Properties of a wiener process with coalescence
A Wiener process with coalescence and its analog are discussed. We prove the existence of an initial distribution with preset final probabilities for this analog and investigate the problem of the existence of such distributions concentrated at a single point or absolutely continuous with respect to...
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| Date: | 2006 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2006
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3469 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509566883594240 |
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| author | Malovichko, T. V. Маловичко, Т. В. |
| author_facet | Malovichko, T. V. Маловичко, Т. В. |
| author_sort | Malovichko, T. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2020-03-18T19:55:25Z |
| description | A Wiener process with coalescence and its analog are discussed. We prove the existence of an initial distribution with preset final probabilities for this analog and investigate the problem of the existence of such distributions concentrated at a single point or absolutely continuous with respect to the Lebesgue measure. The behavior of a semigroup of a Wiener process with coalescence in the two-dimensional case and properties of a Wiener flow with coalescence are studied. |
| first_indexed | 2026-03-24T02:43:09Z |
| format | Article |
| fulltext |
UDK 519.21
T. V. Maloviçko (Nac. texn. un-t Ukra]ny „KPI”, Ky]v)
VLASTYVOSTI VINEROVOHO PROCESU
ZI SKLEGVANNQM
The Wiener process with coalescence and its analog are discussed. The existence of an initial
distribution with preset final probabilities for this analog is proved and the problem of the existence of
such distributions concentrated at a single point or absolutely continuous with respect to the Lebesgue
measure is investigated. The behavior of a semigroup of the Wiener process with coalescence in the
two-dimensional case and properties of the Wiener flow with coalescence are studied.
Rozhlqnuto vineriv proces zi sklegvannqm i joho analoh. Dovedeno isnuvannq poçatkovoho roz-
podilu iz zadanymy final\nymy jmovirnostqmy dlq ostann\oho procesu ta doslidΩeno isnuvan-
nq takyx rozpodiliv, skoncentrovanyx v odnij toçci abo absolgtno neperervnyx vidnosno miry
Lebeha. Vyvçagt\sq povedinka napivhrupy vinerovoho procesu zi sklegvannqm u dvovymirnomu
vypadku ta vlastyvosti vinerovoho potoku zi sklegvannqm.
Vstup. Nexaj W — R-znaçnyj vineriv lyst na R × [ 0; 1 ] , ϕ ∈ ∞C0 ( )R — sfe-
ryçno symetryçna nevid’[mna funkciq z vlastyvistg
ϕ ( )u du
R
∫ = 1
i dlq koΩnoho ε > 0
ϕε( )u = ε ϕ ε− − −1 2 1 1 2/ /( )u , u ∈ R .
U roboti [1] bulo rozhlqnuto rivnqnnq
dx u tε( , ) = ϕε ε( ( , ) ) ( , )x u t q W dq dt−∫
R
,
x uε( , )0 = u,
rozv’qzok xε qkoho ma[ dvi vaΩlyvi vlastyvosti. Perßa vlastyvist\ polqha[ v
tomu, wo xε [ potokom homeomorfizmiv, a druha — v tomu, wo dlq koΩnoho u ∈
∈ R x u t tε( , ); ≥{ }0 [ vinerovym procesom.
Nexaj dlq koΩnoho ε
supp ϕε ⊂ [ – ε ; ε ] .
U roboti [1] bulo dovedeno, wo dlq dovil\nyx toçok u1 , … , un ∈ R vypadkovyj
proces ( xε ( u1, ⋅ ) , … , xε ( un , ⋅ )) ma[ slabku hranycg u prostori C ( [ 0; 1 ], R
n
)
pry ε → 0 + . Cej hranyçnyj proces budemo nazyvaty vinerovym procesom zi
sklegvannqm, oskil\ky vin ma[ taki vlastyvosti. KoΩna joho koordynata [ vine-
rovym procesom, i koΩni dvi koordynaty ruxagt\sq qk nezaleΩni vinerovi pro-
cesy do momentu ]xn\o] zustriçi, a pislq c\oho ruxagt\sq razom.
Qkwo zamist\ ruxu n çastynok u R rozhlqdaty rux odni[] çastynky u pros-
tori R
n, to otryma[mo proces, qkyj povodyt\sq qk vineriv do perßoho momentu,
koly deqki z joho koordynat stagt\ rivnymy. Pislq c\oho momentu vin peretvo-
rg[t\sq na vineriv proces na hiperplowyni, de joho koordynaty zalyßagt\sq
rivnymy. Cq procedura tryva[ doty, doky my ne otryma[mo odnovymirnyj vineriv
proces. Z c\oho momentu rozhlqduvanyj proces bude zbihatysq z cym vinerovym
procesom.
