On a Weak Solution of an Equation for an Evolution Flow with Interaction
We prove that a stochastic differential equation for an evolution flow with interaction whose coefficients do not satisfy the global Lipschitz condition has a weak solution.
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2005
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509774699823104 |
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| author | Karlikova, M. P. Карликова, М. П. Карликова, М. П. |
| author_facet | Karlikova, M. P. Карликова, М. П. Карликова, М. П. |
| author_sort | Karlikova, M. P. |
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| datestamp_date | 2020-03-18T20:01:15Z |
| description | We prove that a stochastic differential equation for an evolution flow with interaction whose coefficients do not satisfy the global Lipschitz condition has a weak solution. |
| first_indexed | 2026-03-24T02:46:27Z |
| format | Article |
| fulltext |
UDK 519.21
M. P. Karlykova (Yn-t matematyky NAN Ukrayn¥, Kyev)
O SLABOM REÍENYY URAVNENYQ
DLQ ∏VOLGCYONNOHO POTOKA
SO VZAYMODEJSTVYEM
We prove that a stochastic differential equation for an evolutionary flow with interaction whose
coefficients do not satisfy the global Lipschitz condition possesses a weak solution.
Dovedeno, wo stoxastyçne dyferencial\ne rivnqnnq dlq evolgcijnoho potoku zi vza[modi[g
z,koefici[ntamy, wo ne zadovol\nqgt\ hlobal\nu umovu Lipßycq, ma[ slabkyj rozv’qzok.
1. Vvedenye. V nastoqwej stat\e rassmatryvaetsq sledugwee uravnenye [1]
dlq potoka so vzaymodejstvyem:
d x ( u, t ) = a ( x ( u, t ), µ t ) d t + b ( x ( u, t ), µ t ) d w ( t ),
(1)
x ( u, 0 ) = u, t ≥ 0, u ∈ R
d
,
hde µ t = µ 0 ° x ( ⋅, t )
–
1
— sluçajnaq mera, opredelqemaq ravenstvom
µ t ( ∆ ) = µ 0 ( u : x ( u, t ) ∈ ∆ ), ∆ ∈ B ( R
d
),
dlq naçal\noj veroqtnostnoj mer¥ µ 0 . Zdes\ B ( R
d
) — borelevskaq σ-alheb-
ra v R
d
. ∏to uravnenye moΩno rassmatryvat\ kak opysanye dvyΩenyq kontynu-
al\noj system¥ çastyc v sluçajnoj srede, kohda traektoryy otdel\n¥x çastyc
zavysqt ot poloΩenyj ostal\n¥x çastyc.
Oboznaçym çerez � prostranstvo veroqtnostn¥x mer v R
d
y opredelym na
nem metryku Vasserßtejna
γ ( µ, ν ) = inf ( , )
( , )κ µ ν
κ
∈ ∫∫ −
+ −Q d
u
u
du d
R
v
v
v
1
.
Zdes\ Q ( µ, ν ) — mnoΩestvo veroqtnostn¥x mer v R
d × R
d
, ymegwyx µ y ν
svoymy proekcyqmy. Yzvestno [2], çto metryka γ sootvetstvuet slaboj sxody-
mosty y ( �, γ ) — polnoe separabel\noe metryçeskoe prostranstvo.
V [1] dokazan sledugwyj fakt.
Teorema 1. Pust\ dlq nekotoroho L > 0 y dlq lgb¥x u1 , u2 ∈ R
d
, ν1 ,
ν2 ∈ �
|| a ( u1 , ν1 ) – a ( u2, ν2 ) || + || b ( u1 , ν1 ) – b ( u2, ν2 ) || ≤ L u u1 2 1 2− +( )γ ν ν( , ) .
Tohda uravnenye (1) ymeet edynstvennoe syl\noe reßenye.
Dlq stoxastyçeskyx dyfferencyal\n¥x uravnenyj bez vzaymodejstvyq
(kohda a y b ne zavysqt ot µ ) suwestvuet druhoe obweprynqtoe uslovye su-
westvovanyq y edynstvennosty syl\noho reßenyq [3]:
suwestvugt C > 0 y posledovatel\nost\ { LN , N ≥ 1 } takye, çto
∀ u ∈ R
d : || a ( u ) || + || b ( u ) || ≤ C u1 +( ) ,
∀ u, v ∈ B ( 0, N ) : || a ( u ) – a ( v ) || + || b ( u ) – b ( v ) || ≤ LN || u – v ||.
