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! | / „ | 2 = ll(/n r(tfc)-
of ip G (L2)x> given by
(Sx
z{xWx{x), (s, •) is a function ofts. Then the power series expansion ofip(t,x) in t will produce the polynomials P„{x). But then the key point is to find such a function ip{t,x). At the present time we can handle functions of the following type: 1. x with p(t) being determined by the above theorem. Here are some examples: ) = {( oo n (z)| < c exp{c'|z|B c } for some constants e, c' > 0. Then £(B) C L2(C,fj,) by Fernique's theorem (see, for example, Kuo 12 and the papers cited there). For m G N and l, let mp € N so that ip £ £mp(E-p). Let r be a real number such that r > In 2/(2 In •K — 2 In 2), and choose rhp to be the least integer greater than y/mpe. Then IMImp.p-r < Pmp,p || 1, La(B,fi) c ££,E x a m p l e 4.2. Let v be a Borel measure in (E*,B(E*)) erty that there exists p > 1 so that for all m € N, [ 2(/x * [D2(w * ! define UF, F G ££,. • Translation )). Proposition 5.7 (Kuo 1 3 ,Lee 1 5 ). (i) For v? € 5oo, A Ta,p | Bn ] of ip relative to Bn enjoys the following integral representation (•) ^t G A f° r * e [0> !]• Then the conditional expectation E[Dtp(-) ht\!Ft] lies in L2(C, fi). By Theorem 6.2, f(y) dy^j € £oo. Differentiate ((F(Z(t)), the equality holds: /"} E^LoM/n) e [^]oo,oo, we have
£G£C,
JB'
is an isomorphism from (L2)x onto the Hilbert space K of holomorphic functions F on .Ec with a reproducing kernel exp[(£,rj)\, £,TJ £ E. Let [EPUx = {
- > [Ep]u,x
-
(L2)x
<-+ \Ep]l -
[E]*u<x,
2
and [E]U:x C (£ )x C [E]*uX is a Gel'fand triplet. Since u G ^+,1/2 satisfies (G3), the exponential function (f>f(x) G [-E]u,x for any £ G EQ- Hence the Sx -transform can be extended to a continuous linear functional on [E]^ x as follows. Definition 3.1 (Sx-transform). For $ G [E]*uX, Sx-transform is defined by
(5Jr*)(O = ((*,0f)>,
^£c,
(3.3)
where ((•, •)) is the bilinear pairing of [£]* x and [£]«,*• • Now we come to the characterization of [£]* x associated with px, X = G, P, in a single statement. Theorem 3.2. Let a measure fix on E* be given. Suppose u G C+,1/2 satisfies conditions (G1)(G2)(G3). Then a C-valued function F on Ec is the Sx-transform of a generalized function in [£]* x if and only if it satisfies the conditions:
41
(a) For any £, rj € EQ, the function F(z£ + rj) is an entire holomorphic function of z e C . (b) There exist constants K,a,p>0
such that
\F(0\
VteEc.
On Gaussian and Poisson White Noises
In the previous two sections, we have constructed the Gel'fand triples in terms of Fock space and Schrodinger representations with relevant transformations, J and Sx- Notice that in the Section 2, we did not introduce any probability measure on E* as the standard white noise theory 19 ' 20,23 . Hence, the property of a measure on E* plays virtually no role in the definition of J-transform. In fact, the essential tools to prove Theorems 2.2 and A.3 are the Cauchy integral formula for entire holomorphic functions of several variables, Legendre transform, dual function, Schwartz kernel theorem, and properties of the nuclear space. • So, does it imply that the considerations of the growth order of holomorphic funcions and associated topologies are sufficient to examine stochastic processes by the generalized function theory on infinite dimensional space ? = > The answer is completely No even for the study of fundamental stochastic objects such as Wiener and Poisson processes. Let us emphasize the following point. It is easy to see that the "flow"
*
f(0,l [ 0 ,t],0,---) \(0,-lM],0,---)
ift>0 ift<0,
is an element of T(Hc)- Then the "tangent vector" $>t is (0,S t ,0, •••) and belongs to TU{EC)* with u(r) = e r and J$t(0 = £(*)• On the other hand, it is known that the Brownian motion B(t) is represented by [)
\-i?(im)
if*
Similarly, the compensated Poisson process is given by
[-fiih.ow
ut<°-
[
'
42
Since characteristic functions ljo^j and l[ti0] are elements of H, B(t) and P(t) — t are in (L2)G and (L2)p, respectively. Hence we obtain UG^UQ1 = B(t) and UP$tUpl = P{t) - t. So the distributional derivative of B{t) with respect to t, so-called Gaussian white noise B(t), has the form B(t) = lf{5t) for each ( 6 E . Similarly, the Poisson white noise P(t) has the expression Pit) r 1 = l[(St) for each t G R. In those cases, since St is in i?*, B(t) and P(f) — 1 belong to [£]* G and [J3]* p , respectively (A function u will be chosen in the proof of Theorem 4.2). The relationships between the vector $t, classical Gaussian and Poisson white noises have been discussed by AsaiKubo-Kuo 8 as follows. Proposition 4.1. It holds that (1) UG$tUGl = B{t), (2) Up4tUp1 = P(t)-l. It can be shown by Theorem 2.2 that the tangent vector <$t of the flow
are generators of the Heisenberg-Weyl algebra Qhw In [as,a*]
=Ss(t)I,
[a„at} =0, [a*,a*} = 0 as in (2.4). • Operators at,a*,Nt,I we have
are generators of the oscilator algebra gos. In fact,
[as, a*] =Ss(t)I, [Ns,at] = -S3(t)at,
[Ns,a*t] = 8s(t)a*,
[a„at] = 0, [a*s,a*t] = 0, [N„Nt] = 0. Now, how do we relate classical white noises with quantum white noises ? By considering the isomorphism Ux, it is possible to see the classical-quantum
43
correspondences and distinguish two different kinds of white noises. This issue has been addressed in the papers by Asai-Kubo-Kuo 8 and Ito-Kubo 13 from the viewpoint of white noise theory. T h e o r e m 4.2. It holds that = (1) Uc{at +atWc ^(*)» where B(t) is considered as a multiplication operator. (2) Up(at +a% + Nt + I)Upl = P(t), where P(t) is considered as a multiplication operator. For the proof of Theorem 4.2, the essential point is to compute matrix elements for B(t) and P(t) — 1, and to notice crucial differences on product laws of exponential vectors, ^(x)^(x)=4>f+v(x),
(4.3)
# ( * ) < ( * ) =#+,+,*(*).
(4.4)
In this paper, our proof will not be based on the differential operator dttG and difference operator dt:p as in the paper 8 , but on Equation (4.3)(4.4) as follows. Proof. Let us start the proof with the Gaussian case X = G, first. (J(at + a t *)e(0)fo) = « ( « . + a j ) e ( 0 , e f o ) » r >
£,* 6 Ec
= W)+r}{t))e{i'n)-
(4-5)
On the other hand, since we have Equation (4.3) and (SGB)(£) = (St,0 = £(t) satisfies the condition (b) with u(r) = exp(r) in Theorem 3.2, (SGB
(4.6)
Due to Equations (4.5)(4.6), we have Uc(at + OI)UQ1 = B(t), where B(t) is considered as a multiplication operator. Hence we get B{t) £ (E)Q. Therefore, we have finished to prove our first assertion. Next consider the Poisson case X = P. (J(ot + a*t + a*tat + /)e(0)fa) = <<(«* + a* +Nt + J ) e ( 0 , e f o ) » r > = m)+v{t)+v(mt)
+ l)e<^
^
€ EC
(4.7)
Note that the function rj£ above makes sense as a member of E
44
satisfies the condition (b) with u(r) — exp(r) in Theorem 3.2. So Equation (4.4) gives (SP(P - l ) 0 f )fo) = « P ( t ) - 1, # < »
p
= m+v(t)+vmt))e&").
(4.8)
1
By Equations (4.7)(4.8), we have UP(at + a;+Nt + I)Up = P(t), where P(t) is considered as a multiplication operator. Hence we get P[t) e (.E)p. Thus we have proved the second claim. • Appendix A
On Characterizations of Generalized and Test Functions
Remark A.l. Theorem 3.2 was first proved by Potthoff-Streit 25 in case of X = G and u(r) = er. It was extended to the case of X — G and u(r) = exp[(l — j3)rT^] by Kondratiev-Streit 16 ' 17 . Moreover, Cochran et al. 9 proved the case when X = G and the growth condition (b) is determined by the exponential generating function Ga{r) = ^ ^f-rn. Asai et al. 4 , 6 ' 7 minimized conditions on sequences {a(n)} of positive real numbers in such a way that Theorem 3.2 holds. Example A.2. The Gel'fand triplet [E]u>x C {L2)x C [E]*uX becomes (1) the Hida-Kubo-Takenaka space19'20'23 if X = G and u(r) = er, and the Ito-Kubo space12 if X = P and u(r) = er, (2) the Kondratiev-Streit space17 if X = G and u(r) = exp[(l + /3)rT+?] for 0?
there exists a constant K > 0 such that
|F(0| < ffu(a|£|ip)1/2' V^Ec-
45
The characterization of [£^]«,x associated with fix, X = G,P, is stated below in a single statement. The proof associated with JIG can be found in our paper 7 . Theorem A.4. Let a measure fix on E* be given. Suppose u £ C+,1/2 satisfies conditions (G1)(G2)(G3). Then a C-valued function F on Ec is the Sx-transform of a generalized function in [E]Uix if and only if it satisfies the conditions: (a) For any £,77 € Ec, the function F(z^ + 77) is an entire holomorphic func-
tion o / z 6 C , (b)' For any constants a, p > 0, there exists a constant K > 0 such that \F{i)\ < KuiaWl,,)1/2,
V£ G Ec.
Remark A.5. Theorem A.4 was proved by Kuo et al. 21 in case of X = G and u(r) = eT. It was extended to the case of X = G and u(r) — exp[(l -f-/?)/"^] by Kondratiev-Streit 17 . Moreover, Asai et al. 3 proved the case when X = G and the growth condition (b)' is determined by the exponential generating function Gi/a(r) = Y2 STnTnT7"™- -A-sai e ^ al. 4,6 ' 7 minimized conditions on sequences {a(n)} of positive real numbers in such a way that Theorem A.4 holds. References 1. N. Asai, A note on general setting of white noise analysis and positive generalized functions, RIMS Kokyuroku (Kyoto) 1139 (2000), 19-29. 2. N. Asai, I. Kubo, and H.-H. Kuo, Bell numbers, log-concavity, and logconvexity, Acta. Appl. Math., 63 (2000), 79-87. 3. N. Asai, I. Kubo, and H.-H. Kuo, Characterization of test functions in CKS-space, in: "Mathematical Physics and Stochastic Analysis: in honor of L. Streit", S. Albeverio et al. (eds.) World Scientific, (2000), pp. 6878. 4. N. Asai, I. Kubo, and H.-H. Kuo, CKS-space in terms of growth functions, in: "Quantum Information II", T. Hida and K. Saito. (eds.) World Scientific, (2000), pp. 17-27. 5. N. Asai, I. Kubo, and H.-H. Kuo, Characterization of Hida measures in white noise analysis, in: "Infinite Dimensional Harmonic Analysis", H. Heyer et al. (eds.) D. M. Grabner, (2000), pp. 70-83. 6. N. Asai, I. Kubo, and H.-H. Kuo, Roles of log-concavity, log-convexity, and growth order in white noise analysis, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 4 (2001), 59-84.
46
7. N. Asai, I. Kubo, and H.-H. Kuo, General characterization theorems and intrinsic topologies in white noise analysis, Hiroshima Math. J., 31 (2001), 299-330. 8. N. Asai, I. Kubo, and H.-H. Kuo, Gaussian and Poisson white noises with related characterization theorems, preprint (2002). 9. W. G. Cochran, H.-H. Kuo, and A. Sengupta, A new class of white noise generalized functions, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 1 (1998), 43-67. 10. R. Gannoun, R. Hachaichi, H. Ouerdiane, and A. Rezgui, Un theoreme de dualite entre espaces de fonctions holomorphes a croissance exponentiele, J. Funct. Anal., 171 (2000), 1-14. 11. R. L. Hudson and K. R. Parthasarathy, Quantum Ito's formula and stochastic evolutions, Comm. Math. Phys., 93 (1984), 301-323. 12. Y. Ito, Generalized Poisson functionals, Prob. Th. Rel. Fields, 77 (1988), 1-28. 13. Y. Ito and I. Kubo, Calculus on Gaussian and Poisson white noises, Nagoya Math. J., I l l (1988), 41-84. 14. Yu. G. Kontratiev, Nuclear spaces of entire functions in problems of infinite dimensional analysis, Soviet Math. Dokl., 22 (1980), 588-592. 15. Yu. G. Kondratiev, P. Leukert, and L. Streit, Wick calculus in white noise analysis, Acta Appl. Math., 44 (1996), 269-294. 16. Yu. G. Kondratiev and L. Streit, A remark about a norm estimate for white noise distributions, Ukrainian Math. J., 44 (1992), 832-835. 17. Yu. G. Kondratiev and L. Streit, Spaces of white noise distributions: Constructions, Descriptions, Applications. I, Reports on Math. Phys., 33 (1993), 341-366. 18. I. Kubo, H.-H. Kuo, and A. Sengupta, White noise analysis on a new space of Hida distributions, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2 (1999), 315-335. 19. I. Kubo and S. Takenaka, Calculus on Gaussian white noise I, II, III, IV, Proc. Japan Acad., 56A (1980), 376-380, 56A (1980), 411-416, 57A (1981), 433-437, 58A (1982), 186-189. 20. H.-H. Kuo, "White Noise Distribution Theory", CRC Press, 1996. 21. H.-H. Kuo, J. Potthoff, and L. Streit, A characterization of white noise test functionals, Nagoya Math. J., 121 (1991), 185-194. 22. Y.-J. Lee, Analytic version of test functionals, Fourier transform, and a characterization of measures in white noise calculus, J. Funct. Anal., 100 (1991), 359-380. 23. N. Obata, "White Noise Calculus and Fock Space", Lecture Notes in Math. 1577, Springer-Verlag, 1994.
47
24. N. Obata, Generalized quantum stochastic processes on Fock space, Publ. of RIMS Kyoto Univ., 31 (1995), 667-702. 25. J. Potthoff and L. Streit, A characterization of Hida distributions, J. Funct. Anal., 101 (1991), 212-229.
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Quantum Information V Eds. T. Hida and K. Saito (pp. 49-55) © 2006 World Scientific Publishing Co.
RENORMALIZATION, ORTHOGONALIZATION, A N D GENERATING FUNCTIONS NOBUHIRO ASM, IZUMI KUBO, AND HUI-HSIUNG KUO
ABSTRACT. Let fj. be a probability measure on the real line with finite moments of all orders. Apply Gram-Schmidt orthogonalization process to the system {1, x, • • • , x",... } to get a sequence {Pn}^L0 of orthogonal polynomials with respect to (i. In this paper we explain a method of deriving a generating function ip(t, x) for fi. The power series expansion of i/)(t, x) in t produces the explicit form of polynomials P n , n > 0. 1. GAUSSIAN MEASURE AND H E R M I T E POLYNOMIALS
Let /i be the Gaussian measure on the real line with mean 0 and variance a2, dn(x) = - ^ L - e - * 2 / 2 " 2 dx. V27roIt is well-known that the Hermite polynomials [n,2]
„i
H
^^=E2kkl(nl2ky^2)k^2k, k=0
v
(1.1)
n = 0,l,2,...,
(1.2)
'
form an orthogonal basis for the space L2((i). generating function et*-«H*,2
Moreover, we have the well-known
= Y;-yHn{x;o2).
(1.3)
n=0
Now we raise the questions: (1) How to derive the Hermite polynomials from the Gaussian measure fi? (2) How to derive the generating function il>(t, x) = etx~" * I2 from the Hermite polynomials or the measure /i? Of course, one can say that the Hermite polynomials can be derived by applying the Gram-Schmidt orthogonalization process to the system {1, x, x2,... , xn,... }. But this derivation is impractical since it does not give the explicit form of the n-th Hermite polynomial Hn(x;cr2). On the other hand, suppose we know the answer to the second question, then we can just expand the generating function as a power series in t to obtain the Hermite polynomials. Therefore, the real problem is to find the generating function 4>(t,x) = etx~° ' I2 in Equation (1.3). The idea to find an answer to this naive problem comes from the multiplicative renormalization introduced by Hida in his book [5]. Consider the function
(1.4)
(The issue of how to find this function will be addressed in Section 3.) Regard x as a random variable with distribution given by n in Equation (1.1) and take the
50 expectation to get Then define the multiplicative renormalization of ip(t, x) by
««•*>- ^
k ""-'""•
(1 5)
'
Thus we have derived the generating function in Equation (1.3) by the multiplicative renormalization of the function in Equation (1.4) with respect to the Gaussian measure fi. Note that ip(t, x) is a smooth function of t and so has a series expansion
w.*) = E ^ * n -
(!-6)
n=0
On the other hand, il>{t,x) can also be expanded as 4>{t,x) =
etxe-aH*>2
- ( 5Z „!*")( 5Z 2"n! t2n) =
SlS("-2*)! 2**! J'""
(L7)
Prom Equations (1.6) and (1.7) we see clearly that Pn(x) is exactly the Hermite polynomial in Equation (1.2). Thus we have derived Hermite polynomials from the Gaussian measure fi via the functions ip(t, x) in Equation (1.4) and ip(t,x) in Equation (1.5). Moreover, it can be easily checked that for any t and s, E^(t,-)ij(s,-) = e^ts. The fact that Efj.ip(t, -)ip(s, •) is a function of the product ts implies that the Hermite polynomials are orthogonal with respect to the Gaussian measure (j,. 2. POISSON MEASURE AND CHARLIER POLYNOMIALS
Next we consider the Poisson measure with parameter A > 0,
c(W) = e " 4
* = 0,1,2
(2.1)
How to proceed to find a complete orthogonal sequence of polynomials for fi and the corresponding generating functions? Consider the function V{t,x) = {l+tf.
(2.2)
(The issue of how to find this function will be addressed in Section 3.) Regard x as a random variable with distribution given by fj, in Equation (2.1) and take the expectation to get E^{t,-)=ext. Hence the multiplicative renormalization of
w>x^i$$r)=*-"<1+tr-
(2 3)
-
51 This function can be expanded as a power series in t as follows:
*..>-(£^)(i*K x
n=0
n=0
'
v
V
fc=0
v
n=0
>
'
'
where px$ = 1 and px>n = x(x — 1) • • • (x — n + 1) for n > 1. In view of Equation (2.4) we define the n-th CharUer polynomial Cn(x;X) by C n (x;A) = £ ( £ ) ( - A ) " - W -
(2-5)
Then we have the equality 00
1>{t, x) = e -
At
fn ( l + t) = Y, ^jCW(i; A). x
(2.6)
n=0
Moreover, we can easily check that for any t and s, = ext°.
E^(t,-W(s,-)
Since E,j.ip(t, -)il>(s, •) is a function of the product ts, the Charlier polynomials are orthogonal with respect to the Poisson measure fj,. Thus the function ip(t,x) in Equation (2.3) is a generating function for the Charlier polynomials. 3. G E N E R A L CASE
Now, consider a general probability measure [/, on the real line R. Assume that J"K |x|" dn(x) < oo for all n = 0,1,2, Then we can apply the Gram-Shcmidt orthogonalization process to the system {1 « 30* 00 * * • • « 00 « * • • } to get a sequence {P„; n = 0 , 1 , 2 , . . . } of orthogonal polynomials such that PQ = 1 and Pn is a polynomial of degree n with leading coefficient 1. Q u e s t i o n : How to find the explicit form of the polynomial
Pn(x)?
Being motivated by the examples in the previous sections, we consider a function of the form oo V{t,x)
= Y,9n{x)tn, n=0
where g„(x) is a polynomial of degree n satisfying the condition limsupHsnll^" < o o . n—>oo
v f v
Let tp(t, x) be the multiplicative renormalization of (p(t, x) defined by
Then we have the following fact. For details, see the papers [1] and [2].
(3.1)
52 Theorem: The function E^iplt, -)ip(s, •) is a function of the product ts if and only ifip(t, x) is a generating function for fi, i.e., it has the series expansion
f^Qn(x)tn,
i>(t,x) =
(3.3)
n=0
where Qn{x) is a polynomial of degree n and the polynomials Qn 's are orthogonal with respect to the measure p,. Let an be the leading coefficient of Qn. Then the polynomials we are looking for in the above question are given by Pn(x) = Qn(x)/an. Moreover, we have the equality oo i>(t,x)
YtanPn(x)tn.
= n=0
Thus here is the A n s w e r to the above Question: Find a function
Gaussian N(0,cr2), a > 0 Poisson Poi(X), A > 0 Gamma T(a), a > — 1 2. (f(t,x) = (l — p{t)x)
ip(t,x)
i>(t,x)
etx
e
(l + t)x
e~xt(l+t)x
tx
Pn(x)
ti-itr2t2
t)-01-^^
(l +
Hermite Charlier Laguerre
with p(t) and c being determined by the above theorem.
Here are some examples: M Uniform on [—1,1] r(/?+1)
x2)3-*
. (1
|x|
(/3>-i,/3^0)
T Vl-X2 '
\x\ < 1
ip(t,x)
Vl-2tx+t2
^l-2tx+fi
2
_(l+t if_
1
(l-2tx+P)t>
(l-2tx+t 2 )P
4-4tx+t2
4-t2 4-4tx+t 2
Pn(x)
Legendre Gegenbauer Chebyshev (first kind)
\VT
\x\<\
l-2tx+t*
l-2tx+t*
Chebyshev (second kind)
Note that the Legendre and Chebyshev of the second kind are the special cases of Gegenbauer with f3 = 1/2 and /3 = 1, respectively. However, although the measure for the Chebyshev of the first kind is a special case of the Gengenbauer with (3 = 0, we cannot obtain the corresponding
53 of Gegenbauer by letting (3 = 0. Moreover, we want to point out that in some books (e.g., page 25 in [3]) the generating function of the Chebyshev polynomials of the first kind is stated in the form:
n ? ^ = £*.<*)*".
^
71=0
where the Chebyshev polynomial Tn(x) of the first kind is defined by Tn(x) = cos(narccosx),
n > 0.
However, our Chebyshev polynomials of the first kind in the above chart are defined (through our method of deriving Pn's from the generating function) by 4-t2 4 - 4te + 1 2
= £f n (*)*".
(3.5)
n=0
By expanding the left-hand side of Equation (3.5) as a power series in t, we can easily check that the polynomials Tn(x) are given by:
fo(x) = 1, i
T„(x) = ^
[[n/2] n/2
J
/
\
2 k
g(-l)*( -•x: ) , 2 \"z.f,'. fc )*»--(l — i
n > l .
The polynomials Tn and Tn are related by Tn(x) = 2n-1fn(x),
T0(x) = fo{x) = 1,
n>\.
With this relationship we see that Equation (3.5) implies Equation (3.4). We mention that another formulation of the generating function for Tn(x) is given by (see e.g., page 89 in [6]) 1-t2 l-2tx + t2
= T0(x) + 2j2Tn(x)tn. n=l
For the derivation of the generating functions in the above chart and further information, see our papers [1] and [2]. Below we give a new example of generating function and the corresponding orthogonal polynomials. Consider the negative binomial distribution fi with parameters 0 < p < 1 and r = 1,2,... given by
K{k}) = ( 7 ) (-1) V ( l - P)k, * = 0,1,2,... . Try the function of the first type, i.e.,
q=
l-p.
Hence the multiplicative renormalization of
E.*(t, xMs, x) = £ ( - - L ^
—r
+
_-L_ _ _ _ _ Z _ _ _ _ !
54 As observed in [1] we have 1 1 • + • 1 - q9{t) 1 - qO(s) 1 / 1 p ~Pr\l-qO(t) In order for Exip(t,x)ip(s,x)
p (1 - q6(t))(l - q0(s)) 1\/ 1 pj\l-q8(s)
1 P/
to be a function of ts, we must have 1 1 = at°. 1 - qO(t) ~ p
Therefore, the function 9(t) is given by 1 + 2atb w
l+patb
Choose a = q/p and b = 1 to get 6{t) = j+hf(t,x)
Then the corresponding functions
and ip{t,x) are given by:
^x)
= (ITJS >
rP{t,x) = (l + t)x(l +
qt)-x-r.
Now, expand the function ip(t,x) as a power series in t to get
ll>(t,x) =
T^Pn(x)tn
n=0
with the polynomial Pn(x) being given by
^)%^E(:)(r;)'"-' We remark that we do not know how to derive these polynomials directly from the Gram-Schmidt orthogonalization process. A c k n o w l e d g e m e n t s . This was research was supported by a grant of MonbuKagaku-Sho (Ministry of Education and Science) to Hiroshima University. HHK is most grateful to Professor I. Kubo and the Mathematics Department of Hiroshima University for their warm hospitality during his visit May 20-August 19, 2001. He also thanks the Academic Frontier in Science of Meijo University and Professors T. Hida and K. Saito for financial supports and the warm hospitality during this conference.
REFERENCES
[1] Asai, N., Kubo, I., and Kuo, H.-H.: Multiplicative renormalization and generating I; Preprint (2002) [2] Asai, N., Kubo, I., and Kuo, H.-H.: Multiplicative renormalization and generating II; (In preparation) [3] Chihara, T. S.: An Introduction to Orthogonal Polynomials. Gordon and Breach, [4] Erdelyi, A. (editor): Higher Transcendental Functions III, Bateman Manuscript McGraw Hill, 1955 [5] Hida, T.: Analysis of Brownian Functional. Carleton Mathematical Lecture Notes
functions functions 1978. Project. 13, 1975
55 [6] Hitotsumatsu, S., Moriguchi, S., and Udagawa, K.: Mathematical Formulas III, Special Functions. Iwanami, 1975 [7] Kuo, H.-H.: White Noise Distribution Theory. CRC Press, 1996 [8] Szego, M.: Orthogonal Polynomials. Coll. Publ. 23, Amer. Math. Soc, 1975 NOBUHIRO ASAI: INTERNATIONAL INSTITUTE FOR ADVANCED STUDIES, KIZU, KYOTO, 619-0225,
JAPAN IZUMI KUBO: DEPARTMENT OF MATHEMATICS, GRADUATE SCHOOL OF SCIENCE, HIROSHIMA
UNIVERSITY, HIGASHI-HIROSHIMA, 739-8526, JAPAN HUI-HSIUNG Kuo:
DEPARTMENT OF MATHEMATICS, LOUISIANA STATE UNIVERSITY, BATON
ROUGE, LA 70803, USA
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Quantum Information V Eds. T. Hida and K. Saito (pp. 57-76) © 2006 World Scientific Publishing Co.
INSIDER TRADING IN CONTINUOUS TIME
EMILIO BARUCCI di Statistica e Matematica Applicata all'Economia, Universita di Pisa, Via Cosimo Ridolfi, 10 - 56124 Pisa, Italy E-mail: ebarucciQec. unipi. it
Dipartimento
Dipartimento
ROBERTO MONTE di Studi Economici, Finanziari e Metodi Universita di Roma "Tor Vergata", Via Columbia, 2 - 00133 Roma, Italy E-mail: [email protected]
Quantitativi,
BARBARA TRIVELLATO Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abbruzzi, 24 - 10129 Torino, Italy E-mail: [email protected]. it We consider a market where two assets are exchanged: a risk-free asset and a risky asset, which pays a continuous dividend. In the market there are three types of agents: an insider trader, a market maker, and a representative noise trader. We study a rational expectations equilibrium model of asset prices by assuming that the insider trader enjoys some private information on the dividend drift. In equilibrium, both market price and the informed trader's strategy are linear combinations of a suitable set of state variables and their estimates. 1
Introduction
We consider a market, which lives forever, where two assets are traded: a riskless asset and a risky one. T h e riskless asset pays a constant interest r a t e r > 0 in continuous time. T h e risky asset, with price P(t), pays a continuous time dividend stream, D(t). We assume t h a t the time series of t h e dividend stream up to the current instant time t is observable by all agents and t h a t the dynamics of D(t) is driven by the stochastic differential equation dD{t) = 6dt + <jDdwD(t),
(1)
where 6 and
58
The representative noise trader, or liquidity trader, models the aggregate effect of all agents who trade in the market with price-inelastic demand. These agents buy and sell the risky asset only for random liquidity needs. Therefore, they introduce a noise component in the market demand. We assume that the representative noise trader trades smoothly. Namely, the representative noise trader's order flow is the "derivative" of his inventory. We assume that the representative noise trader's order flow follows an Ornstein-Uhlenbeck process, which is independent of the information supplied by the dividend history (see 5.1), see also 3 . More precisely, writing U(t) for the current value of the representative noise trader's order flow, we suppose dU(t) = ~auU{t)
+ au dwu(t),
(2)
where au, o~u are positive parameters, constant in time, and wu(t) is a Wiener process independent of wait). The insider trader, or informed trader, enjoys some private information which allows him to know the exact value of the parameter 9 in Equation (1). In addition, he observes U(t). The insider trader trades in the market aiming to exploit all information available to himself, on account of the feedback effect of his trade on the market price defined by the market maker. As a matter of fact, his demand reveals, at least in part, his information to the market maker and therefore is reflected with some noise by the market price. We assume that also the insider trader trades smoothly. Namely, writing Z(t) for the current value of the insider trader's inventory and z(t) for his order flow, we suppose dZ{t) = z(t) dt.
(3)
The insider trader is risky averse. He choices the variation of his order flow and maximizes the expected value of his exponential intertemporal utility over an infinite time horizon, given at a certain time instant his wealth and the state of the economy, by controlling his consumption rate. Hence the insider trader's maximization problem becomes r+oo
sup \ Et,m,Y
Jt
_e-(pS+^c(s))
dg
(4)
where p > 0 and ip are constant parameters, c(t) is the consumption rate, and Et, m ,y [•] is the conditional expectation operator given the time instant t, the state vector of the economy Y, and the informed trader's wealth m. Here, it is important to notice that the informed trader observes the realization of the uncertainty, represented by u>u(t) and wr>(t), before adjusting his trading
59 strategy. As we will show below, this reflects on the choice of his trading strategy. The market maker ignores the exact value of the parameter 9 in (1). He assumes that 9 is a random variable independent of the Wiener process wo{t). The market maker is risky neutral. He updates his believes continuously and sets the price of the risky asset equal to the conditional expected present value of the future dividend stream, given the public information (i.e. the dividend time series), and the aggregate order flow information. In symbols, r+oo -t-oo
P(t) = E
(5) e-rl'-*)D(s) ds\Ft / where (.Ft) t > 0 i s the cr-field generated by the dividend process and by the aggregate order flow. Note that, as l, the market maker observes only the aggregate order flow U(t) + z(t), and not the insider trader's order flow. This fact gives rise to a filtering problem: the marker maker's learning about the insider trader's information from the aggregate order flow and the dividend time series. Similarly to 7 , the equilibrium arises from an utility maximization condition (4) and a market efficiency condition (5). It is a Bayesian Nash equilibrium, in the sense that the market maker postulates the insider trader's strategy when he chooses the rule to update his beliefs, and the insider trader does the same in defining his trading strategy. In equilibrium, conjectures are self-confirming. We will show that there is a linear equilibrium: both updating rule of the market maker and the trading strategy of the insider trader can be chosen linearly. 2
Market Maker's Optimal Filtering
Our first step is to adopt the market maker's point of view. As above discussed, the market maker observes only the dividend flow D(t) and the total order flow U(t) + z(t). On the basis of this information he tries to estimate the parameter 9 in Equation (1) and the informed trader's order flow z(t). The model is described by the system of stochastic differential equations
(
dD(t) = 6dt + aDdwD(t), dU(t) = -auU(t) dt + au dwu{t),
(6)
dZ(t) = z(t)dt, and to manage (6) it is convenient to introduce a matrix notation. Actually, setting yT = (D{t), 9, U(t), z(t), Z{t)) we can write dy(t) = Ay(t) dt + Q dw(t) + k dz(t), (7)
60
where /01 0 00\ 00 0 00 00-av0 0 00 0 00 \00 0 10/
Q
0 0 0 \0
/0\ 0 0 1
0 av 0 0/
(8)
W
and WD(t) Wu(t)
w(t)
(9)
In addition, we write yj(t) = (D(t),z(t) + U(t)) for the observation vector, and yj{t) = (De(t),6e(t),Ue(t),ze(t),Ze(t)) for the market maker's estimate vector of the state vector y(t), given the public information and the total order flow information. We have then Vo(t)
=
MTy(t),
(10)
where MT
10000 00110
and ye(t) = E{y(t)\F?°]
(11)
where T\° is the c-algebra generated by y0(s), for s < t. The market maker assumes that the informed trader adjusts his trading strategy linearly. Namely, from the market maker's point of view, the informed trader's order flow z(t) satisfies the linear stochastic differential equation dz(t) = aTy(t) dt + ajye(t) dt + qT dw(t),
(12)
for some aT = ( a i , a 2 , a 3 , a 4 , a 5 ) ,
aj = (a\,a\,a%,a\,a\),
qT = {qx,q2).
(13)
As a consequence, we show that the market maker may adjust his believes linearly. Namely, he may write the evolution equation for ye(t) in the form dye{t) = Geye(t) dt + He dy0{t),
(14)
where the matrices Ge and He are going to be determined through the optimal filtering procedure. Indeed, substituting (12) into (7), we obtain V2. dy{t) = Aiy{t) dt +fcajye(t) dt + Q{' dw{t)
(15)
61
where A!=A
+ kaT,
Q\/2~Q
and
+ kqT.
(16)
Then, combining (15) and (10) we have dy0{t) = MTAiy(t)
dt + MTkaJye(t)
dt + MTQ\/2
dw(t),
(17)
and, substituting the latter into (14) we can write dye(t) = HeMTA1y{t)dt+(Ge
+ MTkaJ)
ye(t)dt + HeMTQ\/2
Hence, introducing the stacked vector Y(t) = (Yj(t))._1,
dw(t). (18)
given by
on account of (15) and (18), it is immediately seen that Y(t) satisfies the stochastic differential equation dY(t) = AY{t) dt + Q1'2 dw{t), where
On the other hand, from (17), we can formulate the evolution oiy0(t) in terms of Y(t) by writing dy0{t) = BY(t) dt + R1'2
dw(t),
where B = (MTA1,
MTkaJ),
and
B}'2 = MTQ\/2.
(20)
Therefore, the filtering problem becomes (dY(t) = AY(t)dt + Ql/2dw(t), \ dy0{t) = BY(t) dt + R1'2 dw(i).
{
'
It is worth noting that (21) is not a standard Kalman-Bucy linear filtering problem, owing to the occurrence of the same noise both in the state process Y(t) and in the observed y0{t). Nevertheless, it is still possible to filter Y(t)
62
by y0{t) (see 5 ), and we obtain the equation for the optimal estimate Ye(t) = E\Y(t)\J?>], dYe(t) = AYe(t)dt +(Q1/2
(i? 1 / 2 )\t(t)BAR- 1 (jB(Y(t)-Y e (t))dt+R 1/2 dw(tjj
, (22)
where S(t) = E \{Y(t) - Ye(t)) (Y(t) - ye(*))Tl satisfies the Riccati equation dt{t) = (At{t) + t(t)AT - (Q1/2(/?1/2)T
+ Q\ dt t(t)BT)
+
R-1 (Ql'2{R}'2)r
+ t(t)BTY
dt. (23)
Since ye(t) is the market maker's estimate of y(t), it is also observed, ye(t) — E[2/c(OI-7?°]- Therefore,
™-(!$)• and E(£) can be clearly decomposed into four 5 x 5 blocks as follows
where £(£) = E [(j/(t) - ye(t)) (y(t) - ye(t))T].