Metog dano] roboty [ doslidΩennq vlastyvostej vinerovoho procesu zi skle-
gvannqm ta joho bil\ß zruçnoho analoha.
1. DoslidΩennq analoha vinerovoho procesu zi sklegvannqm. Nexaj za-
© T. V. MALOVIÇKO, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4 489
490 T. V. MALOVIÇKO
dano obmeΩenu oblast\ Q v R
2
z kuskovo-hladkog meΩeg ta skinçennu kil\-
kist\ toçok A1 , A2 , … , An na ]] meΩi (dlq zruçnosti poklademo A0 = An ) . Ci
toçky rozbyvagt\ meΩu ∂Q na n duh (rys.B1).
Rys.B1
Rozhlqnemo proces Y ( ⋅ ) = Y x y( , )( )
0 0
⋅ , qkyj startu[ z ( , )x y Q0 0 ∈ ta povo-
dyt\sq, qk dvovymirnyj vineriv do momentu vyxodu na meΩu oblasti. U cej
moment proces opynq[t\sq na odnij iz duh
�
A Ak k−1 i potim povodyt\sq, qk
odnovymirnyj vineriv proces na duzi
�
A Ak k−1 z pohlynannqm na kincqx. (Pid
vinerovym procesom na duzi
�
A Ak k−1 z pohlynannqm na kincqx budemo rozumity
proces h w− ⋅1( ( )), de vidobraΩennq h stavyt\ u vidpovidnist\ koΩnij toçci M
duhy dovΩynu duhy
�
A Mk−1 , a w( )⋅ — vineriv proces na vidrizku
�
[ ];0 1A Ak k− z
pohlynannqm na kincqx.)
Bil\ß stroho cej proces moΩna zadaty takym çynom:
Y tx y( , )( )
0 0
= ÷ ÷ ÷{ } { } { } ( )
( )( ) ( )t t
k
n
w A A w
kw t w t
k k
< ≥
=
∈
+ −∑
−
τ τ τ
τ
τ
1 1
� ,
de w ( ⋅ ) — dvovymirnyj vineriv proces, qkyj startu[ z toçky ( x0, y0 ) , τ — mo-
ment perßoho vyxodu w ( ⋅ ) na meΩu oblasti Q, w x y
k
k k( , )
( ) ( )⋅ , k = 1, n , — neza-
leΩni vinerovi procesy na duhax
�
A Ak k−1 z pohlynannqm v toçkax Ak−1 j Ak ,
qki startugt\ iz toçok ( , )x yk k ∈
�
A Ak k−1 .
NevaΩko dovesty nastupne tverdΩennq.
Lema/1. Proces Y ( ⋅ ) [ markovs\kym.
Vraxovugçy vymirnist\ perexidnyx imovirnostej, moΩemo rozhlqdaty proces
Y ( ⋅ ) iz vypadkovym poçatkovym rozpodilom.
Teorema/1. Nexaj Q — obmeΩena oblast\ v R2
z rehulqrnog kuskovo-
hladkog meΩeg, A1 , A2 , … , An ∈ ∂Q. Todi isnu[ poçatkovyj rozpodil procesu
Y ( ⋅ ), pry qkomu Y ( ⋅ ) zupynq[t\sq v toçkax A1 , A2 , … , An z dovil\nymy zada-
nymy dodatnymy jmovirnostqmy p1 , p2 , … , pn takymy, wo pkk
n
=∑ 1
= 1.
Dovedennq. Nexaj dlq k ∈ { 1, 2, … , n } ψk x( ) — imovirnist\ toho, wo
proces Yx ( )⋅ zupynyt\sq v toçci Ak . Budemo ßukaty potribnyj rozpodil u vy-
hlqdi µ = ai xi
n
i
δ=∑ 1
.
Dlq toho wob µ = ai xi
n
i
δ=∑ 1
, de xi ∈ Q, i = 1, n , zadovol\nqla umovy teo-
remy, neobxidno i dostatn\o vykonannq umovy
ψ j i i
i
n
x a( )
=
∑
1
= pj, j = 1, n . (1)
Nexaj
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4
VLASTYVOSTI VINEROVOHO PROCESU ZI SKLEGVANNQM 491
ϕ j y( ) =
df
0 1 1
1
1
1
1
1
1
, ,
, ,
, .
y A A A A
A y
A A
y A A
yA
A A
y A A
j j j j
j
j j
j j
j
j j
j j
∉
∈
∈
− +
−
−
−
+
+
+
� �
�
�
�
�
�
�
∪
Todi
ψ j x( ) = Mx j wϕ τ( ( )),
de τ — moment perßoho vyxodu w na meΩu oblasti Q . Z neperervnosti ϕj na
∂Q vyplyva[, wo ψj harmoniçna v Q . Oskil\ky za umovog koΩna toçka meΩi
∂Q [ rehulqrnog, to
lim ( )
x Q
x a
j x
∈
→
ψ = ϕ j a( )
dlq dovil\no] toçky a ∈ ∂Q [2].
