Convergence of solutions of stochastic differential equations to the Arratia flow
We consider the solution $x_{\varepsilon}$ of the equation $$dx_{\varepsilon}(u,t) = \int\limits_\mathbb{R}\varphi_{\varepsilon}(x_{\varepsilon}(u,t) - r) W(dr,dt), $$ $$x_{\varepsilon}(u,0) = u,$$ where $W$ is a Wiener sheet on $\mathbb{R} \times [0; 1].$ For the case where $\varphi_{\varepsilon}^...
Збережено в:
| Дата: | 2008 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2008
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3266 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509322024321024 |
|---|---|
| author | Malovichko, T. V. Маловичко, Т. В. Маловичко, Т. В. |
| author_facet | Malovichko, T. V. Маловичко, Т. В. Маловичко, Т. В. |
| author_sort | Malovichko, T. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:49:31Z |
| description | We consider the solution $x_{\varepsilon}$ of the equation
$$dx_{\varepsilon}(u,t) = \int\limits_\mathbb{R}\varphi_{\varepsilon}(x_{\varepsilon}(u,t) - r) W(dr,dt), $$
$$x_{\varepsilon}(u,0) = u,$$
where $W$ is a Wiener sheet on $\mathbb{R} \times [0; 1].$
For the case where $\varphi_{\varepsilon}^2$ converges to
$p \delta(\cdot - a_1) + q \delta(\cdot - a_2),$ i.e., where a boundary function describing the influence
of a random medium is singular more than at one point, we prove that the weak convergence of
$\left(x_{\varepsilon}(u_1, \cdot),...,x_{\varepsilon}(u_d, \cdot) \right)$ to
$\left(X(u_1, \cdot),...,X(u_d, \cdot) \right)$ takes place as $\varepsilon\rightarrow0_+$ (here, $X$ is the Arratia flow). |
| first_indexed | 2026-03-24T02:39:15Z |
| format | Article |
| fulltext |
UDK 519.21
T. V. Malovyçko (Nac. texn. un-t Ukrayn¥ „KPY”, Kyev)
O SXODYMOSTY REÍENYJ STOXASTYÇESKYX
DYFFERENCYAL|NÁX URAVNENYJ K POTOKU ARRAT|Q
We consider the solution xε of the equation
dx u tε ( , ) =
R
∫ ( )ϕε εx u t r W dr dt( , ) – ( , ),
x uε ( , )0 = u,
where W is a Wiener sheet on R × 0 1;[ ]. For the case where ϕε
2
converges to p aδ( – )⋅ 1 +
+ q aδ( – )⋅ 2 , i.e., where a boundary function describing the influence of a random medium is singular
more than at one point, we prove that the weak convergence of x uε ( , )1 ⋅( , … , x udε ( , )⋅ ) to
X u( , )1 ⋅( , … , X ud( , )⋅ ) takes place as ε → 0+ (here, X is the Arratia flow).
Rozhlqnuto rozv’qzok xε rivnqnnq
dx u tε ( , ) =
R
∫ ( )ϕε εx u t r W dr dt( , ) – ( , ),
x uε ( , )0 = u,
de W — vineriv lyst na R × 0 1;[ ]. Dovedeno, wo u vypadku, koly ϕε
2
zbiha[t\sq do p aδ( – )⋅ 1 +
+ q aδ( – )⋅ 2 , tobto hranyçna funkciq, wo opysu[ vplyv vypadkovoho serydovywa, synhulqrna
bil\ß niΩ u odnij toçci, ma[ misce slabka zbiΩnist\ x uε ( , )1 ⋅( , … , x udε ( , )⋅ ) do X u( , )1 ⋅( , …
… , X ud( , )⋅ ) , de X— potik Arrat\q, pry ε → 0+ .
Odnym yz osnovn¥x obæektov, rassmatryvaem¥x v dannoj rabote, qvlqetsq po-
tok Arrat\q [1]. Na yntuytyvnom urovne on moΩet b¥t\ opysan kak semejstvo
vynerovskyx çastyc, startugwyx yz kaΩdoj toçky R, dvyΩuwyxsq nezavysymo
vplot\ do momenta vstreçy, pry kotoroj ony skleyvagtsq y dalee dvyΩutsq
vmeste, takΩe soverßaq brounovskoe dvyΩenye. M¥ Ωe budem pol\zovat\sq
sledugwym eho opredelenyem.
Potokom Arrat\q naz¥vaetsq sluçajn¥j process X u( ){ ; u ∈ }R so znaçe-
nyqmy v C 0 1;[ ]( ) takoj, çto dlq lgb¥x u1 < … < un :
1) X uk( , )⋅ — vynerovskyj process, startugwyj yz toçky uk ,
2) dlq lgboho t ∈ 0 1;[ ]
X u t( , )1 ≤ … ≤ X u tn( , ),
3) na mnoΩestve
f C f un
k k∈ [ ]( ){ =0 1 0; , : ( )R , k = 1, … , n, f t f t tn1 0 1( ) ( ), ;<…< ∈[ ]}
raspredelenye X u( , )1 ⋅( ,<… , X un( , )⋅ ) sovpadaet s raspredelenyem n-mernoho vy-
nerovskoho processa, startugweho yz toçky (u1, … , un) .
