Spain
Jan Seidler Mathematical Institute of the Academy of Sciences, Prague, Czech Republic S. S. Sritharan U.S. Navy, San Diego, California Gianmario Tessitore Universita di Genova, Genoa, Italy
Aubrey Truman University of Wales Swansea, Swansea, Wales, United Kingdom Hisao Watanabe Okayama University of Science, Okayama, Japan
Copyright © 2002 Marcel Dekker, Inc.
Contributors
JiangLun Wu University of Wales Swansea, Swansea, Wales, United Kingdom Nung Kwan Yip Purdue University, West Lafayette, Indiana Jerzy Zabczyk Mathematics Institute of the Polish Academy of Sciences, Warsaw, Poland
Copyright © 2002 Marcel Dekker, Inc.
ix
The SemiMartingale Property of the Square of White Noise Integrators LUIGI ACCARDI Centre Vito Volterra, Universita degli Studi di Roma Tor Vergata, Roma 00133, Italy
ANDREAS BOUKAS American College of Greece, Athens 15342, Greece
Abstract. The abstract commutation relations of the algebra of the square of white noise of Accardi, Lu, and Volovich are shown to be realized by operator processes acting on the Fock space of Accardi and Skeide which is very closely related to the Finite Difference Fock space of Boukas and Feinsilver. The processes are shown to satisfy the necessary conditions for inclusion in the framework of the representation free quantum stochastic calculus of Accardi, Fagnola, and Quaegebeur. The connection between the FiniteDifference operators and the creation, annihilation, and conservation operators on usual symmetric Boson Fock space is further studied.
1. Introduction The "square of white noise" or "SWN" algebra was defined in [ALV 99] as the Lie algebra generated by elements B ( f ) , B^(f), and N(f) satisfying the commutation relations:
where c > 0 and /, g, 4>, tfj are suitable functions. Let D = {z e C/\z\ < 1/2} and let S(R+,D) denote the set of step functions defined on R+ with values in D i.e / € S(R+,D] O / = S"_1
ai £D,IiC [0, +00), Ii H Ij = 0 for i ^ j, i, j = 1, 2, ..., n e N.
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2
Accardi and Boukas
A good candidate for a Fock space, on which the above commutation relation can be realized in the operator sense, was defined in [AS 99] as follows:
DEFINITION 1.1. Let c> 0. The SWN Fock space F is the Hilbert space completion of the linear span of " exponential vectors" tf>(f), f € S(R+, D), under the inner product po
>= e*°
After the rescaling c —>• 2 and a^ —>• ^ the SWN inner product defined above is seen to agree with that of the FiniteDifference Fock space of [Bou 88] and [Fei 87]. To realize the above commutation relations on F we define the SWN operators B(f), B^(f), N(f) by their action on the exponential rectors of F as follows:
DEFINITION 1.2. Let f , g £ S(R+,D). Then
JO
^'e REMARK 1.3 For e sufficiently close to zero, /, g G S(R+,D) implies that g + e/ and e6 g are also in S(R+, D).
2. Matrix elements of the SWN operators PROPOSITION 2.1 Let /,(/», g € S(R+,D). Then

2c T /(J Jo I 4(f>((s)g(s)
= 4c " o
Copyright © 2002 Marcel Dekker, Inc.
1 4(s)g(s)
Square of White Noise Integrators
PROOF: r\
—
r°°
2c Jo
f°°
= 2c Jo
fd>
———=t 1 — 4:(j>g f&
—J^^t 1 — 4(f>g
e=0 r\
/»o
2—  e=0 e^
o
1—
= 2c/
Jo
+4
g
NOTE: By Proposition 2.1 < ^(0),S t (/)^(p) >= < B(f)^(4>),^(g) > i.e B^(f) and B(f) are one the adjoint of the other on the exponential domain e. Similarly < V((g) >= < N(f)i/j(), $(g) > i.e. JV(/) is the adjoint of N(f) .Moreover (in the sense of matrix elements) B(/)V>(0) = JV(/)V>(0) = 0
Copyright © 2002 Marcel Dekker, Inc.
4
Accardi and Boukas
PROPOSITION 2.2 Let g,i,f, € S(R+,D). Then
4c2
ds
r
~g(s)~ (1  4/(
/O
1 —40(s)/(s)
+4c
+8c
PROOF
Copyright © 2002 Marcel Dekker, Inc.
Jo
1—
Square of White Noise Integrators
Oed6] ,00
(l4/0r S +
'Jo 5
^
g7
/~
Jo
g0_
^ /00
14/0
/7_
Jo 14/0
d2
o
(1 
= (0)>
r j*^* r ^0^ 14/0 7o 14/0
Jo
/•oo
+8c /
Tpvrt1)2
g7
^
(fa)o of random variables such that for each n €. D the map t € R+ 4 F(t)n is Borel measurable. A random variable F is "iadapted to At]" if domain (F} = D't,
domain (F*) D D't and Fa't]£ = a't]F£, F*a't]£ = c/t]F*€ for aU a't € A't] and £ € D. A stochastic process (F(t))t>o is "adapted to the nitration (At])t>o if F(t) is tadapted to At] for all t > 0. A "simple" adapted process (F(t))t>o is a process which can be written in the form F(t) = ^—iF(t]e)X[tk,tk+1) f°r some n (f)\?
Jo
which implies that
Since, for all j, k € {1, 2, ..., n}
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Square of White Noise Integrators
15
and
\8k(f)\ 1R be a polynomial which is positive semidefinite on [0, oo) and a2 = — (^^y \t~o] • Then
CFa(f) = exp  / .
JR"
af • op(A)fdx\ , / € 5 J
(3)
defines a Gaussian rcovariant noise field.
Proof, (i) Continuity of Cp follows from if} being C°°, normalization is a consequence of ip(Q) = 0 and positive definiteness can be derived from the fact that ip is a conditionally positive definite1 function in t and thus fRd tf)(f) dx is conditionally positive definite in /. But the exponential of a conditionally positive definite function is positive definite by Schoenberg's ° if
Copyright © 2002 Marcel Dekker, Inc.
z
= Q,t£lRL,z£€,
= l,...,n;n£]N
Local, Relativistic Quantum Vector Fields
25
theorem [13]. rinvariance of Cp follows from the rinvariance of V1 and the invariance of the Lebesgue measure under orthogonal transformations. Likewise, invariance of Cp under translations follows from translation invariance of dx. That Cp(f + h) factors for supp/ fl supp/i = 0 is implied by V>(0) = 0. (ii) The proof that Cpg is the characteristic functional of a Gaussian random field is standard (we note that a by definition is positive semidefinite and commutes with the positive operator p(—A).) Invariance follows from the fact that cr2 commutes with r(g) andp(—A) is invariant under orthogonal transformations. Translation invariance follows as in (i). That Cpg(f + h) factors if supp/ D supp/i = 0 is a consequence of
for /, h as above. • Let D : S —¥ S be a rcovariant partial differential operator with constant coefficients, i.e. Dfg = (Df)g and Dfa = (Df)a V/ e S,g € SO(d),o € JRdAssuming that Dis continuously invertible, the following representation has been obtained [12] for the Fourier transform of the Green's function of D:
with mi € C— (— oo, 0], mj ^ mi for I ^ j and v\ € IN. QE(^} is an LxLmatrix with polynomial entries of order < K = 2(J2k=i vi ~ 1) which fulfills the Euclidean transformation law a,dT^(QE(g~1k)) = QE(k)^g € SO(d). Without loss of generality we assume that QE is prime w.r.t the factors (fc 2 + m 2 ), i.e. that none of them divides all of the polynomial matrix elements of QE and furthermore we impose a "positive mass spectrum" condition mi € (0, oo) for I = 1, . . . , N. Given this representation of D~l(k) we define
and we note that p(t) > 0 as t > 0.
Let tp be a rinvariant Levy characteristic s.t. dip(t)/dtp\t=o = 0 and D as above. We define F as in Proposition 2.2 (i) and F9 as in Proposition 2.2 (ii) with p(t) = p(t, D) as in Equation (5), F9 independent of F (in the stochstic sense) . We set up the crucial SPDE of this work as
DA = F + F9
Copyright © 2002 Marcel Dekker, Inc.
(6)
26
Albeverio et al.
which can be solved pathwisely in oo, see [2] for a related situation. • Here we would like to point out that the difference between A as defined here and the corresponding fields defined in [5] is that in that reference
 £2)
(15)
and the effect of the Gaussian noise F9 is to replace (15) by (8). As we
shall see in the discussion of relativistic scattering theory below, this correction leads to a welldefined particle like asymptotics, for v\ = 1,1 = 1,..., n, of the related relativistic quantum field models. Without this correction (or
Copyright © 2002 Marcel Dekker, Inc.
28
Albeverio et al.
the somewhat ad hoc replacement in [5]) the asymptotics would be dipolelike rather than particle like. Such effects now only occur when some of the v\ are strictly larger than one, cf. [18].
Condition 2.4 We say that the partial differential operator D fulfills the no dipole condition, if in the representation (4) we have v\ = I for I = 1,..., N. We restrict from now on our models to fulfill this condition (cf. [18] for the meaning of this condition).
3
Analytic continuation of the Schwinger functions
In this section we discuss the analytic continuation of the truncated Schwinger functions S% to relativistic truncated Wightman functions W£. A solution to this problem can be obtained by representing 5j as FourierLaplace transform, i.e.
x W^ai...ctn(kl,...,kl)dk1dkn,
(16)
where x\ < ... < x^ and W^ai...an ( the Fourier transform of W^) is a tempered distribution which fulfills the spectral property, i.e. it has support in the cone {(/ci,... ,kn] G JRdn : QJ = Y,\=\h € VQ , j = 1,... ,nl}. Here, VQ~ stands for the closed backward lightcone (that we do not use the forward lightcone for the formulation of the spectral condition as done in some other references, is a matter of convention on the Fourier transform). Under this condition the above integral representation exists. From the general theory of quantum fields it follows that 8% is the analytic continuation W% from points with purely relativistically real time to the Euclidean points of purely imaginary time. Furthermore, it follows from the symmetry and Euclidean covariance of the 5j that W% fulfills the requirements of Poincare covariance (w.r.t. the analytic continuation of the representation r) and locality, see e.g. [32]. Following an idea of [12], we expand the denominator of Eq. (4) into partial fractions (we recall that we assume Condition 2.4 to hold) N
1
b
+ mf) with bi € M uniquely determined and &/ ^ 0. Thus, D1(a;) can be
represented as QE(—id/dx) Y^iLi M—A + m^)~1(a;) and for n > 3
Copyright © 2002 Marcel Dekker, Inc.
Local, Relativistic Quantum Vector Fields
29 N
n X
.
n
b
T\ lr I TT(A + m; / 7P*^ J1H r=l ^j=l
Setting .mn(^ • • • , *n) = J
[(A + m*)1^  x) dx
(19)
we note that a FourierLaplace representation (16)of S%mi mn(xi,...,xn) has been calculated in [2] Proposition 7.8. W^ mij ,m ( & i , . . . , kn) is given by
3=1 1=1
i i=j+i
1=1
(20)
Here 5^(k) = 0(±k°)6(k2 — m2) where 0 is the Heaviside step function and k2 = k° — fc 2 . Furthermore, let
k1),...,(ik0n,kn)).