Rozhlqnemo n poslidovnostej
{ }( ),x ki
k ≥ 0 , i = 1, n ; ∀i ∀k : x Qi
k( ) ∈ , lim ( )
k i
kx
→∞
= Ai , i = 1, n .
Oskil\ky za umovog Ai , i = 1, n , rehulqrni, to
lim ( )( )
k j i
kx
→∞
ψ = ϕ j iA( ) = δij , i, j = 1, n . (2)
Systema
δij j
i
n
a
=
∑
1
= pj
, j = 1, n ,
nevyrodΩena, oskil\ky ]] vyznaçnyk ∆ = 1. Poznaçymo çerez ∆( )k
vyznaçnyk
systemy
ψ j i
k
i
k
i
n
x a( )( ) ( )
=
∑
1
= pj, j = 1, n .
Todi z rivnosti (2) vyplyva[
lim ( )
k
k
→∞
∆ = ∆ = 1.
Tomu
∃ ∈K N ∀ >k K ∃ … ∈! , , ,( )( ) ( ) ( )a a ak k
n
k n
1 2 R : ψ j i
k
i
k
i
n
x a( )( ) ( )
=
∑
1
= pj, j = 1, n ,
1 = pj
j
n
=
∑
1
= ψ j i
k
i
k
i j
n
x a( )( ) ( )
, =
∑
1
= ai
k
i
n
( )
=
∑
1
,
∀ i : lim ( )
k i
ka
→∞
= lim
( )
( )k
i
k
k→∞
∆
∆
=
∆
∆
i = pi .
Oskil\ky vsi pi dodatni, to
∃ ∈N N ∀ >k N : ai
k( ) > 0, i = 1, n .
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4
492 T. V. MALOVIÇKO
Takym çynom, dlq dostatn\o velykyx k mira µk = ai
k
xi
n
i
k
( )
( )δ=∑ 1
zadovol\nq[
vsi umovy teoremy.
Teoremu dovedeno.
Poznaçymo çerez P klas imovirnisnyx mir µ na Q takyx, wo Yµ( )⋅ zupynq-
[t\sq v toçkax A1 , … , An z dodatnymy jmovirnostqmy p1 , … , pn .
Z navedeno] teoremy vyplyva[, wo P mistyt\ bezliç imovirnisnyx mir.
Teorema/2. Klas P mistyt\ miru, absolgtno neperervnu vidnosno miry
Lebeha.
Dovedennq. Qk vydno z dovedennq teoremy 1,
∃ >ε0 0 ∀ ( x1 , … , xn ) ∈
( ( , ))Q B Ak
k
n
∩ ε0
1=
∏ ∃ ! ( a1 , … , an ) : ak x
k
n
k
δ
=
∑
1
∈ P .
Poznaçymo
Gk = Q B Ak∩ ( , )ε0 , k = 1, n , G = Gk
k
n
=
∏
1
.
Pry c\omu ε0 vyberemo tak, wob mnoΩyny Gk ne peretynalysq.
Z formul Kramera, neperervnosti ψk , k = 1, n , ta neperervnosti vyznaçnyka
vyplyva[, wo
ak ( x1 , … , xn ) ∈ C ( G ) .
Nexaj takoΩ λ2n — 2n-vymirna mira Lebeha, a C = ( ( ))λ2
1
n G − .
Rozhlqnemo f : G → P taku, wo
∀ = … ∈
�
x x x Gn( , , )1 : f x( )
�
= a xk x
k
n
k
( )
�
δ
=
∑
1
.
Vvedemo funkcig mnoΩyn
∀ ∈B QB( ) : µ ( B ) =
C f x B d xn
G
( )( ) ( )
� �
λ2∫ .
Todi
∀ ∈B Qk B( ), k ≥ 1, B Bi j∩ = ∅, i ≠ j :
µ Bk
k=
∞
1
∪ =
C f x B d xk
k
n
G
( ) ( )
� �∪
=
∞
∫
1
2λ =
C f x B d x
G k
k n∫ ∑
=
∞
1
2( )( ) ( )
� �
λ =
=
k
k n
G
C f x B d x
=
∞
∑ ∫
1
2( )( ) ( )
� �
λ =
k
kB
=
∞
∑
1
µ( ) ,
tobto µ [ σ-adytyvnog (za pobudovog µ [ nevid’[mnog), a takoΩ
µ ( Q ) =
µ Gk
k
n
=
1
∪ =
C a x x d x
G k
n
k G k n
kk
n∫ ∑
=
( )=
1
2
1
( ) ( ) ( )
� �
∪
÷ λ =
=
C a x d x
G k
n
k n∫ ∑
=1
2( ) ( )
� �
λ = C d x
G
n∫ λ2 ( )
�
= C Gnλ2 ( ) = 1.