Pust\ W — R-znaçn¥j vynerovskyj lyst na R × 0 1;[ ].
Rassmotrym uravnenye
© T. V. MALOVYÇKO, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11 1529
1530 T. V. MALOVYÇKO
dx u tε( , ) =
R
∫ ( )ϕε εx u t r W dr dt( , ) – ( , ),
(1)
x uε( , )0 = u.
V rabote [2] pokazano, çto dlq del\taobraznoj posledovatel\nosty funkcyj
ϕε
2
ymeet mesto slabaq sxodymost\ x uε( , )1 ⋅( ,<… , x udε( , )⋅ ) pry ε → 0 + v pro-
stranstve C
d0 1; ,[ ]( )R k
�
X( )⋅ = X u( , )1 ⋅( ,<… , X ud( , )⋅ ). Naßa cel\ sostoyt v
tom, çtob¥ pokazat\, çto analohyçn¥j rezul\tat ymeet mesto y v tom sluçae,
kohda ϕε
2
sxodytsq v prostranstve ′D k obobwennoj funkcyy p aδ( – )⋅ 1 +
+ q aδ( – )⋅ 2 , t.<e. kohda predel\naq funkcyq, opys¥vagwaq vlyqnye sluçajnoj
sred¥, synhulqrna bolee çem v odnoj toçke. Pryçyna, po kotoroj dannoe utver-
Ωdenye budet ymet\ mesto, zaklgçaetsq v tom, çto matryc¥ dyffuzyy budut
sxodyt\sq k edynyçnoj matryce, esly rasstoqnye meΩdu lgb¥my dvumq koordy-
natamy otlyçno ot nulq y ot a2 – a1. Tohda predel\n¥j process budet vesty se-
bq kak vynerovskyj do tex por, poka rasstoqnye meΩdu kakymy-lybo eho koor-
dynatamy ne budet ravno nulg yly a2 – a1. Kak tol\ko nekotor¥e dve koordy-
nat¥ stanut ravn¥my meΩdu soboj, dyffuzyq v¥rodytsq y πty koordynat¥
skleqtsq. A yz-za nev¥roΩdennosty dyffuzyy na mnoΩestve
v ∈{ Rd : ∃i0 ,
j i j0 0 0
v v– = a2 – a1, ∀ i j, v vi j≠ } predel\n¥j process budet provodyt\ na nem
nulevoe vremq, a potomu budet sovpadat\ po raspredelenyg s koneçnomern¥m
suΩenyem potoka Arrat\q.
Rassmotrym ϕ ∈ C0
∞( )R , qvlqgwugsq symmetryçnoj neotrycatel\noj
funkcyej s tem svojstvom, çto
R
∫ ϕ( )u du = 1.
Dlq kaΩdoho ε > 0 poloΩym
ϕε( )u =
p u a
ε
ϕ
ε
– 1
+
q u a
ε
ϕ
ε
– 2
,
hde p + q = 1, 0 < p < 1, a1 < a2.
Dlq ukazannoj funkcyy ϕε reßenye xε uravnenyq (1) suwestvuet, edyn-
stvenno y qvlqetsq potokom homeomorfyzmov (sm. [3], teorema 4.5.1). Obozna-
çym çerez Ft{ ; t ≥ }0 potok σ-alhebr, poroΩdenn¥x W. Tohda pry kaΩdom
u ∈R x u tε( , ){ ; t ≥ }0 — neprer¥vn¥j Ft -martynhal, pryçem sovmestnaq xa-
rakterystyka πtyx processov s naçal\n¥my toçkamy u1 y u2 ymeet vyd
x u x u tε ε( , ), ( , )1 2⋅ ⋅ =
0
1 2
t
x u s r x u s r dr ds∫ ∫ ( ) ( )
R
ϕ ϕε ε ε ε( , ) – ( , ) – ,
a pry u1 = u2 = u
x u tε( , )⋅ =
0
2
t
x u s r dr ds∫ ∫ ( )
R
ϕε ε( , ) – = t.
Sledovatel\no, sohlasno teorem¥ Levy (sm. [4], hl. II, teorema 6.1) process
x u tε( , ){ ; t ≥ }0 qvlqetsq vynerovskym.
Yssleduem povedenye processov
�
x tε( ){ = x u t x u tdε ε( , ), , ( , )1 …( ); t ∈[ ]}0 1; , hde
naçal\naq toçka
�
u = (u1, … , ud ) fyksyrovana, pry ε → 0 +.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
O SXODYMOSTY REÍENYJ STOXASTYÇESKYX DYFFERENCYAL|NÁX … 1531
Lemma 1. Semejstvo
�
xε{ ; ε > }0 slabo otnosytel\no kompaktno v
C
d0 1; ,[ ]( )R .
Dokazatel\stvo. Dlq proyzvol\noho ε > 0
E
�
xε( )0 = E
�
u =
�
u < + ∞,
E
� �
x t x tε ε( ) – ( )2 1
4
∞ = E max ( , ) – ( , )
1
2 1
4
≤ ≤k d
k kx u t x u tε ε ≤
≤
k
d
k kx u t x u t
=
∑
1
2 1
4E ε ε( , ) – ( , ) = 3 2 1
2d t t( – ) .
