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GAUSSIAN WHITE NOISE CALCULUS OF GENERALIZED EXPANSION
2002,22B(3): 359-368
GAUSSIAN WHITE NOISE CALCULUS OF GENERALIZED EXPANSION 1 Chen Zeqian ( F* ,'t ft ) Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
Abstract A new framework of Gaussian white noise calculus is established, in line with generalized expansion in [3, 4, 7]. A suitable frame of Fock expansion is presented on
Gaussian generalized expansion functionals being introduced here, which provides the integral kernel operator decomposition of the second quantization of Koopman operators for and its dual, creation chaotic dynamical systems, in terms of annihilation operators operators a;.
at
Key words Gaussian white noise, generalized expansion functional, Fock expansion,
chaotic dynamical systems, Koopman operator 2000 MR Subject Classification
1
60H40
Introduction
Classical white noise calculus has a long history: it can be traced back to R.Wiener in 1920s' and was launched out by T.Hida in 1975 with his lecture notes on generalized Brownian functionals. This new approach toward an infinite dimensional analysis is an important and popular theme which is intensively and extensively studied in many works (see for example, [1, 8, 9, 12] and references therein). These investigations are based on the concept of CH-spaces. The CH-spaces are remarkable objects, but the point is that in applications to some problems of Mathematical Physics this framework is not always appropriate. For example, when we consider the problem of rigging for the Koopman operators of chaotic dynamical systems, the CH-spaces are usually not suitable (for details see [2, 13] and references therein). The problem on extensions of Gaussian white noise calculus without using the CH-spaces is a subject of somewhat current interest. In retrospect, L.K.Hua[7] proposed a new approach to generalized functions. He considered formal Fourier series (1)
as periodic generalized functions no matter they are convergent or not, that is, no restrictions being imposed on the scalars an's. Recently, a definite meaning of the series (1) was given by X.Ding and p.LuO[3]. They also present its generalization in the case of Hilbert spaces and certain concrete constructions in some classical function spaces (see [3, 4] for details). 1 Received
December 22, 2000. E-mail: [email protected]
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Following some of their ideas, in this paper the author will give a new class of Gaussian white noise functionals without using the CH-spaces. In some sense, this paper is a natural continuation of the Brussels-Austin groups' work on generalized spectral decompositions for chaotic dynamical systems (see for example [2,13]). The generalized spectral decomposition of operators on a Hilbert space H is achieved by replacing the initial space H by a dual pair (E, E*), where E is a locally convex space, which is a dense subspace of H endowed with a topology, stronger than the Hilbert space topology. Gel'fand[5] was the first to give a precise meaning to the generalized spectral decompositions based on the CH-spaces. However, when the question of the existence of a generalized spectral decomposition of extensions of Koopman operators for dynamical systems is raised, this issue is delicate because the original Gel'fand theory was constructed for operators which admit a spectral theorem, like normal operators, giving a generalized spectrum identical with the Hilbert space spectrum. The Koopman operator of the chaotic dynamical system, however, either does not admit a spectral theorem or the generalized spectrum is very different from the Hilbert space spectrum. The original Gel'fand theory had to be extended to arbitrary dual pairs (E, E*) of linear topological spaces. In this paper, we will develop an appropriate framework of Gaussian white noise calculus for an arbitrary dual pairs (E, E*) of linear topological spaces with an additional assumption that there exists a unique standard Gaussian measure J.L on the a-algebra generated by cylinder sets of E*. We will generalize the ideas put forth in [3, 4, 7] and show how far these generalizations go towards providing a new framework of Gaussian white noise calculus. Our current techniques are slightly different from those found in [1, 8, 9, 12]; they are largely inspired by generalized expansion in [3]. Roughly speaking, a generalized functional is a series of the form
L(: x Q9n :, un), Un
E Ec~n, n = 0,1,2,···
(2)
n
without additional restrictions imposed on the Un's. Here E cannot be a CH-space. We shall give a definite meaning of the formal series (2) and present a frame of Fock expansion on Gaussian generalized expansion functionals of form (2), which provides the integral kernel operator decomposition of the second quantization of Koopman operators for chaotic dynamical systems, in terms of annihilation operators and its dual, creation operators
at
2
a;.
Generalized Expansion Functionals
Let us consider a standard triplet of linear topological spaces H, E, E*, where H is a real separable Hilbert space and E is a linear topological space, which is embedded in H densely and topologically so that there exists a unique standard Gaussian measure J-l on the a-algebra generated by cylinder sets of E*. For example, E = pr limn Hi, is a nuclear countably Hibertian space or E = ind limn H n is a complete (LB)-space (see [10]), where H n = span {ek' 1 :::; k :::; n} for a certain orthonormal basis (ek)~l of H. Unless otherwise stated, E* always carries the strong dual topology or topology of bounded convergence. In the following we denote the pairing of elements from dual spaces by symbol ( , ), the scalar product in H by ( , ). We assume that the pairing between E and E* is compatible with the scalar product in H, in the sense that (u, h) = (u, h), u E E, h E H.
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Let X and Y be two locally convex spaces. We will denote by X @ Y the projective tensor product X @7r Y when there is no of confusion. Denote by X0 n the closed subspace of X0 n == X Q9 ... Q9 X (n times) spanned by x 0n with x running over X. For F E (X0 n)*, let F" be an element in (X 0n )* uniquely determined by
9n which stands for the group of permutations of {I, ... , n}. The symmetrization F of F is defined by F == -;h L(jEQn F", If F == F, it is called symmetric. We will always denote where
(J
E
by ..:¥*0 n the subspace of (X0 n)* consisting of all symmetric elements. If X is a locally convex space over R, its complexification is denoted by Xc. The canonical bilinear form on X* x X, which is an R-bilinear form, is naturally extended to a C-bilinear form on Xc x Xc which is denoted by the same symbol. Denote by Qn the space of all polynomials of the form n
¢(x) == L(: x 0k :,Uk), x E E*, Uk E E~k, k==O
(3)
equipped with the topology as follows: A sequence {¢m} in Qn converges to rP E Qn if and only if lim.; u k1 == Uk in E3 k for every k == 0,1, ... , n, where n
n
¢(x) == L(: x 0k :,Uk), ¢m(x) == L(: x 0k :,Uk)· k==O k==O Define Q == indlim n Qn, the inductive limit of {Q.n}. Then the space Q becomes isomorphic to k k , that is, Q == EB~==o the topological direct sum of via
E3
E3
00
¢(x) == L(: x 0k :, Uk) +----+ k==O
¢ == (Uk)~O·
Note that only a finite number of Uk are nonzero. The notion of convergence of sequences in this topology on Q is the following: A sequence {¢m} of Q converges to ¢ E Q if and only if there exists n so that rP, rPm E Qn for all m == 1,2, ... and lim m rPm == rP in Qn. Evidently, Q is embedded in L 2 (E*, J-L)c densely and topologically. Now we can introduce the dual space Q* with respect to L 2 (E * , J-L)c . As a result we have constructed the triple The (bilinear) dual pairing << , >> between Q and Q* is connected to the (sesquilinear) inner product (( , )) on L 2 (E * , {l)c by «~,¢
»== ((q>,¢)), q>
E L 2 (E * ,{l )c ,
¢ E Q.
