Ukrainian Mathematical Journal, Vol. 47, No. 10, 1995

SOME PROPERTIES OF RANDOM EVOLUTIONS

Ya. I. Eleiko 1 and V . M . Shurenkov 2 UDC 519.21

We study asymptotic properties of matrix-valued random evolutions and consider an example of evolu- tions of this type.

Let x ( t ) be a regenerating process [1] with regenerating times % = 0, "c 1, "~2 . . . . . I: n ... defined in a probabil-

ity space (f2, F, P).

We consider a family of nonnegative m x m-dimensional matrix-valued processes ~ ( t ) , 0 < t < "c, with func-

tional dependence on a small parameter e but statistically independent of the regenerating process x(t), t >_ O. By

using the process ~ ( t ) and the sequence of regeneration times "t 0, "c 1 . . . . . % . . . . we construct a matrix-valued

evolution of the form

N~(t) =

~ I E (t), 0 ~ t ~ "I~l,

~(2:1)~e2(~2 - 1 : l )" '~k(Tk -- gk-1)~k+l ( 1 - Tk), "gk <t--<'gk+l,

(1)

where ~ ( t ) is a sequence of independent copies of the process ~ ( t ) , n = 1, 2 . . . . . By construction, the process

N~(t) is a multiplicative functional.

The sequence z 1, "~2 - "cl . . . . . "ok- %-1 . . . . consists of independent equally distributed random variables. We

also assume that M'c < ~,.

Let us find the asymptotics of random evolutions under consideration as t ~ oo and e --~ 0. By the formula of

total probability, we have

t

M N e ( t ) = M(Ne( t ) , "r > t) + ] M(Ne(u ) ' "r e d u ) M N ( t - u) o

t

= M(~e( t ) , " c > t ) + f M(~e(u) , z e d u ) M ( N e ( t - u ) ) . 0

(2)

Denote

RE(t ) = MNe(t) ,

GE(t ) = M(~e( t ) , "c > t),

1 L'viv University, L'viv.

2 Deceased.

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 10, pp. 1333-1337, October, 1995. Original article submitted July 29, 1993.

0041-5995/95/4710-1519 $12.50 �9 1996 Plenum Publishing Corporation 1519

1520 YA. I. ELEIKO AND V. M. SHURENKOV

Ke(du ) = M ( ~ e ( u ), "c e du) .

Equation (2) is a renewal equation. In the notation introduced above, it takes the form

t

Rs( t ) = Gs( t ) + ~ K e ( d u ) R s ( t - u ) . (3) o

Let K s = M (~s('c)) and let II c II denote the norm of a matrix C. Also let ~ | y be a matrix (X i "yj),j = 1 .... m,

where x i and yj are the ith and j th coordinate of the vectors ~ and y, respectively. The following theorem is

tree:

T h e o r e m 1. Assume that the sequence of matrices Gs( t ) is such that

IIaAt)ll sup sup < o% (4) t>__0 max (1, r )

for some T > 0 and

1 t---~TGs(t) ---> f ( z ) as e-->O, t-->oo, and t ( ) ~ e - 1 ) --+ z, (5)

K s--+ K as ~--->0 (6)

in the operator norm. Moreover, K is an indecomposable matrix with Perron root 1 and ~ e is the Perron

root that corresponds to the matrix Ke In this case, if

lim sup yKe(dy) = O, t ' ~ ~ E

then

1

1 M N e ( t ) 1 f lim 1 = - - ff | g f(c(1 - y))(1 - y)Ve yc/a dy , t ~ , s - . o t ~'+ a

o tO~e - 1)---~ c

where ~ and ~ are, respectively, the right and left positive eigenvectors of the matrix K,

K g = g ,

o o

~ K = ~, (~, V) = 1, and a = g ~ yK(dy)V . o

Proo f Since M N s ( t ) satisfies the renewal equation (3), its solution can be represented in the form

Re(t ) = t

H a ( d u ) G e ( t - u), 0

(7)

where

SOME PROPERTIES OF RANDOM EVOLUTIONS 1521

He(du ) = s K[*(du), r = 0

(8)

