Estimates for the Rate of Convergence in Ordinary Differential Equations under the Action of Random Processes with Fast Time
We study the procedure of averaging in the Cauchy problem for an ordinary differential equation perturbed by a certain Markov ergodic process. We establish several estimates for the rate of convergence of solutions of the original problem to solutions of the averaged one.
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| author | Bondarev, B. V. Kovtun, E. E. Бондарев, Б. В. Ковтун, Е. Е. Бондарев, Б. В. Ковтун, Е. Е. |
| author_facet | Bondarev, B. V. Kovtun, E. E. Бондарев, Б. В. Ковтун, Е. Е. Бондарев, Б. В. Ковтун, Е. Е. |
| author_sort | Bondarev, B. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:00:05Z |
| description | We study the procedure of averaging in the Cauchy problem for an ordinary differential equation perturbed by a certain Markov ergodic process. We establish several estimates for the rate of convergence of solutions of the original problem to solutions of the averaged one. |
| first_indexed | 2026-03-24T02:45:45Z |
| format | Article |
| fulltext |
UDK 519.21
B. V. Bondarev, E. E. Kovtun (Doneck. nac. un-t)
OCENKY SKOROSTY SXODYMOSTY V OBÁKNOVENNÁX
DYFFERENCYAL|NÁX URAVNENYQX,
NAXODQWYXSQ POD VOZDEJSTVYEM
SLUÇAJNÁX PROCESSOV S BÁSTRÁM VREMENEM
We study the procedure of averaging in the Cauchy problem for an ordinary differential equation
perturbed by some ergodic Markov process. We establish some estimates of the rate of convergence of
solutions of the initial problem to solutions of averaged problem.
Vyvça[t\sq procedura userednennq u zadaçi Koßi dlq zvyçajnoho dyferencial\noho rivnqnnq,
zburenoho deqkym erhodyçnym markovs\kym procesom. Vstanovleno deqki ocinky ßvydkosti
zbiΩnosti rozv’qzkiv poçatkovo] zadaçi do rozv’qzkiv useredneno].
1. Ocenka skorosty sblyΩenyq neprer¥vn¥x martynhalov s semejstvom
vynerovskyx processov. Yzvestno, çto lgboj neprer¥vn¥j lokal\n¥j mar-
tynhal, kvadratyçeskaq varyacyq kotoroho stremytsq s veroqtnost\g edynyca
k beskoneçnosty, zamenoj vremeny moΩet b¥t\ preobrazovan v brounovskoe dvy-
Ωenye. Sformulyruem v vyde lemm¥ sledugwyj rezul\tat.
Lemma (teorema 9. 3 [1]). Pust\
µt
t t, ,� 0 0≥{ } — neprer¥vn¥j lokal\n¥j
martynhal, takoj, çto lim ,t t→+∞ [ ]µ µ = + ∞, y dlq kaΩdoho t τ t =
= inf : ,s ts> [ ] >{ }0 µ µ . Tohda process µτt
neotlyçym ot brounovskoho dvy-
Ωenyq Wt
, t ≥ 0. Zdes\ µ µ,[ ]t , t ≥ 0, — kvadratyçeskaq varyacyq martyn-
hala µt
t t, ,� 0 0≥{ }.
V sylu toho, çto [2, c. 120] v sluçae neprer¥vnoho kvadratyçno yntehryrue-
moho martynhala kvadratyçeskaq varyacyq sovpadaet s xarakterystykoj 〈
µ,
µ
〉t
, t ≥ 0, v dal\nejßem budem yspol\zovat\ xarakterystyku 〈
µ, µ
〉t
, t ≥ 0,
vmesto kvadratyçeskoj varyacyy µ µ,[ ]t , t ≥ 0. Otmetym takΩe, çto τt =
= inf : ,s ts> [ ] >{ }0 µ µ — markovskyj moment otnosytel\no semejstva
� 0 0t t, ≥{ } . ∏tot fakt sleduet yz sootnoßenyq τt s≤{ } = t s≤{ }µ µ, y
toho, çto 〈
µ, µ
〉s
, s ≥ 0, — � 0
s
-yzmerymaq velyçyna.
Vvedem process ζt = µ
t – µτt
. V¥qsnym, ymeet ly πtot process martynhal\-
noe svojstvo otnosytel\no nekotor¥x potokov σ-alhebr. Rassmotrym snaçala
potok
� 0 0t t, ≥{ } .
Sleduet otmetyt\, çto, naprymer, v razloΩenyy D. O. Çykyna [3, 4] martyn-
hal µt
, t ≥ 0, — kvadratyçno yntehryruem¥j y predstavym v vyde
µ
t =
0
0
+∞
∫ /{ }M s dstη( ) � – ρ0 = M s ds t
0
0
+∞
∫ /
η( ) � – ρ
0
,
t. e. martynhal µ
t
, t ≥ 0, qvlqetsq rehulqrn¥m, v sylu teorem¥ 2.7 [5] — rav-
nomerno yntehryruem¥m y v sylu zameçanyq k teoreme 2.9 [5] ymeet svojstvo
M µτ
σ/{ }� 0 = µ
min ( τ, σ ) . (1)
V sylu svojstva (1)
M t
sζ /{ }� 0 = µ
s – µ τmin ,t s( ) ≠ ζs
,
© B. V. BONDAREV, E. E. KOVTUN, 2005
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4 435
436 B. V. BONDAREV, E. E. KOVTUN
t. e. sluçajn¥j process ζt
, t ≥ 0, ne qvlqetsq martynhalom otnosytel\no
neub¥vagweho semejstva
� 0 0t t, ≥{ } .
Estestvenno rassmotret\ takΩe potok
� 0 0min , ,t t tτ( ) ≥{ }. V dannom sluçae
oçevydno, çto
M t
s sζ τ/ ( ){ }� 0
min , = M M t
t st sζ τ τ/ ( ) ( ){ }[ ]� �0 0
min , min , = 0 ≠ ζs
,
t. e. martynhal\noe svojstvo kak otnosytel\no potoka
� 0 0t t, ≥{ } , tak y ot-
nosytel\no potoka
� 0 0min , ,t t tτ( ) ≥{ } ne ymeet mesta. Takym obrazom, stremle-
nye poluçyt\ ocenky sxodymosty v supremumnoj metryke, osnov¥vaqs\ na nera-
venstvax typa neravenstva A. N. Kolmohorova, v obwem sluçae vrqd ly obosno-
vano.
Lemma 1. Spravedlyvo ravenstvo
| ζt | = | µt – µτt | = µ µτ τmax , min ,t tt t( ) ( )− . (2)
Dokazatel\stvo. Pust\ χ
(
τt ≥ t
) — yndykator sob¥tyq (
τt ≥ t
), χ
(
τt <
< t
) — yndykator sob¥tyq (
τt
< t
). Tohda
| µt – µτt | = χ τ χ τ µ µτ( ) ( )t t tt t
t
≥ + <[ ] −[ ] =
= µ χ τ µ χ τ µ τ µ χ ττ τt t t t t tt t t t
t t
( ) ( ) ( ) ( )≥ + < − ≥ − < =
= µ χ τ µ χ τ µ τ µ χ ττ τ τ τmin , max , max , min ,( ) ( ) ( ) ( )t t t t t t t tt t t t
t t t t( ) ( ) ( ) ( )≥ + < − ≥ − < ≤
≤ ( ) max , min ,τ µ µτ τt t tt
t t
≥ −( ) ( ) + χ τ µ µτ τ( ) max , min ,t t tt
t t
< −( ) ( ) =
= µ µτ τmax , min ,t tt t( ) ( )−
y obratno
µ µτ τmax , min ,t tt t( ) ( )− = µ µ χ τ χ ττ τmax , min , ( ) ( )t t t tt t
t t( ) ( )− ≥ + <[ ] =
= µ χ τ µ χ τ µ τ µ χ ττ τ τ τmin , max , max , min ,( ) ( ) ( ) ( )t t t t t t t tt t t t
t t t t( ) ( ) ( ) ( )≥ + < − ≥ − < =
= µ χ τ µ χ τ µ τ µ χ ττ τt t t t t tt t t t
t t
( ) ( ) ( ) ( )≥ + < − ≥ − < ≤
≤ χ τ χ τ µ µτ( ) ( )t t tt t
t
≥ + <[ ] −[ ] = | µt – µτt |,
otkuda y sleduet (2).
Lemma 1 dokazana.
Lemma 2. Pust\ τt
= inf : ,s ts> [ ] >{ }0 2µ µ σ , tohda spravedlyva ocenka
sup
0
2
≤ ≤
−
t T
tM
t
µ µτ ≤ sup ,
0≤ ≤
−
t T
tM tµ µ σ . (3)
Dokazatel\stvo. Sohlasno lemme 1 ymeem
M t t
µ µτ−
2
= M t tt t
µ µτ τmax , min ,( ) ( )−
2
=
= M t t
µ τmax ,( )
2 – 2 M t tt t
µ µτ τmax , min ,( ) ( ) + M t t
µ τmin ,( )
2 =
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
OCENKY SKOROSTY SXODYMOSTY V OBÁKNOVENNÁX … 437
= M t t
µ τmax ,( )
2 –
2 0M M t t
t
t t
tµ µτ τ
τ
max , min ,
min ,
( ) ( )
( )/{ }[ ]� + M t t
µ τmin ,( )
2 =
= M t t
µ τmax ,( )
2 –
2 0M Mt t
t
t t
tµ µτ τ
τ
min , max ,
min ,
( ) ( )
( )/{ }[ ]� + M t t
µ τmin ,( )
2 =
= M t t
µ τmax ,( )
2 – 2 2M t t
µ τmin ,( )[ ] + M t t
µ τmin ,( )
2 =
= M t t
µ τmax ,( )
2 – M t t
µ τmin ,( )
2
.
