Diffusion Approximation with Equilibrium for Evolutionary Systems Switched by Semi-Markov Processes
We consider an evolutionary system switched by a semi-Markov process. For this system, we obtain an inhomogeneous diffusion approximation results where the initial process is compensated by the averaging function in the average approximation scheme.
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irk-123456789-1658322020-02-18T01:27:59Z Diffusion Approximation with Equilibrium for Evolutionary Systems Switched by Semi-Markov Processes Korolyuk, V.S. Limnios, N. Статті We consider an evolutionary system switched by a semi-Markov process. For this system, we obtain an inhomogeneous diffusion approximation results where the initial process is compensated by the averaging function in the average approximation scheme. Для систем, що перемикаються иапівмарковськими процесами, одержано результати про неоднорідну дифузійну апроксимацію, де вихідний процес компенсується усередненою функцією в апроксимаційній схемі усереднення. 2005 Article Diffusion Approximation with Equilibrium for Evolutionary Systems Switched by Semi-Markov Processes / V.S. Korolyu, N. Limnios // Український математичний журнал. — 2005. — Т. 57, № 9. — С. 1253–1260. — Бібліогр.: 8 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165832 519.21 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Korolyuk, V.S. Limnios, N. Diffusion Approximation with Equilibrium for Evolutionary Systems Switched by Semi-Markov Processes Український математичний журнал |
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We consider an evolutionary system switched by a semi-Markov process. For this system, we obtain an inhomogeneous diffusion approximation results where the initial process is compensated by the averaging function in the average approximation scheme. |
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Article |
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Korolyuk, V.S. Limnios, N. |
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Korolyuk, V.S. Limnios, N. |
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Korolyuk, V.S. |
| title |
Diffusion Approximation with Equilibrium for Evolutionary Systems Switched by Semi-Markov Processes |
| title_short |
Diffusion Approximation with Equilibrium for Evolutionary Systems Switched by Semi-Markov Processes |
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Diffusion Approximation with Equilibrium for Evolutionary Systems Switched by Semi-Markov Processes |
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Diffusion Approximation with Equilibrium for Evolutionary Systems Switched by Semi-Markov Processes |
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Diffusion Approximation with Equilibrium for Evolutionary Systems Switched by Semi-Markov Processes |
| title_sort |
diffusion approximation with equilibrium for evolutionary systems switched by semi-markov processes |
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Інститут математики НАН України |
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2005 |
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Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/165832 |
| citation_txt |
Diffusion Approximation with Equilibrium for Evolutionary Systems Switched by Semi-Markov Processes / V.S. Korolyu, N. Limnios // Український математичний журнал. — 2005. — Т. 57, № 9. — С. 1253–1260. — Бібліогр.: 8 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT korolyukvs diffusionapproximationwithequilibriumforevolutionarysystemsswitchedbysemimarkovprocesses AT limniosn diffusionapproximationwithequilibriumforevolutionarysystemsswitchedbysemimarkovprocesses |
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2025-07-14T20:05:34Z |
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2025-07-14T20:05:34Z |
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1837654117909004288 |
| fulltext |
UDC 519.21
V. S. Korolyuk (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv),
N. Limnios (Univ. Technol. Compiènge, France)
DIFFUSION APPROXIMATION
WITH EQUILIBRIUM OF EVOLUTIONARY SYSTEMS
SWITCHED BY SEMI-MARKOV PROCESSES
DYFUZIJNA APROKSYMACIQ Z RIVNOVAHOG
EVOLGCIJNYX SYSTEM, WO PEREMYKAGT|SQ
NAPIVMARKOVS|KYMY PROCESAMY
We consider an evolutionary system switched by a semi-Markov process. For this system we obtain a
nonhomogeneous diffusion approximation results where the initial process is compensated by the
averaging function in the average approximation scheme.
