Functional Integrals: Approximate Evaluation and Applications

Mathematics and Its Applications
Volume 249
by
L. A. Yanovich
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
Egorov, A. D. (A~eksandr D~itr1ev1ch)
[Pribl1zhennye metody vych1slen1 fa kont1nual 'nykh integralov. Engl1shl
Functional integrals : approximate evaluat10n and applications by A.D. Egorov, P.I. Sobolevsky, and L.A. Yanov1ch.
p. cm. -- (Mathemat1cs and its appl1cations ; v. 249) Includes bibliographical references and index. ISBN 978-94-010-4773-9 ISBN 978-94-011-1761-6 (eBook) DOI 10.1007/978-94-011-1761-6 1. Linear topological spaces. 2. Integration, Functional.
1. Sobolevskil, P. 1. (Pavel Iosifov1chl II. fAnov1ch, L. A. (Leonid Aleksandrovichl III. T1tle. IV. Ser1es: Mathematics and its appl1cat1ons (Kluwer Academic Publishersl ; v. 249. QA322.E3813 1993 515' .73--dc20 93-9599
ISBN 978-94-010-4773-9
Printed on acid-free paper
This is an updated and revised translation of the original work Approximate Evaluation of Continuallntegrals Nauka and Tekhnika, Minsk © 1985, 1987
All Rights Reserved © 1993 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1993 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.
Contents
1 Backgrounds from Analysis on Linear Topological Spaces 1 1.1 Cylindric Functions, Functional Polynomials, Derivatives 1 1.2 Definition of Functional Integrals with Respect to Measure, Quasi
measure and Pseudomeasure, Relations with Random Process Theory 5
1.3 Characteristic Functionals of Measures 7 1.4 Moments, Semi-invariants, Integrals of Cylindric Functions 11
2 Integrals with Respect to Gaussian Measures and Some Quasimeasures: Exact Formulae, Wick Polynomials, Diagrams 15 2.1 Some Properties of Spaces with Gaussian Measure. Formulae
for Change of Integration Variables 15 2.2 Exact Formulae for Integrals of Special Functionals. Infinitesimal
Change of Measure 20 2.3 Integrals of Variations and of Derivatives of Functionals. Wick
Ordering. Diagrams 26 2.4 Integration with Respect to Gaussian Measure in Particular Spaces 34
3 Integration in Linear Topological Spaces of Some Special Classes 47 3.1 Inductive Limits of Linear Topological Spaces 47 3.2 Projective Limits of Linear Topological Spaces 48 3.3 Generalized Function Spaces 52 3.4 Integrals in Product Spaces 55
4 Approximate Interpolation-Type Formulae 65 4.1 Interpolation of Functionals 65 4.2 Repeated Interpolation. Taylor's Formula 67 4.3 Construction Rules for Divided Difference Operators 68 4.4 Approximate Interpolation Formulae 77
5 Formulae Based on Characteristic Functional Approximations, which Preserve a Given Number of Moments 81 5.1 Approximations of Characteristic Functionals 81 5.2 Reducing the Number of Terms in Approximations 89 5.3 Approximate Formulae 101
6 Integrals with Respect to Gaussian Measures 109 6.1 Formulae of Given Accuracy in Linear Topological Spaces 109 6.2 Formulae Based on Approximations of the Correlation Functional 119
vi
6.3 Stationary Gaussian Measures 128 6.4 Error Estimates for Approximate Formulae Based on Approxi-
mations of the Argument 130 6.5 Formulae which are Exact for Special Kinds of Functionals 1:34 6.6 Convergence of Functional Quadrature Processes 139
7 Integrals with Respect to Conditional Wiener Measure 147 7.1 Approximations of Conditional Wiener Process which Preserve a
Given Number of Moments 147 7.2 Formulae of First Accuracy Degree 155 7.3 Third Accuracy Degree 158 7.4 Arbitrary Accuracy Degree 161
8 Integrals with Respect to Measures which Correspond to Uniform Processes with Independent Increments 167 8.1 Formulae of First, Third and Fifth Accuracy Degrees 168 8.2 Arbitrary Accuracy Degree 176 8.:3 Integrals with Respect to Measures Generated by Multidimen-
sional Processes 189 8.4 Convergence of Composite Formulae 193 8.5 Cubature Formulae for Multiple Probabilistic Integrals 200
9 Approximations which Agree with Diagram Approaches 211 9.1 Formulae which are Exact for Polynomials of Wick Powers 211 9.2 Approximate Integration of Functionals of Wick Exponents 215 9.3 Formulae which are Exact for Diagrams of a Given Type 219 9.4 Approximate Formulae for Integrals with Respect to Quasimeasures 226 9.5 Some Extensions. Composite Formulae 229
10 Approximations of Integrals Based on Interpolation of Measure 235 10.1 Approximations of Integrals with Respect to Ornstein-Uhlenbeck
Measure 235 10.2 Integrals with Respect to Wiener Measure, Conditional Wiener
Measure, and Modular Measure 241 10.3 Formulae Based on Measure Interpolation for Integrals of
Non-Differentiable Functionals 245
11 Integrals with Respect to Measures Generated by Solutions of Stochastic Equations. Integrals Over Manifolds 249 11.1 Approximate Formulae for Integrals with Respect to Measures
Generated by Solutions of Stochastic Equations 249 11.2 Approximations of Integrals with Respect to Measures Generated
by Stochastic Differential Equations over Martingales 25:3
vii
11.3 Formula of Infinitesimal Change of Measure in Integrals with Respect to Measures Generated by Solutions of Ito Equations 260
11.4 Approximate Formulae for Integrals over Manifolds 266
12 Quadrature Formulae for Integrals of Special Form 277 12.1 Formulae Based on Algebraic Interpolation 277 12.2 Formulae Based on Trigonometric Interpolation 282 12.:3 Quadrature Formulae with Equal Coefficients 292 12.4 Tables of Nodes and Coefficients of Quadrature Formula of Highest
Accuracy Degree for Some Integrals :300 12.5 Formulae with the Minimal Residual Estimate :319
13 Evaluation of Integrals by Monte-Carlo Method :327 1:3.1 Definitions and Facts Related to Monte-Carlo Method :327 1:3.2 Estimates for Integrals with Respect to Wiener Measure :3:31 13.:3 Estimation of Integrals with Respect to Arbitrary Gaussian Measure
in Space of Continuous Functions :3:34 13.4 A Sharper Monte-Carlo Estimate of Functional Integrals :338
14 Approximate Formulae for Multiple Integrals with Respect to Gaussian Measure 34:3 14.1 Formulae of Third Accuracy Degree :344 14.2 Formulae of Fifth Accuracy Degree :350 14.:3 Formulae of Seventh Accuracy Degree :357 14.4 Cubature Formulae for Multiple Integrals of a Certain Kind :3.59
15 Some Special Problems of Functional Integration :367 15.1 Application of Functional Integrals to Solution of Certain Kinds
of Equations :367 15.2 Application of Approximations Based on Measure Interpolation
to Evaluation of Ground-State Energy for Certain Quantum Systems :375
15.3 Mean-Square Approximation of Some Classes of Linear Functionals 378 15.4 Exact Formulae for Integrals with Respect to Gaussian and
Conditional Gaussian Measures of Special Types of Functionals 391
Bibliography 401
Index 417
Preface
Functional integration is a relatively new and sufficiently broad area of scientific research. In addition to the ongoing development of the mathematical theory, ex tensive research is being carried out on applications to a wide spectrum of applied problems.
Quantum statistical physics, field theory, solid-state theory, nuclear physics, optics, quantum optics, statistical radiotechnics, radiation physics of high-energy particles, probability theory, stochastic differential equations are some of the areas in which applications are found [1]-[10], and this list steadily grows.
An important condition for the applicability of functional integrals is the existence of efficient evaluation methods. The development of these methods, however, has en countered serious problems due to the fact that the elaboration of many issues from analysis on infinite-dimensional spaces is far from being finished. This is also true in the case of the theory of functional integration and, in particular, the theory of integrals w.r.t. quasimeasures including Feynman integrals. At present, the most e laborated theory deals with functional integration w.r.t. count ably additive measures [11]-[17].
This monograph is mainly devoted to methods of evaluation of functional integrals w.r.t. count ably additive measures and certain quasimeasures on general and concrete spaces and, in particular, of integrals w.r.t. measures generated by random processes and quasimeasures which correspond to fundamental solutions of partial differential equations.
An approximate evaluation of functional integrals was initiated in the papers of Cameron [18], Vladimirov [19], Gelfand and Chentsov [20], devoted to the evaluation of Wiener integrals. More recently, the ideas of these authors have been extended in [21]-[33].
An evaluation of functional integrals is also considered in more physics-oriented papers (see [34]-[39] and the bibliography therein).
Research on some issues of approximate evaluation of integrals w.r.t. Gaussian measures is given in the papers [40]-[58].
Recently, the authors have developed methods of approximate evaluation of inte grals w.r.t. measures which correspond to various random processes including pro cesses with independent increments, of integrals w.r.t. quasimeasures. A number of new results have also been obtained concerning the approximate evaluation of inte grals w.r.t. Gaussian measures. In particular a method has been developed which agrees with the Feynman diagram method; formulae have been constructed which employ various ways for the specification of Gaussian measures; approximations have been constructed for integrals w.r.t. measures on spaces of functions defined on infinite intervals; interpolation formulae have been derived for integrals w.r.t. non Gaussian measures. Formulae have also been obtained for integrals w.r.t. measures generated by the solutions of stochastic differential equations w.r.t. martingales, and w.r.t. measures generated by Gaussian processes on Riemann manifolds. An approx-
ix
x
imate method has been developed for the evaluation of integrals which is based on the formula of infinitesimal change of measure. All these issues comprise the contents of this book.
Most of the approximate formulae considered in here are based on the require ment that they are exact for functional polynomials of a given degree and that they converge to the exact value of the integral. For the construction of these formulae, we use various approximations for the argument of the integrated functional in the general case, and in the case of the measure defined by a random process, we use approximations of the process.
Attention is paid to the construction of approximate formulae for concrete mea sures. In particular, formulae are given for integrals w.r.t. measures which correspond to Wiener, conditional Wiener and other Gaussian processes, the Gamma-process, and Laplace, Poisson and telegraph processes. Integrals w.r.t. measures defined by multidimensional processes and random fields are also considered.
For integrals w.r.t. the Gaussian measure of functionals of special kinds, approx imate formulae in the form of quadrature sums are investigated. An evaluation of integrals w.r.t. Gaussian measure by the Monte-Carlo method is considered.
Approximation expressions for most of the approximate formulae considered con tain multiple integrals; therefore cubature formulae for the evaluation of certain class es of such integrals are obtained. They are constructed based on the formulae of a given degree of accuracy for the corresponding functional integrals, and therefore multiplicity is of no principal importance for their construction.
This monograph considers applications of the constructed approximate formulae to the solution of applied problems, in particular, to the solution of certain integral equations and partial differential equations, to the determination of the energy for the ground state of model quantum systems and, to the evaluation of the expectations for functionals of random processes. Certain extremal problems of approximation theory are solved, and exact formulae are given for the evaluation of integrals w.r.t. conditional and unconditional Gaussian measures of special kinds of functionals most commonly occurring in applications.