Sledovatel\no, semejstvo
�
xε{ ; ε > }0 slabo otnosytel\no kompaktno (sm. [4],
hl. I, teorema 4.3).
Lemma dokazana.
Takym obrazom, yz lgboj posledovatel\nosty
�
x
nε{ } moΩno v¥delyt\ slabo
sxodqwugsq podposledovatel\nost\. Pust\
�
x
nε fi
�
y , n → ∞,
hde
�
x
nε{ } — nekotoraq slabo sxodqwaqsq posledovatel\nost\, pryçem
εn → 0, n → ∞.
Naßa cel\ sostoyt v tom, çtob¥ pokazat\, çto predel\n¥j process
�
y sovpa-
daet po raspredelenyg s koneçnomern¥m suΩenyem
�
X potoka Arrat\q.
Otmetym, çto process¥ yi
qvlqgtsq odnomern¥my vynerovskymy processa-
my kak slab¥e predel¥ odnomern¥x vynerovskyx processov
�
x
n
i
ε .
Lemma 2. Sluçajn¥j process
�
y qvlqetsq martynhalom.
Dokazatel\stvo. PokaΩem snaçala, çto dlq proyzvol\noj ohranyçennoj
neprer¥vnoj funkcyy ψ : Rdm → R
E x t x r x r
n n n
i
mε ε εψ( ) ( ), , ( )
� �
1 …( ) → E y t y r y ri
m( ) ( ), , ( )ψ
� �
1 …( ) , n → ∞.
Oboznaçym
ξn = x t x r x r
n n n
i
mε ε εψ( ) ( ), , ( )
� �
1 …( ),
ξ = y t y r y ri
m( ) ( ), , ( )ψ
� �
1 …( ) .
Poskol\ku
sup
n
nE ξ2 < + ∞,
ξn fi ξ, n → ∞,
to ξn , n ≥ 1, ravnomerno yntehryruem¥ y (sm. [5], teorema 5.4)
Eξn → Eξ , n → ∞.
Dalee, tak kak
�
x
nε — martynhal, to dlq lgb¥x neprer¥vn¥x ohranyçenn¥x
funkcyj ψ i : R
dm → R y dlq proyzvol\n¥x 0 < r1 < … < rm ≤ s < t
E Ey t y r y r y t y r y rm
d
d m
1
1 1 1( ) ( ), , ( ) , , ( ) ( ), , ( )ψ ψ
� � � �
…( ) … …( )( ) =
=
lim ( ) ( ), , ( ) , , ( ) ( ), , ( )
n
m
d
d mx t x r x r x t x r x r
n n n n n n→∞
…( ) … …( )( )E Eε ε ε ε ε εψ ψ1
1 1 1
� � � �
=
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
1532 T. V. MALOVYÇKO
=
lim ( ) ( ), , ( ) , , ( ) ( ), , ( )
n
m
d
d mx s x r x r x s x r x r
n n n n n n→∞
…( ) … …( )( )E Eε ε ε ε ε εψ ψ1
1 1 1
� � � �
=
= E Ey s y r y r y s y r y rm
d
d m
1
1 1 1( ) ( ), , ( ) , , ( ) ( ), , ( )ψ ψ
� � � �
…( ) … …( )( ).
Sledovatel\no,
E
� �
y t y r r s( ) ( ), ≤{ } =
�
y s( ) ,
t.<e.
�
y — martynhal.
Lemma dokazana.
Lemma 3. Ymeet mesto
y yi j
t
, ≤ λ s t y s y si j≤ ={ }: ( ) ( ) + λ s t y s y s a ai j≤ ={ }: ( ) – ( ) –2 1 ,
hde λ — mera Lebeha.
Dokazatel\stvo. Pust\
�
ξn( )⋅ =
x u x u x u s r x u s r dr ds
n n n n n nd i jε ε ε ε ε εϕ ϕ( , ), , ( , ), ( , ) – ( , ) –1
0
⋅ … ⋅ ( ) ( )
⋅
∫ ∫
R
=
=
x x x s r x s r dr ds
n n n n n n
d i j
ε ε ε ε ε εϕ ϕ1
0
( ), , ( ), ( ) – ( ) –⋅ … ⋅ ( ) ( )
⋅
∫ ∫
R
.
Poskol\ku
E
�
ξn( )0 = E ( , , , )u ud1 0… = ( , , , )u ud1 0… < + ∞,
E
� �
ξ ξn nt t( ) – ( )2 1
4
∞
=
=
E max max ( ) – ( ) ; ( ) – ( ) –
1
2 1
4
4
1
2
≤ ≤ ∫ ∫ ( ) ( )
k d
k k
t
t
i jx t x t x s r x s r dr ds
n n n n n nε ε ε ε ε εϕ ϕ
R
≤
≤
k
d
k kx t x t
n n
=
∑
1
2 1
4
E ε ε( ) – ( ) + ( – )t t2 1
4 = 3 2 1
2d t t( – ) + ( – )t t2 1
4 ≤
≤ ( )( – )3 1 2 1
2d t t+ , t1, t2 0 1∈[ ]; ,
semejstvo
�
ξn n; ≥{ }1 slabo otnosytel\no kompaktno v prostranstve C 0 1;[ ]( ,
R
d + )1
. Sledovatel\no, yz posledovatel\nosty
�
ξn n; ≥{ }1 moΩno v¥brat\ ta-
kug podposledovatel\nost\ (dlq udobstva oboznaçenyj otoΩdestvym ee s ys-
xodnoj), çto
�
ξn fi y yd1, , ,…( )θ , n → ∞,
hde θ — slab¥j predel
0
⋅
∫ ∫ ( ) ( )
R
ϕ ϕε ε ε εn n n n
x s r x s r drdsi j( ) – ( ) – .
