Some remarks on a Wiener flow with coalescence
We study properties of a stochastic flow that consists of Brownian particles coalescing at contact time.
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2005
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| author | Dorogovtsev, A. A. Дороговцев, А. А. Дороговцев, А. А. |
| author_facet | Dorogovtsev, A. A. Дороговцев, А. А. Дороговцев, А. А. |
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| description | We study properties of a stochastic flow that consists of Brownian particles coalescing at contact time. |
| first_indexed | 2026-03-24T02:47:06Z |
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UDK 519.21
A. A. Dorohovcev (Yn-t matematyky NAN Ukrayn¥, Kyev)
NEKOTORÁE ZAMEÇANYQ
O VYNEROVSKOM POTOKE SO SKLEYVANYEM
We study properties of a stochastic flow that consists of Brownian particles coalescing at contact time.
Vyvçagt\sq vlastyvosti stoxastyçnoho potoku, wo sklada[t\sq z brounivs\kyx çastynok, qki
skleggt\sq v moment zustriçi.
Sluçajn¥j process, opys¥vagwyj dvyΩenye brounovskyx çastyc, kotor¥e pry
vstreçe skleyvagtsq, no ne zamedlqgt dvyΩenyq, b¥l postroen v [1]. Pryve-
dem2vkratce sootvetstvugwee postroenye. Pust\ W — vynerovskyj lyst na
R × +∞[ ; )0 [2]. Rassmotrym neotrycatel\nug funkcyg ϕ ∈C0
2( )R takug, çto
ϕ ( )u du
R
∫ = 1.
Dlq proyzvol\noho ε > 0 opredelym funkcyg ϕε na R sootnoßenyem
ϕε( )u =
1
1 2
1 2
ε
ϕ
ε/
/u
.
Pust\ dlq kaΩdoho u ∈ R x u tε( , ) — reßenye stoxastyçeskoho dyfferency-
al\noho uravnenyq
d x u tε( , ) = ϕε ε( )( , ) ( , )x u t p W dp dt−∫
R
(1)
s naçal\n¥m uslovyem x u uε( , )0 = . Yzvestno [3], çto pry sdelann¥x predpolo-
Ωenyqx otnosytel\no funkcyy ϕ moΩno v¥brat\ modyfykacyg sluçajnoho
polq { }( , ); ,x u t u tε ∈ ≥R 0 , neprer¥vnug po sovokupnosty peremenn¥x. Slu-
çajn¥j process { }( , );x u t tε ≥ 0 pry fyksyrovannom u ∈ R moΩno rassmatry-
vat\ kak process dvyΩenyq çastyc¥, naxodyvßejsq v naçal\n¥j moment vreme-
ny v toçke u y dvyΩuwejsq pod vlyqnyem sluçajn¥x vozmuwenyj W [2]. Ot-
metym, çto { }( , );x u t tε ≥ 0 qvlqetsq neprer¥vn¥m martynhalom s xarakterys-
tykoj t y, sledovatel\no, vynerovskym processom. Takym obrazom, xε — semej-
stvo vynerovskyx processov. Pry πtom dlq razn¥x naçal\n¥x toçek u1 y u 2
sluçajn¥e process¥ zavysym¥ (v veroqtnostnom sm¥sle), tak kak yz neravenst-
va u1 < u2 sleduet
P{ }: ( , ) ( , )∀ ≥ <t x u t x u t0 1 2ε ε = 1.