(21)
We then define
lli=i m r
+ fc 2 )
(22)
j=i
and
E
"M (23)
and we can now put together these pieces in the following theorem:
Theorem 3.1 The truncated Schwinger functions S% have a FourierLaplace representation (16) with W"J defined in Eqs. (22) and (23). Equivalently,
Copyright © 2002 Marcel Dekker, Inc.
30
Albeverio et al.
5j is the analytic continuation of W^ from purely real relativistic time to purely imaginary Euclidean time. The truncated Wightman functions W^ fulfill the requirements of temperedness, relativistic covariance w.r.t. the representation of the orthochronous, proper Lorentz group f : L+(d) —> Gl(L), locality, spectral property and cluster property. Here f is obtained by analytic continuation of T to a representation of the proper complex Lorentz group over Cd (which contains SO(d) as a real submanifold) and restriction of this representation to the real orthochronous proper Lorentz group. Proof. That S% has a FourierLaplace representation with W£ defined as above for the case n > 3 follows from the related representation for Sntmlt...tmn, the linearity of the FourierLaplace transform and the general formula for differentiation of a FourierLaplace transform. That Wj is tempered can be derived as in [3] or [1]. The formula f or n = 2 can be derived as in the case of the twopoint function of the free field. The properties of W^ now follow from the general formalism of analytic continuation, [32] •
4
Quantum fields with indefinite metric
In this section we show that the Wightman functions constructed in Theorem 3.1 can be considered as vacuum expectation values of some quantum field theory (QFT) with indefinite metric [29]. We first introduce the concept of a QFT with indefinite metric: Let "H be a (separable) Hilbert space and T> C TL a dense domain. Let 77 be a selfadjoint operator on "H s.t. if = 1. n is called the metric operator. We define 0^(2?) as the unital, involutive algebra of (unbounded) Hilbert space operators A : V >• V s.t. A^ = r]A*ri\r, : V >• T> exists. Here A* stands for the Hilbert space adjoint of (A, "D). The canonical topology on 0^(2?) is induced by the seminorms A 4  ($1,77^^2)!) ^i> ^2 € T>. We say that a sequence of Wightman functions {Wn}n£fj0, Wn € S'n (where Sn = S®n and ' stands for the topological dual  in contrast to the previous sections test functions and distributions from now on are complex valued ) fulfills the Hilbert space structure condition (HSSC) if on Sn there is a Hilbert seminorm pn s.t.
\Wj+l(f®h}\ is generated by repeated application of operators (f),f € S on ^Q, ^0 is invariant under the representation U of P+ and U fulfills the spectral condition f^d(rj^i, \}(a}tyi}elp'a da = 0 if p is not in the forward lightcone (here • is the Minkowski inner product); (ii) is Hermitean 01*' (/) = 0(/*) where f* is the (component wise) complex conjugation of f; (f) is local (4>(f) and 4>(h] commute on V if the support of f and h are spacelike separated: (x — y) < 0 for x € supp/, y e snpph); transforms fcovariantly (\J(g)(f>(f)\J(g~ ) = 0(/ i) Vg € P+ with f (x) = 0
2
1
5
g
1
f(g)f(g~ x)); (Hi)
Wn(h ® • • • ® /«) = (*o, ^(/i) • • • ^(/n)*o) Vn € JN, fi € S.
For the proof, a kind of GNSconstmction on an inner product space is performed, see e.g. [36, 29, 25]. It should be noted that the above assignment of a QFT with indefinite metric to a sequence of Wightman functions in general depends on the Hilbert seminorms pn in (24) and therefore is not 'intrinsic' for the sequence of Wightman functions. For an example see [10]. Concerning our models in Theorem 3.1 we now get: Theorem 4.2 The Wightman functions defined in Section 3 fulfill the HSSC (24). In particular, there exists a QFT with indefinite metric (cf. Theorem 41) s.t. the Wightman functions are given as the vacuum expectation values
of that QFT. Proof. By Theorem 4.1 the Wightman functions fulfill all the requirements of Theorem 4.1, except for the HSSC. That also the HSSC holds, can be seen most easily by verifying a uniform continuity property w.r.t. .®n for the truncated Wightman functions W^, as explained above. Here . is some Schwartz norm on S. It has been verified in [1, 4] that there is such a uniform continuity for W^mij...,mn and thus also the linear combinations of these distributions in (23) have this property. But the Fourier transformed Wightman functions of our model are given by the multiplication of the described linear combination by a polynomial Q^(k\, . . . , fc ) and it is thus sufficient to verify that the degree of Q^ in any variable ki is bounded independently of n, since we then can replace the Schwartz norm j. by the n
2
Pl(d) is the semidirect product of L^(rf) with the translation group IK1.
Copyright © 2002 Marcel Dekker, Inc.
32
Albeverio et al.
Schwartz norm (1 + A;2)'/2. for / larger or equal to this uniform degree. That such a uniform bound of the degree in the ki exists is a straight forward consequence of the definitions (10) and (21). •
5
On the construction of asymptotic states and the S'matrix
Here we describe the scattering behavior of the QFT models with indefinite metric constructed in Theorem 4.2. Since the standard axiomatic scattering theory [22, 23, 35, 34] heavily relies on positivity of the Wightman functions, we can not apply these methods here. Let us therefore first consider the general problem of the construction of
asymptotic states for quantum fields with indefinite metric following [1, 17]: Let 0, which in our case are determined by the representation (4) of the partial differential operator D. Let
S and fiin/out :£>•«$
by X*(A;,in) 0 0 \ 0 Xt(fc,loc) 0 0 0 Xt(k,ont) )
(26)
ant fit = JoQf*, fiin/out = fi t o Jin/out. Here JF (^) denotes the (inverse) Fourier transform. For fi € 0, b = 1, QE = 1) these spaces carry a positive semidefinite metric [1]. Finally we want to show that the scattering of the fields in Theorem 5.1 is nontrivial. Given the form factor functional, one can define the 5matrix of the theory via
Copyright © 2002 Marcel Dekker, Inc.
Local, Relativistic Quantum Vector Fields
35
Sr;nr(fl ® ' ' ' ® fr', fr+l ® ' ' ' ® fn)
•°ut(/n)#o)
(32)
where (.,.) = (., 77.) is the indefinite inner product on H. Using the definitions (28)(30) one can verify by an explicit calculation the following corollary:
Corollary 5.2 The Smatrix of the models in Theorem 5.1 is nontrivial (if ij}(t) has a Poisson part). The Fourier transformed, truncated Smatrix is given by S%(k\; £2) = W%(k\, k%) and
. . . , kn) f[btj
j=l
'
II 6m (*')*£*") l=r+i J l=i
(33)
forn > 3 where f c j , . . . , fcj? < 0 and fe° + 1 ,..., fc° > 0. We remark that in and out fields are free fields and fulfill canonical commutation relations, the relations given in Corollary 5.2 suffice to determine the whole scattering matrix. Aknowledgements. We would like to thank C. Becker and R. Gielerak for interesting discussions. The financial support of D.F.G. via Project " Stochastische Analysis und Systeme mit unendlich vielen Ereiheitsgraden" is gratefully acknowledged.
References [1] S. Albeverio, H. Gottschalk: Scattering theory for quantum fields with indefinite metric, Univ. Bonn preprint 2000, to appear in Commun. Math. Phys. [2] S. Albeverio, H. Gottschalk, J.L. Wu, Convoluted generalized white noise, Schwinger functions and their continuation to Wightman functions, Rev. Math Phys., Vol 8, No. 6, p. 763, (1996).
Copyright © 2002 Marcel Dekker, Inc.
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Albeverio et al.
[3] S. Albeverio, H. Gottschalk, J.L. Wu, Models of local relativistic quantum fields with indefinite metric (in all dimensions), Commun. Math. Phys. 184, p. 509, (1997).
[4] S. Albeverio, H. Gottschalk, J.L. Wu, Nontrivial scattering amplitudes for some local relativistic quantum field models with indefinite metric, Phys. Lett. B 405, p. 243 (1997). [5] S. Albeverio, H. Gottschalk, J.L. Wu, Scattering behaviour of quantum vector fields obtained from Euclidean covariant SPDEs, Rep. on Math. Phys. 44 No. 1/2, 2128 (1999).
[6] S. Albeverio, R. HoeghKrohn: Euclidean Markov fields and relativistic quantum fields from stochastic partial differential equations. Phys. Lett.
B177, 175179 (1986). [7] S. Albeverio, R. HoeghKrohn: Quaternionic nonabelian relativistic quantum fields in four space—time dimensions. Phys. Lett. B189, 329336 (1987).
[8] S. Albeverio, R. HoeghKrohn: Construction of interacting local relativistic quantum fields in four spacetime dimensions. Phys. Lett. B200, 108114 (1988), with erratum in ibid. B202, 621 (1988). [9] S. Albeverio, K. Iwata, T. Kolsrud, Random fields as solutions of the inhomogenous quarternionic CauchyRiemann equation. I. Invariance and analytic continuation , Commun. Math. Phys. 132 p. 550, (1990).
[10] H. Araki, On a pathology in indefinite inner product spaces, Commun. Math. Phys. 85, p. 121 (1982). [11] T. Balaban: Ultra violet stability in field theory. The (A) C H —>• H generates a strongly continuous semigroup S on H • B and C are bounded linear operators on H
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Controllability of Stochastic Linear Heat Equations
41
• (O, £, P) is a probability space, j3t, t > 0 is a standard brownian motion and Ft = &{0S '• s € [0, t]} is the filtration generated by (3
• the control u is a ^"predictable square integrable process Q X [t, T] —>
H, in symbols u € l>([t, T] x Q, #"). as it is well known, see [5], with such assumptions there exists a unique mild solution y^ ' of (3) belonging to the space Cp([t,T},L?((l, H)) of all predictable mean square continuous processes. Finally if A generates an x
u
analytic semigroup of contractions (as in the heat equation case) then ys'x'u € L2(tt,D(A)1/2) for all s € (t,T\. In relation to (3) we are concerned with the following controllability properties: Property 1 (Null Controllability) System (3) is null controllable if for allO7)>u = 0 Property 4 (Backward Approximate Controllability) System (4) is approximately controllable if for all e > 0, all 0 < t < T, all x in L2(V,,Ft,V,H) and all rj in L?(Q,FT,y>,H) there exists v € L%,([t,T\ X
ft, H) such that E\p?'*'v  x\2H < e.
Copyright © 2002 Marcel Dekker, Inc.
Controllability of Stochastic Linear Heat Equations
43
To complete this section we notice that the concrete heat equations (1) and (2) can be treated in the framework introduced in this section just letting:
• H = L2(D) . A : H\V) n H&(D) + H, (Af)($
= A e /(£) + a(f)/(£), V£ € V.
€ 2?.