Definition 2.1 Elements ¢ E Q and q> E Q* are called a polynomial functional and a generalized expansion functional, respectively. For the case of that either E == pr limn Hi, is a nuclear countably Hibertian space or E == indlim n Hi, is a complete (LB)-space, it is easy to check that Q is embedded in (E)c densely and topologically and hence (E)c is embedded in Q* densely and topologically.
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By merely repeating the proof of the classical Wiener-Ito-Segal isomorphism theorem (see for example [12, Theorem 2.3.5]), one concludes that ¢ E L2(E*, /-l)c if and only if there exists a unique (un)~==o E f(H c ) such that ¢(x) == E~==o(: x®n :,u n), in the L 2 -sense. In that case,
h nlllunW = II(un)~=ollf(Hc)· 00
114>11
2
=
(4)
n=O
As proved below, a similar assertion is also true for Q*. Theorem 2.1 For each
00
«
< Vk,Uk >, ¢
E
Q,
(5)
k==O
where ¢ and (Uk)k==O are related as in Eq (3), along with convenient Uk == - 0 for k being greater than some n. Conversely, given a sequence (vn)~==o, Vn E E~0n, a generalized expansion functional is defined by (5). Proof By definition, for each n the restriction
«
< Vj,Uj >,
¢ E Qn.
This concludes the proof of the first assertion. Conversely, since Q == ind limn Qn, the functional
L(: x®k :,Vk),
(6)
k==O
which is convergent in Q*, where Vk E E~0k. Conversely, for any sequence Vk E E~0k, k == 0,1, ... the series (6) defines a generalized expansion functional in Q*. Expression (6) is also called the Wiener-Ito expansion of
C TI k==O E*0n oo
.
VIa 00
Given U E
H3 m
L(: X®k :,Vk)
f---t
k==O
and v E
H3 n and 0 :::; k :::; min(m, n), we define the right (left) contraction
U@k V (u@k v) by U Q9k V ==
L
L (u, eu
Q9 ea)(v, eo Q9 ea)e u @ eo,
(7)
uEAm.-k ,OEAn-k aEAk
(U@k V ==
L
L(u,ea@eu)(v,ea@eo)eu@eo,)
uEAm.-k,OEAn-k aEAk
(8)
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It can be proved that the series of the right hand of both (7) and (8) converges in (n+m2k) d 0 Hc an Ilu ®k vii ~ Ilullllvll, Ilu ®k vii ~ Ilullllvll· This yields that (u, v) -+ U ®k V ((u, v) -+ u ®k v) is a continuous bilinear map from H3 m x H3 n -+ H3(n+m-2k) and hence the definition of the contraction is independent on the choice of the basis (ek)~l in H. Moreover, if u E H2 m and v E H2 n we denote by U@kV the symmetrization of u ® k v. By the Wiener product formula (see for example [12, §3.5]), it is easy to check that Q is closed under pointwise multiplication and the multiplication is a continuous bilinear map from Q x Q into Q. Consequently, for every ¢ E Q and E Q* the mapping
-+«
, ¢
»,
Q,
is a continuous linear functional on Q. Hence there exists a unique expansion functional '1J == '1J (, ¢) E Q* such that « '1J,
An appeal to Corollary 2.1 immediately yields that Corollary 2.2 The bilinear map (, ¢) -+ ¢ becomes a separately continuous bilinear map from Q* x Q into Q*. In particular, each E Q*, as multiplication, is a continuous linear operator from Q into Q*. Definition 2.3 For each y E E* we define the translation operator T y by Ty¢(x)
= ¢(x + y),
¢ E Q, x E E*.
By [12, Lemma 4.2.2] we easily conclude that Ty is a continuous operator on Q. Moreover, we have Theorem 2.2 For any x, y E E* the following limit ~( ) -. 1· Toy¢(x) - ¢(x) D y'fJ X - . im () O-tO
exists for all ¢ E Q. In particular, the mapping ¢ -+ D y ¢ is a continuous map from Q into Q and Dy(¢'l/J) = (Dy¢)'lj; + ¢(Dy'lj;), ¢, 'l/J E Q. Proof Suppose that ¢(x) E Q which admits a Wiener-Ito expansion: n
¢(x)
= L(: x 0 k :,Uk),
Uk E E®k,
X
E E*.
k=O
Since: (x + y)0 n := EZ=o : x 0 k : @y0{n-k) for all x, y E E* and every n = 0,1,' .. ,(see for example [12, Lemma 4.2.2]) we follow that n
Dy¢(x) =
L k(: x 0 {k- l ) :, y@lUk). k=O
(9)
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This completes the proof. Theorem 2.3
If the Wiener-Ito expansion of ~ E Q* is given as ~(x) == L~o(: x®k :
,Vk), then for any Y E E*, the dual operator D; of D y is continuous on Q* and 00
D;~(x) == L(: x®(k+l) :,Y®Vk).