This book also sketches the necessary background from analysis on infinite-dimen sional spaces.
We would like to thank our colleagues from the Institute of Mathematics of the Byelorussian Academy of Sciences for fruitful discussions on the scope and the main results of the book, and Dr. N. Korneenko for the translation and TEX setting of the manuscript.
We also wish to express our gratitude to Kluwer Academic Publishers, whose proposal stimulated us to prepare this book.
Chapter 1
Backgrounds from Analysis on Linear Topological Spaces
The book is devoted to functional integrals defined on separable locally convex linear topological spaces (or, briefly, on linear topological spaces). The accepted degree of the generality of the exposition allows to embed into a general scheme the issues of evaluation of functional integrals which are most commonly encountered in literature.
1.1 Cylindric Functions, Functional Polynomials, Derivatives
Let X be a linear topological space; X' is the dual space of linear continuous func tionals on X. For 1 E X' and x E X, the value of 1 on x will be denoted by (1, XI or byl(x).
We would like to mention two classes of functionals on X which are of special importance in functional integration: cylindric functionals and functional polyno mials. A functional F( x) is called cylindric, if it may be represented in the form of F( x) = f( (h, x I,' .. , (In' X I), where f (u) is a function defined on the n-dimensional Eu clidean space Rn, u = (Ul,'" ,un), lj E X', j = 1,2,···,n (n = 1,2,·· .). In general, this representation is not unique. Cylindric functionals are closely related to the defi nition of functional integrals (as we shall see, functional integrals of cylindric function als may be written in the explicit form), and moreover, a wide class of functionals may be approximated by the cylindric ones. Let us consider the simplest example. Let X be a linear topological space with basis {ej}, j = 1,2, ... , i.e., X :1 x = I:i=l (lj, x lej, where the series converges under topology of X, {lj}"j = 1,2,···, is the dual basis in X'. Let further F(x) be a continuous functional on X. Then F(x) = liIDn-+oo Fn(x), where Fn(x) = F(I:i=l(lj,x)ej) is a cylindric functional.
1
2 Cbapter 1.
A functional polynomial of degree N on X is defined to be a functional of form
N
PN(:X) = L Pn(:X), n=O
where Pn(:X) = Pn(:X,"":X) is the homogeneous form which corresponds to an n-linear form Pn(:X1, ... , :xn ) on X x ... xX; Po(:x) is a constant. We shall use functional ~
n
polynomials for the development of approximate methods of evaluation of functional integrals. In spaces of functions defined on a segment T of the real line, the functional polynomials will be of the form
x :x(t1)'" x(tn) dt1 ... dtn,
where ao =const, an(tb' .. , tn) is a (possibly, generalized) kernel. Therefore, if X = C[O,I] is the space of continuous functions on segment [0,1], N = 2, ao = 0, a1(t) == 0, a2(tt,t2) = c5(s - tt}c5(T - t2), where c5(s - t) is Dirac's delta-function, then P2(:x(·)) = :X(S):X(T) is the second degree functional polynomial.
If X is a linear topological space with basis, then, in terms of the previous example, we have
00
(1.1)
( )
1/2 . i:: jPn( ejll"', ejJj2 ::; C < 00, n = 1,2,···, N. 11 ,.'',In=l
Note that if X is the Hilbert space and {Ij}, {ej}, j = 1,2, ... , are the orthonormal bases, then this condition ensures the continuity of the polynomial PN(:X); this fact follows from the estimate derived with the help of Bunyakovsky inequality:
00
= L Pn(ej""',ejn)(lj,,x)"'(ljn'x) ::;cll:X111 .. ·llxn ll· j1,"',jn=1
Here, (l,x) = (I,x) is the scalar product in X. Let us give some information on differentiation of functionals. A functional F( x)
defined on a linear topological space is said to be differentiable at point x along direction a E X if there exists the limit of the expression [F(x + ,Xa) - F(x)]/'x as
Backgrounds from Analysis on Linear Topological Spaces 3
~ --t 0, A E R. This limit is called the first variation of the functional F( z) at point z along direction a and is denoted by SF(z; a). Therefore,
SF(z;-a) = lim(F(z + ~a) - F(Z))/A = dd\ F(z + Aa)1 . >'-+0 A >'=0
By induction, the n-th variation of the functional F( z) along directions al,a2,"',an is defined as follows. If the (n-l)-th variation s(n-l)F(x;all,,·an_d is defined, then
s(n) F(z; all'" an-b an) =
= d~ s(n-l) F(z + Aan ; al,'" an-dl = >'=0
on ( n )
= OAl'" OAn F X + .r; Ajaj >'1=>'3=---=>'n=0
If the variation SF(x; a) at point z E X along any direction a from some subspace Y ~ X may be represented in the form of a linear continuous functional on Y:
SF(z;a) = (F~(z),a) == F~(z)a (z E X,a E Y),
then the element FY( z) is called the Y -derivative or simply derivative if Y = X) of the functional F(x) at point x and is denoted by F~(z) (or FI(Z), if Y = X).
Furthermore, if the n-th variation of the functional along directions al, a2, ... , an E Y ~ X may be represented as an n-linear continuous functional on ~
n
then it is possible to speak about the n-th Y -derivative of the functional F(x) which is denoted by Fi-n)(z); if X = Y then the subscript "Y" is suppressed.
These definitions may also be extended to the case of mappings of X into a linear topological space Z. If A : X --t Z then A~) (z) is an n-linear continuous form defined on Y with its values in Z.
The introduced derivatives have the same properties the ordinary ones. In par ticular, the following formulae are valid:
(AlF(z) + A2G(Z))~ = AlF~(z) + A2G~(Z), All ~2 E R
(F(z)G(z))~ = F~(z)G(z) + F(x)G~(x), (F(Ax))~ = F~(Az)A~(z),
(1.2)
where F : Z --t R, A: X --t Z, A~(x)a E V for a E Y, V is a subspace of space Z. If X is the space of functions x = z(t) on T then we may use still other notations
4 Chapter 1.
for the derivative F'(x) == F'(x;t) == §x(t)F(x(.)). Let us mention a special case (see
[59]) of the last formula from (1.2) (the rule of the differentiation with respect to a parameter)
!F(xT) = \F'(XT),!XT)== (1.3)
= r §F(xT(·)) aXT(t) d - iT .5XT(t) ar t,
where r is a real parameter. The integral in (1.3) is assumed to be the generalized
one, i.e. .5~~:(U) may be a generalized function (i.e a distribution) as well. The
right-side integral from (1.3) becomes an ordinary one, if X is the space L2(T) of functions which are square summable on T.
Integrating formula (1.3) over r from T = 0 to r = 1, we obtain
(1.4)
Finally, we shall give Taylor's formula (see [59]) for functionals F(x) defined on a linear topological space X:
11 (1 r)m-1 + - F(m)(x + ry)ymdr. o (m - I)!
(1.5)
If X is the space of functions on T then formula (1.5) may be represented as follows:
F(x(.) + y(.)) = F(x(.))+
X I1y(tj) dt1 • .. dtmdr. (1.6) j=l
All remarks with respect to formula (1.3) remain also valid for the latter one.
1.2 Definition of Functional Integrals with Re spect to Measure, Quasimeasure and Pseu domeasure. Relations with Random Process Theory
Now we shall give the definition of the functional integral. Let X be a linear topo logical space. A cylindric set on X is defined to be a set Q1l ..... I .. (B) C X of the form:
Q1l ..... I,.(B) =
= {x EX: ((11, x), ... , (In,x)) E BeRn}, (1. 7)
where 1; E X', i = 1,2"", n; B is a Borel set in Rn. A finitely additive set function fL defined on cylindric sets is called a cylindric measure. The cylindric measure fL meets the natural conditions of consistency which are associated with the nonuniqueness of representation of a cylindric set in the form (1.7). For fixed nand 11,"', In, a cylindric measure fL defines the finitely additive set function fLl 1 ..... I .. on the Borel sets of space Rn according to the formula
Two cases should be considered now.
Case 1. fLl 1 ..... I .. is a measure (i.e. a countably additive set function) on Rn for any nand h,···, In E X'. In this case fL is said to be a quasimeasure (d. [15]); and if
F(x) = f((h,x), ... ,(ln,x))
is an arbitrary cylindric functional, such that f( Ul, ... un) is measurable with respect to fLl 1 ..... I .. , then the functional integral w.r.t. quasimeasure fL of functional F(x) is defined by the equation
Ix F(x) fL (dx) = k .. f(u) fLl 1 ..... I .. (du), (1.8)
where the right side is the Lebesque integral over Rn and u = (Ul, ... , un). Case 2. fLl 1 ..... I,. is not a measure, but it defines a generalized function, i.e a
continuous linear functional on the space of test functions defined on Rn. In this case, fL is called a cylindric infinite-dimensional generalized function, or a pseudomeasure. An example of the pseudomeasure is Feynman's measure which is to be considered later. If f(Ul,""Un ) is a test function then eqn. (1.8) defines the integral of the cylindric function
6 Chapter 1.
where the right-side integral should be understood as the value of the generalized function J.Lh, ... ,I,. at f(u):
Ix F(x) J.L (dx) = (f, J.Lh, .. ·,I,.). (1.9)
As may be seen from the definition, the class of cylindric functions integrable w.r.t. a pseudomeasure is narrower than the class of cylindric functions integrable w.r.t. a quasimeasure. Therefore, in concrete cases, one usually encounters the problem of extension of an integral w.r.t. a pseudomeasure to the case of functions f other than test ones.
The next step in defining the functional integral is to extend it to non-cylindric functions. The following situations are possible. A quasi measure may be extended to a count ably additive measure on the u-algebra generated by all cylindric sets of X (in such cases, this algebra will be considered to coincide with Borel's u-algebra of space X). In this case, the functional integral is defined to be the Lebesque integral over X w.r.t. measure J.L.
In the more general case when the quasimeasure is not count ably additive, the functional integral w.r.t. quasimeasure of an arbitrary function is defined as the limit of the integrals of the cylindric functions which approximate this function. Each such passage to the limit requires an individual consideration.
A similar situation arises in the case of pseudomeasures. A pseudomeasure may turn out to be a generalized function defined on some space of test functionals. Then a functional integral of a test functional is defined as:
Ix F(x)J.L(dx) = (F,J.L), (1.10)
i.e. as the value of a linear continuous functional J.L on F. In the general case, an integral w.r.t. pseudomeasure of an arbitrary functional is defined as the limit of integrals of the approximating cylindric functionals. In the latter case, to prove the existence of the integrals may turn out to be very difficult problem. Issues of integration in linear topological spaces and some techniques of proving the existence for the mentioned types of integrals are considered in [15].