Dlq lgboho poloΩytel\noho δ < 1
4 2 1( – )a a , naçynaq s nekotoroho nomera,
supp
R
∫ ⋅ϕ ϕε εn n
r r dr( – ) ( ) � a a a a1 2 1 2– – ; –δ δ+[ ] ∪ – ;δ δ[ ] ∪
∪ a a a a2 1 2 1– – ; –δ δ+[ ],
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
O SXODYMOSTY REÍENYJ STOXASTYÇESKYX DYFFERENCYAL|NÁX … 1533
a znaçyt, dlq lgboho s ∈[ ]0 1;
R
∫ ( ) ( )ϕ ϕε ε ε εn n n n
x s r x s r dri j( ) – ( ) – ≤ h x s x s
n n
i j
δ ε ε( ) – ( )( )
y dlq lgboho t ∈[ ]0 1;
0
t
i j
n n n n
x s r x s r drds∫ ∫ ( ) ( )
R
ϕ ϕε ε ε ε( ) – ( ) – ≤
0
t
i jh x s x s ds
n n∫ ( )δ ε ε( ) – ( ) ,
hde
h u
u a a a a
a a a a
u a a a a
a a a a
u
δ
δ δ
δ δ δ δ
δ δ δ
δ δ δ
( )
,
– – ; –
– ; – – ; – ,
, – ; – – – ; –
; – – – ; ,
=
∈ +[ ]
[ ] +[ ]
∈ ∞( ] +[ ]
[ ] + ∞[ )
+
1
0 2 2 2
2 2 2
1 2 1 2
2 1 2 1
1 2 1 2
2 1 2 1
∪
∪ ∪
∪ ∪
∪ ∪
aa a
u a a a a
u a a
u a a a a
u u
u u
u a a
u a a
2 1
1 2 1 2
2 1
1 2 1 2
1 2
2 1
2 2
2 2
2 2
2 2
2 2
–
, – – ; – – ,
–
–
, – ; – ,
, – ; – ,
– , ; ,
–
, – – ;
δ
δ δ
δ
δ δ
δ
δ δ
δ
δ δ
δ
δ
+ ∈[ ]
+ + ∈ + +[ ]
+ ∈[ ]
+ ∈[ ]
+ + ∈ aa a
u a a
u a a a a
2 1
1 2
2 1 2 12 2
– – ,
–
–
, – ; – .
δ
δ
δ δ
[ ]
+ + ∈ + +[ ]
Poskol\ku funkcyq hδ lypßyceva s konstantoj 1/δ , otobraΩenye
C zd0 1; ,[ ]( )R ' →
0
⋅
∫ ( ) ∈ [ ]( )h z s z s ds Ci j
δ ( ) – ( ) ; ,0 1 R
takΩe lypßecevo s konstantoj 2/δ , a znaçyt, neprer¥vno. Poπtomu
0
⋅ ⋅
∫ ∫ ∫( ) ( ) ( )
h x s x s ds x s r x s r drds
n n n n n n
i j i j
δ ε ε ε ε ε εϕ ϕ( ) – ( ) , ( ) – ( ) –
0 R
fi
fi
0
⋅
∫ ( )
h y s y s dsi j
δ θ( ) – ( ) , , n → ∞.
Vsledstvye toho, çto
f C t f ft t∈ [ ]( ) ∀ ∈[ ] ≤{ }0 1 0 1 02 2 1; , : ; –R
zamknuto, ymeem
P θ δt
i j
t
h y s y s ds t– ( ) – ( ) , ;( ) ≤ ∈[ ]
∫ 0 0 1
0
≥
≥
lim P
n
t
i j
n n n n
x s r x s r dr ds
→∞ ∫ ∫ ( ) ( )
0 R
ϕ ϕε ε ε ε( ) – ( ) – –
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
1534 T. V. MALOVYÇKO
–
0
0 0 1
t
i jh x s x s ds t
n n∫ ( ) ≤ ∈[ ]
δ ε ε( ) – ( ) , ; = 1,
t.<e. s veroqtnost\g 1 dlq vsex t ∈[ ]0 1;
θt ≤ h y s y s dsi j
t
δ ( ) – ( )( )∫
0
.
Perexodq k predelu pry δ → 0, v sylu teorem¥ Lebeha o maΩoryruemoj sxody-
mosty poluçaem, çto s veroqtnost\g 1 dlq vsex t ∈[ ]0 1;
θt ≤ λ s t y s y si j≤ ={ }: ( ) ( ) + λ s t y s y s a ai j≤ ={ }: ( ) – ( ) –2 1 .