(10)
k=O
Proof Since« D;~, ¢ »==« <1>, D y ¢ » for all ¢ E Q, it concludes the proof. Corollary 2.3 Let ¢ E Q be given with the Wiener-Ito expansion ¢(x) == L~o(: x®k : , Uk),
where only a finite number of
are nonzero. Then, for Yl, ... , Yrn E E* one has that
Uk'S
(11) and
[D y1,Dy2] == 0, For ~,'lt E Q* with ~
I"V
{Vk}, W ~oW
I"V
[D;l' D;2] ==
o.
{Wk} we define the Wick product
I"V
{h k},
L
hk ==
~ o
W by
Vi®Wj.
i+j=k
Then one concludes easily that Q* and Q are both closed under the Wick product and in particular, the Wick product
(~,
w)
~ ~
o W is a continuous bilinear map from Q* x Q* to Q*
and Q x Q to Q, respectively.
3
Foce Expansion
We now fix the notation used throughout the rest of the paper. Let T be a topological space equipped with a a-finite Borel measure u and consider the real Hilbert space H == L 2(T, v). A linear manifold E in H is a linear topological space satisfying the following conditions: (HI) E is embedded in H densely and topologically so that there exists a unique standard Gaussian measure J.L on the a-algebra generated by cylinder sets of E*, the strong dual space of E. (H2) For each function
U
E
E there exists a unique continuous function il on T such that
u(t) == u(t) for v-a.e. t E T. ·(H3) For each t E T the evaluation map 8t functional, Le., 8t E E*.
: U
~ u(t), u E E, is a continuous linear
(H4) The map, t ~ 8t , t E T, is continuous with respect to the strong dual topology of E*. Since E is embedded in H densely and topologically, E®n is embedded in H®n densely and topologically for each n == 1,2, ... Then we further suppose that (H5) For 0 ::; k ::; min(m,n), (u,v) ~ u 0k map from E3 m x E3 n ~ E~(n+m-2k). For a fixed t E T the differential operator
a; :
V
((u,v) ~
U
0 k v) is a continuous bilinear
at == D 8t is called, in usual Fock space language,
an annihilation operator and its dual Q* ~ Q* a creation operator. Proposition 3.1 (a) For u E E~(l+n), v E E~(m+n) and W E (E~(l+m))* the maps v ~<
W, U
0
n V
>,
V ~< W, U Q?)n V
>
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are both continuous on E~(m+n). Therefore there exist two elements in (E~(m+n))*, denoted by w 0l U and w 0 l U respectively, such that
0l U,V
>==< w,u 0 n u >» < w 0 l U,V >==< W,U 0 n V > .
(b) For s, t E T, [as, at] == 0, [a;, a;] == o. (c) For every ¢ E Q, (at + a;)Q(x) ==<: x :,8t (d) For ¢, 'ljJ E Q we denote by TJ
Then
TJ
a;l ···8;,8
-t«
)
> Q(x). t1
···8t rn ¢,'l/J »
·
so that
=L 00
17,1/;
(m
+ k)!(l + k)!
k'
k=O
Ul+k
0k vm+k,
where ¢(x) == L~=o <: x 0 k :, Vk > and 'ljJ(x) == L~=o <: x 0 k :, Uk > . Proof The fact (a) is immediate by assumption (H5), while (b) follows from Corollary 2.3. Also, the fact (d) can be proved by the same argument of [12, Lemma 4.3.1] and we omit the details. To prove (c), it suffice to check the identity (a) for ¢(x) ==<: x 0 n :,u >, U E E3 n. First note that if
==<:
x 0 m :, w
then
>,
¢(x)
==<:
x 0 n :, U >, w E E~&;m, U E E2 n,
min(m,n)
(¢
==
L
k!(r)(~)
<: x 0(m+n-2k)
:,W0k U >
.
k=O
Hence one concludes that
On the other hand, a simple application of (9) and (10) yields that at¢(x)
== n <: x 0(n-l)
:,6t®lU
>,
at¢(x)
==<: x 0(n+l) :,8t 0 u > .
Then (c) is immediate and the proof is complete. Proposition 3.2 With each ~ E (E0(l+m))* we shall associate
3 1,m(I\'; )¢ (x ) for ¢
r-o.J
=
f
n=O
(n :,m)! <: x0(l+n) :, I\'; 0 m u n+ m >
(12)
{un}. Then 3l,m(K) is a continuous linear operator from Q into Q* such that (13)
Proof By Proposition 3.1 (a) 3l,m(~) is well defined by (12) and a continuous linear operator from Q into Q*. For the proof of (13) we let 'ljJ {v n } . Then, by definition, r-o.J
(m+n)!(l+n)! n! ",,00 (m+n)!(l+n)! L..Jn=O n! ",,00
L..Jn=O
< ~ 0 m Um+n' Vl+ n > < ~,Vl+n 0 n u m+ n >
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from which (13) follows. The proof is complete. The operator 3l,m(~) is thus defined through two canonical bilinear forms:
This suggests us to employ a formal integral expression:
We call 3l,m(~) an integral kernel operator with the kernel distribution «. The kernel distribution is not uniquely determined due to Proposition 3.1 (b). For the uniqueness we need a "partially symmetrized" kernel. We put
With this notation and (13) we may prove the following assertions easily (see for example [12, Proposition 4.3.4-6]). Proposition 3.3 (1) 3l,m(~) == 3l,m(Sl,m(~)) for r: E (E~(l+m»)*. (2) Let ~ E (E~(l+m»)*. Then 3l,m(~) == 0 if and only if 3l,m(Sl,m(~)) == o. (3) Assume that 3l,m(~) == 3l',m'(K/) t= 0 for ~ E (E~(l+m»)* and ~' E (E~(l'+m'»)*. Then l == l', m == m ', and Sl,m(~) == Sl',m,(~/). For each ~ E (E®(l+m»)* it can be proved that there exists a unique integral kernel operator 3l,m(~)*
from Q into Q* such that
In particular, if 3 l ,m '='l ,m (x)" I-J
-1 -
(~)
T'+m
is given as in (14), then
~(Sl·" ..
Sl , tl , ... ,m t )8t*1 ... 8t*m 881 ···88, ds l··· dSldt l ... dt m· .