We shall mostly consider functional integrals w.r.t. count ably additive probability measures. These integrals are the mathematical objects whose properties are well investigated, moreover, they are intimately related with the random process theory (see [lll). In particular, let x == x(w) be a random element with its values in space X, i.e. a mapping of probability space {n, P} into X, where n is the space of random events and P is the probability, under which the functional (1, x(w)) is a random value for any 1 E X'. If a random element x(w} with its values in X is given, then the probability measure J.L is defined on the u-algebra generated by the cylindric set.s of space X as follows:
= P{w En: ((h, x(w)), 0", (In, x(w))) E BeRn}.
In this case, the equality holds:
Ix F(x)J-L(dx) = 10 F(x(w))P(dw) == EF(x) (1.11)
for any F for which any of the integraIs exist; here E denotes expectation. If X is the space of real functions of argument t E T, then a random element x( w)
is a random function which in denoted as x(t,w) == x(t) == Xt. If t is a real number then x(t) is an ordinary random process. Moreover, if the process is stochastically continuous then the expression x(t) = (8(t - 0), x(o)), i.e. the value of the process at any time moment is defined and J-Ltl, .. ,t" == J-L6(tl _.), .. ,6(t n _.) is an ordinary finite dimensional distribution of the random process. In this case, a measure J-L on space X of sample functions x( t) of the process is completely specified by its definition on the sets of the form
If the process x (t) is a generalized one then the corresponding measure is defined on space X which is duaI to some space Y of test functions, i.e. X = yl, with the measure being specified by its values on cylindric sets of the form Qll, .. ,ln(B), where 11 ,000, In E Y.
In the particular examples of functional integrals, we shall mostly consider mea sures which correspond to some given random processes.
1.3 Characteristic Functionals of Measures
Besides the definition of functional integral by means of the explicit specification of a cylindric measure, we may alternatively define it with the help of the characteristic functional of this measure, i.e. the functional x(l) defined on X' by the equation
(1.12)
It is known [11, 15] that if the characteristic functional of a measure is known, then we may uniquely define the measure of any cylindric set and hence, the cylindric measure. Note that in the case of the pseudomeasure the right-side integral in (1.12) should be understood in terms of eqn.(1.9). The conditions under which a characteristic functional may define a count ably additive measure are known in some cases [11]- [16],[60].
In general, the characteristic functiollals of all cylindric measures considered in the book may be written as follows:
(1.13)
8 Ohapter 1.
where Kn{lI,"', In) is a real symmetric continuous n-linear form on X' X ••• x X' ----....-.-- n
and Un (n = 1,2, ... ) are real or complex parameters. Some Un may be equal to zero here, and the number of the summands may be finite. In many concrete cases, the series in the exponent may be written in the closed form. Let us consider some more restricted classes of functionals of form (1.13).
Let U be a set with a finite measure v defined on it, 1 E X', a E X, p(u) is a mapping from U into X, parameters Un are real and l(x) = (l, x). Then we may define
X(l) = exp {il(a) + E i~n fu [n(p(u))V(dU)} , (1.14)
A possible example of such a functional is
X(l) = exp {il(a) + fug(l(p(u)))v(du)} , (1.15)
where g(z) is a function which is analytic in the vicinity of zero ((1.15) is reduced
to (1.14), if we set Un = ~g(z)1 ,Uo = Ul = 0). The following two-parameter ~ az z=o
family of functionals
X(l) = exp {il(a) + _1_ r [.!..(eia1(P(U)) -1) a - {3 Ju a
- ~(ei,81(P(U» -1)] V(dU)} , (1.16)
where a, {3 are real parameters, a ~ /3, is the special case of (1.15). If /3 ~ 0, we obtain the functional
X(l) = exp {il(a) + ~2 fu [eia1(p(u)) - 1-
- ial(p( u))] v( du)} , (1.17)
which corresponds to Poisson measure, and if a ~ ° and {3 ~ 0, then we obtain the functional
X(l) = exp {il(a) - ~ fu 12(p(u))v(du)} , (1.18)
which corresponds to Gaussian measure. In general, the characteristic functional of a Gaussian measure has the form:
X(l) = exp {im(l) - ~K(l, I)} ,
where m(l) is a linear functional and K(II,12) is a positive definite bilinear one. One more example of functionals of type (1.15) is:
X(l) = exp {il(a) - fu [In(1 - iul(p(u))) + iul(p(u))] v(du)} , (1.19)
We would like to mention separately the case when X is some set of functions z(t) on segment [0, T] C R. Then
l(a) = loT l(t)a(t)dt, l(p(u)) = loT pt(u)l(t)dt, (1.20)
where the functions a(t), Pt(u) may also be the generalized ones. In the latter case, functional (1.15) may be considered as the characteristic functional of a random process z(t) which be represented as follows:
Zt = kPt(u)((du) + a(t), t E [0, TJ, (1.21 )
where ( is a stochastic orthogonal measure,
Another interesting class of functionals is of form (1.15) with
() . b 2 1 ( i>.z i'\z ) (d') 9 z = taz - - z + e - 1 - -- 7r A 2 R 1 + A2 '
(1.22)
where a E R, b ~ 0, 7r(dA) = dM(A) for A < 0 and 7r(dA) = dN('\) for ,\ > 0; M(A), N('\) satisfy the following conditions:
1. M(A) and N(A) are nondecreasing functions on (-00, 0) and (0,00), respec tively;
2. M(-oo)=N(oo)=O;
3. J~E A2dM(A) < 00, J~ A2dN(A) < 00 for any f > O.
If U = [O,T], v(du) = du, then bearing in mind (1.20), we obtain
X(l) = exp {ia loT loT p.(u)l(s) dsdu - ~ loT loT B(t,s)l(t)x
X l(s) dtds + LIoT [exp {i'\ loT p.(u)l(s) dS} - 1-
''\ T ] } -1: A210 p.(u)l(s)ds dU7r(d'\) , (1.23)
where B(t,s) = lTd: Pt(u)P.(u) du, lT2 = b+ JR A27r(d'\). This functional is the char acteristic one for the random process
(1.24)
\0 Chapter 1.
where e. is a homogeneous process with independent increments which satisfies the condition eo = O.
Note that (1.23) for M(A) = N(A) = 0
gives rise to functionals of type (1.18); it gives rise to functionals of type (1.17) for
{ -a if A < 1 b = 0, M(A) = 0, N(A) = 0 th - . o erWlse
and to functionals of type (1.19) for
1001 v b = O,M(A) = O,N(A) = - -exp(--) dv,a > 0.
>.. v a
We shall mention still another characteristic functional which corresponds to the measure which we shall call the Abel measure:
x(l) = exp L + l~(l, I)} . (1.25)
The characteristic functional of the Cauchy measure:
X(l) ~ exp { - t, Q;I{I, ';)I} , (1.26)
where aj are positive numbers, L:~1 aj < 00, {ej}, j = 1,2"", is a basis in X, is an example of the characteristic functional which cannot be represented in form (1.13).
Characteristic functionals of form (1.13) give examples of quasimeasures, when the number of summands in the exponent is finite, the last summand has the index n = 2p, p > 1 and Q2p = (_1)P+1 q2P. In particular, let us mention the case, when
x(l) = exp { - (;~! fu 12P(P(U))V(dU)} , (1.27)
where the notation is the same as for (1.14). For the space of functions on [0, TJ, (1.27) will assume the form
{ q2p T[T ]2P} X(I) = exp - (2p)! 10 10 pt(u)l(t) dt du. (1.28)
Finally, when there is a finite series in the exponent of (1.13) and the last summand enters with the imaginary term then we obtain pseudomeasures, namely, the Feynman measures. For example,
x(l) = exp {-~ fu 12(P(u))v(du)} , (1.29)
in particular,
1.4 Moments, Semiinvariants. Integrals of Cylin dric Functions
This section contains a number of exact formulae for functional integrals w.r.t. both general and particular measures.
The integrals of the form
(1.31)
are called the n-th order moments of the {cylindric} measure p; the integral Mo =
Ix p( dz) = 1 is called the zero-order moment. If the characteristic functional is known, then its moments may be computed by the formula
an n
= (-it a>. ... a>. x(?: >'ili) 1 n J=1 ).1= ... =). .. =0
(1.32)
The moments for the measures with characteristic functionals of form (1.13) do exist; they are finite and easily computable by formula (1.32). Examples of computations of such kind will be encountered later in derivations of approximate formulae.
The functional
is called the n-th order semiinvariant of the measure p. If the measure corresponds to a random process defined on [0, T], then
Kn(lb"', In) = loT ... IoT K n(t1,"', tn)I(t1)" .l(tn) dt1 ... dtn,
where the integral is interpreted as the value of an n-linear continuous form on X' x ... x X'. In this case, Kn(lb ... , In) is called the semiinvariant function of the process. We shall later use semiinvariants for the derivation of approximate formulae.
We shall now present the forms of formulae (1.8)-(1.10) for some particular mea sures. The measures PII,. ..• I .. are deduced from the characteristic functionals with the help of the multidimensional inverse Fourier transform.
For Gaussian measures, we have:
ixf((h,z), ... ,(/n,z))p(dz) =
12 Chapter 1.
x exp {-l(K-1[u - m], [u - m])} d:'u. (1.33)
Here, K-1 is the inverse for matrix K = IIK(li, Ij)ll, i,j = 1,2,,,,, n;
m = (m(ll)"'" m(ln)), m(l) is the mean value of the measure; K(ll, 12) is the cor relation functional of the measure; u = (Ul,···, un), dnu = dUl ... dUn; (.,.) is the inner product for Rn. The special case of this formula for the space of continuous functions is as follows: ix f( x( tl)' ... , x( tn ) )JL( dx) =
= (21!'tn/2(detKtl/2 r f(u)x iRn
X exp {-l(K-1[u - m], [u - mJ)} d:'u. (1.34 )
where K is the matrix with the elements B(ti,tj) (i,j = I,2, .. ·,n); B(t,s) is the correlation function of the measure.