Dlq system, wo peremykagt\sq napivmarkovs\kymy procesamy, oderΩano rezul\taty pro ne-
odnoridnu dyfuzijnu aproksymacig, de vysxidnyj proces kompensu[t\sq userednenog funkci[g
v aproksymacijnij sxemi userednennq.
1. Introduction. Dynamic systems described by evolutionary equation is a classical
topic in stochastic modelling. Asymptotic analysis of such systems is studied by
several authors (see, e.g., [1 – 5]).
The usual asymptotic approach, in the diffusion approximation scheme, consist into
normalize the process about an equilibrium point obtained by a balance condition with
respect to the equilibrium distribution. Another diffusion approximation can be
obtained by considering fluctuation with respect to the average process. In a previous
work we have studied evolutionary systems with Markov switching in two cases [6].
The first case when the average process is a deterministic function and the second case
when the average was a stochastic process.
In the present paper, we compensate the initial process by an averaging
deterministic function instead of an equilibrium point (see, e.g., [6]) and we obtain a
nonhomogeneous diffusion approximation result.
In Section 2 we describe processes implied in our analysis. In Section 3 we present
result (Theorem 1) and in Section 4 the proof of this theorem.
2. Preliminaries. Let E be a Polish space and � its Borel σ-algebra. We call
the measurable space ( E, � ) a standard state space.
The semi-Markov continuous stochastic system is considered in the series scheme
with small series parameter ε > 0, ε → 0, described by a solution of the evolutionary
equation in R
d
d
dt
U t a U t x tε
ε
ε
ε
( ) ( );=
2
. (1)
The velocity function admit the following representation:
a u x a u x a u xε ε( ; ) ( ; ) ( ; )= + 1 , (2)
where u ∈ R
d and x ∈ E.
The semi-Markov switching process x t( ), t ≥ 0, on the standard state space ( E,
� ), is given by the semi-Markov kernel
Q x B t P x B F tx( , , ) ( , ) ( )= , (3)
where x ∈ E, B ∈ �, and t ≥ 0, and supposed to be supposed to be uniformly ergodic
with the stationary distribution π( )B , B ∈ �, satisfying the relation
© V. S. KOROLYUK, N. LIMNIOS, 2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9 1253
1254 V. S. KOROLYUK, N. LIMNIOS
π ρ( ) ( ) ( )dx dx m x
m
= , (4)
where ρ( )B , B ∈ �, is the stationary distribution of the embedded Markov chain xn ,
n ≥ 0, given by the stochastic kernel
P x B x B x xn n( , ) := ∈ =( )+P 1 . (5)
In addition
m x F t dtx( ) ; ( )=
∞
∫
0
, F t F tx x( ) : ( )= −1 , m dx m x
E
: ( ) ( )= ∫ ρ . (6)
It is well-known (see, e.g., [3]) that under some additional conditions the stochastic
system U tε( ), t ≥ 0, converges weakly to the deterministic average process ˆ ( )U t , t ≥
≥ 0, defined by a solution of the average evolutionary equation
d
dt
U t a U tˆ( ) ˆ ˆ( )= ( ) , (7)
with the average velocity
ˆ ( ) : ( ) ( ; )a u dx a u x
E
= ∫ π . (8)
It is natural that the fluctuation of the stochastic system around the average process
can be described by the diffusion process (see [6]). The diffusion approximation
scheme for the semi-Markov continuous stochastic system (1) here considered for the
centered and normalized process
ζ εε ε( ) ( ) ˆ( )t U t U t= −[ ]−1 . (9)
3. Main results. The main result is formulated as follows.
Theorem 1. Let the stochastic evolutionary system (9) be defined by relations
(1) – (9) and the following conditions be fulfilled:
C1) the switching semi-Markov process x t( ), t ≥ 0, is uniformly ergodic with
stationary distribution π( )dx on the compact phase space E ;
C2) the following asymptotic expansions take place:
a u x a x ua x u x( ; ) ( ; ) ( ; ) ( , ; )v v v vv+ = + ′ +ε ε θε
0 ,
a u x a x u x1 1 1( ; ) ( ; ) ( , ; )v v v+ = +ε θε ,
where, for any R > 0,
sup ( , ; )
v
v
≤
≤
∈
→
R
u R
x E
i u xθε 0, ε → 0, i = 0, 1.