K[*(t) =

t

f K~r-1)*(du)KE(t-u), 0

K 1. t ( ) = K~([0, t)),

Ke ~ = { I ' where t > 0 ,

O, where t < 0 ,

where I is the matr ix of the identity operator and O is the null matrix. Since M z < 0% it is well known [2] that

~ H e ( [ O , t ) ) ----> l (ec /a-1) U a " |

as t ---> ~ , e --+ 0, and t 0% - 1) --+ c. W e have

t 1

t~+l f Ha(du)M(~a(t-u ), "C>t-u) - 0

1 1

t~,+ I f dHe(tu)M(~e(t(1 -u)), z > t ( 1 - u ) ) o

1 -- g 1

1 1

+ /.T+ 1 f dHe(tu)M(~e(t(1-u)) , z > t ( 1 - u ) ) . (9) 1 - e 1

If t --~ ~ , e --+ O, and t ()~ - 1) --~ c, then, according to (5),

M(~e(t(1 - u)), "~ > t(1 - u))

q(1 -u ) ) r + f ( c ( 1 - u))

uniformly in u e [0, 1 - E l ] and, for u > 0, H~(tu)/t has the limit

1 i eSC/ads ~ | ~ a o

in the operator no rm as t --> ~ , e --> 0, and t 0~e - 1) -+ c.

Further, under condit ion (4), we have

II M ( ~ A u ) , z > u) II < 13 (max (1, uV))

with some constant [3. Hence, for t > 1/E~, the norm of the second term on the r ight-hand side o f relation (9) is not greater than

1522

1 D 7 II H e ( t ) - H e ( t ( 1 - El))II

By the conditions of the theorem, this expression has the following limit:

1

98[ ~ eYC/adyl[ ~ | ~ll- a

1 - e 1

By the choice of e 1, it can be made as small as desired. Theorem 1 is proved.

Consider an example of stochastic evolution.

Example 1.

YA. I. ELEIKO AND V. M. SHURENKOV

Let ~Ite(t) be a matrix-valued random evolution defined as a solution of a differential equation

dgte(t) = gte(t)Ae(x(t)) (10) dt

with initial condition

Here, sional matrix-valued functions, and 8 is a small parameter. The solution of Eq. (10) can be represented in the form

~ e ( O ) = I.

x(t) is a regenerating process with regeneration times "c 1, "c 2 . . . . . Ae(x(t)) is a family of m x m-dimen-

The solution of Eq. (12) has the form

Representation (13) implies that lution of Eq. (12) as follows:

with boundary condition

dTe(s, t) = re(s ' t )Ae(x ( t ) ) (12) dt

Te(s , s ) = I.

n

Te(s, t) = lim.. H (i + A~(x(ti)Ati)~ (13) 0 max a' i ~ i = 1

S = t o < t 1 < . . . < t n = t , A t i = t i - t i _ I .

(11)

T~(s, t) = Te(s, u)TE(u, t). The solution of Eq. (10) can be represented via the so-

We also consider an auxiliary matrix-valued evolutionary equation

0 = t O < t 1 < . . . < t n = t , A t i = t i - t i _ I .

n

~te(t) = lira, H ( I + A e ( x ( t i ) A t i ~ a t i --+ 0 i = 1 m a x

SOME PROPERTIES OF RANDOM EVOLUTIONS 1523

tC~(t) =

T ~(0, t) ,

Te(0, Zl)Ta('Cl, t),

... T E "c Te( 0, '~I)Tr z2) (Zk-1, k) T (%, t),

t<7:1,

Z 1 < t < ' ~ 2,

q7 k </- _< "(Tk + 1,

k = T ~ If we denote ~e(t) ( % , "c~ + t), then {~(t) are independent copies of the process ~e(t) = Te(O, t),

This follows from the fact that ~ (t) satisfies the differential equation

with boundary condition

d ~ ( t ) = ~ ( t ) A ~ ( x k ( t ) ) dt

~ ( o ) = ~,

where xk ( t ) = x (Zk + t), 0 < t < Zk+1 - Z ~ are independent copies of x( t ) , 0 < t < "c 1.