V£ e 2? 3
Riccati Characterization
In [14] a characterization of null controllable forward systems in terms of a Riccati equation with singular initial datum was proved. Namely:
Definition 3.0.1 A strongly continuous operator valued map P : [0, T] —>• S+(.H") (where T^+(H) is the set of self adjoint nonnegative bounded operators in H) is a solution of:
P'(t) = A*P(t) + P(t)A  P(t)BB*P(t) + C*P(t)C P(0) = +00
(&)
if for each 6 6 (0, T) allt>6 and all z € H P(i)z = S*(t  d)P(5)S(t  6)z+
r*
+ \ S*(ta}[C*P( 0 such that \\L1z\\ < C  1/22 1 for each z 6 Z. Then we can state the analogous of Proposition 4.1
Proposition 4.3 Equation (3) is null controllable if and only if for all 0 < t < T there exists Ct,r > 0 such that and all rj e L 2 (fi,^ r t,P,5"):
s.
(11)
In the same way equation (4) is approximately controllable if and only if for all and all x € L 2 (fi, Tt, P, H) and all t>Q:
dS.
(12)
Proof. We prove the statement for the forward equation being the proof for the backward one identical, the argument is taken from [14]. Let iJf : L\Sl, ft, P, H) )• L2(fi, FT, P, H)
LlX = y*/'0
L*2'T : I^([t, T] x ft, H) 4 L2(fi, ^r, P, ff ) L2u Clearly equation (3) is null controllable if for all 0 < t < T:
Range (l£r) C Range (iff) moreover by (8) : (rt,Ty (L/1
T,r,,0
) ?y — pt
(Tt,T}*
(L,2
) rj
and the claim follows from Lemma 4.2
T,r,,Q
n
The above characterizations immediately yield:
Corollary 4.4 If forward equation (3) is null controllable then it is approximately controllable and the same holds for the backward equation (4).
Copyright © 2002 Marcel Dekker, Inc.
46
Barbu and Tessitore
Finally let us notice that in the particular case of the forward heat equation (1) condition (9) becomes, as it was pointed out in [7] the unique continuation condition:
[p(s,0 = o, V£ e o, vs e [t,T\] => b(*,e) = o, ve € 2?, v* e [*,r]] for any p € Op([t, T], L2(fi, f#(X>))) n L,([t, T] x Q, ^(D)) q 6 4([ f )X*> £) + c(*> £M*, 0) ds + 9( In a similar way condition (9) becomes the observability condition: rji
2
(t, £)d£ < Ct,T® ! f
D
Jt
2 P
(s,
JO
for aU t < T and all (p, g) verifying (13). Identical characterizations involving the solutions of the forward heat equation hold for the controllability of the backward equation. In reality as it will be stated in the next section such characterization is enough to prove the null (and therefore approximate) controllability of the backward heat equation (2) . On the contrary on the side of the controllability of the forward heat equation (1) only partial results can be obtained. For instance we have Example 4.5 (Small Noise Image) Assume that Ker (B*) C Ker (C*) and let (p, q) be a solution of equation (4) in [t, T] with B*p(s) = 0 for all s € [t, T]. Multiplying (4) by B* we get
(B*A*p(s) + B*C*q(s)ds)ds + B*q(s)d(3s = 0 which immediately implies B*q(s) = 0 almost surely in [t, T] X fi and by our assumption C*q(s) = 0 (we prove the above for regular solutions and then extend it by density argument). Therefore for all r e [t,T\:
p(T}= I e(T3^Atq Jr
and computing the conditional expectation:
=0
Vr e [t, T]
(14)
If the deterministic system y' = Ay + Bu, is approximately controllable relation (14) implies p(T) = 0 and by Proposition 4.1 we can conclude that equation (3) is approximately controllable. In particular if {£ : c(£) 7^ 0} C O then equation (1) is approximately controllable.
Copyright © 2002 Marcel Dekker, Inc.
Controllability of Stochastic Linear Heat Equations
47
Example 4.6 (Discretized heat equation in dimension 1) Let H = W1 for some n € N, and let for all v € H:
(Av)i = 2"
(Av)i
= 21 (v2  2ui)
(Av)n = 21 («„_!  2vn)
for some GI, cn € R. Finally fixed any subset
1 and Cxt > 0 independent of rj such that for all mild solution (p, q) of equation (2) in [t, T] with v = 0
E / [e2f>a (0V + 0Vp2) d£da < QrtE / f f
f
V '  *
Jt JD
Copyright © 2002 Marcel Dekker, Inc.
•
i i
J. i
/
&
——
j.
f

Jt J
(16)
Controllability of Stochastic Linear Heat Equations
49
Remark 5.4 The term that does not allow to prove observability of the backward heat equation and consequently null controllability of the forward heat equation is the one involving q. In [1] a trivial computational mistake allowed to prove a Carleman estimate without such a term. Therefore the proof of the null controllability of the forward equation (1) was wrongly claimed.
6
Robust solutions of backward equations
In [4] Da Prato, lannelli and Tubaro (see [4]) proved existence of a solution to equation (3) multiplying it by exp(— C(3i) and then solving pathwise the obtained equation. In [7] FernandezCara, GarridoAtienza and Real proposed to apply the same idea to the backward heat equation (2) with the purpose to deduce unique continuation property for (2) from the well known fact that it holds for deterministic parabolic equations. Although the proof is not satisfactory it is worth, because of its interest, to describe it here. We come back to the general setting. Let (pTi7?'°, ^T'7?>°) be the solution of (4) with v = 0. Define: p(s) = e^/V'"'0;
q(s) = e°'^ (q^'° + C*p^'°) ;
fj = ect^rj
Computing by ltd rule dsp(s) we get:
dsp = ect^(A  ±C2)*ec*P*p(s)ds + q(s}d(3s
p(T] = rj
Then we solve pathwise the equation relative to the above drift. Namely for all ui € Q all r > 0 and all z 6 H we let Z(s, T, u}z be the solution of:
daZ(s, r, u}z = ec*& (A  \C*}*eCfl3*Z(s, T, u)z
Z(T, r, u}z = z
obtaining that p(s) = E (Z(s, T, )rj\Fs) and p(s) = e~c>^ (Z(s, T, The idea is now to deduce (9) from the fact that a similar condition holds for Z(, T, uj) for all o> € Ct. Notice that in the case of the heat equation (1)
 A/ + 2&(Vc  V/) + &Ac  /3,2VC2  i and it is well known that the unique continuation property holds for the deterministic equation driven by the right hand term, for all ui €. fl, see [9].
Copyright © 2002 Marcel Dekker, Inc.
50
Barbu and Tessitore
Coming back to the general framework we assume that C commutates with B (which is always true for the heat equation (1)). If C also commutates with A then Z(S,T, u}z = e^A Consequently, if the deterministic system
is approximately controllable then B*p(s) = 0 for all s, Pa.s. implies that:
By a simple argument we can conclude that 77 = 0. The proof of the approximate controllability of (3) is therefore completed. On the contrary if C do not commutate with A (as in the general heat equation case) Z(T, T) is only measurable with respect to jFy and the above argument can not be applied. In [7] the process (f>(s) = e~c"^TZ(s,T, )rj is introduced observing that
a =E Then the rest of the argument is based on the claim that B*p(s) = 0 for all s € [t,T], Pa.s. implies E (B*(J>(S) \\F^\ F^\ = 0. But this does not seem to be
correct since (j)(s) is only .Frmeasurable. Resuming the argument given in [7] and described in this section allows again to prove the approximate controllability of equation (1) only when c is a constant function.
References [1] V. Barbu, G. Tessitore, Null Controllability of Stochastic Heat Equations with Multiplicative Noise, Preprint S.N.S., Pisa, 1998. [2] V. Barbu, G. Tessitore, On the Carleman estimates for a Stochastic
Heat equation and its applications, manuscript. [3] J.M. Bismut, An Introductory Approach to Duality in Optimal Stochastic Control, SIAM Rev.20, (1978), pp. 6278. [4] G. Da Prato, M. lannelli, L. Tubaro, Some Results on Linear Stochastic Differential Equations in Hilbert Spaces, Rendiconti Accademia dei Lincei, Vol. LXTV (1978), pp. 2229.
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[5] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.
[6] N. El Karoui, R. Mazliak ed., Backward Stochastic Differential tions, Pitmann R.N. in Math. 364, (1997).
Equa
[7] E. FernandezCara, M.J. GarridoAtienza, J. Real, On the Approximate Controllability of a Stochastic Parabolic Equation with Multiplicative Noise, C.R. Acad. Sci. Paris, t. 328, Serie I, (1999), pp. 675680. [8] E. FernandezCara, J. Martin, J. Real, On the Approximate Controllability of Stochastic Stokes Systems, Stoch. Anal. Appl., 17 (1999), pp. 563577.
[9] A.V. Fursikov, O. Yu Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34, (1996), Research Institute of Mathematics, Seoul National University, Korea.
[10] Y. Hu, S. Peng, Adapted Solution of a Backward Semilinear Stochastic Evolution Equation, Stoch. Anal. Appl., 9, (1991), pp.445459. [11] E. Pardoux, S. Peng, Adapted Solution of Backward Stochastic Equations, System and control Letters, 14, (1990), pp.5561.
[12] S. Peng, Backward Stochastic Differential Equation and Exact Controllability of Stochastic Control Systems, Progress in Natural Science, Vol 4, 3, (1994), pp.274283. [13] M. Sirbu, A Riccati Equation Approach to the Null Controllability of Linear Systems, Comm. Appl. Anal.,in print. [14] M. Sirbu, G. Tessitore, Null Controllability of an Infinite Dimensional SDE with State and Controldependent Noise., Preprint Universita di Genova 2000
[15] G. Tessitore, Existence, uniqueness and space regularity of the adapted solutions of a backward SPDE, Stoch. Anal. Appl., 14 (1996), pp. 461486.
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Stochastic Differential Equations for TraceClass Operators and Quantum Continual Measurements ALBERTO BARCHIELLI and ANNA MARIA PAGANONI Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32,120133 Milano, Italy
1
Introduction
The theory of measurements continuous in time in quantum mechanics (quan
tum continual measurements) has been formulated by using the notions of instrument and positive operator valued measure [1, 2, 3, 4, 5, 6], arisen inside the operational approach [1, 7] to quantum mechanics, by using functional integrals [8, 9, 10], by using quantum stochastic differential equations [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] and by using classical stochastic differential equations (SDE's) [12, 13, 14, 15, 16, 17, 18, 4, 19,
20, 21, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 6]. Various types of SDE's are involved, and precisely linear and non linear equations for vectors in Hilbert spaces and for traceclass operators. All such equations contain
either a diffusive part, or a jump one, or both. In Section 2 we introduce a class of linear SDE's for traceclass operators, relevant to the theory of continual measurements, and we recall how such SDE's are related to instruments [1, 36] and master equations [37] and, so, to the general formulation of quantum mechanics. In this paper we do not present the Hilbert space formulation of such SDE's and we make some mathematical simplifications: no time dependence is introduced into the coefficients and only bounded operators on the Hilbert space of the quantum
system are considered; for cases with time dependence see for instance [28, 31, 35] and for examples involving unbounded operators see for instance [13, 14, 19, 21, 29, 38, 32]. In Section 3 we introduce the notion of a posteriori
state [3, 39] and the non linear SDE satisfied by such states; then, we give conditions from which such equation is assured to preserve pure states and 53
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to send any mixed state into a pure one for large times. Finally in Section 4 we review the known results about the existence and uniqueness of invariant measures in the purely diffusive case [40, 33, 35] and we give some concrete examples of physical systems. Other asymptotic results are given in [29, 41]. In the whole presentation we try to underline the open problems.