In addition the velocity functions a u x( ; ) a n d a u x1( ; ) satisfy the global solution
of the equation (1) and (7).
Then the weak convergence for 0 ≤ t ≤ T,
ζ ζε( ) ( )t t⇒ 0 , ε → 0, (10)
takes place. The limit diffusion process ζ0( )t , t ≥ 0, is determined by the generator
of the coupled process ζ0( )t , ˆ ( )U t , t ≥ 0,
Lϕ ϕ ϕ ϕ( ; ) ( , ) ( , ) ( ) ( , ) ˆ( ) ( , )u b u u B u a uu uv v v v v v vv= ′ + ′′ + ′1
2
. (11)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
DIFFUSION APPROXIMATION WITH EQUILIBRIUM OF EVOLUTIONARY SYSTEMS … 1255
Here
b u a ua( ; ) ˆ ( ) ˆ ( )v v v= + ′1 , (12)
ˆ( ) ( ) ( ; )a dx a x
E
v v= ∫ π , ˆ ( ) ( ) ( ; )a dx a x
E
1 1v v= ∫ π ,
where prime and double prime mean first and second derivatives respectively.
The covariance matrix B( )v , v ∈ R
d , is determined by the relations
B B B( ) ( ) ( )v v v= +0 1 , (13)
B dx a x R a x
E
0 02( ) ( ) ˜( ; ) ˜( ; )v v v= ∫ π ,
B dx x a x a x
E
1 2( ) ( ) ( ) ˜( ; ) ˜ ( ; )*v v v= ∫ π µ , (14)
µ( )
( ) ( )
( )
x
m x m x
m x
= −2
22
where a* means transpose of vector a.
Remarks. 1. The particular case µ( )x = 0 correspond to the exponential
distribution F tx( ) = 1 – exp ( )−{ }λ x t . As a corollary in this case, we get the results
given in [6].
2. The limit diffusion process ζ0( )t , t ≥ 0, in nonhomogeneous in time and is
solution of the following SDE
d t a U t a U t t dt B U t dW tζ ζ0
1
0 1 2( ) ˆ( ) ˆ ˆ( ) ( ) ˆ( ) ( )/= ( ) + ′( )[ ] + ( ) ,
where W t( ), t ≥ 0, is the standard Wiener process in Rd .
3. The stationary regime for the average process ˆ ( )U t , t ≥ 0, is realized when he
average velocity ̂ ( )a v has an equilibrium point ρ : ˆ( )a ρ = 0. Then the limit diffusion
process ˆ( )ξ t , t ≥ 0, is of the Ornstein – Uhlenbeck process with the following
generator:
ˆ ( ) ( ) ( ) ( )Lϕ ϕ ϕu b u u B u= ′ + ′′1
2
,
where
b u b ub( ) = +1 0 , b a1 1= ˆ ( )ρ , b a0 = ′ˆ ( )ρ , B B= ( )ρ .
4. Proof. The proof of Theorem 1 is divided on several steps. At first, the
extended Markov chain
ζ ζ ε τε ε
n n= ( )2 , ˆ ˆ ,U Un n
ε ε τ= ( )2 , x xn n= ( )τ , n ≥ 0, (15)
is considered, where τn, n ≥ 0, is the sequence of the Markov renewal moments
(moments of jumps of the semi-Markov process x t( ), t ≥ 0 ), that is,
τ τ θn n n+ += +1 1, n ≥ 0,
F t t x xx n n( ) = ≤ =( )+P θ 1 .