OtΩe, µ — jmovirnisna mira na Q.
ZauvaΩymo, wo µ [ absolgtno neperervnog vidnosno λ2 z wil\nistg
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4
VLASTYVOSTI VINEROVOHO PROCESU ZI SKLEGVANNQM 493
k
n
G k k kk
x u x
=
∑
1
÷ ( ) ( ) ,
de dlq koΩnoho k ≤ n
u xk ( ) = C a x x x x x dx dx dx dx
G
k k k n k k n
ii k≠∏
− + − +∫ … … … …( , , , , , , )1 1 1 1 1 1 .
Zalyßylosq pokazaty, wo µ ∈ P. Dlq koΩnoho j ma[mo
Q
j x d x∫ ψ µ( ) ( ) =
Q
j
k
n
G kx x u x d x
k∫ ∑
=
ψ λ( ) ( ) ( ) ( )
1
2÷ =
= C x a x x x x x dx dx dx dx dx
k
n
Q
j
G
k k k n k k n
ii k= ∏
− + − +∑ ∫ ∫
≠
… … … …
1
1 1 1 1 1 1ψ ( ) ( , , , , , , ) =
= C a x x dx dx dx
k
n
G
k j k n
=
∑ ∫ … …
1
1( ) ( )
�
ψ =
= C a x x x dx dx
G k
n
k n j k n∫ ∑
=
…
…
1
1 1( , , ) ( )ψ =
= C p dx dx
G
j n∫ …1 = pj
.
Teoremu dovedeno.
Vynyka[ pytannq: çy ne moΩna vkazaty toçku v oblasti Q, wob proces
Y ( ⋅ ) , qkyj startu[ z ci[] toçky, nadxodyv do toçok A1 , … , An z dovil\nymy na-
pered zadanymy jmovirnostqmy p1 , … , pn vidpovidno.
U zahal\nomu vypadku vidpovid\ [ nehatyvnog, pro wo svidçyt\ nastupnyj
kontrpryklad.
Pryklad/1. Nexaj oblast\ Q [ vnutrißnistg prqmokutnyka, toçky A1 ,
A2 , A3 , A4 — verßynamy c\oho prqmokutnyka, a A5 leΩyt\ na storoni A 1 A4 ,
pryçomu A A1 5 < A A4 5 . Todi ne isnu[ tako] poçatkovo] toçky, wob proces
nadxodyv do vkazanyx toçok z rivnymy jmovirnostqmy, tobto ne isnu[ tako]
toçky x0 ∈ Q, wo
δx0
∈P pry p1 = … = p5 =
1
5
.
Dovedennq. Prypustymo suprotyvne, a same
∃ ∈x Q0 : ψ k x( )0 =
1
5
, k = 1 5, .
Krim c\oho, prypustymo, wo vidstan\ vid toçky x0 do storony A3 A4 menßa,
niΩ vidstan\ vid x0 do A1 A2.
Nexaj D1 i D2 — seredyny storin A2 A3 ta A1 A4 vidpovidno (rys.B2).
Rys.B2
Poznaçymo
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4
494 T. V. MALOVIÇKO
C1 = { }: ( , )x A A a x A b∈ ≤ <1 2 2ρ ,
C2 = { }: ( , )x A A a x A b∈ ≤ <3 4 3ρ .
Dlq dovil\noho x D D∈ 1 2 spravdΩu[t\sq rivnist\
Px { Y ( ⋅ ) popada[ v C1 } = Px { Y ( ⋅ ) popada[ v C2 } .
Tomu
Px0
{ Y ( ⋅ ) popada[ v C1 } =
=
D D1 2
∫ Px { Y ( ⋅ ) popada[ v C1 } dPx0
{ Y ( ⋅ ) popada[ na D1 x } =
=
D D1 2
∫ Px { Y ( ⋅ ) popada[ v C2 } dPx0
{ Y ( ⋅ ) popada[ na D1 x } =
= Px0
{ Y ( ⋅ ) popada[ v C2 çerez D1 D2 } <
< Px0
{ Y ( ⋅ ) popada[ v C2 } .
OtΩe, jmovirnist\ popadannq na A1 A2 , a potim v toçku A2 menßa za jmovir-
nist\ popadannq na A3 A4 , a potim v toçku A3 .