Thus, ~ge(t) can be represented in the form

We(t) =

~ I ( z I ;::2 ),,~ (t - T1),

) ~ ( ' c 2 - Zl) . .- k - 1 J ~ I ( t - zk ) ,

t_<'~l,

'171 <t_<%2,

~k < t-<'~k+l-

By construction,

and

t < Z 1.

We(t) coincides with the stochastic evolution (1). Denote M ( ~ a ( t ) , z > t ) = GE(t). Assume that

II a~ (t)I1 sup sup < t max{1, t 7'}

1 7"9-G~(t) -'+ f ( z ) as ~--->0, t--->oo, and t (LE-1 ) ---> z,

Ke --+ K as ~---)0

in the operator norm,

is the Perron root of the matrix Ka. Moreover, if Ka = M ( ~ a ( z ) ) , the matrix K is indecomposable and its Perron root is equal to 1, and ~e

lira sup y = 0 ,

1524 YA. I. ELEIKO AND V. M. SHURENKOV

then, according to Theorem 1, we have

1 lim 1 Mt[ta(t) = 1 fi_ | V ~ f(c(1 - y))(1 - y)~'e cy/a dy,

e~0, t ~ t q'+l a 0

t(~.~ -1)~c

Where ~- and g are right and left positive eigenvectors of the matrix K and

a = V f yK(dy) ~. 0

Assume that the fight-hand side of Eq. (10) has the form

Ae(x) = A + 51(~)B1~(x) + 52(~)B~(x) + . . . + 5k(~)B~(x) + o(Sk(~)),

where a sequence of functions 51 (e) . . . . . 5k(e ) forms a scale such that

5i(a) > 0, i = 2 . . . . . k, 5i- l (e) ~--,0

and B~(x) --~ Bi(x ), i = 1, . . . , k. The matrix A has nonnegative nondiagonal elements. Moreover, its Perron root

is equal to zero. Then there exist right and left positive eigenvectors ~ and ~ of the matrix A such that A ~ =

and ~A = 0.

I fwedeno te Ke=MTe(z )and K = M e za, then, by the assumption, K~--~K as e---~0, the matrix K has

the Perron root 1 with fight and left eigenvectors ~- and ~, respectively, and the matrix Ke has the Perron root

Le with right and left eigenvectors ~ and ~ such that ~,e ~ 1, ge ~ ~, and ~E --~ ~ as e ~ 0.

Let us now determine the asymptotics of ~,z - 1. Denote

b i = M f ~B i ( x ( r ) )udr , o

b12

T, S

= M f V f T~ 0 0

b 3 = M ~ ~Bl(X(s))e(Z-s)Av~ds, 0

where V is the generalized inverse matrix for K - I. The following theorem is tree:

Theorem 2. If b 1 r then ~,~- 1 = b l S l ( ~ ) + O(51(E))- I f b 1 =0, then

(a) ~ e - 1 = 52(E)b 2 + 0 (52(E) ) f o r 52(E) = 0 (52 (~ ) ) ,

SOME PROPERTIES OF RANDOM EVOLUTIONS

(b) ~,~- 1 = (b12-b3)812(c) + o (82 (e ) ) for a2(e ) = o(8~(a))

(c) Z c - 1 = ( b l 2 - b 3 +db2)82 (~ ) + o(82(~)) for 82(c ) - d82(e) .

The proof of this theorem is based on the representation

~E-1 = $ e ( K ~ - K ) ~

and the solution

n t

Te(t) = eta + Z ~i(E)f Te(s)B[(x(s))e(t-S)Ads, i = I 0

whence we easily get the required result.

REFERENCES

1525

1. I.N. Kovalenko, N. Yu. Kuznetsov, and V. M. Shurenkov, Random Processes. A Handbook [in Russian], Naukova Dumka, Kiev (1983).

2. V.M. Shurenkov, "Transients in renewal theory," Mat. Sb., 112, No. 1,115-132 (1980).