V dannoj stat\e sdelana pop¥tka perenesty πto uslovye na sluçaj uravne-
nyj so vzaymodejstvyem. Pry πtom vaΩnug rol\ budet yhrat\ meroznaçn¥j
© M. P. KARLYKOVA, 2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7 895
896 M. P. KARLYKOVA
process { µ t , t ≥ 0 }, kotor¥j v sootvetstvyy s [1] budem naz¥vat\ πvolgcyon-
n¥m processom.
2. Slabaq kompaktnost\ semejstva πvolgcyonn¥x meroznaçn¥x pro-
cessov. PreΩde vseho ustanovym nekotor¥e svojstva reßenyj uravnenyq (1).
Lemma 1. Pust\ { a
α, b
α, α ∈ � } udovletvorqgt uslovyg
∃ C > 0 ∀ u ∈ R
d, µ ∈ � : ,|| a
α
( u, µ ) || + || b
α
( u, µ ) || ≤ C u1 +( ) ,
{ x
α, α ∈ � } udovletvorqgt uravnenyqm
d x
α
( u, t ) = a x u t dtt
α α αµ( , ),( ) + b x u t dw tt
α α αµ( , ), ( )( ) ,
x
α
( u, 0 ) = u, µα
t = µα
0 ° x
α
( ⋅, t )
–
1
,
pryçem semejstvo
µ αα
0 , ∈{ }� slabo kompaktno. Tohda:
∀ p > 1 ∃ Cp > 0 : ∀ t ∈ [ 0, 1 ]
1) E sup ( , )
0 1≤ ≤t
p
x u tα ≤ C up
p1 +( );
2) E sup ( , ) ( , )
t s t
p
x u t x u s
≤ ≤ +
−
δ
α α ≤ C up
p p1 2+( ) /δ .
Dokazatel\stvo. 1. Yspol\zuq neravenstvo Burkxol\dera [3], poluçaem
E sup ( , )
0≤ ≤s t
p
x u sα ≤
≤ ′ + ( ) + ( )
∫ ∫
≤ ≤
C u a x u s ds b x u dwp
p
t
s
p
s t
t p
E E
0 0 0
α α α α α
τ
αµ τ µ τ( , ), sup ( , ), ( ) ≤
≤ ′′ + +( ) +
−
+( )
∫ ∫
/
C u x u s ds
p
p
x u s dsp
p
t
p
p t p
E E
0 0
2
2
1
1
1α α( , ) ( , ) ≤
≤ ′′′ + +
∫
≤ ≤
C u x u dsp
p
t
s
p
0 0
1E sup ( , )
τ
α τ ,
otkuda sohlasno lemme Hronuolla
E sup ( , )
0
1
≤ ≤
+( )
s t
p
x u sα ≤ ′′′ +( ) ′′′
C u ep
p C tp1 .
Tohda
E sup ( , )
0 1≤ ≤t
p
x u tα ≤ C up
p1 +( ).
2. Analohyçno,
E sup ( , ) ( , )
t s t
p
x u t x u s
≤ ≤ +
−
δ
α α ≤
≤ ′ ( ) + ( )
−
+
≤ ≤ +
∫ ∫C a x u d b x u dwp
p
t
t
p
t s t t
s p
δ τ µ τ τ µ τ
δ
α α
τ
α
δ
α α
τ
α1 E E( , ), sup ( , ), ( ) ≤
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7
O SLABOM REÍENYY URAVNENYQ DLQ ∏VOLGCYONNOHO POTOKA … 897
≤ ′′ +( ) +
−
+( )
−
+ +
∫ ∫
/
C x u d
p
p
x u dp
p
t
t
p
p
t
t p
δ τ τ τ τ
δ
α
δ
α1 2
2
1
1
1E E( , ) ( , ) ≤
≤ ′′′ +( )/ −
+
∫C x u dp
p
t
t
p
δ τ τ
δ
α2 1 1 ( , ) ≤ C up
p p1 2+( ) /δ .