We have then,
* F = ( E ( 'f), and
Hence, (22) can be rewritten as dYe{t) =
{HeMJAl
Ge + HeMTkaJ )
+ ( ( Q l +l%f]
M
Ye dt
^
) QT1MT [A, (y(t) - „.(*)) dt + Q\'2 dw(t)] ,
63
or, equivalently, 'dye(t)\ dye®) [Ai + kaj - (Qi + Z(t)Aj) MQYXMT (A± + kaj)] ye(t) dt Geye(t)dt 1 (Q1 + E(t)Aj)MQ^ dy0(t) + He dy0(t) where, for brevity, we introduce the temporary notation Qi = MTQ1M. The latter clearly implies He = (Q1 + Z(t)Aj)M(MTQ1My1
(24)
and Ge = (l- HeMT) (Ai + kaj) .
(25)
On the other hand, by straightforward computation, we obtain At(t) + £{t)AT + Q =
(A{L{t) + V{t)A~[ + Qx (HeMr (A^t) + V HeMT ( ^ E ( t ) + Qi) HeQtHj
Qi))T\ J '
and (Q1'2{Rl'2)T+t{t)BT)
R-1 (g 1 / 2 (i? 1 / 2 ) T +i:(i)5 T ) T
J{Q1+Yl{t)Aj) MQ^MT (Q!+E(i)^7) T (HeMT ( A i E ^ + Q i ) ) ^ V HeMT (i4iE(t>+^i) HeQtHj J' It then follows that (23) reduces to dE(t)=(.AiE(t) + E(t)Aj + Qi) dt - (E(t)i4j" + Qi) Af (M T Q!M) _ 1 M T (E(t)Aj" + Qi)T dt. (26) To study (26), let us observe first that the first and the second columns of the matrix E(i)M are given by /E[(D(t)-De(t))(D(t)-De(t))}\ E{(D(t)-De(t))(0-ee(t))} E[(D(t)-De(t))(U(t)-Ue(t))} E[(D(t)-De(t))(z(t)-ze(t))} \E[(D(t)-De(t))(Z(t)-Ze(t))}/
64
and / E [(£/(*) + z(t) - (Ue(t) - ze(t))) (D(t) - De(t))} \ E \(U(t) + z(t) - (US) ~ ze(t))) (6 - 6e(t))} E [(U(t) + z(t) - (Ue(t) - ze(t))) (U(t) - Ue(t))} E [(U(t) + z(t) - (US) ~ *«(«))) (*(*) - zS))} \ E {(U(t) + z(t) - (US) - zS))) (Z(t) - ZS))] ) respectively. Thus, since both processes D(t) and U(t) + z(t) are observed, the matrix T,(t)M vanishes identically. Then, let us introduce the matrix M1
10 0 0 0 0 0 1/\/2 1/V2 0
whose rows are an orthonormal basis for the linear span of the rows of and
MJ =
MT,
'0 0 1/V2 - l / v ^ C 0 1 0 0 0 00 0 0 1
whose rows are an orthonormal basis for the subspace of R 5 which is orthogonal to the above mentioned linear span. Clearly also E(i)M vanishes identically, and the columns of the matrix 00\ 10 0 01/V2 1/V2 0 oi/>/5-- l / > / 2 0 0 0 0 01/
/I 0
M = (M, MJ] =
0 0
0 0
\° 5
constitute an orthonormal basis in R . Now, with respect to the vector y0(t) = MTy(t), equivalent to the observation vector y0(t), Equation (26) becomes dE(t)=(^iE(i) + T,(t)Aj + Qi) dt - {E(t)A[ + Qi) M ( M T Q i M ) _ 1 MT {H(t)A[ + Qi)T dt, (27) and it can be rewritten in the equivalent form d (MTE(t)M) =MT (A^it) -MT(Z(t)Aj
+ Z(t)A]
+ Qi)
Mrdt
+ Qi) M(MTQ1M)~1MT(E(t)Aj
+ Q1)TMTdt.
(28)
65
The latter splits in the four equations d (MTS(i)M) = M T (AiE(t) + Z(t)A[ T
+ Qi) Mdt
+ Qi) M ( M T Q i M ) _ 1 M T ( E ( t ) A j " +Q1)TMdt,
-M (E(t)Aj
(29)
d (MTE(<)MX)
= M T (AiE(t) + E(*)^7 + Qi) T
M
idt
T
-M (E(t)i4}" + Q i ) M ( M Q 1 M ) ~ 1 M T ( E ( t ) ^ ] r + Q i ) T M ± d t , (30) d(MjE(t)M) = M j (i4iE(*) + E(t)A7 + Qi) Mdt - M j (E(t)Aj +Qi) M ( M T Q i M ) - 1 M T ( E ( i ) A [ + Q!) T Mdt,
(31)
d (MjE(t)Mj.) =Ml (i4iE(t) + E(t)4j" + Qi) M X A - M j (E(t)>l7 + < 5 i ) M ( M T Q i M ) _ 1 M T ( E ( t ) A [ + Q1)TM±dt.
(32)
On the other hand, a straightforward computation shows that (29)-(31) vanish identically. Therefore we are reduced to consider only the 3 x 3 differential Riccati equation (32), and it can be shown that it converges to a unique stationary solution for all set of possible parameters, which arises as the solution to the algebraic Riccati equation
M7(i4iE + Ei47 + Q i ) M x -Ml
3
({XAj
+ Qi) M ( M
T
Q!M)
_ 1
M
T
{XAj
+ Qi)T)
Mx
= 0.
Informed Trader's Optimal Trading
As already discussed in the introduction, the informed trader choices the variation of his order flow, dz(t), and in addition he aims to maximize the expected value of his exponential intertemporal utility rate over an infinite time horizon, by controlling the stacked vector, Y(t), and his wealth, m(t), through his current consumption rate, c(t).
66
By (7), the choice of dz(t) influences straightforwardly the state vector of the economy, y(t). Therefore, through (10), also the market maker's observations y0(t) are influenced, and since the insider trader postulates that the market maker adjusts his estimates ye(t) linearly, according to (14), from the insider trader's point of view we can write dye(t) = Geye{t) dt + HeMT (Ay(t) dt + Q dw(t) + k dz(t)).
(33)
Hence, it is immediately seen that the stacked vector Y(t) satisfies dY(t) = AY(t) dt + Q dw(t) + k dz(t),
(34)
where
HeMTAGe)'
Q=\HeMTQj'
l=z
\HeMTk)-
(35)
The informed trader's wealth, m(t), is modeled as the solution to the stochastic differential equation (see 5.2, see also 6 ) dm(t) = rm(t) dt + Z(t)((D{t)
- rP{t)) dt + dP{t)) - c{t) dt,
(36)
where the risky asset price, P(t), which appears in (36), and is not a state variable, is given by
P{t) = E
-
+oo
/
e-Tl—t'>D(s)ds\Flla
On the other hand, since for every s > t we have
E [D(a)\F?] = E W)\W\ + E [D(a) - D{t)\J^'] = D(t) + E [6(s ~t) + aD (wD(s) - wD(t)) \f?°] = D(t) + ee(t)(s-t), being WD(S) — wo{t) future with respect to Tt, we can write P(t) = r~lD{t)
+ r-20e(t)
= pTY(t),
(37)
where pT =
(r-\r-2,0,0,0,0,0,0,0,0).
Therefore, combining (36) with (34) through (37), and noticing that we can also write Z = k^Y,
and
0e =
kjY,
67
where kz and kgc are vectors whose components are all zero except for the row corresponding to the state variables Z and 6e respectively, we obtain dm{t) = (rm(t) - k^Y(t) r
+P QkjY(t)
~pTAY(t))
{r^kjjit) T
dw(t) + P kkjY(t)
-
c{t))dt
dz{t).
(38)
In what follows we show that there exists a linear diffusion evolution for dz{t) which allows the informed trader to maximize his intertemporal utility over the consumption strategies. Notice that the insider trader knows Y{t) and m(t) at each time instant t before adjusting his trading strategy. In other words, he enjoys a complete observation. Therefore (see 8 , 9 ) , it makes sense that he adjusts the variation of his order flow in a feedback form. Moreover, since the state vector of the economy follows an autonomous system of stochastic differential equations, he can choose a stationary Markov policy dz(t) = a(t) dt + q(t) dw(t),
(39)
where a(t) = a(Y(t), m(t)),
q(t) = q(Y(t),m(t)).
(40)
Hence, his optimization problem becomes to compute the value function y(t)y,m)=maxJEt,y,m
f °° _e-(P«+*«=W) ds j ,
where c(t) = c(Y(t), m(t)), and (Y(t),m(t))
(41)
is subject to
dY(t) = {AY{t) + ka(t)) dt+(Q
+ kq(t)) dw(t),
(42)
and dm{t) = (rm{t) - Y{t)Tkz T
T
+Y{t) kzP
(r-lkJcY(t)
-pr
(AY(t) + ka{t))) - c(t)) dt
(Q + kq{t)) dw{t).
(43)
We claim that, when choosing the linear trading strategy given by a(t) = -{kr Lk)-l{kT
LAY{t) + iprkTpz(t))
(44)
and q(t) = -(fc T Lfc)- 1 fc T LQ,
(45)
then the solution to (41) is V*(t,Y,m)
= _ e -(pt+^ T ^+V"-m+A 0 ) ;
(46)
68
for a suitable real constant Ao and a suitable symmetric constant matrix L, corresponding to the optimal consumption c.(t) =
^(t)LY(t)
+ ^rm(t)-ln(r)
+ X0^
To this task, it suffices to show that, when choosing (44) and (45), the function V*(t,Y,m) is a solution to the Bellman equation dtV(t, Y, m) + max UcV(t,
Y, m) - e ^ * ^ \ = 0,
(48)
for c* = argmax UcV(t,
Y, m) - e - ^ + ^ J ,
subject to the condition lim Et,mtY{V(t
+ T,Y(t + T),m(t
+ T))}=0,
(49)
T—>+oo
where Cc is the infinitesimal generator of the diffusion process (42), (43) corresponding to the choice of the control c(t) = c, given by 10
10
T
+Y kz J2 (PT (Q + k) (Q + k)T) d^Yi +\YTkzPT 2
(Q + kg) (Q + kqf pklY
d\m
10
+ '£(AY
+ ka)jdyl
+ (rm - YTkz
{r~lkjY
-pT
(AY + ka)) - c)dm.
Indeed, computing the derivatives dtV* = dYjV* =
-pV, -(YTL)jV*,
dmV* = -i>rV\ ^i,yiv* = (LyyTL-L).i.v, d$j>mV* =
MYTL)jV*,
(50)
69 where we are using V* as a shorthand for V*(t, Y, m), and substituting in (50), we obtain 10
w
= \ £ {($+*«) @+*«)"% (LYYTL - L)i,jv* 10
+i>rYJkz £ (pT (Q + k) (Q + k)T) . (YTL)jV +
I V ,Vy T fc z p T (Q + kg) (Q + kq)1pkjYV*
-Y/(AY
+ ka).(YTL)jV*
j=\
-ipr (rm - YTkz
- pT (AY + ka)) - c) V*.
(r^kjj
(51)
On the other hand, thanks to the properties of the trace functional, 10
((Q + k) (Q + k)T)
£
(LYYTL - L)..
= tr ((Q + kq)T (LYYTL - L) (Q + kq)) = t r ( ( 0 + kqV LYYTL (Q + kq)) - t r ((Q + kq)T L(Q + kq)) = YTL(Q
+ kq)(Q + kq)T LY
-tr((Q
+ kq)T L(Q + kq))-
(52)
Moreover, 10
£ p T (Q + kq) (Q + kq)] (YTL)j = PT(Q + kq) (Q + k)T LY,
(53)
j=l
and 10
(AY + ka). (YTL)j
£
= (AY + ka) LY.
3= 1
Therefore, combining (51) with (52)-(54), and observing that we have ^YTL(Q
+ kq)(Q + kq)T LY
(54)
70
+^rYTkzpT
(Q + kq) (Q + kq)1 LY
+ l^r2YTkzPT
(Q + kq) (Q +
kq)TpkjY
= i y T {^rpkl +L)T (Q + kq) (Q + kqf {^rpkTz + L) Y, we can rewrite CCV* = - | t r ((Q + kqf L (Q + kq)) V* + \YT {^rpkl +L)T (Q + kq) (Q + kq)T (i>rpkTz + L) YV* -YT
(iprpkj +L)T (AY + ka) V*
-ipr (rm - r~1YTkzkJcY)
V* - i>rcV*.
(55)
Now, on account of dZ(t)
=z(t)dt,
substituting (34) into the strategy (39) given by (44) and (45), a straightforward computation yields dz(t) = - ( F L f c ) - 1 (kTLAY(t) dt + iprkTpdZ(t) + kTLQdw(t)) = -{k^Lky1 [kTL (dY(t) - ka(t) dt - (Q + kq(t)) dw(t)) +iprkTpdZ(t) + kTLQdw(t)} = -{kTLkyl (kTLdY(t) + ifjrkTpdZ(t)) + a(t) ds + q(t) dw(t) = -{kTLkylkT (L + i>rpkTz) dY(t) + dz(t) and the latter clearly implies the condition kT (L + iprpk],) = 0.
(56)
Conversely, it is easily seen that starting from kT (L + iprpkTz) dY(t) = 0, taking into account that kl dY(t) = dZ(t), and replacing dY(t) with its expression (34), we end up with dz(t) = -(fcTLfc)-x ({kTLAY(t) + iPrkTpz(t)) dt + kTLQdw(t)) ,
(57)
71 which is our strategy. Therefore, the choice of the strategy (57) is equivalent to set up the condition (56). Hence, thanks to (55) and (56), setting = iprcV*-e-(-pt+'l'c\
I(t,Y,m,c)
(58)
and J(t, Y, m) = - i t r ({Q + k - k(kT Lk)-1kT + \YT -Yr
(^rpkl
+L)TQQ
LQ)T L(Q + k~- jfe(jfcT' Lk)~lVLQ)\
(^rpkj
V*
+ L) YV*
(iprpkj + L)T AYV*
-pV* -iPr(rm-r-1YTkzkJeY)V*,
(59)
Equation (48) becomes max{I(t,Y,m,c)}
+ J(t,Y,m)
= 0,
(60)
c
and we are now in a position to prove the desired result. Maximizing I(t, Y, m, c) with respect to c, the first order condition yields
/.Prom the latter, on account of (46), we obtain (47) and max{I(t,Y,m,c)}
= r (^YTLY
+ i>rm -ln(r)
+ X0 + l) V*.
(62)
On the other hand summing (59) and (62), it is clearly seen that we shall have established Equation (60) if we can prove the existence of suitable Ao, L such that p - r (1 + A0 - ln(r)) + ^ t r ({Q - k(kTLk)-1kTLQ)T -\rYTLY -ipYTkzkJcY
- l-YJ (^rpkj
L (Q + L)T QQ ^rpkTz
+ YT {iprpkTz + L)T AY) = 0.
k{kTLk)-lkTLQ)\ +L)Y (63)
The above equation is the sum of two terms, the first is independent of the state variables, while the second is a quadratic function of them. A solution is one that makes both terms identically equal to zero. This is achieved for
72
the first term by setting the suitable value for the constant Ao. On the other hand, the quadratic term of (63) can be rewritten as
"(-3
rL - \{^rpkTz
+ L)JQQr^rpkrz
+ L)
+ AT{iPrpkTz + L ) ) y = 0,
-ipkzkl
and we are reduced to solve the algebraic Riccati equation -rL - (L + il>rpk]:)TQQT(L + i>rpkTz) -4>{kzk]c + kjkej
+ AT(L + iprpkj) + (L + iprpk];)1A = 0.
(64)
Also in this case, after reducing Equation (64), on account of (56), by exploiting a procedure similar to that in the reduction of Equation (26), it can be shown that there is a solution for all set of possible parameters. Finally, we are only left with the task of proving that lim E t , m , y [V* (t + T,Y(t
+ T),m(t
+ T))} = 0.
T—>+oo
Indeed, from V* {t + At, Y(t + At),m(t
+ At)) - V* (t, Y(t), m (t)) t+At
dV*(s,Y(S),m(s)),
/ applying the Ito formula, we can write V* (t + At, Y(t + At),m(t
+ At)) - V* (t, Y (t) ,m (t))
f t+At
•Jfit
{8SV* (s, Y(s),m
(s)) + Cc. V* (s, Y (s), m (s))) ds
rt+At I-f-£iE
a(Y (s), m (s))VY,mV*
/
(s, Y(s),m
(s))) dw (s),
(65)
where a(Y (t) ,m(t)) denotes the diffusion matrix of the process (Y (t), m (t)) and Vy >m denotes the gradient operator in the state space of (Y (t) ,m(t)). On the other hand, since V* (t, Y, m) is a solution to the Bellman equation (48), for c(t) = c*(t), we have t+At
/
(dsV* (s,Y(s),m(s))
+ £ c .V* (s,Y(s) t+At
/
,m(s)))
e -(p,+Vc*( S )) d s _
ds
73
Now, on account of the latter, applying the expectation operator on both sides of (65), we obtain E t ,y, m [V* (t + At, Y(t + At),m(t 1 -AtEt'Y'm
t+At
/
+ At))} - E«, y , ro [V* (t, Y (t), m (t))} At
e-{ps+4,c>(s))ds
and, passing to the limit as At —* 0, it follows
dEttY,m[V*(t,Y(t),m(t))} dt Finally, from (61), we have e-(Pt+i,c*(t))
=
_rV*
\-(p*+V>c*(t))
^ Y(t),m
(t)),
and therefore E t ,y, m [V* (t, Y (t), m (t))} satisfies the differential equation
-••^i t; -yw."W)i__ r ^,|v.(«,y W , mW )i. The desired result clearly follows. 4
Further Directions of Research
This work is our first approach in studying market models characterized by the presence of an insider trader. Actually, the dividend model (1) that we have assumed is somewhat simple, and does not exclude the possibility of negative dividends. Neverthless, this model is sometimes exploited in literature. For instance, Veronesi in 12 analyses this model as a particular case of his more general dD(t) = 9(t) dt + aD dwD(t),
(66)
where 6(t) is a two-state continuous time Markov process. Therefore, a first natural generalization of our work is to consider (66) as the equation for the dynamics of the dividends. Further directions of research can be explored by assuming rather different models for the dynamics of the dividends. Interesting models are proposed by Campbell and Kyle in 2 . A more realistic approach should incorporate the breakdown of the model as the dividends go to zero. Furthermore, it would be interesting to consider other possibilities for the intertemporal utility rate function of the informed trader.
74
5 5.1
Appendices Representative Noise Trader
The characterization (2) of the representative noise trader's order flow relies upon a simple model of continuous partial adjustment of the representative noise trader's underlying desired inventory. To see this, write S{t) and S*(t) for the noise trader's actual inventory, and the desired one, respectively, and let U(t) be the noise trader's order flow. Then the assumptions that the representative noise trader trades smoothly and is subjected to random liquidity needs are expressed respectively by the equations U(t)dt = dS(t),
(67)
dS*(t) = crdwu(t),
(68)
and
where a is a suitable positive constant, and wa(t) is a Wiener process. Now, to model the idea that the representative noise trader pushes his actual inventory smoothly towards its desired level, we can write U(t) = -a(S(t)-S*(t)),
(69)
where a is a positive parameter. Indeed, by (69), the representative noise trader's actual inventory is always being pushed towards the desired one at a rate proportional to the difference of the two, and the parameter o.\j measures the intensity by which the representative noise trader tries to keep his inventory at the desired level. If otu is large, the representative noise trades is "impatient", and he trades a great deal. On the contrary, if au is small, the representative noise trader is "patient", his trade flow is small. Finally, differentiating (69), and substituting (67) and (68), it easily follows dU(t) = -avU(t)
dt + av dw(t),
where a\j = otua. 5.2
Informed Trader's Wealth
Consider an investor who invests in two assets: a risky assett and a riskless one. The riskless asset, of current value Po(t), pays a constant interest rate r > 0, so that it is characterized by the equation
dP0(t) = rP0(t)dt .
(70)
75
The risky asset, of current value P(t), pays a continuous dividend, D(t). We write Zo{t) and Z(t) for the number of shares of the riskless assett and of the risk assett, respectively, owned by the investor at the time istant t, and we write m(t) for the investor's wealth at t. Then m(t) = Z0{t)P0(t) + Z(t)P(t).
(71)
We assume first that the trading of assets takes place at discrete time istants, say t and t + At. Then, as time flows from t to t + At, the variation in the investor's wealth due to the variations in the prices of the assets and to the dividend reward is Z0{t)P0 (t + At) + Z{t)P (t + At) + Z(t)D {t + At) At. After having observed the above variations, the investor changes the composition of his portfolio, and if there is no infusion or withdrawal of funds (self financing strategy), the new composition is related to the old one by the equation Z0 (t + At) P0 (t + At) + Z (t + At) P(t + At) = Z0(t)P0 (t + At) + Z(t)P (t + At) + Z(t)D (t + At) At. In this case, the total variation in the investor's wealth is given by m(t + At) - m(t) = Z0(t) [Z0 {t + At) - P0(t)] +Z(t)[P{t + At)-P(t)} +Z(t)D (t + At) At. On the other hand, if the investor needs to consume, from the time instant t to the time instant t + At, an amount Ate (t + At) of his wealth, for instance owing to the living expenses, then the total variation is given by m (t + At) ~ m{t) = Z0(t) [Po (t + At) - P0{t)] +Z(t)[P(t + At)-P(t)} +Z(t)D (t + At) At - Ate (t + At).
(72)
The continuous-time analogous of (72) is dm{t) = Z0{t) dPQ{t) + Z(t) dP(t) + Z(t)D{t) dt - c{t) dt, and, combining (73) with (70) and (71), we obtain dm(t) = rm{t) dt + Z(t) [(D(t) - rP(t)) dt + dP(t)] - c(t) dt, as desired.
(73)
76
References 1. K. Back: Insider Trading in Continuous Time, The Review of Financial Studies, 5, 3, 387-409 (1992). 2. J.Y. Campbell, A.S. Kyle: Smart Money, Noise Trading and Stock Price Behaviour, Review of Economic Studies, 60, 1-34 (1993). 3. , J.B. Delong, A. Shleifer, L.H. Summers, R.J.Waldmann: Noise Trader Risk in Financial Markets, Journal of Political Economy 98, 703-738 (1990). 4. G. Gennotte: Optimal Portfolio Choice under Incomplete Information, The Journal of Finance, XLI, 3, 732-746 (1986). 5. G. Gennotte, A.S. Kyle: Intertemporal Information Aggregation, Working Paper, University of California, Berkeley (1991). 6. Karatzas, I., Sheevre, S.E. (1988): Brownian Motion and Stochastic Calculus. Springer-Verlag, New York. 7. A.S. Kyle: Continuous Auctions and Insider Trading, Econometrica, 53, 1315-1335 (1985). 8. W.H. Fleming, R.W. Rishel: Deterministic and Stochastic Optimal Control Springer Verlag, New York, 1982. 9. W.H. Fleming, H.M. Soner: Controlled Markov Processes and Viscosity Solutions Springer Verlag, New York, 1993. 10. R.S. Lipster, A.N. Shiryaev: Statistic of Random Processes I, II, Springer Verlag, New York, 1977. 11. J. Yong, X.Y. Zhou: Stochastic Controls: Hamiltonian Systems and HJB Equations Springer Verlag, Berlin, 1999. 12. P. Veronesi: Stock Market Overreaction to Bad News in Good Times: A Rational Expectation Equilibrium Model The Review of Financial Studies, 12, 5, 975-1007 (1999).
Quantum Information V Eds. T. Hida and K. Saito (pp. 77-87) © 2006 World Scientific Publishing Co.
EXISTENCE, UNIQUENESS, CONSISTENCY A N D D E P E N D E N C Y ON D I F F U S I O N COEFFICIENTS OF GENERALIZED SOLUTIONS OF N O N L I N E A R DIFFUSION EQUATIONS IN COLOMBEAU'S A L G E B R A HIDEO DEGUCHI Department of Mathematics, Hiroshima University Higashi-Hiroshima 739-8526, Japan [email protected] A b s t r a c t . In this paper, we discuss solutions of the Cauchy problem for nonlinear diffusion equations in the framework of generalized functions introduced by Colombeau and extend the results in Biagioni and Oberguggenberger [2] to more general equations. We obtain results on existence and uniqueness of a generalized solution, which is shown to be consistent with the classical solution. Furthermore, we establish the relationship between two generalized solutions corresponding to different diffusion coefficients and initial data.
1.
Introduction
In 1982, Colombeau introduced the algebra Q of generalized functions to deal with the multiplication problem for distributions (see [3], [4]). This algebra Q is a differential algebra which contains the space of distributions V and has the space of smooth functions C°° as a subalgebra. Furthermore, this algebra Q has a better structure than the space V as seen from the following fact. We can not define the product of distributions in V in general, but for a certain class of V, we can define the one with the use of smooth approximations by convolution with mollifiers. In this sense, we have H = H2 in £>'(R), where H is the Heaviside function. On the other hand, H ^ H2 in £ ( R ) . This implies that the algebra Q carries a lot of information not contained in the space V. Besides the usual equality in Q, there exists a weak equality called association (denoted by « ) . This concept brings information of elements of Q down to the level of distribution theory. Indeed, we have H « H2. Also, in the algebra Q we can perform nonlinear operations more general than the multiplication (cf. Section 2). Therefore we can deal with nonlinear differential equations with singular data and coefficients in this setting. For example, generalized solutions of nonlinear parabolic equations with singular data are studied in Colombeau and Langlais [5] and Langlais [8]. But, we are interested in generalized solutions of nonlinear diffusion equations with generalized constants as diffusion coefficients. Generalized solutions of such an equation have been studied by Biagioni and Oberguggenberger [2]. More
78
precisely, they have studied generalized solutions of the Cauchy problem (ut+ uux = fiuxx \M|t=o=wo
in Gs,g([0,T] x R), in £ S , 3 (R),
^'^
where /x is a generalized constant belonging to the algebra Gs,g of generalized functions, which is a modified version of the algebra introduced by Colombeau [3], [4]. This algebra QStg(Rd) contains the space of bounded distributions VLOO(Rd). They formulated the Cauchy problem ( ut + uux = 0, \u|t=o = wo,
0
*• ' '
in Gs,g([0,T} x R ) , in
' '
as fut+u^RiO \w|t=o=«o
in the present setting, where " « " denotes the association relation on Gs,g, and showed that a generalized solution of Eq. (1.1) with ( i « 0 satisfies Eq. (1.3) if it satisfies a boundedness assumption. Furthermore, they showed the existence and uniqueness of a generalized solution of Eq. (1.1) and showed that, if the initial data belongs to L°°(R) and /i « 0, the generalized solution of Eq. (1.1) is associated with the weak solution of Eq. (1.2) which satisfies the entropy condition and, if /i is a positive number, the generalized solution of Eq. (1.1) is associated with the unique classical solution satisfying Eq. (1.1) in the classical sense. In this paper we will study generalized solutions of the Cauchy problem f ut + fit,x,u)ux \u|t=o = «o
+ g(t, x,u)u = [iuxx
in Qs,g([0,T] x R), in G»,g(R),
^ ^. '
where ^ is a generalized constant. Our purpose in this paper is to extend the results in Biagioni and Oberguggenberger [2] to a more general equation (1.4). This paper is organized as follows: We first recall the definition and properties of the Colombeau algebra Gs,g(ty in Section 2. In Section 3, we describe the results which were obtained recently by the author [6], namely, the existence and uniqueness results of a generalized solution of Eq. (1.4) (Theorems 3.1 and 3.2), and consistency results with classical (weak) solutions (Theorems 3.3 and 3.4). In Section 4, we investigate how two generalized solutions ui,U2 of Eq. (1.4), which arise from different diffusion coefficients /ii,/J2 and initial data ito,i,wo,2, are related to each other. In particular, we are interested in the relationship between «i and «2 for the case that both n\ and ^ are associated with 0. Indeed, under certain assumptions on UQ,I and UQ,2, we
79
can show that u\ and u
The Colombeau algebra of generalized functions
We briefly recall the definition and properties of the algebra Qs^g of generalized functions, which is a modified version of the algebra introduced by Colombeau [3], [4]. _ Let n be a nonempty open subset of R d and let fi be its closure. We denote by T>L-*> (CI) the algebra of restrictions to Q of real valued and bounded smooth functions on R d whose derivatives are bounded. Let £S)9[fi] be the algebra of all maps from the interval (0,1] into £>z,°° (H). Thus each element of £SiS[f2] is a family (ue)ee(o,i] of real valued and bounded smooth functions on ^- ^M,s,9[n] is defined by all elements (u £ ) e 6 ( 0 j l ] of £ s , s [n] with the property that for all a £ NQ, there exist J V e N , c > 0 and n > 0 such that sup \D"ue(x)\
< ce~N
for 0 < e < n.
•/Vs,9[n] is defined by all elements (u £ ) eG ( 0i i] of £ s , s [^] with the property that for all a £ NQ and q £ N , there exist c > 0 and n > 0 such that s u p | D > £ ( a ; ) | < ceq
for 0 < e < r\.
The algebra of generalized functions Gs,g(fy is defined by the quotient space &, fl (n) = £M,.,a[fi)/JV.,fl(fi). Here "s" means "simplified" and "g" means "global". The algebra Gs,g(ty on an open subset Cl of R d is defined by the same way. We denote by (we)£e(o,i] a representative of a generalized function u £ C/Sig(f2). Then for generalized functions u,v £ Gs,g(ty and any a € N Q , we can define the partial derivative D%u to be the class of (D%us)£e(0ti] and the product uv to be the class of (u £ u e ) ee (o,i]- Also, for any generalized function u £ Gs,g([0,T] x R), we can define its restriction u | t = 0 to {t = 0} to be the class of (u £ (0,z)) e6 (o,i]. In the algebra Gs,g, we also can define nonlinear operations more general than the multiplication. To see this, we define the following notion. Definition 2.1 We say that a function / € C°°(R d ) is slowly increasing at infinity if for all a € N$ there exist c > 0 and r £ N such that, for all a; £ R d , \Daf(x)\
< c(l + \x\)r.
80
We denote by Ojvr (R d ) the space of slowly increasing functions at infinity. If / € OM(RP) anduj G ^ S ] 5 (R d ) for i — 1 , . . . ,p, we can define a generalized function f{uu • • •, up) G £ s , s (R d ) to be the class of (f(u\,..., u£)) e6 (o,i]For details see [1], [3], [4]. Definition 2.2 A generalized function u G Gs,g(ty is said to be associated with a distribution w G V(0,) if it has a representative (u£)eg(o,i] e £M,S, S [^] such that ue —>w
in V'(n)
as e -> 0.
We denote by u as to if u is associated with u;. In other words, a generalized function u G <7S)S(fi) is associated with a distribution w if u behaves like w on the level of information of distribution theory. Remark 2.3 The algebra £/s,s(Rrf) contains the space of bounded distributions VLoa (R d ) as follows: Let T be an element of VLao (R d ). Then a family (T * Pe{x))ee(o,i] c a n be a representative of T and its class is associated with T, where p is a fixed element of <S(Rd), the space of rapidly decreasing smooth functions, satisfying / p(x) dx = l,
/ xap(x) dx = 0,
for all a G Ng, |a| > 1,
(2.1)
and
In this sense, we obtain an inclusion relation P^oo(R d ) C ^ s , s ( R d ) . Furthermore, in the case T G 2?L°° (R d ) 5 w e have that (T-T*pe)£e(o,i]GATs,9[Rd], by applying Taylor's formula and (2.1). Hence we can take a family (T) ee ( 0 ,i] as a representative of T. Thus T>L°° (R d ) is a subalgebra of £ S i 3 (R d ). Definition 2.4 A generalized function p, G Gs,g(ty is called a generalized constant if it has a representative (ne)ee(o,i] which is constant for each e G (0,1]. We call a generalized constant p, a generalized positive number if it has a representative (/xe)£6(0ii] satisfying that there exist N G N and TJ > 0 such that eN
81 Definition 2.5 We say that a generalized function u G Gs,g{ft) is of bounded type if it has a representative (uf)eg(o,i] G £M,s,g[ty satisfying sup |u£(a;)| < c for 0 < e < 77 for suitable c > 0 and rj > 0. We note that u G i ° ° ( R ) , viewed as an element of £S)ff (R), is of bounded type. Definition 2.6 The support of a generalized function u € Gs,g(ft) is defined by the complement of the largest open subset ft such that (ue\Q>)£^o 1] € We note that u G £ s , 9 (n) has compact support if and only if it has a representative (u £ ) £6 ( 0 ,i] satisfying that the supports of u£(-) are contained in a common compact set for all sufficiently small e G (0,1]. 3.
Existence, uniqueness and consistency results
Here, we describe the results which were obtained recently in [6]. The following theorems establish the existence and uniqueness of a generalized solution of Eq. (1.4). Theorem 3.1 Assume that f,g£ C°°(R 3 ) satisfy the following conditions: for every a G N3,, there exist c > 0 and r G N such that, for all (t, x, u) G R 3 , \Daf(t,x,u)\
+ \u\)r, + \u\Y,
and g > 0. Let u$ G Gs,g(R) and let ju be a generalized positive number. Then for each T > 0 there exists a solution u G Gs,g([0,T] x R) of Eq. (1.4). Theorem 3.2 Assume that f,g are functions as in Theorem 3.1 and that a representative (fJ-E)ee(o,i] °f a generalized positive number ji satisfies / / log - > 1 for any 0 < e < n with n > 0. Then for each T > 0 the solution u G Gs,g([0,T] X R) of bounded type of Eq. (1.4) is unique. Our next results show that the generalized solution is consistent with the classical (weak) solution in the sense of association.
82 Theorem 3.3 Assume that F is an element of C°°(R 3 ) satisfying that for any a £ N § there exist c > 0 and r £ N such that for all (t, x, u) e R 3 \DaF(t,x,u)\
< c ( l + |w|) r ,
(3.1)
and for all (t, x, u) £ R 3 u-Fx(t,x,u)>0.
(3.2)
Furthermore, assume that for all u and (t, x) £ [0, T] x R Fuu > 0, and that there exist positive numbers r and A such that Fuu > A > 0 for bounded u and 0 < t < T. Let UQ E L°°(R) and let /i be as in Theorem 3.2 and n sa 0. Finally, let v £ Gs,g([0,T] x R) be the solution of the Cauchy problem (vt + (F(t,x,v))x=nvxx \v\t=o = uo
in G*,g([0,T] x R), in G,,g(R-)-
,
Then v is associated with the weak entropy solution u of the Cauchy problem ut + (F(t,x,u))x=0, u\t=0 = uo,
0
Theorem 3.4 Let f and g be functions satisfying the conditions given in Theorem 3.1. Let u0 G L°°(R) and let p be a fixed positive real number. Furthermore, let v £ £ s , 9 ([0, T] x R) be the solution of Eq.{\A). Then v is associated with the bounded classical solution u of the equation ut + f{t,x,u)ux
+ g{t,x,u)u
= imxx,
0 < t < T, i £ R
satisfying the initial data uo in the sense that for any continuous function ip on R with compact support, roo
OO
/
u(t,x)ip(x)dx = /
u0(x)ip(x)dx.