Measures which correspond to processes with independent increments are featured by integrals of functionals of the form:
F(x(.)) = fC'vx(h),"" V'x(tn )),
where V'X(tk) = X(tk) - x(tk-d. Thus, for the measure which corresponds to Poisson process with the characteristic functional
x(l) = exp {A iT [ei1(I[u,T]O) - I]dU} ,
l[u,T](r) = {aI" if r E [u,T], otherwise,
the following formula holds:
x(l) = exp {- iT In[I - iO'I(l[u,T]('))]dU},
the following formula is valid:
(1.36)
where f(y) = fooo uy- 1 e-u du is Euler's gamma-function, R'.t is the product of 4he
positive half-axes, and 'Vti = ti - ti-l' We shall also give an example for the case of the quasi measure with characteristic
functional
X(l) = exp { -101 [11 l(t) dtfP dU} , (1.37)
which is the special case of the characteristic functional of form (1.28) for T = 1, (j2p = (2p)!, v(du) = du, Pt(u) = l[u,T](t). Note that (1.35) may also be written in the form:
X(l) = exp { -101101 min(t1 ,···, t2p)1(tl)"'" 1(t2p) dt1 ••• dt 2P } , (1.38)
The following formula holds:
r f('VX(tl)"'" 'Vx(tn))J1(dx) == r f(u) IT S('Vtk,uk - uk_ddnu, (1.39) Jx JRn k=l
where
S( r, u) = ~ r exp[ -rv2p + iuv]dv 27f JR
is the fundamental solution of the parabolic equation
oS = (-1 )p+l 02p S. Or ou2p
Using (1.37), we obtain the following formula (cf. [61]):
L f (1 1 cxl(r)x(r) dr, .. ·, [ cxn(r)x(r) dr) J1(dx) =
= r f(u) {( 27fr n r exp[-i(u,v)-JRn JRn
-!,' (i; v. l' n.(,) dr dr] ~v} ~u, ( 1.40)
where (u, v) = L:k=l UkVk, which is valid under the condition of the measurability of f(Ul,''''Un) and the condition
If(Ul,"', un) exp[-E(U, u)~]1 :::; H(Ul'"'' Un)
for any E > 0, where fRn H(Ul, .. ·,un)dnu < 00. Finally, we shall give a formula which is valid for arbitrary countably additive measures and which approximates the functional integral by ordinary N-fold integrals. Let
x(N) = \II(N)((l~N),x), ... ,(lW),x))
14 Chapter 1.
be some approximation of an element x E X under topology of the space X; let F be continuous on X and let the conditions for the passage to the limit under the sign of the functional integral in the first of the following equations be fulfilled. Then using (1.8) we obtain
Chapter 2
Integrals with Respect to Gaussian Measures and Some Quasimeasures: Exact Formulae, Wick Polynomials, Diagrams
This chapter contains some important relations and exact formulae for integrals w.r.t. Gaussian measures and quasimeasures whose characteristic functional is an exponent of a (2p)-linear form. It considers formulae for infinitesimal change of measure, poly nomials with Wick ordering and Feynman diagram method, both for measures and quasimeasures. Main relations for integrals w.r.t. Gaussian measures in some partic ular spaces are given in detail.
2.1 Some Properties of Spaces with Gaussian Measure. Formulae for Change of Integra tion Variables
The general form for the characteristic functional of a Gaussian measure and for the integral of a cylindric function w.r.t. Gaussian measure in linear topological space are given in chapter 1. From the equation
r exp[i(~, x) - im(OlJ.L(dx) = exp[-~K(COl Jx 2
and relation (1.32), we may derive the central moments for a Gaussian measure:
M2n == M2n(~1 - m(~l)"" '~2n - m(~2n)) = 2n n
= 1 IT[(~j,x) - m(~j)lJ.L(dx) = L IT K(~i2k_P~i2k)' M2n- 1 = 0, (2.1) X j=l k=l
15
16 Chapter 2.
where m( 0 = (~, ml-') = Ix (~, x) JL( dx) is the mean value of the measure and the summation is over all partitions of the numbers {I, 2, ... ,2n} into disdjoint pairs. From (2.1) it follows that M2 = K(~,"l) where K(~,"l) is the correlation functional of the Gaussian measure JL. The general form of the characteristic functional for a Gaussian measure implies that the latter is completely defined by its mean value and correlation functional.
In what follows, we shall consider the mean value of a Gaussian measure to be equal zero. In many cases, we may without any loss of generality consider the correla tion functional to be nondegenerate, i.e., K(~,~) = 0 if and only if ~ = O. Otherwise, we could find the subspace Xo C X where the measure is concentrated, and the correlation functional will no longer be degenerate there. One may take as Xo the subspace in X which is the set of all x such that (~, x) = 0 for all ~ E X' such that K(~,O = O. In this case, the space dual to Xo is the factor space of X' w.r.t. the set of all ~ such that K(~,O = O.
The following construction is described and substantiated in papers [15, 65] in more details. The Hilbert space which is the closure of the set of functionals of form (~, .) (~ E X') in space L2 ( X, JL) will be denoted by H, and 11. E X denotes the Hilbert subspace dual to H whose closure in X is the support of measure JL. For almost all x E X, a functional (a, x)( a E 'H., x E X) is defined. It is specified by the series
00
(a,x) = 2]</>k,x)(a,ek)1i (2.2) k=l
where {</>k}, {ek}, k = 1,2,···, are orthonormal bases in H and 'H., respectively, with </>k E X' for all k and (</>k, ei) = bki' We assume that the space 11. is separable for all spaces X and measures JL considered. In particular, this is true for separable Frechet spaces and for other spaces considered here.
Functionals of the form (2.2) are called measurable linear functionals. Note that 00
(ek'x) = (</>k,X), (a,h) = "'L-(</>k,h)(a,ek) = (a,h)1i k=l
for a, hE 'H.. It is an important property of the considered spaces with Gaussian measure that
they admit the following expansion 00 00
x = "'L-(ek,x)ek = "'L-(</>k,x)ek, (2.3) k=l k=l
which converges under topology of space X for almost all x E X. The notation here is the same as for (2.2). Proofs of the convergence of expansion (2.3) may be found in papers [62,63]. The definition of H implies that (~,"l)H = K(~,"l). Let T define an isomorphism of space 11. into H which assigns a basis {</>d to the basis {ek}. Then
00
Integrals w.r. t. Gaussian Measures and Some Quasimeasures 17
00
00
00
= ~)T-1~, ek)1t(T-177, e,.) = (T-1~, T-1"')1t. "=1
Let us consider transformations of integrals w.r.t. Gaussian measure under change of variables. The following transformation formula for change of variables for an integral defined on a measurable space with an arbitrary measure JL is known (see, e.g. [64]). If 9 : X -+ X is a measurable transform then
Ix F(g(z))JL(dz) = Ix F(z)JLg(dz),
where JLg is a measure on X specified by the equation JLg(B) = JL(g-1(B)) for any measurable set B. The meaning of the formulae considered below is that, under certain constraints on g, the measure JLg(dz) is ofthe form JLg(dz) = p(z)JL(dz) where p( z) is some density functional.
The following transformation formulae for the integrals w.r.t. Gaussian measure under translation and under general linear transformation may be obtained from the above equality of integrals taking into account the equivalence results for Gaussian measures from [11, 65] (see also [66]). They may also be derived by an analogy with the nonlinear transform formulae given below.
Thus, suppose F( z) is a functional to be integrated and a E 1i. Then the transformation formula for an integral w.r.t. Gaussian measure under translation is as follows:
(2.5)
In order to state the next result, we need some definitions. A compact operator A in Hilbert space is called a Hilbert-Schmidt operator, if l:k:l A% < 00, where {A,.Hk = 1,2, ... ) is the sequence of its eigenvalues. On the other hand, if the series l:k'=1 IA,.I converges then this operator is called the operator of trace class. Clearly, any trace class operator is a Hilbert-Schmidt one. The product DA(A) = nk'=1(1- .u,.), where p,.} are the eigenvalues of operator A, is called the Fredholm determinant at point A for operator A. The Fredholm determinant of a trace class operator is finite for any A. It is an entire function of complex variable A with zeroes at points A = A;;l, k = 1,2, ... If A is a Hilbert-Schmidt operator then the product nk:l (1 - .u,.) may diverge. For such operators, the determinant of the form bA(A) = nk:l(l - .u,.)e~k~ is introduced and is called the Carleman determinant.
An operator A in Hilbert space is called positive, if the scalar product (Az, z) 2: 0 for any z from this space. A positive operator U is called the square root of operator
18 Chapter 2.
A, if U2 = A. This is usually written as U = Al/2. Let T* denote an operator dual to operator T.
Let T be a measurable linear transformation T : X -+ X and let its restriction to 1i be a bounded linear operator such that its inverse is limited and the operator A = 1- T*T is of trace class. Then
where 00 1 _ A2
RA(x,z) = L T(ele,z?; Ie=l Ie
{An and {ele} are the eigenvalues and the orthonormal eigenvectors of operator T*T, respectively; DA(l) is the Fredholm determinant of operator A at point A = 1. Let us give two more versions of (2.6):
r F(z)p,(dz) = Dy2(1) r F(Tz)exp[-~<J.>(z,z)]p,(dz), h h 2
where <J.>(x,x) = l:~l(A~ -1)(ele,z)2; {Ale} and {ele} are the eigenvalues and the orthonormal eigenvectors of operator T*T, respectively; DA (l) is the same as in (2.6).
If T = I + S and the operator I - T*T is of trace class, then
Ix F(x)p,(dz) = Dt( -1) Ix F(z + Sz)exp[-W(x)]p,(dz), (2.7)
where \II(z) = l:hi(7]1e + hD(\IIIe,x)2; DA(-l) is the Fredholm determinant of the operator A = S + S* + SS* at point A = -1; hie} and {\IIIe} are the eigenvalues and the orthonormal eigenvectors of operator S, respectively. Note that D A (-1) is finite because A is of trace class.
Consider now the case when A = 1- T*T is a Hilbert-Schmidt operator. Namely, let T = 1+ S, where S is a Hilbert-Schmidt operator, and let the inverse operator T- l exist and satisfy the same conditions. Then
Ix F(z)p,(dx) = Sy2( -1) Ix F(z + Sz)x
1 x exp[-(<J.>(z,z) - trS) - 21ISzll~]p,(dz),
where SA ( -1) is the Carleman determinant at point A = -1;
00
(2.8)
{7]Ie} and {WIe} are the eigenvalues and the orthonormal eigenvectors of operator S, respectively. Formula (2.8) is obtained by a simple transformation of (2.7). Note that
both terms of the expression <1>(z, z )-trS do not exist isolately, because S is a Hilbert Schmidt operator; this expression is meant to be the limit of the corresponding partial sums.
Let T = 1+ S be a nonlinear continuous transformation on X, where I is the identity transformation; let S have the derivative along the subspace H, which is continuous and of trace class in H, and Sz E H. Let further an inverse transform T- 1 exist and have the same properties as T. Then the following relation holds.
f F(z)JL(dz) = f F(z + Sz)Dsl(z)(-l)exp[-<1>(z) - ~IISzll~lJL(dx), (2.9) Jx Jx 2
if any side of this equality exists; here we denote <1>(z) = ~k=1(ele,Sz)(ele,x); {ele} is an orthonormal basis in H; D~( x)( -1) is the Fredholm determinant of operator S'(x) at point -l.
It is enough to prove this formula for the case when F( x) is continuous and bounded for almost all z E X. In this case, we have
f F(z)JL(dz) = lim f F(x(N)JL(dx)) = Jx N-+oo Jx
N N pN(dv) = rr(21rt1/2exp(-v~/2)dvle,x(N) = l.:(ele,x)ele.
1e=1 1e=1
Let us change variables in the last integral over RN :
N
Vie = Ule + (S(l.: Uie;) , ele)1t, k = 1,2"", N. i=1
The elements of the Jacobian det IloVIe/OUil! for this transformation are
N
OVIe/OUi = 0lei + (S'(l.: Ujej)ei, ele)1t, j=1
where S'(~f=1 ujej) is the derivative of operator S at point ~f=1 ujej. After this change of variables followed by passing to integration over X we obtain
20 Cbapter 2.
For almost all z E X, the following limit relations hold as N ---+ 00 (see [66]):
N
N N
I)Sz(N), ek)1t(ek, z) ---+ 'L(Sz, ek)(ek, z), k=l k=l
N
det 118ik + (S'(z(N»)ei, ek)1t1l ---+ DSI(.,) ( -1),
and the desired formula follows. If S' is a Hilbert-Schmidt operator then formula (2.9) will be extended as follows:
Ix F(z),.,,(dz) = Ix F(z + SZ)DSI(",)( -l)x
X exp[-(<1)(z) - trS'(z)) - ~IISzll~l,.,,(dz), where <1)(z) = Lk:l(ek, Sz)(ek' z); {ek} is an orthonormal basis in 'H, and 8SI(.,) ( -1) is the Carleman determinant of the operator S' (z ). A note concerning the terms in the expression <1)(z) - trS'(z) similar to the one related to formula (2.8) is also valid.