Lemma dokazana.
Nam potrebuetsq kryteryj slaboj kompaktnosty semejstva sluçajn¥x πle-
mentov v C ( [ 0, 1 ], � ) yz [1]. Rassmotrym dve posledovatel\nosty funkcyj
{ fn , n ≥ 1 } y { gN , n ≥ 1 } v Cb ( R
d
) takye, çto:
1) f1 ≡ 1 ∀ n ≥ 2 : fn ∈ C0 ( R
d
);
2) ∀ n ≥ 1 : max
R
d
fn ≤ 1;
3) ∀ ϕ ∈ C0 ( R
d
) : max
R
d
ϕ ≤ 1 ∃ f knk
, ≥{ }1 :
max
R
d k
fnϕ − → 0, k → ∞;
4) ∀ x ∈ R
d
, n ≥ 1 : 0 ≤ gn ( x ) ≤ 1;
5) ∀ x ∈ R
d
, || x || ≤ n : gn ( x ) = 0 ;
6) ∀ x ∈ R
d
, || x || ≥ n + 1 : gn ( x ) = 1.
Lemma 2 [1]. Semejstvo { ξα , α ∈ � } sluçajn¥x πlementov v C ( [ 0, 1 ],
� ) slabo kompaktno tohda y tol\ko tohda, kohda:
1) dlq kaΩdoho k ≥ 1 semejstvo sluçajn¥x processov ξ αα , ,fk ∈{ }�
slabo kompaktno v C ( [ 0, 1 ] );
2) semejstvo ξ αα , , ,g kk ∈ ≥{ }� 1 slabo kompaktno v C ( [ 0, 1 ] );
3) ∀ t ∈ [ 0, 1 ] ∀ ε > 0 :
sup ( ),
α
αξ ε
∈
>{ }
�
P t gk , k → ∞.
Zdes\ yspol\zovano oboznaçenye
〈 µ, f 〉 =
R
d
f u du∫ ( ) ( )µ .
Dalee budem v¥byrat\ fk udovletvorqgwymy uslovyg Lypßyca s postoqn-
n¥my Ak, gk — s obwej postoqnnoj A.
Pryvedenn¥j kryteryj pozvolqet ustanovyt\ slabug kompaktnost\ semej-
stva πvolgcyonn¥x processov, sootvetstvugwyx uravnenyg (1), v termynax eho
koπffycyentov.
Teorema 2. V uslovyqx lemm¥ 1 semejstvo meroznaçn¥x processov { µα
⋅ ,
α ∈ � } slabo kompaktno v C ( [ 0, 1 ], � ).
Dokazatel\stvo. Proverym uslovyq lemm¥ 2.
1. Dlq slaboj kompaktnosty semejstva sluçajn¥x processov { ξ
α, α ∈ � }
v C ( [ 0, 1 ] ) dostatoçno, çtob¥ [4]
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7
898 M. P. KARLYKOVA
sup ( )
α
αξ
∈
≥{ }
�
P 0 N → 0, N → ∞,
sup sup ( ) ( )
α δ
α α
δ
ξ ξ ε
∈ ≤ ≤ +
− >
�
1
P
t s t
t s → 0, δ → 0 +.
V rassmatryvaemom sluçae
P µα
0 , f Nk ≥{ } =
÷ µα
0 , f Nk ≥{ } → 0, N → ∞,
ravnomerno po α, tak kak
µα
0 , fk ≤ sup
R
d
fk ≤ 1.
Ocenym teper\ s pomow\g lemm¥ 2 dlq p > 2
E sup , ,
t s t
s k t k
p
f f
≤ ≤ +
−
δ
α αµ µ =
= E sup ( , ) ( , ) ( )
t s t
k k
p
d
f x u s f x u t du
≤ ≤ +
∫ ( ) − ( )( )
δ
α α αµ
R
0 ≤
≤ E EA x u t x u s duk
K t s t
p
k
∫
≤ ≤ +
−
sup ( , ) ( , ) ( )
δ
α α αµ0 ≤
≤ A C u duk
p
K
p
p p
k
∫ +( ) /1 2
0δ µα ( ) ≤ A uk
p
u K
p p
k
sup
∈
+( ) /1 2δ ,
hde Kk — kompakt, na kotorom sosredotoçena fk . Otsgda
sup sup , ,
α δ
α α
δ
µ µ ε
∈ ≤ ≤ +
− >
�
1
P
t s t
s k t kf f ≤
≤
1 1 2
δ ε
δ
A uk
p
u K
p
p
pk
sup ∈ +( ) / → 0, δ → 0.