PokaΩem, çto θ — xarakterystyka y. Lehko vydet\, çto θ — vozrastagwyj
process. Kak y pry dokazatel\stve lemm¥<2, nesloΩno proveryt\, çto process
m y yij i j=def – θ
qvlqetsq martynhalom, a tak kak
y t y ti j( ) ( ) = m tij ( ) + θt ,
to
y yi j
t
, = θt ≤ λ s t y s y si j≤ ={ }: ( ) ( ) + λ s t y s y s a ai j≤ ={ }: ( ) – ( ) –2 1 .
Lemma dokazana.
Rassmotrym sluçajn¥j process
� � �
z t y t t t( ) ( ) ( – ) ; ;= ∧ + ∨( ) ∈[ ]{ }σ ω σ 0 0 1 ,
hde σ — moment pervoho v¥xoda
�
y yz mnoΩestva
G = R
d
d i ju u i j u u\ ( , , ):1 … ∃ ≠ ={ }
yly 1, esly dlq vsex t ∈[ ]0 1;
�
y t G( ) ∈ ,
a
�
ω — nezavysym¥j ot
�
y d-mern¥j vynerovskyj process, startugwyj yz 0.
M¥ xotym pokazat\, çto
�
z qvlqetsq d-mern¥m vynerovskym processom, ot-
kuda budet sledovat\, çto
�
y do v¥xoda yz G vedet sebq tak Ωe, kak vynerov-
skyj process. Snaçala dokaΩem sledugwee vspomohatel\noe utverΩdenye.
Lemma 4. Pust\ 0 ≤ s < t ≤ 1. Tohda
E y t t z r r si j( ) ( – ) ( );∧ ∨( ) ≤{ }σ ω σ 0
�
= y s si j( ) ( – )∧ ∨( )σ ω σ 0 p.n.
Dokazatel\stvo. Yzmerymost\ y s si j( ) ( – )∧ ( )σ ω σ ⁄ 0 otnosytel\no σ
�
z r( ){ ;
r s≤ } oçevydna. Ostalos\ pokazat\ tol\ko, çto
∀m1, m2 ∈N , 0 < t1 < … < tm1
≤ s, 0 < r1 < … < rm2
≤ s,
∀ …A Am1 1
, , , B Bm
d
1 2
, , ( )… ∈B R :
E y t ti j( ) ( – )∧ ∨( )σ ω σ 0
1 � �
y t A y t Am m( ) , , ( )1 1 1 1
∧ ∈ … ∧ ∈{ }σ σ ×
×
1 � �
ω σ ω σ( – ) , , ( – )r B r Bm m1 10 0
2 2
∨( ) ∈ … ∨( ) ∈{ } =
=
E y s si j
y t A y t Am m
( ) ( – )
( ) , , ( )
∧ ∨( ) ∧ ∈ … ∧ ∈{ }σ ω σ σ σ0
1 1 1 1
1 � � ×
×
1 � �
ω σ ω σ( – ) , , ( – )r B r Bm m1 10 0
2 2
∨( ) ∈ … ∨( ) ∈{ }.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
O SXODYMOSTY REÍENYJ STOXASTYÇESKYX DYFFERENCYAL|NÁX … 1535
V kaçestve mnoΩestv Bi dostatoçno rassmotret\ tol\ko mnoΩestva, mera
Lebeha hranyc¥ kotor¥x ravna 0. Budem pryblyΩat\ σ markovskymy momenta-
my σk , hde
σ ωk ( ) = l
k2
, esly
l
k
– 1
2
< σ ω( ) ≤ l
k2
, l = 1 2; k
, ω ∈Ω .
Tohda po teoreme Duba (sm. [4], hlava I, teorema 6.11)
l s
j
k
r l Bk k
t l
∈ [ )
∈
∑
N∩
�
1 2 2
2 1 1;
–
–E ω
ω
1 ×
×
E 1 1
σ σσ
k k
l
i
y t Ay t
=
∧ ∈{ }∧
2
1 1
( ) ( )
� =
=
l s
j
k
r l Bk k
s l
∈ [ )
∈
∑
N∩
�
1 2 2
2 1 1;
–
–E ω
ω
1
E 1 1
σ σσ
k k
l
i
y t Ay s
=
∧ ∈{ }∧
2
1 1
( ) ( )
� .
Perexodq k predelu po k, poluçaem trebuemoe ravenstvo dlq m1 = m2 = 1.
Obwyj sluçaj dokaz¥vaetsq analohyçno.
Lemma dokazana.
Lemma 5. Sluçajn¥j process
�
z qvlqetsq martynhalom.
Dokazatel\stvo. Analohyçno dokazatel\stvu pred¥duwej lemm¥ poluça-
em, çto
E
� �
y t z r r s( ) ( );∧ ≤{ }σ =
�
y s( )∧ σ p.n.,
E
� �
ω σ( ) ( );t z r r s∧ ∨( ) ≤{ }0 =
�
ω σ( – )s ∨( )0 p.n.
Sledovatel\no,
�
z( )⋅ — martynhal.