Theorem 3.1 Let ~ E (E~(l+m»)*. Then 3l,m(~) can be extended to a continuous linear operator from L 2(E * , {l )c into Q*. Furthermore, if 3l,m(~) admits an extension to a bounded operator on L 2(E * , {l)c, then Sl,m(~) == 0 or l == m == o. Namely, except scalar operators no integral kernel operator admits an extension to a bounded operator on £2(E*, {l)c. Proof By Theorem 2.1 the right hand of (12) is well defined for ¢ {un} in L 2(E * , {l)c . Thus 3l,m(~) can be defined on £2(E*, {l)c with values in Q*. The continuity of 3l,m(~) from L 2(E * , {l )c into Q* concludes from Corollary 2.1. This proves the first assertion. "'-I
For the proof of the second assertion we merely need to repeat the proof of [12, Proposition 4.5.5] and omit the details. Now we are in a position to state Fock expansion for chaotic dynamical systems. Recall that the observable phase functions of dynamical systems evolve according to the corresponding Koopman operator
v j(x) ==
j(Sx),
(15)
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where S is an endomorphism or an ~utomorphism of the measure space (T, v), and f is a phase function in £2 (T, v). The idea of using operator theory for the study of dynamical systems can be traced to Koopman and von Neumann in 1930s' and was extensively used thereafter because the corresponding Koopman operator carries all dynamical information. Koopman showed in 1931 that for any measure preserving dynamic S, the operator V in (15) is isometric on the space of square integrable phase functions, the converse of which was proved by Goodrich, Gustafson and Misra in [6]. In the case of invertible dynamics the Koopman operator is unitary. We consult [11] for the details. During the last ten years the Brussels-Austin groups (see [2, 13] and references therein) has demonstrated that for the chaotic dynamical systems there exist spectral decompositions of the evolution in terms of resonance and resonance states which appear as eigenvalues and eigenprojections of the evolution operator. These new spectral decompositions define an extension of the evolution to suitable dual pairs (E, E*) of linear topological spaces with E c £2 C E*. This procedure is referred to as rigging, and the triple
E C £2 C E* is called a rigged Hilbert space. Summarizing for the reader's convenience, a dual pair (E, E*) of linear topological spaces with E C £2 C E* constitutes a rigged Hilbert space for the linear endomorphism V of the Hilbert £2 if the following conditions are satisfied: (i) E is a dense subspace of £2.
E is complete and its topology is stronger than the one induced by £2. (iii) E is invariant with respect to the adjoint V* of V, i.e., V* E c E. (iv) The adjoint V* is continuous on E. The choice of the test function space E depends on the specific operator V and generally is not a nuclear CH-space (see [2, 13] for details). Theorem 3.2 Let V be the Koopman operator of a chaotic dynamical system and let I'(V) and df(V) the second quantization and differential second quantization of V, respectively. Then, (ii)
r(V)?
=
f: ~!3n,n([I0n
n
0 (V - I)0 ]*Tn )¢
(16)
n=O
for all ¢ E £2(E*, /-L)e, where the right hand of (16) converges in Q*, and
df(V)
== 3 1,1([1 0
11 ]*7).
(17)
Here t« E (E@2n)* is defined by < Tn,U@n @v@n >==< ii,» .>", u,» E E. Proof By Theorem 3.1 the right hand of (16) is well defined. By definition one has that r(V)¢u == ¢vu for U E He. Hence,
«r(v)¢u,¢v
»== e,
U,V
E He.
Then the verification of (16) is a simple computation. Since
L 00
df(V)¢(x)
==
n=O
<: x@n
:"n(V)u n >,
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I'O(V)
== 0,
I'n(V)
==
L [Ok ® V ® [0(n-k-l),
Vo1.22 Ser.B
n-l
n ~ 1,
k=O
one concludes that
«dr(V)¢u,¢v
»==< Vu,v > e, u,v
E He.
This proves the identity (17) and completes the proof. By Theorem 3.1, except that S is the identity operator on T the Fock expansion of the Koopman operator is always an infinite series of unbounded operator from L 2 (E*, J-l)e into Q*. Also, dr (V) is often viewed as a unbounded operator defined on a dense subspace of L 2 (E* , J-l). However, by (17) dr(V) can be extended to a continuous linear operator on L 2 (E * , J-l) with values in Q*. Acknonledgement
I am very grateful to Dr.X.Wang for helpful discussion. References
2 3 4 5 6 7 8 9 10 11 12 13
Albeverio S, Daletsky Yu L, Kondratiev Yu G. Non-Gaussian infinite dimensional analysis. J Funct Anal, 1996, 138: 311-350 Antoniou I, Suchanecki Z. Spectral decomposition and extended formulation of unstable dynamical systems. In: Atmanspacher H, Ruhnau E, eds. Time, Temporality, Now. Springer, 1998. 293-325 Ding Xiaqi, Luo Peizhu. Generalized expansion in Hilbert space. Acta Math Sci, 1999, 19: 241-250 Ding Xiaqi, Luo Peizhu. Weak convergence of some series. Acta Math Sci, 2000, 20B: 433-442 Gel'fand I M, Vilenkin N Ya, Generalized functions IV. New York: Academic Press, 1964 Goodrich K, Gustafson K, Misra B. On converse to Koopman's lemma. Physica A, 1980, 102: 379-388 Hua Luoken. Introduction to generalized functions. Math Achievements, 1963, 6(4) (in Chinese) Huang Zhiyuan, Van Jiaan. Introduction to infinite dimensional stochastic analysis. Beijing: Academic Press, 1999(in Chinese) Kondratiev Y G, Sila J L, Streit L. Generalized appel system. Methods Punct Anal & Topology, 1997, 3: 28-61 Kothe G. Topological vector spaces I. Berlin: Springer-Verlag, 1983 Lasota A, Mackey M C. Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics. 2th Edition. New York: Springer-Verlag, 1994 Obata N. White noise calculus and Fock space. Berlin: Springer-Verlag, 1994 Suchanecki Z, Antoniou I, Tasaki S, Bandtlow 0 F. Rigged Hilbert spaces for chaotic dynamical systems. J Math Phys, 1996, 37(11): 5837-5847
GAUSSIAN WHITE NOISE CALCULUS OF GENERALIZED EXPANSION 1 Chen Zeqian ( F* ,'t ft ) Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
Abstract A new framework of Gaussian white noise calculus is established, in line with generalized expansion in [3, 4, 7]. A suitable frame of Fock expansion is presented on
Gaussian generalized expansion functionals being introduced here, which provides the integral kernel operator decomposition of the second quantization of Koopman operators for and its dual, creation chaotic dynamical systems, in terms of annihilation operators operators a;.