(3.5)
J-oo Remark 3.5 It is well-known (Oleinik [11]) that Eq. (3.4) has the unique weak entropy solution under some condition on F. In Theorem 3.3 we need, in addition to the conditions on F as in Oleinik [11], only the requirement that F is smooth and polynomially bounded, together with all derivatives, to obtain the result that the generalized solution of Eq. (3.3) is associated with the weak entropy solution of Eq. (3.4).
83
4.
Dependency of generalized solutions on diffusion coefficients and initial data
In this section, we will investigate the relationship between two generalized solutions of the Cauchy problem ut + (F(t,x,u))x=/J,UXX in Qs,g([0,T] x R ) , .^^ u\t=o = u0 in Qs,g{R), which arise from different diffusion coefficients and initial data. For this purpose we need the following lemmas. Lemma 4.1 Assume that u is a continuous function in [0,T] x R d having continuous derivatives Ut,uXi and uXiXj for i, j = 1 , . . . , d, and that it satisfies the growth condition \u(t,x)\ < M ( l + |z|) for some constant M > 0. Furthermore, assume that u satisfies the equation d
d
ut - ] P a,ij(t,x)uXiXj
+ y"lOj(t,x)u Xi + a(t,x)u
i,j=l
=
f(t,x)
i=l
for 0 < t < T, that the moduli of the coefficients aij, a, do not exceed c and that a(t,x) > — ao, where c and ao <we nonnegative constants. Then the estimate sup
\u(t,x)\ < (sup \u(0,x)\+T
(t,x)e[0,T]xR d
xeRd
is valid if Y?i,j=i aH(*>x)&tj
sup (t,x)€[0,T]xRd
>0 for all £ =
fa,...,£d)
\f(t,x)\)exp(a0T) € Rd.
Proof. The assertion of Lemma 4.1 can be shown by a slight modification of the proof for [7], Chapter 1, Theorem 2.5.1 Lemma 4.2 Assume that F satisfies conditions (3.1) and (3.2). Let uo £ <7s,a(R) and let n be a generalized positive number. Furthermore, let (u£)se(o,i] S £M,«,g[[0,T]xR] be a representative of a solution u €
0
(42)
for respective representatives (uQ)se(oi] and (^ e ) ee (o,i] of UQ and fi. Then the inequality sup (t,i)6[0,T]xR
\fJ.sK(t,x)\
< c(l + T)(l + sup K ( z ) | r + »e sup |(u§)'(aO| z€R.
16R
holds for each e € (0,1], where c > 0 and r £ N depend only on F.
84
Proof. We use an argument similar to the one in the proof for Proposition 2.1 in Marcati and Natalini [9]. By Lemma 4.1, the conditions on F and Eq. (4.2), we have the inequality \ue(t,x)\
sup
< SUP\UQ(X)\
(t,i)6[0,T]xR
(4.3)
x6R
for each e 6 (0,1]. Multiplying the first equation in Eq. (4.2) by we have the equation
Fu(t,x,u£),
(F(t,x,ue))t - Ft(t, x,u£) + Fu{t,x,u£)(F(t,x,u£))x e s = n {(F(t,x,u ))xx Fxx{t,x,ue) -2Fxu(t, x,u£)u% -Fuu(t, x,u£)(uex)2}.
(4.4)
Differentiating the first equation in Eq. (4.2) with respect to x, we have the equation (u£x)t + Fxx(t, x, u£) + 2Fxu(t, x, u£)u£x +Fuu(t,x,u£)(u£x)2
+ Fu(t,x,u£)(u£x)x
=fi£(u£x)xx.
(4.5)
From Eqs. (4.4) and (4.5) it follows that (F(t,x,u£))t
- Ft(t,x, u£) + Fu(t,
= fi£{(F(t,x,u£))xx
x,u£)(F(t,x,u£))x
+ (u£x)t + Fu(t,x,u£)(u£x)x
-
n£{u£x)xx}.
Hence we have the equation (F(t,x, u£) - n£u£x)t - Ft(t, x,ue)+ = »e(F(t,x,uE)-n£u£x)xx. Put w£ = F(t,x,u£) equation
Fu(t,x,u£)(F(t,x,u£)
- j/<)x (4.6)
— n£u£x. Then by Eq. (4.6) the function w£ satisfies the
w\ + Fu(t, x, u£)w% = n£w£xx + Ft(t, x, u£). Applying Lemma 4.1, we obtain the inequality sup (t,x)e[0,T]xB.
\w£{t,x)\ < ( s u p | w e ( 0 , z ) | + T
sup
xGR
(J,i)£[0,T]xR
\Ft(t,x,u£)\).
Therefore from the conditions on F and inequality (4.3) the assertion follows. I Let |/|* be the pseudo-norm introduced by Lax [10] as y
|/|» = sup I /f
f(x)dx
y€R\Jo
for locally integrable functions / on R, but possibly infinite. Now we cite the following lemma from [2] (for the proof, see [2], Lemma 3.2).
85 Lemma 4.3 / / |/ e |» ->• 0 as s ->• 0 then fe - • 0 in £>'(R). The converse is true if {/e}eg(o,i] w a bounded subset of L°°(R) and the supports of the functions fe are contained in a common compact set. The following proposition follows from Lemma 4.1 and a similar argument to the proof for Proposition 5.2 in [6]. Proposition 4.4 Assume thatF satisfies conditions (3.1) and (3.2). Fori — 1,2, letuoj £ -L°°(R) and let ^i > 0. Furthermore, letui be a bounded classical solution of the equation ut + (F(t, x,u))x
= HiUxx,
0
satisfying the initial data uo,i in the sense of Eq. (3.5) for i = 1,2 and assume that |u 0 ,i — wo,2|* is finite. Then for each T > 0, sup \m{t, •)
-u2(t,-)\*
0
We now turn to a comparison between two generalized solutions of Eq. (4.1), which arise from different diffusion coefficients and initial data. Theorem 4.5 Assume that F satisfies conditions (3.1) and (3.2). For i = 1,2, letuoti £ Qs^g(H) have compact support, and let u^^ andu'0i be of bounded type. Furthermore, let /x, be a generalized positive number of bounded type satisfying fii Ifi2 »* 1- Finally, letui £ £/»,9([0,T] x R) be the solution constructed in Theorem 3.1 of Eq. (4.1) with fi = /i$ and «o = wo,i for i = 1,2, respectively. Then we obtain ui « u2 if uo,i « t*o,2Proof. By the assumption of compact support on UQ^, we have that uo,i has a representative (uo,;)£€(o,i] s u c n that the supports of UQ t(-) are contained in a common compact set for all sufficiently small e £ (0,1]. Hence, from the assumptions that t*o,i is of bounded type and that uo,i ~ wo,2, we can apply Lemma 4.3 to a family (UQ1 — UQ2)ee(o,i] a n d see that \ue0 x — u^2\* —* 0 as e —> 0. Furthermore, from the proof for Theorem 3.1, we can see that the solution U; e Gs,g([0, T] x R) constructed in Theorem 3.1 of Eq. (4.1) has a representative (wf)ee(o,i] £ £M,S, 3 [[0, T] X R] which satisfies Eq. (4.2) with / / = \i\ and u% = UQ i: where (/4)ee(o,i] is a representative of /z;, for i = 1,2. Hence, by Lemma 4.2 and the assumptions that uo,2, u'0 2 and \i2 are of bounded type, we obtain the uniform boundedness of [i\dxu\ for sufficiently small e € (0,1]. Also, the assumption \i\jii 0 as e —> 0. Therefore from Proposition 4.4 the assertion follows. I
86
Remark 4.6 In the case UQ,I = uo,2, we can drop the assumption of compact support on uo,i for i = 1,2. Remark 4.7 In Theorem 4.5, we assume that fii has a representative (/4 )ee(o,i] satisfying the inequality /if > M for all sufficiently small e € (0,1] and i = 1,2 with a suitable constant M > 0 in place of the assumption that Hi is of bounded type. Then the assertion of Theorem 4.5 holds under the assumption Hi w /J 2 • Remark 4.8 We can take f(t,x,u) = u and g(t,x,u) = 0 in Theorems 3.1,3.2 and 3.4, and F(t,x,u) — u2/2 in Theorems 3.3 and 4.5. Hence, our results include the ones in Biagioni and Oberguggenberger [2]. Acknowledgments The author is most grateful to Professor Takeyuki Hida and Professor Kimiaki Saito giving him the opportunity to talk in the conference. He also would like to express his hearty thanks to Professor Izumi Kubo of Hiroshima University for the guidance concerning this paper. References 1. H. A. Biagioni, A nonlinear theory of generalized functions, Lecture Notes in Mathematics, Vol. 1421, Springer-Verlag, Berlin, 1990. 2. H. A. Biagioni and M. Oberguggenberger, Generalized solutions to Burgers' equation, J. Differential Equations 97 (1992), 263-287. 3. J. F. Colombeau, New generalized functions and multiplication of distributions, North-Holland Math. Studies, Vol. 84, North-Holland, Amsterdam, 1984. 4. J. F. Colombeau, Elementary introduction to new generalized functions, North-Holland Math. Studies, Vol. 113, North-Holland, Amsterdam, 1985. 5. J. F. Colombeau and M. Langlais, Generalized solutions of nonlinear parabolic equations with distributions as initial conditions, J. Math. Anal. Appl. 145 (1990), 186-196. 6. H. Deguchi, Generalized solutions of nonlinear diffusion equations, Hiroshima Math. J. 32 (2002), 125-143. 7. O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Translations Math. Monographs, Vol. 23, Amer. Math. Soc, Providence, RI, 1968.
87
8. M. Langlais, Generalized functions solutions of monotone and semilinear parabolic equations, Mh. Math. 110 (1990), 117-136. 9. P. Marcati and R. Natalini, Convergence of the pseudo-viscosity approximation for conservation laws, Nonlinear Anal. 23 (1994), 621-628. 10. P. D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. Pure Appl. Math. 7 (1954), 159-193. 11. O. A. Oleinik, Discontinuous solutions of non-linear differential equations, Amer. Math. Soc. Transl. (2) 26 (1963), 95-172.
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Quantum Information V Eds. T. Hida and K. Saito (pp. 89-101) © 2006 World Scientific Publishing Co.
O N MATHEMATICAL T R E A T M E N T OF Q U A N T U M COMMUNICATION GATE ON FOCK SPACE
WOLFGANG FREUDENBERG Fakultat
Brandenburgische Technische Universitdt Cottbus, Germany 1, Institut fur Mathematik, PF 101344, D-03013 Cottbus, Germany freudenbergQMath. TU-Cottbus.DE
MASANORI OHYA AND NOBORU WATANABE Department
of Information
E-mail:
Sciences, Tokyo Chiba 278-8510, [email protected] and
Universiy of Sciences, Noda Japan [email protected]
City,
In usual computer, there exists an upper bound of computational speed because of irreversibility of logical gate. In order to avoid this demerit, Fredkin and Toffoli 4 proposed a conservative logical gate. Based on their work, Milburn 5 constructed a physical model of reversible quantum logical gate with beam splittings and a Kerr medium. This model is called FTM (Fredkin - Toffoli - Milburn gate) in this paper.This FTM gate was described by the quantum channel and the efficiency of information transmission of the FTM gate was discussed in u . FTM gate is using a photon number state as an input state for control gate. The photon number state might be difficult to realize physically. In this paper, we introduced a new device on symmetric Fock space in order to avoid this difficulty. In Section 1, we briefly review quantum channels and beam splittings. In Section 2, we explain the quantum channel for FTM gate In Section 3, we intruduced a new device on symmetric Fock space and discuss the truth table for our gate.
1. Quantum channels Let (B(Hi),6(Wi))and (B(H 2 ),S(W 2 )) be input and output systems, respectively, where H(Hk) is the set of all bounded linear operators on a separable Hilbert space Hk and &{Hk) is the set of all density operators on Hk (fc = 1,2). Quantum channel A* is a mapping from G(Hi) to &(%)• (1) A* is linear if A*(\Pl + (1 - X)p2) = XA* (Pl) + (1 - A)A* (p2) holds for any px,p2 6 &{H\) and any A e [0,1]. (2) A* is completely positive (C.P.) if A* is linear and its dual A :
90
B(W 2 ) - • B(Wi) satisfies n
i,j—1
for any n e N , any { A J C B(W 2 ) and any {Ai} c B(Wi), where the dual map A of A* is defined by trA*(p)B = trpA(B), V/» € 6(Wi), VB e B(W 2 ). (1) Almost all physical transformation can be described by the CP channel 6
8
9
Let /Ci and /C2 be two Hilbert spaces expressing noise and loss systems, respectively. Quantum communication process including the influence of noise and loss is denoted by the following scheme 7 : Let p be an input state in &CHi), £ be a noise state in & (fCi).
I &{Hi)Bp
•
p € A*P e 6 (7Y2)
I Loss ®(Wi) 7* I 6(Wi®/Ci)
A* , ^ n*
6(W 2 ) T a* 6(W 2 ®/C 2 )
The above maps 7*, a* are given as l*(p)=P®H,
pGS(Wi),
a* W = t r ^ f f ,
(2) (3)
The map II* is a channel from S ( Hi®ICi)to & (?i-2
( p » 0 = ( a * ° n * o 7 *) (p)
(4)
for any p e &(7ii). Based on this scheme, the attenuation channel and the noisy quantum channel are constructed as follows: (1) Attenuation channel A*> was formulated such as A5(p)= t r i l l s (p®£ 0 ) = trK2Vo(p®\0)(0\)V*,
(5)
91 where £ 0 = |0)(0| i s the vacuum state in ©(/Ci), Vo is a mapping from Hi ® K-i to W2 ® £2 given by Vodm) ® |0» = f ^ C f l J ) ® K - j ) ,
(6)
\ni) is the rii photon number state vector in Hi and a and /? are complex numbers satisfying \a\ + |/3| = 1. In particular, for the coherent input state p = \9) (9\ <8> |0) (0| e © (Hi®ICi), we obtain the output state of lip by Yl*0 (\9) (9\ ® |0) (0|) = \Q9) {a6\ ® | - 0 0 ) ( - 0 0 | . 10) <0|
I \0) (0\—\^
\<*e) (a9\
I
\-m(-&\ Fig 1.1 Beam Splitting rig Lifting £Q from © (H) to © {H®K,) in the sense of Accardi and Ohya 1 is denoted by £; (\9) {9\) = \a9) (a9\ ® \/30) (0O\. £Q (or Ilo) is called a beam splitting. Based on liftings, the beam splitting was studied by Accardi - Ohya and Fichtner - Preudenberg - Libsher 3 . (2) Noisy quantum channel A* with a noise state £ is defined by
A*(p)=trz2n*(p®0 = trK2V(p®QV\
(8)
where £ = |mi)(mi| is the rrii photon number state in ©(/Ci) and V is a mapping from Hi ® /Ci to Hi ® IC2 denoted by ni+mi
V(\m)®\mi))=
Yl j
Cf;i,mi|j>®|n1+m1-j),
92
C" 1 '" 11 ^ XQ
(9) rKni-jJIO'-rJI^x-j+r)!
,mi-j+2r /'_S\™i+J-2r
("*)'
if and L are constants given by K = min{rai,j}, L = max{mi — j , 0}. In particular for the coherent input state p = \9) (6\ ® \K) {K\ £ S (Wi<8>/Ci), we obtain the output state of II* by IT (\9) (0\ ® |/e)
+ an)
(-06
+
an\
\K) (K\
I |0> (°\ —• \ —^ \a0 + PK) (aO + i
0K\
\-(36 + aK)(-/3e + aK\ Fig 1.2 Generalized Beam Splitting II* II* was defined by Ohya - Watanabe beam splitting.
10
, which is called a generalized
2. Quantum channel for Fredkin-Toffoli-Milburn gate In usual computer, we could not determine two inputs for the logical gates AND and OR after we know the output for these gates. This property is called an irreversibility of logical gate. This property leads to the loss of information and the heat generation. Thus there exists an upper bound of computational speed. Fredkin and Toffoli proposed a conservative gate, by which any logical gate is realized and it is shown to be a reversible gate in the sense that there is no loss of information. This gate was developed by Milburn as a quantum gate with quantum input and output. We call this gate Fredkin-ToffoliMilburn (FTM) gate here. Recently, we reformulate a quantum channel for the FTM gate and we rigorously study the conservation of information for FTM gate n . The FTM gate is composed of two input gates 1^ I 2 and one control gate C. Two inputs come to the first beam splitter and one spliting input passes through the control gate made from an optical Kerr device, then two spliting inputs come in the second beam splitter and appear as two outputs
93 (Fig.2.1). Two beam splitters and the optical Kerr medium are needed to describe the gate. h
Optical Kerr Device C, _
^
-&s fiS,(T? = 0.5)
BSin = 0.5)
N
^X
O,
o,
Fig 2.1 FTM gate (1) Beam splitters: (a) Based on 10 , let V\ be a mapping from Hi ®H2 to Hi ® H2 with transmission rate r]x given by
ni+ri2
Vi (M ® |n2» = Y, C^'n2\j)®\ni+n2-j)
(10)
for any photon number state vectors \ni)
KB
si {Pi ®p2) = Vi (pi ® p2) V{
(11)
for any states p1 ® p2 € S(7ii ®H2). In particular, for an input state in two gates Ii and I2 given by the tensor product of two coherent states Pi ® P2 = |#i)(SiI <S> 1^2)(&21, I I B S 1 ( P 1 ® p 2 ) is written as
94
n*BSl(Pl®P2)
(12) (b) Let V2 be a mapping from H\ ® H2 to Hi ® H2 with transmission rate 7/2 given by "1+712
V2 (|m> ® |n 2 )) = J ]
^"Mni+na-i)
|J>
(13)
for any photon number state vectors |ni) ® |n 2 ) € H i ® H 2 - The quantum channel I I ^ S 2 expressing the second beam splitter (beam splitter 2) is defined by n
s S 2 (Pi ® P2) = V2 (Pi ® Pa) ^2*
(14)
for any states Pi <8>p2 € ©(Hi®H2). In particular, for coherent input states Pi ® P2 - l^i)(^il ® \62){62\, n3 S2 (/>! ®p 2 ) i s written as n
BS2(Pl ® P2) = V^201 - y / l - ^ y (\APl - V 1 - ^2^2
® V 1 - »fe0i + V ^ } ( V 1 - ^ ^ + V%6,2 • (15) (2) Optical Kerr medium: The interaction Hamiltonian in the optical Kerr medium is given by the number operators iVi and Nc for the input system 1 and the Kerr medium, respectively, such as Hint = h\ (iVi
(16)
where h is the Planck constant divided by 27r, \ 1S a constant proportional to the susceptibility of the medium and I2 is the identity operator on H2Let T be the passing time of a beam through the Kerr medium and put VF = hxT, a parameter exhibiting the power of the Kerr effect. Then the unitary operator UK describing the evolution for time T in the Kerr medium is given by
UK = exp (-iV¥ (Ni ®I2® Nc)) •
(17)
95
We assume that an initial (input) state of the control gate is the n photon number state £ = |ra) (n|, a quantum channel A^ representing the optical Kerr effect is given by Hk(Pi®P2®Q
= UK(p1®p2®S)UZ
(18)
for any state p1 ® p 2 ® £ € 6 (Hi ® H2®IC). In particular, for an initial state px ® p 2 ®£ - \6i) (8i\® \02) (62\ ® \n) (n\, A*K(px ® p2 ® £) is denoted by
=
exp ®|02)(02|®|n)(n|.
(19)
Using the above channels, the quantum channel for the whole FTM gate is constructed as follows: Let both one input and output gates be described by Hi, another input and output gates be described by Hi and the control gate be done by K., all of which are Fock spaces. For a total state px (g>p2®£ of two input states and a control state, the quantum channels ABS1, A*BS2 from 6{Hi ®H2®K) to G(Hi ®H2®K) are written by VBSk{Pi®P2®0
= 1tt*BSk{p1®p2)®(,
(A; = 1,2),
(20)
Therefore, the whole quantum channel Ap T M of the FTM gate is denned by A.FTM = h*BS2 oh*K0
K*BS1.
(21)
In particular, for an initial state p a ® p2 ® £ = |#i) (9i\ ® \92) (62\ ® \n) (n|, Ap T M (p2 ® p2®0 is obtained by A
FTM(PI®P2®£)
\V„0l + Vn02) (Hjl + VnH ® \vndi + p,J2) (vn6i + p„0 2 | ® \n) (n\
(22)
where
»k = \ {exp (-iy/fk) + l} , yk = | {exp (-iyffk)
- l} ,
(23) (k = 0,1,2,-••).
(24)
If y/F satisfies the conditions y/~F = 0 or y/F = (2k + 1) -K (k = 0,1,2, • • • ) , then one can obtain a complete truth table in FTM gate.
96 However, it might be difficult to realize the photon number state \n) {n\ for the input of the Kerr medium physically. In stead of the Kerr medium, we introduce new device related to symmetric Fock space. We contruct a quantum logical gate mathematically with this new device in the next section. 3. Quantum logical gate on symmetric Fock space In this section, we reformulate beam splittings on symmetric Fock space and we introduce a new operator instead of the Kerr medium on that space. We discuss the mathematical formulation of quantum logical gate by means of beam splittings and the new operator. Let G be a complete separable metric space and Q be a Borel c-algebra of G. v is called a locally finite diffuse measure on the measurable space (G,G) if v satisfies the conditions (1) v{K) < oo for bounded K G Q and (2) v ({x}) = 0 for any x G G. We denote the set of all finite integer valued measures
(K) = n} .
Let M be a u-algebra generated by Mx,n- F is the a-finite measure on (G, Q) defined by
F{Y) = i y (p0) + Y, A / !y f I > A »n (rfzi • ••<**»). n = l n'
JG
\j=l
"
/
where l y is the characteristic function of a set Y, ip0 is an empty configulation in M and 6X. is a Dirac measure in Xj. M = L2 (M,M,F) is called a (symmetric) Fock space. We define an exponetal vector exp. : M —> C generated by a given function g : G —+ C such that [1
(
Let / and g be functions from G to C and
ex
P / (&) • exPg (
exp/.p (
97 One can observe that exp g € M if and only if g £ L2 (G, v). Vc : dom(Vc) —> A42 is the compound Hida-Malliavan derivative 2 , 3 given on a dense domain dom (Dc) containing the exponential vectors by X>c* (
,iPl,ip2eM).
In particular, for an exponential vector exp„ with g G L2 (G, v) one has D c exp 5 = expg ® exp p . The compound Skorohod integral Sc : dom (Sc) —> .M is given on the dense domain dom (5 C ) C .A4 ® A1 containing tensor products of exponential vectors by
S c $ ( v ) = 5 ^ $ (£>, V - (p) ($ e dom (5 C ), y> € M). In particular, for tensor products of exponential vectors exp g ® exp h we have Sc (exp 9
(g, h G L2 (G, v)) .
Let T be a linear operator on l? (G, v) with ||T|| < 1. Then the operator r (T) called second quantization of T is the bounded operator on Ai satisfying T (T) exp s = exp T g . Clearly,
r(T 1 )r(T 2 ) = r(T 1 r 2 ), r (r*) = r (T)*. For a function * : M —» C we denote by O^ the multiplication operator by * , i.e., 0$9 (
beam splittings
€M,
on Fock space
Let a, j3 be measurable mappings from G to C satisfying a \a(x)\2
+ \P(x)\2=l,
(xeG).
We intoduce an unitary operator Vayp, V* ^ : M ® M -^ M ® M defined by Va,0 = (Oexpa ® 0«x P/J ) ^ ,
98 Then one can obtain
{Va,p$) (
Yl
exp
« (^i) exP/3 (Vi ~ ^1)
x exp_y5 (£ 2 ) exp a {
* (^1 +
for $ e M ® X and tp^tpz € M. Let .4 s i (W) be the set of all bounded operators on M. and S (A) be the set of all normal states on A. £a,p '• A ® A ->.A ® .A defined by £»w8 (C) = VlpCVa,p,
VC€A®A
is the lifting in the sense of Accardi and Ohya £a,p given by
l
and the dual map £* g of
is the CP channel from &{A® A) to &{A® A). Using the exponetial vectors, one can denote a coherent state 0* by 0f (A) = ( e x P / , Aexpf)
e"11'1'2,
V/ e L 2 (G, 1/), VA € A
In particular, for the input coherent states rj0
£* ^ is called a generalized beam splitting on Fock space because it also hold the same properties satisfied by the generated beam splitting II* in Section 1. Now we introduce a self-adjoint unitary operator U, which denotes a new device instead of the Kerr medium, defined by £($)(^2)EE (-1)1^1
$ ( ^
2
)
for 6 M ® M and ip1,tp2e G, where \
^2 (-A) = wi ® K (U(A®I)U") = •±2 \m\
[ JM
F{dv)\1>{
99 for any A G A, i> £ M (ip ^ 0) and / G L2 {G, v). If K is given by the vacuum state 6 , then the output state w2 is equals to uj\ and if K is given by one particle state, that is, K(-) = jhp {ip, mip) with ip f^f (where Mj is the set of one-particle states), then LO2 is obtained by 9~. Let M0 (resp. M e ) be the set of
Wl G A
where Ai and A2 are given by /AI = PF/M0^(^)IV'MI2,
lA2 = 1 ^ / M e F ( ^ ) | V ' M I 2 . Two output states 0/3 (•) = a;2 <8> % (£a2,02 ((") ®-0) a n d % (') = w2
^2Qa2(a1f+l31g)+P2(-01f+alg)
+ +
X2Q-02(cif+01g)+a2(-01f+a1g)
where w2 = A i 0 ~ ( a i / + / 3 l 9 ) + A 2 0 Q l / + / ? l 9 and % = »h = 0 " ^ / + a i 9 . 3.2. Complete
truth table for the new logical gate
In this section, we show the complete truth table giving by the above logical gate on Fock space. We put u>i = 6* and rj1 = 09. If we assume the case (1) of Ai = 0 and A2 = 1, then one has W3 = „
_
Q(aia2-0102)fH»201+&i02)gj 0(-aij32-"2/3i)/+(-/M2+"i"2)s
and if we assume the case (2) of Ai = 1 and A2 = 0, then one has Uj3
—
(){-aia2-0102)f+{a201+a102)g
„
_
Q(ai02-a20i)f+(~0102+ociO'2)g_
For example, we have the complete truth tables for the following two cases (I) and (II): (I) When a\ = a 2 = P\ = P2 = -4= are satisfied, two output
100 states of the new logical gate become UJ3 = 83 and r}3 = Q~* under the case (1) and u>3 = 0~' and r)3 = 99 under the case (2). (II) When a^ — ^-J- and Pk = ^ w i t n lk,&k& [0, 2TT] hold ajo^ = /?i^2> o n e h a s 7i +72 = ^2 -<5i and two output states of the new logical gate become LJ3 = 69 and rj3 = 9~' under the case (1) and W3 = 6~' and rj3 = 69 under the case (2). The new logical gate treats three initial states u0, r]0 and K corresponding to two input gates I\, I2 and the control gate C, respectively. The true T and false F of the input state UJ (resp. rj) are described by two different states UJT and wF (resp. r)T and r)F), that is; True O coherent state UJT — 6* (resp. r)T = 89), False <=> vacuum state.w^ = 0° (resp. rjF =9°). Moreover, the truth state KT and the false state KF are denoted by the control states of the case (1) and (2), respectively. When the initial control state K is KF under the above case (I) or (II), the final states of the new logical gate corresponding to two input gates 0\, O2 are obtained as the following truth table:
h
h
T F T F
T T F F
C F F F F
0, T F T F
O2
T T F F
When the initial control state K is KT under the above case (I) or (II), the final states of the new logical gate corresponding to two input gates 0\, O2 are obtained as the following truth table:
h
h
T F T F
T T F F
C T T T T
Oi
O2
T T F F
T F T F
It means that the new logical gate performs the complete truth table. Further results will be appear in our joint paper 12 .
101
References 1. L.Accardi and M.Ohya, Compound channels, transition expectations, and liftings, Appl. Math. Optim., 39, 33-59, 1999. 2. K.H. Fichtner and W. Freudenberg, Remarks on stochastic calculus on the Fock space, Quantum Probability and Related Topics VII, World Scientific Publishing Co., 305-323, 1991. 3. K.H. Fichtner, W. Freudenberg and V. Liebscher, Beam splittings and time evolutions of Boson systems, Fakultat fur Mathematikfi und Informatik, Math/ Inf/96/ 39, Jena, 105, 1996. 4. E.Fredkin and T. Toffoli, Conservative logic, International Journal of Theoretical Physics , 21 , 219-253, 1982. 5. G.J. Milburn, Quantum optical Fredkin gate, Physical Review Letters , 62, 2124-2127, 1989. 6. M.Ohya, Quantum ergodic channels in operator algebras, J. Math. Anal. Appl., 84, 318 - 327, 1981. 7. M.Ohya, On compound state and mutual information in quantum information theory, IEEE Trans. Information Theory, 29, 770 - 777, 1983. 8. M. Ohya, Some aspects of quantum information theory and their applications to irreversible processes, Rep. Math. Phys., 27, 19 - 47, 1989. 9. M. Ohya and D. Petz, Quantum Entropy and Its Use, Springer, 1993. 10. M. Ohya and N. Watanabe, Construction and analysis of a mathematical model in quantum communication processes, Electronics and Communications in Japan, Part 1, 68, No.2, 29-34, 1985. 11. M. Ohya and N. Watanabe, On mathematical treatment of optical Fredkin Toffoli - Milburn gate, Physica D, 120, 206-213, 1998. 12. W. Freudenberg, M. Ohya and N. Watanabe, Generalized Fock space approach to Fredkin - Toffoli - Milburn gate, SUT preprint.
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Quantum Information V Eds. T. Hida and K. Saito (pp. 103-119) © 2006 World Scientific Publishing Co.
A F R O N T I E R OF W H I T E NOISE ANALYSIS
TAKEYUKI HIDA Meijo University Nagoya 468-8502, Japan E-mail: [email protected]
White Noise Analysis, both classical and quantum, has developed extensively in these years, however it has been focussed mainly on the case of Gaussian white noise. Now, it seems to be a good time for us to pay more attention to non Gaussian white noise. There is, of course, much similarity between Gaussian and non Gaussian cases, but we are interested in the analysis that depends on dissimilarity between them. A significant direction will be proposed in this paper, aiming at the path-wise analysis of functionals of additive processes which are elemental processes.
AMS Subject Classification (2000) 60H40 1. Introduction Having had a very quick review of the analysis of (Gaussian) white noise functionals, A Frontier of White Noise Analysis will be proposed. When the white noise theory is discussed, functionals in question are defined on the measure space (E*, ii) of white noise, where E* is the space of generalized functions on R and /i is the standard Gaussian measure, called the white noise measure. The Hilbert space (L2) = L2(E*,(i) used to be taken as the starting point and at the same time it is considered as a milestone. To see the idea of our approach, we may say some words. If we are allowed to take a reluxed view of mathematical rigor, we will express functions and operations in the visualized forms in terms of B(t), that is white noise, to have representations of random complex phenomena. Then, one can think of how to analyze given functionals with variables 5(t)'s. Our aim is to investigate mathematically random evolutional complex systems by using the analysis of functionals of white noise, in particular we
104
shall discuss those systems which are actually observed in various applications. For the study the following steps are in order: Reduction
—»
synthesis
—>
analysis
The idea is that first the innovation of the random system is constructed, then functionals of the given innolvation are to express the original random system, and finally those functionals are to be analysed. The innovation can often be constructed by the variations of the given random evolutional phenomena, and sometimes, like in the communication theory, it is given in advance. Our attention will be focussed on the analysis of functionals of the innovations. Of course, the cases where the innovation can actually be constructed are more attractive. There are other cases where the innovation is given in advance by some ways or others, and those are also important and have been well investigated. The essential part of the analysis comes from the white noise theory, which is the central way of the analysis of functionals of the innovation. There naturally arises an infinite dimensional analysis. The last step is the application of the analysis. Many applications to quantum dynamics have been known; now application to bioscience would be the most fruitful area, although only part of it will be illustrated in what follows.
2. Background 2.1.
Reduction
To fix the idea let us observe the case where the random complex system is taken to be a stochastic process X(t). Levy's stochastic infinitesima equation for a stochastic process X(t) is expressed in the form 6X(t) = $ ( * ( * ) , 3
105
contains the same information as that newly gained by the X(t) during the infinitesimal time interval [t, t + dt). If such an equation is obtained, then the pair ($, Y{t)) can completely characterize the probabilistic structure of the given process X(t). Under mild assumptions the innovation may be considered as the time dereivative of an additive process Z(t). Further it may be assumed that the additive process has no fixed discontinuity and has stationary independent increments. Tacitly it is assumed there is no non-random part. Then, the Levy decomposition of such an additive process shows that Z(t) = X0(t) +
X^t),
where Xo(t) is Gaussian, in fact, a Brownian motion up to constant, and where X\ (t) is a compound Poisson process involving mutually independent Poisson processes with various jumps. Thus, a Brownian motion and Poisson processes with various jumps are all elemental additive processes. As a generalization of the stochastic infinitesimal equation for X(t), one can introduce a stochastic variational equation for random field X(C) parameterized by an ovaloid C:
8X{C) = $ ( X ( C ' ) , C < C,Y(s),s
e
C,C,6C),
where C < C means that C" is in the inside of C. The system {Y(s), s G C} is the innovation which is understood in the similar sense to the case of X(t). The two equations above have only a formal significance, however we can give rigorous meaning to the equations with some additional assumptions and the interpretations to the notations introduced there. The results obtained at present are, of course, far from the general theory, however one is given a guideline of the approach to those random complex evolutional systems in line with the innovation theory and hence with the white noise theory. As in the case of X(t) we can consider elemental random fields, or equivalently elemental noises with multi-dimensional parameter.
106
3. Gaussian s y s t e m s 3.1. Gaussian
processes
First we discuss a Gaussian process X(t),t £T,T being an interval of R1, say T = [0, oo). Assume that it is separable and has no remote past. Then, the innovation which is to be a white noise B(t) can be constructed explicitly. The original idea of such an approach to Gaussian processes can be found in the P. Levy's 1956 paper (Proceedings of the third Berkeley Symposium). Under the assumption that the process has unit multiplicity and under other mild assumptions, the Gaussian process X(t) has innovation B(t) which is a white noise such that
X(t) = [ Jo
F(t,u)B(u)du,
where F(t, u) is a sure (non-random) kernel function. This is the so-called the canonical representation of X(t) and F(t,u) is the canonical kernel. This might seem to be rather elementary, howeveer such an understanding is not correct. Profound structure behind this formula would lead us to a deep insight as we shall see later. On the other hand, take a Brownian motion and a kernel function G{t, u) of Volterra type. And we are given a Gaussian process Z(t) expressed as a stochastic integral Z(t) = I Jo
G(t,u)B{u)du.