Lemma dokazana.
Lemma 6. Dlq lgb¥x i, j
λ s t y s y s a ai j≤ ={ }: ( ) – ( ) –2 1 = 0 p.n.,
hde λ — mera Lebeha.
Dokazatel\stvo. Oboznaçym
α = a2 – a1,
l = ( , ): –v v v v1 2 2 1 ={ }α ,
lh = ( , ): – ( – ; )v v v v1 2 2 1 ∈ +{ }α αh h , h > 0.
Tohda dlq lgboho h > 0
Eλ s t y s y s a ai j≤ ={ }: ( ) – ( ) –2 1 = Eλ s t y s y s li j≤ ( ) ∈{ }: ( ), ( ) ≤
≤ Eλ s t y s y s li j
h≤ ( ) ∈{ }: ( ), ( ) = E
0
t
y s y s li j
h
dt∫ ( ) ∈{ }1
( ), ( )
=
=
0
t
i j
hy s y s l dt∫ ( ) ∈{ }P ( ), ( ) ≤
0
t
n
i j
hx s x s l dt
n n∫ →∞
( ) ∈{ }lim P ε ε( ), ( ) .
Pust\
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
1536 T. V. MALOVYÇKO
f ( , )v v1 2 =
0
1
2
2
1 2
1 2
2
1 2
1 2 1 2
, – – ,
– – , – ( – ; ),
( – – ), – .
v v
v v v v
v v v v
≤
+( ) ∈ +
≥ +
α
α α α
α α
h
h h h
h h
Po formule Yto
E( , ) ( ), ( )u u
i j
i j n n
f x t x tε ε( ) = f u u( , )1 2 + E( , ) ( – ; ) ( ) – ( )u u
t
h h
i j
i j n n
x s x s ds
0
∫ + ( )1 α α ε ε –
–
E( , ) ( – ; ) ( ) – ( ) ( ) – ( ) –u u
t
h h
i j i j
i j n n n n n n
x s x s x s r x s r drds
0
∫ ∫+ ( ) ( ) ( )1 α α ε ε ε ε ε εϕ ϕ
R
.
Budem rassmatryvat\ tol\ko
ε < α
2D
,
hde D = diam supp ϕ.
Tohda pry h < α / 2 y x si
ε( ) – x sj
ε ( ) ∈ (α – h; α + h)
R
∫ ( ) ( )ϕ ϕε ε ε εx s r x s r dri j( ) – ( ) – ≤ 1
2
.
Dejstvytel\no, poskol\ku
ϕε( )u =
p u a
ε
ϕ
ε
– 1
+
q u a
ε
ϕ
ε
– 2
,
to
R
∫ ( ) ( )ϕ ϕε ε ε εx s r x s r dri j( ) – ( ) – =
=
pq r
x s x s a a
r dr
i j
R
∫ +
ϕ
ε
ϕε ε–
( ) – ( ) –
( )1 2 ≤ pq ≤ 1
2
vsledstvye toho, çto
x s x si j
ε ε
ε
( ) – ( )
> α
ε2
> D,
x s x si j
ε ε α
ε
( ) – ( ) +
>
2α
ε
– h
>
3
2
α
ε
> 3D.
Tohda
E( , ) ( ), ( )u u
i j
i j n n
f x t x tε ε( ) – f u u( – )1 2 = E( , ) ( – ; ) ( ) – ( )u u
t
h h
i j
i j n n
x s x s ds
0
∫ + ( )1 α α ε ε –
–
E( , ) ( – ; ) ( ) – ( ) ( ) – ( ) –u u
t
h h
i j i j
i j n n n n n n
x s x s x s r x s r drds
0
∫ ∫+ ( ) ( ) ( )1 α α ε ε ε ε ε εϕ ϕ
R
≥
≥ 1
2
0
E( , ) ( – ; ) ( ) – ( )u u
t
h h
i j
i j n n
x s x s ds∫ + ( )1 α α ε ε =
= 1
2
0
t
u u
i j
i j n n
x s x s h h ds∫ ∈ +{ }P( , ) ( ) – ( ) ( – ; )ε ε α α =
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
O SXODYMOSTY REÍENYJ STOXASTYÇESKYX DYFFERENCYAL|NÁX … 1537
= 1
2
0
t
u u
i j
hi j n n
x s x s l ds∫ ( ) ∈{ }P( , ) ( ), ( ) )ε ε .
Poskol\ku funkcyq f neotrycatel\na, to
1
2
0
t
u u
i j
hi j n n
x s x s l ds∫ ∈{ }P( , ) ( ) – ( ) )ε ε ≤ E( , ) ( ), ( )u u
i j
i j n n
f x t x tε ε( ) =
= E( , ) ( ) – ( ) ( – ; )
( ) – ( ) –u u
i j
x t x t h hi j n n i jx t x t h
n n
1
2
2
ε ε α α
α
ε ε
+( ) ∈ +{ }1 +
+ E( , ) ( ) – ( )
( ) – ( ) –u u
i j
x t x t hi j n n i jh x t x t
n n
2 ε ε α
α
ε ε
( ) ≥ +{ }1 ≤
≤ 2 2h <+< 4
2 2 2h x t x tu u
i j
i j n n
E( , ) ( ) ( )ε ε α( ) + ( ) +( ) = 2 2h <+< 4 2 2h t + α .