V rabote [1] dokazano, çto dlq proyzvol\n¥x u1 < u2 < … < un
, n ≥ 1, slu-
çajn¥e process¥ { }( , ), , ( , ); [ ; ]x u t x u t t Tnε ε1 0… ∈ sxodqtsq po raspredele-
nyg22pry ε → +0 v prostranstve C T n([ ; ], )0 R dlq kaΩdoho T. Predel\noe
raspredelenye na C n([ ; ), )0 + ∞ R oboznaçym çerez µu un1… . Semejstvo
{ }; , , ,µu u nn
u u n
1 1 1… … ∈ ≥R qvlqetsq sohlasovann¥m semejstvom koneçnomer-
n¥x raspredelenyj y, sohlasno teoreme Kolmohorova, emu sootvetstvuet slu-
çajn¥j process { ( , ); }x u u⋅ ∈R so znaçenyqmy v C([ ; ))0 + ∞ . Pry πtom mer¥
µu un1… na C n([ ; ), )0 + ∞ R mohut b¥t\ polnost\g oxarakteryzovan¥ sledug-
wym naborom svojstv:
1) na mnoΩestve G ⊂ C n([ ; ), )0 + ∞ R vyda
© A. A. DOROHOVCEV, 2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10 1327
1328 A. A. DOROHOVCEV
G = { : ( ) , , , ,
�
f f u k nk k0 1= = …
∀ ≥ < < … <t f t f t f tn0 1 2: ( ) ( ) ( )}
mera µu un1… sovpadaet s raspredelenyem standartnoho n-mernoho vynerovsko-
ho processa, startovavßeho yz toçky ( , , )u un1 … ;
2) odnomern¥e raspredelenyq µu un1… (t. e. sootvetstvugwye v¥boru lgboj
odnoj koordynat¥) qvlqgtsq vynerovskymy meramy;
3) µu un
G
1… ( ) = 1, hde G — zam¥kanye G v C n([ ; ), )0 + ∞ R v topolohyy
ravnomernoj sxodymosty na kompaktax.
Otmetym, çto process x ymeet rqd xoroßyx svojstv. Tak, spravedlyva sle-
dugwaq teorema.
Teorema�1. { ( , ); }x u u⋅ ∈R — odnorodn¥j markovskyj process v C([ ; ))0 + ∞ ,
neprer¥vn¥j po veroqtnosty.
Dokazatel\stvo. Postroym predpolahaemug perexodnug funkcyg dlq
processa x. Pust\ f C∈ + ∞([ ; ))0 , u ≥ 0, w — standartn¥j odnomern¥j vyne-
rovskyj process, ∆ — borelevskoe podmnoΩestvo C([ ; ))0 + ∞ . Opredelym slu-
çajn¥j moment vremeny
τ = inf : ( ) ( ) ( ){ }t w t u f f t+ + =0 .
PoloΩym
w tf
u ( ) =
w t u f t
f t t
( ) ( ), ,
( ), .
+ + ≤
≥
0 τ
τ
Pust\ teper\
P fu( , )∆ = P{ }wf
u ∈ ∆ .
To, çto Pu po vtoromu arhumentu qvlqetsq veroqtnostnoj meroj, oçevydno.
DokaΩem yzmerymost\ Pu po pervomu arhumentu. Dlq πtoho rassmotrym na
C([ ; ))0 2+ ∞ operacyg skleyvanyq funkcyj. Pust\
A = { }( , ) ([ ; )) : ( ) ( )f g C f g∈ + ∞ ≤0 0 02 .
Zametym, çto A — borelevskoe podmnoΩestvo C([ ; ))0 2+ ∞ . Dlq ( , )f g A∈
opredelym
t0 = inf : ( ) ( ){ }t g t f t= ,
g tf ( ) =
g t t t
f t t t
( ), ,
( ), .
≤
≥
0
0
OtobraΩenye
C([ ; ))0 2+ ∞ ' ( f, g ) � gf ∈ C([ ; ))0 + ∞
qvlqetsq yzmerym¥m po sovokupnosty peremenn¥x. Dejstvytel\no, opredelym
v C([ ; )) [ ; )0 02+ ∞ × + ∞ mnoΩestvo
B = { }( , , ) : ( ) ( )f g t f t g t= .
Lehko proveryt\, çto B — zamknutoe mnoΩestvo v C([ ; )) [ ; )0 02+ ∞ × + ∞ . Sle-
dovatel\no [4], otobraΩenye
( f, g ) � τ f g, = inf : ( ) ( ){ }t f t g t=
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10
NEKOTORÁE ZAMEÇANYQ O VYNEROVSKOM POTOKE SO SKLEYVANYEM 1329
qvlqetsq yzmerym¥m kak debgt mnoΩestva B . S uçetom πtoho fakta yzmery-
most\ operacyy ( f, g ) � gf oçevydna.