V sylu toho, çto min (
t, τt
) y max (
t, τt ) — markovskye moment¥ otnosytel\no
potoka
� 0 0t t, ≥{ } , yz predstavlenyq µt
2 = 〈
µ, µ
〉t
+ vt
, hde sluçajn¥j pro-
cess vt yzmerym otnosytel\no potoka
� 0 0t t, ≥{ } y qvlqetsq martynhalom s
nulev¥m srednym, ymeem
µ τmax ,t t( )
2 = µ µ τ, max ,t t( ) +
vmax ,t tτ( ),
µ τmin ,t t( )
2 = µ µ τ, min ,t t( ) + vmin ,t tτ( ) ,
a otsgda v sylu soxranenyq martynhal\noho svojstva v markovskye moment¥
vremeny centryrovann¥m processom vt
M t t
µ τmax ,( )
2 – M t t
µ τmin ,( )
2 = M
µ µ τ, max ,t t( ) – M
µ µ τ, min ,t t( ) .
Dalee,
M
µ µ τ, max ,t t( ) – M
µ µ τ, min ,t t( ) ≤ M t tt t
µ µ µ µτ τ, ,max , min ,( ) ( )− =
= M t tt t t tt t
µ µ µ µ χ τ χ ττ τ, , ( ) ( )max , min ,( ) ( )− ≥ + <[ ] ≤
≤ M t tt t t tt t
µ µ χ τ µ µ χ ττ τ, ( ) , ( )max , min ,( ) ( )≥ − ≥ +
+ M t tt t t tt t
µ µ χ τ µ µ χ ττ τ, ( ) , ( )max , min ,( ) ( )< − < ≤
≤ M t t
t t t tµ µ χ τ µ µ χ ττ, ( ) , ( )≥ − ≥ +
+ M t tt t tt t
µ µ χ τ µ µ χ ττ, ( ) , ( )< − < = M
t tµ µ µ µτ, ,− . (4)
Yz (4) v sylu toho, çto µ µ, tt
= σ2t , ymeem (3).
Vvedem „b¥stroe” vremq t / ε (
ε > 0 — malaq velyçyna). Pust\ µεt =
= εµ εt / , Wt
ε = ε εWt / — semejstvo standartn¥x vynerovskyx processov, po-
luçenn¥x yz brounovskoho dvyΩenyq Wt
, t ≥ 0, kotoroe, v svog oçered\,
poluçaetsq yz martynhala pry sootvetstvugwej zamene vremeny.
Yz lemm 1 y 2 v¥tekaet sledugwyj rezul\tat.
Teorema 1. Pust\
µt
t t, ,� 0 0≥{ } — neprer¥vn¥j lokal\n¥j martynhal,
takoj, çto lim ,t t→+∞ [ ]µ µ = +
∞ y, krome toho, τt = inf : ,s ts> [ ] >{ }0 2µ µ σ
dlq kaΩdoho t. Tohda v¥polnqetsq neravenstvo
sup
0
2
≤ ≤
−
t T
t tM Wµ σε ε = sup
0
2
≤ ≤
/ /−
t T
t tM Wεµ σ εε ε ≤
≤ sup ,
0
2
≤ ≤
/ −
t T
tM tε µ µ σε . (5)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
438 B. V. BONDAREV, E. E. KOVTUN
Dokazatel\stvo neobxodymo provesty lyß\ dlq sluçaq σ2 > 0, σ2 ≠ 1, tak
kak pry σ2 = 1 vse uΩe obosnovano. Sluçaj σ2 > 0, σ2 ≠ 1 poluçaem, esly
snaçala vmesto martynhala µt rassmotret\ martynhal µt = µt / σ , dlq
kotoroho vse uslovyq, dostatoçn¥e dlq spravedlyvosty teorem¥ 1, ymegt
mesto, t. e. v dannom sluçae spravedlyva ocenka
sup
0
2
≤ ≤
−
t T
t tM Wµε ε = sup
0
2
≤ ≤
/ /−
t T
t tM Wε µ εε ε ≤
≤ sup ,
0≤ ≤
/ −
t T
tM tε µ µ σε .
Yz posledneho umnoΩenyem na σ2 > 0, σ2 ≠ 1, poluçaem (5).
Teorema dokazana.
2. Usrednenye v stoxastyçeskyx systemax. Ocenka skorosty sxodymos-
ty. Rassmotrym uravnenye
d ξ ε ( t ) = ε a ( ξ ε ( t ) ) f ( η ( t ) ) d t, ξ ε ( 0 ) = ξ0 . (6)
Zdes\ process η ( t ), t ≥ 0, udovletvorqet uravnenyg
d η ( t ) = b ( η ( t ) ) + σ ( η ( t ) ) d W ( t ), η ( 0 ) = η0 . (7)
Budem sçytat\ v¥polnenn¥my sledugwye uslovyq:
A) koπffycyent¥ uravnenyq (6) udovletvorqgt teoreme suwestvovanyq y
edynstvennosty reßenyq [6]; krome toho, ymeet mesto ocenka
| a ( x ) | ≤ C̃ < + ∞,
suwestvuet proyzvodnaq ′a xx ( ), dlq vsex x opredelen¥ g ( x ) =
0
x dy
a y∫ ( )
y
g
–
1
( x );
B) otnosytel\no koπffycyentov b ( x ), σ ( x ) budem predpolahat\, çto su-
westvugt neprer¥vn¥e proyzvodn¥e ′b xx ( ) , ′σx x( ), ′′b xxx ( ) , ′′σx x x( ), pryçem
proyzvodn¥e starßyx porqdkov ohranyçen¥, a takΩe v¥polnqgtsq uslovyq
′b xx ( ) ≤ – λ, 0 < λ < + ∞,
0 < γ ≤ σ
2
( x ) ≤
1
γ
, γ ∈ ( 0, 1 ].
Pust\ U ( x ) — reßenye uravnenyq
f ( x ) – f = b x
dU x
d x
( )
( )
+
1
2
2
2
2σ ( )
( )
x
d U x
d x
, (8)
hde ρ ( x ) — reßenye soprqΩennoho uravnenyq
d
d x
x x
2
2
21
2
σ ρ( ) ( )
–
d
d x
b x x( ) ( )ρ[ ] = 0,
−∞
+∞
∫ ρ( )x d x = 1. (9)
Zdes\
−∞
+∞
∫ f x x d x( ) ( )ρ = f . (10)
Yz (9) sleduet
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
OCENKY SKOROSTY SXODYMOSTY V OBÁKNOVENNÁX … 439
d
d x
x x
2
2
21
2
σ ρ( ) ( )
–
d
d x
b x
x
x x
2 1
22
2( )
( )
( ) ( )
σ
σ ρ
= 0,
−∞
+∞
∫ ρ( )x d x = 1. (11)
Pust\
V ( x ) =
1
2
2σ ρ( ) ( )x x ,
tohda yz (11) ymeem
d
d x
V x( )[ ] –
2
2
b x
x
V x
( )
( )
( )
σ
= 0,
−∞
+∞
∫ ρ( )x d x = 1,
otkuda
ρ ( x ) =
2 2
2
0
2
C
x
b y
y
dy
x
σ σ( )
exp
( )
( )∫
.
Normyruq poslednee sootnoßenye, okonçatel\no poluçaem
ρ ( x ) =
1 2 1 2
2
0
2 2
0
2
1
σ σ σ σ( )
exp
( )
( ) ( )
exp
( )
( )x
b y
y
dy
x
b y
y
dy d x
x x
∫ ∫ ∫
−∞
+∞ −
.
Yz (10) v sylu teorem¥ Fredhol\ma [7] sleduet, çto reßenye (8) suwestvuet.
Pust\ ρ ( x, t, y ) — plotnost\ veroqtnosty perexoda (pry naloΩenn¥x ohra-
nyçenyqx na koπffycyent¥ ona suwestvuet (sm. [2])), pryçem
lim ( , , )t x t y→+∞ ρ = ρ ( y ). Dejstvytel\no, πrhodyçnost\ v sylu uslovyj B) sle-
duet yz rezul\tatov rabot¥ [8], tak kak pry | x | ≥ R > | b ( 0 ) | / λ ymeem x b ( x ) ≤
≤ – λ x
2 + | b ( 0 ) | | x | ≤ – λR b x−( )( )0 , t. e. uslovyq sootvetstvugwej teorem¥
yz rabot¥ [8] v¥polnen¥.