Sledovatel\no,
µ αα
⋅ ∈{ }, ,fk � slabo kompaktn¥ v C ( [ 0, 1 ] ).
2. Ymeem µα
0 , gk ≤ 1, poπtomu pervoe uslovye slaboj kompaktnosty v
C ( [ 0, 1 ] ) dlq
µ αα
⋅ ∈ ≥{ }, , ,g kk � 1 v¥polneno. Dalee,
sup , ,
t s t
s k t kg g
≤ ≤ +
−
δ
α αµ µ ≤ A x u s x u t du
Kε
α α αµ∫ −( , ) ( , ) ( )0 +
ε
2
,
hde Kε — takoj kompakt, çto
∀ α ∈ � : µα
ε0 R
d K\( ) <
ε
2
.
Tohda, kak y v pred¥duwem punkte,
1
δ
µ µ ε
δ
α αP sup , ,
t s t
s k t kg g
≤ ≤ +
− >
≤
≤
1
20δ
µ ε
δ
α α α
ε
P sup ( , ) ( , ) ( )
t s t K
x u s x u t du
≤ ≤ +
∫ − >
→ 0, δ → 0,
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7
O SLABOM REÍENYY URAVNENYQ DLQ ∏VOLGCYONNOHO POTOKA … 899
ravnomerno po α. Sledovatel\no,
µ αα
⋅ ∈ ≥{ }, , ,g kk � 1 slabo kompaktno v
C ( [ 0, 1 ] ).
3. Proverym tret\e uslovye lemm¥ 2:
P µ εα
t kg, >{ } ≤ P µ εα α
0 u x u t k: ( , ) ≥( ) >{ } ≤
≤ P µ εα α
0 2
u u N x u t k: , ( , )≥ ≥( ) >{ } +
+ P µ εα α
0 2
u u N x u t k: , ( , )≤ ≥( ) >{ } ≤
≤ P µ εα
0 2
u u N: ≥( ) >{ } + P u N
x u t du
k
≤∫
>
α αµ ε( , ) ( )
2
0
2 2
≤
≤
÷ µ εα
0 2
u u N: ≥( ) >{ } +
C u du
k
u N2
2
0
2
1
≤∫ +( )µ
ε
α ( )
.
Poslednee neravenstvo sleduet yz lemm¥ 1.
V¥byraq N tak, çtob¥ dlq vsex α ∈ � v¥polnqlos\
µα
0 u u N: ≥( ) <
ε
2
,
poluçaem
sup ,
α
αµ ε
∈
>{ }
�
P t kg → 0, k → ∞.
Teorema dokazana.
Teorema 3. Pust\ v¥polnen¥ uslovyq lemm¥ 1. Tohda dlq lgboho nabora
{ u1 , u2 , … , uN } ⊂ R
d semejstvo
x u x u x uN
α α α α( , ), ( , ), , ( , ),1 2⋅ ⋅ … ⋅ ∈{ }�
slabo kompaktno v C ( [ 0, 1 ], R
d
N
).
Dokazatel\stvo sleduet yz ocenok lemm¥ 1 analohyçno dokazatel\stvu
teorem¥ 2.
3. Suwestvovanye slaboho reßenyq. Pust\ a y b udovletvorqgt uslo-
vyg A): ∃ C > 0, { LN , n ≥ 1 },
∀ u ∈ R
d, µ ∈ � : || a ( u, µ ) || + || b ( u, µ ) || ≤ C u1 +( ) ,
∀ u, v ∈ R
d, µ, ν ∈ � : || a ( u, µ ) – a ( v, ν ) || + || b ( u, µ ) – b ( v, ν ) || ≤
≤ L uN − +( )v γ µ ν( , ) .
