ＨｮＭＹＮＩｅＨｾＭｭﾻｾＨｭＩ＠

,f2Ji

m=-oo

ｾ］Ｍｯｯ＠

On the other hand, i t follows from (2.9),(5.10), Proposition 5.1 ( i i ) and (6.2) that for any n,m E Z

(6.5)

(6.6)

E(n-m) - E(n-1-m) + B1 (E(n-m) + E(n-m-l»

=

fTI e- i (n-m)B{1_e iB +B

1 (1+e i9 )}h(e iB )d9 -TI

00

L

^Y1

ＨｮＭＮｬｮｅＨｾＭｭＩ＠

ｾ］Ｍｯｯ＠

= fll e-i(n-m)B(1_e2iB)f1 liO_P1(dt)h(eiO_)d9

-TI -1 1-te

Therefore, by substituting (6.5) and (6.6) into (6.3) and (6.4), respectively, we conclude from Corollary 4.1 (i) that for any n E Z

X(n) - X(n-1) + B1 (X(n) + X(n-1» + (y

1*X)(n)

=

¹

^ｾ＠ (I

^H e-i (n-mH) (Xl

L. dO)!;(m)

J2i

^{m=-oo -TI}

J2TI

a. s. ,

which completes the proof of Theorem 6.1. (Q.E.D.)

Definition 6.1. We call the stochastic difference equation (6.1) the first KMO-Langevin equation associated with X.

As the converse of Theorem 6.1, we wil \ show

normalized Gaussian white noise ｾ＠

=

ＨｾＨｮＩ［ｮｅｚＩ＠ on a probability space (Q, t,P). there exists a unique real stationary Gaussian proc-ess X = (X(n);nEZ) on (Q.t.P) with reflection positivity such that X satisfies the first KMO-Langevin equation (6.1) ; the covariance

function of this X coincides with

Proof By Theorem 5.1, we get a bounded Borel measure

associated wi th

a

lakes the form (5.4). Choose an

,9.2(Z)-funclion E = h as a canonical representalion kernel lo define a G

real stationary Gaussian process X

=

(X(n);nEZ):

0:)

X(n)

= ＨｾＲｔｉＩＭＱ＠

^[

ｅＨｮＭｭＩｾＨｭＩ＠

m=-o:)

Then i t follows from Theorem 6.1 that this X is our desired process.

To prove lhe uniqueness of such a process X, let y:::

( y ( n ) ; n E Z ) b e an 0 l her rca I s l a t ion a ^I' y G a u s s ian pro c e s son ( ｾｬ＠ , 1, P ) the same")

satisfying ceqUation (6.1>. Fix any m E Z . By multiplying

both hand Sides of equation (6.1) by and then

summing up with respect to n , we can observe from (2.10), Corollary 4.1, (5.10), Proposition 5.1 ( i i ) and (6.2) that

m

Y(m)

=

(j2i)-1 [

ｅＨｭＭｮＩｾＨｭＩ＠

n=-o:)

which implies Y

=

^{X .} (Q.E.D.>

Example G.1. Let Xp

=

(Xp(n);nEZ) be a real stationary Gaussian process wi th the covariance function np given in (3.2)

(pE(-l,)» . We nole that each Xp has the simple Markov

property, and in parlicular Xo represents a normalized Gaussian

whi te noise

a.

The first KMO-Langevin equation of Xp lakes the simplest form

(6.13) X (n) - X (n-1) p p

=

-B(l)(X (n)+X (n-l» p p p +

｡ＨｬＩｾＨｮＩ＠

p

a.s.(n E Z) •

(1) (1)

where the pair (cx p .Bp ) was given by (4.21). In case p

=

⁰

the above form (6.13) for the white noise ( becomes trivial:

(6.14) ｾＨｮＩ＠ - ｾＨｮＭＱＩ＠

=

ＭＨｾＨｮＩＫｾＨｮＭｬﾻ＠ + ＲｾＨｮＩ＠ a.s.(n E Z) . Example 6.2. Let X

=

(X(n);nEZ) be a real stationary Gauss-ian process with the covarGauss-iance function R of the form (4.22).