Netrudno zametyt\, çto reßenyem uravnenyq (8) qvlqetsq funkcyq [3]
U ( x ) =
0
+∞
−∞
+∞
∫ ∫ −[ ]f y f x t y d y dt( ) ( , , )ρ =
0
+∞
∫ ( ) −[ ]M f t f dtxη ( ) ,
hde
ηx t( ) = x +
0
t
xb s ds∫ ( )η ( ) +
0
t
x s dW t∫ ( )σ η ( ) ( ) . (12)
Poskol\ku dlq plotnosty ρ ( x, t, y ) spravedlyvo sootnoßenye (obratnoe urav-
nenye Kolmohorova)
∂ρ
∂
( , , )x t y
t
= b x
x t y
x
( )
( , , )∂ρ
∂
+
1
2
2
2
2σ ∂ ρ
∂
( )
( , , )
x
x t y
x
,
to
b x
dU x
d x
( )
( )
+
1
2
2
2
2σ ( )
( )
x
d U x
d x
= − −[ ]
+∞
−∞
+∞
∫ ∫
0
f y f
x t y
t
dt d y( )
( , , )∂ρ
∂
=
= –
−∞
+∞
∫ −[ ]f y f y d y( ) ( )ρ + f x f( ) −[ ] = f x f( ) −[ ].
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
440 B. V. BONDAREV, E. E. KOVTUN
Pust\ f ( x ) ymeet ohranyçennug proyzvodnug, tohda
′U xx ( ) = − ′( )[ ]
+∞
∫
0
M f t
d t
dx
dtx x
xη η
( )
( )
. (13)
Dyfferencyruq (12) po x, ymeem [9]
d t
dx
xη ( )
=
= exp ( ) ( ) ( ) ( )
0 0
2
0
1
2
t
x x
t
x x
t
x xb s ds s ds s dW s∫ ∫ ∫′ ( ) − ′ ( )[ ] + ′ ( )
η σ η σ η . (14)
Podstavlqq (14) v (13), poluçaem
′U xx ( ) = − ′( )[ ]
+∞
∫
0
M f tx xη ( ) ×
× exp ( ) ( ) ( ) ( )
0 0
2
0
1
2
t
x x
t
x x
t
x xb s ds s ds s dW s dt∫ ∫ ∫′ ( ) − ′ ( )[ ] + ′ ( )
η σ η σ η .
V sylu toho, çto ′f xx ( ) ≤ C < + ∞, y v sylu uslovyq B) ′b xx ( ) ≤ – λ, λ > 0,
ymeem
′U xx ( ) ≤
≤
0 0 0
2
0
1
2
+∞
∫ ∫ ∫ ∫− − ′ ( )[ ] + ′ ( )
CM ds s ds s dW s dt
t t
x x
t
x xexp ( ) ( ) ( )λ σ η σ η ≤
C
λ
, (15)
tak kak
M s ds s dW s
t
x x
t
x xexp ( ) ( ) ( )− ′ ( )[ ] + ′ ( )
∫ ∫1
2
0
2
0
σ η σ η = 1.
PokaΩem, çto v sdelann¥x otnosytel\no koπffycyentov predpoloΩenyqx su-
westvuet ravnomernaq po t ≥ 0 ocenka sverxu dlq
m ( t ) = M η
2
( t ).
Yspol\zuq formulu Yto, naxodym
η
2
( t ) = η0
2 +
0
2
t
s b s ds∫ ( )η η( ) ( ) +
0
2
s
s ds∫ ( )σ η( ) +
0
2
s
s s dW s∫ ( )η σ η( ) ( ) ( ).
Otsgda
m ( t ) = η0
2 +
0
2
t
M s b s ds∫ ( )η η( ) ( ) +
0
2
s
M s ds∫ ( )σ η( )
y
d m ( t ) = 2 M η ( t ) b ( η ( t ) ) d t + M σ
2
( η ( t ) ) d t ≤
≤ – 2 λ m ( t ) d t + 2 λ | b ( 0 ) | M | η ( t ) | + M σ
2
( η ( t ) ) d t ≤
≤ – 2 λ m ( t ) d t + 2 λ | b ( 0 ) | m t dt( ) +
1
γ
dt .
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
OCENKY SKOROSTY SXODYMOSTY V OBÁKNOVENNÁX … 441
UmnoΩaq obe çasty posledneho neravenstva na exp { – 2 λ t }, poluçaem
dl m tt2λ ( ) ≤ l b m t dtt2 2 0
1λ λ
γ
( ) ( ) +
,
otkuda
l m tt2λ ( ) ≤ Mη0
2 + 2 λ | b ( 0 ) |
0
2
t
sl m s ds∫ λ ( ) +
l t2 1
2
λ
γλ
−
≤
≤ Mη0
2 + 2 λ | b ( 0 ) |
0
2
t
s sl m s l ds∫ λ λ( ) +
l t2 1
2
λ
γλ
−
.
Pust\ V ( t ) = l t2λ m ( t ), tohda yz posledneho neravenstva ymeem
V ( t ) ≤ Mη0
2 +
l t2 1
2
λ
γλ
−
+ 2 λ | b ( 0 ) |
0
t
sl V s ds∫ λ ( ) .
Pust\
Z ( t ) = Mη0
2 +
l t2 1
2
λ
γλ
−
+ 2 λ | b ( 0 ) |
0
t
sl V s ds∫ λ ( ) ,
tohda
dZ t
dt
( )
=
l t2λ
γ
+ 2 0λ λb l V tt( ) ( ) ≤
l t2λ
γ
+ 2 0λ λb l Z tt( ) ( ) .
Yz posledneho neravenstva poluçaem
dZ t
Z t
( )
( )2
≤
l
Z t
t2
2
λ
γ ( )
+ λ λb l t( )0 ≤
l
M l
t
t
2
0
2 22 1 2
λ
λγ η γλ+ − /( ) ( )
+ λ λb l t( )0 ≤
≤
l
M l l
t
t t
λ
λ λγ η γλ2 1 20
2 2 2− −+ − /( ) ( )
+ λ λb l t( )0 ≤
≤
l
M
tλ
γ η2 0
2
+ λ λb l t( )0 ,
esly Mη0
2 ≤ 1 / ( 2 λ γ ), y
dZ t
Z t
( )
( )2
≤ l tλ λ
γ2
+ λ λb l t( )0 ,
esly Mη0
2 > 1 / ( 2 λ γ ).
Takym obrazom, v sluçae, kohda Mη0
2 ≤ 1 / ( 2 λ γ ), ymeem
Z t( ) ≤ Mη0
2 +
1
2
2 0
1
0
2γ η
λ
λ
λ
M
b
l t
+
−
( ) ,
otkuda
M tη2( ) ≤ V t l t( ) −2λ ≤ Z t l t( ) −2λ ≤
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
442 B. V. BONDAREV, E. E. KOVTUN
≤ l M
M
b
lt
t
− + +
−
2
0
2
0
2
2
1
2
2 0
1λ
λ
η
γ η
λ
λ
( ) ≤
≤ M
M
bη
γ η
λ
λ0
2
0
2
2
1
2
2 0
1+ +
( ) ≤ M
M
bη
γ λ η
0
2
0
2
1
2
2 0+ +
( ) ,
lybo
M tη2( ) ≤ V t l t( ) −2λ ≤ Z t l t( ) −2λ ≤
≤ l M b
lt
t
− + +
−
2
0
2
2
2
2 0
1λ
λ
η λ
γ
λ
λ
( ) ≤ M bη
λγ0
2
2
1
2
2 0+ +
( ) ,
esly Mη0
2 > 1 / ( 2 λ γ ).
Yz poluçenn¥x ocenok oçevydn¥m obrazom sleduet ocenka
M tη2( ) ≤ 4 2 00
2
2
M bη +[ ]( ) = C0 < + ∞. (16)
Dlq processa η ( t ), u kotoroho naçal\noe raspredelenye sovpadaet s πrho-
dyçeskym, oçevydno
Mη0
2 = M tη2( ) =
=
−∞
+∞
−∞
+∞ −
∫ ∫ ∫ ∫
x
x
b y
y
dy
x
b y
y
dy dx dx
x x
2
2
0
2 2
0
2
1
1 2 1 2
σ σ σ σ( )
exp
( )
( ) ( )
exp
( )
( )
=
= C1 < + ∞. (17)
Esly koπffycyent a ( x ) uravnenyq (6) takoj, çto
| a ( x ) | ≤ C̃ < + ∞, g ( x ) =
0
x
dy
a y∫ ( )
,
pryçem dlq vsex x funkcyy g ( x ) y g
–
1
( x ) opredelen¥, to netrudno zametyt\,
çto
dg
tξ
εε
= f dt + f t f dtη
ε
−
(18)
s naçal\n¥m uslovyem g ( ξ0 ).
Rassmotrym narqdu s (18) dyfferencyal
d g ( Z0 ( t ) ) = f dt (19)
s naçal\n¥m uslovyem g ( ξ0 ). Yz (18) y (19) ymeem
g
tξ
εε
– g ( Z0 ( t ) ) = ε η
ε
0
t
f t f dt
/
∫ ( )( ) −[ ] .