2
Linear SDE's and instruments
Let us denote by S,(A\\A^) the space of bounded linear operators from the Banach space A\ to the Banach space Ai and let us set £(Ai) = £(Ai', Ai). Let % be a separable complex Hilbert space and let us consider the spaces of operators £(%), 2 e C("H). According to [5], Def. 2 and Theor. 2, 72..* is a quasiinstrument. Then, we introduce the following bounded operators on 3
oo
i=0
j=l
..
C, = 1
_^
* 1
I_
(1)
The adjoint operators of £, A are generators of normcontinuous quantum dynamical semigroups [37]. We set also
K,[p]
= £Q[P] + £i[p] — 7: D%p — — pT)0 FtLet us now consider the following linear SDE for £(%) valued regular right continuous (RRC) processes:
KK] ds + f; / (Lj(rs + =1J°
(3)
(j[os}(y)os~}N(dy,ds), with nonrandom initial condition p € X('H); all the integrals are defined in the weak, or a(Z(U),£(U)}} topology of 1(%). The problem of the existence of a solution of eq. (3) can be reduced to a problem for SDE's in H. It is possible, in many ways, to find a larger probability space, where more Wiener and Poisson processes live, and to construct a linear SDE for a process tpt m % f°r which existence and uniqueness of the solution can be proved by standard means and such that the process at, denned by (a,at] = E[(^t\atpt)} Va € £(H), satisfies eq. (3) [28, 31]. Prom this representation one obtains that there exists a solution of eq. (3) such that:
i. the map p \> o~t(u>) is completely positive; ii. if p € S(H), then 0ti = Tr{ oo for all initial points from M \ T), then, if we call MQ the closure of the positive orbit of XQ, all the points in MQ are recurrent and all the points in M\MQ are transient. Moreover there exists an unique invariant probability measure fj, and Supp/^ = MQ. Let us observe that Theorems 4 and 5 deal with different cases, in fact the number m of fields entering the diffusive part of the SDE is 1 for Theorem 5, while for Theorem 4 one necessarily needs m > d. In order to check the hypotheses of Theorem 4 we have firstly to prove the ellipticity of the diffusion matrix on M but a finite set of points; the check of the ellipticity can be reduced to some properties of the operators Lj . Firstly, the tangent space to M in a point p is Tp = {T € 2(%) : r = r*, pr + rp = T}; moreover for p € M it is possible to write p = IV'XV'I with ij) £~H, V> = 1 and then it is possible to prove that we have also T\^(^\ = {T € £(?0 : T = \ip}((f>\ + ^{Vi € H, ( € "H, 4> 7^ 0, {4>\ijj) = 0 there exists j such that
Re{0L^) + 0.
(21)
For what concerns the application of Theorem 5 to eq. (16), one has to study the equation p't = B\(pt). If the initial condition is po — V'o){V'o,
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ipo € "H, HV'oll = 1) then the solution of such an equation can be written as
Pt=
IV'tXV'tl M , 2no i
HV'tll
i
L t
Vt =e Vo
(22)
In [33] some concrete examples are studied with LI selfadjoint and, in particular, with LI onedimensional projection. Note that eq. (16) is invariant for a change of sign of the Lj's and the Wiener processes; so, to check the hypotheses of Theorem 5 also the equation pt = —B\(pt) can be considered. Even for "H = C2, there are some physically interesting examples. Let us consider a twolevel atom of resonance frequency UJQ, stimulated by a monochromatic laser of frequency w; the interaction between the atom and the electromagnetic field is mediated only by the absorption/emission process. It is known that the detection schemes of the fluorescence light known as homodyne/heterodyne detection and direct detection can be treated by SDE's as eq. (12), of diffusive and jump type respectively [26, 27, 22]. In the case of an homodyne/heterodyne detection scheme with a local oscillator in resonance with the stimulating laser, the equation for the a posteriori states turns out to be of the form (12), with only the diffusive part and with
Lj = (eja}az + i(Aa)cr_  i(aA} 0, ft = 2{aA) > 0. In [35] we have proved that, if the complex numbers (&j\a) are not all proportional to a fixed one, then eq. (21) is satisfied in the whole manifold M but in the point po = I
I, where the condition (20) is satisfied; so, in
this case, the hypotheses of Theorem 4 are fulfilled and there exists a unique
invariant measure p, which has Supp // = M. In [26] the problem of the invariant measure for a twolevel atom is studied by means of numerical simulations. The authors consider a heterodyne detection scheme which satisfy all the hypotheses of Theorem 4, as we have shown in [35], and a homodyne scheme which only in a particular case can
be handled via Theorem 4; however, Theorem 5 can be applied. The model is characterized by m = I, LI = e"1^ Hcr_, 0 € [0,2Tr), (aA) = ift/2. In
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this case the solution of the equation p't = &i(pt) can be easily computed via
eq. (22) and one obtains that the hypotheses of Theorem 5 are satisfied with r = 0, XQ = po = I 
1 . This means that there exists a unique invariant
probability measure fj, with support MQ implicitly given in Theorem 5. Prom the structure of the fields A and BI, we conjecture that the support of n should be the whole M, apart from exceptional cases such as the one studied in [35], which corresponds to Aw = 0, = Tr/2 and for which MQ reduces to a circumference (for "H = C2, M can be represented as a spherical surface). Let us end by discussing the connections between the invariant measure for eq. (12) and the equilibrium state of the master equation (9). The following considerations apply both to the diffusive and to the jump case; we assume that H is finite dimensional and that there exists a unique invariant probability measure p, and that Supp^u =: MQ C M. By definition of invariant measure we have
/ W(f(Pt)}^p} = ! f(p)n(dp)
JM
(24)
JMO
for every bounded measurable complex function / on M; Ep is the expectation in the case the initial condition for the process is p. Moreover, by the uniqueness of /j, we have ergodicity [48], i.e. rr\
lim 1 /
TS+OO T Jo
f(pt)dt = I f(p)n(dp) JMo
a.s.
(25)
By applying these two facts to the function f(p) = (a, p), where a € £(%), and by recalling that W(pt\ = r]t satisfies the master equation (9), we obtain that the equation jC[rj\ = 0 has a unique solution solution 7?eq in *?("H) , given
by ??eq= /
w(dp);
(26)
JMo
moreover for every initial condition 1 [T 7p / Ptdt = %q
a.s.
L JQ
(27)
Another quantity of interest is _D 2 (a;p) = {(a* — (a, p))(a — (a*,p)),p) = (a*a,p) —  (a, p) 2 , a € £(%), p € £(%), which is a quantum analogue of the variance. Note that we have the decomposition \{a,p  7 f c q )  ( d p )
MO
(28)
and, by the ergodicity, we obtain
lim i / D\apt)dt= f 1 JQ
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D\ap)v(Ap}.
J M() M()
(29)
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For the case m = 1, LI = L\, the asymptotic behaviour of D2(Li;pt) has been studied in [33].
References [1] E. B. Davies, Quantum Theory of Open Systems (Academic Press, London, 1976). [2] A. Barchielli, L. Lanz, G. M. Prosperi, Statistics of continuous trajectories in quantum mechanics: Operationvalued stochastic processes, Found. Phys. 13 (1983) 779812.
[3] V. P. Belavkin, Theory of the control of observable quantum systems, Automat. Remote Control 44 (1983) 178188. [4] A. Barchielli, V. P. Belavkin, Measurements continuous in time and a posteriori states in quantum mechanics, J. Phys. A: Math. Gen. 24 (1991) 14951514. [5] A. Barchielli, A. M. Paganoni, A note on a formula of LevyKhinchin type in quantum probability, Nagoya Math. J. 141 (1996) 2943.
[6] A. Barchielli, Entropy and information gain in quantum continual measurements, preprint n. 430/P, September 2000, Mathematical Department, Politecnico di Milano. [7] K. Kraus, States, Effects and Operations, Lecture Notes in Physics 190 (Springer, Berlin, 1980). [8] A. Barchielli, L. Lanz, G. M. Prosperi, A model for the macroscopic description and continual observation in quantum mechanics, Nuovo Cimento 72 B (1982) 79121.
[9] G. Lupieri, Generalized stochastic processes and continual observations in quantum mechanics, J. Math. Phys. 24 (1983) 23292339. [10] S. Albeverio, V. N. Kolokol'tsov, O. G. Smolianov, Continuous quantum measurement: local and global approaches, Rev. Math. Phys. 9 (1997) 907920.
[11] A. Barchielli, G. Lupieri, Quantum stochastic calculus, operation valued stochastic processes and continual measurements in quantum mechanics, J. Math. Phys. 26 (1985) 22222230.
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[12] V. P. Belavkin, Nondemolition measurements, nonlinear filtering and dynamic programming of quantum stochastic processes. In A. Blaquiere (ed.), Modelling and Control of Systems, Lecture Notes in Control and Information Sciences 121 (Springer, Berlin, 1988) pp. 245265. [13] V. P. Belavkin, A new wave equation for a continuous nondemolition measurement, Phys. Lett. A 140 (1989) 355358.
[14] V. P. Belavkin, P. Staszewski, A quantum particle undergoing continuous observation, Phys. Lett. A 140 (1989) 359362.
[15] V. P. Belavkin, A continuous counting observation and posterior quantum dynamics, J. Phys. A: Math. Gen. 22 (1989) L1109L1114. [16] V. P. Belavkin, A stochastic posterior Schrodinger equation for counting nondemolition measurement, Lett. Math. Phys. 20 (1990) 8589. [17] V. P. Belavkin, A stochastic calculus of quantum inputoutput processes and quantum nondemolition filtering, Current Problems in Mathematics, Newest Results 36 (1991) 26252647.
[18] V. P. Belavkin, Continuous nondemolition observation, quantum filtering and optimal estimation. In C. Bendjaballah, O. Hirota, S. Reynaud (eds.), Quantum Aspects of Optical Communications, Lect. Notes Phys. vol. 378 (Springer, Berlin, 1991) pp. 151163. [19] V. P. Belavkin, Quantum continual measurements and a posteriori collapse on CCR, Commun. Math. Phys. 146 (1992) 611635. [20] V. P. Belavkin, Quantum stochastic calculus and quantum nonlinear filtering, J. Multiv. Anal. 42 (1992) 171201. [21] V. P. Belavkin, P. Staszewski, Nondemolition observation of a free quantum particle, Phys. Rev. A 45 (1992) 13471356. [22] A. Barchielli, A. M. Paganoni, Detection theory in quantum optics: stochastic representation, Quantum Semiclass. Opt. 8 (1996) 133156.
[23] A. Barchielli, On the quantum theory of direct detection. In O. Hirota, A. S. Holevo, C. M. Caves (eds.), Quantum Communication, Computing, and Measurement (Plenum Press, New York, 1997) pp. 243252.
[24] A. Barchielli, Stochastic differential equations and a posteriori states in quantum mechanics, Int. J. Theor. Phys. 32 (1993) 22212232.
[25] A. Barchielli, On the quantum theory of measurements continuous in time, Rep. Math. Phys. 33 (1993) 2134.