Assume that G(t,u) is smooth on the domain 0 < u < t < oo. Then, we have Theorem 3.1 The variation 5Z(t) of the Gaussian process Z(t) is defined and is given by 6Z(t) = G{t,t)B(t)dt
+ dt J Jo
Gt(t,u)B(u)du,
where Gt(t,u) — ^G(t,u). The B{t) is the innovation of Z(t) if and only if G(t, u) is the canonical kernel. Proof. The formula for the variation of Z(t) is obtained easily. If G is not a canonical kernel, then the sigma field B t ( X ) is strictly smaller than B t (jB),
107
in particular the B(t) is not expressed as a function of the Z(s), s < t + 0. Note that if, in particular, G(t,u) is of the form f(t)g(u), then Z(t) is a Markov process and there is always given a canonical representation. Hence B(t) is the innovation. Remark. In the variational equation for X(t), the two terms in the right hand side are of different order as dt tend to zero, so that two terms seem to be discrimiated. But in reality, the problem like that is not so simple. As a result of having obtained the innovation, we can define the partial derivative denoted by dt: dB(t)' Since it is defined by the knowledge of the process Z(s),s canonical kernel F(t,u) should be obtained by F(t,u) = duZ(t),
< t + 0, the
u
We now come to the analysis of nonlinear functionals of white noise B(t),t £ R1. The collection of those functionals with finite variance forms a Hilbert space (L2). The direct sum decomposition of (L 2 ) into the subspaces Hn,n > 0, is obtained:
(L2) = 0W„. The decomposition stands for the Fock space. The Hn is called the space of homogeneous chaos of degree n. The time propagation is particularly important. It is expressed as a one-parameter unitary group Ut,t € R, acting on (L 2 ) determined by UtB{s) = B(s +1). Appealing to the Hellinger-Hahn theorem, it is shown that Ut has 1) simple multiplicity on Hi, and 2) countable multiplicity on Hn,
n>2.
In addition we can associate a symmetric L 2 (i? n )-function with a functional
F£L2(Rn),
108
where A means symmetric. Such an isomorphism can be obtained by the S-transform: (SV)(0 = / exp[< x,£ >}if(x)d^(x), J(sy
which is an infinite dimensional analogue of the Laplace transform. This property has been applied to the electrical engineering and to biological problems. Two cases are given below. Examples of application to identification of input-output systems. 1. Nonlinear input-output systems in electrical engineering. 2. K. Naka's method of identification of retina. Such a method of analysis is called the Wiener expansion. In reality, the Hellinger-Hahn theory is applied to identify the specral density function associated to each cyclic subspace or to each network, since the spectrum can well determine the structure of a cyclic subspace. Now one may think of its generalizations. If one wishes to carry on the so-called causal analysis, where time propagation is expressed explicitly in terms of 5(r.)'s (without smearing), then he is led to get the kernel function by applying differential opereators ? as well as their powers to the given random phenomena. It is noted that, to realize this idea, enough tools from analysis are provided for this purpose. Actually, the space (£)* of generalized white noise functionals is defined in such a way that
(s) c (L2) c (sy, where (S) is like an infinite dimensional analogue of the Schwartz space of test functions and (S)* is the dual space. The canonical bilinear form that connects (S) and (S)* is denoted by (,). The isomorphism between the subspace "Hn and the symmetric L2(Rn) can be generalized by the generalized 5-transform, which is well defined on (5)* since exp[< x,£ >] belongs to (S). Functions of the differential operators including exponential functions, and their adjoint operators play important roles.
109
Having established the generalization of white noise analysis, we have the following theorem which will be useful in applictions. Suppose a given random complex system is expressed as a generalized white noise functional ip. We wish to identify the system in terms of the kernel functions (generalized functions). Theorem 3.2. The kernel function (generalized function) Fn of degree n associated with ij) € (S)* can be obtained by the formula (dtA2---dtnip,l)
=
F(t1,t2,---,tn).
This result is useful in the study of engineering communication systems. In particular, for the system where the input signal, that is taken to be white noise, can be controled. The dt corresponds to the infinitesimal change of the signal, while d* stands for the interference by an instantenious noise. Next application is a randomization of the Lotka-Volterra equation. Attempts towards this direction are wide and numerous, however we take a particular case, still giving us a suggestion (see e.g. [14]). It is known that the famous Lotka-Volterra equation is derived by the variational calculus. Assume that there are n species of population size Nr, r = 1,2,..., n. Volterra is clever enough to take Xr = J0 Nr(s)ds and introduce the notion of the vital action as an integral of Lagrangian functional with variables Xr. Then, he applied the variation to obtain the Euler equation by the usual method. As a result the Lotka-Volterra equation is derived. Through the reduction procedure, one is allowed to randomise the linear term of the Lagrangian function so as to be fitting for the random environment. The randomized equations are now expressed in the form dN ~
= fr(N)+dtNr,
N = (Nu...,Nr),r
= l,2,...n,
where each term is random. Thus, we are given a stochastic differential equation with a special biological meaning. @@ Before we consider the reversibility or caisality of random evolutional phenomena some operators are prepared. Set
A(t)* = j F(t - u)d*udu, where the kernel F is taken to be a function only of t — u, since stationarity and causality in time for the action is assumed in most interesting cases.
110
Note that 9* is a creation operator standing for the action that comes from the fluctuation involved in the phenomena. As in the representation of Gaussian processes, the kernel F may or may not be canonical. This is seen when we form a Gaussian process X(t) = A(t)*l. When F is taken to be canonical, then the operator is fitting for the prediction. In contrast with this case, there is a choice of the kernel F so as to be fitting for the backward operations, that is future values will determine the innovation. If the exponentials of A(t)* is introduced, it is interesting to see coherent families of random variables parameterized by t : {exp[A(s)*]-l,s
the family is fitting for the study of nonlinear predictions or of the symmetry (asymmetry) of evolution.
3.2. Gaussian
random
fields
There are various Leitmotive to discuss random fields, among others in quantum field theory and biological science where fluctuation plays a dominant role. The basic tool is, of course, the innovation. To fix the idea we consider a Gaussian random field X(C) parameterized by a smooth convex contour in R2 that runs through a certain class C which is topologized by the usual method using the Euclidean metric. Denote by W(u),u £ R2, a two dimensional parameter white noise. Let (C) denote the domain enclosed by the contour C. Assume that a Gaussian random field X{C) is expressed as a stochastic integral of the form: X{C)=
f
F(C,u)W(u)du,
where F(C, u) be a kernel function which is locally square integrable in u. For convenience we assume that F(C, u) is smooth in (C, u). The integral is a causal representation of the X(C). The canonical property can be defined as a generalization to a random field as in the case of a Gaussian process. The stochastic variational equation for this X(C) is of the form 5X{C)=
f F(C,s)5n{s)W{s)ds+ JC
f J(C)
6F(C,u)W(u)du.
Ill
In a similar manner to the case of a process X(t), but somewhat complicated manner, we can form the innovation {W(s), s E C}. Example. A variational equation of Langevin type. Given a stochastic variational equation SX(C) = -X(C)
f k5n(s)ds + X0 [ v(s)d*8n(s)ds,C <= C, Jc Jc where C is taken to be a class of concentric circles, v is a given continuous function and d* is the adjoint operator of the differential operator ds. Applying the 5-ransform to the equation, we can solve the transformed equation by appealing to the classical theory of functional analysis. Then, applying the inverse transform 5 - 1 , the solution is given: X(C) = X0 I
exp[-kp(C,u)}d*v(u)du,
J(C)
where p is the Euclidean distance. Once the innovation is obtained, the above example suggests that one can think of possibility of application of the theory to the biological systems, where X(C) is a mathematical model of random phenomena that varies as C changes in a space-time region being interfered with by fluctuation that occurs at every point in (C). For example, the kernel functions are obtained by the same idea appeared in Theorem 3.2. As for the question on how to obtain the innovation or generalized innovation from more general class of random fields may be discussed by referring to the literature [9],
112
4. Functionals of Poisson noise 4.1. Poisson
noise
Having been suggested by the Levy decomposition of an additive process, a Poisson process P(t) comes after Brownian motion. Poisson process is another kind of elemental additive process. Taking its time derivative P(t) we have a Poisson white noise. It is a generalized stationary stochastic process with independent value at every point. For convenience we may assume that t runs through the whole real line. In fact, it is easy to define such a noise. The characteristic functional of the centered Poisson white noise is of the form oo
(e*M - l)dt], /
-oo
where £ £ E, and where A is the intensity. There is the associated measure space (E*,^p), L (E*,nP) = (L2)P is defined.
and the Hilbert space
2
We now pause to write some Leitmotive of the analysis of Poisson functionals. The first one goes back to 1940's. N. Wiener and his collaborators discussed functionals of Poisson process motivated by the research of biological objects. See [14], [16] . There have been, of course, many attempts in this direction. However we have found many interesting applications of white noise analysis. Indeed, various results of the analysis on (L 2 )p have been obtained, however most of them have been studied by analogy with the Gaussian case or its modifications, so far as the construction of the space of generalized functionals is concerned. Here, we only note that the (£ 2 )p admits the direct sum decomposition of the form
n
The subspace HpiTl is formed by the Poisson Charlier polynomials of degree n. Those polynomials are defined as follows. Let p(k,X) = j^e~x,k = 0,1,2,... be the Poisson distribution with
113
intensity A. Then A /2
>(fc A) "n r( i)i r A p{kX) ' ,
(k.\\
D {k,x)~ Pn
where A/(fc) = f(k + 1) - f(k).
Then, we have
^Pn(A;, X)pm{k, A) = 5 n>m
and the addition formula (-a-b)n
,.
,
,
L.
^ m=0
(-a)m(-6)n-m v
...
, ,,
'
There might occur a misunderstanding regarding the functionals of Poisson noise, even in the case of linear functional. The following example would illustrate this fact (see [1]). Let a stochastic process X(t) be given by an integral X(t)=
[ F{t,u)P(u)du. Jo It seems to be simply a linear functional of P{t), however there are two ways of understanding the meaning of the integral; one is defined i) in the Hilbert space by taking P(t)dt to be a random measure. Another way is to define the integral ii) for each sample function of P(t) (the path-wise integral). This can be done if the kernel is a smooth function of u over the interval [0, t\. Assume that F(t, t) never vanishes and that it is not a canonical kernel, that is, it is not a kernel function of an invertible integral operator. Then, we can claim that for the integral in the first sense X(t) has less information compared to P(t). Because there is a linear function of P(s),s < t which is orthogonal to X(s),s < t. On the other hand, if X(t) is defined in the second sense, we can prove Proposition. Under the assumptions stated above, if the X(t) above is defined sample function-wise, we have the following equality for sigmafields: Bt(X)=Bt(P),t>0.
114 Proof. By assumption it is easy to see that X(t) and P(t) share the jump points, which means the information is fully transferred from P(t) to X(t). This proves the equality. The above argument tells us that we are led to introduce a space (P) of random variables that come from separable stochastic processes for which existence of variance is not expected. This sounds to be a vague statement, however we can rigorously defined by using a Lebesgue space without atoms, and others. There the topology is defined by either the almost sure convergent or the convergence in probability, and there is no need to think of mean square topology. On the space (P) filtering and prediction for strictly stationary process can naturally be discussed. For further idea we may refer to the literatures [15] and [16], where one can see further profound idea of N. Wiener. Another short remark is that the Poisson roise, together with randomized intensity can serve in quantum optics as we can see in KlauderDudarshan [11].
4.2.
Proposed framework innovations
for a stochastic
analysis
of
We now pause to propose a new framework of stochastic analysis, having been suggested by the examples discussed in the last subsection. It is clear that one should think of functionals of paths of a stochastic process and operations acting on paths themselves. There is often required to calculate outside of L 2 -space; for one thing, the process in question may not have finite variance. More crucially, operations like jump finding or subordination must be discussed outside of the L 2 -space. Thus, the basic space is taken to be generated by innovations; a Brownian motion and compound Poisson processes. The topology is the convergence almost everywhere. Note that almost everywhere quasi-convergence is considered within this framework. Such a space will be denoted by (P). Both the spaces (L2) and (P) as well as their generalizations are the place where our game is played.®
115
4.3. Multi-dimensional
parameter
Poisson
noise
It is quite natural that we come to an introduction of a multi-parameter Poisson white noise, denoted by {V(u)}, which is a generalization of {P(t)}. Start with the characteristic functional Cp(£) which is to be the expectation of exp[i < V,£ >}:
CP(0
= exp[A /
(e^W -
l)dt%
JR*
where £ 6 E with a nuclear space E C L2(Rd). A probability measure /ip defined on the space E*. It can be shown that a stochastic bilinear form < x,f > with x £ E*, £ € E can be defined a.e. /xp. In particular, if / is taken to be the indicator function ID of a domain D, then the characteristic functional shows that < X,ID > is a random variable on the probability space (E*,[ip) and is subject to the Poisson distribution with intensity A|D|, where \D\ denotes the volume of D. To fix the idea, consider the case d = 2. The paper [16] by Wiener and Rosenblueth gave a Leitmotiv for the following observation. Set X — X(x) = < x, ID >. Theorem 4.1. 1) The random variable X(x) expresses the number of the singular points, with each of which a delta function is associated as a sample function. 2) Under the condition that X(x) involves delta functions as many as n, those points are equally distributed over D. Proof. The characteristic functional (p(z) of X(x) is given by
^(z)=exp[A|D|(e«-l)]. This proves the assertions. Remark. More precise meanings of the assertion 2) can be illustrated by taking D to be either a disc or a square when d = 2. We then come to a study of linear functionals of a Poisson noise. In
116 view of the biological applications alluded to in the Introduction, it is of fundamental importance to have a functional of Poisson noise parameterized by a point on the surface describing the observed results (see e.g. [16]). Theorem 4.2. Let a random field X{C) parameterized by a contour C be given by a stochastic integral X(C)=
[
G(C,u)V(u)du,
J(C)
where the kernel G(C, u) is continuous in (C, u). Assume that G(C, s) never vanishes on C for every C. Then, the V(u) is the innovation. Proof. The variation 5X(C) exists and it involves the term / G(C, s)Sn(s)V(s)ds, Jc where {Sn(s)} determines the variation SC of C. Here is used the same technique as in the case of [9], so that the values V(s), s G C, are determined by taking various SC's. This shows that the V(s) is obtained by the X(C) according to the infinitesimal change of C. Hence V(s) is the innovation. Here are three remarks worth to be mentioned. 1) Suppose one is permitted to take single variation, then it is impossible to form V(s), but one may use conformal transformations acting on C to have the values of the innovation V(s). 2) In the Poisson case one can see a significant difference on getting the innovation from the case of a representation of a Gaussian process. Indeed the canonical property of a representation can not be defined in a same manner. A theorem regarding this question can be given, but in an abstaract manner. 3) There are cases where not the real innovation, but B(t) is obtained up to plus or minus sign. For example, if one is permitted to use some nonlinear operations acting on sample functions, it is possible to form such an innovation from a non-canonical representation of a Gaussian process ( Si Si [12]), although the proof needs a profound property of a Brownian motion. Then, there remains a question on how to determine the sign to get a reasonable innovation. There are literature (see P. Levy [11, Chapt. VI]), however we prefer the use of a certain subgroup of the infinite dimensional rotation group.
117 4.4. Compound
Poisson
noise.
As soon as we come to a compound Poisson process, which is a more general innovation, the second order moment may not exist, so that we have to come to the space (P). The Levy decomposition of an additive process, with which we are now concerned, is expressed in the form tu {uPdu(t) - ^_—jdn(u)) + oB{t),
/ where Pdu (t) is a random measure of the set of Poisson processes, and where dn[u) is the Levy measure such that
/ ih?dn{u) < °°The decomposition of a Compound Poisson process into the individual elemental Poisson processes with different jumps can be carried out in the space (P) with the use of the quasi-convergence (see [11, Chapt.V]) . We are now ready to discuss the analysis acting on sample functions of a compound Poisson process. A generalization of the Proposition in the last subsection to the case of compound Poisson white noise is not difficult in a formal way without paying much attention. However, we wish to pause at this moment to consider carefully about how to find a jump point of Z(t) with the height u designated in adavance. This question is heavily depending on the computability or measurement problem. Questions related to this problem shall be discussed in the separate paper.
5. Concluding remarks A Brownian motion and each Poisson process as the component of the compound Poisson process seem to be elemental. Indeed, this is true in a sense. On the other hand, there is another aspect. Indeed, we know that the inverse function of the Maximum of a Brownian motion is a stable process, which is a compound Poisson process ( see. [12]). There the -L2-technique is no more available (see [10]). There is another surprising result. A Poisson noise can eventually be derived from a Browniann motion (Cockran-Kuo-Sengupta), certainly not by the L2 method. This may be illustrated in the following manner. In terms
118 of the probability distribution, it is shown that a certain generalized (Gaussian) white noise functional has the same distribution as that of a Poisson white noise. There arises a question on how to find concrete operations (variational calculus may be involved there) acting on the sample functions of B(£)'s to have a Poisson white noise. We need some more examples to propose a problem to give a good interpretation to such phenomena.
6. Addenda 1. Between two kinds of noises Gaussian and Poisson type, there are many similarities, but dissimilarities are important and interesting. Thus, it is worth mentioning the significance of the properties enjoyed by Poisson noise. 2. Measurement problem occurs in the case of compound Poisson process. Acknowledgements The author is grateful to Professor Si Si whose cooperation on the study of Poisson noise is highly appreciated. References 1. L. Accaxdi and Si Si, Innovation of some stochastic processes and random fields. Volterra Center Notes, 2002. to appear. 2. L. Accardi et al, Selected papers of Takeyuki Hida. World Scientific Pub. Co. 2001. 3. T. Hida, Stationary stochastic processes. Princeton Univ. Press. 1970. 4. T. Hida, Analysis of Brownian functional. Carleton Univ. Math. Notes, no. 13, 1975. 5. T. Hida, Brownian motion. Springer-Verlag. 1980. 6. T. Hida et al, White Noise. An infinite dimensional calculus. Kluwer Academic Pub. Co. 1993. 7. T. Hida et al, Functional word in a protein I. Overlapping words. Proc. Japan Academy 72 B (1996) 85-90. 8. T. Hida et al. ed. Advanced mathematical approach to biology. World Scientific Pub. Co. 1997. Artcles by Tom Ray, Ken. Naka and T. Hida. 9. T. Hida and Si Si, Innovation for random fields. Infinite Dimensional Analysis, Quantum Probability and Related Topics. 1 (1998), 499-509. 10. T. Hida, White Noise Analysis: A New Frontier. Volterra Center Notes. N.499. January 2002.
119 11. Jhn R. Klauder and E.C.G. Dudarshan, Fundamentals of Quantum Optics. Math. Phys. Monograph series, W.A. Benjamin, Inc. 1968. 12. P. Levy, Processus stochastiques et mouvement brownien. Gauthier-Villars. 1948. 2eme ed. 1964. 13. Si Si, Random fields of Poisson type. Preprint. 2002. 14. V. Volterra, Principes de biologie mathematique. Acta Biotheoretica (Leiden) III, Parte I, (1937), Opere Mat. di V. Volterra 5 (1962) 414-447. 15. N. Wiener, Generalized harmonic analysis. Acta Math. 55 (1930), 117-258. 16. N. Wiener and A. Rosenblueth, The mathematical formulationof the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle. Archivos del Instituto de Cardiologia de Mexico. 16 (1946), 205 - 265. 17. K. Yosida, Functional analysis. 6th ed. Springer-Verlag. 1980.
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Quantum Information V Eds. T. Hida and K. Saito (pp. 121-144) © 2006 World Scientific Publishing Co.
AN INTERACTING FOCK SPACE WITH PERIODIC JACOBI PARAMETER OBTAINED FROM REGULAR GRAPHS IN LARGE SCALE LIMIT AKIHITO HORA Department of Environmental and Mathematical Sciences Faculty of Environmental Science and Technology Okayama University Okayama 700-8530 Japan E-mail: [email protected] NOBUAKI OBATA Graduate School of Information Sciences Tohoku University Sendai 980-8579 Japan E-mail: [email protected] Asymptotic spectral analysis of the adjacency matrix of a large regular graph is formulated within algebraic or quantum probability theory. We prove a quantum central limit theorem for the quantum components of the adjacency matrices of growing regular graphs under a weaker condition. A new example of growing regular graphs is constructed, for which the limit is described in terms of an interacting Fock space whose Jacobi parameter is periodic. The central limit measure is obtained from the periodic continued fraction expansion of the Cauchy transform.
1
Introduction
Let {Q„ = (V("\ E^)} be a growing family of regular graphs (always assumed to be connected), where the growing parameter v runs over an infinite directed set. The degree of Qv is denoted by K(U). For each graph Qv we fix an origin x0 G VM and consider the stratification induced by the natural distance function on the graph: V(u)
=
oo (J yiu)^
y(u)
=
{ y e
y(u).
Q{x^y)
= n}
(1)
n=0
(Vii = 0 may occur.) The adjacency matrix Av of Qv admits a quantum decomposition: Av = At + A~,
(2)
which is canonically induced from the stratification (1). We are interested in asymptotic behavior of the quantum components as v —> oo and study it from
122
the viewpoint of algebraic probability theory. According to the stratification (1), we define
*P = \VP\-1/a E *-.
0)
where Sx stands for the indicator function of the singlet {x} and those functions form a complete orthonormal basis of V.v = £2(V^). One can expect easily that the quantum components Af behave like the annihilation and creation operators on a "Fock space" spanned by the "number vectors" 4>n , where n runs over 0 , 1 , 2 , . . . whenever V„ ' ^ 0. In the limit as v -» oo this guess is realized concretely in terms of quantum central limit theorem, where the limit is described by an interacting Fock space. To be precise we need some statistical assumptions on how the regular graphs Qv grow as v -> oo. For x G Vn we put u+(x) = \{yEV^1;y~x}\,
u.(x)
= \{y € V& ; y ~ x}\.
(4)
These are the numbers of points in the upper or lower stratum connecting with x, respectively. The average and variance of u-(x) over V„ are defined by
u ^ i e ' r 1 E w-(*)'
respectively. We consider the following five conditions: ( A l ) uy(x) + CJ_ (x) = K(U) for all x G V^"\ in other words, there is no edge lying in a stratum; (A2) lim,, K(V) = oo; (A3) for each n > 0 there exists a limit uin = lim,, w„ ' < oo; (A4) lim„ c{n] = 0 for all n > 0; (A5) for each n > 1 we have Wn = sup Wiv) < oo,
Wiv) = max{w_ (x); x G V ^ } .
With these notations we may claim the following
123
Theorem 1.1 Let {Qv = (V^"\E^)} be a growing family of regular graphs satisfying conditions (A1)-(A5). Let (Y, {Xn},B+, B~) be the interacting Fock space associated with {Xn} given by Xn = u\... un, XQ — 1. Then, lim /*<">, - ^ = ... - £ = *<*A Ao/ds /or a// j , k > 0 and /or any c/ioice *y, *<; »'n the right hand side are number The proof is deferred in Section 5. A Hashimoto [9], where a growing family of the assumptions (Al), (A2), (A5) and
= <¥,-, B«« . . . B"¥ f c >r
(5)
o/ e i , . . . , e m 6 { ± } , m > 1. #ere vectors ofT. similar result was first obtained by Cayley graphs was discussed under
(A3") for each n there exist constant numbers w n > 0 and Cn > 0 independent of v such that \{x 6 V^
; u-(x) jt wn}\ <
CnK{u)n-1
holds for all n > 1 and v. In fact, Hashimoto [9] proved convergence of the matrix elements (5) with respect to "coherent vectors" as well as "number vectors" under an assumption slightly stronger than (A5), and clarified Gauss-Poisson interpolation investigated in Hashimoto [8]. Later on we formulated in Hashimoto-Hora-Obata [10] and Hora-Obata [17] Hashimoto's theorem for general regular graphs and proved Theorem 1.1 under assumptions (Al), (A2), (A3") and (A5). Note that (A3") implies (A3) and (A4), but not conversely. By virtue of (2) asymptotics of the adjacency matrix Av follows immediately from Theorem 1.1 (classical reduction). We obtain the following Theorem 1.2 Let {Qv = (V^,E^)} and (T, {Xn},B+,B~) be the same as in Theorem 1.1. Let \i be the probability measure corresponding to (T,{Xn},B+,B-). Then, it holds that ]to(*£\(-£=)m*P)
= f x™»(dx),
m = 0,l,2,....
In particular, the moments of odd orders vanish and fi is symmetric. It is a natural question to characterize the class of probability measures appearing as in Theorem 1.2. In Hashimoto-Hora-Obata [10] we examined some Cayley graphs with our method. The standard Gaussian measure is obtained from the lattices ZN and the Wigner semi-circle law from the homogeneous trees associated with the free groups F/v. These are prototypes of
124
classical and free central limit theorems, see e.g., Hiai-Petz [12], VoiculescuDykema-Nica [21]. From the Coxeter groups with the off-diagonal elements of the Coxeter matrix being > 3 the Wigner semi-circle law is obtained, see Fendler [7] for a different derivation. From the symmetric group &N with Coxeter generators the standard Gaussian measure is obtained. The same occurs when Syv is equipped with all the transpositions as a set of generators. On the other hand, the Wigner semi-circle law is obtained from 6jv equipped with the generators {(12), (13),..., (IN)}, s e e Biane [2]. No "natural" example of growing Cayley graphs is known, from which another probability measure is obtained. However, beyond Cayley graphs there are interesting examples. In Section 3 we construct a new example of growing regular graphs for which the limit is described in terms of an interacting Fock space with a periodic Jacobi parameter. Thus the Cauchy transform (also called the Stieltjes transform) of the corresponding probability measure admits a periodic continued fraction expansion. Similar probability measures are derived by Bozejko [3] from a deformation of convolution products called the r-free convolution though the range of parameter is different. When condition (Al) is removed, the situation becomes full of variety. For example, the probability measures obtained in the limit are no longer symmetric. We know two examples from distance regular graphs: From a growing family of Hamming graphs the standard Gaussian measure and Poisson measures are obtained, see Hashimoto-Obata-Tabei [11]. From a growing family of Johnson graphs an exponential distribution and geometric distributions appear, see Hashimoto-Hora-Obata [10]. While, these limit distributions were first investigated by Hora [13] with a classical method. A general strategy of investigating the limit distribution has been discussed for a growing family of distance regular graphs, see Hashimoto-Hora-Obata [10]. Another type of limit procedure has been also studied by Hashimoto [8] and Hora [14,15,16]. 2
Preliminaries
For the sake of the readers' convenience we assemble some basic notion and notation used in Theorems 1.1 and 1.2. 2.1
Adjacency Matrix
Let Q = {V,E) be a regular graph of degree 1 < K < oo. When x,y € V are connected by an edge, we write x ~ y. That x ~ x never occurs. By assumption for each x € V the number \{y € V; y ~ x } | = K is constant. In this paper, unless otherwise specified, all graphs are assumed to be connected.
125
Then, for any pair x, y £ V there is a walk XQ ~ xi ~ • • • ~ x„ such that x = xo and y = xn. In that case n is called the length of the walk. The length of the shortest walk connecting x and y is denoted by 8{x,y) and is called the distance between them. By definition d(x,x) = 0. Note that d(x,y) = 1 if and only if x ~ y. Let A = (j4ij/)x,yev be the adjacency matrix of Q, namely, A is a symmetric matrix defined by 1,
x~y
w
^ H :10, I:JL^ otherwise
The adjacency matrix is identified with a bounded operator acting on £2(V):
Af(x) = ^2f(y),
f££2(V).
x£V,
y~x
Note that ||.A(| = K. For each x £ V denote by Sx the indicator function of the singlet {x}. Then {6X ; x eV} forms a complete orthonormal basis of (?(V). Obviously, A6x = ^5y,
x£V.
(7)
y~x
2.2 Stratification and Quantum Decomposition We fix a point xo 6 V as an origin of the graph. Then, the graph is stratified into a disjoint union of strata: oo
V = ( J Vn,
Vn = {x e V ; <9(x0,x) = n},
(8)
n=0
where Ki = 0 may occur. Obviously, \V0\ = 1, \Vi\ = K, and \Vn\ < /c(/c-l) n _ 1 for n > 2. By the triangle inequality we see that if x £ Vn and x ~ y, then i/ e Vn_i UV„U Vn+i. In this paper we avoid the case of y £ Vn, that is, we assume throughout the following condition: (Al) there is no edge lying in a common stratum. We assign to each edge x ~ y of the graph Q = (V, E) an orientation compatible with the stratification, i.e., in such a way that x -< y if x £ Vn and y £ Vn+i. Then we define
(A+)yx = [Ayx 10
= 1
iiyyx
>
otherwise,
( 4 - ) x _ fAy* yx
[0
= 1
if
y^x> otherwise,
, 9)
126
or equivalently,
A+5X = J2 Sv>
A s
( 10 )
~ * = J2 V y-ix
yyx
Then we come to a quantum decomposition of A: A = A+ + A~,
(A+)* = A~.
(11)
The former relation is checked by (7) and the latter by definition (9). 2.3
Interacting Fock Space
We refer to Accardi-Bozejko [1] for more details. Let Ao = 1, Ai, A2, • • • > 0 be a sequence of nonnegative numbers and assume that if Am = 0 occurs for some m > 1 then A„ = 0 for all n > m. According as An > 0 for all n or ATO = 0 occurs for some m > 1, we define a Hilbert space of infinite dimension or of finite dimension: mo — 1
00
r = £ec*„,
r= £ ec*n,
n=0
n=0
where m 0 is the first number such that ATOo = 0, and {*„} is an orthonormal basis. We call \P n the n-th number vector. The creation operator B+ and the annihilation operator B~ are defined by ' A„+i B+*„ = W-pi*„+i,
B - * o = 0,
B-9n
n>0,
= J-^-9n-1,
n>l.
In the case when T is of finite dimension we tacitly understand that B+^mo-i = 0. Equipped with the natural domains, B± become closed operators which are mutually adjoint. Then T({An}) = (r, {A„}, B+, B~) is called an interacting Fock space associated with {A n }. By simple computation we have B + i ? - # o = 0,
B+B'Vn
= - ^ - *n,
n>l,
(12)
An-1
B-B+*n=^±i*„,
n>0,
(13)
B+"$0 = 7Al*n,
n > 0.
(14)
127
2.4
Orthogonal Polynomials
Let (J, be a probability measure on R with finite moments of all orders, i.e., / \x\mn{dx) < oo,
m = 0,1,2, ••• ,
JR
and {P n } the associated orthogonal polynomials normalized in such a way that Pn(x) = xn + Then there exists uniquely a pair of sequences o^, 0:2, • • • € R and wi, w2, • • • > 0 such that Po(x) = 1, Pi{x) = x-a1} xPn(x) = Pn+i{x) + a„ + 1 P„(x) +w„P„_i(x),
n>\
(15)
The pair { a n } , {uJn} is called the Szego-Jacobi parameter. When the probability measure /i is supported by a finite set of exactly mo points, the orthogonal polynomials {P„} terminate at n = m 0 - 1 and the Szego-Jacobi parameter becomes a pair of finite sequences ct\,..., amo and u>i..., u>mo_i, where the last numbers are determined by (15) with Pn+i = 0. Note also that \i is symmetric if and only if an = 0 for all n > 1. Theorem 2.1 (Accardi-Bozejko [1]) Let {Pn} be the orthogonal polynomials with respect to fi with Szego-Jacobi parameters {an}, {w n }. Let r({A n }) be an interacting Fock space associated with A 0 = 1,
A n = UJ1U2 ...u)n,
Then there exists an isometry U from T({\n}) mined by U*0 = Po,
UB+U'Pn
= Pn+1,
n > 1.
(16)
into L 2 (R,/i) uniquely deter-
Q = U(B+ + B~+
aN+1)U\
where Q is the multiplication operator by x densely defined in L2 (R, n) and ctN+i is the operator defined by aN+i^n = o„+i*nIn fact, the isometry U is uniquely specified by i/X^^n *-> P n - A question of when U is a unitary, or equivalently when the polynomials span a dense subspace in L 2 (R, n) is related to the so-called determinate moment problem, see e.g., [6,20,22]. By Theorem 2.1, given an interacting Fock space (T, {\n},B+,B~), there exists a probability measure n such that (*0,(B+ + S - + aiV+i)m*o)= / JR
xmp{dx).
128 This fj, is unique if the corresponding moment problem is determinate. There is a formula linking the Cauchy transform (also called the Stieltjes transfrom) of fj, and the Szego-Jacobi parameter {an}, {un}: G„W
= /" M*!> = _ J Jn z — x
21
£
f^L_
z — ot\ — z — « 2 — z — 0:3— z — a 4
.
(17)
If /x is supported by a bounded interval, G^z) is holomorphic on {\z\ > r} for some r > 0 and the continued fraction converges. Conversely, if both {«„}, {w n } are bounded sequence, then the continued fraction converges uniformly on {\z\ > r'} for some r' > 0 and there exists a unique probability mesure (j, such that (17) holds. In this case n is supported by a bounded interval, see e.g., [6,22]. Unless fj, is supported by a bounded interval, (17) is still useful but the situation becomes complicated, see [22]. Remark 2.2 The Cauchy transform is defined for every probability measure fi on R without assuming existence of moments, and becomes a holomorphic function on {lm(z) ^ 0}. In fact, the integral in (17) converges absolutely and uniformly on every compact subset in that domain, see e.g., [5,22]. 3 3.1
N e w Examples Construction of Regular Graphs with Periodic Parameters
Theorem 3.1 Let a,b,k> u\ = 1,
1 be integers. Define K = abk and
u2 = a,
u>3 = b,
W4 = a,
u>s = b, —
(18)
Then there exists a n-regular graph Q = (V, E) which admits a stratification V = U^Lo ^« such *^a* u , - ( x ) = w n for x e Hw n > 1. P R O O F . We shall explicitly construct a K-regular graph having the desired property.
1° Let Vo and V\ consist of a single point XQ (origin) and of K points, respectively. We draw edges connecting each point in V\ and XQ. Then XQ has K edges. 2° We construct V2 and edges connecting between V\ and V2. The number of points in V2 is determined by counting such edges. Since each x € Vi must have K - 1 edges connecting with points in V2 and each y € V2 has a edges connecting with points in V\ by request, we have the relation: ( « - l ) | K i | =o|V 2 |.
129 Thus,
m = ^-A
(19)
which is an integer for K = abk. We must prove that the points in V\ and those in V2 can be connected by edges in such a way that each point y € V2 has o edges and each x e Vi has n — 1 edges. This is possible by looking at \V1\ = K=-X
a
a,
|Vj| = ^ - ^ a
= - x ( K - 1). a
We can divide Vj and V2 into «/o = bk subsets: bk
bk
Vl = \Jv}i\
V* = \JV2i)
with
|Vi{<)| = a, \V&\ = K - 1 .