Esly
Y
t
ε ε
= g
tξ
εε
, V0 ( t ) = g ( Z0 ( t ) ),
to
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OCENKY SKOROSTY SXODYMOSTY V OBÁKNOVENNÁX … 443
ξ
εε
t
= g Y
t−
1
ε ε
, Z0 ( t ) = g
–
1
( V0 ( t ) ),
otkuda pry 0 ≤ θ ≤ 1 poluçaem
ξ
εε
t
– Z0 ( t ) = g Y
t−
1
ε ε
– g
–
1
( V0 ( t ) ) =
= a g V t Y
t
V t Y
t
V t− +
−
−
1
0 0 0( ) ( ) ( )θ
ε εε ε ,
a tak kak
| a ( x ) | ≤ C̃ < + ∞,
to
sup ( )
0
0
2
≤ ≤
−
t T
M
t
Z tξ
εε ≤ ˜ sup ( )C M g
t
g Z t
t T
2
0
0
2
≤ ≤
− ( )ξ
εε ≤
≤ ˜ supC M f t f dt
t T
t
2
0 0
2
≤ ≤
/
∫ ( )( ) −[ ]
ε η
ε
. (20)
Pust\ g
–
1
( Y0 ( t ) ) = Z0 ( t ), tohda
d Z0 ( t ) = d g
–
1
( Y0 ( t ) ) =
f
g g Y t
dt
x′ ( )( )−1
0( )
= a Z t f dt0( )( ) , Z0 ( 0 ) = x.
Yz rassuΩdenyj, pryvedenn¥x v¥ße, v¥tekaet sledugwyj rezul\tat.
Teorema 2. Pust\ v¥polnen¥ uslovyq A) y B). Tohda spravedlyva ocenka
sup ( )
0
0
2
≤ ≤
−
t T
M
t
Z tξ
εε ≤
≤ ˜ ( )C
C
M M b T
C2 2
2
2 0
2
0
2
2 2
24 4 0ε
λ
η η ε
γλ
+ +[ ]
+
, (21)
esly verno (16), lybo
sup ( )
0
0
2
≤ ≤
−
t T
M
t
Z tξ
εε ≤ C̃ C
C
T
C2 2
1
2
2
2
24ε
λ
ε
γλ
+
, (22)
esly verno (17). Zdes\ Z0 ( t ) — reßenye zadaçy
d Z0 ( t ) = a Z t f dt0( )( ) , Z0 ( 0) = ξ0 .
Dokazatel\stvo. Prymenqq formulu Yto k funkcyy U ( x ), posle ynteh-
ryrovanyq ymeem
U
tη
ε
– U ( η ( 0 ) ) =
0
t
LU s ds
/
∫ ( )
ε
η( ) +
0
t
xs U s dW s
/
∫ ( ) ′ ( )
ε
σ η η( ) ( ) ( ),
otkuda s uçetom (8)
ε η
ε
0
t
f t f dt
/
∫ ( )( ) −[ ] = ε η η
ε
U U
t
0( )( ) −
+
+ ε
0
t
xs U s dW s
/
∫ ( ) ′ ( )
ε
σ η η( ) ( ) ( ). (23)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
444 B. V. BONDAREV, E. E. KOVTUN
Yz (23) s uçetom (13) y (14) poluçaem ocenku
M f t f dt
t T
t
sup
0 0
2
≤ ≤
/
∫ ( )( ) −[ ]
ε η
ε
≤
≤ ε
λ
η η2
2
2 0
2
0
2
2
4 4 0
C
M M b+ +[ ]
( ) + ε
γλ
T
C 2
2 , (24)
a s uçetom (15), (17) ymeem
M f t f dt
t T
t
sup
0 0
2
≤ ≤
/
∫ ( )( ) −[ ]
ε η
ε
≤ ε
λ
2
1
2
24C
C
+ ε
γλ
T
C 2
2 . (25)
Yz (24) y (20) sleduet (21) (analohyçno yz (25) y (20) sleduet (22)).
Yssledovanye system¥ uravnenyj typa (6), (7) predstavlqet ynteres na vre-
mennom yntervale porqdka 1 / ε
2
. Tak, esly na vremennom yntervale porqdka
1 / ε predel\n¥m znaçenyem budet nesluçajn¥j process Z0 ( t ), to vo vremeny
t / ε
2
predel\n¥m processom dlq system¥ vyda (6) uΩe budet nekotor¥j stoxas-
tyçeskyj process. PredpoloΩym sledugwee. Pust\ dana systema
d ξ ε ( t ) = a ( ξ ε ( t ) ) η ( t ) d t, ξ ε ( 0 ) = 0, (26)
hde πrhodyçeskyj process η ( t ), t ≥ 0, udovletvorqet uravnenyg
d η ( t ) = b ( η ( t ) ) + σ ( η ( t ) ) d W ( t ), η ( 0 ) = η0
. (27)
Zdes\ η0 — nezavysymaq ot η ( t ) sluçajnaq velyçyna, raspredelenye kotoroj
sovpadaet s πrhodyçeskym raspredelenyem.
Otnosytel\no koπffycyenta σ ( x ) budem predpolahat\ v¥polnenn¥my us-
lovyq: suwestvugt neprer¥vn¥e proyzvodn¥e ′σx x( ), ′′σx x x( ), pryçem proyz-
vodn¥e starßyx porqdkov ohranyçen¥,
b ( x ) = – λ x, λ > 0,
0 < γ ≤ σ
2
( x ) ≤
1
γ
, γ ∈ ( 0, 1 ].
Pust\ U ( x ) — reßenye uravnenyq
x = – λx
dU x
d x
( )
+
1
2
2
2
2σ ( )
( )
x
d U x
d x
. (28)
Plotnost\ ynvaryantnoho raspredelenyq, oçevydno, ymeet vyd
ρ ( x ) =
1 2 1 2
2
0
2 2
0
2
1
σ
λ
σ σ
λ
σ( )
exp
( ) ( )
exp
( )x
y
y
dy
x
y
y
dy dx
x x
∫ ∫ ∫−
−
−∞
+∞ −
.
Oçevydno takΩe, çto
−∞
+∞
∫ x x dxρ( ) = 0.
Netrudno ubedyt\sq v tom, çto
U ( x ) = –
0
+∞
∫ M t dtxη ( ) = –
x
λ
.
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OCENKY SKOROSTY SXODYMOSTY V OBÁKNOVENNÁX … 445
Dejstvytel\no,
M txη ( ) = x – λ η
0
t
xM s ds∫ ( ) ,
otkuda M txη ( ) = x l
–
λ
t
. Yz (26) sleduet
ξ
εε
t
2
=
0
2
t
a
s s
dt∫
ξ
ε
εη
εε . (29)
Pust\ snova
g ( x ) =
0
x
d y
a y∫ ( )
,
tohda yz (29) ymeem
g
tξ
εε 2
= ε η
ε
0
2t
s ds
/
∫ ( ) .
Ocenym skorost\ sblyΩenyq processa
ζε ( t ) = ε η
ε
0
2t
s ds
/
∫ ( )
s nekotor¥m semejstvom vynerovskyx processov. Analohyçno (23) s uçetom (28)
ymeem
ζε ( t ) = ε η
ε
0
2t
s ds
/
∫ ( ) = ε η0 – εη
ε
t
2
– ε σ η
λ
ε
0
2
1
t
s dW s
/
∫ ( )( ) ( ). (30)
Analohyçno (8) rassmotrym uravnenye
σ σ
λ
2
0
2
2
( )x −
= – λx
dU x
d x
( )
+
1
2
2
2
2σ ( )
( )
x
d U x
d x
,
(31)
σ0
2 =
−∞
+∞
∫ σ ρ2( ) ( )x x d x .
Reßenye (31) zapyßem v vyde
U ( x ) =
0
2
0
2
2
+∞
∫
( )( ) −
M
t
dtxσ η σ
λ
.
Po predpoloΩenyg suwestvuet ′σx x( ), ′σx x( ) ≤ C < + ∞, tohda
′U xx ( ) ≤
≤
0 0 0
2
0
1
2
+∞
∫ ∫ ∫ ∫− − ′ ( )[ ] + ′ ( )
C
M ds s ds s dW s dt
t t
x x
t
x xγ
λ σ η σ ηexp ( ) ( ) ( ) ≤
C
λ γ
.
Oçevydno takΩe, çto
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
446 B. V. BONDAREV, E. E. KOVTUN
sup ( )
t
t
≥0
η =
−∞
+∞
∫ x x dxρ( ) =
=
−∞
+∞
−∞
+∞ −
∫ ∫ ∫ ∫−
−
x
x
y
y
dy
x
y
y
dy dx dx
x x
1 2 1 2
2
0
2 2
0
2
1
σ
λ
σ σ
λ
σ( )
exp
( ) ( )
exp
( )
= C1 < + ∞.