Lemma 3. Pust\ a, b udovletvorqgt uslovyg A). Pust\, krome toho,
{ x
α, α ∈ � } udovletvorqgt uravnenyg (1) s naçal\n¥my meramy µα
0 , pryçem
semejstvo µ αα
0 , ∈{ }� slabo kompaktno. Tohda dlq lgboho kompakta
K ⊂ R
d
∀ ε > 0 ∃ C ( ε ) > 0 ∀ u, v ∈ K, t ∈ [ 0, 1 ] :
E x u t x tα α( , ) ( , )− v ≤ ε + C u( )ε − v .
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7
900 M. P. KARLYKOVA
Dokazatel\stvo. Pust\ N > 0. Oboznaçym σα
u = inf t x u t≥{ 0: ( , )α ≥
≥ N u1 +( )}, σ
α
u,v = σα
u ∧ σα
v . Tohda v sylu lemm¥ 1
P σα
u,v ≤{ }1 ≤ P sup ( , )
0 1
1
≤ ≤
≥ +( )
t
x u t N uα +
+
P sup ( , )
0 1
1
≤ ≤
≥ +( )
t
x t Nα v v ≤
C u
N u
2
2
2 2
1
1
+( )
+( )
+
C
N
2
2
2 2
1
1
+( )
+( )
v
v
≤
2 2
2
C
N
.
Dalee
E x u t x tu u
α α α ασ σ, ,, ,∧( ) − ∧( )v vv
2
≤
≤
3 Eu a x u s a x s ds
t
s s
u
− +
( ) − ( )
∧
∫v v
v
2
0
2
σ
α α α α
α
µ µ
,
( , ), ( , ), +
+
E
0
2t
s s
u
b x u s b x s dw s
∧
∫ ( ) − ( )( )
σ
α α α α
α
µ µ
,
( , ), ( , ), ( )
v
v ≤
≤ 3 22 2
0
2
u L x u s x s dsN
t
u u− + ∧ − ∧( )
∫v vv vE α α α ασ σ( , ) ,, , ,
otkuda sohlasno lemme Hronuolla
E x u t x tu u
α α α ασ σ, ,, ,∧( ) − ∧( )v vv
2
≤ C N u eC N t( ) ( )− v 2
.
Tohda
E x u t x tα α( , ) ( , )− v ≤
E x u t x tu u u
α α α α ασ σ σ, ,, , ,∧( ) − ∧( ) >{ }v v vv ÷ 1 +
+ E x u t x t u
α α ασ( , ) ( , ) ,− ≤{ }v v÷ 1 ≤
≤ C N u eC N t( ) ( )− v 2 +
C u
C
N
2
2 2 2
22
2+ +( )v ≤
≤ C N e uC N( ) ( ) − v +
C
N
u1 + +( )v .
V¥byraq N tak, çtob¥ v¥polnqlos\
C
N
u1 + +( )v < ε,
poluçaem trebuemoe neravenstvo.
Lemma dokazana.
Lemma 4. Pust\ µ0 =
k
N
k uc
k=∑ 1
δ , hde ck > 0, k = 1, … , N,
k
N
kc=∑ 1
= 1.
Pust\, krome toho, a, b udovletvorqgt uslovyg A). Tohda uravnenye (1)
ymeet edynstvennoe reßenye.
Dokazatel\stvo. Uravnenye (1) v dannom sluçae ymeet vyd
d x u tk( , ) = a x u t c dtk
i
N
i x u ti
( , ), ( , )
=
∑
1
δ + b x u t c dw tk
i
N
i x u ti
( , ), ( )( , )
=
∑
1
δ , (2)
k = 1, … , N;
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7
O SLABOM REÍENYY URAVNENYQ DLQ ∏VOLGCYONNOHO POTOKA … 901
d x u t( , ) = a x u t c dt
i
N
i x u ti
( , ), ( , )
=
∑
1
δ + b x u t c dw t
i
N
i x u ti
( , ), ( )( , )
=
∑
1
δ , u ∈ R
d
.