For each i, we draw edges between Vj and V2 in such a way that any pair x 6 V^' and y e V2(i) is connected. For distinct i,j there is no edge connecting between V^' and V2 . In this way, each x € V\ has K edges with tj- (x) = 1 and each y £ V2 has a edges connecting with points in V\. 3° We construct V3 and edges connecting between V2 and F3. The number of points in V3 is determined by the relation: (K - o)|V 3 | = 6|V3|. Hence, in view of (19) we have l _ « ( « ~ ! ) ( « ~ Q)
Since V2| =
K(K-1)
a
=
K(K-1)
, ab
X b,
.
\V3\ =
K(«-1)
r—^x(«-o), ab
a similar argument as in 2° allows us to draw edges between Vi and V3 in such a way that each point in V2 has K — a edges and each point in V3 has b edges. In total each point in y € V2 has K edges with ui-(y) = a. 4° This procedure can be applied repeatedly and we obtain a /t-regular graph having the desired property, see Figure 1. In fact, the number of points
130
in each stratum is given by |Vb| = l, ,
^
|Vi| = « , ^(/C-Q)(/C-6)N\"~1
«(«-!)
=
~a~~ {
|VWl1 =
a~b
^6
[
)
n > 1,
'
ab
n-l
)
n > 1.
Figure 1. Construction Procedure: o = 2, 6 = 3, K = 6
Remark 3.2 There are three trivial cases (i) K = 1; (ii) K = a > 2 and 6 = 1; (ii) « = 6 > 2 and a = 1. Except these cases the K-regular graph constructed in Theorem 3.1 has infinitely many strata. By modifying the above proof we obtain the following Theorem 3.3 Let ai,..., o m , k > 1 be integers. Define K = aia,2 • • • amk and Wi = 1,
W2=aij
W j m + i + i = Oi,
•••! 3 > 0.
I^m+l
— 0>m,
l < i < m .
Then there exists a K-regular graph Q = (V, E) which admits a stratification V = U^Lo ^n suc^ ^ a i u-(x) = w « for x € Vn, n > 1. 3.2
Limit
Distribution
Let a, 6 > 1 be integers fixed. For each integer k > 1 let Qk be the abkregular graph constructed in Theorem 3.1. As is easily verified, {Qk} fulfills
131 conditions (Al)-(A5) and becomes an example of Theorem 1.1. Here the interacting Fock space describing the limit is determined by the parameter: Ao = 1,
A n = u>i.. .u)n,
where u>„ is a periodic sequence given in (18). We are interested in the corresponding probability measure (i. Lemma 3.4 The Cauchy transform of \x is given by (2b-l)z2 + a-b- v /^_2(q G {Z)
»
=
+
^
+
(a-&)2
2z{(6-l)z* + a - 6 + l ) }
'
(20)
where Im (z) > 0 and Re (z) > 0. P R O O F . By general theory mentioned at the end of §2.4, the Cauchy transform of n is given by the continued fraction: G„(*) = !
-
-
b
- 2
*
.
(21)
By using the periodicity, it is not hard to obtain a compact expression of G„(z) as in (20). I By applying the Stieltjes inversion formula [5,22] we come to the following Theorem 3.5 Let x(x) be the indicator function of [-y/Z-y/b,-\y/a-y/b\]u[\y/^-y/b\,y/a+y/b]. and define ( N - V2(a + paAX)
b)x*~x*-(a-b)*
~ 2TT|X|{(6 - l)x* + a - b + 1} *
W
'
Then fi(dx) is given as follows: (1)
Ifl
+
pa,b(x)dx.
(2) If b = a or b = a + 1, p,(dx) = (3) Ifb>a
patb(x)dx.
+ 2,
li(dx) = - 11 - / & _ 1 _ a ) ^ _ 1 \ j ( * € where £ = ^ ( 6 - 1 - a)/(6 - 1).
+
^-«)( d x ) + Pa,6(x)d«.
132
Remark 3.6 In [3] Bozejko introduced a one-parameter deformation of the free product called the r-free convolution, where r runs over [0,1]. The Cauchy transform of the central limit measure is given by a periodic continued fraction as in (21) with a = r, b = 1. 4 4-1
Remarks on Conditions ( A 1 ) - ( A 5 ) Statistical Quantities for Graphs
Conditions (A3) and (A4) for n = 0,1 are automatically satisfied because of structure of the stratification. In fact, ujg = 0, u\ = 1 and <70 = o~\ — 0 for all v. When Vn±% and Vn+i = 0 occurs, we understand that wo = 0,
u>i = 1,
UJ2 > 1,
.-.,
wn = / t > l ,
w n + i = • • • = ().
(22)
Otherwise, w0 = 0,
<Ji = 1,
u)n > 1,
n > 2.
Note also that if Vn ^ 0 and for some n > 1, we have uin > 1. In fact, every x G V„ is connected with at least one point in Vn-i4-2
How Graph Grows
Roughly speaking, under conditions (A1)-(A5) the graph grows upwards by adding new points and new vertices. Proposition 4.1 If {Gv} satisfies conditions (Al), (A2) and (A3), then for each n > 1 there exists u0 = vo(n) such that V„ ^ 0 for all v > u0. In particular, wn > 1 for alln> 1. P R O O F . We prove by contradiction. Suppose that there exist n > 1 and vi
Proposition 4.2 If {Gv} satisfies (Al), (A2) and (A5), then for each n > 1 there exists UQ = vo{n) such that every x £ V„_i has an edge connecting with a point in a upper stratum whenever v > v$. In particular, Vn v > v0.
¥" 0 f0T °^
133
P R O O F . By induction on n. The assertion for n = 1 is clear. Assume the assertion holds up to n — 1, where n > 1. By (Al) we have J+HX)=K{U)-J^(X),
xeVt\,
^>^o-
By (A5), V{+\x) > «(!/) - W^i Since Wn-\
> K{V) -
Wn-l.
is independent of v, we see by (A2) that lim min{w+ (x); x € V^-xl = oo.
In particular, there exists v\ > VQ such that m i n f ^ (x); x e V^_\ } > 1,
v > vx.
Hence if v > vi, every x e V^jj possesses an edge connecting with a point in an upper stratum. In that case, obviously, Vkv) £ 0. I
4-3
Condition Equivalent to (A3) and (A4)
For a growing family of regular graphs {Qv = {V^"\E^)} lowing condition:
consider the fol-
(A3') for each n there exists a constant number un independent of v such that tol{,6^iM,)^}|=1 |V„ M |
We then come to Proposition 4.3 Under (Al), (A3), (A4). PROOF.
(A2) and (A5), we have equivalence:
(A31)
(=*>) Divide Vn(l/) into two parts:
Ug = {xe VM ; «_(*) = un)
U^s = {xe V™ •
W _( X ) +
„„},
134
where the index n is omitted for simplicity. The average of ui-(x) is given by n
|T/(")|
=
^
T-T- ^ W_(x)H ^^r- V^ U)-(x) ,y(f), L^, ^ > |y(l>)| Z ^ *• > *€!/%' ' ' x€U\ns
=
r(") 7-r-U'nH
r-r-
>
U)-.[x).
x€U l ^
»ing
Since a;_(z) < Wn for x € Vw by (A5), we see that
,
ls l
+
r £
+ r )
^ -* ( -^i)* m" " i#ff^ " " Applying Lemma 5.2 and (A3'), we obtain \U(v)
^ = 0. lim 3 —J-V-
(24)
lima4? ) =w n )
(25)
,.
I ^ SI
and hence
which proves (A3). We next consider the variance. By Minkowski's inequality, we obtain 1/2
l|v
J
" Levi"
s { i £ ( - w - ^ r + l j ^ E n.-S")'}"'. where the first term is estimated by using \U!-{X) - U)n\ < U>-(x) + Un < Wn + Wn
and the second term is a sum of a constant independent of x. Then (\U(v) l \ 1 / 2 V | »n
| '
Taking (24) and (25) into account, we obtain lim„CT-n = 0, which is (A4).
135
{<=) Let n > 1 be fixed. By (A3), for any e > 0 there exists u0 such that |w^") - w „ | < e, If x e V„
y > ^0.
satisfies |u>_(x) - w„| > 2e, we have |w_(a:) - w M | > \u-{x) - w „ | - |w„ -
W M|
> e.
Hence |{s 6 Vjjv); |h>-(a) - ^ 1 > *}\ M
|v» |
|{» € V™ ; |o;.(a) - wP\ > e}\
<
|v„ (v) |
-
By Chebyshev's inequality and (A4) we have K * € V < ^ M - ^ > 2 . ) | W
IV„ I
< (
^ y
- V « /
We prove that u)n is an integer. Suppose otherwise. Then, since u-(x) is always an integer, we can choose a sufficiently small e > 0 such that VM={xeVM;)u,-(x)-LJn\>2e}. But this contradicts (26) and we conclude u>„ to be an integer. Since U(x) and u>n are all integers, we may choose a sufficiently small e > 0 such that \{x e V^ ; u-(x) / QJn}\ = \{x e V^y); |n_(aQ - o ; n | > 2e}| \V^V)\ |V n w | As is shown in (26), the right hand side tends to 0 as v -> oo. Therefore
^ K ^ e ) ; . (,)^ n }| = A
|V„ H |
which proves (23).
I
4-4 (A4) is Necessary for an Interacting Fock Space in the Limit Lemma 4.4 Let Q = (V, E) be a regular graph with stratification V = U^Lo Vn satisfying (Al). Let A = A+ + A~ be the quantum decomposition of the adjacency matrix. Then for n > 0 we have ( $ n , A - A + $ n ) = ||A+# n || 2 = i ^ t i i ( w a +1
+al+1)
,
($n>A+^-*n) = p - $ „ | | 2 = l ^ i i ((K-Wn_x)a+aLi).
(27)
(28)
136
The proof is a direct computation and is omitted, see also (39), (40). Proposition 4.5 Let {Qv = (V"("),£("))} be a growing family of regular graphs satisfying (A1)-(A3). Assume that there exists an interacting Fock space (T,{\n},B+,B~) such that = (*n,B-B+*n)
(29)
l i m / * M , - ^ - ^ _ * M \ = (*n,S+B-*n)
(30)
]ua(*M,-£=-£=*&)
"
\
\/K{V)
V«(I/)
/
hold for all n > 0. Then T is necessarily infinite dimensional and {Gv} fulfills condition (A4). PROOF.
In view of (12) and (13) we obtain lim(«M - ^ = - ^ = * M )
=
^±i
1 = 2 lta/e-^T-m*) Anr -1 K V (") v^i")
" \
/
(31)
(32)
~
On the other hand, with the help of Lemma 4.4 the left hand sides are written in terms of statistical quantities depending on v. We begin with (31). Since
^±i
= lim
An
/$(*), 5
ft gw\
v^M VKM
" \
* K(I/)|F„ 'I
X
I (33)
y
applying Lemma 5.1 and conditions (A2), (A3), we obtain
" K{V)\V!IV)\
V
4+A
K{y
}>
Un+1
(34)
Recall that u)n > 1 for all n > 1, see Proposition 4.1. Then (33) becomes An+1 . 1 ,. (i/)2 un+ i + limCT;YI > —— = = w lim "v n+i + A„ U>n+1 "
n>0,
(35)
which guarantees also that the limit lim,, o^+i 'Si «exists. Moreover, it is clear that A„ > 0 for all n > 0. Namely, T is of infinite dimension.
137
We next consider (32). In a similar manner as above, we see from Lemma 5.1 that
- ^ - = lim/$<,">, AXA„_i
%-*W\
* \ " 'v^)V^)
Since both ]im„ tt>„-i
anc
^ lim^i-i An
A n -i
are
=w„,
/
convergent, we see from (34) that n>l.
Finally, combining (35) and (36), we obtain lim„ <7„ = 0 for n > 1. 5 5.i
(36) I
Proof of Theorem 1.1 Estimate of Strata
Let (/ = (V, E) be a K-regular graph with stratification V = U^L0 Vn. We assume (Al) is satisfied, namely there is no edge lying in a stratum. Lemma 5.1 Let n > 0 and assume Vn ^ 0. Then, un+1\Vn+1\ = K\Vn\(l-^y
(37)
PROOF. Suppose first that Vn+i ^ 0. The number of edges whose endpoints lying in Vn is n\Vn\. Dividing these edges into two parts, we have
*\Vn\ = Y^w+(x)+ ][>_(*) *€V„
=
£ 3/6V„ + ,
x£V„
W_(p)+ XI W-(«)=Wn+l|V n+ l|+W„|V„|.
(38)
i£V„
This proves the assertion. (37) is valid also for n = 0 since we have put UIQ = 0. If the stratification terminates at finite steps, say, V = VQ U V\ U • • • U Vn and y n + i = 0, then w n+ i is not defined but at a tacit understanding ojn+± \Vn+i \ = 0 we have (38). I
138
Lemma 5.2 Let n > 1 and assume that Vn ^ 0. Then, wx > 1, ... , w„ > 1 and \Vn\=
+0(Kn-1),
""
where 0{nn~l)
is a polynomial in K of degree (n — 1). PROOF. An immediate consequence from Lemma 37.
I
5.2 Estimate of Error Terms In this subsection too wefixv so that this suffix is omitted for notational simplicity. Explicit actions of A* on the number vectors follow directly from definitions (3) and (10). We have 1/2 —r=9 0>„+l . T. ) n- " W •^ IITTS V
V«
*n+i
V K\ n\
+ ^n/m/2
E
(Mv)-Wn+i)*v,
n > 0,
(39)
«>1,
(40)
^-('-^(Wf*+
( < c [ ^| ) 1 / 2
E
("»-!-"-(*))*„
^ * o = 0.
(41)
In order to express the above actions in a unified manner we need some notation:
^-U&J 5
» =W (ZnJ~uT72 E ("-(») " w «)^> K - i l ) 1 / 2 J/GV„
5
"
=
<42)
• "*'•
( K |v; +1 |)i/2 E (<"» - u-(*))**'
n > 1, n
£ °-
(44) ( 45 )
139
Then, (39) and (40) are unified as follows: - ^ $ n = 7 ^ + e $ n + e + S;+€,
€ = ±,
U > 0,
(46)
where n + e stands for n ± 1 according as e = ± . Setting 7 l ! * - i = SZi = 0, we can involve (41) in (46) too. We next consider repeated action of A±. Suppose we are given m > 1 and € i , . . . , e m € {±}. Then, applying (46) repeatedly, we obtain _
:Z_ $ A£ • • •
V2
— ~«i
A£ * «
—
Vm
* ,
, ,
'n+«i 7n+«i+«2 ' ' ' ' n + f i + " 4 e m *n+«xH
|-£m
Here we assumed that n + ei > 0,
n + £i + e2 > 0,
...,
n + ex + e2 + • • • + e m > 0.
(48)
If a negative number appears among the above, we have Ae™ A" ^ — . . . — $„ = 0 . In fact, let A; be the first number such that n + ei + e2 + • • • + £fe < 0. Then n + ei+e2-\ hefc_i = Oande* = - , and hence A^-1 ...A£l$n is a constant multiple of $ 0 and A(kAtk~x ...Aei$n = 0. We must estimate of the error term of (47). For A; > 1 we set Wk = max{w_(;r); x € Vk).
(49)
Obviously, Wk < K. Then, for n > 1 and q > 0 we define M„ )? by _ (max{WklWk2...Wk
M n9
'
I < kuk2l...,kq
\l,
q>l, 9 = 0.
Lemma 5.3 Let e i , . . . , e m 6 { ± } , m > 1, be given arbitrarily. Let p and q be the numbers of + and — in { e i , . . . ,em}, respectively. Then for any n > 1 with n+ p — q>0 we have Af-m
$
J.
Ai\
—
—S+
.
,,
M +M
- *" -
(^-m\Vn\\ri(
(JWTJ
l 2
l
/
IV I \ V2 \Vn\ X"
Ui^J
....
•
(51)
140 PROOF. It is sufficient to prove the assertion under (48). If otherwise, the left hand side of (51) vanishes and the assertion is trivial. Note first Am
A 1
'
* c+
Sn =
W• • • 7*
=
1
y^,
, v
W^wr* 5
(w_(^
£ m
e
A» Wn)
" W''' ^
nv2 £ ("-(»>" w »)^ m • • • ^ V
fdy
(52)
We use a new notation. According as e = ± , we set y->z=< [y)~z
€ = -.
For y,z eV we put w(2/;ei,--.,e m ;.z) = |fe,...A-i)6r-1;|,4:l4:J..>-;\.1^z}|. This counts the walks from y to z along edges with directions e j , . . . , c m . Then (52) becomes A6" A' 1 + K-m/2
=
17iv
IW2 X ,
W^n-lU
Z2
(u-(y)-Un)w(y;e1,...,em;z)5z.
y g V n ,eV. + p _,
Therefore, I
^
Ae'
i4> n+p
+
\ _
KT m / 2
1 1/2
- " ^ ' " v^ " / " |VWP-,I x^3
H
HK-il) 1 / 2
(w_(?/)-w n )w(y; c i , . . . , e m ; z).
(53)
»6V„ 2 6 V n + p _ ,
Let y e Vn be fixed. Then ]T
ty(j/;ei,...,c m ;z)
(54)
z€V„+p-,
coincides with the number of walks from y to a certain point along edges with directions t\,... ,em. Consider an intermediate point z € Vk in such a walk. The number of edges from z with + direction is given by K — LJ-(Z), which is bounded by K uniformly. On the other hand, the edges with - direction is given by u-(z) and bounded by Wk as defined in (49). Thus (54) is obtained
141
by a product of such numbers. Remind that + direction appears p times, direction q times, and any walk starting from y e Vn to a certain point along edges with direction e i , . . . , e m contained in V0 U V\ U • • • U Vn+P. Then, by using (50) we have
z€Vn+p_,
The right hand side is independent of y € Vn. Now we come to an estimate of (53). In fact, ^n+p-qi <
rz '''
rz
n
K~m'2
KPMn+p,q
172 {K\vn^\)^
vn+p-q\^
K, --m - m//22
„PM
2p—m
—
\Vn\
1 3 l w -(2/)- w »l »evn /
1/2
\Vn +p-q\
\
l 2
l
1/2
\Vn\ *\Vn-
This proves inequality (51). Lemma 5.4 Let t\,... ,em G { ± } , m > 1, be given arbitrarily. Let p and q be the numbers of + and — among { e i , . . . , e m } , respectively. Then for any n > 0 with n + p — q > 0, $
Atl 2p—m
\Vn\ \Vn+p-q
< anMn+Ptq I —
1/2
1/2
K\Vn+l\)
(55)
PROOF. By definition (45) we have SQ = 0. Hence for n = 0 the left hand side of (55) vanishes and the assertion obviously holds. Suppose n > 1. We note the identity:
which is verified by (44) and (45). Then (55) follows from (51).
I
142
5.3
Proof of Theorem 1.1
When we consider a growing family of graphs Qu, quantities introduced in the previous subsection depend on the growing parameter v. Inserting (47) into the left hand side of (5), one obtains: $ 3»
'
Atl $(") s/lt(v)
\/K{V)
In+ti /n+e 1 +£ 2 • • " in+Ci-i
\-em \ ^ j
J ^n+«iH
l-cm /
(56) The coefficients 7^ depends on f. Explicit expressions of 7^ being given in (42) and (43), with the help of Lemma 5.1 and condition (A3) we come to lim7+ = y/u^,
lim7 n = y W t - i •
(57)
Therefore, in order to prove that the second term of (56) vanishes as u -> 00 it is sufficient to show that
(*t\-VK{i/)
lim " \
A'k+i
?«»
= 0.
y/K[u)
(58)
By Lemmas 5.3 and 5.4 we have Aim
c2p—m
Ail
< <7 n M ra + P]9
\vn
1/2
Vn+p-q\
^-M—(TWTJ
*i.
(^r •« l^^i;
•(60)
(The left hand side vanishes unless j = n + p — q.) The constant numbers in the right hand sides depend on v. We examine them one by one. By (A5) we have sup M$p,g
< 00.
It follows from Lemma 5.2 that AC2p—m \Vn\
\vn.•+P-9I
= 0(K2p-m+n-{n+p-q))
= 0(KP+q-m)
= 0(1).
143
Since 0(1) is uniform in v by (A3), K{v)2P~m sup • u
< 00.
|v£i.
Similarly, sup
\vkv)\
- < oo,
lim
|KM|
*(*0|tffil
V
= 0.
Therefore, the right hand sides of (59) and (60) vanish by (A4) in the limit as v -» oo. Consequently, only the first term of (56) contributes to the limit and we come to
IM/."-#-...-£_.") =
>-1^i-'yn+(1ln+e1+e2
• • • 7n+e1 + --+em0j,n+«H
|-£m •
(61)
Using (57) we come to the final form, which is equal to <*,•,£<»... B e i *„>. The verification is straightforward by definition of the interacting Fock space (r, {A„},B + , B~). Thus we have completed the proof. Acknowledgments We thank Professor M. Bozejko for stimulating conversation. This work is supported by JSPS Grant-in-Aid for Scientific Research No. 12440036. References 1. L. Accardi and M. Bozejko: Interacting Fock spaces and Gaussianization of probability measures, Infin. Dimen. Anal. Quantum Probab. 1 (1998), 663-670. 2. Ph. Biane: Permutation model for semi-circular systems and quantum random walks, Pac. J. Math. 171 (1995), 373-387. 3. M. Bozejko: Deformed free probability of Voiculescu, RIMS Kokyuroku 1227 (2001), 96-114. 4. M. Bozejko, B. Kiimmerer and R. Speicher: q-Gaussian processes: Noncommutative and classical aspects, Commun. Math. Phys. 185 (1997), 129-154.
144 5. T. S. Chihara: "An Introduction to Orthogonal Polynomials," Gordon and Breach, 1978. 6. P. Deift: "Orthogonal Polynomials and Random Matrices: A RiemannHilbert Approach," Courant Lect. Notes Vol. 3, Amer. Math. Soc, 1998. 7. G. Fendler: Central limit theorems for Coxeter systems and Artin systems of extra large type, preprint 2000. 8. Y. Hashimoto: Deformations of the semicircle law derived from random walks on free groups, Prob. Math. Stat. 18 (1998), 399-410. 9. Y. Hashimoto: Quantum decomposition in discrete groups and interacting Fock spaces, Infin. Dimen. Anal. Quantum Probab. 4 (2001), 277-287. 10. Y. Hashimoto, A. Hora and N. Obata: Central limit theorems for large graphs: Method of quantum decomposition, preprint, 2001. 11. Y. Hashimoto, N. Obata and N. Tabei: A quantum aspect of asymptotic spectral analysis of large Hamming graphs, in "Quantum Information III (T. Hida and K. Saito, Eds.)," pp. 45-57, World Scientific, 2001. 12. F. Hiai and D. Petz: "The Semicircle Law, Free Random Variables and Entropy," Amer. Math. Soc, 2000. 13. A. Hora: Central limit theorems and asymptotic spectral analysis on large graphs, Infin. Dimen. Anal. Quantum Probab. 1 (1998), 221-246. 14. A. Hora: Gibbs state on a distance-regular graph and its application to a scaling limit of the spectral distributions of discrete Laplacians, Probab. Theory Relat. Fields 118 (2000), 115-130. 15. A. Hora: A noncommutative version of Kerov's Gaussian limit for the Plancherel measure of the symmetric group, preprint, 2001. 16. A. Hora: Scaling limit for Gibbs states of the Johnson graphs, preprint, 2002. 17. A. Hora and N. Obata: Quantum decomposition and quantum central limit theorem, to appear in "Fundamental Problems in Quantum Physics (Ed. S. Tasaki)," World Scientific. 18. S. Kerov: Gaussian limit for the Plancherel measure of the symmetric group, C. R. Acad. Sci. Paris 316 Serie I (1993), 303-308. 19. W. Mlotkowski: A.-free probability, preprint, 2001. 20. J. A. Shohat and J. D. Tamarkin: "The Problem of Moments," Amer. Math. Soc, 1943. 21. D. Voiculescu, K. Dykema and A. Nica: "Free Random Variables," CRM Monograph Series, Amer. Math. Soc, 1992. 22. H. S. Wall: "Analytic Theory of Continued Fractions," AMS Chelsea Pub., 1948.
Quantum Information V Eds. T. Hida and K. Saito (pp. 145-158) © 2006 World Scientific Publishing Co.
E R R O R E X P O N E N T S OF CODINGS FOR STATIONARY GAUSSIAN CHANNELS
SHUNSUKE IHARA School of Informatics and Sciences, Nagoya University, Nagoya 464-8601, Japan E-mail: [email protected] We are interested in the probability of error to transmit a message over a stationary Gaussian channel without feedback. If the entropy of the message is less than the channel capacity, the optimum probability of error tends to zero as the blocklength goes to infinity. In this paper, we investigate the exponent in the probability of error and show that, if the entropy of the message is less than the capacity, the optimum probability of error tends to zero exponentially fast. To investigate such an asymptotic behavior as the error exponent, large deviation theorems play important roles.
1
Gaussian Channels
We are interested in the probability of error to transmit a message over a stationary Gaussian channel. The channel is presented by Yn = Xn + Zn,
n = l,2,...,
(1)
where X = {Xn} and Y = {Yn} are the channel input and the channel output, respectively, and the noise Z = {Zn} is a regular stationary Gaussian process with spectral density function (SDF) g(X). We may assume that the expectations of stationary processes are zero. Then the covariance of Z is given in the form r einXg(X)d\. J—n We assume that X and Z are mutually independent, meaning that the channel is without feedback. Moreover, we assume that an average power constraint E[ZkZk+n]=
2 hmsupiy>[X f c ]
n—*oo
^
(2)
k=l
is imposed on the inputs, where A > 0 is a constant. For each n, a message £„ is a random variable such that P(£n=m)
= —,
meMn
= {l,2,...,Mn},
(3)
146
where Mn is an integer. We assume that the limit linin^oo n _ 1 logM n exists. Then the entropy rate H(£) of £ = {£„} is equal to H(0=
Hm -H{£n)= n—*oo n
lim - l o g M „ . n—*oo ft
Let us define encoding and decoding schemes. The encoder is a mapping fn '• -Mn —* R"^ a n d the decoder is a mapping ipn : Rn —> M.n. The received message £ n is given by
where Y" = (Yi,..., Yn) is the output signal correspondint to the input signal X? = „(£„). For each coding scheme (tp,i>) = {(fn,ipn)}, en = en((pn, ipn) = P(£n ± f n ) is called the probability of error and (<£>„, tpn) is said to be an (n, Mn, e n ) code. Definition 1 A rate R is said to be achievable if there exists a sequence (
lim - l o g M „
(4)
n—»oo n
and lim en(ipn,ipn)
= 0.
(5)
n—*oo
The maximum achievable rate CCod = sup{i?; i?is achievable} is called the (coding) capacity of the channel. The mutual information /(£, rj) between random variables £ and 77 is defined as
HZ,V) = E
dfi£ x fin
if fic„
147 Definition 2 The (information) capacity dnf as
of the channel (1) is defined
Cinf
=SUp7(X,Y), x where the supremum is taken for all input signals X = {Xn} constraint (2) and Y is the output corresponding to X.
satisfying the
Let a > 0 be a constant determined by
f
(a-g(X))dX = A,
JX;g(X)
and define a SDF /o(A) by /o(A)=max(a-s(A),0).
(6)
Let Xo be a stationary Gaussian process with SDF /o(A) and independent of the noise Z, and YQ the corresponding output. Note that Xo satisfies (2). It is well known (cf. [4, 7]) that the information capacity is given by
1 f f (X)+g(X) Cinf = 7I(X0,Y0) = - J_^log —0 - ^ dX. The following result is known as the coding theorem for the channel (1). Theorem 1 For the Gaussian channel (1) under the constraint (2),
C„,
=C
' r^W^tiW,;, 4TT
y_T
( T)
g(X)
Theorem 1 means that, if the entropy rate H(£) = limn^oon - 1 logM„ is less than the capacity, there exists a coding scheme (
suc
h that (4) and
l i m s u p - l o g e n ^ n ^ n ) < ~r n—+oo
(8)
ri
are satisfied, then R and r are said to be an r-achievable rate and an Rachivable error exponent, respectively. The maximum r-achievable rate Ccod(^) = sup{i?; R is r-achievable}
148 is called the r-capacity of the channel. The maximum i?-achievable error exponent Fe(R) is defined by Fe(R) = sup{r; r is i?-achievable}. The function Fe(R), 0 < R < Ccod, may be called the reliability function. It seems to be a hard task to establish a formula for the r-capacity Ccod(r) or the reliability function Fe(R). The main aim of the paper is to give a lower bound for Ccod(r) and show that, if the entropy rate H(£) is less than the capacity, the optimum probability of error goes to zero exponentially fast as n —> oo. To state our result, for the SDF 51(A) of the noise Z and the SDF /o(A) given by (6), we introduce the following functions: X>
^^UTTX-'-^TTX-}' 7V[X) fa;)_I_J_
-2
^j_^xfQ{\)+9{\) r fo(X)g(X)
4, L(-foW+9(X))2dX'
W(x) = ^ J'logxf^9{X) 4TT
J^ x(x-l)
+
(9)
r /o(*) + g(A) dX
x-l +
^
dX
g(\)
r
(10)
/0(A)g(A)
~^L(xfow+9(wdX'
(11)
where V(x) and W(x) are denned for all x such that xfo(X) + g(X) > 0 on [—7T,7T].
Theorem 2 For each r > 0, c(e/me #(r) G (—1,0) and £(r) < 1 by U{0{r))=r
(12)
and
v
«'» " T W
(13)
TTien t/ie r-capacity Ccod(r) of the Gaussian channel (1) subject to the constraint (2) is bounded by Ccod(r) > W(t(r)).
(14)
149 Remark 1 Since U(x) is strictly decreasing on (—1,0], l i m ^ - i U(x) = oo and linia;_,o U(x) = 0, we see that 6(r) € (—1,0) is uniquely determined by (12) and limr_>o 9{r) = 0. Since the function V(x) is strictly decreasing and V(l) = 0, we know that t{r) < 1 is uniquely determined by (13) and lim r _ot(r) = 1. The function W(x) is decreasing if x < 0 and increasing if x > 0. Clearly lim W{t{r)) = W(l) = i - £
log / 0 ( A ] ( y ( A ) dX = Ccod.
(15)
It follows from (14) and (15) that lim Ccod(r)
=Ccod.
r—»0
This implies that Fe{R)>0,
0
In other words, if the entropy of the message is less than the capacity, the optimum probability of error tends to zero exponentially fast. The proof of Theorem 2 will be given in Section 3. For the proof we apply a large deviation theorem to stationary Gaussian processes. Large deviations properties for quadratic forms of Gaussian processes are studied in Section 2. Various applications of large deviation theorem in information theory have been discussed in [5]. 2
Large Deviation Theorem
We consider a sequence £ = {£„} of random variables. We put A„(0) = log£[exp(0Cn)],
e&R,
(16)
and define the logarithmic moment generating function A(#) as A(0) = lim -A„(n0),
(17)
n—>oo TL
if the limit exists. We note that A„(0) and A(8) are convex functions. Let V be the set of all 6 e R such that the limit (17) exists and the function A(-) is of C 1 class in a neighborhood of 9, and define a set V by V = {A'(0); 6 G V}. In the following we assume that T>?4>,
(V)° ± <j>,
where (V')° is the interior of V. The function \I>(#) is defined by *(0) = 9M{9) - A(9),
6eV.
(18)
150 Let us fix 9* G V such that A'(0*) 6 (X>')° and 9* j= 0, and consider a half-line Ti = {xe R, 9*(x-a*)
>0},
where a* = A'(9*). Clearly a* G II (II denotes the closure of II) and #(0*) = inf{*(6»); A'(0) G II}. 0
For our purpose the following large deviation theorem (cf. [3]) is useful. Proposition 1 Assume that the condition (18) is satisfied. (i) For any measurable set A C II, lim s u p - l o g P(C„ &A)<
-#(0*).
(ii) Let A be an open set such that A n (V)° ^ <j). Then, for any 9 €.T> such thatA'(9)eAn(V')°, l i m i n f - l o g P « n G A) > n—»oo
-$(9).
n
(iii) Let A C IT be an open set such that AD (a* — 5, a* + 5) =fc
n—>oo n
/n particular, lim - log P(Cn G n ) = lim - log P(C„ € IT) = - # ( 0 * ) .
(19)
Large deviations for quadratic forms of stationary Gaussian processes have been studied [1, 2]. Let X = {Xn}, Y = {Yn} and Z = {Zn} be stationary Gaussian processes with SDF's /i(A), 5(A) and /2(A), respectively, and assume that Y and Z are regular. The n-dimensional random variable Yj" has the probability density function qn(y) = (27r)-"/ 2 |T„(s)r 1 / 2 exp j - I ^ ) - ^ ) J ,
V G «",
where rn(5)=kfc(ff)l l
,
-
Sj,k=l,...,n
tjk(g)=
r
e^-k^g(X)d\,
J_n
denotes the Toeplitz matrix. Let us apply the large deviation theorem to (n = ^logqn(Z?-xr[),
n = l,2,...,
(20)
151
where x = {xn} is a sample path of X = {Xn} and x" = (xi,...,xn). The large deviation property for the sequence C — { d } has played important roles to study the string matching probability of stochastic processes [6]. We note that the entropy h(Y) = lim n _oo n~ 1 /i(Y 1 n ) (per unit time) of the Gaussian process Y is equal to h(Y) = ^
f
log{47r2ep(A)} dX,
(21)
where ^(Yj™) denotes the differential entropy of YJ1 (cf. [7]). Let 0 be the set of all 9 such that functions fi(X)(g(\) + 0/2(A)) - 1 , / 2 (A)( 5 (A) + 0 / 2 ( A ) ) - \ h(X)f2(X)(g(X) + 6f2(X))-2 and log(5(A) + 6f2(X)) are integrable on [—n,n\. Proposition 2 Let £ = {£„} be the random sequence given by (20). Then, for almost all sample paths x = {xn}, the assertions in Proposition 1 are true with functions A(#) and ^(9), 9 £ 0 , given by A(0) = -01og(27r) + 1=2 r log 5 (A) dX
-^£log( 5 (A) + W )),A-A£_^, A
+
0
9 f
A(A)
2-^L9(X) + 9f2(X)dX>
_j_
r
/i(A)
4vr 7_w 5(A) + 0/2(A)
(22)
g r +
/I(A)/ 2 (A)
4TT y_w (5(A) + 9f2{X)f ^
[Z6)
and
The proof will be given in Appendix. We consider two special cases. The following two corollaries will play important roles to prove our main results.
152
Corollary 1 Let Y — {Yn} and Z = {Zn} be regular stationary Gaussian processes with the common SDF g(X), and £ = {Cn} be a random sequence given by C„ = ilogg n (Z 1 n ),
n = l,2,....
