10A-8179 649 MULTIPLICATIVE STOCHASTIC PROCESSES INVOLVING THE

TIME-DERIVATIVE OF A MA.. (U) STATE UNIV OF NEW YORK ATBUFFALO DEPT OF CHEMISTRY H F ARNOLDUS ET AL. JUL 86

1'UNCLASSFIE USUFFALO/DC/86/TR-11 N99614-86-K-0043 F/O 12/1 N

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MICOCPY ESLTO 136 CARNATONA Hk u , TN IIAR I'A

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OFFICE OF NAVAL RESEARCH

Contract N00014-86-K-0043

TECHNICAL REPORT No. 11

Multiplicative Stochastic Processes Involving the Time-Derivative

of a Markov Process

by

Henk F. Arnoldus and Thomas F. George

Prepared for Publication D T ICY in .LECTE

(0 Journal of Mathematical Physics AUG 1 1 86Departments of Chemistry and Physics

State University of New York at Buffalo JD.TBuffalo, New York 14260

*July 1986

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Multiplicative Stochastic Processes Involving the Time-Derivative of a Markov Process

12. PERSONAL AUTHORS) Henk F. Arnoldus and Thomas F. George

13& TYPE OF REPORT 13b. TIME COVERED 14. OATE OF REPRT tYr. Mo.. Day) 1S. PAGE COUNT

FROM _ TO J July 1986 2414. SUPPLEMENTARY NOTATION

Prepared for publication in Journal of Mathematical Physics

17. COSATI CODES IEC Ee M5 kg i 6tygrrf 4Rcen~ 4 O'0 st LD GROUP sue GRk IN ATIPMULTIPLICATIVE ATOMIC RESPONSE

MARKOV PROCESS PHASE-LOCKED RADIATION

1.ABSTRACT (Coftlifue on eVuerse if Receaaar and ideft tiy by block nuonlberp

The characteristic functional of the derivative 0(t) of a Markov process *(t) and

the related multiplicative process o(t), which obeys the stochastic differential equation

i6(t) = (A + (t)B)o(t), have been studied. Exact equations for the marginal character-

istic functional and the marginal average of a(t) are derived. The first equation is

applied to obtain a set of equations for the marginal moments of o(t) in terms of the

prescribed properties of 0(t). It is illustrated by an example how these equations

can be solved, and it is shown in general that (t) is delta-correlated, with a smooth

background. The equation of motion for the marginal average of a(t) is solved for various

cases, and it is shown how closed-form analytical expressions for the average <o(t)> can

20. OISTRISUTION/AVAILASILITY OF ABSTRACT 21 ABSTRACT SECURITY CLASSIFICATION be obtained.

UNCLASSIFIEO/UNLIMITED 9 SAME AS RPT 9 OTIC USERS C3 Unclassified

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Dr. David L. Nelson (202) 696-4410

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Journal of Mathematical Physics, in press

MULTIPLICATIVE STOCHASTIC PROCESSES INVOLVING THE TIME-DERIVATIVEOF A MARKOV PROCESS

Henk F. Arnoldus and Thomas F. GeorgeDepartments of Physics & Astronomy and Chemistry

239 Fronczak HallState University of New York at Buffalo

Buffalo, New York 14260

ABSTRACT

The characteristic functional of the derivative #(t) of a Markov

process 0(t) and the related multiplicative prpcess 4(t), which obeys the

stochastic differential equation ib(t) = (A + *(t)B)O(t), have been studied.

Exact equations for the marginal characteristic functional and the marginal

average of o(t) are derived. The first equation is applied to obtain a set

of equations for the marginal moments of 4(t) in terms of the prescribed

properties of *(t). It is illustrated by an example how these equations can/,

be solved, and it is shown in general that 4(t) is delta-correlated, with a

smooth background. The equation of motion for the marginal average of a(t)

is solved for various cases, and it is shown how closed-form analytical

expresssions for the average <0(t)> can be obtained.