(3)
Na x ( uk , t ), k = 1, … , N, ymeem systemu uravnenyj (2), koπffycyent¥ ko-
toroj
˜ ( , , , )a u u uk N1 2 … = a u ck
i
N
i ui
,
=
∑
1
δ ,
˜ ( , , , )b u u uk N1 2 … = b u ck
i
N
i ui
,
=
∑
1
δ
udovletvorqgt uslovyg lynejnoho rosta y lokal\nomu uslovyg Lypßyca.
Sledovatel\no, πta systema ymeet edynstvennoe reßenye. Krome toho, yz ne-
prer¥vnosty x ( ui , ⋅ ) sleduet, çto
µ⋅ =
i
N
i x uc
i
=
⋅∑
1
δ ( , ) ∈ C ( [ 0, 1 ], � ).
Dlq ostal\n¥x u ∉ { u1 , … , un } koπffycyent¥ uravnenyq (3) takΩe budut
udovletvorqt\ lokal\nomu uslovyg Lypßyca y uslovyg lynejnoho rosta, a
krome toho, budut neprer¥vn¥ po t. Sledovatel\no, (3) takΩe budet ymet\
reßenye.
Lemma dokazana.
Lemma 5. Pust\ � 1 , �2 , … , �N — poln¥e metryçeskye prostranstva.
Semejstvo sluçajn¥x πlementov
ξ ξ ξ αα α α
1 2, , , ,…( ) ∈{ }N � slabo kompaktno
v,,,,�1 × �2 × … × �N tohda y tol\ko tohda, kohda
ξ αα
1 , ∈{ }� , …
… ,
ξ αα
N , ∈{ }� slabo kompaktn¥ v �1 , … , �N sootvetstvenno.
Dokazatel\stvo standartno y poπtomu ne pryvodytsq.
Analohyçno sluçag uravnenyq bez vzaymodejstvyq vvedem sledugwee
opredelenye.
Opredelenye. Uravnenye ( 1 ) ymeet slaboe reßenye, esly najdutsq:
veroqtnostnoe prostranstvo ( Ω , F, P ), neub¥vagwee semejstvo σ-alhebr
Ft t, ,∈[ ]{ }0 1 , F t-sohlasovann¥j neprer¥vn¥j po ( u, t ) sluçajn¥j process
˜( , )x u t y vynerovskyj Ft-martynhal w̃ takye, çto v¥polneno (1).
Teorema 4. Pust\ koπffycyent¥ a, b udovletvorqgt uslovyg A ). Toh-
da uravnenye (1) ymeet slaboe reßenye.
Dokazatel\stvo. Pust\ µ0
n
— dyskretn¥e mer¥ y µ0
n ⇒ µ0 , n → ∞ . So-
hlasno lemme 4 dlq nyx suwestvugt reßenyq uravnenyq (1). Oboznaçym yx
x
n
( ⋅, t ). V sylu teorem¥ 2 semejstvo meroznaçn¥x processov µ⋅ ≥{ }n n, 1 slabo
kompaktno v C ( [ 0, 1 ], � ), a sohlasno teoreme 3 semejstvo x u x un n( , ), ( , )1 2⋅ ⋅{ , …
… , x u nn
N( , ),⋅ ≥ }1 slabo kompaktno v C ( [ 0, 1 ], R
d
N
) dlq lgb¥x { u1 , u2 , …
… , uN } ⊂ R
d
. Pust\ (ne bolee çem sçetnoe) mnoΩestvo U — mnoΩestvo toçek
v R
d
, v kotor¥x sosredotoçen¥ µ0
n
, n ≥ 1. Tohda sohlasno lemme 5 dlq lgboho
nabora { u1 , u2 , … , uN } ⊂ U semejstvo µ⋅ ⋅ ⋅ … ⋅{ n n n n
Nx u x u x u, ( , ), ( , ), , ( , )1 2 ,
w n⋅ ≥ }, 1 slabo kompaktno v C ( [ 0, 1 ], � × R
d
N× R
d
). Yspol\zuq dyaho-
nal\n¥j metod Kantora, moΩno v¥brat\ podposledovatel\nost\ yndeksov nk ,
k ≥ 1, takug, çto
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7
902 M. P. KARLYKOVA
µ⋅ ⋅⋅ … ⋅( ) ≥{ }n n n
N
k k kx u x u w n, ( , ), , ( , ), ,1 1
slabo sxodytsq dlq lgboho nabora { u1 , u2 , … , uN } ⊂ U. Oboznaçym ee tak Ωe,
kak ysxodnug. Rassmotrym predel πtoj posledovatel\nosty
µ t tt y u t u U t w t, , , ˜( , ), , , , ˜ , ,∈[ ] ∈ ∈[ ] ∈[ ]( )0 1 0 1 0 1 .