Teper\ otmetym, çto
P fu( , )∆ = M I∆( )( )w u f+ .
Poπtomu yzmerymost\ P fu( , )∆ po f qvlqetsq sledstvyem standartn¥x arhu-
mentov teoryy mer¥ [4]. Proverym, çto semejstvo { },P uu ∈R udovletvorqet
uravnenyg Çepmena – Kolmohorova. Rassmotrym f C∈ + ∞([ ; ])0 y çysla u1 ≥
≥ 0 y u2 ≥ 0. Dlq opysanyq qder Pu1
, Pu2
y Pu u1 2+ voz\mem dva nezavysym¥x
standartn¥x vynerovskyx processa w1 y w2 . Tohda dlq proyzvol\noho bore-
levskoho ∆ v C([ ; ))0 + ∞
P h P f dhu u
C
2 1
0
( , ) ( , )
([ ; ))
∆
+ ∞
∫ = M I∆( )( ) ( ( ))w u u f w u f f2 2 1 00
1 1
+ + + + + .
Dlq toho çtob¥ v¥raΩenye v pravoj çasty b¥lo bolee prost¥m, vvedem v ras-
smotrenye sluçajn¥e moment¥
τ1 = inf : ( ) ( ){ }t w t u w t2 2 1+ = ,
τ2 = inf : ( ) ( ) ( ){ }t w t u f f t1 1 0+ + = .
Postroym nov¥j sluçajn¥j process w sledugwym obrazom:
w t( ) =
w t t
w t u t
2 1
1 2 1
( ), ,
( ) , .
≤
− ≥
τ
τ
Process { }( );w t t ≥ 0 qvlqetsq standartn¥m vynerovskym processom. Dejst-
vytel\no, τ1 — moment ostanovky otnosytel\no potoka σ-alhebr {F t =
= σ ( ( ), ( ), ), }w s w s s t t1 2 0≤ ≥ . Poπtomu v sylu stroho markovskoho svojstva [3]
process¥ w s w2 1 2 1( ) ( )+ −τ τ y w s w1 1 1 1( ) ( )+ −τ τ , s ≥ 0, qvlqgtsq standart-
n¥my vynerovskymy processamy, ne zavysqwymy ot σ-alhebr¥
Fτ1
. Sledova-
tel\no, process
w t( ) = w w w t wt t2 2 1 1 11
1 1
( ) ( ) ( ( ) ( )){ } { }I I≤ >+ + −[ ]τ ττ τ =
= w t w u w t wt t2 1 1 2 1 1 11 1
( ) ( ) ( ) ( ){ } { }I I≤ >+ − + −[ ]τ ττ τ =
= w w t ut t2 1 21
1 1
( ) ( ){ } { }I I≤ >+ −( )τ τ
qvlqetsq standartn¥m vynerovskym processom. Teper\ nesloΩno proveryt\,
çto
( )( ) ( ( ))w u u f w u f f2 2 1 00
1 1
+ + + + + = ( )( )w u u f f+ + +2 1 0 .
Tem sam¥m
P h P f dhu u
C
2 1
0
( , ) ( , )
([ ; ))
∆
+ ∞
∫ =
= P{( ) }( )w u u f f+ + + ∈2 1 0 ∆ = P fu u1 2+ ( , )∆ .