Sledovatel\no,
Eλ s t y s y s a ai j≤ ={ }: ( ) – ( ) –2 1 ≤
≤
0
t
n
i j
hx s x s l dt
n n∫ →∞
( ) ∈{ }lim ( ), ( )P ε ε ≤ lim sup ( ), ( )
n
t
i j
hx s x s l dt
n n→∞
∫ ( ) ∈{ }
0
P ε ε ≤
≤ 2 2h <+< 4 2 2h t + α → 0, h → 0 +.
Takym obrazom,
λ s t y s y s a ai j≤ ={ }: ( ) – ( ) –2 1 = 0 p.n.
Lemma dokazana.
Sledstvye. Ymeet mesto
y yi j
t
, ≤ λ s t y s y si j≤ ={ }: ( ) ( ) .
Lemma 7. Process
�
z qvlqetsq vynerovskym.
Dokazatel\stvo. Najdem xarakterystyku
�
z :
z t z ti j( ) ( ) = y t t y t ti i j j( ) ( – ) ( ) ( – )∧ + ∨( )( ) ∧ + ∨( )( )σ ω σ σ ω σ0 0 =
= y t y ti j( ) ( )∧ ∧σ σ <+< ω σ ω σi jt t( – ) ( – )∨( ) ∨( )0 0 +
+ y t ti j( ) ( – )∧ ∨( )σ ω σ 0 <+< y t tj i( ) ( – )∧ ∨( )σ ω σ 0 =
= y yi j
t
,
∧σ
<+< ( – )t ijσ δ∨( )0 <+< θij t( ) =
= ( )t ij∧ σ δ <+< ( – )t ijσ δ∨( )0 <+< θij t( ) = t ijδ <+< θij t( ) ,
tak kak pry i = j
y yi j
t
, = t
v sylu toho, çto yi
— vynerovskyj process, a pry i ≠ j
y yi j
t
,
∧σ
≤ λ σs t y s y si j≤ ∧ ={ }: ( ) ( ) = 0.
PokaΩem, çto θij ( )⋅ — martynhal otnosytel\no potoka σ-alhebr Fs{ =
= σ
�
z r( ){ ; r ≤ s ∧ }σ ; s ≥ }0 .
Po postroenyg
θij t( ) = y t ti j( ) ( – )∧ ∨( )σ ω σ 0 <+< y t tj i( ) ( – )∧ ∨( )σ ω σ 0 <+
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
1538 T. V. MALOVYÇKO
+< m tij ( )∧ σ + ˆ ( – )m tij σ ∨( )0 ,
hde
m tij ( ) = y t y ti j( ) ( ) – y yi j
t
, ,
ˆ ( )m tij = ω ωi jt t( ) ( ) – t ijδ .
Process¥ mij ( )⋅ y ˆ ( )mij ⋅ — neprer¥vn¥e martynhal¥ otnosytel\no potokov
σ-alhebr σ
�
y r( ){{ ; r ≤ s}; s ≥ 0} y σ ω
�
( )r{{ ; r ≤ s}; s ≥ 0} sootvetstvenno. Po
lemme<4 yi( )⋅ ∧ σ ω σj ( – )⋅ ∨( )0 qvlqetsq martynhalom otnosytel\no Fs{ ; s ≥
≥ 0} . Sledovatel\no, θij ( )⋅ — martynhal.
Takym obrazom,
�
z t = t I,
y sohlasno teoreme Levy (sm. [4], hlava II, teorema 6.1)
�
z — d-mern¥j vynerov-
skyj process.
Lemma dokazana.
Teorema. Semejstvo
�
xε{ ⋅( ) = x uε( , )1 ⋅( , … , x udε( , )⋅ )} slabo sxodytsq pry
ε → 0 + v prostranstve C
d0 1; ,[ ]( )R k
�
X( )⋅ = X u( , )1 ⋅( , … , X ud( , )⋅ ), hde X—
potok Arrat\q.
Dokazatel\stvo. Ostalos\ pokazat\, çto lgbaq predel\naq v slabom
sm¥sle toçka
�
y semejstva
�
xε{ ; ε > 0} ravna po raspredelenyg
�
X( )⋅ .
Kak uΩe otmeçalos\, yi
— odnomern¥j vynerovskyj process dlq lgboho i.
Poskol\ku dlq vsex ε > 0
P
�
xε ∈{ }G = 1,
hde
G = Rd
d i ju u i j u u\ ( , , ):1 … ∃ ≠ ={ },
G =
� �
f C f u i d f t G td
i i∈ [ ]( ) = = ∈ ∈[ ]{ }0 1 0 1 0 1; , : ( ) , , , ( ) , ;R ,
a mnoΩestvo G zamknuto, to v sylu xarakteryzacyy slaboj sxodymosty
P
�
y ∈{ }G = 1.
Tak kak do momenta pervoho v¥xoda yz mnoΩestva G process
�
y sovpadaet s
vynerovskym processom
�
z , ohranyçenye raspredelenyq
�
y na G sovpadaet s
ohranyçenyem vynerovskoj mer¥ na πto Ωe mnoΩestvo.