PreΩde vseho pokaΩem, çto w̃⋅ — vynerovskyj martynhal otnosytel\no
potoka σ -alhebr Ft = σ µ˜ ( ), , ˜( , ), ,w s y u s u U s ts ∈ ≤{ }. Dlq lgboj ϕ ∈
∈ Cb ( R
d
k
(
1
+
m
) × �
k
) y τ1 , τ2 , … , τk ≤ s, u1 , … , um ∈ U rassmotrym
Ee w w y u y ui w t w s
k m kk
λ
τ τϕ τ τ µ µ τ τ˜ ( ) ˜ ( ) ˜ ( ), , ˜ ( ), , , , ˜( , ), , ˜( , )−( ) … … …( )1 1 11
=
= lim ( ), , ( ), , , , ( , ), , ( , )( ) ( )
n
i w t w s
k
n n n n
m ke w w x u x u
k→∞
−( ) … … …( )E λ
τ τϕ τ τ µ µ τ τ1 1 11
=
= e w w x u x ut s
n k
n n n n
m kk
− −
→∞
/ … … …( )λ
τ τϕ τ τ µ µ τ τ
2
1
2
1 1 1
( ) lim ( ), , ( ), , , , ( , ), , ( , )E =
= e w w y u y ut s
k m kk
− − / … … …( )λ
τ τϕ τ τ µ µ τ τ
2
1
2
1 1 1
( ) ˜ ( ), , ˜ ( ), , , , ˜( , ), , ˜( , )E , s ≤ t,
otkuda sleduet, çto w̃ — vynerovskyj Ft - martynhal.
Rassmotrym uravnenye dlq x̃ :
dx u t˜( , ) = a x u t dtt˜( , ), µ( ) + b x u t dw tt˜( , ), ˜ ( )µ( ) ,
(4)
˜( , )x u 0 = u.
Otmetym, çto πto uravnenye bez vzaymodejstvyq, tak kak process µt{ } sejças
yzvesten. V sylu uslovyq A) ono ymeet edynstvennoe reßenye, neprer¥vnoe po
u, t [3]. Dlq u ∈ U v¥raΩenyq
x
n
( u, t ) – u –
0
t
n
s
na x u s ds∫ ( )( , ), µ –
0
t
n
s
nb x u s dw s∫ ( )( , ), ( )µ
slabo sxodqtsq k
˜( , )y u t – u –
0
t
sa y u s ds∫ ( )˜( , ), µ –
0
t
sb y u s dw s∫ ( )˜( , ), ˜ ( )µ .
S druhoj storon¥, πty v¥raΩenyq ravn¥ 0, poπtomu v sylu edynstvennosty re-
ßenyq (4) dlq u ∈ U ˜( , )y u t = ˜( , )x u t p. n.
Zametym dalee, çto tak kak µ0
n ⇒ µ0 , n → ∞, to
γ µ µt x t, ˜( , )0
1� ⋅( )− =
lim , ˜( , )
k
t
k x t
→∞
−⋅( )γ µ µ0
1�
s veroqtnost\g 1. V svog oçered\ yz-za sovmestnoj slaboj sxodymosty µ
n
y x
n
na πlementax yz U
E γ µ µt
k x t, ˜( , )0
1� ⋅( )− = E γ µ µt
k y t, ˜( , )0
1� ⋅( )− = lim , ( , )
k
t
n k nx t
→∞
−⋅( )E γ µ µ0
1� .