A Ccesioti For

NTIS CRA&IDTIC TAB ElU,,anno'nced 5Jjrt~fca.tiun

By ......Di-A ibution I

Availability Codes,/ - Avail ardfor

(I, LT ED Disi Spc lPACS: 2.50, 5.40L.-J,).4J

, ,,.".". - ' 'w , " ", , .'. , " ",% ' * m '% . . _ _ . - , "' -' ,.,.' .'- ', - . ,' '. ,'. .-- _ _ ,. " ',' ' ' .- ,",-

2

I. INTRODUCTION

The equation of motion for the density operator of an atom in a finite-

bandwidth laser field or the equation for the regression of the atomic

dipole correlations assumes the general form1'2

do (1.1

i dt (A + (t)B)o

where A and B are linear operators in Liouville space, which act on the

Liouville vector c(t). Here 0(t) represents the laser phase, which is

considered to be a real-valued stochastic process. The fluctutating phase

broalens the laser line, but the atom responds to the instantaneous

3frequency shift 0(t), which is the time-derivative of the laser phase. The

process 0(t) is again a stochastic process, and via Eq. (1.1) the state of

the atom or the correlation functions 0(t) become stochastic quantities.

The issue in quantum optics is then to solve the multiplicative stochastic

differential equation (1.1) for the average <o(t)>. The first solution was

4obtained by Fox, who assumed the process #(t) to be Gaussian white noise,

which corresponds to a diffusive Gaussian phase *(t) (the Wiener-L~vy

process). This result was generalized to a Gaussian process 0(t) with a

finite correlation time and an exponentially decaying correlation

function5 - 7 (the Ornstein-Uhlenbeck process), and to a process #(t), which

0 8,9is again diffusive, but not Gaussian (the independent-increment process).

Furthermore, Eq. (1.1) can be solved for <a(t)> if we have 0(t) as a Markovrandom-jump process,1013 which models a multimode laser.14 '1 5

In these examples the solvability of the problem relies on the Gaussian

property of 4(t), or hinges on the prescribed stochastics of 0(t). This

." implies that the process 0(t) is actually considered to be the driving

3

process. For a single-mode laser in general, however, the phase

fluctuations 4(t) are specified rather than the derivative *(t) of this

process. A prime example would be the atomic response to phase-locked16 17

radiation, as it is generated for instance by some ring lasers. In this

paper we shall develop a general method to solve Eq. (1.1) for the case that

#(t) is a given Markov process. The formal theory will be exemplified by a

specific choice for #(t), which models phase-locked radiation. Furthermore,

we shall study the time-derivative of *(t) itself and extract the

stochastics of (t) from the properties of *(t).

II. THE STOCHASTICS OF *(t)

Let us define the phase 0(t) as a homogeneous Markov process. 18 Then

its stoch stics is fixed by the probability distribution P(*,t) and the

conditional probability distribution P (0), which has the

significance of the probability density for the occurrence of *(t+T) *2 if

*(t) *. #"For a homogeneous process this is independent of t by

definition. The higher-order statistics is now determined by the Markov

19property. From the obvious relation

fd,' Pt-t 0 (,I')P(*',t 0) P(s,t), t Z t0 (2.1)

it follows that it is sufficient to prescribe the probability distribution

P(#,t) for a single time-point tO only. The time-evolution towards t > t

can then be found from Eq. (2.1) and Pt -t0(18

The conditional probability distribution obeys the Master equation 18

T PT(*31#1) - fd,2 1W(*3 * 2) -a( 2 )6(43 - 2)}Pr(*2t*i) (2.2)

with W(O'js) 0 as the transition rate of the process from 0 to *' and

4

a(#) - Sdo' W(0'10) (2.3)

which is the loss-rate of *, independent of the final value *'. The initial

condition for Eq. (2.2) reads

Po(s31#1 ) - 6(l3-* ) (2.4)

so a gi--en W($'I) determines P(T43J*1) for every T 1 0. Hence the

stochastics of a homogeneous Markov process *(t) is fixed as soon as P(,,t0)

and W(#'J*) are prescribed. These functions will from now on be assumed to

be given.