Prymenqq formulu Yto k funkcyy U ( x ), posle yntehryrovanyq poluçaem
U
tη
ε2
– U ( η ( 0 ) ) =
0
2t
LU s ds
/
∫ ( )
ε
η( ) +
0
2t
xs U s dW s
/
∫ ( ) ′ ( )
ε
σ η η( ) ( ) ( ) ,
otkuda s uçetom (31)
ε σ η σ
λ
ε
2
0
2
0
2
2
2t
t
dt
/
∫
( )( ) −
= ε η η
ε
2
20U U
t( )( ) −
+
+ ε σ η η
ε
2
0
2t
xs U s dW s
/
∫ ( ) ′ ( )( ) ( ) ( ). (32)
Yz (32) s uçetom stacyonarnosty processa η ( t ) y poluçenn¥x ocenok ymeem
sup
0
2
0
2
0
2
2
2
≤ ≤
/
∫
( )( ) −
t T
t
M
t
dtε σ η σ
λ
ε
≤ 2 2
1ε C + ε
λγ
T
C
.
Yz (30) v sylu teorem¥ sleduet ocenka
sup ( ) ( )
0
0
2
≤ ≤
−
t T
M t W tζ σ
λε ε ≤ 2 2
1ε C + ε
λγ
TC
C+
2 1 .
Teorema 3. Pust\ dlq koπffycyentov uravnenyq (26) v¥polneno uslovye
A), a otnosytel\no koπffycyentov b ( x ), σ ( x ) uravnenyq (27) budem predpo-
lahat\ v¥polnenn¥my uslovyq: b ( x ) = – λ x , λ > 0, suwestvugt neprer¥vn¥e
proyzvodn¥e ′σx x( ), ′′σx x x( ), pryçem proyzvodn¥e starßyx porqdkov
ohranyçen¥,
0 < γ ≤ σ
2
( x ) ≤
1
γ
, γ ∈ ( 0, 1 ], σ0
2 =
−∞
+∞
∫ σ ρ2( ) ( )x x d x ,
krome toho, naçal\noe raspredelenye processa η ( t ) sovpadaet s πrhodyçeskym.
Tohda spravedlyva ocenka
sup ( )
0
2
0
≤ ≤
−
t T
M
t
Z tξ
εε ε ≤ 4 2
2
2 1ε
λ
C̃
C
C+
+ 2ε
λγ
T
CC̃
, (33)
hde
dZ tε
0( ) = ′ ( ) ( )a Z t a Z t dtx ε ε
σ
λ
0 0 0
2
22
( ) ( ) +
σ
λ ε ε
0 0a Z t dW t( ) ( )( ) , Zε
0 0( ) = 0.
Dokazatel\stvo. Pust\ snova
g ( x ) =
0
x
dy
a y∫ ( )
,
krome toho,
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
OCENKY SKOROSTY SXODYMOSTY V OBÁKNOVENNÁX … 447
Yε ( t ) = g
tξ
εε 2
, V tε
0( ) = g Z tε
0( )( ),
hde
g
tξ
εε 2
= ε η
ε
0
2t
s ds
/
∫ ( ) , g Z tε
0( )( ) =
σ
λ ε
0 W t( ).
Tohda, oçevydno, ymeet mesto ocenka
sup ( )
0
2
0
≤ ≤
− ( )
t T
M g
t
g Z tξ
εε ε ≤ 2 2
1ε C + ε
λγ
TC
C+
2 1 .
Poskol\ku
ξ
εε
t
2
= g
–
1
( Yε ( t ) ), Z tε
0( ) = g V t− ( )1 0
ε ( ) ,
to
sup ( )
0
2
0
≤ ≤
−
t T
M
t
Z tξ
εε ε = sup ( ) ( )
0
1 1 0
≤ ≤
− −( ) − ( )
t T
M g Y t g V tε ε ≤
≤ ˜ sup ( )C M g
t
g Z t
t T0
2
0
≤ ≤
− ( )ξ
εε ε ≤ C̃ C
TC
C2 22
1 1ε ε
λγ
+ +
.
Dalee, tak kak
dZ tε
0( ) = dg V t− ( )1 0
ε ( ) = ′ ( )( ) ( )( )− −a g V t a g V t dtx
1 0 1 0 0
2
22ε ε
σ
λ
( ) ( ) +
+
σ
λ ε ε
0 1 0a g V t dW t− ( )( )( ) ( ) = ′ ( ) ( )a Z t a Z t dtx ε ε
σ
λ
0 0 0
2
22
( ) ( ) +
σ
λ ε ε
0 0a Z t dW t( ) ( )( ) ,
to (33) takΩe ymeet mesto.
Teorema 3 dokazana.
3. Usrednenye v peryodyçeskyx sredax. Rassmotrym uravnenye
d ξ ε ( t ) = ε a ( ξ ε ( t ) ) f ( η ( t ) ) d t, ξ ε ( 0 ) = ξ0 . (34)
Budem sçytat\ v¥polnenn¥my uslovyq:
A1
) koπffycyent¥ uravnenyq (34) udovletvorqgt teoreme suwestvovanyq
y edynstvennosty reßenyq [6, c. 26], krome toho, ymeet mesto ocenka
| a ( x ) | ≤ C̃ < + ∞,
0
x
dy
a y∫ ( )
< + ∞, | x | < + ∞,
y suwestvuet proyzvodnaq ′a xx ( ).
Pust\
d η ( t ) = b ( η ( t ) ) d t + σ ( η ( t ) ) d W ( t ), t ≥ 0, (35)
s nesluçajn¥m naçal\n¥m uslovyem η0 . Otnosytel\no koπffycyentov urav-
nenyq (35) budem predpolahat\ v¥polnenn¥my uslovyq:
B1
) γ ≤ σ
2
( x ) ≤
1
γ
, γ ∈ ( 0, 1 ],
funkcyy b ( x ), σ ( x ) peryodyçn¥ s peryodom 1, koπffycyent¥ b ( x ), σ
2
( x )
ymegt ohranyçenn¥e hel\derov¥ proyzvodn¥e.
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
448 B. V. BONDAREV, E. E. KOVTUN
Funkcyq f ( x ) peryodyçna po x s peryodom 1. Budem takΩe predpolahat\,
çto | f ( x ) | ≤ C < + ∞, proyzvodn¥e pervoho porqdka funkcyy f ( x ) ohranyçen¥
postoqnnoj K y ravnomerno neprer¥vn¥ po x. ∏ty ohranyçenyq nazovem uslo-
vyem (S). Zametym, çto pry v¥polnenyy uslovyq B1
) suwestvuet plotnost\ ve-
roqtnosty perexoda u processa η ( s ) [10, c. 371] y ymeet mesto πksponencyal\no
b¥straq sxodymost\ k πrhodyçeskomu raspredelenyg [10, c. 373], a ymenno, yme-
et mesto ocenka
sup ( ) ( ) ( )
η
η ρ
0 0
1
M f t f x x dx( ) − ∫ ≤ K C l
–
δ
t
, (36)
hde poloΩytel\n¥e postoqnn¥e K, δ zavysqt ot koπffycyentov b ( x ), σ ( x ).
Rassmotrym zadaçu
1
2
2
2
2σ ( )x
d U
d x
+ b x
dU
d x
( ) = f ( x ) – f , (37)
U ( x ) = U ( x + 1 ), U ′ ( x ) = U ′ ( x + 1 ).
Pust\
ϑ( )x = exp
( )
( )
0
2
2
x
b y d y
y∫
σ
,
0
1
2∫ b y d y
y
( )
( )σ
≠ 0,
tohda
ρ ( x ) =
ϑ ϑ ϑ
σ ϑ ϑ
ϑ ϑ ϑ
σ ϑ ϑ
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
x x y d y
x x x
x x y d y
x x x
dxx
x
x
x
+
− +[ ]
+
− +[ ]
+ − + −
−
∫ ∫ ∫1
1
1
1
1 1
2
0
1
1 1
2
1
(38)
budet plotnost\g πrhodyçeskoj mer¥, sootvetstvugwej reßenyg uravnenyq
(35), pryçem ρ ( x )-reßenye peryodyçeskoj zadaçy
L*
ρ =
1
2
2 2
2
d x x
d x
ρ σ( ) ( )( )
–
d x b x
d x
ρ( ) ( )( )
= 0,
ρ ( x ) = ρ ( x + 1 ),
odnoznaçno opredelqetsq uslovyem normyrovky
0
1
∫ ρ( )x dx = 1.
Pust\
0
1
2∫ b y d y
y
( )
( )σ
= 0,
tohda
ρ ( x ) = ϑ
σ
ϑ
σ
( )
( )
( )
( )
x
x
y d y
y2
0
1
2
1
∫
−
. (39)
Pust\
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OCENKY SKOROSTY SXODYMOSTY V OBÁKNOVENNÁX … 449
0
1
∫ f x x dx( ) ( )ρ = f .
Poskol\ku
0
1
∫ −[ ]f x f x dx( ) ( )ρ = f – f = 0,
hde ρ ( x ) opredeleno lybo v (38), lybo v (39) v zavysymosty ot toho, ymeet mesto
0
1
2∫ b y d y
y
( )
( )σ
≠ 0 yly
0
1
2∫ b y d y
y
( )
( )σ
= 0,
v sylu teorem¥ Fredhol\ma [7] reßenye U ( x ) zadaçy (37) suwestvuet y, kak
netrudno zametyt\, proyzvodnaq ot reßenyq
ψ ( x ) =
dU
dx
x( ) = x
x
y f y f y d y
x x
+
∫ −[ ]( )
+ −
/
1 22
1
ϑ σ
ϑ ϑ
( ) ( ) ( )
( ) ( )
, esly
0
1
2∫ b y d y
y
( )
( )σ
≠ 0, (40)
lybo
ψ ( x ) = Cϑ−1( )x + ψ0 ( x ), esly
0
1
2∫ b y d y
y
( )
( )σ
= 0,
hde ψ0 ( x ) — nekotoroe çastnoe reßenye (37).