(25)
Then the assertions in Proposition 1 are true with functions h(9) = -9h(Z)+9--l-\og(l
+ 9),
(26)
and 9(9) = U(0),
6 > -1,
(28)
where U(0) is the function of (9). Proof. By putting /i(A) = 0 and / 2 (A) = 5(A) in (22), (23) and (24), we obtain (26), (27) and (28). Corollary 2 Let X — {Xn} and X = {Xn} be stationary Gaussian processes with SDF /o(A) and let Y = {Yn} and Z — {Zn} be regular stationary Gaussian processes with SDF g(X). Let £ = {£„} be a random sequence defined by
n = l,2,...,
(29)
where x = {xn} and z = {zn} are sample paths of X and Z, respectively. Then the assertions in Proposition 1 are true with functions
A(9) = -9h(Z)
+
- + -J_^ log g
m + g W
<*-&L
9MX)+g(X)
dA
'
(30)
_J_ f fo(X)+g(X) ^J-,0fo(X)+g(X) = -h(Z)
+
6 r /o(A)(/ 0 (A)+ g (A)) 4 7 r i _ 7 r (6f0(X) + g(\))*
+ V(6)
(31)
and 9(9) = W(9),
(32)
153 where 0 € © and 0 is the set of all 6 such that functions log(p(A) + 0/o(A)), /o(A)(5(A) + efo(X))-1 and /o(A)5(A)(5(A) + 0/o(A))- 2 are integrable, and V{6) and W{9) are functions of (10) and (11). Proof. Note that the SDF of the process X + Z = {Xn + Zn} is / 0 (A) + g(X). Hence, by replacing /i(A) and /2(A) with /o(A) +(A) and g(X), respectively, in (22), (23) and (24), we have (30), (31) and (32). 3
P r o o f of M a i n R e s u l t s
To prove Theorem 2 we constract a coding scheme by using the so-called random coding method. Proof of Theorem 2. We define a coding scheme (?, ip) — {(fn, V'n)} a s follows. Let X = {Xn} be a stationary Gaussian process with SDF /o(A) given by (6) and X<m) = {Xnm)}, m = 1,2,..., be independent copies of X. We assume that by observing X^ we get a realization x ^ = {a;„ '} of X^"1). For each n, the encoder tpn is defined by
m G Mn = {1,2,..., Mn},
(33)
where the integer Mn will be specified later. The decoder ipn is defined by Mvi)
= m,
y^eRn,
(34)
if there exists a unique m € M.n such that \\ogqn{y1
- (i(™»)?) > -c(r),
(35)
where
«r)=~h^-wBm
(36)
and 6(r) is given by (12); otherwise 1>n(Vi) = 0-
(37)
We put An,m
= {y^eRn;-
n
logqn(y? - (x™)?)
>
-c(r)}.
To evaluate the probability of error, we may assume that <^n(l) = (a^ 1 ))" is input to the channel, so that the corresponding output is F 1 " = (x (1) )5 l + Z1ri.
(38)
154 Receiving the output signal y™, the error occurs when (39)
Vi $ A i , i ,
or Mn
Vi e ( J A,,TO.
(40)
m=2
Since ZJ* = Yf - ( i ( 1 ) ) ? , the error of type (39) occurs with probability P{Y? i AnA) = P (j- logqn(Z?)
< -c(r)\
.
Using (12), (19), (27), (28) and (36), we have lim - logP (- logqn(Z?) < -c(r)) = -U(6(r)) = -r. (41) n—»oo n \n j To evaluate the probability of the error of type (40), let realizations x^ = {x^} of the input X^ = {Xn1]} and z = {zn} of the noise Z = {Zn} be fixed. We denote by y = {yn} the corresponding output, where Vn
=
%n
i
Zn.
We apply Proposition 1 to the process {£„} given by Cn = ^log<7„(yr - ( X W ) J ) ,
(m^l).
(42)
Then the logarithmic moment generating function A(6) of {£„} is given by (30). It is easy to see that A'(t(r)) = -h(Z)
+ ^-^L- =
-c(r).
Therefore, using (19) and (32), we have lim - logP(£„ > -c{r)) = -W{t{r)).
(43)
n—>oo n
Since W(0) = 0, W(l) = 7(X, Y), and W{6) is increasing on (0, oo), we know that 0<W(t(r))<7{X,Y). Let R < W(t(r)) be a rate and define Mn by Mn=\enR].
(44)
155 Since M,
P ( (J { i log <*„(!/? - (X^)ni) > -c(r) \ m = 2 *•
- ,
< MnP Q logg„(j/r - (X ( m ) )?) > -c(r)\
,
it follows from (42), (43) and (44) that limsup 1 l o g P ( ( J {- log 9 n (y? - (X
(45)
This means that, for sufficiently large n, there exists a set {(a:^OT^)™}m=i)...,M„ of realiztions of {(X( m ))?} m=i,...,M„ for which ViiAn,m, Vro^l, (46) or equivalently the error of the type (40) does not occur. Therefore, by (41), we conclude that any rate R less than W(t{r)) is r-achievable. Thus we have (14). Appendix
Proof of Proposition 2
To prove Proposition 2, we use analogous arguments we have adopted in [6]. We need some properties concerning the asymptotic behavior of Toeplitz matrices. It is convenient to use the notion of asymptotic equivalence of sequences {^4„} and {Bn} of matrices, where An and Bn are n-dimendional square matrices. We define norms of matrix by
\An\2 = - V | a « | 2 ,
| K H 2 = max
{ ( A l A ^ ^ - T ^ ^ l ) ,
where a,j is the (i, j) component of An. We say that {An} and {Bn} are asymptotically equivalent and denote An ~ Bn, if there exists a constant K < oo such that IIA.H, \\Bn\\
Vn,
156
and lim \An-Bn\
= 0.
n—>oo
Concerning the asymptotic behavior of Toeplitz matrices, the following properties are known. lim - Tr Tn(V) = f lim - log \Tn(g)\ = ~
(47)
C log(27r5(A)) dX,
(48)
2nTn(
T (5)_1
"
(49)
~ 4i T " ( 9 _ 1 ) '
(5°)
where 5(A) is a SDF of regular stationary process, T n (g _ 1 ) is the Toeplitz matrix of l/g(X) and
lim
-E[{AnX?,X?)]=2n
f
n—KX>n
y{X)f{X)dX.
7_7r
We are now in a position to prove Proposition 2. Proof of Proposition 2.
Since {2^)-ne'2\Tn{g)\'9/\2^)-n'2\Tn{f2T112
E[exp(neCn)} =
jRne^^-9-(Tn{g)-\z-x),{z-x))-l-(Tn{h)-lz,z)\dz,
x one can easily show that
(2*)-ne/2\Tn(g)\-9/2\Tn(h)\-V2\An,e\-1/2
£[ex P (n0Cn)] =
x exp < -(Tn(g)'1x,
{Bnfi - In)x) \ ,
where An,e = Tn(f2)-1
+ 9Tn(g)-\
Bn,e = 6{Tn(f2)-lTn{g)
+
9In)-\
157
and In is the n-dimensional identity matrix. Therefore, for the function A„(#) defined by (16), we have -An(n6) n
= -°-log(27r) - Llog 2 in
\Tn(g)\ - ~ log |T„(/ 2 )| - J - log \An,6\ In In
+7r(Tn(g)-1x, (Bn,e - In)x). In Using (48), (49) and (50), we have 4TT2
(51)
\f2
g)
and lim -{log |T n (/ 2 )| + log\A n , e \} = - L r n-+oo n
log f 1 + ^ )
27T ./_„.
V
dA.
(52)
g J
We can also show that Bn $ — In ~ — 77- T n 2TT "\g + ef2 so that
T^)-1^,* - J„) ~ --LT„ ,/ * n ( ^ nTn f-2—) ~ - - U 8TT3 \gJ \g + 8hJ toP n\g + ef J2
Therefore, using Lemma 1, we have lim V n t e ) - 1 * , (Bn,e - In)x) = ~
f
„/_ff,
m
dA.
(53)
It follows from (21), (48), (51), (52) and (53) that A(0) = lim
-An(n9)
n—>oo n
_e_ r 4TT
A (A)
^
7_T 5(A) ++ 0/2 (A)
"«
«+•
•d\.
g(\) + ef2(X)
Thus we have obtained (22). Eq. (23) is an easy consequence of (22) and Eq. (24) follows from (22) and (23).
158 References [1] B. Bercu, F. Gamboa and A. Rouault, Large deviations for quadratic forms of Gaussian stationary processes. Stochastic Proc. Appl. 7 1 , 75-90 (1997). [2] W. Bryc and A. Dembo, Large deviations for quadratic functionals of Gaussian processes, J. Theoretical Prob. 10, 307-332 (1997). [3] A. Dembo and O. Zeitouni, Large Deviation Techniques and Applications (Jones and Bartlett Pub., Boston, 1992). [4] S. Ihara, Information Theory for Continuous Systems (World Scientific, Singapore, 1993). [5] S. Ihara, Large deviation theorems for Gaussian processes and their applications in information theory. Acta Appl. Math. 63, 165-174 (2000). [6] S. Ihara and M. Kubo, The asymptotics of string matching probabilities for Gaussian random sequences, Nagoya Math. J. 166, (2002). [7] M. S. Pinsker, Information and Information Stability of Random Variables and Processes (Holden-Day, San Francisco, 1964).
Quantum Information V Eds. T. Hida and K. Saito (pp. 159-180) © 2006 World Scientific Publishing Co.
W H I T E NOISE ANALYSIS O N CLASSICAL W I E N E R SPACE REVISITED YUH-JIA LEE Department of Applied Mathematics, National University of No. 700 Kaohsiung University Rd., Kaohsiung, TAIWAN E-mail: [email protected]
Kaohsiung, 811
H S I N - H U N G SHIH Center
of General Education, E-mail:
Kun Shan University of Technology, TAIWAN 710 kitty@mathl. math. ncku. edu. tw
Tainan,
The white noise calculus on the abstract Wiener space (C',L 2 [0,1]) are reformulated using a class £ of entire functions of exponential growth as test functionals {£ is shown to be equivalent to the Yan-Meyer space). The generalized Wiener functionals, are defined and studied via their functional representation. Examples, such as multiple Wiener integrals, additive renormalization, multiplicative renormalization and conditional expectation, etc. are given; the Clark formula is discussed and Ito formula is reproved for generalized functional of Gaussian processes, including Brownian motion and Brownian bridge.
l Introduction The theory of generalized functions of infinite variables has been formulated in terms of Malliavin calculus and Hida calculus. The former, introduced by Malliavin 23 , studied the calculus of generalized Wiener functionals and their applications on the classical Wiener space {C,B{C),n) (see also Watanabe 26 ), where C is the space of continuous functions defined on [0,1] and vanishing at 0, /J, the Wiener measure on C and B{C) is the Borel field of C; while the latter, also known as the white noise analysis initiated by T. Hida 4 , investigates the calculus of generalized white noise functionals on the white noise space (S',B(S'),u) (see also Hida-Kuo-Pottohoff-Streit 5 ), where S' is the space of tempered distributions and v the standard Gaussian measure on S' and B(S') is the Borel field of 5 ' . Later, Hida calculus was reformulated on an arbitrary abstract Wiener space by Lee 15,16 . As C is also regarded as an abstract Wiener space which has the space C of Cameron-Martin functions as its reproducing kernel Hilbert space, it is desirable to reformulate Hida calculus on C. A primary report was
160 given in Lee 20 , there Hida's theory on the alternative abstract Wiener space (C',L2[0,1]) was reformulated by using the space £ of mean of exponential type entire functions denned on the complexification of .^[0,1] as the class of test functionals. The choice of the alternative abstract Wiener space instead of the original classical Wiener space is due to the reason that the white noise derivative dt = Di„ „ is defined to be the Prechet derivative in the direction of l[ (j i] which is in L2P, 1] \ C. Since the space £ is not complete, it is also desirable to find a complete replacement of £. In this paper we show that the space £ of analytic version of Yan-Meyer space 24 is what we need for reformulating Hida's theory on /^[0,1]. The generalized Wiener functionals are then defined and studied via their functional representations. Examples, such as generalized multiple Wiener integrals, additive renormalization, multiplicative renormalization and conditional expectation, etc. and many others are given; the Clark formula is discussed and Ito formula is reproved for generalized functional of Gaussian processes, including the Brownian motion and the Brownian bridge.
2 A Gel'fand triple on t h e classical Wiener space Let C be the collection of real-valued continuous functions x which is defined on [0,1] and satisfies x(0)=0 and C the subclass of C consisting of absolutely continuous function x whose derivative x satisfies / \x(t)\2dt< 00. Jo Then C is a Banach space with the sup-norm | • ^ and C is a Hilbert space with norm | • |o = \ / ( - , -)o and the inner product (•, -)o defined by (x, y)0 := / x(t)y(t) dt Jo for x, y £ C. The space C is usually called the Cameron-Martin space and it is well known that (C',C) forms an abstract Wiener space (AWS, for abbreviation). The Wiener measure /i is then realized as the abstract Wiener measure with variance parameter t = 1. It is easy to see that the pair (C, L2) is also an abstract Wiener space, where L2 = L2 [0,1] with the | • |2-norm. The Wiener measure fi is then extended to a measure on L/2[0,1], still denoted by /x, in such way that for any Borel subset E of L2, fJ>(E) = fi(E C\ C). The dual space C* of C, may be identified as a dense subspace of C given as follows: C* = {x € C : x is of bounded variation and right continuous with x(l) = 0}.
161 Under this identification, the C —C* pairing is given by (x,y) = -
x(t)dy(t), x£C,y£C*. Jo Similarly, the dual space L2[0, 1] of L2[0, 1] is identified as a subspace of C given as follows: L2 = {x £ C* : x is absolutely continuous s.t. x £ L2, i ( l ) = 0 and £(0) = 0}. By the above identification, we have
(x, y)2 = - I Jo
x(t)y(t)dt
for x £ L2 and y £ L2, where (x, y)i denotes the L2 — L\ pairing. We sum up the above results into a Lemma for future applications. Lemma 2.1. (a) If we identify the dual space of C by itself, then C* and 1/2 can be identified as subspaces of C'. Furthermore, the following inclusive relations hold in such a way that the smaller space are densely embedded in the larger space: L*2cC*=C'cCc
L2.
(b) For x £ C and y £ L\, the following identity holds: (x, y) = (x, y)2 = (x, y)Q. Example 2.2. 1. (The representation of the Brownian
motion)20:
For each t £ [0,1], let bt(s) = sAt, 0 < s < 1. Then bt 6 C* and the Brownian motion {B(t) : 0 < t < 1} is represented by
*(t,x) = «(*) = { £ $ ; x£C x £ E- , p
p>l.
2. (The representation of the Brownian bridge)21: A nature representation of Brownian bridge {Bb(t) : 0 < t < 1} is given by Bb(t,x)
= B(t,x) - tB(l,x)
= (x, fr),
where fit £ C* such that (x, f3t) = (x, bt—tbi) for all x £ C. However the above representation is not adapted. An adapted representation is given as follows.
162 For each t e [0,1], define ft(s) = (t - l ) l n ( l - s At) and ft(s) = 0, 0 < s < 1. Then ft € C* and the Brownian bridge {£<,(*) : 0 < t < 1} is then represented by
B 6( t,x)=*(*)={;*'§)• * f F
>l
[ (x, ft), x <E £ L p , p > 1. Let K be a bounded linear operator on L2[0,1] defined by Kx(t) = / ( s A t ) x ( s ) d s , x € L 2 [ 0 , l ] . Then K is a positive self-adjoint compact operator having a complete orthonormal basis (CONS, for abbreviation) consisting of functions {en(t) = •y/2 sin(n — l/2)ivt: n = 1,2,...} with the corresponding eigenvalues given by { l / ( ( n - 1/2)TT) 2 : n = 1 , 2 , . . . } . For n e N, let
/„(*) = VKen(t),
te[0,l].
Observe that, for each n e N and for t G [0,1], e n ( l - i) = ( - l ) n + 1 fn(t), then {/„ : n £ N} C C*. Moreover, {/„ : n e N} is also a CONS of L 2 J0,1]. Thus, {/„ : n e N} is a CONS of C. Let A be the inverse operator of VK on C. Then v4 is a self-adjoint operator densely defined on C with Afn= Xn fn for each n € N, where A„ = (n — 1/2) 7r For any p > 0, let Ep be the domain of Ap. Then .Ep is a real Hilbert space with the norm \x\p = \Apx\o (C = Eo) and {(1/A^) fn : n € N} forms a CONS of Ep. The increasing family {| • \p : p > 0} of norms are compatible and comparable, and the embedding from Ep+a into Ep is of Hilbert-Schmidt type whenever a > 1/2. Next, let E-p be the completion of C with respect to the norm |x|_ p = | J 4 _ P X | O . Then E-p is a Hilbert spaces with a CONS {A£ fn-n€ N}. Identify x G E* with the element Y^=\ (x> fn) fn in E_p, where (•, •) is the E*-Ep pairing, then E-p becomes the dual space of Ep. Set E — (~lp>o Ep and endow E with the projective limit topology induced by Ep's. Then E is a nuclear space with the dual E* = Up>o E-p and E C C C E* forms a Gel'fand triple.. Observe that, for x € C and p > 1, oo
N-p=
E n=l
/•!
K2P{*,fn)l=
X(t)2dt<
\x\l.
JQ
It follows that C C £ - p for all p > 1. Furthermore, by the denseness of C in C and in E-p, we have the following chain of continuous inclusion:
EcEpcEgcE1=L*2cC*
CC' cCcL2
= E-1cE_qc
E-p C E*,
163
where p > q > 1 and Li = Z,2[0,1]. For notational convenience, we will use the notation (•, •) to stand for all the dual pairings of E*-E, E-P-Ep (p > 1), and C-C*. For p > 1, (C',E-P) is an AWS. Let /x and (J—P, p > 1, be the abstract Wiener measures of C and E^p respectively. Then the measurable support of H-p is contained in C and , for any integrable complex-valued function
JE-P
I
(p(x)fx(dx).
JC
The following theorem characterize the space Ep Theorem 2.3 (Lee-Shih 2 2 ). (i) For p G N, Ep consists of all the functions x £ C with the properties: (i) x,x,... ,a;(p) are absolutely continuous with x^p+1^ G L2IO, 1] and (ii) 2fc ar( )(0) = rr( 2fc+1 )(l) = 0 for k = 0 , 1 , . . . ,[p/2]. Moreover, |x| 2 = /„* \x(P+1\t)\2
dt and
(x, y) =
— / x(t) dy(t) Jo
for 1 6 I 2 and y G Ep.
(ii) The nuclear space E consists of all real-valued infinitely differentiable functions x defined on [0,1] such that x(2fc)(0) = x^ 2fe+1 ^(l) = 0 for each keNU{0}.
3 The test and generalized Wiener functionals For a fixed Banach space B with the | • |B-norm, let Bc be the complexification of B with the | • Is^-norm of Bc defined by \x + iy\sc = sup{||e i e (x-My)||B c : 9 G [0,27r]} for x,y G B, where ((x+iyUg = |Z|B + |2/IB- Now, we consider the AWS (C, C) and (C, E-p) (p > 1). Let B be either C or £ _ p . Denote by £(B) the class of those functions
IM| £m( B)=
snp{\^(z)\e-m^:z€Bc}.
Let 5 m (B) = {
164
£{B) with the inductive limit topology induced by the family {£m(B)}. Then £{B) becomes a locally convex topological algebra. For notational simplicity, we denote £m(E-p), £m(C), £(E-P), and £{C) respectively by £m,v, £m,o, £P, and £Q. Then we have the following chain of continuous inclusions: £00 = f~lp>i £pc£pC£q
L2(C, fi)
asp>q>l,
where
J f(x)
:hi---hn:(x)
n(dx),
(3.1)
where Dn denotes the n-th Frechet derivative in the directions of C and :hi---hn:(x)= Jc Yl%i (x + iV' hi) Kdy) (Lee 14 ). When hi = h2 = • • • = hn = x £ C, and T is n—linear Hilbert-Schmidt operator defined on C, we define Txn = L2(C,/i) - lim / T(Pk(x) + i Pk(y))n fc—00 Jc
fi(dy),
Pk(z) = Y?j=i (2> fj) fj f° r z £ &*• Then the well-known Wiener-Ito decomposition of a function / € L2(C,n) can be reformulated as follows. 00
/(*)=
E l ••Dn(^*f)(0)xn: „ n!
a.e. (/x).
n=0
Moreover,
/ \f(x)\2»(dx) = E ^ H ^ ^ X ^ W v n=0 P
Let T(,4 ) be the second quantization of Ap on and let 00
L2(C,^L)
( A = v^
7-U
)
m
for / e L 2 (C,/x). For m € N and p > 1, define the || • || m)P -norm on L2(C,n) by \\m,p
•
l|r(^)(Qm/)|U2(c,M).
165
Denote by (E)m,P the class of functions / in L2{C,p) so that ||/|| m ,p < +°°Then {{E)mjP : m £ N} is an increasing sequence of Hilbert spaces of which the inner products induced by the || • || miP -norms. Let (E)p be the inductive limit of {(£')m,p} and (E) = D p >i (E)p, equipped with the projective limit topology. Then the following chain of continuous inclusions hold:
(E)c(E)pc(E)qcL2(C,v) for p > q > 1. Note that \i * f may be extended to a function defined on £L PiC . Using this fact, we shall see that every member of (E) enjoys an analytic version. Theorem 3.1 (Lee 1 9 ). Every member f of (E) admits a unique analytic function f £ £<x> defined on E* such that f = / fi-almost everywhere on E*. In fact, /(*) =
E
f Dn(fx*f)(z
-, n
+ iy)nfi(dy),
z e E*c,
(3.2)
n=0 - JC
where the series converges absolutely and uniformly on bounded subsets of E*. The second quantization T(AP) of Ap enjoys an integral representation as follows (see Lee 19 ): for / s L2(C), we have T(AP)f(x) =
Urn / (|i * f)(A»Pk(x) + iAvpk{y)) >x(dy) fc—oo Jc
=
lim
f f f(APPk(x)+iAPPk(y)
fe—oo Jc
+
z)f,(dyMdz),fi),
Jc
where the limit is taken in L2(C,/J). Employing such a representation, we have the following growth estimates. Theorem 3.2 (Lee 1 9 , Lee-Shih 2 2 ). (i) Let f be in (E). For anyp > 1, let mp e N so that f e {E)mpiP.
Then
||/|k m p , p < Cmp \\f\\mp,p, where cm = Jc emlyl°° fi(dy) for any m £ N. (ii) Let
f\\£mp,P,
where f3mp,p is a constant depending only on p and mp.
166 (Hi) SQO C (E) and the mapping f —> / is a homeomorphism from (E) onto c-oo*
Denote the dual spaces of So, £P (p > 1), and £oo respectively by ££, £*, and £^ which are topolozied by the weak*-topologies. Then we have the following chain of continuous inclusions: for p > q > 1,
5oc c £p c £q c s0 c
L2(C, M)
c £0* c £*q c e; c C -
Members of ££, will be referred as the generalized Wiener functionals (in short, GWF). Let ((•,•)) stand for the dual pairing of £Q-£Q-, £p-£P, and ££,foo, respectively, for notational convenience. The exponential vector functional £(77) associated with rj e Ec which is given by
£(77)= exp{(-,77)-i jf
fi(tfdt\.
Then e(rj) S £oo- We note that for any / e L2(C,fi), (/x* /)(??) = ((/, £(??))). Definition 3.3. TTie S-transform SF of F £ £^ is defined as a complexvalued functional on Ec by SF(rj)=
((Fein))),
n € Ec.
Denote the dual space of (E) by (E)* which is endowed with the weak*topology and let ((•, -))ym denote the dual pairing of (E)* and (E). Since (E) is dense in (E)p for any p > 1, (E) becomes the reduced topological projective limit of {(E)p). Then, by [Theorem 6, pp 290] 10 , (E)* = U p >i [E)*p which is endowed with the inductive limit topology. By Theorem 3.2, the embedding j : £(*, —» (E) is continuous. So, for any G e (E)*, G o j e ££a. Thus (E)* can be identified as a subspace of ££,. Conversely, for F G £^, define F on (E) by ((F, f))ym
:= ((F, /)),
/
G
(E),
where / is the analytic version of / on E*. It follows from Theorem 3.2 that
Fe(E)*. A functional G on Ec is analytic if it satisfies the following two conditions: (A-l) for all n,(j> £ Ec, the one complex variable mapping C 9 A \—> G(rj+X
167 Theorem 3.4 (The characterization theorem 2 2 ). Let F e £^ be fixed. Then the S-transform SF of F is an analytic function on Ec and there exists p>l so that for any m £ N , the number oo m 2 ™
"m,-p(f):=
E n=0
\,i-j
j-^\\DnSF(0)fHSHEp)
is finite. Conversely, suppose that G is an analytic function defined on Ec and satisfies the condition (A-2) for some p > 1. Let q be sufficiently large so that e 2 • Y^Li ^j < 1- Then there exists a unique F £ S^ such that nm-q{F) < +oo and SF = G, where Xj = (j - (1/2))TT for j £ N. For
<
||M*V>||£
\
—
11 r
I
"• / II I II c - m p . p
r l|C-mp,p
< c m \\tpWs —
""p
llr
,
M^Triptp'
where k is the least integer greater than e m p . Remark 3.6. Combining Theorem 3.3 with Proposition 3.4 we see that S(£oo) = £00c
A(EC) =
S{S^),
where A(EC) is the space of all analytic functions on Ec. Applying Theorem 3.3, one obtains Proposition 3.7. (i) SF = 0iffF = 0 for F £ £^. a total subset of £"oo.
Equivalently, the class {s(n) : n £ E} is
(it)
4 Examples of generalized Wiener functionals Let (C, B) be an AWS with B = C or B - E-p, p = 0 , 1 , 2 , . . . and let | • | B denote the corresponding norm. Unlike Hida's original approach, examples of GWF's given in this section will be defined by their linear functional forms instead of their ^-transforms, for more details we refer the reader to Lee 15 .
168
E x a m p l e 4 . 1 . Denote by Llxp(B,fi) defining on (B, B(B)) such that
the space of all measurable functions /
f(x) em ^B fi(dx) < oo
for all m G N.
/. Llxp(B,[i) is regarded as a subspace of £(B)* by identifying each / G Lg Xp (5,/x) with the functional Gf denned by
((Gf,
f(x)^x)^dx),
JB
for
e™\*\-p v{dx)
satisfying the prop-
(4.1)
Then the measurable support of v is contained in E-p. define
For > G £oo,
JE-v
Then v G f^,. For example, fi and Ot(x,dy) = /j,(dy — e *x) (the transition measure of Omstein-Uhlenbeck process) are GWF's in £Q . Moreover, we have Theorem 4.3 (Lee 1 8 ). Let F be a positive generalized function in £^a, that is, ((F, (p)) > 0 for (p G £oo with tp > 0. Then there exists a unique finite measure uF on (E*, B(E*)) so that {{F,
f
JE*
Moreover, the measurable support of vF is contained in E-p for some p > 1 and for all m G N, the condition (4.1) is satisfied, Example 4.4. (The generalized coordinate functionals) Let h€ B* and denote h(x) = (x, h). Then ~h G L2(fi) C ££, and
((h, V>» = / (x, h) V(x) li{dx) = (h, Dfi * V>(0)), JB
169
where Dfi * i>(0) denotes the Prechet derivative of fi. * ip at 0. For y £ B, choose {yn} C E so that yn —> y in 5 . Then, for ? € £oo, lim
(x,yn)ip(x)fi{dx)
n—»oo /.->
=
lim (yn, Dfi* ip(0)) = (y, Dfi * y(0)). n—>oo
This leads the following definition:
((j7> v » = &/> -°A* * v(o)) for v? e foo. Thenyef^. In particular, the white noise B(i) associated with the Brownian motion may be defined by be defined by ((B(t),
vGSoo,
where /i t = ftbt = l [t>1] . Then 5 ( i ) € £J\ Example 4.5. (The multiplicative renormalization) Let h GC and a S C, then we have 2
e-±<*
\h\l, f c «*(*.fc> y ,( a; ) /i (d a ;) =/i¥ ,( a /i). is
Define : e Q/l := e-i Q 2 l , , £'e a ' 1 . Then we have ((:eal:,
(4.2)
(4.2) holds not only for h £ C but also for h £ B. Defining : eal : by (4.2), then : ea~h :€ <%,. In particular, let y = ht = l[t,i]. Then we have ((: exp{a5(t)} :, ip)) = (fi *
€ C and (f €
/ : hi • • • hn : (x)
•
(4.3)
Note that the right hand side of (4.3) makes sense also for hj £ B. This leads to define additive renormalization of y~i • • • yn for yj £ B given as follows: ((: 2/1 • • • Vn ••, V» = Dn{n * ip)(0)yi •••ynThen :yi---yn:£ S^. In particular, for any positive times {£1,£2, • • • , t n } C [0,1], not necessarily distinct, define the renormalization of B(ti) • • • B(tn) by ((: B(ti) • • • B(tn) :,
•••htn
170 Example 4.7. Let T be a trace class operator defined on C, then {Tx, x)o can be defined a.e.(/j) on C and / c (Tx, x)0 fi(dx) = Tr(Ts), where Ts denote the symmetrization of T. For
:€ ££,. In particular, when T = I, we have # : A B(tfdt:,
cp\
= tracer [£>2(M * ip)(0)}.
Example 4.8. (Generalized multiple Wiener integrals) For / € L 2 ([0,1]"), then we have
f In(f)(x)V(x)ti(dx)
Je
= (([
J[o,i}n
f(tx,.. .,tn)dB(t{).
f(h,..., J\z [0,1]"
..dB(tn),
tn)DnS
• • • htndh
---din (4.4)
for (p G £oo. Observe that the the mapping ^lv given by *v(*i,...,«n)=
Dn((i*ip)(0)btl---btn
lies in £ ® n . If / is a generalized function in (E*)®n, then we define
«/„(/),
{{6M
may be de-
* » = vmo eA~m) IAX~ Wofh{x) -u}h)Kdx)-
for h e C and tp € £oo. It is clearly a positive generalized function in ££,.
171
5 Calculus of generalized W i e n e r functionals • Multiplication
by test
functionals
Let F € £^ and ip e £OQ. Then for any
For z G E, F £ £*, define the translation T Z .F by
((TZF,
z e
r_^»e^lo
for
Differentiation For any z £ E and for any ? G £00, define ((DzF
((F,z
It is easy to see that DZF is the unique generalized function G in ££, satisfying SG(y) = (x, D(SF)(y)), for y € E. For any x € E*, the mapping £>x is a continuous linear operator from £<„ into itself. Let Dx denote the adjoint operator of Dx, i.e. {{D*xF,
?£ £00.
holds for r],€,&E.
Proposition 5.1. Let F £ £^ and z € E. Then zF=
DZF + D*ZF
in£^.
Let dt — Dht and (?t* = D\ , where ht = l[t,i]- Then we have Proposition 5.2. (i) For z eC and
/ Jo
z{t) dt
xeC.
and F £ ££,,
D*ZF=
[
z{t)d*tFdt,
Jo where the integral exists in the sense of Pettis integral.
172
The combination of Proposition 5.1 and Proposition 5.2 yields the following well-known relation: B(t) =
dt+dt.
Example 5.3 (Lee 1 5 ). Let
/ / y(s)z(s)ip(x)dsfi(dx). Jc Jo Symbolically, we have dsy= y(s). In particular,
Then we have Dzy = (y,z). dsB(t)= St(s). Similarly, for yi,... ,yn £ E*, we have
D z :y\---yn
:= ,n_1y
J2 (*Mi)> z">1 : ^ ( 2 ) ' " ' ^ ( " )
:
where •K rums through all permutations of { 1 , . . . , n}. Symbolically , we write ds : m • • -yn : =
(n
_ 1 ) , J2 y*d)(s)
:
y*V) • • '&(«) : •
In particular, da : B^)
• • • B(tn) : = _ 1 _ £
5K(1) (s) : B^2)
• • • BK(n)
:.
Example 5.4 (Lee 1 5 ' 2 0 ). Let T be a symmetric bounded linear operator on C. For (p £ £oo and z £ E, apply Example 4.8 and the integration by parts formula, one sees that ((Dz : (Tx, x)0 :,
f 2Tz(x)
^dx)
2Tx(s)z(s)ip(x)dsfi(dx).
Then ds :(Tx,x)0:=
2Tx(s).
In particular, when Tx(t) = JQ f(s) x(s) ds for / £ 1*2 [0,1], we have
: f f(u)B(u)) Jo
du: J = 2/(s)B(s).
173 Example 5.5 (Lee 1 5 ' 2 0 ). Forti,t2,t3,...,tn, d*tld^...dll=
we have :B(t1)B(t2)...B(tn):
Let / be twice Frechet differentiable at x in C, then the restriction of D2 f(x) to C is of trace class on C by Goodman's Theorem (see Kuo 12 ). Define the Laplacian A/(x) of / at x by A/(x) = T r ( D 2 / ( x ) | c ) - Then the number operator Nf(x) is defined by Nf(x) = — A/(x) + (x, Df(x)). Then N is a self-adjoint operator on £(*,. Moreover , we have Definition 5.6 (Lee 1 5 ). Let F G £^ and
and A*F - J,,1
d;d;Fdt.
Transform
For a, /? e C and tp £ £ define the Fourier-Wiener transform J-a,p
:,
y € E*
(b) F(y) = -iD*y6,
y€E*
(c) T '• Btl • • • Btn := (—i)ndf • • • dtnSo, where So denotes the delta function concentrated at 0 £ E.
174
(d)
T : exp ( | jf B(t)2d?j
• Conditional
: =: exp ( i - jf B(t) 2 d/) :, (a ± 1)
expectation
Let £ „ = 0"({B(t,) : 0 = t0 < h < t2 • • • < tn = 1}). Theorem 5.9 (Huang-Lee 9 , Lee 2 0 ). Let
E[
It is easy to see that E[ip \ Bn ] £ SoTheorem 5.10 (Huang-Lee 9 ). Lett £ [0,1] and Bt be the a-field generated by B(u); 0 < u < t. Then, for (p G So, the conditional expectation E[<£> | Bt] of (p relative to Bt has an integral representation E[
J
Qty)n(dy)
for a.e. (/i) x in C, where Ptx(s) = x(s A t) for s £ [0,1] and Qt = I — PtMoreover, E[ tp \ Bt ] £ S0. Applying Theorem 5.9 and 5.10 we define the conditional expectation E[*|B„] and E[*|B t ] for a GWF $ as follows. Definition 5.11. For a GWF $ £ ££, we define «E[*|B n ], ^)>o = P , E[¥»|B„])>0; and «E[*|Bt],
6 The white noise integration • The Kubo-Takenaka formula A £ o - v a l u e d stochastic process X — {X(t) : t £ [0,1]} is called a generalized stochastic process. Definition 6.1. A generalized stochastic process {X(t) : t £ [0,1]} is called nonanticipating with respect to the filtration {B(t) : t £ [0,1]}, where B(t) — o~{B(u) : 0 < u < i], if for each t, X(t) is measurable with respect to B(t) in the sense that for all t £ [0,1], E[X(t)\B(t)} = X(t).