at
Key words Gaussian white noise, generalized expansion functional, Fock expansion,
chaotic dynamical systems, Koopman operator 2000 MR Subject Classification
1
60H40
Introduction
Classical white noise calculus has a long history: it can be traced back to R.Wiener in 1920s' and was launched out by T.Hida in 1975 with his lecture notes on generalized Brownian functionals. This new approach toward an infinite dimensional analysis is an important and popular theme which is intensively and extensively studied in many works (see for example, [1, 8, 9, 12] and references therein). These investigations are based on the concept of CH-spaces. The CH-spaces are remarkable objects, but the point is that in applications to some problems of Mathematical Physics this framework is not always appropriate. For example, when we consider the problem of rigging for the Koopman operators of chaotic dynamical systems, the CH-spaces are usually not suitable (for details see [2, 13] and references therein). The problem on extensions of Gaussian white noise calculus without using the CH-spaces is a subject of somewhat current interest. In retrospect, L.K.Hua[7] proposed a new approach to generalized functions. He considered formal Fourier series (1)
as periodic generalized functions no matter they are convergent or not, that is, no restrictions being imposed on the scalars an's. Recently, a definite meaning of the series (1) was given by X.Ding and p.LuO[3]. They also present its generalization in the case of Hilbert spaces and certain concrete constructions in some classical function spaces (see [3, 4] for details). 1 Received
December 22, 2000. E-mail: [email protected]
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Following some of their ideas, in this paper the author will give a new class of Gaussian white noise functionals without using the CH-spaces. In some sense, this paper is a natural continuation of the Brussels-Austin groups' work on generalized spectral decompositions for chaotic dynamical systems (see for example [2,13]). The generalized spectral decomposition of operators on a Hilbert space H is achieved by replacing the initial space H by a dual pair (E, E*), where E is a locally convex space, which is a dense subspace of H endowed with a topology, stronger than the Hilbert space topology. Gel'fand[5] was the first to give a precise meaning to the generalized spectral decompositions based on the CH-spaces. However, when the question of the existence of a generalized spectral decomposition of extensions of Koopman operators for dynamical systems is raised, this issue is delicate because the original Gel'fand theory was constructed for operators which admit a spectral theorem, like normal operators, giving a generalized spectrum identical with the Hilbert space spectrum. The Koopman operator of the chaotic dynamical system, however, either does not admit a spectral theorem or the generalized spectrum is very different from the Hilbert space spectrum. The original Gel'fand theory had to be extended to arbitrary dual pairs (E, E*) of linear topological spaces. In this paper, we will develop an appropriate framework of Gaussian white noise calculus for an arbitrary dual pairs (E, E*) of linear topological spaces with an additional assumption that there exists a unique standard Gaussian measure J.L on the a-algebra generated by cylinder sets of E*. We will generalize the ideas put forth in [3, 4, 7] and show how far these generalizations go towards providing a new framework of Gaussian white noise calculus. Our current techniques are slightly different from those found in [1, 8, 9, 12]; they are largely inspired by generalized expansion in [3]. Roughly speaking, a generalized functional is a series of the form
L(: x Q9n :, un), Un
E Ec~n, n = 0,1,2,···
(2)
n
without additional restrictions imposed on the Un's. Here E cannot be a CH-space. We shall give a definite meaning of the formal series (2) and present a frame of Fock expansion on Gaussian generalized expansion functionals of form (2), which provides the integral kernel operator decomposition of the second quantization of Koopman operators for chaotic dynamical systems, in terms of annihilation operators and its dual, creation operators
at
2
a;.
Generalized Expansion Functionals
Let us consider a standard triplet of linear topological spaces H, E, E*, where H is a real separable Hilbert space and E is a linear topological space, which is embedded in H densely and topologically so that there exists a unique standard Gaussian measure J-l on the a-algebra generated by cylinder sets of E*. For example, E = pr limn Hi, is a nuclear countably Hibertian space or E = ind limn H n is a complete (LB)-space (see [10]), where H n = span {ek' 1 :::; k :::; n} for a certain orthonormal basis (ek)~l of H. Unless otherwise stated, E* always carries the strong dual topology or topology of bounded convergence. In the following we denote the pairing of elements from dual spaces by symbol ( , ), the scalar product in H by ( , ). We assume that the pairing between E and E* is compatible with the scalar product in H, in the sense that (u, h) = (u, h), u E E, h E H.
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Let X and Y be two locally convex spaces. We will denote by X @ Y the projective tensor product X @7r Y when there is no of confusion. Denote by X0 n the closed subspace of X0 n == X Q9 ... Q9 X (n times) spanned by x 0n with x running over X. For F E (X0 n)*, let F" be an element in (X 0n )* uniquely determined by
9n which stands for the group of permutations of {I, ... , n}. The symmetrization F of F is defined by F == -;h L(jEQn F", If F == F, it is called symmetric. We will always denote where
(J
E
by ..:¥*0 n the subspace of (X0 n)* consisting of all symmetric elements. If X is a locally convex space over R, its complexification is denoted by Xc. The canonical bilinear form on X* x X, which is an R-bilinear form, is naturally extended to a C-bilinear form on Xc x Xc which is denoted by the same symbol. Denote by Qn the space of all polynomials of the form n
¢(x) == L(: x 0k :,Uk), x E E*, Uk E E~k, k==O
(3)
equipped with the topology as follows: A sequence {¢m} in Qn converges to rP E Qn if and only if lim.; u k1 == Uk in E3 k for every k == 0,1, ... , n, where n
n
¢(x) == L(: x 0k :,Uk), ¢m(x) == L(: x 0k :,Uk)· k==O k==O Define Q == indlim n Qn, the inductive limit of {Q.n}. Then the space Q becomes isomorphic to k k , that is, Q == EB~==o the topological direct sum of via
E3
E3
00
¢(x) == L(: x 0k :, Uk) +----+ k==O
¢ == (Uk)~O·
Note that only a finite number of Uk are nonzero. The notion of convergence of sequences in this topology on Q is the following: A sequence {¢m} of Q converges to ¢ E Q if and only if there exists n so that rP, rPm E Qn for all m == 1,2, ... and lim m rPm == rP in Qn. Evidently, Q is embedded in L 2 (E*, J-L)c densely and topologically. Now we can introduce the dual space Q* with respect to L 2 (E * , J-L)c . As a result we have constructed the triple The (bilinear) dual pairing << , >> between Q and Q* is connected to the (sesquilinear) inner product (( , )) on L 2 (E * , {l)c by «~,¢
»== ((q>,¢)), q>
E L 2 (E * ,{l )c ,
¢ E Q.