III. THE CHARACTERISTIC FUNCTIONAL

A convenient way to represent the stochastic properties of a stochastic

process is by means of its characteristic functional.9 '2 0 Since we are

concerned with the process *(t), we define

t

Zt[k) - <exp(-ifds *(s)k(s))> , t 2 to , (3.1)to

which is a functional of the test function k(t). Here the angle brackets

denote an average over the stochastic process *(t) or 0(t), whatever is

prescribed. A general method to evaluate Z t[k] for the case where 0(t) is a

homogeneous Markov process has been given by van Kampen.2 1

Knowledge of the characteristic functional Z t[k] determines completely

the stochastics of *(t), which can be seen as follows. Choose k(s) as the

sequence of 8-functions

n

k(s) 6(s-tE)kz tE > to (3.2)

and take t - - in (3.1). Then we find

I.).a- ....... ............. ..............

5

Z.[k] - <exp(ikn*(tn) + ... + ikI(t1))> (3.3)

which is the moment-generating function of *(t). If we write zn(k n,t;

... ;klt 1 ) for Z.[k, then we can obtain the moments of (t) *(t)

according to

<*(t )> _ ()n Z " n(k nt " ;kl'tl)0

n kn k .a I (f;

(3.4)and the probability distributions by

Pn n' tn; 2n) n f--- dn ... d(2w) n

-ikn n- ... -ik 1 1 (n e* nnt n; ... ;kl~t 1

(3.5)

where we have introduced Pn in order to distinguish from the probability

distributions for *(t) itself.

IV. THE MARGINAL AVERAGE

A. GENERAL

The exponential in Eq. (3.1) is a functional of both k(t) and *(t), so

it depends on the values of *(t) in the complete interval [t0,tl. After the

average has been taken it will be only a functional of k(t). The general

attempt to evaluate averages of a functional is to derive an equation for

the average. For subsequently solving this equation for functionals which

involve Markov processes, this scheme is most conveniently carried out by an

22intermediate introduction of Burshtein's marginal averages. Since in our

problem the stochastics of *(t) is assumed to be given, the appropriate

marginal characteristic functional, which is related to Z t[k), should be

defined as

--.. \-X. - * * -- .

6

t

Qt[,0 k] - <6(#(t)-# 0 ) exp(-i fds *(s)k(s))> (4.1)to

for t a to. The initial value is then

Qt0[0,k] - <6(0(t0 )-*0 )> - P(#0,t0) 1 (4.2)

and Z t[k] follows from Qt[o0k] according to

Zt[k] - fd*0 Qt[$0,k] . (4.3)

For t - t we find with Eq. (4.2) Z[k] = fd0 P( o,t0) = 1, in agreement0

with Eq. (3.1).

In order to derive an equation for the time-evolution of Q 10k],

we first increase t by a small amount At > 0. This gives

Qt+Att*okl = <6($(t+At)-*0 ) exp(-i( (t+At)-*(t))k(t))

t

exp(-i fds *(s)k(s))> (4.4)

to

Subsequently, we expand the exponential functional of *(s) in a series, and

we take the average in (4.4) term by term. Hereafter, we apply the Master

equation (2.2) for Pt+At(flYo0 and take the limit At - 0. This yields an

equation for the marginal average, and explicitly we find

Q 09k] fd (W( 0 1) -a(

-i(,o0-O)k(t)x e Qt[O,k] . (4.5)

The Markov process *(t) is characterized by P(0t 0) and W(001), which

respectively determine the initial value and the time-evolution of Q [o0k].

-pI".1 '''"' . ' '" " ." . ". .. .". " '- " . ". " ". °. , . " . _ ' ". ". " . . . '

7

For a specific choice of W(4o), we have to solve Eq. (4.5), after which

the characteristic functional Z t[k] can be obtained from Eq. (4.3).

Notice the resemblance between the result (4.5) and the Master equation

(2.2). If we multiply Eq. (2.2) by P(41,t0 ), take Tr t - t0 and apply the

relation (2.1), we find

S- P( 0 ,t) = fd4 {W(4 0 10) - a(4)6(0 0-0)) P(O,t) , (4.6)

which is the Master equation for P(*ot). This equation is identical to Eq.

(4.5), including the initial condition (4.2), if we set k(t) 0 0. On the

other hand, it follows from Eq. (4.1) that Qt[$o,kI = <6(0(t)-O 0 )> = P(o,t)

if we take k(t) = 0 , so that in this case Eq. (4.5) should indeed reduce to

- Eq. (4.6).