It follows from Proposition 5.1 ( i ) that

(6.15)

where (6.16)

o

poq n 1

n-2 2 POq1 (ql-l)

for for for

n E {-1.-2.···}

n E {O.l}

n E {2.3.···}

We note that (i) in Example 4.2 is a special case q ₁

=

⁰

in Example 4.2. Therefore. we see from Theorem 6.1 that X satisfies the following KMO-Langevin equation:

(6.17)

of (ii)

a.s.(n E Z) • where the pair (0:1.B1) was given by (4.26) or (4.29). It deserves

mention that the second term on the right hand side of

equation (6.17) depends upon the whole past of X in case q1 ｾ＠ 0 • i.e. a

1 ^;<! 0 .

In this final section we will prove a couple of relations between our objects - the first KMO-Langevin data (al,BI,y

l ) e LI ' the outer function h of X and other important quantities.

The physical meaning of these relations will be explained in de t a i I I ate r (s e eRe mar k s 7. 2 ... 7 . 4 ) .

a normalized Gaussian while noise. By Theorem 6.2, we obtain a real stationary Gaussian process X = (X(n);nEZ) as the unique solution of the first KMO-Langevin equation:

(7 . I )

a.s.(neZ) , where is given by (5.10). This process has the covariance function R = Ho E .9.1

(Z) , vii th We wi 11 begin wi th

Lemma 7.1. The following limit exists:

(7.2) D _ lim 2 ¹ N^E « [ ^N X ( n » ² ) =

ｎｾｯｯ＠ n=O

Proof. For any N EN, N

E ( ( [ X(n»2» = (N+I)H(O) n=O

= (N+I)H(O)

00

[ H(n) _ H(O) n=O 2

N-l N

+ 2 [ ⁽ [ R(n-m»

n=O m=n+1 N n

+ 2 [ ⁽ [ RUn) n=1 9.=1

Thercfore, we have thc assertion, noline lhal R E

ｾ｝ＨｚＩ＠

(Q.E.D.)

Definition 7.1. In view of the definition of diffusion con-stant for the continuous-time case (cf.(2.30) in [10]), the above limit D is called the diffusion constant of the process X.

As a discrete analogue of Theorem 2.1 in [10], we will show

Theorem 7.1.

(i) For any 0 E (-R,R)

( i i )

where

(7.3)

( i i i ) D =

( i v) D =

(v)

1 =

2 lim h(eiT ) TJ--JI

= JI(fR

1

13

1 ^<1+e

iO) + 1_eie

+ 2JIY1

un 1-

^{2do)-1 .}

-R

0:2 1 2(213

1)2

C

R(O) 13 1 'Y1 2131 213

1

1 J1

fl

l+t

R(O) -1 -1 1_tud(dt)P1 (du)

Proof. By noting (5.11), we see from Corollary 4.1 that (7.4) 211m h(cie

) = OJ..-JI

Therefore, 0) follows from Corollary 4.] 0),(5.10) and Proposi-tion 5.1 ( i i ) .

By using Corollary 4.1 0 ) , (5.1'0) and Proposition 5.1 ( i i ) again, we see from (2.7) that

2

(Xl iO

ie ｾ＠

-2

l\ ⁽⁰⁾ =

2Jl1

13 1 ⁽¹ + e ) + 1-e + 2 JTY 1 (0 )

I

a . e . 0 E ( - )1 , 11) •

By integrating both hand sides with respect to

e,

we have ( i i ) . We now compute the diffusion constant D. By (3.1) and C3.7), we have

(7.5)

(7.6)

(7.7)

ro

L

HCn)

n=O

⁼ f

¹ -1 ^{1=t(J(dt) 1}

R(O)

= a

([O,ro»

c

Hence, appealing to the result in [10]' Theorem 2.1<iii), we get

2 2

(X _c (Xl

= = ^,

4132

8132

c 1

which completes the proof of ( i i i ) . (iv) is an immediate consequence of ( i i ) and ( i i i ) .

Now, we proceed to the proof of (v). We first claim

( 7 • 8 ) ^E(O)

(7. g) E(n) ₌ - - E ( O ) -1-13 1 - -213 1 ^n-1

L

^{E(m) ___ 1_}

1+13 1 1+13 1 ^{m=l (n} 2. 1) •

By Theorems 4.2 and 4.3, we have (7.10)

IX)

..l... ("'

211 L

n=O

E(n)z n = h(z) (z E U

1(0» . In particular,

E(O) = 2nh(Z)l

z^{=0 .}

The l' e for e, by t a kin g z

=

⁰ (r e s p. n

=

^{0 ) 1 nTh e 0 rem ·t. 1 (I' C S P .}

Proposition 5.1(1», we get (7.8). Furthermore, by using Theorem 4.1 again, we see from (7.10) that for any n E N

and so

n

E(n) - E(O)

=

^{[(E(m) - E(m-I»}

m=l

n-l n

=

^-8₁ (E(0)+E(n)+2 [ E(m» - [(Y1*E)(m)

m=1 m=1

which implies (7.9).