Zametym, çto esly
0
1
∫ ψ( )x dx = 0,
to naxoΩdenye v qvnom vyde funkcyy U ( x ) neobqzatel\no, dostatoçno znat\
lyß\ ocenku ee modulq, kotoraq neposredstvenno sleduet yz neravenstva
sup ( )
0 1≤ ≤x
U x ≤
0
1
∫ ψ( )x dx ≤ sup ( )
0 1≤ ≤x
xψ = D1 < + ∞. (41)
V sylu (37) ymeem
ε η
ε
0
t
f s f ds
/
∫ ( ) −[ ]( ) = ερ η η
ε
U U
t
( )0 − ( )
+ ε σ η ψ η
ε
0
t
s s dW s
/
∫ ( ) ( )( ) ( ) ( ),
otkuda
sup ( )
0 0≤ ≤
/
∫ ( ) −[ ]
t T
t
M f s f dsε η
ε
≤ sup ( ) ( ) ( )
0 0≤ ≤
/
∫ ( ) ( )
t T
t
M s s dW sε σ η ψ η
ε
+ ε 2 D1 ≤
≤ ε 2 D1 + T
D1
γ
ε .
V sylu (37) takΩe ymeem
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
450 B. V. BONDAREV, E. E. KOVTUN
ε η
ε
0
t
f s f ds
/
∫ ( ) −[ ]( ) =
= ε η η
ε
U U
t
( )0 − ( )
+ ε σ η ψ η
ε
0
t
s s dW s
/
∫ ( ) ( )( ) ( ) ( ),
t. e.
ε η
ε
0
t
f s f ds
/
∫ ( ) −[ ]( ) = ρεt + µεt .
V sylu (41) oçevydno, çto
sup
0 1≤ ≤t
tρ
ε ≤ ε 2 D1 .
Takym obrazom,
ε η µ
ε
ε
0
t
tf s f ds
/
∫ ( ) −[ ] −( ) ≤ ε 2 D1 , (42)
sup ( )
0 0≤ ≤
/
∫ ( ) −[ ]
t T
t
M f s f dsε η
ε
≤ ε ε
γ
2 1 1D D
T+
. (43)
Poskol\ku
µεt = ε σ η ψ η
ε
0
t
s s dW s
/
∫ ( ) ( )( ) ( ) ( ),
to
〈 µ, µ 〉t =
0
2
t
s s ds∫ ( ) ( )[ ]σ η ψ η( ) ( ) → + ∞, t → + ∞,
s veroqtnost\g 1. Poslednee oçevydno, tak kak v sylu uslovyq (36) s veroqt-
nost\g 1
lim ,
n→+∞ /ε µ µ ε1 = lim ( ) ( )
ε
ε
ε σ η ψ η
→
/
∫ ( ) ( )[ ]
0
0
1
2s s ds =
=
0
1
2 2∫ σ ψ ρ( ) ( ) ( )x x x d x .
V spravedlyvosty posledneho moΩno takΩe ubedyt\sq putem sledugwyx ras-
suΩdenyj. Dlq funkcyy f ( x ) = σ ψ2 2( ) ( )x x najdem reßenye zadaçy
d
d x
ψ
+
2
2
b x
x
x
( )
( )
( )
σ
ψ =
2 2 2
0
2
2
σ ψ σ
σ
( ) ( )
( )
x x
x
−[ ]
, σ0
2 =
0
1
2 2∫ σ ψ ρ( ) ( ) ( )x x x d x , (44)
ψ( )x = ψ( )x +1 .
Oçevydno, çto ono budet ymet\ vyd (40), a ymenno
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
OCENKY SKOROSTY SXODYMOSTY V OBÁKNOVENNÁX … 451
ψ( )x =
dU
d x
x( ) = x
x
y x x y d y
x x
+
∫ −[ ]( )
+ −
/
1 2 2
0
2 22
1
ϑ σ ψ σ σ
ϑ ϑ
( ) ( ) ( ) ( )
( ) ( )
,
(45)
sup ( )
0 1≤ ≤x
xψ = D2 .
Dalee yz (37), prymenqq formulu Yto y yntehryruq, poluçaem
1
0
2 2
0
2
n
s s ds ds
n
∫ ( ) ( ) −[ ]σ η ψ η σ( ) ( ) =
=
U n U
n
η η( ) ( )( ) − 0 –
1
0
n
s
dU s
ds
dW s
n
∫ ( ) ( )σ η η
( )
( )
( ).
V sylu uslovyq (36) ymeem
Mη4( )n + η0
4 ≤ η0
4 +
0
1
4∫ x x dxρ( ) + K l
–
δ
n
,
otkuda v sylu toho, çto
M
n
s
dU s
ds
dW s
n
1
0
4
∫ ( ) ( )
σ η η
( )
( )
( ) ≤
const
n2 ,
okonçatel\no naxodym
n
n
P
n
s s ds ds
=
+∞
∑ ∫ ( ) ( ) −[ ] >
1 0
2 2
0
21 σ η ψ η σ ε( ) ( ) ≤
n n=
+∞
∑
1
2 4
const
ε
< + ∞,
t. e. ymeet mesto stremlenye velyçyn¥
0
2 2
n
s s ds∫ ( ) ( )σ η ψ η( ) ( )
k beskoneçnosty pry n → + ∞ s veroqtnost\g 1.
Yz (37), prymenqq formulu Yto y yntehryruq, takΩe ymeem
ε σ η ψ η σ
ε
0
1
2 2
0
2
/
∫ ( ) ( ) −[ ]( ) ( )s s ds ds =
= ε η
ε
ηU U
1
0
−
( ) – ε σ η ηε
0
1/
∫ ( ) ( )
( )
( )
( )s
dU s
ds
dW s .
Otsgda v sylu toho, çto pry v¥polnenyy uslovyq (36)
M η
ε
1
+ | η |0 ≤ | η0 | +
0
1
∫ x x d xρ( ) + K l
–
δ
/
ε
,
a
sup ( )
0 1≤ ≤x
xψ = D2 < + ∞,
poluçaem
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
452 B. V. BONDAREV, E. E. KOVTUN
M U Uε η
ε
η1
0
−
( ) ≤ ε D2 η ρ δ ε
0
0
1
+ +∫ − /x x d x Kl( )
y
M s
dU s
ds
dW sε σ η ηε
0
1/
∫ ( ) ( )
( )
( )
( ) ≤ ε
γ
D2
,
M s s ds dsε σ η ψ η σ
ε
0
1
2 2
0
2
/
∫ ( ) ( ) −[ ]( ) ( ) = (46)
= εε η ρ δ εD x x d x Kl2 0
0
1
+ +∫ − /( ) + ε
γ
D2
.
Teorema 4. Pust\ koπffycyent¥ uravnenyq (35) udovletvorqgt uslovyqm
A1
), B1
), funkcyq f ( x ) neprer¥vnaq, 1-peryodyçeskaq. Tohda esly
0
1
∫ ψ( )x d x = 0,
0
1
∫ ψ( )x d x = 0,
to spravedlyva ocenka
sup ( )
0 1 0
0
≤ ≤
/
/∫ ( ) −[ ] −
t
t
tM f s f ds Wε η σ ε
ε
ε ≤
≤ D x x d x Kl2 0
0
1 1 2
ε η ρ δ ε+ +∫ − /
/
( ) + ε
γ
4 2
4
D
+ ε2 D1 , (47)
hde
0
1
2 2∫ σ ψ ρ( ) ( ) ( )x x x dt = σ0
2
,
funkcyy ψ( )x y ψ ( x ) v¥pysan¥ v sootnoßenyqx (40) y (45), postoqnn¥e
D1 , D2 opredelen¥ v (41), (45),
f =
0
1
∫ f x x dx( ) ( )ρ , σ0
2 =
0
1
2 2∫ σ ψ ρ( ) ( ) ( )x x x dt ,
ρ ( x ) opredeleno v (38) lybo v (39).
Spravedlyvost\ ocenky (47) sleduet yz ocenok (42), (46) y teorem¥ 1.
Pust\
ξ
εε
t
= ξ0 +
0
t
a
s
f
s
ds∫
ξ
ε
η
εε , (48)
| a ( x ) | ≤ C < + ∞, dlq lgboho x opredelena funkcyq g ( x ) y obratnaq k nej
g
–
1
( x ):
g ( x ) =
0
x
dy
a y∫ ( )
, Yε ( t ) = g
tξ
εε
, ξ
εε
t
= g
–
1
( Yε ( t ) ).