Later we shall consider change of integration variables for Wiener integrals, both for the general case and for some special cases.
2.2 Exact Formulae for Integrals of Special Func tionals. Infinitesimal Change of Measure
First, we give two relations which we shall use in the derivation of approximate formulae for integrals w.r.t. Gaussian measure and in the analysis of its convergence (see [58]).
Let a functional F( z) be continuous almost everywhere on X and let it satisfy the condition IF(z)1 ~ <1)(z), where <1)(z) is a nonnegative summable functional such that <1) (z(N») is nondecreasing as N ---+ 00. Then
(2.10)
N
pN (du) = II (211't1/ 2 exp[-uV2]duk. k=l
Integrals w.r.t. Gaussian Measures and Some Quasimeasures 21
(2.11)
where :c(N) = L:~=l(ek,:c)ekj w(N)(u) = L:~=l Ukekj {ek},k = 1,2,···, is an orthonor mal basis in 1i..
Let us proceed now to the evaluation of integrals w.r.t. Gaussian measure for par ticular functionals. First we shall evaluate integrals of functionals which are functions of measurable linear functionals.
Theorem 2.1 Let al,· .. , an be linearly independent elements from 1{. Then the fol lowing equality holds
Ix F((ab :c), ... , (an, :C))JL(d:c) = (211"tn/2[det Atl/2 JJln exp[-l(A-lu,u)]F(u)d"u,
(2.12) if any of these integrals exists, where A is a matrix with elements aij = (ai, aj)1t i,j=1,2,···,n, dnU=dul···dun.
Proof. We shall make use of formula (2.5) for the transformation of integrals un der translation, wherein we set a = -,\ L:k=l Vkak, with ,\ and Vk being real numbers, and F(:c) = 1. Then we have
n
= exp[,\2/211 L vkakll~] = exp[,\2 /2(Av, v)]. k=l
This equality is also valid for ,\ = -i. This, together with the following formula:
implies that
= (211"t/2(detAtl/2 k .. e-i(U,V)exp[-l(Au,u)]d"u,
i.e. formula (2.12) is valid for F(u) = e-i(u,v). Let now the function F(u) be specified by its Fourier transform, i.e. F( u) = fR" e-i(v,u) f( v )d"v. Let us multiply the previous relation by f( v) and integrate it over space Rn w.r.t. the variables Vb· .. , vn. After changing the order of the integration we obtain (2.12). The validity of the formula for arbitrary functions F(u) is verified by passage to the limit.
22 Chapter 2.
Let us mention two special cases of the obtained formula. For n = 1 we have:
Ix F((a,x))p(dx) = (27rt1/ 2 1I a lli/ ~ exp[- 211:~I~JF(U)dU. Let aI, ... ,an be orthonormal elements of space Rn. Then
Consider now evaluation of integrals of the functionals which are functions of quadratic functionals on X :
00
A(x,x) = L akj(ek,x)(ej,x), (2.13) k,j=l
where ak.i = (Aek, ej); A is a self-adjoint trace class operator on H; {ek}, k = 1,2, ... , is an orthonormal basis in H. The double series in (2.13) converges for almost all x E X. The functional A(x,x) does not depend on the choice of basis {ek} and may be written as follows
00
A(x,x) = L Ak(ek,x)2, k,j=l
where {Ak} and {ek} are the eigenvalues and the orthonormal eigenvectorr; of operator A, respectively.
The following integrals are easy to evaluate with the help of (2.12):
IxA(x,x)p(dx) = trA,
Let us prove the equality
where D A (A) is the Fredholm determinant of operator A at point A; Re A < All (AI is the largest eigenvalue) and
In fact, ,\ ,\ 00 r exp[-A(x,x)Jp(dx) = r exp[- LAk(ek,x?Jp(dx) =
Jx 2 Jx 2 k=l
00
= II (1 - AAkt 1j2 = [DA (AW 1j2 .
We may similarly evaluate an integral of the more general form:
where 9 E 1i, BA(2A) = (I - 2AAt1, Re A < (2A1t1 (AI is the largest eigenvalue); b is a numeric parameter; I is the identity operator.
Now formula (2.14) will be exploited in the proof of a more general result.
Theorem 2.2 Let A1(x,x),A2 (x,x), .. ·,An(x,x) be quadratic functionals of form (2.13) and let 91,92,'" ,9m be linearly independent elements of space 1i. Then the following equality holds:
Ix F(A1(x, x), .. ·, An(x, x); (91, x), .. ·, (9m, x ))J1(dx) =
= r p(u; v)F(u; v) dnudmv iRn+m
(2.15)
1 x exp[-2(C-1(~)v,v)] ~C
C(O is a matrix with elements Ckj = ([I+2iA(e)]-1 9k ,9jh-l; DA(e)(A) is the Fredholm determinant of the operator A(O = L:k=l ekAk at point A; u = (Ul"'" un), V =
(Vl'···'Vn ).
Proof of this theorem is similar to that of'Theorem 2.1, i.e., first, we prove the validity offormula (2.15) with the help of (2.14) for the functions F(u;v) which are specified by their Fourier transforms; and then we prove the formula for arbitrary functions by passage to the limit.
Consider special cases of formula (2.15). If F(u; v) does not depend on u, i.e., F(u; v) = F(u), then we obtain formula
(2.12). suppose that F(u; v) = F(v), i.e., it does not depend on u, then
24
JRn
Chapter 2.
D AW( -2i) is the Fredholm determinant of the operator A(~) = 2:;:=16.A,. at point A = -2i. Consider still another example:
Ix exp[~A(x,X)]F((9'X))/L(dx) =
= [211'DA(A)b(,\)t1/2 In exp[- 2~:)]F(V) dv,
where b('\) = ([/ - AA]-lg,g)1t. Let us turn our attention to the formulae of infinitesimal change of measure (or
measure interpolation) for functional integrals. For integrals w.r.t. Gaussian measure /L over the space X = X([O, T]) of functions defined on segment [0, T] of the real line, the interpolation formula is of the form ([68]):
Ix f(x)/L(dx) = Ix f(:C)/Lo(dx)+
+ ~ r1 rT rT(B _ Bo)(t,r) r f"(xjt,r)/L.(dx)dtdrds, 2 Jo Jo Jo Jx
where /L. is a Gaussian measure on X with correlation function
B.(t,r) = sB(t,r) + (1- s)Bo(t,r),
B(t,r) is the correlation function of measure /L, Bo(t,r) is the correlation function of Gaussian measure /Lo, (B - Bo)(t,r) = B(t,r) - Bo(t,r), f"(xjt,r) is the second variational derivative of the functional f(x)j the mean values of the measures are assumed to be zero.
The following theorem (see [69]) generalizes the measure interpolation formula to integrals w.r.t. quasimeasures with characteristic functionals of the form:
(2.16)
where K (~, ... ,~) is the value of a symmetric 2p-linear form over X' X ••• x X' at point (~y!,),~)j (-l)P+1K(~, ... ,~) 2: 0 for all ~ E X'.
Let /Lu be a quasi measure on X with its characteristic functional in the form of (2.16), where K(~, ... ,~) is substituted by Ku(~,""~) = uK(~, .. ·,~) + (1- u)Ko(~,'" ,~), Ko(~,'" ,~) is a (2p)-linear symmetric functional which corresponds to a given quasimeasure /Lo' Let f(n}(x) denote an n-th order functional derivative for functional f (x) j define
Tr[(K - Ko)f(n}(x)] =
00
L (K - KO)({3ip,,·,{3i,,)f(n}(X)ai, "'ai",
where {a;}, {{3;}, i = 1,2"", are the basis in X and the dual basis in X', respec tively.
Theorem 2.3 Let functional Tr[(K - Ko)f(2P}(X)] be defined and continuous on X. Suppose that the sequence of cylindric functionals
n
Tr[(K - Ko)f(2P}(L (13k, x)ak)]' n = 1,2, ... , k=1
which are integrable w.r.t. quasimeasures Pu for any u E [0,1], converges to the func tional Tr[(K - Ko)f(2P}(x)]. Then the following relation holds:
Ix f(x)p(dx) = Ix f(x)po(dx) + (2~)! 11 Ix Tr[(K - Ko)f(2P}(x)]Pu(dx) duo (2.17)
Proof. First we prove the theorem for cylindric functionals
In this case,
where r (-ly
n 2p
X L II YilKu(eill " " ei2P)]any. ;, ,.··,j,p=1 1e=1
Let us use the evident identity
f g(Z)pl(Z)anZ - f g(z)po(z)anz = fl dd f g(z)pu(z)anzdu. (2.18) JR" JR" Jo U JRn
Differentiating of Pu(z), we obtain:
Substitute now the obtained expression into the integral w.r.t. dnz on the right side of (2.18). Then integrating this integral by parts with the help of the identity
26 Chapter 2.
= [g( (6, z),· .. , (en, z) ))(2P)a1 a 2 ••• a2p,
we obtain formula (2.17) for cylindric functionals. Once again, passage to the limit proves this formula for arbitrary functionals f( z) that satisfy the hypothesis of the theorem.
Note that in the case of the function space X = X([O, T]) we have:
rT (2p) rT = Jo ... Jo (B-Bo)(t1, .. ·,t2P)f2P(Z;t1, .. ·,t2P)d2Pt.
In the case of p = 1, the theorem implies the corresponding result for integrals w.r.t. Gaussian measures; in this case we need not assume the existence of a basis in X, because we may use, for the similar purposes, the bases {<Pi}, {ei}, i = 1,2, ... , in spaces Hand 'H, respectively. In particular,
00
Tr[(K - Ko)f"(x)) = L (K - KO)(<Pi,<pj)f"(x)eiej. i,j=1
2.3 Integrals of Variations and of Derivatives of Functionals. Wick Ordering. Diagrams
Let functional F( z) be differentiable along direction a E 'H at any point x E X and let
d sup!>.I<.l dA F(x + Aa)1
be summable for some f > O. Then the following relation holds:
Ix 8F(z; a)JL(dx) = Ix F(z)(a,z)JL(dz)
The latter relation follows from the identity
Ix F(x + Aa)JL(dz) = exp[- ~21Iall~) Ix F(x) exp[A(a, x))JL(dz)
(which is a modification of formula (2.5)), if we differentiate it w.r.t. ,\ and set ,\ = O. Moreover, the above condition enables differentiation under the integral sign.