B. INDEPENDENT INCREMENTS

In order to display the usefulness and applicability of the marginal-

functional approach, we consider an example. Let us specify the transition

rate by

W(0 )= 0 w(*0-4) , y>0 , (4.7)

where the function w(n) is normalized as

fdn w(n) I1 (4.8)

The stochastic process 4(t) will be defined on the real axis, with -- < * <

. The assertion (4.7) states that the probability for a transition * 0

depends only on the phase difference 0 0 , and from Eq. (2.3) we find that

a(#) -, so that the total loss-rate for * is independent of *. This is a

diffusion process, and it is commonly referred to as the independent-

increment process. As an initial condition for the probability

8

distribution, we take

P(*,t0 ) = 6W (4.9)

Comparison of the Master equations for P (410') and P(O,t) then shows that

the probability distribution and the conditional probability distribution

are related according to

Ptt 0(1[$) = P(O-*ot) (4.10)

The Master equation (4.6) for P(.,t) can be solved by Fourier

transformation with respect to 0. If we write

P(P,t) = <eP(t)> = fdO ei O P(O,t) , (4.11)

which has P(p,t O) = 1 as the initial condition, then the solution of Eq.

(4.6) is immediately seen to be

Y( (P)-l)t-t )

P(p,t) = e , t to (4.12)

in terms of the Fourier transform w(p) of w(#). Note that w(O) = 1, as a

result of the normalization (4.7). Along the very same lines we can solve

Eq. (4.5) for the Fourier transform Ot [p,k]. We obtain

tt

Ot[pk = exp(-y f ds f d (l-eio(p-k(s)))w(o)) (4.13)tO --

after which the characteristic functional follows from

Z tk = t [O,k] , (4.14)

which yields the familiar result 8

9

V. THE MARGINAL MOMENTS

A. GENERAL

If we take k(s) as the sequence of delta functions (3.2) in the

definition (4.1) of the marginal characteristic functional, it assumes the

form

6n

Q [0 k] = <6((t)- 0)exp(i k£ *(t£) O(t-t£))> , (5.1)t 0' 0

with *(t) = 0(t) and 0(t) the unit-step function. Just as we can find the

moments <j(t n ) ... P(t1 )> of *(t) from Z [k], we can obtain the marginal

moments <6(0(t)-o 0 )(t n) ... ,(tl)> from Qt[0,k]. Obviously the integral

over 0 of the marginal moments yields the moments. The characteristic

functional Zt [k] becomes independent of t if t > t for all E, but Q[0,k]

remains time-dependent. This is due to the appearance of 6(0(t)-o0).

Furthermore, the time t is a dynamical variable in Eq. (4.5), so that care

should be exercised in the time-ordering. The marginal moments follow from

Q[ 00 by differentiation, according to

<6(0(t)-0 0),P(t n ) .. (tl1)> O)(t-t n ) ... O)(t-t ) =

( -)n '' . k t[' = "'" = ki = O" (5.2)

Equation (4.5) for Q [0,k implies an equation for the marginalt

moments. First, we note that

exp(-i(o 0- )k(t)}Qt[4,k ] =

n

<6(0(t)-O)exp(i > k {(.0-.)6(t-t) + i(t£)0(t-t)})> (5.3)£= I

.. . ~ ' ~ - - -. -

K 10

After substituting this expression in the right-hand side of Eq. (4.5),

differentiating with respect to k, "'" ,k1 , setting k =" k = 0 and

integrating over time, we obtain

S<6(¢(t)-¢O) Wt n ) .. (t 1)> O(t-t n ) ... o(t-tl 1 = d¢ (W(€0010 - a(€)6(O 0-0))

tSf dt' <60t)¢{¢- o)6(t'-t )+ ,(tn)O(t'-t )

to

((0O-,)6(t'-tl) + *(tl)O(t'-t)> (5.4)

When we set t > tz for all , we have a Master-like equation for <S( (t)- 0

*(t) ... (t )>, and the lower-order marginal moments <6(0(t)-O)(tm) ...