Definition 2.1 Elements ¢ E Q and q> E Q* are called a polynomial functional and a generalized expansion functional, respectively. For the case of that either E == pr limn Hi, is a nuclear countably Hibertian space or E == indlim n Hi, is a complete (LB)-space, it is easy to check that Q is embedded in (E)c densely and topologically and hence (E)c is embedded in Q* densely and topologically.
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By merely repeating the proof of the classical Wiener-Ito-Segal isomorphism theorem (see for example [12, Theorem 2.3.5]), one concludes that ¢ E L2(E*, /-l)c if and only if there exists a unique (un)~==o E f(H c ) such that ¢(x) == E~==o(: x®n :,u n), in the L 2 -sense. In that case,
h nlllunW = II(un)~=ollf(Hc)· 00
114>11
2
=
(4)
n=O
As proved below, a similar assertion is also true for Q*. Theorem 2.1 For each
00
«
< Vk,Uk >, ¢
E
Q,
(5)
k==O
where ¢ and (Uk)k==O are related as in Eq (3), along with convenient Uk == - 0 for k being greater than some n. Conversely, given a sequence (vn)~==o, Vn E E~0n, a generalized expansion functional is defined by (5). Proof By definition, for each n the restriction
«
< Vj,Uj >,
¢ E Qn.
This concludes the proof of the first assertion. Conversely, since Q == ind limn Qn, the functional
L(: x®k :,Vk),
(6)
k==O
which is convergent in Q*, where Vk E E~0k. Conversely, for any sequence Vk E E~0k, k == 0,1, ... the series (6) defines a generalized expansion functional in Q*. Expression (6) is also called the Wiener-Ito expansion of
C TI k==O E*0n oo
.
VIa 00
Given U E
H3 m
L(: X®k :,Vk)
f---t
k==O
and v E
H3 n and 0 :::; k :::; min(m, n), we define the right (left) contraction
U@k V (u@k v) by U Q9k V ==
L
L (u, eu
Q9 ea)(v, eo Q9 ea)e u @ eo,
(7)
uEAm.-k ,OEAn-k aEAk
(U@k V ==
L
L(u,ea@eu)(v,ea@eo)eu@eo,)
uEAm.-k,OEAn-k aEAk
(8)
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It can be proved that the series of the right hand of both (7) and (8) converges in (n+m2k) d 0 Hc an Ilu ®k vii ~ Ilullllvll, Ilu ®k vii ~ Ilullllvll· This yields that (u, v) -+ U ®k V ((u, v) -+ u ®k v) is a continuous bilinear map from H3 m x H3 n -+ H3(n+m-2k) and hence the definition of the contraction is independent on the choice of the basis (ek)~l in H. Moreover, if u E H2 m and v E H2 n we denote by U@kV the symmetrization of u ® k v. By the Wiener product formula (see for example [12, §3.5]), it is easy to check that Q is closed under pointwise multiplication and the multiplication is a continuous bilinear map from Q x Q into Q. Consequently, for every ¢ E Q and E Q* the mapping
-+«
, ¢
»,
Q,
is a continuous linear functional on Q. Hence there exists a unique expansion functional '1J == '1J (, ¢) E Q* such that « '1J,
, ¢
,¢) = ¢ = ¢. Definition 2.2 The product ¢ = ¢ is defined uniquely by the formula:
«
¢, sp
»=«
, ¢
»,
E Q*, ¢,
An appeal to Corollary 2.1 immediately yields that Corollary 2.2 The bilinear map (, ¢) -+ ¢ becomes a separately continuous bilinear map from Q* x Q into Q*. In particular, each E Q*, as multiplication, is a continuous linear operator from Q into Q*. Definition 2.3 For each y E E* we define the translation operator T y by Ty¢(x)
= ¢(x + y),
¢ E Q, x E E*.
By [12, Lemma 4.2.2] we easily conclude that Ty is a continuous operator on Q. Moreover, we have Theorem 2.2 For any x, y E E* the following limit ~( ) -. 1· Toy¢(x) - ¢(x) D y'fJ X - . im () O-tO
exists for all ¢ E Q. In particular, the mapping ¢ -+ D y ¢ is a continuous map from Q into Q and Dy(¢'l/J) = (Dy¢)'lj; + ¢(Dy'lj;), ¢, 'l/J E Q. Proof Suppose that ¢(x) E Q which admits a Wiener-Ito expansion: n
¢(x)
= L(: x 0 k :,Uk),
Uk E E®k,
X
E E*.
k=O
Since: (x + y)0 n := EZ=o : x 0 k : @y0{n-k) for all x, y E E* and every n = 0,1,' .. ,(see for example [12, Lemma 4.2.2]) we follow that n
Dy¢(x) =
L k(: x 0 {k- l ) :, y@lUk). k=O
(9)
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This completes the proof. Theorem 2.3
If the Wiener-Ito expansion of ~ E Q* is given as ~(x) == L~o(: x®k :
,Vk), then for any Y E E*, the dual operator D; of D y is continuous on Q* and 00
D;~(x) == L(: x®(k+l) :,Y®Vk).