Sledovatel\no, semejstvo { }Pu udovletvorqet uravnenyg Çepmena – Kolmo-
horova. Postroym na R, kak na parametryçeskom mnoΩestve, odnorodn¥j mar-
kovskyj process s perexodn¥my veroqtnostqmy { }Pu . Oboznaçym çerez µ v ras-
predelenye v C([ ; ))0 + ∞ vynerovskoho processa, startovavßeho yz v . Zametym,
çto dlq proyzvol\n¥x u ≥ 0, v ∈ R, analohyçno dokazannomu ranee,
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10
1330 A. A. DOROHOVCEV
P h dhu
C
( , ) ( )
([ ; ))
∆ µv
0 + ∞
∫ = µv+u( )∆ . (2)
Rassmotrym teper\ semejstvo koneçnomern¥x raspredelenyj vyda
νv v1 1… × …×
n n( )∆ ∆ =
= µv v -v v -v1
1
2 1
2
1
1
1 1 2 1( ) ( , ) ( , )dh P h dh P h
n n
n
n n
∆ ∆ ∆
∆∫ ∫ ∫…
−
−
− . (3)
Zdes\ v v1 ≤…≤ n , ∆ ∆1, ,… n — borelevskye podmnoΩestva C([ ; ))0 + ∞ . V sylu
(2) y toho fakta, çto { }Pu udovletvorqgt uravnenyg Çepmena — Kolmoho-
rova, semejstvo
{ }νv v1… n
qvlqetsq semejstvom sohlasovann¥x koneçnomern¥x
raspredelenyj, kotoromu sootvetstvuet markovskyj process na R so znaçenyq-
my v C([ ; ))0 + ∞ . Otmetym teper\, çto mer¥
{ }νv v1… n
sovpadagt s koneçnomer-
n¥my raspredelenyqmy processa x. ∏to sleduet yz postroenyq qder { }Pu y
opredelenyq (3). Sledovatel\no, x qvlqetsq markovskym processom.
Rassmotrym dlq v1 ≤ v2 y proyzvol\noho T > 0 velyçynu
ηv v1 2
=
sup ( , ) ( , )
[ ; ]0
1 2
T
x t x tv v− .
Velyçyna ηv v1 2
ymeet takoe Ωe raspredelenye, kak
′ηv v1 2
= sup ˜ ( ) ( )
[ ; ]
( )
0
2 1
T
w t + −v v ,
hde process w̃ poluçen yz standartnoho vynerovskoho processa w sledugwym
obrazom:
˜ ( )w t =
2
2 1
w t t
t
( ), ,
( ), ,
≤
− − ≥
τ
τv v
τ = inf : ( ) ( )t w t2 2 1= − −{ }v v .
Sledovatel\no,
∀ ε > 0 :
P{ }η εv v1 2
> =
P{ }′ >η εv v1 2
→ 0, v2 – v1 → 0 + .
Teorema dokazana.
Teorema�2. Process x ymeet cadlág modyfykacyg kak sluçajn¥j process,
zadann¥j na R y prynymagwyj znaçenyq v prostranstve C([ ; ])0 1 .
Dokazatel\stvo. Dlq kaΩdoho n ≥ 1 oboznaçym çerez An mnoΩestvo çy-
sel { / };k kn2 ∈Z . V¥berem posledovatel\nost\ çysel { };t nn ≥ 1 monotonno
ub¥vagwej k 0 tak, çtob¥
P ∃ ∀ ≥ ∀ ∈ − + − <
∈
∈ −
n n n r A n n x r s x r s
nn
t
s t
n
n
0 0
0
0 1
1∩ [ ; ]: sup ( , ) ( , )
[ ; ]
[ ; ]
τ
τ = 1. (4)
Takug posledovatel\nost\ moΩno v¥brat\ v sylu lemm¥ Borelq – Kantelly s
uçetom toho fakta, çto pry kaΩdom r sluçajn¥j process { }( , ) ; [ ; ]x r s r s− ∈ 0 1
qvlqetsq standartn¥m vynerovskym processom.
Dalee, dlq kaΩdoho u ∈ R opredelym sluçajnug funkcyg ˜ ( , )x u ⋅ sledug-
wym obrazom. Pust\ { },0 1 11= <…< = ≥s s nn nmn
— posledovatel\nost\ vlo-
Ωenn¥x razbyenyj otrezka [ 0; 1 ] takaq, çto
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10
NEKOTORÁE ZAMEÇANYQ O VYNEROVSKOM POTOKE SO SKLEYVANYEM 1331
∀ n ≥ 1 : max ( )
, ,k m nk nk
n
s s
= … − + −
0 1 1 ≤ tn .
Dlq kaΩdoho çysla snk poloΩym
˜ ( , )x u snk = lim ( , )
r u
r A
nkx r s
→ +
∈
.