(t I)> with m < n appear as inhomogenous terms. Hence, Eq. (5.4) should be

solved successively for n = 1, n = 2... We note that Eq. (5.4)

provides an explicit expression for <(t) ... *(tl)> in terms of the lower-

order marginal moments after an integration over 00 Indeed, from the

property

fd,0 {W( 0 10) - a(0)6(€ 0-0)) = 0 , (5.5)

the term with <n((t')-0)0(t ) ... (t )> on the right-hand side of Eq.n 1

(5.4) vanishes after an integration over 0"

B. LOWEST ORDERS

In order to exhibit clearly the structure of the equation for the

marginal moments, we consider the cases n = 1 and n = 2 in some more detail.

After a slight rearrangement, Eq. (5.4) for n = 1 can be written as

<6(W(t)-¢O)WtI)> = fd, {W(010) -a()6(00-0))

t

x(O {(O)p(,t 1 + fdt' <6(0(t')-O)(t )}

t (5.6)

........-..-.-.,.................. ...-...................................... "..

U~U. •-.. * U .. ' U . U - * . . .*i. *, % '..i

11

for t 1 t . This integral equation in time is equivalent to the

differential equation

t <6(O(t)-#O)*(tl)> . fd, w(0014) - a(#)S(00-#))

- <6(#(t)-#),(ti )> ,(5.7)

.. together with the initial condition

<6(0(t 1 )-0O)(t1 )> = fd, {W(00 14) - a(O)6(O- )(o 0 -¢)P(O,t1 ) . (5.8)

The equation for the first marginal average <6(0(t)-O)4(t 1 )> is identical

to the Master equation (2.2), but with a different initial value.

*Integration of (5.8) over 0 yields

<*(tl)> = fdod 0 W($01*)(* 0-.)P(O,t1) , (5.9)

which expresses explicitly the average of <*(tl)> in the given functions

W(0010) and P(s,tl). With the aid of the Master equation, we can cast (5.9)

in the form

<*(t)> = fdO * I P(0,tl)

d <O(t )> (5.10)=dt 1 1

as it should be.

The solution of Eq. (5.6) for <6(#(t)-*O)0(tl)> provides the input for

the explicit expression for the two-time correlation function, which becomes

<*(t 2)(t1)> = fdofd,0 {W(4 0 10) - a(016(00-01 }

S((0O- 0)26(t 2- t I)P(O,t 2)

+ (0 O-0)(<6(O(t 2)-o)*(t 1)>E(t2- t 1

+ <6(o(t I)-o)*(t2)>()(tC-t 2) .(.1€ € i , " " " € " " " ' " " " " - " " " " " " . . . . . " " . . . ..2.

- 1 1 T -+

12

The appearance of 6(t2 -t1 ) shows that the time-derivative of any Markov

process is 6-correlated with a continuous background.

C. RANDOM JUMPS

Equation (5.11) for instance might seem awkward, but it is really

straightforward in its application. Let us illustrate this with an example.

Consider the random-jump process 0(t), defined as a stationary process with

transition rate

w(0') yP(O) , y>0 , (5.12)

in terms of an arbitrary probability distribution P(O). Eq. (5.12) is

equivalent to the statement that the probability for a transition 0' * is

independent of the initial value 0,13. From Eq. (5.9) we immediately derive

<0(t , (5.13)

which is, in view of (5.10), necessary for a stationary process. From (2.3)

we obtain a(*) = y, and the solution of Eq. (5.7), with initial value (5.8),

is readily found to be

--y(t-t)

<6(0(t)-O)(t1 )> - yP(0)( 00-b1 ) e 1 t t1 . (5.14)

Here we have introduced the moments of P(O) as

bn -f fdo cn P(,) ,(5.15)

which are parameters of the process *(t). Solution (5.14) can be

substituted into Eq. (5.11), which gives the correlation function

2 -yJt2 -t1<*(t Wtt)>= y(b2 -b2){26(t1 -t2 ) - y e } , (5.16)

for all t I , t2 . From (5.15) it follows that

4.