Yz (48) sleduet
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
OCENKY SKOROSTY SXODYMOSTY V OBÁKNOVENNÁX … 453
d Yε ( t ) = f
t
dt
ε
yly
Yε ( t ) = g ( ξ0 ) + ε η
ε
0
t
f s ds
/
∫ ( )( ) . (49)
Narqdu s (49) rassmotrym process
Y0 ( t ) = g ( ξ0 ) +
0
t
f ds∫ , (50)
a takΩe uravnenye
ξ0 ( t ) = ξ0 +
0
0
t
a s f ds∫ ( )ξ ( ) .
Netrudno zametyt\, çto
Y0 ( t ) = g ( ξ0 ( t ) ), ξ0 ( t ) = g
–
1
( Y0 ( t ) ).
Yz (49) y (50) v sylu (43) sleduet ocenka
sup ( ) ( )
0
0
≤ ≤
−
t T
M Y t Y tε ≤ sup ( )
0 0 0≤ ≤
/
∫ ∫( ) −
t T
t t
M f s ds f dsε η
ε
≤
≤ sup ( )
0 0≤ ≤
/
∫ ( ) −[ ]
t T
t
M f s f dsε η
ε
≤ ε ε
γ
2 1 1D D
T+
.
Dalee, oçevydno, çto ymeet mesto ocenka skorosty sxodymosty k usrednennomu
uravnenyg
sup ( )
0
0
≤ ≤
−
t T
M
t
tξ
ε
ξε = sup ( ) ( )
0
1
0
1
≤ ≤
− −( ) − ( )
t T
M g Y t g Y tε ≤
≤ C M Y t Y t
t T
sup ( ) ( )
0
0
≤ ≤
−ε ≤ C D D
Tε ε
γ
2 1 1+
.
Yzvestno, çto v stoxastyçeskom varyante pryncypa usrednenyq N. N. Boholg-
bova osoboe znaçenye ymeet vtoroe pryblyΩenye [11 – 13]. Pry dovol\no ßyro-
kyx ohranyçenyqx pokazano, çto normyrovannaq raznost\
ξ ε ξ
ε
ε t t/( ) − 0( )
v sla-
bom sm¥sle sxodytsq k nekotoromu haussovskomu processu. Predstavlqet yn-
teres ocenka skorosty sxodymosty v takoj procedure. Zapyßem yntehral\noe
sootnoßenye dlq ζε ( t ). Netrudno ubedyt\sq v tom, çto ymeet mesto ravenstvo
ξ
εε
t
– ξ0 ( t ) = g Y t− ( )1
ε ( ) – g Y t− ( )1
0( ) =
= a g Y t Y t Y t Y t Y t− ( ) + −[ ]( ) −[ ]1
0 0 0( ) ( ) ( ) ( ) ( )θ ε ε ,
(51)
ξ ε ξ
ε
ε t t/( ) − 0( )
=
Y t Y t
a g Y tε
ε
( ) ( )
( )
−[ ] ( )−0 1
0 +
+
Y t Y t
a g Y t Y t Y t a g Y tε
εε
θ( ) ( )
( ) ( ) ( ) ( )
−[ ] ( ) + −[ ]( ) − ( )[ ]− −0 1
0 0
1
0 .
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
454 B. V. BONDAREV, E. E. KOVTUN
Dalee,
Y t Y t
a g Y t Y t Y t a g Y tε
εε
θ( ) ( )
( ) ( ) ( ) ( )
−[ ] ( ) + −[ ]( ) − ( )[ ]− −0 1
0 0
1
0 ≤
≤ LC
Y t Y tε
ε
( ) ( )−[ ]0
2
.
Narqdu s (51) rassmotrym ravenstvo
Vε ( t ) = σ ε ξε0 0W a tt / ( )( ) . (52)
Yz (51) y (52) sleduet
sup
( )
( )
0
0
≤ ≤
/( ) − −
t T
M
t t
V t
ξ ε ξ
ε
ε
ε =
= sup
( ) ( )
( )
0
0
0 0
≤ ≤
−[ ] − ( )/
t T
tM
Y t Y t
W a tε
εε
σ ε ξ + LC M
Y t Y t
t T
sup
( ) ( )
0
0
2
≤ ≤
−[ ]ε
ε
.
(53)
Poskol\ku
sup ( )
0 1≤ ≤x
xψ = D1
, sup ( )
0 1≤ ≤x
xψ = D2
, f =
0
1
∫ f x x dx( ) ( )ρ ,
a
ε η
ε
0
t
f s f ds
/
∫ ( ) −[ ]( ) =
= ε η
ε
ηU
t
U
−
( )0 – ε σ η ηε
0
t
s
dU s
dx
dW s
/
∫ ( ) ( )
( )
( )
( ), (54)
y v sylu ohranyçennosty σ
2
( x ),
dU x
dx
( )
ymeem
sup ( )
( )
( )
0 0
2
≤ ≤
/
∫ ( ) ( )
t T
t
M s
dU s
dx
dW sε σ η ηε
≤ ε
γ
T D1
2 1
, (55)
to yz (54) y (55) poluçaem
sup ( )
0 0
2
≤ ≤
/
∫ ( ) −[ ]
t T
t
M f s f dsε η
ε
≤ ε
γ
2
1
2
1
24
1
D TD+
. (56)
Yz (53) y (56) sleduet
sup
( )
( )
0
0
≤ ≤
/( ) − −
t T
M
t t
V t
ξ ε ξ
ε
ε
ε ≤
≤ sup
( ) ( )
0
0
0
≤ ≤
−[ ] − /
t T
tM
Y t Y t
W Cε
εε
σ ε + LC D TDε
γ
2
1
2
1
24
1+
. (57)
Yz pryvedenn¥x v¥kladok sleduet takaq teorema.
Teorema 5. Pust\ dlq koπffycyentov uravnenyj (34) y (35) v¥polnen¥
uslovyq A1
), B1
),
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
OCENKY SKOROSTY SXODYMOSTY V OBÁKNOVENNÁX … 455
f =
0
1
∫ f x x dx( ) ( )ρ ,
0
1
2 2∫ σ ψ ρ( ) ( ) ( )x x x dt = σ0
2
,
funkcyy ψ( )x y ψ ( x ) v¥pysan¥ v sootnoßenyqx (44) y (40),
0
1
∫ ψ( )x dx = 0,
0
1
∫ ψ( )x dx = 0,
postoqnn¥e D1
, D2
opredelen¥ sootnoßenyqmy (41), (44), ρ ( x ) opredeleno
v (38). Tohda spravedlyva ocenka
sup
( )
( )
0
0
0 0
≤ ≤
/( ) − − ( )/
t T
tM
t t
W a t
ξ ε ξ
ε
σ ε ξε
ε ≤
≤ ε ε4
1C C( ) + LC D TDε
γ
2
1
2
1
24
1+
+
+ D x x d x Kl2 0
0
1 1 2
ε η ρ δ ε+ +∫ − /
/
( ) + ε
γ
4 2
4
D
+ ε2 1D , (58)
hde nesluçajn¥j process ξ0 ( t ) qvlqetsq reßenyem uravnenyq
ξ0 ( t ) = ξ0 +
0
0
t
a s f ds∫ ( )ξ ( ) .
Neravenstvo (58) sleduet yz (57) s uçetom (47).
V zaklgçenye otmetym, çto poluçenn¥j rezul\tat ne protyvoreçyt yzvest-
nomu faktu [13], v sylu kotoroho normyrovannaq raznost\
ζε ( t ) =
ξ ε ξ
ε
ε t t/( ) − 0( )
slabo sxodytsq pry ε → 0 k nekotoromu haussovskomu processu, stoxastyçes-
kyj dyfferencyal kotoroho ymeet vyd
d ζ ( t ) = γ ( t ) ζ ( t ) d t + α ( t ) d Wt , ζ ( 0 ) = 0. (59)
Dejstvytel\no, pust\
Vε ( t ) = σ ε ξε0 0W a tt / ( )( ) ,
tohda
d Vε ( t ) = σ ε ξε0 0W da tt / ( )( ) + a t d Wtξ σ ε ε0 0( )( ) / =
= ′ ( ) ( ) /a t a t f W dtx tξ ξ σ ε ε0 0 0( ) ( ) + a t d Wtξ σ ε ε0 0( )( ) / ,
otkuda
d Vε ( t ) = ′ ( )a t f V t dtx ξ ε0( ) ( ) + σ ξ ε ε0 0a t d Wt( )( ) / , Vε ( 0 ) = 0,
t. e. (59) ymeet mesto.
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
456 B. V. BONDAREV, E. E. KOVTUN
Kak otmeçalos\ v [12, c. 303], v sluçae, esly f = 0, za vremq 0,
T
ε
pro-
cess ξ ε ( t ) ne otojdet na zametnoe rasstoqnye ot naçal\noho poloΩenyq. Oka-
z¥vaetsq, çto v πtom sluçae peremewenyq porqdka 1 proysxodqt za vremenn¥e
ynterval¥ porqdka 1 / ε
2
. Po-vydymomu, na πtot fakt vperv¥e b¥lo obraweno
vnymanye v rabote R. L. Stratonovyça [13]. Tam Ωe na fyzyçeskom urovne stro-
hosty b¥lo ustanovleno, çto semejstvo processov ξ
εε
t
2
pry nekotor¥x us-
lovyqx sxodytsq k nekotoromu dyffuzyonnomu processu, y v¥çyslen¥ xarak-
terystyky predel\noho processa. Strohoe obosnovanye πtoho utverΩdenyq b¥-
lo dano R. Z. Xas\mynskym [14]. Pry suwestvenno menee ohranyçytel\n¥x pred-
poloΩenyqx dokazatel\stvo pryvedeno v rabote A. N. Borodyna [15].