Let us introduce the following families of semigroups:
Γt xx u U tε εϕ ϕ( ) ( ) ( )= ( ),
(16)
U ux
dε( )0 = ∈R ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1256 V. S. KOROLYUK, N. LIMNIOS
where U tx
ε( ), t ≥ 0, is a solution of the evolutionary system
d
dt
U t a U t xx x
ε
ε
ε( ) ( );= ( ) , x ∈ E,
and, similarly,
ˆ ( ) ˆ( )A U ttϕ ϕv = ( ),
(17)
ˆ( )U d0 = ∈v R ,
where ˆ ( )U t , t ≥ 0, is a solution of the average evolutionary system (7).
It is worth noticing that the generators of semigroups (16) and (17) are respectively:
ΓΓε εϕ ϕ( ) ( ) ( ; ) ( )x u a u x u= ′ ,
ˆ ( ) ˆ( ) ( )Aϕ ϕv v v= ′a .
The following generators will be also used:
ΓΓ( ) ( ) ( ; ) ( )x u a u x xϕ ϕ= ′ ,
˜ ( ) ( ) ˜( ; ) ( )ΓΓ x u a u x xϕ ϕ= ′ , ˜( ; ) : ( ; ) ˆ( )a u x a u x a u= − .
The main object in asymptotic analysis with semi-Markov processes is the
compensating operator of the extended embedded Markov chain (15) which is given
ere in the next lemma.
Lemma 1. The compensating operator of the extended embedded Markov chain
(15) is determined by the relation
L
ε
ε
ε
ε
ε
εϕ ε ϕ ϕ( , , ) ( ) ( ) ( , ) ( ) ˆ ( , , ) ( , , )u x q x F dt x A P u x u xx t t t
v v v v v= −
−
∞
∫2
0
2 2 2Γ Γ , (18)
where the semigroup Γt xε( , )v , t ≥ 0, is defined by the generator
ΓΓε εϕ ε ϕ( , ) ( ) ( ; ) ( )x u a u x uv v= + ′ , (19)
a u x a u x a u x a u xε
ε εε ε( ; ) : ( ; ) ( ; ) ( ; )= = +− −1 1
1 , (20)
the semigroup Γt
ε( )v , t ≥ 0, is defined by the generator
ΓΓε ϕ ε ϕ ε ϕ( ) ( ) ˆ ( ) ( ) : ˆ( ) ( )v v vu u a u= − = − ′− −1 1A . (21)
It is worth noticing that the generator ΓΓ
ε ( , )x v in (19) can be transformed by using
condition C2 of Theorem 1, as follows:
ΓΓ ΓΓε
εε( , ) ( , )x xv v= −1 , (22)
ΓΓ
ε ϕ( , ) ( )x uv : =
: = a u x uε ε ϕ( ; ) ( )v + ′ = a x u( ; ) ( )v ′ϕ + ε ϕb u x u( , ; ) ( )v ′ + θ ϕε ( , ; ) ( )v u x u ,
where by definition
b u x( , ; )v = a x1( ; )v + ua x′v v( ; ) .
Proof of Lemma 1. The proof of this lemma is based on the conditional
expectation of the extended embedded Markov chain (15) which is calculated by using
(1) – (9)
E ϕ ζ ζε ε ε ε
n n n n n nU x u U x x+ + +( ) = = =[ ]1 1 1, ˆ , , ˆ ,v =
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
DIFFUSION APPROXIMATION WITH EQUILIBRIUM OF EVOLUTIONARY SYSTEMS … 1257
=
0
1
0 0
2 2∞
−∫ ∫ ∫+ ( ) − ( )
F dt u a U s x ds a U s dsx
t
x
s
t
( ) ( ); ˆ ˆ ( )E ϕ ε
ε
ε
ε
,
v v v+ ( )
= + = =
∫ +
0
1
2
0
ε
ε εε
t
n x n na U s ds x U u U x xˆ ˆ ( ) , ( ) , ˆ , =
=
0
2 2 2
∞
∫ F dt x A P u xx t t t
( ) ( , ) ( ) ˆ ( , , )Γ Γε
ε
ε
ε
ε ϕv v v = : F
ε ϕ( ) ( , , )x P u xv .