Zdes\ A = Ann=
∞
1∪ y predel suwestvuet dlq kaΩdoho u ∈ R y vsex ω yz vero-
qtnostnoho prostranstva v sylu monotonnosty x po prostranstvennoj pere-
mennoj. Otmetym teper\, çto v sylu (4) poluçenn¥e sluçajn¥e funkcyy ravno-
merno neprer¥vn¥ na mnoΩestve { }; ,s k m nnk n0 1≤ ≤ ≥ pry vsex u y pry vsex
ω yz mnoΩestva polnoj veroqtnosty Ω1 , ne zavysqweho ot u . Poπtomu
˜ ( , )x u ⋅ pry vsex u ∈ R y ω ∈ Ω1 prodolΩaetsq odnoznaçno do neprer¥vnoj
funkcyy na [ 0; 1 ] . DokaΩem, çto dlq proyzvol\noho ω ∈ Ω1 x̃ qvlqetsq
cadlág funkcyej, zadannoj na R y prynymagwej znaçenyq v C([ ; ])0 1 . Dlq
πtoho zametym, çto po postroenyg dlq proyzvol\noho l ≥ 1 y ω ∈ Ω1 funk-
cyy { }˜ ( , ); [ ; ]x u u l l⋅ ∈ − obrazugt ravnostepenno ravnomerno neprer¥vnoe se-
mejstvo. Pry πtom v sylu monotonnosty x po r y opredelenyq x̃ dlq kaΩdo-
ho çysla snk
˜ ( , )x u snk =
lim ˜ ( , )
v
v
→ +u nkx s ,
∃
→ −
lim ˜ ( , )
v
v
u nkx s = ˜ ( , )x u snk− .
Yz ravnostepennoj ravnomernoj neprer¥vnosty sleduet, çto v C([ ; ])0 1
˜ ( , )x u ⋅ =
lim ˜ ( , )
v
v
→ +
⋅
u
x ,
∃ ⋅
→ −
lim ˜ ( , )
v
v
u
x = ˜ ( , )x u− ⋅ .
Proverym, çto x̃ qvlqetsq modyfykacyej x. Dejstvytel\no, v sylu stoxa-
styçeskoj neprer¥vnosty x y opredelenyq x̃ dlq lgboho u ∈ R
P{ }, , : ( , ) ˜( , )∀ ≥ ∀ = … =n k m x u s x u sn nk nk1 0 = 1.
Poskol\ku x u( , )⋅ y ˜( , )x u ⋅ — neprer¥vn¥e sluçajn¥e funkcyy, to
x u( , )⋅ = ˜( , )x u ⋅ p. n.
Teorema dokazana.
Nemnoho vydoyzmenyv dokazatel\stvo pryvedennoj teorem¥, moΩno dokazat\
suwestvovanye cadlág modyfykacyy x kak sluçajnoho processa so znaçenyqmy
v C([ ; ))0 + ∞ .
Rassmotrym teper\ dlq cadlág modyfykacyy processa x sluçajn¥j process
m ( t ) = – inf{ }: ( , ) ( , )u x u t x t≤ =0 0 , t ≥ 0.
Odnomern¥e raspredelenyq processa m lehko v¥çyslqgtsq. A ymenno, ∀ t >
> 0, a ≥ 0 :
P{ }( )m t a≥ = P{ }( , ) ( , )x a t x t− = 0 .
Odnomern¥e sluçajn¥e process¥ x ( 0, ⋅ ) y x ( – a, ⋅ ) takov¥, çto yx raznost\
x ( 0, ⋅ ) – x ( – a, ⋅ ) predstavlqet soboj vynerovskyj process s dyspersyej 2, v
moment vremeny221 startovavßyj yz a y ravn¥j220 posle pervoho popadanyq
v220. Poπtomu
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10
1332 A. A. DOROHOVCEV
P{ }( )m t a≥ = 2
2 2
2
e dxx t
a t
−
+ ∞
∫ /
/ π
= P sup ( )
[ ; ]0 2t
w s
a>
,
hde w — standartn¥j vynerovskyj process. Otmetym, çto nesmotrq na to, çto
odnomern¥e raspredelenyq sluçajn¥x processov m y
′m t( ) = sup ( )
[ ; ]0
2
t
w s , t ≥ 0,
sovpadagt, πty process¥ ymegt razlyçn¥e raspredelenyq (kak sluçajn¥e pro-
cess¥). Dlq toho çtob¥ ubedyt\sq v πtom, rassmotrym sluçajn¥e moment¥
vremeny, svqzann¥e s m y m ′ :
τa = inf : ( ){ }t m t a≥ ,
′τa = inf : ( ){ }t m t a′ ≥ .