Ytak, pust\ f = 0, tohda reßenye predel\noho uravnenyq ymeet vyd ξ0 ( t ) =
= ξ0 . Yz uravnenyq (34) sleduet
ξ
εε
t
2
= ξ0 +
0
2 2
1
t
a
s
f
s
d s∫
ξ
ε ε
η
εε . (60)
Pust\ | a ( x ) | ≤ C̃ < + ∞. PredpoloΩym takΩe, çto dlq lgboho x opredelena
funkcyq g ( x ) y obratnaq k nej g
–
1
( x ):
g ( x ) =
0
x
d y
a y∫ ( )
, Yε ( t ) = g
tξ
εε 2
, ξ
εε
t
2
= g Y t− ( )1
ε ( ) .
Yz (60) sleduet
d Yε ( t ) =
1
2ε ε
f
t
dt
,
yly
Yε ( t ) = g ( ξ0 ) + ε η
ε
0
2t
f s ds
/
∫ ( )( ) . (61)
Narqdu s (61) rassmotrym
Y t0
ε ( ) = g ( ξ0 ) + σ ε ε0 2W
t / (62)
y, sootvetstvenno,
ξε0( )t = g Y t− ( )1
0
ε ( ) . (63)
Tohda
sup ( )
0
2 0
≤ ≤
−
t T
M
t
tξ
ε
ξε
ε = sup ( ) ( )
0
1
0
1
≤ ≤
− −( ) − ( )
t T
M g Y t g Y tε
ε ≤
≤ ˜ sup ( ) ( )C M Y t Y t
t T0
0
≤ ≤
−ε
ε ≤ ε ε
γ
C̃ D D
T
2 1 1+
. (64)
Dyfferencyruq (62) s uçetom (63), ymeem
d tξε0( ) = dg Y t− ( )1
0
ε ( ) =
=
′ ( )( )
( )( ) ( )( )
−
−
−a g Y t
a g Y t
a g Y t dt
x
1
0
2 1
0
3 1
0
0
2
2
ε
ε
ε σ( )
( )
( ) + a g Y t W
t
− ( )( ) /
1
0 0 2
ε
εσ ε( ) .
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OCENKY SKOROSTY SXODYMOSTY V OBÁKNOVENNÁX … 457
Otsgda sleduet
d tξε0( ) = ′ ( ) ( )a t a t dtx ξ ξ σε ε
0 0
0
2
2
( ) ( ) + a t d W
t
ξ σ εε
ε0 0 2( )( ) / . (65)
Teorema 6. V uslovyqx teorem¥ 1 pry f = 0 sluçajn¥j process ξ
εε
t
2
pry ε → 0 sblyΩaetsq s dyffuzyonn¥m processom ξε0( )t , qvlqgwymsq re-
ßenyem (65), pryçem skorost\ sblyΩenyq zadaetsq neravenstvom (64).
1. ÇΩun K., Uyl\qms R. Vvedenye v stoxastyçeskoe yntehryrovanye. – M.: Myr, 1987. – 152 s.
2. Hyxman Y. Y., Skoroxod A. V. Stoxastyçeskye dyfferencyal\n¥e uravnenyq y yx prylo-
Ωenyq. – Kyev: Nauk. dumka, 1982. – 611 s.
3. Lypcer R. Í., Íyrqev A. N. Martynhal¥ y predel\n¥e teorem¥ dlq sluçajn¥x processov
// Ytohy nauky y texnyky. Sovr. probl. matematyky. Fundam. napravlenyq. – 1989. – 45 . –
S.T159 – 251.
4. Çykyn D. O. Funkcyonal\naq predel\naq teorema dlq stacyonarn¥x processov: martyn-
hal\n¥j podxod // Teoryq veroqtnostej y ee prymenenyq. – 1989. – 14, # 4. – S.T731 – 741.
5. Lypcer R. Í., Íyrqev A. N. Statystyka sluçajn¥x processov. – M.: Nauka, 1974. – 696 s.
6. Xas\mynskyj R. Z. Ustojçyvost\ system dyfferencyal\n¥x uravnenyj pry sluçajn¥x voz-
muwenyqx yx parametrov. – M.: Nauka, 1969. – 368 s.
7. Bers L., DΩon F., Íexter M. Uravnenyq s çastn¥my proyzvodn¥my. – M.: Myr, 1966. –
351Ts.
8. Veretennykov A. G. Ob ocenkax skorosty peremeßyvanyq dlq stoxastyçeskyx dyfferen-
cyal\n¥x uravnenyj // Teoryq veroqtnostej y ee prymenenyq. – 1987. – 32,T#T2. –
S.T299 – 308.
9. Hyxman Y. Y., Skoroxod A. V. Vvedenye v teoryg sluçajn¥x processov. – M.: Nauka, 1977. –
568 s.
10. Bensoussan A., Lions J.-L., Papanicolau G. Asymptotic analysis for periodic structures. – North-
Holland Publ. Comp., 1978. – 700 p.
11. Skoroxod A. V Asymptotyçeskye metod¥ teoryy stoxastyçeskyx dyfferencyal\n¥x urav-
nenyj. – Kyev: Nauk. dumka, 1987. – 328 s.
12. Ventcel\ A. D., Frejdlyn M. Y. Fluktuacyy v dynamyçeskyx systemax pod vozdejstvyem
mal¥x sluçajn¥x vozmuwenyj. – M.: Nauka, 1979. – 434 s.
13. Stratonovyç R. L. Uslovn¥e markovskye process¥ y yx prymenenyq v teoryy optymal\no-
ho upravlenyq. – M.: Yzd-vo Mosk. un-ta, 1966. – 319 s.
14. Xas\mynskyj R. Z. O sluçajn¥x processax, opredelqem¥x dyfferencyal\n¥my uravneny-
qmy s mal¥m parametrom // Teoryq veroqtnostej y ee prymenenyq. – 1966. – 11, # 2. –
S.T240 – 259.
15. Borodyn A. N. Predel\naq teorema dlq reßenyq dyfferencyal\n¥x uravnenyj so sluçaj-
noj pravoj çast\g // Tam Ωe. – 1977. – 22, # 3. – S. 498 – 511.
Poluçeno 03.11.2003
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
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| id | umjimathkievua-article-3611 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:45:45Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d5/8b48ef50588ce5544187214cab784ed5.pdf |
| spelling | umjimathkievua-article-36112020-03-18T20:00:05Z Estimates for the Rate of Convergence in Ordinary Differential Equations under the Action of Random Processes with Fast Time Оценки скорости сходимости в обыкновенных дифференциальных уравнениях, находящихся под воздействием случайных процессов с быстрым временем Bondarev, B. V. Kovtun, E. E. Бондарев, Б. В. Ковтун, Е. Е. Бондарев, Б. В. Ковтун, Е. Е. We study the procedure of averaging in the Cauchy problem for an ordinary differential equation perturbed by a certain Markov ergodic process. We establish several estimates for the rate of convergence of solutions of the original problem to solutions of the averaged one. Вивчається процедура усереднення у задачі Коші для звичайного диференціального рівняння, збуреного деяким ергодичним марковським процесом. Встановлено деякі оцінки швидкості збіжності розв'язків початкової задачі до розв'язків усередненої. Institute of Mathematics, NAS of Ukraine 2005-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3611 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 4 (2005); 435–457 Український математичний журнал; Том 57 № 4 (2005); 435–457 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3611/3953 https://umj.imath.kiev.ua/index.php/umj/article/view/3611/3954 Copyright (c) 2005 Bondarev B. V.; Kovtun E. E. |
| spellingShingle | Bondarev, B. V. Kovtun, E. E. Бондарев, Б. В. Ковтун, Е. Е. Бондарев, Б. В. Ковтун, Е. Е. Estimates for the Rate of Convergence in Ordinary Differential Equations under the Action of Random Processes with Fast Time |
| title | Estimates for the Rate of Convergence in Ordinary Differential Equations under the Action of Random Processes with Fast Time |
| title_alt | Оценки скорости сходимости в обыкновенных дифференциальных уравнениях, находящихся под воздействием случайных процессов с быстрым временем |
| title_full | Estimates for the Rate of Convergence in Ordinary Differential Equations under the Action of Random Processes with Fast Time |
| title_fullStr | Estimates for the Rate of Convergence in Ordinary Differential Equations under the Action of Random Processes with Fast Time |
| title_full_unstemmed | Estimates for the Rate of Convergence in Ordinary Differential Equations under the Action of Random Processes with Fast Time |
| title_short | Estimates for the Rate of Convergence in Ordinary Differential Equations under the Action of Random Processes with Fast Time |
| title_sort | estimates for the rate of convergence in ordinary differential equations under the action of random processes with fast time |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3611 |
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