- - -.- I

13

b-b 2 > 0 (5.17)

so that for t1 t2 the correlation (5.16) is negative. For t1 = t2 the 6-

function dominates the negative term, so that <*(t 1 ) 2> is positive, as it

should be.

VI. THE MULTIPLICATIVE PROCESS

So far we have considered the stochastics of *(t) itself, and its

characteristic functional. In this section we shall generalize the method,

in order to solve the multiplicative equation (1.1). To this end we write

the formal solution of (1.1) for the stochastic vector o(t) as

-iA(t-t 0) t

o(t) = e T exp[-lfds *(s)A(s)]o(t 0 (6.1)to0

where T is the time-ordering operator and R(t) is defined as

A~)=eiA(t-t 0 ) B -iA(t-t 0 ) (62()=eB e .(6.2)

In close analogy to the definition of Qt[00,k] in Eq. (4.1), we now

introduce the marginal average of a(t) by

=(ovt) f <6(*(t)-sO)O(t)> . (6.3)

Then we substitute the expression (6.1) for a(t) and replace t by t + At,

which gives a similar formula as Eq. (4.4). That this can also be done for

the time-ordered exponential is sometimes referred to as the semi-group

property of the evolution operator. Along the same lines that led to Eq.

(4.5) we now find

. ai t C( 0 ,t) - AC(4o,t)

+ ifd, {W(,010) a()6(0-)e -1(40-t)B , (6.4)

,4

14

or equivalently

(i-1 A + ia(o 0 )) C(O0,t) = ifd W( 010)e i(o,t) o (6.5)

Notice that the operator B appears in the exponential, rather than 9(t),

as could be expected by analogy with the characteristic functional. For a

given stochastic process 0(t), e.g., a given W(010') and P(o,t 0 ), we have to

solve Eq. (6.4) with the initial condition

C(O0,t0) = <6(0(t 0 )-¢ 0 )o(t 0 )> , (6.6)

- after which <o(t)> follows from

<(t)> = fd 0 C(€ot) (6.7)

For a given non-stochastic state 0(t 0 ), the initial condition reduces to

C(Oott = P(Oolt 0 ) 0(t 0 ) , (6.8)

which differs from (6.6) by the fact that there are no initial correlations.

This means that the process a(t) has no memory to times smaller than to, and

consequently its evolution for t a t0 is completely determined by its

initial state 0(t 0 ). It was emphasized by Arnoldus and Nienhuis 13 that the

common choice (o0 ,t0) P(Oot 0 ) <o(t0 )> is merely an approximation which

only holds for small correlation times of 0(t).

VII. SOLUTIONS

A. INDEPENDENT INCREMENTS

Equation (6.5) for the marginal average of a(t) can be solved for the

independent-increment process with the same procedure as in Section IV,

where we obtained the characteristic functional. If we adopt the Fourier

transform

* ' ' -. ", -£ - , .. - . .%b . , -.. . ,- *..- .. . --. -. - . .....- . ,- ,% .,.',,,. ,,,...,-,

15

(p,t) = fde ipo C(#,t) - <eiP*(t)o(t)> , (7.1)

where the second equality follows after application of Eq. (6.3), then

<o(t)> can be found from

<o(t)> = (Ot) (7.2)

4 With the technique of Section IV we can find &(p,t), and if we differentiate

the result with respect to time, we find

ati - (p, t) = (A - iQ(p)) (p,t) , (7.3)

with

Q(p) = YJdn (1-eig(p-B)) w(r) . (7.4)

The operator Q(p) accounts for the phase fluctuations. If we set p = 0 in

Eq. (7.3), we achieve the equation for <o(t)>, with solution

-i(A-iQ(0))(t-t 0

<(t)> = e <o(t0 )> , (7.5)

for t 2 to. We note that <o(t)> can be expressed in terms of <o(t 0 )> for

this process, so that there are no initial correlations for the diffusion

process. The process *(t) has no memory, and with the results of Section V

it can be shown that *(t) is indeed delta-correlated. This means that

<O(tn) . (t1 )> for all n contains only 6-functions, which implies the4...

factorization in (7.5).