Next we prove the key formula (7.11) 2nfl(0)

IX) IX)

-

Ｑｾｾ＠

^{[ ( [ R(m+k)y}₁^{(k» .}

1 m=O k=O

By substituting (7.8) and (7.9) into (2.13,), we have 1-8 ^IX)

2nfl(0) = E(0)2 + ---lE(O) [ E(n)

I+Bl n=l

2£31 ^IX) n-l

- - - [ ( [ E(.Q»E(n) 1+8 1 n=1 .Q=l

IX) n

-

ｾ＠

^{[ ( [ (y}₁*E)(.Q»E(n) . 1+ 1 n=1 .Q=1

On the other hand, we see from (2.11>,(2.13) and (G.2) thal

and

=

=

= And so

00 n-1 ^{00 00}

[ ( [ ｅＨｾﾻｅＨｮＩ＠ =

n=l ｾ］Ｑ＠

L ( L ｅＨｾＫｭﾻｅＨｾＩ＠

ｾ］Ｑ＠ m=l

00 00

= L ( L ｅＨｭＫｾＩｅＨｾＩ＠

- E(O)ECm+l»

m=l ｾ］ｏ＠

00 00

= 2n

L

R(m) - E(O)

L

E(m)

m=l m=l

00 n

Ｑｾ＠ ( L

^{(Y 1}

*

^E)

ＨｾＩ＠

^{) E (} n ) n=l ｾ］ｬ＠

00 00 00

L ^{L ( L} ｅＨｾＫｭＩｅＨｾＭｫﾻｙＱ＠

(k) m=O k=O ｾ］｝＠

00 00 00

L L

^Y₁^Ck){

L

ｅＨｾＫｭＩｅＨｾＭｫＩ＠ - E(m)E(-k)}

m=O k=O ｾ］ｏ＠

00 00 00

2n [ ( [ Rcm+k)Yl (k»

-

^Y₁ ^{(O)E(O) [ ECm)}

m=O k=O m=O

2nR(0)

1-131 ⁰⁰

=

^{E(0)2 +}

ｾｅＨｏＩ＠

^{[ E(n)} 1 n=1

213 ^{00 00}

- _ 1 (2n

L

R(n) - E(O)

L

E(m»

1+13 1 n=l m=l

00 00 00

- l!B

^{{2n [ (}

^L

^R(m+k)Y₁ ^{(k» - Y}₁ (O)E(O) [ E(m)}

1 m=O k=O m=O

? ⁰⁰ y

1(0) ｾ＠

=

ｅＨｏＩｾ＠

+ E(O)

L

E(n) + 1+13 E(O) ^L E(m)

n=l 1 m=O

4nBl ⁰⁰

- ｾ＠ L

^R(n)

-1 n=l

00 00

2J1 " L ( L " R ( m+ k ) Y 1 (k) ) 1+13 1 ^{m=O k=O}

which impi ies (7.11).

We are now ready to show

(7.12)

co co

ｒｾｏＩ＠

[ ( [ R(m+k)y 1 (k»

m=O k=O

which is proved in the following manner.

Dy (7.10).

co

[ E(n)

=

^{2n lim hex) .}

n=O xtl

And so by (4.11)

(7.13)

10

[ E(n) = n=O

Combining this with (7.8). we have

Yl (0) co

(1+ l+B )E(O) [ E(n)

1 n=O

On the other hand, by Lemma 7.1 and Theorem 7.1 ( i i i ) ,

co

[ H(n)

=

n=1

Therefore, by combining these wilh (7.1 1 ). we see that

and so

R (0) =

co co

- _ｾ＠ [( L

^R(m+k)Y₁ ^(k»

1+ 1 m=O k=O

2

R(O)

(X 1 co co

= --;) -

L ( L

R(m+k)Yl(k» •

ＴｾＱ＠

^{m=O k=O}

which, together with Theorem 7.1 ( i i ) , . iIT',Jiies (7.12).