The next step in the asymptotic analysis is to construct the asymptotic expansion of
the compensating operator with respect to ε.
Lemma 2. The compensating operator (18) – (21) admits the following
asymptotic representation of test function ϕ ∈ C d d
0
3 2,
R R×( ):
L
ε ϕ( , , )u xv = ε ϕ− ⋅ ⋅2 Q x( , , ) + ε ϕ− ⋅ ⋅1 ˜ ( , ) ( , , )ΓΓ x P uv +
+
L0 ( , ) ( , , ) ˆ ( , , )x P P uv vϕ ϕ⋅ ⋅ + ⋅ ⋅[ ]A + θ ϕε
l u x( , , )v , (23)
with the negligible term
sup ( , , )
x E
l u x
∈
→θ ϕε v 0 , ε → 0.
Here, by definition,
Q x q x P I xϕ ϕ( ) ( ) ( )= −[ ] , (24)
is the generator of the associated Markov process x t0( ), t ≥ 0, with the intensity
function
q x
m x
( ) :
( )
= 1 , m x F t dtx( ) : ( )=
∞
∫
0
.
The generator ˜ ( , )ΓΓ x v , and the operator L0( , )x v are defined as follows:
˜ ( , ) ( ) ˜( ; ) ( )ΓΓ x u a x uv vϕ ϕ= ′ , (25)
and
L0 1
1
2
( , ) ( ) ( , ; ) ( ) ( ; ) ( )x u b u x u B x uv v vϕ ϕ ϕ= ′ + ′′ , (26)
b u x a x u a x( , ; ) : ( ; ) ( ; )v v vv= + ′1 , (27)
B x x a x a x1 2( ; ) : ( ) ˜( ; ) ˜ ( ; )*v v v= µ , (28)
µ2
2( ) :
( )
( )
x
m x
m x
= , m x t F dtx2
0
2( ) : ( )=
∞
∫ . (29)
Proof. At the beginning the compensating operator is transformed as follows:
L F
ε εε ε= + −[ ]− −2 2Q q x x I P( ) ( ) . (30)
Now, the following algebraic identity is used:
abc – 1 = (a – 1) + (b – 1) + (c – 1) + (a – 1) (b – 1) + (a – 1) (c – 1) +
+ (b – 1) (c – 1) + (a – 1) (b – 1) (c – 1). (31)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1258 V. S. KOROLYUK, N. LIMNIOS
Setting
a x
t
: ( , )= Γε
ε
2 v ,
b
t
: ( )= Γε
ε
2 v , c A
t
: ˆ= ε2 , (32)
the terms in (30) with (31) and (32) are transformed by using the integral equation for
semigroup
F xa
ε( ) : =
0
2
∞
∫ −[ ]F dt x Ix t
( ) ( , )Γε
ε v =
ε ε
ε
ε2
0
2ΓΓ ( , ) ( ) ( , )x F t x dtx t
v v
∞
∫ Γ =
= ε
ε2 m x x( ) ( , )ΓΓ v +
ε ε
ε
ε4 2
0
2ΓΓ ( , ) ( ) ( , )x F t x dtx t
v v[ ]
∞
∫ Γ =
= ε
ε2 m x x( ) ( , )ΓΓ v +
ε ε4 2 2
2
m x
x
( )
( , )ΓΓ v[ ] +
ε ε ε6 3
3ΓΓ ( , ) ( )x F xav[ ] ,
where
F t F s dsx
k
t
x
k( ) ( )( ) : ( )+
∞
= ∫1 , F t F tx x
( )( ) : ( )1 = ,
F x F t x dta x t3
0
3
2
ε
ε
ε( ) : ( ) ( , )=
∞
∫ Γ v .