The functional Hermite polynomials may be defined by the equality:
H(n)[z;all,,·,an) = {),\ ~{),\ G[z;all ... ,an)! ' 1 n ).1= ... ).,,=0
where 1 n n
G[x; all"', an) = exp[-2" .~ AiAj(ai, ajht + ?= '\i(ai, x))' ',3=1 .=1
ai E H; the functional G[x; , aI, ... , an] is defined for almost all x E X. The following recurrence relation holds for functional Hermite polynomials:
n-l
- "'(a' a )",H(n-2)[x' al ... a, .... a 1] L.J " n n. "''', n- i=l
(here the "hat" over ai means that ai must be omitted); it is verified by an immediate calculation with the help of the above-mentioned definition of functional Hermite polynomials. The explicit form of them is as follows:
n _ n
X L II(ahp_"ahp)1i II (aiq,x), i, #"'#in=l p=l q=2_+l
where (28 )!! = 2 ·4· .. (28), l·J is the floor function. The first three Hermite polynomials are:
H(l)[x;a] = (a,x),
H(2)[x;aI,a2,a3] = (al,x)(a2,x)(a3,x)-
- (aI, a3)1i( a2, x) - (aI, a2)1i( a3, x) - (a2, a3)1i( al, x).
Note also the special case of al = a2 = ... = an = a :
H(n)[x' a] = H(n)[x' a ... a] = ,- '"
Ln/2J (-l)in ! . . = ~ (2j)!!(n_2j)!llall~(a,xt-23.
(2.19)
is summable for some € > O. The formula follows immediately from the definition of the n-th order variation, the definition of Hermite polynomials given at the beginning
28 Chapter 2.
of the section, and relation (2.5) or its modification also given at the beginning of the section.
Consider now the notion of Wick ordering. This term was introduced to denote the process of reordering of birth and death operators in Fock spaces in quantum field theory. More recently it was noticed that the Fock space may be realized as the space L2(X,JL) of functionals square summable w.r.t. Gaussian measure with zero mean, and the orthogonalization process for polynomials in this space corresponds to the reordering process(see [67, 68]).
Thus, if a monomial nj=1 (aj, x) is given, then its Wick ordering (or Wick product) is specified by:
n
: TI(aj,x):= H(nl[x;al, ... ,an]. j=1
The expression : (a, x)n : is called the n-th Wick power. The Wick ordering of an arbitrary formal series (finite or infinite) of monomials ~c"nj=1(aj",x) is defined by:
" " : LC" TI(aj",x):= LCv : TI(aj", x) :. j=1 j=1
Hence, 00 ),n
: exp[),(a, x)] := L I" : (a, x)" : . n=O n.
This functional we call Wick exponent. Note that Wick ordering is defined for a fixed Gaussian measure JL (as well as the Hermite polynomial), therefore, sometimes the notation: :" is used to stress this fact.
For ell'" ,en E X', bearing in mind (2.4), let us define:
In particular,
: (e,x)":= Hfl[x;e] =
j=1 J .. n J.
(2.20)
In general, Wick ordering is defined for Gaussian measures with nonzero mean (see [67]); if measures JL and v are given with their respective means m" and m", then the following equality holds (it may be considered as the definition):
r
= LC;Hi-~~KJm" - ml';el: (e,z)i :,,= i=O
~ L(r~2J r! (t )i (t )r-i-2; =L...J L...J .'.'( _ ·-2.)': I"Z :" I"m,,-ml' X
i=O ;=0 ~.J. r ~ J.
where C; = r!j(i!(r - i)!). Consider the following integrals:
(2.21 )
For their evaluation, the Feynman diagram method may be used, which was originally proposed in quantum field theory (see, e.g. [4, 68]). This method gives an easy way to write down the value of integral (2.21) and it works as follows. To each value of j, we assign a point, i.e., a vertex of the diagram. Each j-th vertex casts n; rays. To obtain a diagram, we have to join all the emanating rays into pairs. If there is an unpaired ray then integral (2.21) is equal to zero. Otherwise, a multiplicative term K(ei,e;) is assigned to each connected pair of rays (here i = j is possible). To each such diagram, we assign the product of all such terms. Integral (2.21) is equal to the sum of all possible diagrams. We shall demonstrate this approach for the following integral:
II = Ix (el> z)2(~2' Z)(~3' z)3JL(dz)
Let us draw all possible kinds of diagrams:
o _. -0 There are three diagrams of this kind. Their contribution to the value of the integral
is 3K(~I' ~dK(~2' ~3)K(6, ~3).
There are six diagrams of this kind. Their contribution to the value of the integral
is 6K(~I' 6)K(~I' 6)K(6, 6)·
<:C_.~>
30 Chapter 2.
There are six diagrams of this kind. Their contribution to the value of the integral is 6K2(e1,ea)K(e2,ea). Hence the integral equals to
1 = 3K(eb edK(e2, ea)K(ea, 6)+
6K(e1, e2)K(eb e3)K(e3, e3) + 6K2(e1, e3)K(~2' ~3).
The generalization of this method to the calculation of integrals of the form
m nj 1 II: II(~jk'X} :p,(dx) X j=l k=l
is the substance of the so-called Wick theorem (see [68]). In the latter case, a vertex of the diagram is assigned to each j and a ray emanating from j-th vertex is assigned to each ejk, k = 1,2 ... , nj. All concerning the evaluation of integral (2.21) remains valid here, bearing in mind (the essence of the Wick theorem) that if a diagram has two connected rays emanating from the same vertex then the contribution of the diagram is zero.
For example, let us evaluate the integral
12 = Ix : (~l,xh :: (~2,xh: p,(dx)
Only diagrams of the shape <=> give a nonzero contribution; there are two of them with the total contribution of 2K2(~1' ~2). The diagrams of the
shape 0 0 give no contribution. Hence, 12 = 2K2(ebe2). In particular, the diagram approach enables to evaluate easily
Ix: (e,x}n:: (TJ,x}m: p,(dx) = Dmnn!Kn(e,TJ),
Ix : exp[a:(~,x}l : p,(dx) = l. This approach is also applicable to the evaluation of integrals
m nj 1 II: II(ajk,x): p,(dx). X j=l k=l
(2.22)
The only difference is that a multiplicative term (ajk1 , ajk l )1i is assigned to a connec tion of rays k1 and k2•
Let us consider now Wick orderings and the corresponding diagram approach for integrals w.r.t. quasi measure over linear space X with the characteristic functional (2.16). First, we obtain the moments of the quasimeasure:
Integrals w.r. t. Gaussian Measures and Some Quasimeasures
I
;=1
31
(2.23)
where the summation is over all partitions of the 2pl indices {I, 2, ... ,2pl} into 1 groups of 2p elements: {il' ... ' i2p}, {i 2p+1, ... , i2p}, ... , {i2p(I-1)+1, ... , i2p!}; if m ~ 2pl then the integral of ITj=l (~j, x) equals to zero.
Wick monomials may be evaluated from their generating function
G(~;x) =: exp((~,x)) .- exp((~,x))[Jxexp((~,x))dJLl-l = exp((~,x) - (2~)!K(~, ... ,O)
with the help of the following formula:
and are as follows:
n " n
X L IIK(~i2p(/-l)tl'~hp,) II (~iq'x). il#···#in=ll=l q=2p>+1
In particular,
L~J (1» 1 . (' )n._ ~ - n. K'(' ')(' )n-2p , . <",x .- L.. ((2 )1)' I( -2)1 <", ••• ,<" <",x • ,=0 p. s. n ps.
Note that ((2p)!)"s! = ((2p)!s)! .. . !, where (ms)! ... ! = m(2m)(3m)··. (sm). If p = 1, we have 2" s! = (2s)!! = 2 ·4 .... (2s), and hence the obtained expression for Wick power coincides with that of for Gaussian measures given earlier. Let us mention some special cases of Wick powers for a quasi measure:
: (~,x)k : : (~, x)2p :
: (~,x)4p:
(~,X)k, k = 1,2, ... ,2p-l; (~, X)2p - K(~, ... , ~), (~, X)4p - ~K(~, ... , ~)(~, X)2p+ ~K2(' ') + ((2p)!)22! <", •.• , <" •
To evaluate integrals of products of Wick powers, we may use a diagram method similar to the diagram summation method used for Gaussian measures. We demon strate it for the cases of products of two and three Wick powers with similar arbitrary
32 ChapteI2.
exponents. First, we precompute the following integrals of Wick exponents using (2.16) and the generating function:
Ix: exp(o:({,:c)):: exp{,8(l1,:C)): p,{d:c) =
We have:
(2.24)
Clearly, while computing the derivatives for 0:1 = 0:2 = 0:3 = 0, nonzero contributions will arise only from the summands with the products
(iii) (10;2) (li3) ~~~ IT K{{l,'" ,{b{2,". ,{2,{3,"" {3)
i 11i2 ,i3
with Eil iii = Ei2 ki2 = Ei31i3 = m, where iii + ki2 + li3 = 2p. The above implies the following diagram summation rule for the computation of nonzero contributions into the value of integral (2.24); namely, take the prediagram
which corresponds to Wick product rr~=l : ({,., :c)m :, where each vertex emanates m rays. The diagram is obtained by binding the rays into {2p)-element groups. If there are free (unbound) rays or all rays of some {2p)-element group emanate from the same vertex, then this diagram gives the zero contribution. Below, there is an example of a nonzero diagram for p = 4, m = 8.
This diagram contributes the multiplicative term
The contribution of the diagram depends only on the number of vertex rays bound into a packet and it does not change, if we interchange two rays of a vertex.
In general, analytic expressions for the number of different diagrams are difficult to derive because there are many various diagrams and many elements. This can only be done for specific diagrams or for small p. In particular, we may find that
(2p-j) (p)
(............... ) 1 P)2 2(..--"'-. xKe, .. ·,e,'T/, .. ·,'T/ + "2 (C2p K e, .. ·,e,'T/, .. ·,'T/)·
All stated above may be extended to quasimeasures with characteristic functional of the form:
where Kn(~"" ,~) is an n-linear form on X' x ... x X', (-1)m+1K2m(~' ... ,~) ~ 0 for all ~ E X', i is the imaginary unity.
Consider a special case, where only forms K2(e,~) and K4(~,~,~,e) are nonzero:
2n n
M2n = h II(ej,x}JL(dx) = E E*K2(6'llek2) x .. · x j=l m=O
x K2( ek2m-l' ek2m )K4( ek2.n+l' ... , ek2m+4) x
x K4 ( ek2m+6' ... , ek2m+&) ... K4( ek2n-31 ek2n-2' ek2n-ll ek2n);
the summation in 2:* is over all partitions of the set {1,2, ... ,2n} into disdjoint 4- and 2-element subsets (including only 2-element and only 4-element ones); if such a partition is impossible, then the integral equals to zero.
34 Chapter 2.
: (~,:c)4:= L>;C~M?): (~,:c)n-; :4, ;=0
where M?) is the j-th order moment of a Gaussian quasimeasure with correlation
functional K2(~' .,,), : (~, :c)~-; is the Wick power w.r.t. quasi measure with correlation functional exp[~K4(~'" ., ~)], i is the imaginary unity. All possible connections of rays in pairs and quadruples are used in diagrams. The contribution to the value of an integral of products of Wick powers arises only from the diagrams such that their rays from vertices are connected in pairs and quadruples, there are no pairs and quadruples of rays from the same vertex, and there are no free rays. For example,
Ix : (~,:c)3:: (~,:c)3: p(d:c) = 6 x ~ + 3 x U = 6K:(~,.,,) + 3K2(~,.,,)K4(~'~'''''''').