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[26] H. M. Wiseman, G. J. Milburn, Interpretation of quantum jump and diffusion processes illustrated on the Bloch sphere, Phys. Rev. A 47 (1993) 16521666.
[27] A. Barchielli, Some stochastic differential equations in quantum optics and measurement theory: the case of counting processes. In L. Diosi, B. Lukacs (eds.), Stochastic Evolution of Quantum States in Open Systems and in Measurement Processes (World Scientific, Singapore, 1994), pp. 114. [28] A. Barchielli, A. S. Holevo, Constructing quantum measurement processes via classical stochastic calculus, Stochastic Process. Appl. 58 (1995) 293317. [29] V. N. Kolokol'tsov, Scattering theory for the Belavkin equation describing a quantum particle with continuously observed coordinate, J. Math. Phys. 36 (1995) 27412760. [30]
A. Barchielli, Some stochastic differential equations in quantum optics and measurement theory: the case of diffusive processes, in C. Cecchini (ed.), Contributions in Probability  In memory of Alberto Frigerio (Forum, Udine, 1996), pp. 4355.
[31]
A. Barchielli, A. M. Paganoni, F. Zucca, On stochastic differential equations and semigroups of probability operators in quantum probability, Stochastic Process. Appl. 73 (1998) 6986.
[32] A. Barchielli, F. Zucca, On a class of stochastic differential equations used in quantum optics. Rendiconti del Seminario Matematico e Fisico di Milano, Vol. LXVI (1996) (Due Erre Grafica, Milano, 1998) pp. 355376.
[33]
V. N. Kolokoltsov, Long time behavior of continuously observed and controlled quantum systems (a study of the Belavkin quantum filtering equation). Quantum Probability Communications, Vol. X (World Scientific, Singapore, 1998) pp. 229243.
[34]
A. C. Doherty, S. M. Tan, A. S. Parkins, D. F. Walls, State determination in continuous measurements, Phys. Rev. A 60 (1999) 23802392.
[35] A. Barchielli, A. M. Paganoni, On the asymptotic behaviour of some stochastic differential equations for quantum states, preprint n. 420/P, July 2000, Mathematical Department, Politecnico di Milano.
[36]
M. Ozawa, Quantum measuring processes of continuous observables, J. Math. Phys. 25 (1984) 7987.
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[37] G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48 (1976) 119130. [38] A. S. Holevo, On dissipative stochastic equations in a Hilbert space, Probab. Theory Relat. Fields 104 (1996) 483500.
[39] M. Ozawa, Conditional probability and a posteriori states in quantum mechanics, Publ. Res. List. Math. Sc. Kyoto Univ. 21 (1985) 279295. [40] A. M. Paganoni, On a class of stochastic differential equations in quantum theories, Ph. D. Thesis (1998), Math. Dept., Univ. Milano. [41] V. N. Kolokoltsov, A note on the long time asymptotics of the Brow
nian motion with application to the theory of quantum measurement, Potential Anal. 7 (1997) 759764. [42] M. Ozawa, On information gain by quantum measurements of continuous observables, J. Math. Phys. 27 (1986) 759763.
[43] M. Ozawa, Mathematical characterizations of measurement statistics. In V. P. Belavkin, O. Hirota, R. L. Hudson (Eds.), Quantum Communications and Measurement (Nottingham, 1994) (Plenum, New York, 1995) pp. 109117. [44] M. Metivier, Semimartingales, a Course on Stochastic Processes (W. de Gruyter, Berlin, New York, 1982). [45] N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes (North Holland, Amsterdam, 1989).
[46] W. Kliemann, Recurrence and invariant measures for degenerate diffusions, Ann. Probab. 15 (1997) 690707. [47] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, (Cambridge Univ. Press, Cambridge, 1992). [48] G. Da Prato, J. Zabczyk, Ergodicity for Infinite Dimensional Systmes (Cambridge University Press, Cambridge, 1996).
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Invariant Measures of Diffusion Processes: Regularity, Existence, and Uniqueness Problems VLADIMIR I. BOGACHEV Moscow State University, Moscow 119899, Russia
MICHAEL ROCKNER Fakultat fur Mathematik, Universitat Bielefeld, D33615 Bielefeld, Germany
1. INTRODUCTION This work gives a survey of the results obtained in a series of papers [1], [5]  [17]. This survey has had as a starting point our lectures at the conference on stochastic partial differential equations in Levico (January 2000) with a continuation at the conference on stochastic analysis in Bielefeld (August 2000) and in a series of the first author's lectures in Kyoto and Osaka in October 2000. It is our pleasure to thank G. Da Prato, M. Fukushima, N. Krylov, I. Shigekawa, and L. Tubaro. The principal object of the above cited papers has been the elliptic equation for a measure /j, on a finite or infinite dimensional linear space or manifold. The main point is that (1.1) is just a symbolic expression which is understood as the equality
= 0
(1.2)
for all functions
bfj, = 0. Then: (i) If fj, is nonnegative, then (detA)1^^ has a density which belongs to
(ii) // A is locally Holder continuous and nondegenerate, then fj, has a density which belongs to L[ (fi, dx) for every r € [1, d'}. oc
If b is bounded, then assertion (i) in the above corollaries follows from the main result in [33] (later it was proved also in [34], [40]). Remark 2.2. Assertion (i) of Theorem 2.1 for nonnegative measures extends to the case when \i is a (7finite nonnegative Borel measure on fi (not necessarily locally bounded). Let us consider the equation L*A b[i = 0, which makes sense via (1.2) also for crfinite measures provided that a^,bl 6 L}OC(£I,IJL). One can find a probability measure /wo such that fj, = /juo, where / is //omeasurable and f ( x ) > 0. Let a^ := /a y , OQ := fb1, AQ = (a bo = (bo)i d. Suppose that either b1 € L^oc(£l, dx) and V €. L^oc(Cl, p) or b1 € lJjoc(£l, p) . Assume
that L*A bp, = 0. Then, p, has a density in Hf^c (J1) that is locally Holder continuous. Corollary 2.4. In the situation of Theorem 2.3, let b € L (Sl,dx} and let Q be the continuous version of the density of p. Then: (i) // p is nonnegative, then, for every compact set K contained in a connected open set U with compact closure in Cl, one has sup Q < C inf g, K K where the constant C depends only on \Q?3\\HP II and K. In particular, if Q is not identically zero on U, then Q > 0 on U. (ii) If g is strictly positive on the boundary of a bounded open set U C U C Q, then g is strictly positive on U. 1
ploc
It is clear from the example in Introduction that this corollary is not valid under the condition bl € I^oc(f2, p), which is an alternative condition for regularity. Remark 2.5. In Theorem 2.3, one cannot omit the hypotheses that A is locally uniformly nondegenerate and a*7 £ Hf^ . Indeed, if A and b vanish at a point XQ, then Dirac's measure at XQ satisfies our elliptic equation. In particular, if it is not given in advance that p is absolutely continuous, then one cannot take an arbitrary Lebesgue version of A. In order to see that p cannot be more regular than A, let us take a probability measure p
with a smooth density that satisfies L*Ib /J. = 0, e.g., let fj, be the standard Gaussian measure and bo(x) = —x. Now, if g is any Borel function with 1 < 9 < 2, then the measure gp satisfies our elliptic equation with A = g~l I and b = g~lbo. In particular, we obtain an example, where A and b are Holder continuous and A is uniformly nondegenerate, but the density of /j, is not weakly differentiable. Also, the condition p > d is essential for the membership of p in a Sobolev class even if A = I (see an example in [6]). However, as we shall see below, if fi is a probability measure on Rd, then the condition 6 € L 2 (ffi d , fj,) implies that p = gdx with y^ € In the case when the coefficients ay' are infinitely differentiable, the above regularity results can be precised (although even the case A = I does not exhibit essential qualitative difference) . In the formulation we shall use the fractional Sobolev classes Hf^(Q,), s € (—00, +00).
Theorem 2.6. Suppose that ali € C°°(£l) and that A is nondegenerate. Let H be a locally finite measure on fi such that L*A b/j, = 0, where b1 e L}oc(£l, fi) . Then: (i) fj. € flf^1~d(p~1)/p~e(J2) for every p > I and e > 0. In particular, fj, = gdx, where g € L^oc(O,,dx) for every p e [l,d/(d — 1)), since in this case ld(p l ) / p > 0.
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(ii) 7/6* € L^oc(Q,,fJ,), where 1 < 7 < d, then Q := dfj,/dx € H for every p € [1, d/(d — 7 + 1)). In particular, g € L^oc(fl, dx) for all p € (iii) If 7 > d and either b1 €. L^oc(fi,dx) or b1 € Lfoc(fl,p), then g := dp/dx € Hiol(£l], hence Q has a version in C fl ~ d / 7 (fi). Let us now turn to global estimates of logarithmic gradients of solutions. Given a probability measure fj, on Rd, we denote by T(fj,) the closure of the space {V 0 outside 0, then the measure f dx is not a unique solution of (1.1) with A= I and b(x) = f ' ( x ) / f ( x ) . Indeed, let g(x) = f ( x ) / 2 if x < 0 and g(x) = cf(x) if x > 0 where c is such that g is also a probability density. Then g' /g = f ' / f and gdx satisfies the same equation. A typical example is the function f ( x ) = cx2e~x . One might think that this nonuniqueness is due to the singularity of b at the origin,
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and in the one dimensional case this is indeed so. However, if d > 2, then even smoothness of b does not guarantee the uniqueness. Let us consider the following example from [14]. Example 3.1. Let ft (a) = xi  2rc exp[sf /2  x%/2], 1
2
b (x) = x d such that a^\uR € HaR'l(UR), bl\uR e LaR(Uft). Assume, in addition, that there exists a function V G (72(R• co
as a; —>• oo.
(3.1)
Then, there exists a unique probability measure ^ on Rd such that \b\ € LIOC([I) and L*Ab[j, = 0. Moreover, fj, admits a continuous strictly positive density g such that Q\uR^i £ HaR'^(Up,i) for every R> 1. Corollary 3.3. Suppose that A = (a) is a continuous mapping on W with 1
7
1
d
values in the space of nonnegative symmetric linear operators on R and that there exists a function V € C2(Rd) such that (3.1) is fulfilled. Then: (i) if b is continuous, then there exists a probability measure fj, such that
£V = °; (ii) if A is nondegenerate and b is measurable and locally
bounded, then there exists a probability measure /i which has a density in the class L^ and satisfies the equation L*Ab/j, = 0.
Corollary 3.4. Let A be a uniformly bounded measurable mapping with values in the space of nonnegative symmetric operators on Rd and let b be a locally bounded measurable mapping such that
lim (b(x),x) = —oo. Then the assertion (i) of the previous corollary is true provided that A and b are continuous, and assertion (ii) is true provided that A is locally uniformly nondegenerate.