Ocenym
E γ µ µt
n k nx t, ( , )0
1� ⋅( )− ≤ ε +
E
K
n nx u t x t du d
ε
κ
2
∫∫ −( , ) ( , ) ( , )v v ,
hde Kε — takoj kompakt, çto
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7
O SLABOM REÍENYY URAVNENYQ DLQ ∏VOLGCYONNOHO POTOKA … 903
∀ n : µ0
n d KR \( ) <
ε
2
,
κ ∈ Q n kµ µ0 0,( ) . V¥berem κ tak, çtob¥
Rd
u
u
du d∫∫ −
+ −
v
v
v
1
κ( , ) ≤ 2 0 0γ µ µn k,( ).
Tohda sohlasno lemme 3
E γ µ µt
n k nx t, ( , )0
1� ⋅( )− ≤ 2 ε +
2 1 0 0C u
u K
n k( ) sup ,
,
ε γ µ µ
εv
v
∈
+ −( ) ( ).
V¥byraq snaçala ε, zatem k, a zatem n, poluçaem
E γ µ µt x t, ˜( , )0
1� ⋅( )− = 0,
otkuda sleduet, çto µt = µ0
1� ˜( , )x t⋅ −
, t. e. x̃ qvlqetsq slab¥m reßenyem
uravnenyq (1).
Teorema dokazana.
1. Dorogovtsev A. A., Kotelenez P. Stochastic flows with interaction and random measures. – Kluwer
Acad. Publ., 2004. – 186 p.
2. Dudley R. M. Real analysis and probability. – Cambridge Univ. Press, 2002. – 566 p.
3. Kunita H. Stochastic flows and stochastic differential equations. – Cambridge Univ. Press, 1997. –
360 p.
4. Byllynhsly P. Sxodymost\ veroqtnostn¥x mer. – M.: Nauka, 1977. – 352 s.
Poluçeno 30.04.2004
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 7
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| id | umjimathkievua-article-3651 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:46:27Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/44/d9f9efca7c665001e5d958a640ca9844.pdf |
| spelling | umjimathkievua-article-36512020-03-18T20:01:15Z On a Weak Solution of an Equation for an Evolution Flow with Interaction О слабом решении уравнения для эволюционного потока со взаимодействием Karlikova, M. P. Карликова, М. П. Карликова, М. П. We prove that a stochastic differential equation for an evolution flow with interaction whose coefficients do not satisfy the global Lipschitz condition has a weak solution. Доведено, що стохастичне диференціальне рівняння для еволюційного потоку зі взаємодією з коефіцієнтами, що не задовольняють глобальну умову Ліпшиця, має слабкий розв'язок. Institute of Mathematics, NAS of Ukraine 2005-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3651 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 7 (2005); 895–903 Український математичний журнал; Том 57 № 7 (2005); 895–903 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3651/4032 https://umj.imath.kiev.ua/index.php/umj/article/view/3651/4033 Copyright (c) 2005 Karlikova M. P. |
| spellingShingle | Karlikova, M. P. Карликова, М. П. Карликова, М. П. On a Weak Solution of an Equation for an Evolution Flow with Interaction |
| title | On a Weak Solution of an Equation for an Evolution Flow with Interaction |
| title_alt | О слабом решении уравнения для эволюционного потока со взаимодействием |
| title_full | On a Weak Solution of an Equation for an Evolution Flow with Interaction |
| title_fullStr | On a Weak Solution of an Equation for an Evolution Flow with Interaction |
| title_full_unstemmed | On a Weak Solution of an Equation for an Evolution Flow with Interaction |
| title_short | On a Weak Solution of an Equation for an Evolution Flow with Interaction |
| title_sort | on a weak solution of an equation for an evolution flow with interaction |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3651 |
| work_keys_str_mv | AT karlikovamp onaweaksolutionofanequationforanevolutionflowwithinteraction AT karlikovamp onaweaksolutionofanequationforanevolutionflowwithinteraction AT karlikovamp onaweaksolutionofanequationforanevolutionflowwithinteraction AT karlikovamp oslabomrešeniiuravneniâdlâévolûcionnogopotokasovzaimodejstviem AT karlikovamp oslabomrešeniiuravneniâdlâévolûcionnogopotokasovzaimodejstviem AT karlikovamp oslabomrešeniiuravneniâdlâévolûcionnogopotokasovzaimodejstviem |