(10)
k=O
Proof Since« D;~, ¢ »==« <1>, D y ¢ » for all ¢ E Q, it concludes the proof. Corollary 2.3 Let ¢ E Q be given with the Wiener-Ito expansion ¢(x) == L~o(: x®k : , Uk),
where only a finite number of
are nonzero. Then, for Yl, ... , Yrn E E* one has that
Uk'S
(11) and
[D y1,Dy2] == 0, For ~,'lt E Q* with ~
I"V
{Vk}, W ~oW
I"V
[D;l' D;2] ==
o.
{Wk} we define the Wick product
I"V
{h k},
L
hk ==
~ o
W by
Vi®Wj.
i+j=k
Then one concludes easily that Q* and Q are both closed under the Wick product and in particular, the Wick product
(~,
w)
~ ~
o W is a continuous bilinear map from Q* x Q* to Q*
and Q x Q to Q, respectively.
3
Foce Expansion
We now fix the notation used throughout the rest of the paper. Let T be a topological space equipped with a a-finite Borel measure u and consider the real Hilbert space H == L 2(T, v). A linear manifold E in H is a linear topological space satisfying the following conditions: (HI) E is embedded in H densely and topologically so that there exists a unique standard Gaussian measure J.L on the a-algebra generated by cylinder sets of E*, the strong dual space of E. (H2) For each function
U
E
E there exists a unique continuous function il on T such that
u(t) == u(t) for v-a.e. t E T. ·(H3) For each t E T the evaluation map 8t functional, Le., 8t E E*.
: U
~ u(t), u E E, is a continuous linear
(H4) The map, t ~ 8t , t E T, is continuous with respect to the strong dual topology of E*. Since E is embedded in H densely and topologically, E®n is embedded in H®n densely and topologically for each n == 1,2, ... Then we further suppose that (H5) For 0 ::; k ::; min(m,n), (u,v) ~ u 0k map from E3 m x E3 n ~ E~(n+m-2k). For a fixed t E T the differential operator
a; :
V
((u,v) ~
U
0 k v) is a continuous bilinear
at == D 8t is called, in usual Fock space language,
an annihilation operator and its dual Q* ~ Q* a creation operator. Proposition 3.1 (a) For u E E~(l+n), v E E~(m+n) and W E (E~(l+m))* the maps v ~<
W, U
0
n V
>,
V ~< W, U Q?)n V
>
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are both continuous on E~(m+n). Therefore there exist two elements in (E~(m+n))*, denoted by w 0l U and w 0 l U respectively, such that
0l U,V
>==< w,u 0 n u >» < w 0 l U,V >==< W,U 0 n V > .
(b) For s, t E T, [as, at] == 0, [a;, a;] == o. (c) For every ¢ E Q, (at + a;)Q(x) ==<: x :,8t (d) For ¢, 'ljJ E Q we denote by TJ
Then
TJ
a;l ···8;,8
-t«
)
> Q(x). t1
···8t rn ¢,'l/J »
·
so that
=L 00
17,1/;
(m
+ k)!(l + k)!
k'
k=O
Ul+k
0k vm+k,
where ¢(x) == L~=o <: x 0 k :, Vk > and 'ljJ(x) == L~=o <: x 0 k :, Uk > . Proof The fact (a) is immediate by assumption (H5), while (b) follows from Corollary 2.3. Also, the fact (d) can be proved by the same argument of [12, Lemma 4.3.1] and we omit the details. To prove (c), it suffice to check the identity (a) for ¢(x) ==<: x 0 n :,u >, U E E3 n. First note that if
==<:
x 0 m :, w
then
>,
¢(x)
==<:
x 0 n :, U >, w E E~&;m, U E E2 n,
min(m,n)
(¢
==
L
k!(r)(~)
<: x 0(m+n-2k)
:,W0k U >
.
k=O
Hence one concludes that
On the other hand, a simple application of (9) and (10) yields that at¢(x)
== n <: x 0(n-l)
:,6t®lU
>,
at¢(x)
==<: x 0(n+l) :,8t 0 u > .
Then (c) is immediate and the proof is complete. Proposition 3.2 With each ~ E (E0(l+m))* we shall associate
3 1,m(I\'; )¢ (x ) for ¢
r-o.J
=
f
n=O
(n :,m)! <: x0(l+n) :, I\'; 0 m u n+ m >
(12)
{un}. Then 3l,m(K) is a continuous linear operator from Q into Q* such that (13)
Proof By Proposition 3.1 (a) 3l,m(~) is well defined by (12) and a continuous linear operator from Q into Q*. For the proof of (13) we let 'ljJ {v n } . Then, by definition, r-o.J
(m+n)!(l+n)! n! ",,00 (m+n)!(l+n)! L..Jn=O n! ",,00
L..Jn=O
< ~ 0 m Um+n' Vl+ n > < ~,Vl+n 0 n u m+ n >
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from which (13) follows. The proof is complete. The operator 3l,m(~) is thus defined through two canonical bilinear forms:
This suggests us to employ a formal integral expression:
We call 3l,m(~) an integral kernel operator with the kernel distribution «. The kernel distribution is not uniquely determined due to Proposition 3.1 (b). For the uniqueness we need a "partially symmetrized" kernel. We put
With this notation and (13) we may prove the following assertions easily (see for example [12, Proposition 4.3.4-6]). Proposition 3.3 (1) 3l,m(~) == 3l,m(Sl,m(~)) for r: E (E~(l+m»)*. (2) Let ~ E (E~(l+m»)*. Then 3l,m(~) == 0 if and only if 3l,m(Sl,m(~)) == o. (3) Assume that 3l,m(~) == 3l',m'(K/) t= 0 for ~ E (E~(l+m»)* and ~' E (E~(l'+m'»)*. Then l == l', m == m ', and Sl,m(~) == Sl',m,(~/). For each ~ E (E®(l+m»)* it can be proved that there exists a unique integral kernel operator 3l,m(~)*
from Q into Q* such that
In particular, if 3 l ,m '='l ,m (x)" I-J
-1 -
(~)
T'+m
is given as in (14), then
~(Sl·" ..
Sl , tl , ... ,m t )8t*1 ... 8t*m 881 ···88, ds l··· dSldt l ... dt m· .