2.4 Integration with Respect to Gaussian Mea sure in Particular Spaces
We demonstrate some relations considered in previous sections for concrete spaces.
a). X = H is the Hilbert space. In this case, HI is usually identified with Hand the correlation functional of the measure is in the form of K (u, v) = (K u, v), where u, v E H, K is a positive trace class operator in H called the correlation operator of Gaussian measure p.
The scalar product in H is as follows:
where K 1/ 2 is the square root of operator K. The space H is the completion of H w.r.t. norm IlvllH = IIKl/2VIIH and the space 'H is the set {K1/ 2H} with scalar product (a, b)1-£ = (K- I / 2a, K-1 / 2b)H, where K- I / 2 is the operator inverse to K I / 2; the isomorphism from 'H into H mentioned in Sect. 2.1 is in the form of T = K -1/2 K- 1 / 2, where K -1/2 is the closure of operator K-1 / 2 in H. The basis {ek} used in formulae (2.2),(2.3) may be taken to be {K I / 2o:k } where {O:k} is any orthonormal basis in H. If O:k = <Pk are the eigenvectors of operator K then we may take ek =
y'Xk<Pk' where Ak are the corresponding eigenvalues of operator K. Then the formulae (2.2),(2.3) will assume the form of
00
00
x = L(x, ")H,,· "=1
medskip b). X = C[a, bj is the space of all functions continuous on segment [a, bj. In
this case, a linear continuous functional is in the form of (~, x) = f: x(t)d~(t) and the correlation functional of the measure is K(~,1J) = f:f: B(t,s)d~(t)d1J(t), where ~(t),1J(t) E C'[a,bj = Vo[a,bj, Vo[a,bj is the space of functions ~(t) of bounded varia tion on [a,bj which satisfy the condition ~(a) = 0, ~(t) = [~(t + 0) + ~(t - 0)l/2 for t E ( a, b); B( t, s) = fqa,b] x( t )x( s )p( dx) is the correlation function of the measure. In this case, the space 1i is the closure of the linear hull of eigenfunctions {,,(t)} of kernel B(t,s) w.r.t. scalar product
where {A,,} are the corresponding eigenvalues of kernel B(t, s). The basis of 1i consists of the functions:
e,,(t) = [i:,,,,(t), k = 1,2, ...
The decompositions (2.2) and (2.3) assume the form:
00 1 b b
(a,x) = E A" (1 a(t),,(t)dt)(l x(t),,(t)dt),
x(t) = fJlb x(t),,(t) dt),,(t). "=1 a
(2.25 )
(2.26)
where B 1/ 2 ( t, u) is the kernel of an operator which is the square root of the operator with kernel B(t,s); {o:,,(u)} is an orthonormal basis in L2[a,bj.
We would like to mention that the following relation takes place:
(B(t,·),a(·))'Ji = a(t). (2.27)
It is easily verified that the functions A -1/2 f; ,,( s) ds, k 1,2, ... , constitute a basis in space H and the isomorphism T from Section 2.1 is specified by the following relations:
T([i:,,,,(t)) = A- 1 / 2l ,,(s)ds,k = 1,2, ...
We note also that if g(t) E L2[a,bj then
a(t) = 1b B1/ 2(t, u)g(u) du E 1i. (2.28)
36 Chapter 2.
Now we consider some issues, mentioned earlier, for the space C[a, b]. Starting with formula (2.19), we obtain its corollaries under various conditions of differentiability ofF(x).
Let a functional F(x) have an H-derivative Fu(x) = Fu(x;t), i.e., 8F(x;a) =
(Fu(x; .), a('))1i' By formula (2.27), (Fu(x; .), B(t, '))1i = (Fu(x; t), and using (2.25) we may easily find that (B(t,.),x(.)) = x(t). Therefore, for n = 1, formula (2.19) becomes
r F~(x;t)p,(dx) = r F(x)x(t)p,(dx). Jc~~ Jc~~
In a similar way, using (2.19) we may prove a more general equality
where the integrand on the right side is the n-th order H-derivative and
If the functional has a C[a, b]-derivative F(n)(x; Sl, ... , sn), then we obtain:
= r F(x)H1n)[x; t1, ... , tn]p,(dx). JC[a,b]
Now we shall give several special cases of formula (2.15) in space 0[0,1]. Let Ak(t, s), k = 1,2, ... ,m, be symmetric functions defined on the unit square and let they be the kernels of the operators of type (2.13). Let further pj(t), k = 1,2, ... , n, be bounded variation functions. Then the following equality holds:
r F (1 1 X(t)dP1(t), ... ,1 1 x(t)dpn(t), JC [O,I] 0 0
r1 r1 x(t)x(s)d;.A1(t,S), ... , r1 r1 x(t)x(s)d;.Am(t,s))p,(dx) = Jo Jo ' Jo Jo '
= (27rtm - n / 2 r [det C . DB (2i)t1/2X JRn t 2m e
x exp {-i(v,O - ~(C-1U,U)} F(u,v)redmv~u,
where the elements of the matrix a = C(O are Ckj(e) = Ii Ii B1 (t, S; e) dpk(t)dpj( s), where
{41Ic(t)} and {/LIc} are the orthonormal eigenfunctions and the eigenvalues, respec tively, of the kernel
Let al(t), ... , an(t) be linearly independent functions on segment [0,1]' then the following formula holds:
where DB~(2il is the Fredholm determinant of the kernel
at point A = 2i.
c). X = am[O, 1J is the space of m times continuously differentiable functions on [0, 1] which satisfy the conditions
x(O) = X'(O) = ... = x(ml(o) = 0,
t t t
A linear continuous functional on am[O, IJ is defined by the equality
where x(ml(t) is the m-th order derivative of x(t). The correlation functional is K({,"l) = f~f~ Bm(t,s)d{(t)d"l(s), where Bm(t,s) = fCm[O,l]X(ml(t)x(ml(s)/L(dx) is a function continuous w.r.t. both variables. Any function x(t) E am[o, 1) may be represented in the form of
1 r x(t) = (m _ I)! 10 (t - u)m-lx(ml(u) duo
This fact may be easily proved by integration by parts. The correlation function is
38 Chapter 2.
Let {,. (t nand {>."}, k = 1, 2, .. " be the orthonormal eigenfunctions and the eigenvalues of the kernel Bm(t, s), respectively. Then the functions {e,.(t) = A<p,.(t)} with
<p,.(t) = (m ~ I)! l(t -u)m-1 ,.(u) du
constitute an orthonormal basis in 11.. In this case, 11. is the closure of the linear hull of the set {<p,.(tn, k = 1,2" .. w.r.t. norm which corresponds to scalar product
formulae (2.2) and (2.3) assume the following forms, respectively:
00 r1 ( r1 (a, a:) = "f(Jo a(m)(t)<p,.m)(t) dt)(Jo a:(m)(t)<p~m)(t) dt)/A,.,
a:(t) = ~(l a:(m)(t)<p~m)(t) dt)<p,.(t).
Let us consider several examples of measures for the above-mentioned spaces and give some additional specific information.
1. The Wiener measure in the space ColO, 1] of continuous functions which satisfy the condition a:(0) = 0 is the Gaussian measure with nonzero mean and correlation function B(t,s) = min(t,s). The eigenvalues and the orthonormal eigenfunctions of the kernel min( t, s) are
4 . 2k - 1 A,. = 1I'2(2k _ 1)2' <P,.(t) = v'2 sm -2 -lI't, k = 1,2, ... ,
respectively, and may be obtained as the solution of the Sturm-Liouville problem -AU"(t) = u(t), u(O) = u'(I) = O. Note that min(t,s) is the Green function for the problem -u"(t) = f(t), u(O) = u'(I) = O. For the case considered, the space 11. generated by the kernel coincides with the space of functions of the form a(t) =
J~ 9(S) ds, where 9(S) E L2 [0, 1], whose scalar product (a, b)1t = J~ a'(t)b'(t) dt. For {a,.(s)k:l} being the complete orthonormal system in L2[0, 1], the functions e,.(t) = J~ a,. (s ) ds form the basis in 11.. The above-mentioned eigenfunctions <p,. (t) may be taken for the a,.(t). A measurable linear functional is of the form
00 101 2k - 1 (a,a:) = L2 a'(s)dsin--lI'sx
i=1 0 2
10 1 2k - 1 101 a:(s) sin --lI'S ds == a'(s) da:(s).
o 2 0
For almost all x(t)E Co [0, 1] under Wiener measure there is the following expan sIOn:
00 r1 t x(t) = E(io u/e(t)dx(t)) 10 u/e(s)ds,
which converges under uniform norm. Formula (1.34) is simplified in the case of Wiener measure for m = ° :
= [(211'tt 1(t 2 - td··· (tn - tn_dt1/ 2 X
x r F(Ull".'1Ln)exp{-~ [u~+ iRn 2 tl
(U2- U l)2 (Un- Un-d2 ]}d d + + ... + Ul'" Un. t2 -t1 tn -tn- 1
The characteristic functional for the Wiener measure X(e) = J~[{(t) - {(I)j2dt may be written in the form of (1.18), if we set U = [0,1], a = 0, lI(du) = du, p(u) = Pt(u) = l[u,I](t).
We shall give the formula for the nonlinear change of the integration variable in spaces with Wiener measure.
Let Tx(t) = x(t) + S(xlt), where the function S(xlt) of the argument t (when x is fixed) is such that it is in C[O, 1], x(O) = S(xIO) = 0, and it has the derivative 1tS(xlt) which is square summable on [0,1]. By the definition of norm in 11. we have:
The following equality also holds:
00 1 d 2: (e/e' S(tlx))(e/e' x) = r -d [S(xlt)] dx(t). /e=1 io t
The derivative S' of the transform S(xlt) along the direction 11. is given by the relation:
d~ S{(x + Ah1t}I>.=0 = 11 ! [S'(xlt)] ! h(t) dt,
where S'(xlt), h(t) E 'H.. Therefore, the sufficient requirements to be imposed onto the transform S(xlt) are the existence of a function S.(xlt) E L2 [0, 1] such that
~ S{(x + Ah)lt}I>.=o = 101 S.(xlt)h(t) dt
40 Ohapter 2.
for h(t) E 0[0,1] and the existence of the Fredholm determinant of the kernel S*(zlt). We obtain in the result:
r F[z(.)]dwz = r F[z(.) + S(zl·)]Ds.(z)( -1)x lco[o,1] lco[o,1]
2. The conditional Wiener measure in space 0 0 [0, 1] is the Gaussian measure with zero mean and correlation function B(t, s) = min(t, s) - ts. The eigenvalues and the corresponding orthonormal eigenfunctions of the kernel min( t, s) - ts are Ale = ('Irk t2 and </>Ie(t) = V2sin bt, k = 1,2, ... , respectively. The kernel min(t, s) - ts is the Green function for the problem -u"(t) = f(t), u(O) = u(l) = O. The mentioned eigenvalues and eigenfunctions of this kernel may be obtained as the solutions of the Sturm-Liouville problem -AUIl(t) = u(t), u(O) = u(l) = O.