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Theorem 3.2 applies to the situation where A = I and b(x) = —k(x)x, where k € Lfoc(Rd) is such that /c(a;)rc2 —>• +00 as \x\ —> oo. More generally, if A = /, then it suffices to have the weaker estimate limsup(a;, x)7"1 2(7 — 1) +n + ( b ( x ) , x ) \ = — oo 
J
"
for some 7 > 1 (then the function V(x) = (x,x)^ can be used). Another example covered by Theorem 3.2 is this: A = I and for some 7 > 1 one has (b(x), x) < — r < — d + 2 — 27 outside some ball (of course, we assume that 6 is locally integrable in power bigger than d). Then we take the function V(x) = (x,x)i. If there is a Lyapunov function, then one can obtain certain estimates of the integrals with respect to the solutions. Corollary 3.5. Suppose that in the situation of Theorem 3.2 there exist a
constant C and a nonnegative measurable function 0 such that one has a. e. L^,bV < C — 0. Then, one has
Qdfj, < C for any probability measure fi
satisfying (1.1). Let us now pass to the uniqueness results which do not require any Lyapunov functions. Let us recall that at the beginning of this section we have considered an example which shows that the best possible local smoothness conditions do not ensure uniqueness. The next result yields uniqueness, but does not ensure existence.
Theorem 3.6. Let p, and v be two Borel probability measures on W* such that L*T bfj, = L*Ibv = 0, where b is a Borel vector field which is in L2(/i) and L?(v). Then fj, = v. Let us note also the following fact. Theorem 3.7. Suppose that a e Hf£(lBL ) and V € if (M ), where p> d, and that A is locally uniformly nondegenerate. Suppose also that equation (1.1) has the following additional property: if a bounded measure p on M.d satisfies this equation, then \fj,\ does also. Then this equation has at most one solution in the class of probability measures. ij
d
oc
d
Proof. Let fj, and v be two different probability measures satisfying (1.1). Then 77 := fj,— v also satisfies this equation. By our assumption, the measure  ?7 is also a solution. We know from Section 2 that T; has a strictly positive continuous density. This is, however, impossible, since the measure 77 has a continuous density which must vanish at some point, since its integral is zero. D It is clear from Example 3.1 that not always equation (1.1) has the above mentioned property (of course, for bounded domains or not necessarily bounded measures, simpler examples exist: it is enough just to note that the absolute value of a harmonic function may not be harmonic) . However, it is important that the invariant measures of the Markov semigroups
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have the property mentioned in the previous theorem. As we shall now see, the uniqueness problem has interesting and deep connections with Markov semigroups. Let us recall that a semigroup (Tt)t>o of bounded linear operators on the space Ll(fj,), where fj, is a probability measure on R d , is called subMarkovian if it takes nonnegative functions to nonnegative ones. If, in addition, it sends 1 to 1, then it is called Markovian. The same terminology is used for semigroups on the space B00(^d) of bounded Borel functions. The measure /j, is said to be subinvariant with respect to (Tt)t>o if, for every bounded nonnegative Borel function /, one has
f Ttf dn < f f dn,
Vt>0.
(3.2)
If one has the equality in (3.2) instead of the inequality, then fj, is called invariant for (Tt)t>oThe following important result of W. Stannat [39] exhibits a deep relation
between solutions of (1.1) and diffusion semigroups with invariant measures.
Theorem 3.8. Let a G Hf£(E. ), where p > d, be continuous, let b G L^oc(M.d,dx), and let A be nondegenerate. Suppose that there exists a probability measure fj, on Rd which satisfies equation (1.1). Then: (i) There exij
d
ists a strongly continuous subMarkovian semigroup (T^)t>o on ^(p.) such that its generator coincides with LA,I on (70° (Rd) and [i is subinvariant for
(r/*)t>o; this semigroup is characterized by the following property: if L is its generator, then, for any A > 0 and f G C0°(Ro is the only continuous semigroup on L^(^i) whose generator coincides with LA,b on (7o°(]Rd). An equivalent condition: for some (hence for all) A G (0, +00), (L ,b  A)(Cg°(R )) is dense in L(^}. In general, the measure ^ from assertion (i) is indeed only subinvariant d
l
A
and there are different semigroups (Tt)t>o whose generators extend LA,I
on
d
Co°(R ). The next result links its invariance with uniqueness. Theorem 3.9. Suppose that the measure \a from assertion (i) is invariant for the associated semigroup (T^)t>Q. Then \i is the only probability measure satisfying (1.1). In the case of infinitely differentiable coefficients, the assertion of the previous theorem was proved by Varadhan [44] who posed the problem for non smooth 6. This problem has been given a positive solution in [1]. Now let us return to Example 3.1 and consider two different probability measures n\ and ^2 satisfying the equation LJ bfj, = 0. By Theorem 3.8,
there exist strongly continuous semigroups (Tt )t>o and (Tt )t>o on L 1 (/^i) and L1^), respectively, such that their generators coincide with L/,6 on
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and the measures n\ and ^ are subinvariant with respect to these semigroups. However, by Theorem 3.9, both measures are not invariant. Moreover, it follows by assertion (ii) of Theorem 3.8 that the semigroups (T£ )t>o
and (Tf
)t>o
on Ll(n\) and Ll(no, then the semigroup is unique and the solution of the original elliptic equation is unique (in the class of all probability measures). However, a natural question is whether there exist other invariant probabilities for the semigroup (a priori it could happen that an invariant measure does not satisfy the elliptic equation). Another natural question is whether the semigroup may have invariant measures in the case when the measure n is not invariant and whether all these invariant measures satisfy the elliptic equation. However, to make these questions unambigous, we have to choose a natural version of the semigroup. Indeed, the semigroup (T/*)t>o acts on Ll(fJ.) and admits many Borel versions . Suppose we set Tf(Q) = /(O) for all Borel functions /. Then Dirac's measure is invariant for such a version. Of course, we can exclude this by considering only absolutely continuous invariant measures, but there is a better option described in the next theorem. Namely, it turns out that every function T^f has a continuous modification which can be explicitly written by means of certain subprobability densities p(t, x, y), t > 0.
Theorem 3.11. Suppose that p, is a probability measure on Rd which satisfies the equation L*A bp, = 0, where A and b are the same as in Theorem 3.8. Assume, in addition, thatp > d + 2. Then there exist unique subprobability
kernels Kt(,dy), t > 0, on Md such that Kt(x,dy) = p(t,x,y)dy, where p(t,x,y) is locally Holder continuous on (0, +00) x Kd x Rd and, for every function f G L1(lRd, /z), the function
x 14 Ktf(x) := / f(y)p(t, x, y) dy is a /^version ofT^f
such that (t, x) i4 K t f ( x ) is continuous on (0, +00)
X
TTj>d
IK. .
Corollary 3.12.
Suppose that in the situation of Theorem 3.11,
there exists
a bounded measure v on R^ that is invariant with respect to the natural
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version of (Tf)t>o given by the kernels (Kt)t>o> ie,
v(dy] = I Kt(x, dy) v(dx],
W > 0.
Then L*A bv — 0. If, in addition, [i is invariant for ^Tt}t>o, then v = constju. Proof. The fact that such a measure v satisfies the equation L*A bi> = 0 has been shown in the proof of Corollary 4.3 in [7] (the proof of this fact does not use the invariance of p). In order to prove the last claim, we may assume that v is nonnegative, since positive and negative parts of v are also invariant for the semigroup (see [1]). Hence it suffices to consider probability measures v. It remains now to apply Theorem 3.9. D
4. INFINITE DIMENSIONAL CASE Let us turn to the infinite dimensional case. The main idea is the same as in the finite dimensional case: given an operator L on some class K, of test functions on the space X (which may be a linear space or a manifold), we say that a measure fj, on X satisfies the equation
L> = 0
(4.1)
l
if Jjf € L (n] for every
du, = 0. Jx
(4.2)
Typical problems related to this equation are essentially the same as in the finite dimensional case: existence, uniqueness and symmetry of solutions as well as their differential properties. However, the formulation of the latter problem requires now new concepts due to the absence of any canonical measure like Lebesgue measure. There are three main directions along which one can investigate such differential properties: 1) choose some reference measure (typically, Gaussian) and study densities of solutions with respect to this measure; 2) directly investigate the differentiablity of solutions in a suitable sense, e.g., in the sense of Fomin (i.e., in the sense of the integration by parts, which is a direct infinite dimensional analog of Sobolev's approach); 3) study the properties of conditional measures. In some cases, these approaches are strongly related, in others not. We shall consider several results and examples illustrating this. Let us consider the case when X is a locally convex space with the dual .X"*. In view of further applications, the reader may assume that X is either a Hilbert space or the countable product of real lines, i.e. X = R3, where S is a countable set, say, the integer lattice in Rm. In the latter case, the dual is the space RQ of all finite sequences. Set
...,/„),V>
(X, {ln}} ••= fall, ..., ln), *!> € qf°(Rn), n e or on its subclass
The investigation of the equation L*^ = Q for measures on infinite dimensional spaces has been considerably influenced by the wellknown work of Shigekawa [38]. In this work, the following situation has been considered. Let H be a separable Hilbert space continuously embedded into X such that there exists a centered Radon Gaussian measure 7 on X with the CameronMartin space H. Denote by  • \jj the Hilbert norm of H. Let {en} be an orthonormal basis in H and let ln € X* be such that ln(&k) = $nk Assume that v: X —> H is a Borel mapping. Set bn(x) = ln(x) + (v(x),en)H.
(4.5)
It is readily seen that if v = 0, then the measure 7 satisfies the equation L*7 = 0 in the above sense. This follows by the integration by parts formula, since, as one can verify, (3ln = ~ln It is instructive to have in mind the situation where X = W°, H = /2 and 7 is the countable product of the standard Gaussian measure on the line. Let us take for {en} the standard basis in Z2; then ln is just the nth coordinate function and bn(x) = —xn + vn(x). Shigekawa [38] proved the following result.
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Theorem 4.1. Let sup \v(x)\n < oo and let L be defined by (4.4), (4.5).
Then the equation L*p, = 0 with respect to the class FC%°(X, {ln}) (or with respect to the class PCQ°(X,{ln})) has a solution p, which is a probability measure absolutely continuous with respect to 7. In addition, this solution is unique in the class of probability measures absolutely continuous with respect to 7. To be more precise, it has been proved in [38] that the closure of L generates a strongly continuous semigroup (Tt)t>o on L1(7) and that this semigroup has a unique invariant probability fj, in the class of the probability measures absolutely continuous with respect to 7. However, one can show that if L*fi = 0 with respect to the class J:C^°(X, {ln}) and p, o and, conversely, if /^ o, then one has bn € L2(p,) and L*p, = 0, i.e., Theorem 4.1 is equivalent to the original result of Shigekawa. In the same paper, Shigekawa raised the problem whether the uniqueness statement extends to the class of all probability measures. A non ambiguous formulation of this problem is concerned with the elliptic equation rather than the semigroup, since, in typical cases, the semigroup (Tt)t>o has no canonical Borel version on the space of bounded Borel functions and its different Borel versions may have different invariant measures. For example, let XQ be a Borel linear subspace in X such that 7(X"o) = 1. If we redefine Ttf by setting Ttf(x) — f(x) if a; € X\XQ, then we obtain a version of (Tt)t>o for which every Borel measure on X\Xo is invariant. Such a version may be obtained by the expression used in [38] via a Wiener process in X for a suitably chosen Wiener process. During several years this problem remained open until it was given a positive solution in [10], where the following more general result has been proved. Theorem 4.2. Let ^ be a probability measure on X such that l € L (n}, \v\H € L2(fj,) and L*fj, = 0 with respect to the class fC%°(X, {ln}) Then fj, is absolutely continuous with respect to 7. In Theorem 4.1, one has g := dfj,/d"f € ZJ2>1(7), in particular, Q € L2(~f). This is not true in Theorem 4.2, where one only has ^/Q € .ff2'1^) (see, e.g., [3], [37] for a detailed discussion of Gaussian Sobolev spaces). Shigekawa's result (in the semigroup setting) has been recently extended by Hino [29], who, in particular, has proved that the same conclusion is true if only exp($v#) € Ll(^) for some 6 > 2. It should be noted that if this more general condition of Hino is satisfied, then the corresponding invariant measure satisfies the equation L*/J, = 0. There have been intensive investigations of the solvability of more general equations for probability measures that are absolutely continuous with respect to a fixed Gaussian measure (see [4], [20], [21], and the references therein). However, in applications there are situations when there is no measure with respect to which all solutions are absolutely continuous. For example, this is the case when there are many mutually singular solutions. In 2
n
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such examples, it would be useful to know, at least, that all solutions have logarithmic derivatives along the vectors en. However, sufficient conditions for that are only known in rather special situations (see [1], [5], [11], [12] for some partial results). In the above cited papers, some positive results have been also obtained in the case of non constant diffusion coefficients. However, in this case there is an interesting example constructed by Tolmachev [41] which shows that, in general, there might be no logarithmic derivatives along any nonzero h.