175 A generalized stochastic process {X(t) : t € [0,1]} is called a step process if there exist a partition {0 = to < t\ < • • • < tn = 1} such that X(t) = Xi for t £ [ti,ti+\). For a generalized nonanticipative step process X(t), we define the generalized stochastic integral with respect to Brownian motion in Ito sense by d
f
X{t)dB{i) := J2 Xi-iWU)
-Bfa-i)).
i=l
Jo
If X{t) is a nonanticipative continuous generalized process satisfying
I
^ \\X(t)\\*n
0
m
dt
for all m € N. We define
f X(t)dB(t) := lim £ *(«,_!)(£&) - 5(^-0) in S^, JO
ll^nll—0
^
where A n = {0 = t0 < t\ < • • • < tn = 1} is any partition of [0,1] and ||A n || is the mesh of A n . Theorem 6.2 (Huang-Lee 9 ). Let {X(t) : t £ [0,1]} be a nonanticipative continuous generalized process. If d
/ \\X(t)\\l.|2 Jo
n
dt < oo for all m € N,
then
f X(t)dB{t)= Jo
[ d*tX{t)dt Jo
Remark 6.3. When X(t) is further assumed to be satisfied the conditions: X(t)€£2(C,/i)and l
Jc Jo
2 \X(t\x)\2dtfi(dx)
< +oo,
the above theorem is exactly the well-known Kubo-Takenaka formula (see Kubo 6 and Kubo-Takenaka 7 ' 8 ). • The Clark formula for GWF's The representation of functional of Brownian motion by stochastic integral with respect to Brownian motion, known as the Clark formula, was first
176 studied in Clark 1 and later in Faria-Oliveira-Streit 2 and in Karatzas-OconeLi 11 . We shall reformulate of the Clark formula on the classical Wiener space for a class of GWF's. Note that even for rj G Ec, Pt(r))(s)(= r](s/\t)) lies in C* but not necessarily lies in Ec. Thus, we restrict our consideration only to the GWF's in £Q. Lemma 6.4 (Lee-Shih 2 2 ). For F G £% and r) G Ec, the mapping SF{Pt(rj)) is absolutely continuous as a function of one variable with respect tote [0,1]. For any TJ G EC, Lemma 6.4 implies that (d/dt) SF(Pt(r))) exists almost everywhere in (0,1) and (d/dt) SF(Pt(r])) is Lebesgue integrable so that SF(Pb(r,))-SF(Pa(r,))
= J
±SF(Pt(r,))dt,
0 < a < b < 1.
In particular, SF{r,) = ((F,l)) + £
±SF(Pt{r,))dt.
For any t G [0,1], let TF(t, •) be a complex-valued mapping on Ec given by TF(t,V) = jtSF(Pt(r,)),
r, G Ec.
Then Tp(t, •) is analytic on Ec. Applying Theorem 3.4, there exists a unique generalized function Kp(t) in ££, such that SKp(t)(r)) = Tp(t, rj) for each rj G Ec. The integral JQ Kp(t) dt exists in E^. Here, for a £^-valued function G on [0,1], the integral J 0 G(t) dt is defined as n
ttm V(fc-*<_!) G(0 in £^, provided that the limit exists, where A = {0 = to < ti < • • • < tn = 1} is any partition of [0,1], ||A|| is the mesh of A, and t^s are arbitrarily taken over [ti, U-i]. Moreover, for any ip G £oo,
[ KF(t)dt,
//oo
JO
Consequently, SF(ri) = ((F, 1)) + S ( f It follows that we have
KF(t) dt J (TI) for F G £0* and n G Ec.
177 Theorem 6.5 (Lee-Shih 2 2 ). Let F be in £Q. Then there exists a unique KF(t) £ SZo so that S(KF(t))(V)
=
±S(E[F\Ft]){V)
for any t £ [0,1] and r\ 6 Ec. Moreover, the integral f0 KF(t) dt exists in ££, and F=
((F,l)) + / Jo
KF(t)dt.
Let
Jo
By a direct computation, the equality (6.1) becomes / T,{t)S(p[D
f Jo
f,(t)S(D
= £ jtS
Jo
a.e. (fi) x on C,
where the integral is the Wiener-Ito stochastic integral. • Generalized ltd formula for Gaussian Processes For / £ S, the Schwartz space on R, and a nonzero h S C , jf f(h(x))
f°°
V27T J-oo
e-Wy2W2of(y)Sp(iyh)dy,
178
where f is the Fourier transform of / . Since the mapping y i-» -(i/2)y IMo S
tpeSvo,
(6.2)
where Gh,v(y) = l / v ^ e - * 1 / 2 ^ ! ? (Sip)(iyh). Then F(h) € £^. Next consider the Gaussian process Z(t) represented by Zt(x) = (x,zt) with zt € C. Assume that the L2P, l]-valued function zt is differentiable with derivative kt, i.e. lim
Zt+e
~ ** = * t in X2[0,1] ,
(6.3)
We shall derive the Ito formula for F(Z(t)) for any tempered distribution F on R. By (6.2), for 0 < t < 1, F(Z(t)) is defined as a generalized function in ^oby
pxz(i)),
F(Z(b)) = F(Z(a)) + J" D*kt F'(Z(t))dt + i jf r(t)F"(Z(t)) dt,
(6.4)
where 0 < a < b < 1. Remark 6.8. The term Ja D£ F'(Z(t))dt tegral. For example,
is interpreted as a stochastic in-
(a) If Z(t) is the Brownian motion, then zt(s) = s At , r(t) = t and kt(s) = l[ t l ](s). In this case, we have rb
rb
I Dlt F'(Z(t))dt = J F'(Z(t)dZ(t). Ja
Ja
(b) If Z(t) is the Brownian bridge, then zt(s) = s A t - st , r(t) = t(l — t) and kt(s) = l[t,i](«) — s- I n this case f D*kt F'(Z(t))dt Ja
= f Ja
F'(Z(t)dZ(t+)
+ f tf"(X{t))dt Ja
in S(L2)*
179 for 0 < a < b < 1, where the second term represents the forward integral denned as follows: Let {Yt : a < t < b},0 < a < b < 1 be a. continuous £ o - v a m e d process. We define /
YtdX(t+):=
Urn f > * ; ~ ^ - J ^
in 5 ^
if this limit exists, where r = {a — to < t\ < • • • < tn = b} (see Kuo 13 ). Using the forward integral, then, for and F € C 2 (R), Ito formula becomes F{Z{b))=
F(Z(a)) + j " F'(Z(t))dZ(t+)
+ \ j " F"(Z(t))dt.
(6.5)
(c) If Z(t) is a semimartingale version of the Brownian bridge represented by Z(t,x) = (x,zt) with J3t(s) = ( t - l ) l n ( l - s A t ) a n d ^ ( s ) = 0, 0 < s < 1. Then r(t) = t(l - t) and kt = l[t,i](s) + ln(l - s At) for s € [0,1], and for 0 < t < 1. In this case the equation (6.5) holds with
/ F'(Z(t))dZ(t+)= J F'(Z(t))dZ(t). Ja
Ja
References 1. J. M. C. Clark: The representation of functionals of Brownian motion by stochastic integrals, Ann. Math. Stat. 41 (1970), 1281-1295; 42 (1971), 1778 2. M. de Faria, M. J. Oliveira, and L. Streit: A generalized Clark-Ocone formula, Preprint, 1999. 3. L. Gross: Abstract Wiener space, In "Proceedings 5th Berkerley Symp. Math. Stat. Prob." Vol.2 (1965), 31-42 4. T. Hida: Analysis of Brownian Functionals, Carleton Mathematical Lecture Notes 13, 1975 5. T. Hida, H.-H. Kuo, J. Pottohoff, and L. Streit: White Noise: An Infinite Dimensional Calculus, Kluwer Academic Pubblishers, 1993 6. I. Kubo: Ito formula for generalized Brownian functionals, Lecture Notes in Control and Information Sciences 49, 156-166, Springer-Verlag, 1983 7. I. Kubo and S. Takenaka: Calculus on Gaussian white noise III, Proc. Japan Acad. 57A (1981), 433-437 8. I. Kubo and S. Takenaka: Calculus on Gaussian white noise IV, Proc. Japan Acad. 58A(1982), 186-18
180
9. H.-C. Huang and Y.-J. Lee: Conditional expectation of generalized Wiener functionals, 2001, Preprint 10. G. Kothe: Topological Vector Space I, Springer-Verlag, New York/ Heidelberg/ Berlin, 1966 11. I. Karatzas, D. Ocone, and J. Li: An extension of Clark's formula, Stochastic and Stochastic Reports 37 (1991), 127-131 12. H.-H. Kuo: Gaussian Measures in Banach Spaces, Lectures Notes in Math. Vol. 463, 1975 13. H.-H. Kuo: White Noise Distribution Theory, CRC Press, 1996 14. Y.-J. Lee: Sharp inequalities and regularity of heat semigroup on infinite dimensional space, J. Fund. Anal. 71 (1987), 69-87 15. Y.-J. Lee: Generalized functions on infinite dimensional spaces and its application to white noise calculus, J. Fund. Anal. 82 (1989), 429-464 16. Y.-J. Lee: A reformulation of white noise calculus, in "White Noise Analysis- Mathematics and Applications", Edited by T. Hida, H.H. Kuo, J. Potthoff and L. Streit, 274-293, World Scientific (1990). 17. Y.-J. Lee: Analytic version of test functionals, Fourier transform and a characterization of measures in white noise calculus, J. Fund. Anal. 100 (1991), 359-380 18. Y.-J. Lee: Positive generalized functions on infinite dimensional spaces, In " Stochastic Process, a Festschrift in Honour of Gopinath Kallianpur", Springer-Verlag, 1993, 225-234 19. Y.-J. Lee: Integral representation of second quantization and its application to white noise analysis, J. Fund. Anal. 133 (1995), 253-276 20. Y.-J. Lee: Generalized white noise functionals on classical Wiener space, J. Korean Math. Soc. 35 No.3 (1998), 613-635 21. Y.-J. Lee and C.-C. Huang: Ito formula for functional of Brownian bridge, preprint 22. Y.-J. Lee and H.-H. Shih: The Clark formula of generalized Wiener functionals, In "Quantum Information IV", to appear. 23. P. Malliavin: Stochastic calculus of variation and hypoelliptic operators, Proc. Int. Sym. S.D.E. Kyto, 1976, Kinokuniya,1978, 195-263 24. P. A. Meyer and J.-A. Yan: Les "fonctions caracteeristiques" des distribution sur l'espace de Wiener, Sem. Probab. XXV, Lecture Notes in Math. 1485 (1991), 61-78 25. D. Ocone: Malliavin's calculus and stochastic integral representations of functionals of diffusion process, Stochastics. 12 (1984), 161-185 26. S. Watanabe: Lectures on Stochastic Differential Equations and Malliavin Calculus, Tata Inst, of Fundamental Research, Bombay, 1984
Quantum Information V Eds. T. Hida and K. Saito (pp. 181-191) © 2006 World Scientific Publishing Co.
F R A C T I O N A L B R O W N I A N M O T I O N S A N D T H E LEVY LAPLACIAN *
KENJIRO NISHI, KIMIAKI SAITO Department
E-mails:
of Information Sciences Meijo University Tempaku, Nagoya 468-8502, Japan [email protected]; [email protected] ALL A N U S H. T S O I Department of Mathematics University of Missouri Columbia, MO 65211, U.S.A. E-mail:[email protected]
In this paper we give a relationship between fractional Brownian motions and the Levy Laplacian acting on a space of white noise distributions introducing an operator changing the white noise B{t) by e iB (*>. In addition, we give a stochastic expression of the semigroup generated by the powers of the Levy Laplacian through the operator.
1. Introduction The Levy Laplacian was introduced by P. Levy 22 . Hida 7 discussed this Laplacian in his theory of white noise functionals. Accardi, Gibilisco, Volovich 2 obtained an important relationship between this Laplacian and the Yang-Mills equations. It has been studied by many authors ( see Refs. 1,3-6,8,12,19,20,21,23-25,28-30,32-38, etc). In the previous paper 36 we extended the Levy Laplacian to a self-adjoint operator densely defined on a Hilbert space and also gave a relationship between the Laplacian and an infinite dimensional Ornstein-Uhlenbeck process through an operator changing the white noise B{t) by elB^\ "This work is supported by the Joint Research Project "Quantum Information Theoretical Approach to Life Science" for the Academic Frontier in Science promoted by the Ministry of Education in Japan.
182
The purpose of this paper is to give a relationship between the fractional Brownian motions and the Levy Laplacian acting on white noise distributions, through an infinite dimensional fractional Brownian motion. We also give a stochastic expression of the semigroup generated by the power of the Levy Laplacian. The paper is organized as follows. In Section 2 we summarize some basic definitions and results in the white noise analysis. In Section 3, following our previous paper 3 6 , we explain a self-adjointness of the Levy Laplacian and give an equi-continuous semigroup of class (Co) generated by the powers of the Levy Laplacian. In Section 4, following our previous papers 34>35, we explain a relationship between the Levy Laplacian and the number operator introducing an operator changing the white noise B(t) by elB^\ In the last section we give an infinite dimensional fractional Brownian motion and a relationship to the Levy Laplacian. A stochastic expression of a semigroup generated by the power of the Levy Laplacian is also obtained through the operator in Section 4. 2. Preliminaries In this section we assemble some basic notations of white noise analysis following Refs. 10, 16, 19 and 26. We take the space E* = <S'(R) of tempered distributions with the standard Gaussian measure fj, such that J
exp{t(x,0} dti(x) = exp ( - ^ | § ) ,
£ € £ = 5(R),
where | • |o is the L 2 (R)-norm, and (•,•) is the canonical bilinear form on E* x E. Let A — — (d/du)2+u2+l. This is a densely defined self-adjoint operator 2 on L (R) and there exists an orthonormal basis {e„; v > 0} for L 2 (R) such that Aev = 2(i/ + l)e„. We define the norm | • \p by | / | p = \Apf\0 for / G E and p G R, and let Ep be the completion of E with respect to the norm | • \p. Then Ep ia a real separable Hilbert space with the norm | • | p and the dual space E'p of Ep is the same as £ L P (see Ref. 16). Let E be the projective limit space of {Ep;p > 0} and E* the dual space of E. Then E becomes a nuclear space with the Gel'fand triple E C L2(R) C E*. We denote the complexifications of L 2 (R), E and Ep by L ^ R ) , Ec and Ec,P, respectively.
183 The space (L 2 ) = L2(E*,/j.) of complex-valued square-integrable functional denned on E* admits the well-known Wiener-Ito decomposition: oo
(i 2 ) = 0ffn, 71=0
where Hn is the space of multiple Wiener integrals of order n € N and Ho = C. Let Lc(R)® n denote the n-fold symmetric tensor product of L2C(R). If ip G (L2) is represented by
Mlo= £n!|/„|g =0
where | • |o means also the norm of L2-;(R)lS>™. For p G R, let ||<£>||p = ||r(A) p (^||o, where T(A) is the second quantization operator of A. If p > 0, let (E)p be the domain of Y(A)P. If p < 0, let (Z?)p be the completion of (L 2 ) with respect to the norm || • || p . Then (E)p, p G R, is a Hilbert space with the norm || • || p . It is easy to see that for p > 0, the dual space (E)p of (E)p is given by [E)_p. Moreover, for any p G R, we have the decomposition oo
(£)„ = ©*#>, n=0
where # 4 is the completion of { I n ( / ) ; / G £ § " } with respect to || • || p . Here E®n is the n-fold symmetric tensor product of EQ. We also have Hn = {In(/); / G E^p} for any p G R, where £ § ^ is also the n-fold symmetric tensor product of Ec,p- The norm ||?||p of > = £^°=o !«(/«) € ( £ ) p is given by / oo
\ 1/2
IMIp=(£nl|/»l2J
- /»€£§£,
where the norm of i?®" is denoted also by | • | p . The projective limit space (E) of spaces (E)p, p G R is a nuclear space. The inductive limit space (E)* of spaces (E)p,p G R is nothing but the dual space of (E). The space (E)* is called the space of generalized white noise junctionals. We denote by
184
(E)* x (E). Then we have OO
n=0
for any $ = £ ~ = 0 I n ( F n ) G (£)* and ^ - £ ~ = 0 I n ( / » ) G (E), where the canonical bilinear form on (E®n)* x ( £ ^ n ) is denoted also by (•,•). Since exp(-,£) G (E), the S-transform is defined on (£)* by 5 [ * ] ( 0 = exp ( - | « , 0 ) « * , e x p ( - , 0 » ,
£ G £C-
3. Self-adjointness of the Levy Laplacian The .S-transform F G S[(E)*] has a property that for any £,77 G i?c the function F(£ + zr\) is an entire function of z G C. Hence we have the series expansion: 00
n
n=0
where F^™) (£) : EQ, X • • • X F C —> C is a continuous n-linear functional. Fix a finite interval T in R. Take an orthonormal basis {C„}$£L0 C E for L {T) satisfying the equally dense and uniform boundedness property (see Refs. 7,10,9,20,22,25,28 and 30). Let VL denote the set of all $ G (E)* such that the limit 2
JV-l
*LS[$](.Q = ^
Jf £ W ( 0 ( C n , Cn) n=0
exists for any £ G F c and is in S[(E)*\. The Levy Laplacian A/, is denned by AL$ = 5_1AL5$ for $ € X>L- We denote the set of all functional $ G T>i such that 5[$](?7) — 0 for all 77 G F with suppfa) C T c by 2?J. Take a white noise distribution $ whose S-transform S[$](£) is given by S[$](0= /
/ ( u ) e i a i l ( u i ) . . . eia^{u^du,
(3.1)
185 / e £ § n , o t € R , k = l , 2 , . . . ,n. Using the Wick ordering : by /
: we write the generalized white noise functional
/ ( « i , . . . ,un):eiaiX^Ul)---eia"x^)
: du,
(3.2)
This functional is in 2?£ and an eigenfunction of the operator A L as follows. Theorem 3.1. 36 A generalized white noise functional $ as in (3.2) satisfies the equation AL* = ~ X > 2 * .
(3.3)
Put D n = J J ^ / ( u ) : f[ ete("-> : du € X>£; / e Ljj(R) & n n L^(R)® n 1 for each n £ N U {0}. Then D n is a linear subspace of (E)-p for any p > Y5, (see 21 ) and A L is a linear operator from D„ into itself such that | | A i $ | | _ p = Tyj||$||_ p for any $ € D n . Define a space D„ by the completion of D n in (E)-p with respect to || • ||_ p . Then for each n € NU{0}, D n becomes a Hilbert space with the inner product of (E)_p. For each n S N U {0}, the operator A^ becomes a continuous linear operator A/, from D n into itself satisfying | | A L $ | | _ P = — | | $ | | _ p for any $ G D „ .
N
(
\
2 £
Put a^v (n) = ^2t=0 (i^i)
{
oo
and define a space
E_ P ) JV
by
oo
5^*Be(E)*;5^aAr(n)||$B||lp
n=0
with the norm j | | • |||- P ,JV given by
(
oo
\ V2
^a N (n)||$ n || 2 _ p n=0
oo
, $ = 53$ n GE_ P l J v /
n=0
186 for each N e N U {0} and p > ^ . Then for any N e N and p > ^ , E_ P i i V is in ( £ ) - ? and is a Hilbert space with respect to the norm ||| • |||_ p ,;vWith the projective limit space E_ P i 0 0 = Dw=i E-P,N, ing inclusion relations:
we have the follow-
E_p,oo C • • • C E_ P i i V + i C E_p,iv C • • • C E _ P i l c ( £ ) - P . For any n € NU {0}, D„ is included in E_p,oo. The operator Az, can be extended to a continuous linear operator defined on E_ Pi 2 into E _ P J I , denoted by the same notation AL, satisfying |||A£$|]|_ P) ./v < |||$|||_ p ,./v +1 , N = 1,2,3,... , $ € E_ P i 0 0 . With these properties, we have the following: Theorem 3.2. 36 The operator A^ becomes a self-adjoint operator densely defined on E_ Pi ;v for each N > 1 and p > j%. Let 7 > 0. Define a linear operator G] on E_ P i 0 0 by oo n=0
for each t > 0 and $ = £ £ L 0 $ „ € E_p,oo, $ n e D „ , n = 0 , l , 2 P r o p o s i t i o n 3 . 1 . 35 Let 7 > 0 a n d p > ^ . Then {GJ;t > 0} is an equicontinuous semigroup of class (Co) generated by - ( - A / , ) 7 as a continuous linear operator from E_p,oo into itself.
4. A relationship between the Levy Laplacian and t h e number operator Put
{
00
00
$ = £ l „ ( / „ ) € (£)_,; S[*](e*) = £ > * ) ® n , / n ) n=0
"=0
exists in 5[E_ P and define an operator K on £_ p by if[$]=S-1[5[$](e«)].
187 The operator K implies a relationship between AL and the number operator Jf on (E)* given by oo
tf$ = £ n I „ ( / „ ) for 4 = £ ~ = o M / » ) 6 ( £ ) * , n=0
as follows.
Proposition 4.1.
36
For any $ € £_ p we ftai/e
ALK[S] = -±KM*]]Put
{
OO
OO
n=0
n=0
s u p p ( / n ) c T,n = 0 , 1 , 2 , . . . } for g > 0 and N G N . Define a space [£^]q,jv by the completion of [E]g^ with respect to the norm || • ||T™— given by / oo
\ 1/2
for ? = £ ^ l 0 I n ( / n ) € (-B)- Then [J5]9,JV is a Hilbert space with norm || • ||[ g ] N- It is easily checked that [£],,# C {E)q for any q > 0. Put = C\q>o [E]q,N with the projective limit topology and also put [£]co,oo = rW>i [^]OO,JV with the projective limit topology. The following fact is proved in Ref. 35. [•EJOO.JV
Proposition 4.2. 3 5 Let p > ^ . Then the operator K is a continuous linear operator from [-E]<x>,oo into E_ P i 0 0 .
188 5. An infinite dimensional fractional Brownian motion and the Levy Laplacian Let {B2(t); t > 0}, k = 0 , 1 , 2 , . . . , be an independent sequence consisting of fractional Brownian motions with the parameter 0 < 7 < 2, which has the covariance function:
The operator K implies a relationship between the semi-group {G~l; t > 0} and an infinite dimensional stochastic process {U^\t > 0} starting at x e E* given by
u x)=exp
^
x+
exp
dB 7 (s)
{-w\){ ^\Io (w\
, t > 0,
where {B 7 (t);t > 0} is an infinite dimensional fractional Brownian motion defined by 00
k=0
Then we have the following: Theorem 5.1. For any t > 0 and
(5-1)
Proof. For ip = exp{(-,r?) — \{r),rf)}, rj G Ec we can check that e-&"
= E[
x€E*.
Since a linear span of exp{(-,77) — \{r),rj)}, 77 € Ec is dense in (E) and [E]ca,oo is in (E), we obtain (5.1). • Let fJ^(x, A) be a transition function given by
lil(x,A) =e-t(WY\A{x),
xeE*,
t > 0,
where A is a bounded Borel set. We can define the function n1(-,A) in (L2) for each t > 0 and bounded Borel set A. Then we have the following: Theorem 5.2. For any t > 0 and
GjK[p\{-) = K
I v{y)vl{-,dy) JE'
(5.2)
189 Proof.
For any ip = J^LQ
/
I n ( / n ) S (E) we have
= £
e^i^M/nX*).
n=0
^
Hence, for any tp = Y^=o M / n ) € £ - P n (-E).
we
Set
oo
K
JE-
fiy^U-^y)
On the other hand, since S[K[
f o r an
y V =
oo
S[GlK[p\\{£) = ^ e - f e l ^ e ' f " , / ^ . n=0
T h u s we obtain (5.2).
•
References 1. Accardi, L. and Bogachev, V.: The Omstein-Uhlenbeck process associated with the Levy Laplacian and its Dirichlet form, Prob. Math. Stat. 17 (1997), 95-114. 2. Accardi, L., Gibilisco, P. and Volovich, I.V.: Yang-Mills gauge fields as harmonic functions for the Levy Laplacian, Russian J. Math. Phys. 2 (1994), 235-250. 3. Accardi, L., Smolyanov, O. G.: Trace formulae for Levy-Gaussian measures and their application, in Mathematical Approach to Fluctuations Vol. II, pp. 31-47, World Scientific, 1995. 4. Chung, D. M., Ji, U. C. and Saito, K.: Cauchy problems associated with the Levy Laplacian in white noise analysis, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 2 (1999), 131-153. 5. Feller, M. N.: Infinite-dimensional elliptic equations and operators of Levy type, Russian Math. Surveys 4 1 (1986), 119-170. 6. Hasegawa K.: Levy's Functional Analysis in terms of an infinite dimensional Brownian motion I, Osaka J. Math, 19 (1982), 405-428. . 7. Hida, T.: Analysis of Brownian Functionals, Carleton Math. Lect. Notes, No. 13, Carleton University, Ottawa, 1975. 8. Hida, T.: A role of the Levy Laplacian in the causal calculus of generalized white noise functionals, Stochastic Processes, pp. 131-139, Springer-Verlag, 1993. 9. Hida, T., Kuo, H.-H. and Obata, N.: Transformations for white noise functionals, J. Fund. Anal. I l l (1993), 259-277. 10. Hida, T., Kuo, H.-H., Potthoff, J. and Streit, L.: White Noise: An Infinite Dimensional Calculus, Kluwer Academic, 1993.
190 11. Hida, T., Obata, N. and Saito, K.: Infinite dimensional rotations and Laplacian in terms of white noise calculus, Nagoya Math. J. 128 (1992), 65-93. 12. Hida, T. and Saito, K.: White noise analysis and the Levy Laplacian, in Stochastic Processes in Physics and Engineering, S. Albeverio et al. Eds., pp. 177-184, 1988. 13. Hille, E and Phillips R. S.: Functional Analysis and Semi-Groups, AMS Colloq. Publ. Vol. 31, Amer. Math. Soc, 1957. 14. Ito, K.: Stochastic analysis in infinite dimensions, in Proc. International Conference on Stochastic Analysis, pp. 187-197, Academic Press, 1978. 15. Kubo, I.: A direct setting of white noise calculus, in Stochastic Analysis on Infinite Dimensional Spaces, pp. 152-166, Pitman Res. Notes in Math. Vol. 310, 1994. 16. Kubo, I. and Takenaka, S.: Calculus on Gaussian white noise I-IV, Proc. Japan Acad. 56A (1980) 376-380; 56A (1980) 411-416; 57A (1981) 433436; 58A (1982) 186-189. 17. Kuo, H.-H.: On Laplacian operators of generalized Brownian functionals, in Lecture Notes in Math. Vol. 1203, pp. 119-128, Springer-Verlag, 1986. 18. Kuo, H.-H.: Lectures on white noise calculus, Soochow J. Math.18 (1992), 229-300. 19. Kuo, H.-H.: White Noise Distribution Theory, CRC Press, 1996. 20. Kuo, H.-H., Obata, N. and Saito, K.: Levy Laplacian of generalized functions on a nuclear space, J. Fund. Anal. 94 (1990), 74-92. 21. Kuo, H.-H., Obata, N. and Saito, K.: Diagonalization of the Levy Laplacian and related stable processes, to appear in Infin. Dimen. Anal. Quantum Probab. Rel. Top. 5 (2002). 22. Levy, P.: Lecons d'Analyse Fonctionnelle, Gauthier-Villars, Paris, 1922. 23. R.Leandre and I. A. Volovich: The stochastic Levy Laplacian and Yang-Mills equation on manifolds, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 4 (2001) 161-172. 24. Nishi, K., Saito, K. and Tsoi, A. H.: A stochastic expression of a semi-group generated by the Levy Laplacian, in Quantum Information III, T. Hida and K. Saito, Eds., pp. 105-117, World Scientific, 2000. 25. Obata, N.: A characterization of the Levy Laplacian in terms of infinite dimensional rotation groups, Nagoya Math. J. 118 (1990), 111-132. 26. Obata, N.: White Noise Calculus and Fock Space, Lect. Notes in Math. Vol. 1577, Springer-Verlag, 1994. 27. Obata, N.: Integral kernel operators on Fock space —Generalizations and applications to quantum dynamics, Acta Appl. Math. 4 7 (1997), 49-77. 28. Obata, N.: Quadratic quantum white noises and Levy Laplacian, Nonlinear Analysis 47 (2001), 2437-2448. 29. Obata.N. and Saito, K.: Cauchy processes and the Levy Laplacian, to appear in Non-Commutativity, Infinite-Dimensionality, and Probability at the Crossroads, N. Obata, T. Matsui and A. Hora, Eds., World Scientific, 2002. 30. Polishchuk, E. M.: Continual Means and Boundary Value Problems in Function Spaces, Birkhauser, Basel/Boston/Berlin, 1988. 31. Potthoff, J. and Streit, L.: A characterization of Hida distributions, J. Fund.
191 Anal. 101 (1991), 212-229. 32. Saito, K.: Ito's formula and Levy's Laplacian I and II, Nagoya Math. J. 108 (1987), 67-76; ibid. 123 (1991), 153-169. 33. Saito, K.: A (Cb)-group generated by the Levy Laplacian II, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 1 (1998) 425-437. 34. Saito, K.: A stochastic process generated by the Levy Laplacian, Acta Appl. Math. 6 3 (2000), 363-373. 35. Saito, K.: The Levy Laplacian and stable processes, Chaos, Solitons and Fractals 12 (2001), 2865-2872. 36. Saito, K., Tsoi, A. H.: The Levy Laplacian as a self-adjoint operator, in Quantum Information, T. Hida and K. Saito, Eds., pp. 159-171, World Scientific, 1999. 37. Saito, K., Tsoi, A. H.: The Levy Laplacian acting on Poisson noise functionals, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 2 (1999), 503-510. 38. Saito, K., Tsoi, A. H.: Stochastic processes generated by functions of the Levy Laplacian, in Quantum Information II, T. Hida and K. Saito, Eds., pp. 183-194, World Scientific, 2000. 39. Yosida, K.: Functional Analysis (3rd Edition), Springer-Verlag, 1971.
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Quantum Information V Eds. T. Hida and K. Saito (pp. 193-201) © 2006 World Scientific Publishing Co.
J U M P F I N D I N G OF A STABLE P R O C E S S
SI SI Faculty
of Information Science and Technology Aichi Prefectural University Nagakute-cho, Aichi-ken, 480-1198, Japan ALLANUS TSOI Department of Mathematics University of Missouri Columbia, MO 65211, U.S.A. W I N W I N HTAY
Department of Computational Mathematics University of Computer Studies Yangon, Myanmar
The functional product of stochastic processes in terms of subordination is discussed under a new setup of white noise analysis. A compound Poisson noise, involving all kind of Poisson noise with different heights of jumps, can be obtained from a Brownian motion by deriving its maximum and minimum processes. In this sense, a relationship between Gaussian and Poisson white noise is discovered. We propose a method of j u m p finding, in other words, a technique how to obtain j u m p points and their heights (jump sizes) even in the case of different heights.
1. Introduction As a background, we first review some general theory of stochastic analysis arising from additive processes. Then a typical case arises from Brownian motion by taking its maximum to obtain a stable process with exponent a = \. We recall Levy's decomposition of stable process from which we can prove that all compound Poisson processes with different height of jumps, in other words different jump sizes, and some stable process from a Brownian motion. Subordination, introduced by Bochner, is the concept of replacing time
194 with random time. It is viewed, in this paper, as a functional product of processes defined as: X • Y(t) = X(Y(t)), where Y(t) is an increasing stochastic process. We first show that the characteristic function of a single Poisson process can be obtained from the characteristic function of compound Poisson process. Next, by using path-wise theory we are able to find the jump points of a Poisson process if it has the same jump size, say u. Finally, even if it has different jump sizes, say Uj, the jump points and the jump sizes Uj are obtained with the assumption that Uj are linearly independent on Z. 2. Background In this section we discuss some stochastic analysis arising from additive processes. Our approach is an innovation approach. Innovation is the most basic concept in the analysis of random evolutional complex systems and it is a stochastic process with independent values at every point. A stochastic process X(t) is said to be a process with independent values at every point if C(fi + &) = C ( 6 ) C ( & ) , whenever &(*)&(*) = 0 As is well known, the innovation of a stochastic process, say X(t), is a stochastic process which has independent values at every point t obtained in the framework of causal calculus and it contains the same information as X. Next we recall an important theorem: Theorem 2.1 (Gel'fand [2]) The functional defined by
Ctf) = exp
Jfiamt
(2.1)
is a characteristic functional of a generalized stochastic process urith independent values at every point if the continuous function f(x) has the form f{x)=
I (eiXx - a(\)(l •mix) /|A|>0
+ i\x)) da(\) + a0 + iaxx - — a x2. 2! 2
where a is a positive ve measure measure such such that that f
da(X) + [
X2da(X) < oo,
(2.2)
195 <22 is a positive number, a(A) is a function on the space of complex numbers such that a(A) — 1 has a zero of the third order at X = 0, and ao, a% are any numbers. The following proposition follows immediately from the above theorem. Proposition 2.2 If X is an innovation of the Gel'fand type with the characteristic functional C(£) of the form (2.1) where a(X) = 1, then X can be decomposed as X = X\ + X2 + X3, where X\,Xi and X3 are compound Poisson process, Gaussian process and a deterministic part respectively. As is seen from (2.2), X is stationary, and so the fixed discontinuity part is naturally missing. Note that the above proposition coincides with Levy's decomposition for an additive process (ref [3]). Let Y be an additive process with stationary increments. Then Y can be decomposed as Y = Y\ + Y2, where Y\ is a compound Poisson process and Y2 is Gaussian, and we may ignore the deterministic part. Y\(t) is the collection of all the jump processes Yu{t) for all jump sizes u, and we have Yi(t) = f(uYdu(t)
-
m{u)dn{u))
where dn(u) is the Levy measure and m(u)dn(u) is the mean of uYdu. Y\(t) is quasi convergence. In this paper we give an analytical method of singling out a Yu{t) for a particular u, and we call this procedure 'jump finding'. Note that
Y2=Y-Y1 is the continuous part of Y, and is Gaussian. A typical example for an additive process is given in the following: Let B(t) be a Brownian motion, M{t) be the maximum of Brownian motion, (i.e. M(t) = maxs
196 Given t, £ > 0. If M(t) > £, we see that there exists r < t such that B(t) attains the value £ for the first time at r. It is well known that r is a Markov time. Theorem 2.3 (P. Levy [9]) The maximum of a Brownian motion has the probability distribution;
P[M(t) <x} = J-^
f e-*d£, x > 0.
M(t)
(2.3)
We see that the maximum M{t) of Brownian motion and its inverse function T(x) are linked by the following relation: x > M(t)
and t < T(x-)
are equivalent
(2.4)
and T ( x - ) < t < T(x+)
is equivalent to M(t) = x.
(2.5)
Prom the above facts and from the probability distribution of M(t), we obtain the following probability distribution of T(x) : Theorem 2.4(PLevy [9]) P\T(x)
x > Q,t > 0.
(2.6)
Proposition 2.5 T(x) is an additive and increasing stable process with exponent a = | . Proof. Additivity comes from the fact that a Brownian motion satisfies the strong Markov property. Remark We can play a similar game to the minimum of a Brownian motion. Summing up all the above, we have the following theorem. Theorem 2.6 All members of compound Poisson processes can be constructed from a Brownian motion.