Zametym, çto
τa = inf : ( , ) ( , ){ }t x a t x t− = 0 .
S uçetom opysanyq koneçnomern¥x raspredelenyj processa x (sm. [1]) y sym-
metryy spravedlyvo ravenstvo
P{ }τ τ1 2= =
1
2
.
Dejstvytel\no, sob¥tye τ1 = τ2 nastupaet tohda y tol\ko tohda, kohda sklejka
x ( – 1, ⋅ ) y x ( – 2, ⋅ ) proysxodyt ne pozΩe sklejky x ( 0, ⋅ ) y x ( – 1, ⋅ ) .
S druhoj storon¥, v sylu neprer¥vnosty vynerovskoho processa
P{ }′ = ′τ τ1 2 = 0.
V pryvedennom rassuΩdenyy yspol\zovano to, çto x, rassmatryvaem¥j kak
potok sluçajn¥x otobraΩenyj, moΩet skleyvat\ razlyçn¥e toçky. Svojstva x,
kak stoxastyçeskoj poluhrupp¥, planyruetsq yssledovat\ v otdel\noj stat\e.
Zdes\ m¥ pryvedem lyß\ utverΩdenyq ob otobraΩenyqx x ( ⋅ , t ) pry fyksyro-
vannom t .
Teorema�3. Pust\ t ≥ 0. Tohda veroqtnost\ toho, çto x ( ⋅ , t ) , kak otob-
raΩenye yz R v R, razr¥vno, ravna221.
Dokazatel\stvo. Rassmotrym vnaçale suΩenye x ( ⋅ , t ) na proyzvol\n¥j
otrezok [ a; b ] . Opredelym sluçajn¥e velyçyn¥
ζ n =
k
n
x a
k
n
b a t x a
k
n
b a t
=
−
∑ + + −
− + −
0
1 21
( ), ( ), .
Poskol\ku x ( ⋅ , t ) — neub¥vagwaq funkcyq, to
ζ n ≥ ζ n + 1 ≥ 0, n ≥ 1.
Pry πtom
Mζn = n w tnM + ( )2 ,
hde wn
+
— standartn¥j vynerovskyj process, startovavßyj yz ( )/b a n− y rav-
n¥j 0 posle popadanyq v 0. Poπtomu
Mζn = n x
t
e e dxx b a n t x b a n t2 2 2
0
1
2
2 2
π
− − − − + −
+∞
−( )∫ ( ( )/ ) / ( ( )/ ) / .
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10
NEKOTORÁE ZAMEÇANYQ O VYNEROVSKOM POTOKE SO SKLEYVANYEM 1333
Po teoreme Lebeha o maΩoryruemoj sxodymosty
Mζn →
b a
t
x
t
e dxx t− −
+∞
∫2
1
2
3 2
0
2
π
/ , n → ∞ .
Vospol\zuemsq teper\ teoremoj Lebeha o maΩoryruemoj sxodymosty po otno-
ßenyg k posledovatel\nosty { };ζn n ≥ 1 . Ymeem
M lim
n n→∞
ζ =
b a
t
x
t
e dxx t− −
+∞
∫2
1
2
3 2
0
2
π
/ .
Poπtomu s poloΩytel\noj veroqtnost\g
ζ ∞ = lim
n n→∞
ζ > 0. (5)
Poskol\ku funkcyq x ( ⋅ , t ) monotonna, to na mnoΩestve tex ω, dlq kotor¥x
spravedlyvo (5), x ( ⋅ , t ) ymeet skaçky na [ a, b ] .