Taking into account (22) the following expansion is obtained:
F x m x x
m x
x xa a
ε
ε ε
εε ε ε θ( ) ( ) ( , )
( )
( , ) ( , )= + [ ] +Γ Γv v v2 2 2 2
2
, (33)
with the negligible term
θ εε ε ε
a ax x F x( , ) : ( , ) ( )v v= [ ]Γ
3
3 ,
on test function ϕ ∈ C d
0
3
R( ) .
Similarly, the asymptotic expansion can be obtained for the next two terms in (31),
(32)
F xb
ε( ) :=
0
2
∞
∫ −[ ]F dt Ix t
( ) ( )Γε
ε v =
− + [ ] +ε ε ε θεm x
m x
xb( ) ˆ ( )
( ) ˆ ( ) ( , )A Av v v3 2 2 3
2
,
(34)
with the negligible term
θ εε ε
b bx F x( , ) : ˆ ( ) ( )v v= [ ]A
3
3 ,
F x F t dtb x t3
0
3
2
ε
ε
ε( ) : ( ) ( )( )=
∞
∫ Γ v ,
on test function ϕ ∈ C d
0
3
R( ) .
Analogously,
F x F dt A I m x xc x t c
ε
ε
εε ε θ( ) : ( ) ˆ ( ) ˆ ( ) ( )= −[ ] = +
∞
∫
0
2 2
2 A v , (35)
with the negligible term
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
DIFFUSION APPROXIMATION WITH EQUILIBRIUM OF EVOLUTIONARY SYSTEMS … 1259
θε ε
c cx F x( ) : ˆ ( ) ( )= [ ]A v
3
2 , F x F dt A dtc x t2
0
2
2
ε
ε( ) : ( ) ˆ( )=
∞
∫ ,
on test function ϕ ∈ C d
0
2
R( ).
At last we analyze the next term
F xab
ε ( ) : =
0
2 2
∞
∫ −[ ] −[ ]F dt x I Ix t t
( ) ( , ) ( )Γ Γε
ε
ε
εv v = Γ Γε ε ε( , ) ( ) ( )x F xabv v 1 ,
F xab1
ε ( ) : =
0 0 0
2 2∞
∫ ∫ ∫
F dt x ds dsx
t
s
t
s( ) ( , ) ( )
ε
ε
ε
εΓ Γv v =
=
2 4
0
2
2 2ε ε
ε
ε
ε
∞
∫ F dt xx t t
( )( ) ( , ) ( )Γ Γv v + ε θε4
ab x( ) .
Hence, by (21) and (22), we get
F x m x x xab ab
ε
ε
εε ε θ( ) : ( ) ( , ) ˆ ( ) ( )= − +2
2
2ΓΓ ΓΓv v , (36)
with negligible term θε
ab x( ) on the test function ϕ ∈ C d3
R( ) .
It can be easily verified that the last three terms in (31), (32) are negligible on test
functions ϕ ∈ C d d3 2,
R R×( ). As a consequence, gathering the extensions (33) –
(36), the asymptotic extension (23) – (28) for the compensating operator is obtained.
In the next step in the proof of Theorem 1, the limit generator (11) is calculated by
using a solution of singular perturbation problem for the compensating operator (23)
(see, e.g., [4, 6]).
Lemma 3. A solution of singular perturbation problem for the generator (23)
L L
ε ε εϕ ϕ θ( , , ) ( , ) ( , , )u x u u xLv v v= + , (37)
on test function ϕε ( , , )u xv = ϕ( , )u v + εϕ1( , , )u xv + ε ϕ2
2( , , )u xv , and negligible
term θ
ε
L u x( , , )v , is realized by the generator L given in Theorem 1, formulae
(11) – (13).
Proof. According to [4, p. 51] (Lemma 3.3), the limit generator in (37) is
represented as follows:
L LΠ Π Π Π Π Π Π= + +˜ ( , ) ˜ ( , ) ( , ) ˆΓΓ ΓΓx PR x P x P Pv v v0 0 A ,
where the projector Π is defined as follows:
Πϕ π ϕ( ) ( ) ( )x dx x
E
= ∫ .