197 3. Stable process and jump rinding We applied the concept "subordination" , introduced by S. Bochner to construct the stochastic model for X-ray data in [13]. The emission of signals from the black hole at random times can naturally be assumed to obey the law of exponential holding time. Thus we can think of the emitting time series as a compound Poisson process which is a particular type of increasing stable process. We first breifly recall the concept of subordination and introduce the method of jump finding. A) Subordination Let X(t) be an additive stochastic process with independent stationary increments. The parameter t is replaced by a random time Y(t), where Y(t) is an increasing random function with y(0) = 0. Then we can define a new stochastic process Z(t) by the functional product defined as:
Z(t) =
X(Y(t)),
where the role of the time t is replaced by the increasing additive process Y(t). We view Z(t) as the observation of a stochastic process X(t) at random time Y(t). Namely, X(t) is an information source and Z(t) denotes the output measured at the instant Y(t). We now consider the particular type of functional product of Brownian motion and the increasing stable process with exponent j , i.e. the process T(x). Let B(t, w) be a Brownian motion and T(x, w1) be the inverse function of M(t), the maximal function of Brownian motion. Define Y(t,w) = B(T(x,w'),w) with w - (w,w') where w e O(P) and w' 6 Q'(P') and so w e (fi x £l')(P x P'). Then it can be shown that Y(t, w) is a Cauchy process. B) J u m p finding Prom the above theorems and propositions, we know that the process T{x) is a compound Poisson process. Intutively for any fixed |u| > 0 we can find the points Xj,x\ < x^ < ... at which T(x) jumps with size u. We
198
propose the following analytic method to single out the Poisson process Pu(t) with jump size u and to locate the jump points Xj. The charactereistic function of an increasing stable process Xa, 0 < a < 1 with exponent a is £[ e -i<x«,0]
=
I I
exp
(>«« - l) dtdn(u)
(3.1)
where dn(u) = -^p^du. Next we replace T with Xi, which is an increasing stable process of exponent 5. Let the characteristic function of Xa(t) be tpa{z). Let \ogipa = "4>a, 0 < a < 1. Recall that, in the case of increasing stable process Xa(t) of exponent a, we have the logarithm of the characteristic functional of Xa as:
*a = j j^m-D^dt. Take £(t) = Sz(t), then it becomes {eizu - 1)
/
du |u|a+1'
By applying a variation of the Riesz theorem, we consider: f ^ u _
1
)
^ U _
f
{
i
z
' u _
1
)
d u
=
r ^
i z
» ^ i z u _
1 )
d u
Let z" vary then we obtain (elzu — l)i u .i+i which corresponds to a single Poisson with jump size u. Thus we obtain the characteristic function of a single Poisson. Remark: In the above, instead of using the L2 technique, the construction is done by the operation acting on sample functions. 4. Poisson noise : path-wise theory Consider the processes X(t) and X(t):
X(t)= f F(t,u)P(u)du,
199 and
X(t) = JG(t,u)Za(u)du, where P(u) is a Poisson noise, and Za(u) is a symmetric stable process with exponent a. Here the kernels F(t,u) and G(t, u) are smooth in the variable u. We can see that they are always canonical, provided that F(t,t) ^
0,G(t,t)^0. We are now going to deal with the problem of finding jump points and jump sizes of a Poisson path. A ) The characteristic functional In the last section we have introduced the method to obtain the characteristic functional of a single Poisson process from the characteristic function of an increasing stable process Xa, 0 < a <1. B) The case of a single j u m p size u Let P(u, t) be a Poisson path with a single jump size u and with countably many jumps almost surely at Oj = Oj(w), where the domain is [0, T]. Then it can be expressd as N
P(u,t) = uY,Saj(t)-
(4-1)
1
Its Laplace transform is given by
/ ( A ) = f eXtP{u,t)dt N
= u^e°jA,
0 < ai < a-i < ... < aN
(4.2)
i=i
in which N is finite almost surely since T is finite. By letting A —• oo, u and a?j are obtained and again letting A —> 0, N and all the a^-'s are obtained. That is, we have obtained the jump points aj's at which the jumps occur.
200 C) T h e case of multiple j u m p sizes u, Consider a Poisson p a t h P(u,t) with different j u m p sizes { « , } , with 0 < U\ < v.2 < ... < « M , t h e n it can be expressed as M
Ni
p(u,t) = Y,Y,Ui5"<M i=l
(4-3)
j=l
with an < Oj2 < ... < ajjVi- Its Laplace transform is given by M Ni / ( A ) = ^ ^ K / ^ ,
(4.4)
in which M and iVj are finite almost surely since T is finite as in t h e above case. Assume t h a t the j u m p sizes {ui} are linearly independent over Z. (We need appropriate condition t h a t guarantees the number of t h e u^s are being finite.) Apply the same method as above, one recovers t h e j u m p sizes Ui, j u m p t i m e points a y and Ni by letting A t e n d t o infinity and A t e n d t o zero. Acknowledgement T h e authors are grateful to the organizers of t h e Q u a n t u m Information Conference who gave the oppotunity t o present t h e results reported in this paper.
References 1. S. Bochner.Harmonic Analysis and the Theory of Probability, Univ. of California press, 1955. 2. I. M. Gel'fand and N. Y. Vilenkin. Generalized Functions Vol. 4, Academic Press, 1964. 3. T. Hida,Stationary Stochastic Processes,Math. Notes, Princeton Univ. Press. 1970. 4. T. Hida, Brownian Motion, Springer-Verlag. 1980. 5. T. Hida and M. Hitsuda, Gaussian Processes. American Math. Soc.Translations of Mathematical Monographs vol.12. 1993. 6. T. Hida and Si Si, Innovation for random fields. Infinite Dimensional Analysis, Quantum Probability and Related Topics. 1 (1998), 499-509. 7. P. Levy,Theorie de L'addition des variables aleatoires, Gauthier-Villars, Paris.1937. 8. P. Levy, Sur certains processus stochastiques homogenes, Composito math., t. 7, 1939, p. 283-339.
201 9. P. Levy, Processus stochastques et mouvement brownien. 2eme ed. GauthierVillars, 1965. 10. M. Oda, Fluctuation in Astrophysical Phenomena, Proceeding of the HAS Workshop, Mathematical Approach to Fluctuations Vol. I edited by T. Hida, 1992 p 115-137. 11. Si Si, Random fields and multiple Markov properties. Proc. of the second conference on Unconventional Models of Computation, Springer-Verlag. 2001. 12. Si Si, Innovation of the Levy Brownian motion. Volterra Center Notes, N.475, May, 2001. 13. Si Si and Win Win Htay, Entropy in subordination and Filtering. Acta Applicandae Mathematicae Vol. 63, Nos 1-3, 2000. 14. Win Win Htay, Optimalities for random functions: lee-Wiener's network and non-canonical representations of stationary Gaussian processes, Nagoya Math. J. Vol. 149(1998), p 9-17.
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Quantum Information V Eds. T. Hida and K. Saito (pp. 203-217) © 2006 World Scientific Publishing Co.
O N E N T R O P Y P R O D U C T I O N OF A O N E - D I M E N S I O N A L LATTICE C O N D U C T O R SHUICHI TASAKI Advanced
Institute for Complex Systems and Department School of Science and Engineering, Waseda 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, E-mail: [email protected]
of Applied University, JAPAN
Physics,
Along the line of previous works, the relative entropy change is investigated for one-dimensional quantum conductors. Firstly, an explicit expression of the relative entropy between the present and initial states is derived. Then, the entropy production at a thermodynamically normal steady state is shown t o be positive and, in the linear transport regime, to reduce to an expression consistent with linear nonequilibrium thermodynamics. The validity of identification of the relative entropy change with thermodynamic entropy production is investigated and such an interpretation is shown to be allowed only when the state in question is close to the steady state.
1
Introduction
The understanding of irreversible phenomena is a longstanding problem of statistical mechanics. One of promising approaches is the one based on infinitely extended dynamical systems 1,2 ' 3 . For such systems, not only equilibrium properties, but also nonequilibrium properties have been rigorously investigated. This approach includes analytical studies of nonequilibrium steady states, e.g., of harmonic crystals 4 ' 5 , a one-dimensional gas 6 , unharmonic chains 7 , an isotropic XF-chain 8 , a one-dimensional quantum conductor 9 and an interacting fermion-spin system 10 . Entropy production has been studied rigorously as well 11 ' 12 ' 14 ' 13 , where the relative entropy between the present and initial states is related to the thermodynamic entropy. For a finite-dimensional system weakly coupled with a single reservoir of inverse temperature /?, Spohn and Lebowitz 15 derived a relation between thermodynamic and relative entropies in the van Hove limit. In terms of a finite dimensional density matrix p(t) describing the system state at time t, the thermodynamic entropy production crth(t) is given by ath(t) = -kBjtS(p(t)\Pf3)
,
(1)
where S {p{t)\pp) is the relative entropy 1 ' 16 ' 17 ' 18 ' 19 defined by S(p(t)\pfs) = -Tr(p(t){lnp(t)-hip0})
,
(2)
204
and pp is the equilibrium state of inverse temperature (3. Ojima, Hasegawa and Ichiyanagi 11 , based on the idea of Ichiyanagi 20 , derived a similar formula relating the entropy production and the relative entropy for an infinitely extended driven system:
,
(3)
where cjt is the state at time t, LJQ is the initial equilibrium state and S (uit\<^o) is the C* generalization of the relative entropy 18 ' 19 ' 1 ' 16,17 . Ojima 12 generalized this formula to the case where the system is coupled with a set of several reservoirs at different equilibria. Nonnegativity of the entropy production and convergence of the entropy production to the steady-state value was investigated as well. Recently, Jaksic and Pillet 14 rediscovered and extended his results, and pointed out that the relative entropy change reduces to the steady-state entropy production introduced by Ruelle 13 . They obtained a condition for strict positivity of the entropy production as well 10 . In this article, along the line of previous works 11 ' 12 ' 14 ' 13 , the relative entropy change is investigated for one-dimensional quantum conductors, where a class of initial states evolve unidirectionally towards thermodynamically normal steady states w+oo9- Firstly, applying the results of Ojima et al. 11 ' 12 and Jaksic-Pillet 14 , we derive an explicit expression of the relative entropy S {(jJt\^in) between the state u>t at time t and the initial state u>in. Then, the entropy production at a thermodynamically normal steady state ui+cx, is shown to be positive and, in the linear transport regime, to reduce to an expression consistent with linear nonequilibrium thermodynamics. Those results suggest that the time derivative of the relative entropy: —kBj[S (ivt\win) corresponds to the thermodynamic entropy production at time t. The possibility is investigated based on a thermodynamic consideration and we conclude that such an interpretation is allowed only when the state u>t is close to the steady state The rest of the article is arranged as follows: The previous results are summarized in the next section (Proposition 1). In Sec,3, after briefly explaining the C* generalization of the relative entropy, we explicitly calculate the relative entropy of the one-dimensional conductor (Proposition 2). Its positivity at the steady state u>+00 is shown (Proposition 3) and its expression in the linear transport regime is given (Corollary 4). In Sec.4, based on a physical argument, we investigate the validity of the identification of the relative entropy change with thermodynamic entropy production (Remark 5). The last section is devoted to the summary.
205
2
Model and Nonequilibrium Steady States
The system in question consists of electrons on an infinitely extended chain interacting with a localized potential and is described by a C*-algebra. 2.1
C* algebra
The basic dynamical variables are creation and annihilation operators, c!and Cj<(7 respectively, of an electron at site j(£ Z) with spin a(= ± ) . They satisfy the canonical anticommutation relations (CAR): [cjt„,ck,T]+ = [c* CT ,4 )T ] + = 0 ,
[c j ) f f ,4 i T ] + = 5jk5aTl
,
(4)
where [A, B]+ = AB + BA is the anticommutator, 0 the null element and 1 the unit. The collection A of dynamical variables is a C* algebra called the CAR algebra 1 , namely it is a Banach *-algebra with C* norm generated by +oo
B(f,g)
= J2
E
{fi'"cJ.« +9i,«ci«}
,
(5)
where the sequences {/,>} and {gj,a} are square summable. The physical states are defined as normalized and positive continuous linear functionals u over the algebra A, i.e., continuous linear functionals satisfying (i) w(l) = 1 and (ii) u(A*A) > 0 (vyl € A). The time-evolution automorphism at : A —» A is generated via a truncated Hamiltonian in a standard way1. Let A\ be a C* subalgebra of A generated by BA(f,g) = £
lim i[HA,A] .
(6)
A—>+oo
is well-defined, where HA = Ylt=-A ^i the local Hamiltonian hj given by h
i = E
1S t n e
truncated Hamiltonian and hj
[-7 {c*j,^i+i,
00
7(> 0) and €j stand for, respectively, the strength of the electron transfer and the localized potential with suppej C [1,L]. The derivation S generates a strongly continuous time-evolution automorphism: at = eSt (see e.g., Theorem 6.2.4 of Ref.fl]) with 8 the closure of 5.
206
2.2
Dynamical conditions and initial states
The "first quantized" Schrodinger operator corresponding to the automorphism oct is assumed to admit a complete set of outgoing scattering states and have no bound state. The outgoing state ipq(j) (—n < q < TT) is the solution of the eigenvalue equation corresponding to an eigenvalue Eq = —2j cos q: - 7 Mi +1) + V»,(i - 1 ) } + tjMi) with the outgoing boundary condition: ^q(j)
-> - = {e™ + Rqe-igj}
,
= ErfgU).
(8)
when j - • -oo(+oo) for q > 0(< 0) , (9)
where Rq is the reflection amplitude. Initial states are prepared in the following way: Firstly, the chain is divided into three: ( - o o , - A f ] , [-M + 1,N - 1] and [AT,+oo) with M > 0 and N > L. The two semiinfinite segments serve as reservoirs and the rest as a finite subsystem. Corresponding to this division, the algebra A is decomposed into a tensor product of the three subalgebras AL, AS and AR: A = AL ® As <8> AR. The Hamiltonian H is, then, represented as a sum of a left-reservoir part HL, a right-reservoir part HR, a subsystem part Hs and a reservoir-subsystem interaction Vint: H = HL + HR + Hs + Vint- The number operator is decomposed as well: TV" = NL + NR + Ns- Then, an initial state <jjin is given by a tensor product Uin = W i (g> CJS <8> Wfl ,
(10)
where U>L stands for a left-reservoir equilibrium state over AL with inverse temperature 0L and chemical potential HL corresponding to the Hamiltonian HL and the number operator NL, WR for a right-reservoir equilibrium state over AR with inverse temperature /3R and chemical potential HR corresponding to HR and NR, and CJS for an arbitrary state over As- The equilibrium states CJL and LJR are defined as KMS (Kubo-Martin-Schwinger) states and, since As is finite dimensional, the state u>s is represented by a density matrix. We remark that, when the density matrix representing u>s is invertible, the initial state u>in is characterized as a KMS state. Let 5^ be a symmetric derivation defined on a dense set V by 5U(A)
-
lim
i [-fa
{H$
- HLN£)
- PR ( # £
- URN*)
+DS,A]
(11)
where -M-l
#£=£*;, j=—A
+A
-Af
# £ = ! > ; N£=Y,nj, j=N
j=—A
+A
K=J2"i j=N
(12)
207
and Ds is a bounded self-adjoint observable such that exp(Ds) corresponds to the density matrix representing u>s, then 8U generates a strongly continuous group a" = exp (s 5U), where the bar on the generator stands for the closure. As easily seen, the initial state u)in is a KMS state at temperature —1 with respect to a", namely it satisfies the KMS boundary condition: w in (Aa^B))
= win (o?(B)A)
(13)
for all elements A, B of some dense subset of A- u>in is the unique KMS state as a" is a tensor product of a free evolution on AL ® AR and an evolution on the finite dimensional subalgebra As- (For the uniqueness of the KMS state for free fermions, see e.g., Example 5.3.2 of Ref.[l]).
2.3
Summary of the previous results
Proposition 1: [9] For t —+ ±oo, the initial state w;n weakly evolves towards the unique quasifree states u>±oo: limt_±oo Wjn (at(A)) = UJ±OO(A) (VA G .4), irrespective of the choice of separating points M, N and the initial subsystem state wg. The states u>±oo are fully characterized by the two-point functions:
w+oo(C*ffCjv0
= <w f'dq {FL (Eq) iPqur^W)+FR
(Eg) v- 9 orv-- 9 (/)},
Jo UJ-oo{c*aCj>a>)
= U+00(c*iaCj-al)*
,
where FL{E) = l / { e ^ ( £ - ^ ) + 1} and FR(E) = l / f e " " ^ - ^ ) + 1} are Fermi distribution functions for the left and right reservoirs, respectively. At the steady state u>+00, the two-probe Landauer-type formula hold for the particle flow and the energy flow: (jf-H^+oc = u, +0o (jf_ l b .) = J*
^\Tg{E)\2
(Jf-1{j)+oo
™E\Tq{E)\2
= u>+oc (jf_ 1 ( ,) = j T
{FL(E) - FR(E)} {FL(E) - FR(E)}
(14) (15)
where (• • -)+oo stands for the average with respect to w + 0 0 , |T q | 2 = 1 — \Rq\2 is the transmission coefficient, q(E) = cos~1{—E/(2j)}, and JJLUJ and Jf_ustand for, respectively, the particle-flow and energy-flow operators from the
208
0.05
-0.05
Figure 1. Averaged particle flow (J^jr1)t vs time t, for N = 0, M = 1, ej = 0,7 = 1/2 and 0fi = 1-3, & = l , W l = 1-3,/x£ = 1.
0' - l)th to the j t h sites:" Jj-l\j
— *7 2s iCj,aCJ-l,
~ C3-l,aCJ,"i
'
Jf-W = - H 2 Yl { C i+i,^-i.- - c;-i,a9,-+i>ff} + ejJJLm •
(16)
(17)
N . B . I : Although the proof of Ref.[9] suggests polynomial convergence, averages of certain observables converge much faster as shown in Fig.l. For an interacting fermion-spin system, Jaksic and Pillet showed10 the existence of a dense set of observables, averages of which converge exponentially fast. 3 3.1
Relative Entropy and Its Change Relative entropy of states over C* algebra
Generalization of the relative entropy (2) to states over a C* algebra is carried out with the aid of GNS (Gelfand-Naimark-Segal) representation of C* algebras and Tomita-Takesaki theory of von Neumann algebras. We summarize the outline following Ref.[ll]. For a given C* algebra A, there exist a Hilbert space K., a vector Q, £ IC and a *-morphism n : A —> B(K.) from A to a set B{K) of all bounded linear operators on /C, such that (i) u(A) = (fi,7r(i4) 0.) (VA G A) and (ii) the set " T h e expression of the energy current is different from that in Ref.[9]. This is due to the difference of the definition of the local energy.
209 {7T(J4)0|A £ .4} is dense in K. (cyclicity of the state ft). The triple (IC, ft, n) is called the GNS representation. A set of all B £ B{K) which commute with every element of iv(A) is denoted as TT(A)' (commutant of ir{A)). n(A)' is again an algebra. Let M. be a double commutant of ir(A): M = n(A)", then _M" = M.. An algebra like M. is called a von Neumann algebra. Given a von Neumann algebra M. C B(IC), a vector ft £ K is called separating if ACl = 0 for A € M. implies A = 0. If a vector ft is separating and cyclic with respect to M, there exist antilinear operators S and F satisfying
SACl = A*Q
fA
FA'ft = A'*Q. (VA' e M') .
e M) ,
(18)
The closure S of S admits a polar decomposition: S = JA1/2
(19)
where A = S*S is positive and self-adjoint, and J is an antilinear involution. Moreover, they satisfy JMJ = M' and A^'-MA - '* = M. This is the outline of Tomita-Takesaki theory. The set V = {AJAJQ\A
G M} C K ,
(20)
is called the natural positive cone, where the bar stands for the closure. For two vectors $,£1 £ V which are both cyclic and separating, one defines an operator S*(n by S*,fiAft = A*V .
(AeM)
(21)
Araki 18 defined the relative entropy of ^ and ft by 5U(fi|*) = ( * , l n A » , n * ) .
(22)
s caue
where A*,n = 5£ n ^ . n * d the relative modular operator and 5*,n is the closure of Sq,tQ. For any faithful states uii and o»2 on a C* algebra, when both of them are represented by separating and cyclic vectors, $ and ft respectively, belonging to the same natural positive cone in a GNS representation, their relative entropy SA{U)2\UI) is defined by SA(u2\o>i) = SA(n\V)
.
(23)
This definition by Araki is slightly different from that previously discussed and the C* counterpart S(u}i\u)2) to (2) is given by 1 S(CJI\(J2)
= -SA(CJ2\U>I)
.
(24)
210 3.2
Relative entropy for the one dimensional conductor
The relative entropy change of the one-dimensional conductor can be calculated by applying the results of Ojima et al. 11 ' 12 or of Jaksic-Pillet 14 . Here we follow the latter. P r o p o s i t i o n 2: For the one-dimensional quantum conductor, when the initial subsystem state is described by an invertible density matrix exp(Z)s), the relative entropy between the present and initial states is given by
kBS(wt\win) = J dt' | ^ 1 + ^ l | + kB(Ds)t - kB(Ds)o (25) where operators JqL and JR correspond to heat flows out of the left and right reservoirs, respectively, JL = J-M-I\-M
~ VLJ-MI-M+I
(26)
JR = ~{JN-I\N
~ ^RJN-I\N}
(27)
'
TL = l/(fcs/?L) and TR = l/(fcs/?ij) are temperatures, respectively, of the left and right reservoirs and we have abbreviated wt(- • •) = (•••)*• Proof. First we remind that the initial state is the unique KMS state at temperature —1 with respect to a strongly continuous one-parameter group CTw _ e5„s ^cf t n e j-emaj-k a t t n e e n d 0 f j n Sec.2.2). Thus assumption (A.l) of Ref.[14] is satisfied. Next we consider another strongly continuous group a\ = exp (t So) generated by So defined on V: 50(A)= limi[Ht + H^,A]
(28)
A—>oo
where the truncated left and right reservoir Hamiltonians, respectively H£ and HR, are given in (12). Because the generators Jo a n d Sw commute with each other on a dense set V: 5W5QA = 5Q5UA (*A £ T>), the state Win o ct\ Ms again a (
(29)
and V belongs to the domain V of 8W. Or assumption (A.2) of Ref.[14] holds.
211
Therefore Theorem 1.1 of Ref.[14] gives S(wt|w in ) = - / dt'ut, {SU(V)) Jo and the desired result (25) follows from
Ut{&(Ds)) = 3.3
j
t
(30)
•
Positivity of entropy production at steady state w+oc
From (25), one finds that the relative entropy change is given by i i \ u dS(ut\uin) o-R(ut\u)in) = -kB—K-jt
(Jl)t = — ^
(JR)t Y
, . , n .. B{HDs))t •
k
. . (31)
At the steady state w + 0 0 , it has the following properties: Proposition 3: The steady-state relative entropy change aRl = <7fl(w+00|a>in) is „st
Wj,)+oo
\JR)+OO
fn0\
- - £ f I W { ^ ~ ^ f } W*> "ftW> • (33) crR is nonnegative and vanishes only when Tj, = T R and /i£ = fiR, or when both reservoirs are in equilibrium. Moreover, the heat flows satisfy - ( J £ > + 0 0 - {JR)+00 J
e
= V(j;_1{j)+00
,
(34)
where V = —{HL — f -R)/ is the voltage difference between the two reservoirs and Jf_nj = —e JfLuj the electric current operator.
212
Corollary 4: Let T0 = (TL + TR)/2 be the mean temperature of the reservoirs, A T = TL — TR the temperature difference, fi0 = (HL + HR)/1 the mean chemical potential and V = —(m - HR)/e the potential difference, and let O(AT/T 0 ) =0(e\V\/fi0) = 77 < 1, then CTR is of order rj2 and is given by V
AT
+ 0(V3),
(35)
where J*_1{j = ~eJf__lb is the electric current, Jq._^ = Jf_^ - fJ.0Jf_1{j is the heat flow in the linear regime, G, Li, Li are constants given by 9
G
4/_>IW(-§|),
*--;/><*-«>iw(-^).
(36)
<m
L =
* I £dE {E -m? ]T"E)? {-?W)
and FQ{E) = < exp ( f ~ff ) + 1 f
1S a
(3s
Fermi distribution function.
2
Note that, when terms of order rj are neglected, the heat flow and the electric current are given by 9 AT (j;_ l b )+oc = GV + L x — ,
AT ( ^ _ l b ) + o o = LXV + L2— .
(39)
N . B . 2 Spohn-Lebowitz 15 and Ojima-Hasegawa-Ichiyanagi11 derived the relation between the relative entropy and thermodynamic entropy production (such as (1) and (3)) by comparing the relative entropy with the thermodynamic expression of the entropy production. Similar considerations were given by Jaksic-Pillet 14 (cf. also Ruelle's work 13 ). The same relation can be obtained in a slightly different way: Since the state wt at time t is locally different from the initial state u>in, the latter is regarded as a local equilibrium approximate to Wf In this view, the relative entropy S(u>t\u>in) is considered to be the difference between the Gibbs and a coarse-grained entropies. Then, the arguments given by Dorfman-van Beijeren21 and Breymann-Tel-Vollmer 22 leads to the same relation.
213 N . B . 3 Proposition 3 and Corollary 4 imply that the identification of the steady-state relative entropy change with thermodynamic entropy production is consistent with thermodynamics. Eq.(32) is the expression expected in thermodynamics (see also the next section). The positivity of a^ is consistent with the second law of thermodynamics 23 . The relation (34) can be interpreted as the equality of the net heat flow into the reservoirs (left-hand side) to the Joule heat generated by the electric current (right-hand side) 23 . And (35) agrees with the expression of the entropy production known in the linear nonequilibrium thermodynamics 23 . Proof. The invariance of the steady state w + 0 0 implies u>+00(5(Ds)) = 0, which leads to (32). Eq.(33) immediately follows from (14), (15) and (32). Because of (33) and an inequality
-(x-y)(—^— v
'\ex
+1
^ - 1 >0, ev + lf ~
where the equality holds only when x = y, cr^ is found to be nonnegative and vanishes only when the two reservoirs are in equilibrium with each other. The relation (34) immediately follows from the definition of heat flow operators J£ and J^. This proves Proposition 3. Corollary 4 can be easily obtained by a straightforward calculation.
4
Thermodynamic Assessment of Relative Entropy Production
It is interesting to rewrite (25) as
Ss(t) - 5,(0) = J*dt' |
^
+^ }
- kBS(u,t\u;in) ,
(40)
where Sa(t) = —/cs(.D,s)t. Observing that exp(-Ds) corresponds to the density matrix representing the initial subsystem state u>s, Ss (t) may be regarded as the subsystem entropy. Then, (40) is interpreted as the entropy balance equa"*"
rt
f (Jq) i
(Jq) i 1
tion: the change of Ss(t) is the sum of entropy flow JQ dt' < j r ' + £ ' > and the entropy production — kBS{wt\u>in). However, this interpretation is not always correct: Remark 5: Suppose that the following assumptions are satisfied: 1. Entropy of the finite dimensional subsystem exists and is finite.
214
2. Reservoirs remain to be in equilibrium and any change of their states can be regarded as a quasi-static process. 3. At steady states, the rate of entropy change is constant in time. Then, one has (a) For arbitrary t, the relative entropy change (TR(ujt\^in) cannot be identified with thermodynamic entropy production. (b) At steady states u>st, the relative entropy change agrees with the thermodynamic entropy production ath ^ I s t e a d y state = °"fl( w »t\^in)
(41)
N . B . 4 Because o-R(wt|wjn) converges to the steady state value for t —> +00, for large but finite t, (TR(cjt\uJin)
Or
-
—S(wt\Uin)
can be identified approximately with the thermodynamic entropy production. Explanation: Let Ss, SL and SR be entropy changes per time of the finite dimensional subsystem, right reservoir and left reservoir, respectively, then their sum is the entropy production of the total system: o-th{t) = Ss(t) + SL(t) + SR(t)
(42)
Because of Assumption 2 and Clausius equality 23 , which is valid in the quasistatic processes, the entropy changes of the reservoirs are given by (JR)t TR
A
'
SL
(JVlt = -Vpl ,
(43)
or aih{t) = S . { t ) - ^ - ^ .
(44)
where {• • -)t stands for the average with respect to u>tFrom (31), one finds that the thermodynamic entropy production is given by aR{iujt\ijJin) if and only if Ss(t) — Ss(t). Now we show that the latter is not the case in general. Let ws be an initial subsystem state where all states appear with equal probability, then the corresponding Ds is given by Ds = — {(M + N — 1) log4}l because the subsystem consists of M + N — 1
215
sites and four states are allowed per site. Therefore, Ss(t) and its steady-state value JS S (+OO) are given by 5 s (+oo) = Ss(t) =Ss(0)
= (M + N- l)kB log4
(45)
independently of the reservoir temperatures, Ti, TR, and chemical potentials, PL, PR- On the other hand, because of the results explained in Proposition 1, when the two reservoirs have the same temperature and chemical potential, i.e., TL = TR = To and PL — PR = Po, the steady state w+oo is an equilibrium state with temperature To and chemical potential fio- When ej = 0, the equilibrium state is translationally invariant and, as is well known 23 , the equilibrium entropy 5 | 9 of the subsystem of length M + N — 1 is given by S? = (M+N-l)f f
dq
[log(l + e-*C*.-")) + ^it'jt
, (46)
where /3o = l/(fcsTo) and Eq = —2jcosq. In this case, Ss(+oo) should be equal to Sf1. However, it is not the case and, thus, Ss(t) cannot be identified with the subsystem entropy. This shows (a). The statement (b) follows almost immediately. Because of Assumption 1, there exits a bounded function Ss(t) such that Ss(t) — Ss(0) = J*0 dt'Ss(t') and, due to Assumption 3, Ss is constant at a steady state. This leads to Ss = 0 at the steady state. Then (44) leads to u
th
I steady state
n->
q->
'
where (• • -)st = w3t(- • •) stands for the steady-state average. Because of ujst(5(Ds)) = 0, (32) leads to the same expression for crR.{wst\win)- This shows (b). 5
Summary
For the one-dimensional conductor, the relative entropy has been investigated. At a thermodynamically normal steady state w+oc>, the negative time derivative of the relative entropy can be identified with thermodynamic entropy production. But it is not the case for the state at arbitrary time t. Let us look through the relative entropy at the other steady state w_ 00 . Since u>_oo are at-invariant, (31) leads to 7fst — rrnl, ,
I, •
^_
&R = <7R(V-oo\Vin) —
\"'L)-00
ji
(JR)-OO
Tf,
/ .ry\
•
(47)
216 As a result of the time reversal symmetry, the average flows at UJ_OO a r e opposite t o those at w + 0 0 and one has aRf = ~aR ^ 0 where t h e equality holds only if t h e two reservoirs are in equilibrium with each other. As shown before 9 , if one s t a r t s from an initial state close t o w-oo, it unidirectionally evolves towards w + 0 0 . And <JR(ujt\^in) = — kB^iS(u}t\uJin) changes its sign as time goes on, because of effj < 0 and aRl > 0. This again implies t h e invalidity of t h e identification of the relative entropy change with t h e r m o d y n a m i c entropy production as the latter is always nonnegative due to the second law of thermodynamics. Recently, for a wider class of q u a n t u m C* dynamical systems, we have shown t h e existence of the steady states, their K M S characterizations and a q u a n t u m analog t o the fluctuation theorem, which will be reported elsewhere 2 4 . Acknowledgments T h e a u t h o r expresses his gratitude to Professor T . Hida and Professor K. Saito (Meijo University) for their kind invitation and hospitality at The fourth International Conference on Quantum Information, 27 February-1 March, 2001, Meijo University. He t h a n k s Professor I. Ojima, Professor T. Matsui and Professor H. van Beijeren for informing him of several i m p o r t a n t references (refs.10-14,21,22) on entropy as well as for stimulative discussions. Also he is grateful t o Professors L. Accardi, T . Arimitsu, T . Hida, N. O b a t a , M. Ohya, K. Saito, S. Sasa, A. Shimizu and I. Volovich for fruitful discussions and valuable comments. Also he t h a n k s Mr. H. Nishimura for drawing F i g . l . This work is partly supported by Grant-in-Aid for Scientific Research (C) from the J a p a n Society of the Promotion of Science. References 1. O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics vol.1 (Springer, New York, 1987); vol.2, (Springer, New York, 1997). 2. LP. Cornfeld, S.V. Fomin and Ya.G. Sinai, Ergodic Theory, (Springer, New York, 1982); L.A. Bunimovich et al., Dynamical Systems, Ergodic Theory and Applications, Encyclopedia of Mathematical Sciences 100, (Springer, Berlin, 2000). 3. D. Ruelle, Statistical Mechanics: Rigorous Results, (Benjamin, Reading, 1969); Ya. G. Sinai, The Theory of Phase Transitions: Rigorous Results, (Pergamon, Oxford, 1982). 4. H. Spohn and J.L. Lebowitz, Commun. math. Phys. 54, 97 (1977) and references therein.
217 5. J. Bafaluy and J.M. Rubi, Physica A153, 129 (1988); ibid. 153, 147 (1988). 6. J. Farmer, S. Goldstein and E.R. Speer, J. Stat. Phys. 34, 263 (1984). 7. J.-P. Eckmann, C.-A. Pillet and L. Rey-Bellet, Commun. math. Phys. 201, 657 (1999); J. Stat. Phys. 95, 305 (1999); L. Rey-Bellet and L.E. Thomas, Commun. math. Phys. 215, 1 (2000) and references therein. 8. T.G. Ho and H. Araki, Proc. Steklov Math. Institute, 228 (2000) 191. 9. S. Tasaki, Chaos, Solitons and Fractals 12 2657 (2001); Statistical Physics M. Tokuyama and H. E. Stanley eds., 356 (AIP Press, New York, 2000); Quantum Information HIT. Hida and K. Saito eds., 157 (World Scientific, Singapore,2001). 10. V. Jaksic, C.-A. Pillet, Commun. Math. Phys. 226 (2002) 131. 11. I. Ojima, H. Hasegawa and M. Ichiyanagi, J. Stat. Phys. 50 633 (1988). 12. I. Ojima, J. Stat. Phys. 56 203 (1989); Quantum Aspects of Optical Communications, (LNP 378,Springer,1991). 13. D. Ruelle, Entropy production in quantum spin systems, math-phys/0006006, 2000. 14. V. Jaksic and C.-A. Pillet, Commun. math. Phys. 217, 285 (2001). 15. H. Spohn and J.L. Lebowitz, Adv. Chem. Phys. 38, 109 (1979). 16. M. Ohya and D. Petz, Quantum Information and Its Use, (Springer, Berlin, 1993). 17. R.S. Ingarden, A. Kossakowski and M. Ohya, Information Dynamics and Open Systems, (Dordrecht, Kluwer, 1997). 18. H. Araki, Publ. RIMS, Kyoto Univ. 11 809 (1976); 13 173 (1979). 19. A. Uhlmann, Commun. Math. Phys. 54 21 (1977). 20. M. Ichiyanagi, J. Phys. Soc. Japan 55 2093 (1986). 21. J.R. Dorfman and H. van Beijeren, Physica A 240, 12 (1997). 22. W. Breymann, T. Tel and J. VoUmer, Phys. Rev. Lett. 77, 2945 (1996); T. Tel and J. Vollmer, Multibaker Maps and the Lorentz Gas in Hard Ball Systems and the Lorentz Gas, ed. D. Szasz, Encyc. Math. Sci. 101, (Springer, Berlin, 2000). 23. H.B. Callen, Thermodynamics, (Wiley, New York, 1979). 24. S. Tasaki and T. Matsui, in preparation.
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