Dlq dovil\noho x ∈ D1 D2
Px { Y ( ⋅ ) popada[ na A2 D1 } = Px { Y ( ⋅ ) popada[ na D1 A3 } ,
Px0
{ Y ( ⋅ ) popada[ na A2 D1 } =
=
D D1 2
∫ Px { Y ( ⋅ ) popada[ na A2 D1 } dPx0
{ Y ( ⋅ ) popada[ na D1 x } =
=
D D1 2
∫ Px { Y ( ⋅ ) popada[ na D1 A3 } dPx0
{ Y ( ⋅ ) popada[ na D1 x } =
= Px0
{ Y ( ⋅ ) popada[ na D1 A3 çerez D1 D2 } <
< Px0
{ Y ( ⋅ ) popada[ na D1 A3 raniße, niΩ na D1 D2 } +
+ Px0
{ Y ( ⋅ ) popada[ na D1 A3 çerez D1 D2 } =
= Px0
{ Y ( ⋅ ) popada[ na D1 A3 } .
Analohiçno, jmovirnist\ popadannq do { x ∈ A2 A3 : a ≤ ρ ( x, A2 ) < b } menßa
za jmovirnist\ popadannq z x0 do { x ∈ A2 A3 : a ≤ ρ ( x, A3 ) < b } pry 0 ≤ a <
< b ≤ A A2 3 2/ .
OtΩe, jmovirnist\ popadannq v toçku A2 çerez A2 A3 menßa za jmovirnist\
popadannq v toçku A3 çerez A2 A3 .
Zvidsy vyplyva[, wo ψ2 ( x0 ) < ψ3 ( x0 ) . Pryjßly do supereçnosti.
Analohiçno dovodyt\sq, wo vidstan\ vid toçky x0 do storony A3 A4 ne moΩe
buty bil\ßog za vidstan\ vid x0 do A1 A2 .
Takym çynom, oskil\ky ψ2 ( x0 ) = ψ3 ( x0 ) , to toçka x0 [ rivnoviddalenog vid
storin A1 A2 ta A3 A4.
Dlq koΩno] toçky x0 ∈ D1 D2
ψ2 ( x0 ) = ψ3 ( x0 ) ,
Px0
{ Y ( ⋅ ) popada[ na A1 A2 } = Px0
{ Y ( ⋅ ) popada[ na A3 A4 }
zavdqky symetri] vinerovoho procesu.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4
VLASTYVOSTI VINEROVOHO PROCESU ZI SKLEGVANNQM 495
Rys.B3
Vyberemo (rys.B3)
′ ∈A A A5 1 4 : ′A A5 4 = A A1 5 ,
i nexaj
ϕ ( x ) =
A A x
A A
1 5
1 5
−
.
Todi
Px0
{ Y ( ⋅ ) popada[ v A1 çerez A1 A4 } =
=
0 1 5; A A[ ]
∫ ϕ ( x ) dPx0
{ Y ( ⋅ ) popada[ do { y ∈ A1 A5 : ρ ( y, A1 ) < x }} =
=
0 1 5; A A[ ]
∫ ϕ ( x ) dPx0
{ Y ( ⋅ ) popada[ do { y ∈ A A4 5′ : ρ ( y, A4 ) < x }} <
<
0 4 5; A A′[ ]
∫ A A x
A A
4 5
4 5
−
dPx0
{ Y ( ⋅ ) popada[ do { y ∈ A A4 5′ : ρ ( y, A4 ) < x }} <
<
0 4 5; A A[ ]
∫ A A x
A A
4 5
4 5
−
dPx0
{ Y ( ⋅ ) popada[ do { y ∈ A4 A5 : ρ ( y, A4 ) < x }} =
= Px0
{ Y ( ⋅ ) popada[ v A4 çerez A1 A4 } .
Dali, dlq dovil\no] toçky x0 ∈ D1 D2
ψ1 ( x0 ) = Px0
{ Y ( ⋅ ) popada[ v A1 çerez A1 A2 } +
+ Px0
{ Y ( ⋅ ) popada[ v A1 çerez A1 A4 } <
< Px0
{ Y ( ⋅ ) popada[ v A4 çerez A3 A4 } +
+ Px0
{ Y ( ⋅ ) popada[ v A4 çerez A1 A4 } =
= ψ4 ( x0 ) ,
tobto
ψ1 ( x0 ) < ψ4 ( x0 ) , x0 ∈ D1 D2 .
Takym çynom, ne isnu[ tako] toçky x0 ∈ Q , wo
ψk ( x0 ) =
1
5
, k = 1 5, .
Teoremu dovedeno.
Dlq vypadku tr\ox toçok na meΩi ma[ misce nastupnyj rezul\tat.