Let us mention two existence results in infinite dimensions. Theorem 4.3. Let X be a separable Hilbert space with an orthonormal basis 00
{en} and let an > 0 be such that ^2 an < °° Suppose that B: X —)• X is n=l a mapping such that, for every n, the function Bn := (B, en) is continuous on every ball with respect to the weak topology. Assume also that
(B(x), x) —> —oo
as (x, x) —>• oo.
Then, there exists a probability measure fj, on X which satisfies the equation L*[i = 0 with respect to the class ^^(X, {l }), where l (x) = (x, e ) and n
n
n
n=l
In many applications, however, one has only the functions Bn, but there is no mapping B with the coordinates Bn. Such examples will be considered below. The previous theorem has the following modification suitable for those examples. We recall that a function G: X 4 [0,+oo] on a topological space X is called compact if the sets {G < c}, c 6 R1, are compact.
Theorem 4.4. Suppose that 6: X —>• [0, +00] is compact and is finite on the finite dimensional spaces E spanned by e\,..., e . Let A > 0 and B be functions on X which are continuous on the sets {Q < c}, c G R1, as well as on the subspaces Ej. Assume that there exist C € (0, +00) and a nonnegative function V on X such that, for every n, the restriction of V to En is compact and twice continuously differentiate and one has n
n
n
n
n
^[Aj(x)dl.V(x)+dejV(x)Bj(x)} 0 such that
/ u'Hudx < —A / («') dx,
\fu € span{??«}•
(49)
Here (and below) u' denotes derivative with respect to x € (0,1). We assume that if> is a locally bounded Borel function and that W® is a "Wiener process with covariance Q" in L2(0,1) or also a cylindrical Wiener process. The classical equation is obtained if H = A, ^(x) = x. We make no assumptions concerning the solvability of (4.8) and consider only the elliptic equations for the corresponding invariant measures. Let us take for XQ the Sobolev space HQ' (0,1) of functions u with u' € L2(0,1) and ti(0) = w(l) = 0. This is a Hubert space with the norm uxo :== «'2, where  • 2 is the norm in L 2 (0,1), compactly embedded into L (0,1). Let us set un = (u, ry n )2, where (•, • )2 is the inner product in L2(0,1), and
Bn(u) = \nun  (ip(u)u', rin}2 = \nun + (*(«), rfn}v
Vu e XQ, (4.10)
where
with some locally bounded Borel function ip. Note that in this case there is no mapping B: XQ —)• XQ with the components Bn. oo
Finally, suppose that £} a22 < oo. Let us consider the operator n=l
n=l
n=l
r
on J C^°(X, {^n})> where Z n («) = un. This operator arises if we consider the oo
process W®(t) = ^ Oinwn(t)i]n^ where {wn(t}} is a sequence of independent 71 = 1
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standard real Wiener processes, i.e., Qrjn = o^r)n. This is the so called time white noise case, i.e., W®(t) is a Wiener process in £2(0, 1). Proposition 4.5. Let\V(y)\ < Ci+c2\y\d, whered 0 such that S(t)D C B, for all t > TO This implies that all the interesting dynamics of the system
takes place inside B. However, a smaller set usually describes more appropriately the asymptotic behaviour (i.e., as t 4 +00) of solutions. This set is
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termed the global attractor, a crucial concept for the study of a large number
of models and whose theory has been widely developed over the last three decades (see, among many others, Temam [52], Hale [35], Ladyzhenskaya [39], Babin and Vishik [4], Robinson [46]). Definition 1 The compact set A C X
is said to be the global attrac
tor associated to the dynamical system ({S(t)}t>o,X) if is invariant (i.e., S(t)A = A, for all t > 0) and the minimal (with respect to inclusion) attracting (uniformly of every bounded subset in X) set, that is, given D C X
bounded,
lim dist(S(t)D,A) \ v ' > / = 0, where dist denotes the Hausdorff semimetric defined as dist(A, B] = d(a, b), for A, B C X. The existence of a compact absorbing set is the simplest hyphotesis ensuring the existence of the global attractor:
Theorem 2 (See Temam [52], Hale [35], Ladyzhenskaya [39]) Suppose there exists a compact absorbing set B C X. Then, the omega limit set of B,
A = A(B) = nT>0 Ut>T S(t)B, is the global attractor associated to S(t). Moreover, if B is connected, so is
A. In general, an attractor exhibits a complex, fractal structure which makes non trivial the description of the dynamics "on the attractor". Thus, results giving information about the asymptotic behaviour of the system from
the dynamics and/or the structure of the attractor become very interesting (Vishik [53], Langa and Robinson [40], Eden et al. [28]). And much more interesting when we are able to prove that the attractor is a finite dimensional set, that is, it is homeomorphic to a subset of Kd, for some d € N. The idea behind this kind of results is deep, as we can then hope that all the complex dynamics of the system only depend on a finite number of degrees of freedom (see Eden et al. [28], Robinson [46]).
3 The stochastic case: difficulties Consider now the following stochastic partial differential equation
u(0,x) = UQ(X), (2)
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where ^jjj£ is the derivative (in the Ito's sense) of a one dimensional twosided Wiener process {Wt : O —> M}teR We can consider this problem as a random perturbation of (1). Note that we add a (very special) nonautonomous term to the equation in (1). Thus, if we want to interpret it as a dynamical system, as the final time t is so important as the initial time s, it is necessary to describe the evolution of solutions u(t], starting at UQ at time s. Thus, we have to introduce a family of two timedependent processes {S(t, s)}t>s, with S(t, S)UQ = u(t;s,uo). Existence of attractors for nonautonomous systems has been studied by several authors (Sell [50], Haraux [35], Chepyzhov and Vishik [17]). But none of these works suit our purposes, due to the large fluctuations of the sample paths, t 4 Wt(u), of a Wiener process. Indeed, examples show that there does not exist an absorbing set for t —>• +00 in this case: every solution goes out from every bounded set with positive probability (Scheutzow [51]). On the other hand, (2) is a stochastic equation, a more complex object than just something nonautonomous, so that its solutions are stochastic processes u(t, o>), with a strong dependence of u> € £1. Important problems on the measurability of trajectories and sets appear. A few years ago, we realized that there were two crucial problems in trying to extend the theory of global attractors to the stochastic case: i) How to define a dynamical system from a stochastic differential equation. ii) How to deal with general unbounded nonautonomous terms in dissipative
equations leading to the non existence of bounded absorbing sets. For the first point, the way from stochastic analysis to dynamical systems was opened around twenty years ago when some authors proved that solutions of some stochastic equations can be described by a two timedependent group of transformations, defined as stochastic flows, XtlS(u) : X —> X (Elworthy [29]). From this concept, Arnold and his collaborators (Arnold [2] and the references therein) have developed the theory of random dynamical systems, which has become the proper framework in which the theory of random attractors has been stated. However, there still exist some unsolved problems in this theory in relation with random attractors. Indeed, for stochastic partial diferential equations the existence of stochastic flows has been proved with a relatively narrow generality (Flandoli [30]). Essentially, only cases in which the equation can be reduced to a deterministic one by a means of a time variable change can be treated, and then some deterministic techniques can be used (Crauel and Flandoli[21], Inikeller and Schmalfuss [36], Keller and Schmalfuss [37]; see also Capinski and Cutland [10] for a more general case). For ii), new concepts of absorbtion, attraction and invariance have to be introduced: the random attractor will be a measurable family of compact
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sets, invariant (as a family) for the corresponding random dynamical system and attracting when the initial time goes to — oo (see the next Section).
4
Concepts of attraction: a comparison study
In this section, we introduce different concepts of global attractors for some stochastic ordinary and partial differential equations. We will remark the differences and relationship among them. But note that we shall only pay attention to those concepts in which the attractor is defined as a subset of the phase space (just as in the deterministic case), so that it "lives" where the dynamics of trajectories holds. Recently, Crauel and Flandoli [21] (see also Schmalfuss [48]) have introduced the concept of attractor for some stochastic partial differential equations, and this has been successfully used in the analysis of qualitative properties for many equations (see, among others, Schmalfuss [?] , Crauel et al. [20], SchenkHoppe [47], Flandoli and Schmalfuss [32]). This concept has to be understood within the framework of the theory of random dynamical systems (Arnold [2]): Let (fl, F, P) be a probability space and {9t : O —>• O, t € R} a family of measure preserving transformations such that (t, u) i4 OfU is measurable, OQ = id, Ot+s = OtQs, for all s, t • X such that P — a.s.
i) X is continuous or differentiable. A map D : fi —>• 2X is said to be a closed (compact) random set if D(u) is closed (compact) for P — a.s. and if w i— » d(x, D(u)) is measurable for any x € X (this coincides with the definition of measurable multifunction in Castaing and Valadier [16]). Definition 3 A closed random set {K(u)}u£fi (for short, K(u)) is said to absorb the set B C X if P — a.s. there exists tg(w) > 0 such that for all t > tB(u)
C Kw.
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Definition 4 Finally, a compact random set A(u) is said to be a random attractor associated to the RDS >p if P — a.s. i) (t,ui)A(ui) = A(Qtu},yt > 0 (invariance) and
ii) for all B C X bounded (and nonrandom) lim dist(0t>n
This is the concept used in, for example, Crauel and Flandoli [21], Crauel et al. [20]. In Crauel [22] it is proved that random attractors are unique even if we use the notion of attracting compact sets K C X instead of bounded sets as in ii). It is worth pointing out that the measurability of A(u) also holds with respect to the past of the system J 0.