A special case arises if we take

Yw(n) = y 6(n) + X 6"(r)) , X > 0 , (7.6)

:. - -) ?- ): '.-.i-)- --- "€ 2 -) ::--; -,-- ) -- - . -:"- ;-: -.-- :- - -:.- -..9.-".-...".)-" - " ".

16

where the primes on the 6-function denote differentiation with respect to

its argument. It is easy to check that this process is the Wiener-Levy

process, or the phase-diffusion process. If we substitute (7.6) into

(4.12), we find that P(*,t) is Gaussian, and obviously this is the only

Gaussian limit of the diffusion process. The operator Q(p) in (7.4) reduces

to

Q(p) = X(p-B)2 , (7.7)

and the equation for <0(t)> becomes

i- <o(t)> = (A-iAB 2)<o(t)> , (7.8)

4which is the result of Fox

B. ORNSTEIN-UHLENBECK PROCESS

The diffusion process has no memory and is essentially non-stationary.

The initial distribution P(#,t0) = 6() diffuses over the whole *-axis,

-- < # < -. The inclusion of a finite memory-time can stabilize this

process. Let us define the transition probability as

W($010) - a(O)6( 0 -) X 6"(0o-0) + y * 6,(0o-) , X > 0Y> o

(7.9)

Then the Master equation (4.6) for P(*,t) becomes the Fokker-Planck

equation 18

a2

P(4,t) = (X __ + * 0) P(Ot) (7.10)at 80 2'•

which has the solution for t *

P(O) - (2,a2) " e a /2 , a2 -/' (7.11)

17

This P(O), together with W(*O[j) from (7.9), defines a stationary Gaussian

Markov process, the Ornstein-Uhlenbeck process. In the limit y * 0 and A

2finite (so a - ) the process 0(t) reduces to the Wiener-L~vy process from

the previous paragraph. From (7.11) we see that 0(t) is centered around * =2

0. The distribution is Gaussian with a variance a around the average * -

0. The preference for -f 0 expresses that this process can be considered

as a model for phase-locked radiation.

With the specific choice (7.9) for the transition rate, the Master

equation (6.5) assumes the form of a second-order partial differential

equation. We obtain

(i- - A + iAB2 ) Y(0,t) iyo 2(2iB + ) a + (iB +at ao a* a,'(7.12)

In the limit y -* 0 and A - yo2 finite, we recover (the Fourier inverse of)

Eq. (7.3) with Q(p) from Eq. (7.7).

In order to obtain a solution of Eq. (7.12), we start with a Fourier

transform with respect to #. The transformed equation then reads

(i - A + iB 2)&(P,t) -iy(o2 (p-2B)p + (p-B) )-1}(p,t) , (7.13)

which is still a partial differential equation. Since we are interested in

&(0,t) - <o(t)>, the obvious approach2 3 would be a Taylor expansion around

p - 0. This yields however a cumbersome inhomogeneous four-term recurrence

relation for the Taylor coefficients. This can be avoided by the

transformation15

(p,t) - (p) &(p,t) , (7.14)

which defines j(p,t). The Fourier transform of the probability distribution

*" is explicitly

a"% . - '... % **l~ .

18

P(p) <eiP#(t)> e- 2p2 (7.15)

and in particular we have P(0) = 1. The equation for g(p,t) becomes

a 2a 2

(t- - A + iB 2)j(Pt) = -iy{p ap - B(o 2 + p))g(Pt) , (7.16)

and it has to be solved for

i(O't) = <o(t)> (7.17)

Let us define the Taylor coefficients v (t) by the expansion

i(p't) = ff (t ) (7.18)

n=0

which can be inverted as

I (t) = <o(t) (-!-) •ip(t) - IWO2> (7.19)n ap p=0

Substitution of (7.18) into (7.16) then gives the equation for the Taylor

coefficients

a2 2(i- - A + iXB + iyn)w (t) yB(no in- (t) - 1 t)) (7.20).at nnn-l'n"

which has to be solved for

10 (t) = <o(t)> (7.21)

Equation (7.20) looks like a homogeneous three-term recurrence relation, but

it will be shown below that the time-derivative a/at gives rise to an

inhomogeneous contribution. Notice that for n - 0 Eq.(7.20) reduces to a

two-term relation between i 0(t) and r 1(t) only.