Teorema/3. Nexaj Q — obmeΩena odnozv’qzna oblast\ v R
2
z rehulqrnog
kuskovo-hladkog meΩeg, a A1 , A2 , A3 — toçky na meΩi Q . Todi isnu[ taka
toçka x0 ∈ Q , wo pry poçatkovomu poloΩenni x0 proces Y ( ⋅ ) zupynq[t\sq
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4
496 T. V. MALOVIÇKO
v toçkax A1 , A2 , A3 z dovil\nymy napered zadanymy dodatnymy jmovirnostq-
my p1 , p2 , p3 takymy, wo pkk=∑ 1
3
= 1.
Dovedennq. Rozhlqnemo trykutnyk ∆ ′ ′ ′A A A1 2 3 v R
3, de ′A1 1 0 0( ; ; ),
′A2 0 1 0( ; ; ), ′A3 0 0 1( ; ; ). Isnu[ neperervne vidobraΩennq g zamykannq oblasti Q
na ∆ ′ ′ ′A A A1 2 3 take, wo
∀ x ∈
�
A Ak k−1 :
g x A
A A
k
k k
( ) ′
′ ′−1
=
�
�
x A
A A
k
k k−1
, k = 1 3, ,
pryçomu g [ bi[kci[g ta zberiha[ meΩi.
Poznaçymo
ψ =
ψ
ψ
ψ
1
2
3
, f = ψ � g−1,
de ψk ma[ toj samyj sens, wo i v teoremi 1.
Oskil\ky
∀ x ∈
�
A Ak k−1 : ψk x+1( ) = 0, ψk x−1( ) =
�
�
x A
A A
k
k k−1
, ψk x( ) = 1 –
�
�
x A
A A
k
k k−1
,
to
∀ x ∈ ∂Q : ψ ( x ) = g ( x ) .
Zvidsy, a takoΩ iz toho, wo
ψ ( x ) = f ( g ( x )) ,
vyplyva[
∀ ∈∂ ′ ′ ′y A A A( )∆ 1 2 3 : f ( y ) = y .
Nexaj g0 — rux, wo vidobraΩa[ ∆ ′ ′ ′A A A1 2 3 na ∆ ′′ ′′ ′′A A A1 2 3 , de
′′ − −
A1
2
2
6
6
0; ; , ′′ −
A2
2
2
6
6
0; ; , ′′
A3 0 6
3
0; ; ,
tobto
g x0( )
�
= Ax b
� �
−
( A i
�
b moΩna lehko obçyslyty), i
g x y z T
1( )( ; ; ) = ( )( ; )x y T , g x y T
2( )( ; ) = ( )( ; ; )x y T0 .
Dali, nexaj f0 : R
2 → R
2
i
f0 = g g f g g1 0 0
1
2� � � �− .
ZvuΩennq f0 na ∆ ′′ ′′ ′′A A A1 2 3 [ neperervnym i vza[mno odnoznaçnym vidobraΩen-
nqm c\oho trykutnyka na sebe, pryçomu
∀ ∈∂ ′′ ′′ ′′z A A A( )∆ 1 2 3 : f0 ( z ) = z .
Z ostann\oho faktu, a takoΩ iz toho, wo ∆ ′′ ′′ ′′A A A1 2 3 [ kompaktom u R
2, a vi-
dobraΩennq f0 : R
2 → R
2
neperervne, vyplyva[ [3], wo
∆ ′′ ′′ ′′A A A1 2 3 ⊂ f A A A0 1 2 3[ ]∆ ′′ ′′ ′′ .
Dlq dovil\nyx dodatnyx çysel p1 , p2 , p3 takyx, wo p1 + p2 + p3 = 1, toç-
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VLASTYVOSTI VINEROVOHO PROCESU ZI SKLEGVANNQM 497
ka B ( p1 , p2 , p3 ) [ vnutrißn\og toçkog dlq ∆ ′ ′ ′A A A1 2 3 . Todi z toho, wo vidobra-
Ωennq f0 zberiha[ meΩi, vyplyva[
∃ ∈ ′′ ′′ ′′z A A A0 1 2 3int ( )∆ : f z0 0( ) = g g p p p T
1 0 1 2 3( ( ))( ; ; ) .
Poznaçymo
y0 =df g g z0
1
2 0
− ( ( )) ∈ int ( )∆ ′ ′ ′A A A1 2 3 ,
todi
f ( y0 ) = f g g z( ( ( )))0
1
2 0
− = g g f z0
1
2 0 0
− ( ( ( ))) = ( ; ; )p p p T
1 2 3 .
Oskil\ky g [ bi[kci[g ta zberiha[ meΩi, to
∃ x0 ∈ Q : g ( x0 ) = y0 .