Nexl we claim (7.14)

0) 0)

[ ( [ H(m+k)y}(k»

m=O k=O ₌

f l fl

^l+t 1-tuO(dt)p1 (du) -1 -1

By (3.1) and Proposilion 5.1(i) , the leIl hand side of (7.14)

=

=

0) 0)

[ ( [ R(m+k»YI(k) k=O m=O

0) 1 k

[(f ｬｾｬｏＨ､ｬＩｙＱ＠

^(k)

k=O -1 1 ⁰⁾

= f ([

Y (k)lk)O(dl) -1 k=O 1 1-t

_ fl 2 fl PI

(du) o(dl) - « l - l ) I - l u ) I - t '

-1 -1

which implies (7.14). Thus, we conclude from (7.12) and (7.14)

that (v) holds. (Q.E.D.)

Before we go into lhe explanation of the physical meaning of Theorem 7.1, we wI]] consider the simplest

Example 7.1. Lel Xp be lhe same stochastic p,'occss as in Example 6.1, and D be lhe diffusion conslanl of

p and Lemma 7.1,

(7.15)

<7.16)

H (0) _p

=

¹

= t +p Dp 2<l-p)'

By combining lhese with (4.21), we see that

(7.19)

2

By (3.2)

(7.20)

R (0) p

which. together with Theorem 7.1 (ii) or Theorem 7.1 (iv). imply (7.21)

In addition, we see from (3.2),(4.19) and (4.21) that a remarkable relation between R a n d E holds:

(7.22) R (n)

=

p

p p

1+8(1)

---'P''---li ( ) ( 1 ) ' n

j2iiex p p

(n 2. 0) .

We will return to the general case and give some characteri-zation of the simple Markovian property. As a discrete analogue of Theorem 2.2 in [10], we can see from Theorem 7.1 that

Theorem 7.2.

( i )

( i i ) The following four statements are equivalent;

C

(a) ^{8 1 , Y 1} 1 2B1 ⁼ (b) _{Y I =} 0 (c) _PI = 0

(d) X = X _p with some p(E(-l,l»

.

Remark 7.1. As we have seen in Theorem 2 . 2 i n [10], the relation (7.22) characterizes the simple Markovian property for

the continuous-time processes. However, this is no longer true

for the present discrete-time processes. We will give such an example. Let X be a real stationary Gaussian process discussed

in the case (i) of Example 6.2 such that

(7.23)

a = a =

1

1 2 2<1_p2) 1

It then follows from Theorem 4.3 and (4.27) the canonical representation kernel E of X becomes

(7.24)

which implies the desired relation (7.25) R ⁽ⁿ⁾

= ＭＭＭＭＭＭＭＭｬｾＭＭＭＭＭＭＭＭｅＨｮＩ＠ ＲＨｬＭｐｾＩｪ＠

2<1-pi)

(n 2. 0) •

Finally we will give three remarks concerning the physical meaning of Theorem 7.1 (cf. [10] for the continuous-time case).

Remark 7.2. In relation (i) in Theorem 7.1, the left hand side denotes a complex mobility of the system X described by equation (7.1), which represents the response of the system X to the external force et

1

a.

On the other hand, the right hand side is determined by the outer funciton of X, which represents the thermal fluctuation of the system in equilibrium without the

external force. The relation ( i ) in Theorem 7.1 mieht be said to be the generalized first fluctuation-dissipation theorem.

Remark 7.3. We are now concerned with relation (ii) in Theorem 7.1. The fluctuation power of a randow force ala in

equation (7.1) is While, the posi tive constant i s expressed in terms of the drift coefficient representing the

systematic part of equation (7.1). And from the physical point of view we can regard R(O) as the absolute constant k T ,where k and T denote the Boltzman constant and absolute temperature of

the system in equilibrium, respectively. This leads us to think of as the generalized friction constant, and the relation (ii) itself might be said to be the generalized second fluctllation-dissipation theorem.

Remark 7.4. For the Markov process Xp in Example 7.1, we found that the diffusion constant Dp is inversely proportional to the friction constant 8(1). This relation (7.20) is

p

analogous to the classical Einstein relation valid for the

Ornstein-Uhlenbeck Brownian motion with continuous time (see(2.29) in [10]). For this reason, we call relation (7.20) for

x

_p

Einstein relation. In a general system described by equation the

(7.1) with ｾＱ＠ ｾ＠ 0 , howeve0 we found a significant deviation (iv) in Theorem 7.1 from the Einstein relation (7.20) with ｾＱ＠

=

^{0 ,}

and obtained the formula (v) in Theorem 7.1 expressing the degree of such a deviation. In view of the analogous fact in the

continuous-time case (Theorem 2.1 in [10]), we call the relation (iv) in Theorem 7.1 the generalized Einstein relation.

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