Rassmotrym teper\ dva otrezka: [ a; a + 1 ] y [ b; b + 1 ] . Pust\ ζ∞( )a y
ζ∞( )b — sluçajn¥e velyçyn¥, postroenn¥e po πtym otrezkam tak, kak πto b¥lo
sdelano v¥ße. Tohda s uçetom opysanyq koneçnomern¥x raspredelenyj proces-
sa x
∀ α, β ∈ R : P P P{ } { } { }( ) , ( ) ( ) ( )ζ α ζ β ζ α ζ β∞ ∞ ∞ ∞< < − < <a b a b → 0, (6)
b a− → + ∞ ,
P{ }( )ζ α∞ <a = P{ }( )ζ α∞ <b . (7)
Sootnoßenye (6) ostaetsq spravedlyv¥m y v tom sluçae, kohda otrezky s lev¥-
my koncamy v a y b ymegt razlyçnug (no fyksyrovannug) dlynu. Teper\ moΩ-
no postroyt\ posledovatel\nost\ { };a nn ≥ 1 takug, çto dlq sluçajn¥x vely-
çyn { }( );ζ a nn ≥ 1 , sootvetstvugwyx otrezkam { }[ ; ];a a nn n+ ≥1 1 , budet spra-
vedlyvo sootnoßenye
P { }( )ζ an
n
>
=
∞
0
1
∪ = 1.
Teorema dokazana.
Otmetym, çto v [5] pokazan rezul\tat, kotor¥j prymenytel\no k processu x
moΩno sformulyrovat\ sledugwym obrazom. Dlq proyzvol\n¥x t > 0 y u ∈ R
na mnoΩestve veroqtnosty221 spravedlyvo vklgçenye
x ( u, t ) ∈ { }( , );x r t r ∈Q .
Odnako, poskol\ku ukazannoe mnoΩestvo veroqtnosty221, voobwe hovorq, zavy-
syt ot u, otsgda ne sleduet v¥polnenye utverΩdenyq teorem¥23.
1. Dorogovtsev A. A. One Brownian stochastic flow // Theory Stochast. Processes. – 2004. – 10 (26),
# 3-4. – P. 21 – 25.
2. Kotelenez P. A class of quasilinear stochastic partial differential equations of McKean - Vlasov
type with mass conservation // Probab. Theory Related Fields. – 1995. – 102. – P. 159 – 188.
3. Kunita H. Stochastic flows and stochastic differential equations. – Cambridge Univ. Press., 1990. –
346 p.
4. Dellacherie C. Capacites et processus stochastiques. – Berlin: Springer, 1980.
5. Darling R. W. R. Constructing nonhomeomorphic stochastic flows // Mem. AMS. – 1987. – 10,
# 376. – 98 p.
Poluçeno 01.12.2004
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 10
|
| id | umjimathkievua-article-3688 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:47:06Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f9/8dfdced6a992cbb9d258d930795021f9.pdf |
| spelling | umjimathkievua-article-36882020-03-18T20:02:18Z Some remarks on a Wiener flow with coalescence Некоторые замечания о винеровском потоке со склеиванием Dorogovtsev, A. A. Дороговцев, А. А. Дороговцев, А. А. We study properties of a stochastic flow that consists of Brownian particles coalescing at contact time. Вивчаються властивості стохастичного потоку, що складається з броунівських частинок, які склеюються в момент зустрічі. Institute of Mathematics, NAS of Ukraine 2005-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3688 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 10 (2005); 1327–1333 Український математичний журнал; Том 57 № 10 (2005); 1327–1333 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3688/4102 https://umj.imath.kiev.ua/index.php/umj/article/view/3688/4103 Copyright (c) 2005 Dorogovtsev A. A. |
| spellingShingle | Dorogovtsev, A. A. Дороговцев, А. А. Дороговцев, А. А. Some remarks on a Wiener flow with coalescence |
| title | Some remarks on a Wiener flow with coalescence |
| title_alt | Некоторые замечания о винеровском потоке со склеиванием |
| title_full | Some remarks on a Wiener flow with coalescence |
| title_fullStr | Some remarks on a Wiener flow with coalescence |
| title_full_unstemmed | Some remarks on a Wiener flow with coalescence |
| title_short | Some remarks on a Wiener flow with coalescence |
| title_sort | some remarks on a wiener flow with coalescence |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3688 |
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