Zvidsy vyplyva[
ψ ( x0 ) = f ( g ( x0 ) ) = f ( y0 ) = ( ; ; )p p p T
1 2 3 ,
tobto
∃ x0 ∈ Q : ψk ( x0 ) = pk , k = 1 3, .
Teoremu dovedeno.
2. Povedinka vinerovoho procesu zi sklegvannqm. Povernemos\ do vinero-
voho procesu zi sklegvannqm. Joho moΩna zadaty takym çynom. Nexaj w1 ,
w2 , … , wn — nezaleΩni vinerovi procesy v R, pryçomu
w1 ( 0 ) < w2 ( 0 ) < … < wn ( 0 ) .
Rozhlqnemo X ( ⋅ ) = ( X1 ( ⋅ ) , … , Xn ( ⋅ )) , de
X1 ( t ) = w1 ( t ) , t ≥ 0,
τ1 = inf { t > 0 : w2 ( t ) = w1 ( t )} ,
a pry k = 2, n
Xk ( t ) =
w t t
X t t
k k
k k
( ), ,
( ), ,
<
≥
−
− −
τ
τ
1
1 1
τk = inf { t > 0 : wk + 1 ( t ) = Xk ( t )} .
Formal\no vineriv proces zi sklegvannqm zada[t\sq takym çynom. Nexaj
Wu u
n
k1, ,
( )
… — vinerova mira na R
n, wo vidpovida[ poçatkovomu poloΩenng
( u1 , … , un ) , a κu u
n
n1, ,
( )
… — rozpodil n-vymirnoho vinerovoho procesu zi skleg-
vannqm, wo startu[ iz ( u1 , … , un ) . Budemo vvaΩaty, wo u1 < … < un .
Rozhlqnemo dlq dovil\noho δ > 0 u prostori C n([ ; ], )0 1 R mnoΩynu
Gδ
( )n = { }: ( ) , , , , ( ) , [ ; ]( )
� �
f f u i n f t G ti i
n0 1 0 1= = … ∈ ∈δ ,
de Gδ
( )n = { }: ,
�
x x x i jn
i j∈ − > ≠R 2δ .
Todi κu u
n n
n1, ,
( ) ( )( )… G = 1, de G ( )n
— zamykannq mnoΩyny Gδδ
( )n
>0∪ , pry-
çomu
∀ δ > 0 ∀ ∈A C nB( )([ ; ], )0 1 R : κ δu u
n n
n
A
1, ,
( ) ( )( )… G ∩ = W Au u
n n
k1, ,
( ) ( )( )… Gδ ∩ ,
a na hranyci mnoΩyny G ( )n
∀ k ∈ { 1, 2, … , n – 1 } , 1 ≤ i1 < i2 < … < ik ≤ n ,
∀ … ∈A A Ci ik1
0 1, , ([ ; ], )( )B R B:
κu u
n
n
A
1, ,
( ) ( )… = κu u
n
i i ii i kk
A A A
1 1 2, ,
( ) ( )… × × …× ,
de
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4
498 T. V. MALOVIÇKO
A = B B Bn1 2× × … × , Bj =
A j i i
C j i i
j k
k
, { , , },
([ ; ], ), { , , }.
∈ …
∉ …
1
10 1 R
Pry c\omu κu
( )1
zbiha[t\sq z Wu
( )1 .
Lema/2. Proces X ( ⋅ ) [ vinerovym zi sklegvannqm v R
n
.
Dovedennq. Po-perße, pokaΩemo, wo koΩna koordynata Xk ( ⋅ ) [ vinerovym
procesom. Dlq X1 ( ⋅ ) ce tak za pobudovog, pryçomu X1 ( ⋅ ) ta w2 ( ⋅ ) [ nezaleΩ-
nymy. Prypustymo, wo Xk ( ⋅ ) — vineriv proces, ne zaleΩnyj vid wk + 1 ( ⋅ ) . Todi
Xk + 1 ( t ) =
w t X tk t k tk k+ < ≥+1( ) ( ){ } { }÷ ÷τ τ .
Dvovymirnyj proces
w k( )( )⋅ =
X
w
k
k
( )
( )
⋅
⋅
+1
[ vinerovym. A oskil\ky operator
U =
1
2
1
2
1
2
1
2
−
unitarnyj, to proces
˜ ( )( )w k ⋅ = Uw k( )( )⋅ =
1
2
1
2
1
1
X w
X w
k k
k k
( ) ( )
( ) ( )
⋅ + ⋅( )
⋅ − ⋅( )
+
+
takoΩ [ vinerovym, pryçomu τk — moment perßoho popadannq dlq ˜ ( )( )w k
2 ⋅ v 0,
a ˜ ( )( )w k
1 ⋅ i ˜ ( )( )w k
2 ⋅ [ nezaleΩnymy. Todi
Xk + 1 ( t ) = w t X tk t k tk k+ < ≥+1( ) ( ){ } { }÷ ÷τ τ =
=
˜ ( ) ˜ ( )( ) ( )
{ }
w t w tk k
t k
1 2
2
−
<÷ τ +
˜ ( ) ˜ ( )( ) ( )
{ }
w t w tk k
t k
1 2
2
+
≥÷ τ =
=
˜ ( )( )w tk
1
2
–
˜ ( )( )
{ } { }( )w tk
t tk k2
2
÷ ÷< ≥−τ τ
.