The space 1i for the conditional Wiener measure coincides with the space of functions of the form
a(t) = l g(s) ds - t 101 g(s) ds; g(s) E L2[0, 1]
with scalar product (a, b)1t = J; a'(t)b'(t) dt. Further information will be supplied in the sequel when we shall discuss approxi
mate evaluation of integrals w.r.t. conditional Wiener measure.
3. The Gaussian measure in 0[0, 1] which corresponds to the oscillatory process, or the Ornstein-Uhlenbeck process, has zero mean and correlation function B( t, s) = exp[-It - sll. The eigenvalues and the corresponding orthonormal eigenfunctions of the kernel exp[ -It - s I] are, respectively, Ale = 1+2",~ and
[2(1 + W~)] 1/2. [ 1 ] </>Ie(t) = 2 sm WIe(t--)+b/2 ,k=1,2, ... ,
3+wle 2
where Wle are the positive roots of the equation tan W = -2w/(1 - w2). Note that the kernel exp[-It - sl]/2 is the Green function of the problem -u"(t) + u(t) = f(t), u(O) = u'(O), u'(l) = -u(I).
4. The three considered examples of measures are the special cases of the Gaussian measure with the correlation function
B(t, s) = p[min(t, s )]q[max(t, s )],
where p( u), q( u) are continuously differentiable functions which satisfy the conditions q(u) > 0, p(u) ~ 0, p'(u)q(u) - p(u)q'(u) > 0 on (0,1). In particular,
exp[-It - sl] = exp[min(t,s)]exp[-max(t,s)]. We give two more examples of the correlation functions of this type:
B(t,s) = _l-sin(min(t,s)cos(max(t,s) -I), cos 1
B(t,s) = -hI sh(min(t,s)sh(l - max(t,s)), s 1
which are the Green functions for the problems -ull(t)-u(t) = f(t), u(O) = u'(l) = 0 and ull(t) - u(t) = f(t), u(O) = u(l) = 0, respectively.
5. A special case of the Gaussian measure in Cm [O,l] is the measure with the correlation function
B(t, s) = [(m -1)!]-2l1" (t - u)m-l(s - v)m-l min(u, v) dudv =
lmin(t.") = (m!t2 0 (t - u)m(s - u)m duo
The space 1l which corresponds to this measure is the space of Cm [O,I]-functions
of the form a(t) = f~g(s)ds, where g(s) E L~m)[O,l], with scalar product (a,b)'}t = f~ a(m+l)(s)b(m+l)(s) ds. Here b(s) E 1l and L~m)[O, 1] is the space of functions which
have m-th order derivatives from L2[O, 1], with scalar product
An orthonormal basis in 1l is formed by the functions
( ) 2V2 1 lt )m-l . (2k - 1 ) e,. t = (k ) ( )' (t-u sm --7l'U du, k = 1,2,,,·, 2-171'm-l.o 2
and a measurable linear functional may be written as follows:
d). Let X = Cm be the space of m-variate functions x(t) defined on a subset Tm of the Euclidean space Rm (t E T m C Rm) and let O:i( t), i = 1,2"", n, be linearly independent and continuous on T m functions. Then
42 (;na]pter 2.
JTm
A = [r B(t,s)ai(t)aj(s)dtds]n. JTmxTm i,j=l
If {a;(t)}i=l are the orthonormal eigenfunctions of the integral equation
r B(t,s)y(s) ds = >.y(t), JTm
(2.30)
then A = [>'i8;3T ·-1' where 8;3· is the Cronecker delta, >'i is the eigenvalue of equation ',3-
(2.30) which corresponds to the eigenfunction a;(t), and formula (2.29) assumes the form:
= (211"tn / 2(II >'let1/ 2 r exp -- L -.!!. F(u)d"u. n [ 1 n u2 ]
1e=1 Jan 2 1e=1 >'Ie (2.31)
Let (;rn be the space (;2 of two-variate functions defined on the unit square Q = {O ~ tl ~ 1, 0 ~ t2 ~ I} and let JL = W be the Wiener measure with correlation function
(2.32)
The functions
4> () 2· 2i - 1 . 2j - 1 .. 1 2 ij t = sm -2-1I"t1 sm -2-1I"t2 , ',} = , , ... ,
are the orthonormal on T2 eigenfunctions of kernel (2.32) and
16 >'ij = (2i -1)2(2j -1)211"4
are the corresponding eigenvalues. The equality
1 :c2(t)dt=lL~.t [1 <Pi;(t):c(t)dt]2 Q ',3=1 Q
is valid for almost all (under Wiener measure) functions :c( t) E (;2. This representation and formula (2.31) imply
r exp[>'/2 r :c2(t) dtJdw:c = 8(>.), JC2 JQ
where 00 00 ( 16A ) -1/2
8(A) = IUl 1 - (2i - 1)2(2j - 1)271"4 '
and A is a numeric parameter. We may similarly prove the equality
r exp[~ r x2(t) dt]dw.x = 5(A), 1C2 21Q
where
j=l JX J7I"
for the case of the conditional Wiener measure with the correlation function
e). Consider a space of real-valued sequences 1p. The elements of 1p are the numeric sequences x = (Xl, X2, . .. ,xn, ... ) such that
00
10=1
Any continuous linear functional defined on this space for p > 1 may be written as follows:
00
f(x) = L fkXk, (2.33) 10=1
where f = (fl,f2, ... .In' ... ) E 1q and Ilfll = (2:~1 1!klq)l/q < 00, lip + 1/q = 1. The space dual for 1p is isometric to space 1q.
Any continuous linear functional defined on II may also be written in the form of (2.33) and the dual space for it is isometric to the space 100 of all bounded sequences x = (Xl,X2, ... ,Xn, ... ) with norm Ilxll = SUP1<k<00 IXkl·
The spaces lp, 1 ::; p < 00, are separable and 100 is not. The correlation functional of a Gaussian measure with zero mean in space 1p, 1 ::;
P < 00, is as follows [16]: 00
i,j=:l
where {s;j} is an infinite symmetric positive definite matrix, 2:~1 s~~2 < 00; ~ =
(~1' ~2' ... '~n' . .. ) and "7 = ("71, "72, ... ,"7n, ... ) are the elements of space 1q. The correlation functional K (~, "7) is considerably simpler, if the infinite matrix
S = {s;J is diagonal. Then the bilinear form K(~,"7) becomes as follows:
00 K(~, "7) = L Sii~i"7j.
;=1
44 Chapter 2.
It is this case that will be considered below. The space H is the space lp completed w.r.t. norm 1171IIH = VK(71,71) with scalar
product (e,71)H = K(e,71)· The vectors {~}OO , where vector {4>k}k°-1 is 4>k = (0, ... ,0,1,0, ... ) (only
v·U k=1 - the k-th coordinate is equal to unity), constitute the orthogonal basis in H. In fact, K (Jk, .Jb) = 8ij, and the Fourier coefficients Ck for the element
are
lie -e nll1- = L Skke~ ~ k=n+1
And since e E lq then L:k:n+1 lekl 2 -+ ° as n grows, i.e., lie - enllh- -+ ° as n -+ 00.
The space 11. is defined here as the closure of the linear hull of the set { JSkk4>kHo=1 w.r.t. norm which corresponds to scalar product
It may be immediately proved that 1i is the subspace of lp. In fact, if z E 1i, i.e., 2
its norm IIZII?i = (L:k:1 ;:; )1/2 is finite, then, as we show below, its norm will also be finite in space lp.
The convergence of the series L:k:1;!; implies that S(n) = L:k:1;}; -+ ° as
n -+ 00 and I;#.. < 1 for large k. Therefore, v·U
for p 2:: 1, hence
starting from some value of n. On the other hand, ~k:1 st'2 < 00, i.e., st'2 < 1, or
s"k:t2 > 1 for large k, hence 00
S(n) > L Iz"IP, le=n+1
i.e., ~k:n+1 Iz"IP ---+ 0 as n ---+ 00. Hence, the norm Ilzlllp of a given element z is finite and therefore, z E lp.
Clearly, the elements {y'skk<l>Ie}k:1 constitute an orthonormal basis in 11.. In fact,
and the Fourier coefficients Cle of an element z = (Xl, X2, • •. ,:en, ... ) are
and
as n ---+ 00.
f). Let X = Rn be the n-dimensional Euclidean space. Any linear functional (a, x) defined on Rn is represented by a linear form
n
(a, x) = L alexle· 1e=1
If {ele}i is a basis of Rn then a symmetric bilinear functional K ({, TJ )is represented as follows:
n
K({,TJ) = L bijeiTJj, (2.34) i,j=l
where { = ~i=l {iei, TJ = ~i:1TJiei and B = Ilbijll is a symmetric matrix. Therefore, any Gaussian measure in n-dimensional vector space Rn is specified by
a vector m = (mt, ... ,mn ), called its mean value, and a matrix B = Ilbijll called its correlation matrix.
If the correlation matrix B is nondegenerate then the Gaussian measure is con centrated on the whole space Rn. Otherwise, if the rank of the correlation matrix is equal to 1 < n, then the measure will be concentrated in an I-dimensional hyperplane of~.
The cylindric sets in Rn are of the form:
(2.35)
where at. ... , an E Rn, Al is a Borel set in RI (I :::; n). (The case 1 > n need not be considered, as the elements at. ... ,al would be linearly dependent in the latter case,
46 Chapter 2.
and this cylindric set would coincide with some other one whose I will no longer be greater than the dimension n of the space).
Similarly to the general case, the measure of the set (2.35) is specified by the Gaussian density from formula (1.33), where K(ai, aj) is specified by (2.34).
For example, if bo = bol X bo2 X .•• x bon, where boi are Borel sets of the real axis, then
(where ek = (0, ... ,0,1,0, ... ,0) with the unity in the k-th position) and
where B is the correlation matrix and
n
(B-l(x - m), (x - m)) = 2:: b~;l)(Xi - mi)(Xj - mj)), i,j=l
b~;l) are the elements of the matrix inverse for B. Therefore, an integral over Rn w.r.t. Gaussian measure J.l with mean
m = (ml, ... ,mn ) and correlation matrix B = Ilbijll is as follows:
r F(x)J.l(dx) = JRn
In particular, an integral w.r.t. Gaussian measure over the real axis R is of the form:
1 JOO 1 (u_m)2 -- e- 2 b F(u) du, .)2ib -00
i.e., it is an integral over the real axis of a function F(u) with weight vk-be-!(U-;)2 , which is the density of the normal distribution with mean m and variance b.
The above-mentioned formulae are simplified considerably for the n-dimensional space. For example, in the case of zero mean the translation formula takes the form:
Chapter 3
Integration in Linear Topological Spaces of Some Special Classes
A number of natura

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Functional Integrals: Approximate Evaluation and Applications

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