(*) ' V
In this equation •
A is the infinitesimal generator of a strongly continuous semigroup St,
t > 0, of linear bounded operators on H, • •
F is a Borel mapping from H to H, B is a linear bounded operator from a real separable Hilbert space K
to H, • Wt, t > 0 is a Kvalued standard cylindrical Wiener process. 105
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Our basic assumption is the following oo
tr SsBB*S*ds < +00. /
If (Al) holds, then the Gaussian measure /j, = Af(Q, 0,
which is a unique mild solution to the linear equation corresponding to (*) (F = 0) (see e.g. [DZ; S]). The process Z is Gaussian and Markovian and it is well known that under (Al) the transition semigroup (Rt) of Z,
Rt(x) := E(4>(Zf}}, is a positivity preserving Cosemigroup of contractions in Lf(H^)^ for all 1 < p < oo. An important example is the so called Malliavin process which is a solution of (*) with A = — ^, F = 0 and a HilbertSchmidt operator B, and then Qoo = BB*. The generator LM (the Malliavin generator) of its transition semigroup (R^1) is known in quantum physics as Number Operator. Let us recall some remarkable properties of (R^) like hypercontractivity ([N]) and Logarithmic Sobolev Inequality for LM ([Gr 1], [Si]). Moreover,
(R^) is symmetric (in L?(H,fj)). We do not assume that the corresponding OU semigroup is associated to a Dirichlet form (in particular symmetric). Applications to nonreversible systems and recently also to Mathematical Finance ([M]) provide some motivation here. We make weak assumptions on nonlinear term, which ensure the existence of martingale solution ([DZ; S]) and correspond to compactness or Girsanov methods. In the latter case our basic assumption on F is the following F : H > im B
is the Borel function and for a certain 6 > 0 < +00,
where B~l means the pseudoinverse ofB. By the Fernique theorem, functions F with B~1F of linear growth satisfy the exponential integrability condition
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in (Fl). Extensions of the Fernique theorem and conditions for (Fl) to hold have been given e.g. in [AMsSh] and [I/]. For a solution X f , 0 < t < T, of equation (*) we define a family of operators (Pt)o = L(p + G0 0) on L (H,(j,) and dom(Lp] = dom(L}. Moreover, Lp = L + G, where G is the unique extension of GO to an Lbounded operator with domain dom(L} and fort>Q Pt = Vt, for all^cL2 (H, p) . F
2
Proof. This follows from a result in [V; P] on Miyadera perturbations and the last equality is shown by approximation. Hyperboundedness
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First, recall that for t > 0 the condition im Ql/2 = im Q^
(A2) is equivalent to
(3.4)
5 0 (i) 1 (3.5)
\\Rt\\p^q=l
and
Xqq(t,p),
where (3.6)
q(t,p):=l
p1
(Recall that always \\S0(t)\\ < 1.) The theorem below says that the semigroup (Pt) has a similar property with hypercontractivity replaced by hyperboundedness. Theorem 3.3. Assume (Al) and (F2). (i) If for a certain to > 0, (A2) holds, then for every t > to, p > 1, q> 1 the operator
is bounded for q < q(t,p) and unbounded for q > q(t,p), where q(t,p) is denned by (3.6). (ii) Conversely, if for a certain to > 0 and qo > PQ > 1
Pto : Lpo (H, n) »• Lqo (H, fi)
is bounded,
then (A2) holds for all t>t0. Invariant measures with densities
The theorem below is a counterpart of Thm. 2.1 ([ChG; E, Thm. 5]), which was proved by compactness method. Part (a) follows from Thm. 3.3 and [H; P]. In the proof of (c), (d) we use [Da, Thm. 7.3].
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Theorem 3.4. Assume (Al) and (F2). If for a certain t0 > 0, (A2) holds, then (a) there exists an invariant measure v for (Pt), which is absolutely continuous w.r.t. j,
(c) p(x) > 0 for p, a.a. x; (d) v is a unique invariant measure for (Pt) in the class of probability measures absolutely continuous w.r.t. p.. Proposition 3.5. If (Al), (A3) below and (F2) are satisfied, then all the statements of Theorem 3.4 hold and moreover for each p € (1, oo) there exist constants Mp > 0, \p > 0 such that
(3.7)
\\Pt I w^llp^Mpe^IMIp,
JH
for all
Logarithmic Sobolev inequality
It has been proved in [ChG; N] that the OU generator L satisfies the Logarithmic Sobolev Inequality (LSI) (3.10) below iff imQ^CimQ1/2,
(A3)
which is stronger than (A2) and it is equivalent to the following condition (e.g. [DZ; S, Prop. B.I]): There exists a > 0 such that
\\Q1/2x\\>a\\Ql^x\\,
(3.8)
for all z e ff.
Define
(3.9)
a := sup{a > 0 : (3.8) holds}.
Theorem 3.6. Assume (Al), (F2) and let (3 be as in (F2). I. If (A3) hoWs, then for every p > 1
(3.10)
/
JH
where (j>p := sgn 4>\4>\p~l and c(p), 7(p) are suitable constants dependent
on a and f3.
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II. Converse]/, if (3.10) holds for somepo > 1 and certain constants c(po) > 0 and ~f(po) > 0, then (A3) is satisfied.
4. The semigroup (Pt) — the case of general F
In this Section we assume (Al) and (A3). Let a be the constant corresponding to (A3) via (3.9). The nonlinear term F in equation (*) is required to satisfy the following condition (Fla) 2 (Fla) F satisfies (Fl) for a certain 6 > —%.
The LSI, established in the case of bounded F, enables us to obtain crucial estimates related to (Fl). Thanks to these estimates, we prove by approximation that for general F, (Pt) is a hyperbounded Cosemigroup in V(H, //) for p > po, po being given explicitly. As a corollary we get a result on invariant measure analogous to the previous one for F bounded. In the particular case of A = — 5, a similar result was obtained in [H; E] by Dirichlet forms approach and the hyperboundedness of (Pt) was proved by direct tedious calculations. For gradient systems (see [DZ; E]), i.e. where Pt is symmetric w.r.t. its own invariant measure, the same Z^regularity of invariant density as this in Cor. 4.2 has been given in a different setting in [L] and [AMsSh]. Finally, an LSI is also proved. Theorem 4.1. Assume (Al), (A3), (Fla) and let a and 6, K be the constants corresponding to (3.9) and (Fla), (Fl), respectively. Then for each p € (1, oo) such that
we have (a) (Pt) is a Cosemigroup on LP(H, (i) and its generator Lp is an extension of LQF. Moreover, 2
/i l)t], (b)
t>0.
For each t > 0, P is a bounded operator from L (H, /u) to L (H, JJL) for P '(H,riforallp>• im Ql2 . ft := /
is a Borel function and for 6 > —r a
JH
then (a) for every p > 2, (P^) is a Cosemigroup in LP(H, /JL) and its generator
LF D L°F; (b) dom2(Lp) = doni2(L); (c) for p > 2, the generator L^ satisfies the LSI (3.10). Corollary 4.4 (The case of symmetric OU). Assume (Al), (A3) and let Rt = Rf in L2(H, a). If (Fla) holds with 6 > Jr, then (i) dom^Lp) = dom2(L) and dom L is characterized in [DG] and [ChG;
M]; (ii) statement (c) of Thm. 4.3 holds. Remark 4.5. Uniqueness of invariant density obtained in Thm. 3.4 and Cor. 4.2 is obviously a weaker statement than uniqueness of invariant measure as in Thm. 2.1. Some results concerning the latter were obtained e.g. in [BR, 1] and [AlBR]. In particular, it follows from [BR, 1] that the uniqueness holds under the assumptions of [Sh; E] and [vV]. We refer to the survey [MaSe, 2] for uniqueness results. Remark 4.6. Let the operator Q in equation (*) have a bounded inverse and assume (Al). Then obviously (A3) holds and by [DZ; S, Cor. 9.23] the condition (F) is also satisfied. Consequently, we can use to (*) both the compactness (Thm. 2.1) and Girsanov method (Thm. 3.4 and Cor. 4.2). Moreover, if F is continuous and has a linear growth then, by [MaSe, 1] and the first part of Thm. 2.1, (Pt) is strongly Feller and irreducible, and consequently we obtain the uniqueness of invariant measure for (*).
5. Examples
We consider the simplest case of system (*) which satisfies (Al) and (F2): (5.1)
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dXt = AXtdt + bdt + bdwt.
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However, in Example 1 the unique invariant measure for (5.1) is singular w.r.t. n and in Example 2 there is no invariant measure for (5.1). By virtue of Thm. 3.4, in both the examples for no t > 0 does (A2) hold. Equivalently, for no t > 0 can Rt and Pt be hyperbounded in LP(H, fj,). An example similar to Example 1, but not so explicit, has also been given in [F; L]. In Example 3 we present a model (*) (with nonconstant F) which satisfies precisely the assumptions of Thm. 3.4. That is for some to > 0 the condition (A2) is satisfied but (A2) does not hold for 0 < t < to. Equivalently the corresponding OU semigroup (Rt) is hypercontractive for t > to but it is not hyperbounded for 0 < t < to. This cannot happen when (Rt) is symmetric or H has finite dimension. Moreover, (A3) is not satisfied here. It should be mentioned that Example 3 is of some importance in Mathematical Finance
Recall that if (St) is a stable semigroup (i.e. linit_>.00 Stx = 0, for all x € H), then (5.1) has an invariant measure v iff (Al) holds and ,00 00
(5.2)
there exists
/
,T ,
Stbdt := T lim /
Jo
Stbdt,
and then
>°° Jo ,.00
where a^ := := /I
Stbdt. ,
Jo By the CameronMartin Thm., v is absolutely continuous w.r.t. fj, = A/"(0, iff (5.3)
aooei
Example L Here H = L 2 (0,oo), the operator
f\ A=^
with dom( A) = H1 (0, oo) ,
On
generates the semigroup of left shift
x € H, b(9) = exp(02/2), 0 > 0,
S(t}x(6] = x(t + 6>),
and it; is a one dimensional Wiener process. Then Q = b ® b and
/
Jo
\\Stb\\'2dt= !
tiStQS;dt=f
Jo
Jo
I
Jt
es2dsdt00
I
Jo
fOO
Ssbds = I
Jo
Ssbu(s)ds.
Then we have for every 6 > 0
and hence the Laplace transform of the function [0, oo) B s > e~s / 2 [1— u(s)] vanishes identically, which implies that u(s) = 1. But u(s) = 1 £ L2(0, oo),
a contradiction. Therefore the measures N(a00,Q00) and N(Q,Q) are singular. Example 2. Consider equation (5.1) in Example 1, where b is now replaced
by 6(0) = (0 +I)3/2,
0>0.
Then (Al) is satisfied. Let oo
/
poo
Ssbds(6) = \
Jo
(s + 0 + l)3/2ds = 2(6 + I)1/2.
Then / ^ L2(0, oo), and hence (5.2) does not hold and there is no invariant measure for (5.1). Example 3. Consider the equation
(5.4)
dXt= \AXt + bf(Xt) dt + bdwt
in the space H = L 2 (0,1), where A = j^ with dom(A) = {x € Hl(Q, 1) : x(l) = 0} generates the semigroup (St)
Let it; be one dimensional Wiener process, / S Bb(H) and 6 € H, b ^ 0. Then Qoo = Qi and (A2) holds for t > 1. Hence for < > 1 the corresponding OU semigroup (Rt) is hypercontractive and (Ft) is hyperbounded in
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by Thm. 3.3. For simplicity take 6 = 1. Then im Q