Theorem 3.1 Let ~ E (E~(l+m»)*. Then 3l,m(~) can be extended to a continuous linear operator from L 2(E * , {l )c into Q*. Furthermore, if 3l,m(~) admits an extension to a bounded operator on L 2(E * , {l)c, then Sl,m(~) == 0 or l == m == o. Namely, except scalar operators no integral kernel operator admits an extension to a bounded operator on £2(E*, {l)c. Proof By Theorem 2.1 the right hand of (12) is well defined for ¢ {un} in L 2(E * , {l)c . Thus 3l,m(~) can be defined on £2(E*, {l)c with values in Q*. The continuity of 3l,m(~) from L 2(E * , {l )c into Q* concludes from Corollary 2.1. This proves the first assertion. "'-I
For the proof of the second assertion we merely need to repeat the proof of [12, Proposition 4.5.5] and omit the details. Now we are in a position to state Fock expansion for chaotic dynamical systems. Recall that the observable phase functions of dynamical systems evolve according to the corresponding Koopman operator
v j(x) ==
j(Sx),
(15)
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where S is an endomorphism or an ~utomorphism of the measure space (T, v), and f is a phase function in £2 (T, v). The idea of using operator theory for the study of dynamical systems can be traced to Koopman and von Neumann in 1930s' and was extensively used thereafter because the corresponding Koopman operator carries all dynamical information. Koopman showed in 1931 that for any measure preserving dynamic S, the operator V in (15) is isometric on the space of square integrable phase functions, the converse of which was proved by Goodrich, Gustafson and Misra in [6]. In the case of invertible dynamics the Koopman operator is unitary. We consult [11] for the details. During the last ten years the Brussels-Austin groups (see [2, 13] and references therein) has demonstrated that for the chaotic dynamical systems there exist spectral decompositions of the evolution in terms of resonance and resonance states which appear as eigenvalues and eigenprojections of the evolution operator. These new spectral decompositions define an extension of the evolution to suitable dual pairs (E, E*) of linear topological spaces with E c £2 C E*. This procedure is referred to as rigging, and the triple
E C £2 C E* is called a rigged Hilbert space. Summarizing for the reader's convenience, a dual pair (E, E*) of linear topological spaces with E C £2 C E* constitutes a rigged Hilbert space for the linear endomorphism V of the Hilbert £2 if the following conditions are satisfied: (i) E is a dense subspace of £2.
E is complete and its topology is stronger than the one induced by £2. (iii) E is invariant with respect to the adjoint V* of V, i.e., V* E c E. (iv) The adjoint V* is continuous on E. The choice of the test function space E depends on the specific operator V and generally is not a nuclear CH-space (see [2, 13] for details). Theorem 3.2 Let V be the Koopman operator of a chaotic dynamical system and let I'(V) and df(V) the second quantization and differential second quantization of V, respectively. Then, (ii)
r(V)?
=
f: ~!3n,n([I0n
n
0 (V - I)0 ]*Tn )¢
(16)
n=O
for all ¢ E £2(E*, /-L)e, where the right hand of (16) converges in Q*, and
df(V)
== 3 1,1([1 0
11 ]*7).
(17)
Here t« E (E@2n)* is defined by < Tn,U@n @v@n >==< ii,» .>", u,» E E. Proof By Theorem 3.1 the right hand of (16) is well defined. By definition one has that r(V)¢u == ¢vu for U E He. Hence,
«r(v)¢u,¢v
»== e
U,V
E He.
Then the verification of (16) is a simple computation. Since
L 00
df(V)¢(x)
==
n=O
<: x@n
:"n(V)u n >,
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I'O(V)
== 0,
I'n(V)
==
L [Ok ® V ® [0(n-k-l),
Vo1.22 Ser.B
n-l
n ~ 1,
k=O
one concludes that
«dr(V)¢u,¢v
»==< Vu,v > e
E He.
This proves the identity (17) and completes the proof. By Theorem 3.1, except that S is the identity operator on T the Fock expansion of the Koopman operator is always an infinite series of unbounded operator from L 2 (E*, J-l)e into Q*. Also, dr (V) is often viewed as a unbounded operator defined on a dense subspace of L 2 (E* , J-l). However, by (17) dr(V) can be extended to a continuous linear operator on L 2 (E * , J-l) with values in Q*. Acknonledgement
I am very grateful to Dr.X.Wang for helpful discussion. References
2 3 4 5 6 7 8 9 10 11 12 13
Albeverio S, Daletsky Yu L, Kondratiev Yu G. Non-Gaussian infinite dimensional analysis. J Funct Anal, 1996, 138: 311-350 Antoniou I, Suchanecki Z. Spectral decomposition and extended formulation of unstable dynamical systems. In: Atmanspacher H, Ruhnau E, eds. Time, Temporality, Now. Springer, 1998. 293-325 Ding Xiaqi, Luo Peizhu. Generalized expansion in Hilbert space. Acta Math Sci, 1999, 19: 241-250 Ding Xiaqi, Luo Peizhu. Weak convergence of some series. Acta Math Sci, 2000, 20B: 433-442 Gel'fand I M, Vilenkin N Ya, Generalized functions IV. New York: Academic Press, 1964 Goodrich K, Gustafson K, Misra B. On converse to Koopman's lemma. Physica A, 1980, 102: 379-388 Hua Luoken. Introduction to generalized functions. Math Achievements, 1963, 6(4) (in Chinese) Huang Zhiyuan, Van Jiaan. Introduction to infinite dimensional stochastic analysis. Beijing: Academic Press, 1999(in Chinese) Kondratiev Y G, Sila J L, Streit L. Generalized appel system. Methods Punct Anal & Topology, 1997, 3: 28-61 Kothe G. Topological vector spaces I. Berlin: Springer-Verlag, 1983 Lasota A, Mackey M C. Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics. 2th Edition. New York: Springer-Verlag, 1994 Obata N. White noise calculus and Fock space. Berlin: Springer-Verlag, 1994 Suchanecki Z, Antoniou I, Tasaki S, Bandtlow 0 F. Rigged Hilbert spaces for chaotic dynamical systems. J Math Phys, 1996, 37(11): 5837-5847