Let us calculate
L1Π = Π Π˜ ( , ) ˜ ( , ) ( )ΓΓ ΓΓx PR x P uv v0 ϕ = Π
˜ ( , ) ˜ ( , ) ( )ΓΓ ΓΓx PR x uv v0 ϕ =
= Π
˜ ( , ) ˜( ; ) ( )ΓΓ x PR a x uv v0 ′ϕ .
By the definition of potential operator R0 [4], we have
QR R Q I0 0= = −Π ,
or
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1260 V. S. KOROLYUK, N. LIMNIOS
q x P I R I( ) −[ ] = −0 Π ,
hence, PR0 = R0 + m x( )[ Π – I ].
So, we can write
L1Π = Π
˜ ( , ) ( ) ˜( ; ) ( )ΓΓ x R m x I a x uv v0 −[ ] ′ϕ =
= Π ˜ ( , ) ˜( ; ) ( )ΓΓ x R a x uv v0 ′ϕ – Π ˜( ; ) ( ) ˜ ( ; ) ( )*a x m x a x uv v ′′ϕ =
= Π ˜( ; ) ˜( ; ) ( )a x R a x uv v0 ′′ϕ – Π m x a x a x u( ) ˜( ; ) ˜ ( ; ) ( )*v v ′′ϕ .
Hence, the first term is
L1 0 0
1
2
1
2
ϕ ϕ ϕ( ) ( ) ( ) ( ) ( )u B u A u= ′′ − ′′v v , (38)
where
B dx a x R a x
E
0 02( ) : ( ) ˜( ; ) ˜( ; )v v v= ∫ π ,
A dx m x a x a x
E
0 2( ) : ( ) ( ) ˜( ; ) ˜ ( ; )*v v v= ∫ π .
The next term is
Π Π Π ΠL L0 1 0
1
2
( , ) ( ) ( , ; ) ( ) ( ; ) ( ) ( ) ( )x P u b u x u B x u uv v v vϕ ϕ ϕ ϕ= ′ + ′′ = ,
where
L0 1
1
2
( ) ( ) ( , ) ( ) ( ) ( )v v vϕ ϕ ϕu b u u B u= ′ + ′′ , (39)
and
B dx x B x
E
1 2 1( ) : ( ) ( ) ( ; )v v= ∫ π µ ,
functions B x1( ; )v , b u( , )v , b u x( , ; )v and µ2( )x are defined respectively in (28),
(12), (27) and (29).
Hence, setting together (38) and (39) we obtain the generator L of Theorem 1.
The last step of the proof concerns the relative compactness of the probability
measures of the processes where it can be realized by the standard scheme as it is given
in [7] and [8].
1. Skorokhod A. V. Asymptotic theory of stochastic differential equations. – Providence: Amer. Math.
Soc., 1981.
2. Sobsczyk K. Stochastic differential equations. – Dordrecht: Kluwer, 1991.
3. Korolyuk V. S., Swishchuk A. Evolution of systems in random media. – CRC Press, 1995.
4. Korolyuk V. S., Korolyuk V. V. Stochastic models of systems. – Kluwer Acad. Publ., 1999.
5. Korolyuk V. S., Limnios N. Average and diffusion approximation for evolutionary systems in an
asymptotic split phase space // Ann. Appl. Probab. – 2004. – 14(1). – P. 489 – 516.
6. Korolyuk V. S., Limnios N. Diffusion approximation of evolutionary systems with equilibrium in
asymptotic split phase space // Theory Probab. and Math. Statist. – 2002. – 69.
7. Korolyuk V. S., Limnios N. Poisson approximation of homogeneous stochastic additive functionals
with semi-Markov switching // Ibid. – 2002. – 64. – P. 75 – 84.
8. Korolyuk V. S., Limnios N. Poisson approximation of increment processes with Markov switching
// Theory Probab. Appl. – 2004. – 49(4). – P. 1 – 18.
Received 17.06.2005
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