Sledovatel\no,
� �
y Xd= .
Teorema dokazana.
1. Arratia R. A. Brownian motion on the line: PhD dissertation. – Univ. Wisconsin, Madison, 1984.
2. Dorogovtsev A. A. One Brownian stochastic flow // Theory Stochast. Process. – 2004. – 10, # 3-4.
– P. 21-25.
3. Kunita H. Stochastic flows and stochastic differential equations // Text. Monogr. Cambridge Stud.
Adv. Math. – 1990. – 24. – 346 p.
4. Vatanabπ S., Ykπda N. Stoxastyçeskye dyfferencyal\n¥e uravnenyq y dyffuzyonn¥e
process¥. – M.: Nauka, 1986. – 448 s.
5. Byllynhsly P. Sxodymost\ veroqtnostn¥x mer. – M.: Nauka, 1977. – 352 s.
Poluçeno 22.05.07,
posle dorabotky — 07.09.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
|
| id | umjimathkievua-article-3266 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:39:15Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f4/588118a8f88d03948a3971e35a7043f4.pdf |
| spelling | umjimathkievua-article-32662020-03-18T19:49:31Z Convergence of solutions of stochastic differential equations to the Arratia flow O сходимости решений стохастических дифференциальных уравнений к потоку Арратья Malovichko, T. V. Маловичко, Т. В. Маловичко, Т. В. We consider the solution $x_{\varepsilon}$ of the equation $$dx_{\varepsilon}(u,t) = \int\limits_\mathbb{R}\varphi_{\varepsilon}(x_{\varepsilon}(u,t) - r) W(dr,dt), $$ $$x_{\varepsilon}(u,0) = u,$$ where $W$ is a Wiener sheet on $\mathbb{R} \times [0; 1].$ For the case where $\varphi_{\varepsilon}^2$ converges to $p \delta(\cdot - a_1) + q \delta(\cdot - a_2),$ i.e., where a boundary function describing the influence of a random medium is singular more than at one point, we prove that the weak convergence of $\left(x_{\varepsilon}(u_1, \cdot),...,x_{\varepsilon}(u_d, \cdot) \right)$ to $\left(X(u_1, \cdot),...,X(u_d, \cdot) \right)$ takes place as $\varepsilon\rightarrow0_+$ (here, $X$ is the Arratia flow). Розглянуто розв'язок $x_{\varepsilon}$ рівняння $$dx_{\varepsilon}(u,t) = \int\limits_\mathbb{R}\varphi_{\varepsilon}(x_{\varepsilon}(u,t) - r) W(dr,dt), $$ $$x_{\varepsilon}(u,0) = u,$$ де $W$ — вінерів лист на $\mathbb{R} \times [0; 1].$ Доведено, що у випадку, коли $\varphi_{\varepsilon}^2$ збігається до $p \delta(\cdot - a_1) + q \delta(\cdot - a_2),$ тобто гранична функція, що описує вплив випадкового серидовища, сингулярна більш ніж у одній точці, має місце слабка збіжність $\left(x_{\varepsilon}(u_1, \cdot),...,x_{\varepsilon}(u_d, \cdot) \right)$ до $\left(X(u_1, \cdot),...,X(u_d, \cdot) \right)$, де $X$— потік Арратья, при $\varepsilon\rightarrow0_+.$ Institute of Mathematics, NAS of Ukraine 2008-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3266 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 11 (2008); 1529–1538 Український математичний журнал; Том 60 № 11 (2008); 1529–1538 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3266/3278 https://umj.imath.kiev.ua/index.php/umj/article/view/3266/3279 Copyright (c) 2008 Malovichko T. V. |
| spellingShingle | Malovichko, T. V. Маловичко, Т. В. Маловичко, Т. В. Convergence of solutions of stochastic differential equations to the Arratia flow |
| title | Convergence of solutions of stochastic differential equations to the Arratia flow |
| title_alt | O сходимости решений стохастических дифференциальных уравнений к потоку Арратья |
| title_full | Convergence of solutions of stochastic differential equations to the Arratia flow |
| title_fullStr | Convergence of solutions of stochastic differential equations to the Arratia flow |
| title_full_unstemmed | Convergence of solutions of stochastic differential equations to the Arratia flow |
| title_short | Convergence of solutions of stochastic differential equations to the Arratia flow |
| title_sort | convergence of solutions of stochastic differential equations to the arratia flow |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3266 |
| work_keys_str_mv | AT malovichkotv convergenceofsolutionsofstochasticdifferentialequationstothearratiaflow AT malovičkotv convergenceofsolutionsofstochasticdifferentialequationstothearratiaflow AT malovičkotv convergenceofsolutionsofstochasticdifferentialequationstothearratiaflow AT malovichkotv oshodimostirešenijstohastičeskihdifferencialʹnyhuravnenijkpotokuarratʹâ AT malovičkotv oshodimostirešenijstohastičeskihdifferencialʹnyhuravnenijkpotokuarratʹâ AT malovičkotv oshodimostirešenijstohastičeskihdifferencialʹnyhuravnenijkpotokuarratʹâ |