Eq. (7.20) is most easily solved in the Laplace domain. If we

introduce

19

n (W) - f dt e 0 n(t) , (7.22)tO

then (7.20) attains the form

A + iXB2 + iyn)i(w) - (B(no2;n-l(W) - ; n+l)) - inn (t 0 (7.23)

Here the initial values wn(t 0 ) for n = 0, 1, 2, ... appear as inhomogeneous

terms. The set wn(t0) for all n represents the initial correlations of a(t)

on t = to, and they connect the time-evolution of <o(t)> for t > t0 to its

recent past.13 In other words, Eq. (7.23) relates the set % (t) for t > tn0

to the initial set v (t 0).

Equation (7.23) can be solved for an arbitrary initial set n (t 0) by* 24

standard techniques, but the solution is not transparent. In order to

elucidate the structure of the solution, we assume a non-stochastic initial

state o(t0 ). From Eq. (7.1) we then find at t - tO

(p,tO ) = f(p) o(t0 ) , (7.24)

and from Eq. (7.14) we obtain

i(p,t0) = o(t0 ) . (7.25)

Hence at t = to the vector i(p,t 0) is independent of p, and therefore the

expansion coefficients are simply

Wn(t) = an,O(t 0 ) (7.26)

Then only the recurrence relation for n - 0 is inhomogeneous, and the

solution of (7.23) for 10 (W) - <;(W)> is readily found to be

6(w)> i 2 (t0 . (7.27)W- A + lAB 2 + R(w)

20

The effect of the finite correlation time, e.g., the deviation from the

Wiener-Lkvy limit, is accounted for by the operator

K(w) - yB 210

-A + LB 2 + li + yB 2-2 yB2 2 B

w- A + iXB + 2iy + B 3--- yB (7.28)

which indeed vanishes for y -* 0, A finite. In this limit, Eq. (7.27) is the

Laplace transform of Eq. (7.8).

The explicit expression (7.27) provides the exact solution for

,ituations where the initial state is non-stochastic and for cases where the

solution is independent of the initial state. As an example from quantum

optics, we mention that Eq. (7.27) with o(t0) = 1, A = 0 and B = 1

represents the laser spectral profile. Another example pertains to the

long-time behavior of the solution <a(t)>. If there is any damping in the

system, which might be caused by the stochastic fluctutations itself, then

the solution for t >> t0 will become independent of the initial state. If

we indicate by - the solution <a(t)> for t - , then - obviously obeys the

equation

(A - lAB 2 K(O)) a = 0 (7.29)

For the problem of atomic fluorescence in a strong laser field, this is the

equation for the atomic steady-state density matrix, which determines the

fluorescence yield. There, the solution 0 of Eq. (7.29) is unique.

VIII. CONCLUSIONS

Solving a multiplicative stochastic process o(t) for its average is

rarely feasible by analytical methods. This is mainly due to the finite

21

correlation time of the driving process 4(t), which prohibits the

factorization of the average of a product into the product of the averages.

Averages of a functional of *(t) might factorize if the process is delta-

correlated. For Markov processes, however, we can simulate a 6-correlation

by the introduction of the marginal average C(#0,t) - <6(*(t)-4 0 )a(t)>. The

combination of the multiplication by 6(4(t)-* 0 ) and the Markov property of

the probability distributions of *(t) then gives rise to a factorization-

like result for the formal expression for the average. Along the same lines

as in a factorization assumption, we can now derive exact equations for

C( 0,t). In this paper we have studied Eq. (1.1), where we considered the

stochastics of *(t) to be given. The equation of motion for the marginal

average turned out to be Eq. (6.4). The applicability of this equation was

illustrated by some examples.

ACKNOWLEDGMENTS

This research was supported by the National Science Foundation under

Grant CHE-8519053, the Air Force Office of Scientific Research (AFSC),

United States Air Force, under Contract'F49620-86-C-0009, and the Office of

Naval Research. The United States Government is authorized to reproduce and

distribute reprints for governmental purposes notwithstanding any copyright

notation hereon.

* ' .# -

22IREFERENCES

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4. . .

23

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