Proces
˜ ( )( )
{ } { }( )w k
k k2 ⋅ ⋅ − ⋅< ≥÷ ÷τ τ utvorg[t\sq z procesu ˜ ( )( )w k
2 ⋅ vidbyttqm vid-
nosno osi abscys u moment perßoho popadannq v 0. Iz stroho markovs\ko]
vlastyvosti ta symetri] vinerovoho procesu vyplyva[, wo proces
ˆ ( )( )w k
2 ⋅ =
˜ ( )( )
{ } { }( )w k
k k2 ⋅ ⋅ − ⋅< ≥÷ ÷τ τ
takoΩ [ vinerovym. Do toho Ω ostannij proces ne zaleΩyt\ vid ˜ ( )( )w k
1 ⋅ . A os-
kil\ky
U w kˆ ( )( ) ⋅ = U
w
w
k
k
˜ ( )
ˆ ( )
( )
( )
1
2
⋅
⋅
=
1
2 1 2
1
˜ ( ) ˆ ( )
( )
( ) ( )w w
X
k k
k
⋅ + ⋅( )
⋅
+
,
to Xk+ ⋅1( ) [ vinerovym procesom, ne zaleΩnym vid wk+ ⋅2( ) . Za indukci[g otry-
ma[mo, wo Xk ( )⋅ — vineriv proces dlq dovil\noho k .
Bezposeredn\o z pobudovy vyplyva[
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VLASTYVOSTI VINEROVOHO PROCESU ZI SKLEGVANNQM 499
Pu u
n
n
X
1, ,
( ){ }( )… ⋅ ∈G = Pu u nn
X t X t X t t
1 1 2 0 1, , { }( ) ( ) ( ), [ ; ]… ≤ ≤…≤ ∈ = 1,
∀ δ > 0 ∀ ∈A C nB( )([ ; ], )0 1 R : Pu u
n n
n
X A
1, ,
( ) ( ){ }( )… ⋅ ∈Gδ ∩ =
= Pu u
n
n
n
n
w w A
1 1, ,
( ) ( ){( ) }( ), , ( )… ⋅ … ⋅ ∈Gδ ∩ = W Au u
n n
n1, ,
( ) ( )( )… Gδ ∩ ,
oskil\ky do momentu perßo] zustriçi bud\-qkyx koordynat proces X ( ⋅ ) zbiha-
[t\sq z n-vymirnym vinerovym procesom.
Rozhlqnemo, nareßti, dlq dovil\nyx 1 ≤ i1 < i2 < … < ik ≤ n proces
˜ ( )X ⋅ = ( )˜ ( ), , ˜ ( )X Xk1 ⋅ … ⋅ , de
˜ ( ) ( )X t X ti1 1
= , t ≥ 0,
˜ inf : ( ) ( ){ }τ1 0
2 1
= > =t X t X ti i ,
a pry j k= 2,
˜ ( )X tj =
X t t
X t t
i j
j j
j
( ), ˜ ,
˜ ( ), ˜ ,
<
≥
−
− −
τ
τ
1
1 1
τ̃ j = inf : ( ) ˜ ( ){ }t X t X ti jj
> =
+
0
1
.
Oskil\ky Xij
( )⋅ [ vinerovymy procesamy, to za tako] pobudovy proces
˜ ( )X ⋅
cilkom analohiçnyj procesu X ( ⋅ ) , za vynqtkom rozmirnosti. Dlq dovil\nyx
Ai1
, … , A Cik
∈ ([ ; ], )0 1 R
Pu un
X A
1, , { }( )… ⋅ ∈ =
= Pu u i i i in k n
X X A A
1 1 1, , {( ) ( )}, , ( )… … ⋅ ∈ × … × =
= Pu u i ii i kk
X A A
1 1, , { ( )}˜ ( )… ⋅ ∈ × …× ,
de
A = B Bn1 × …× , Bj =
A j i i
C j i i
j k
k
, { , , },
([ ; ], ), { , , }.
∈ …
∉ …
1
10 1 R
OtΩe, X ( ⋅ ) [ vinerovym procesom zi sklegvannqm.
Lemu dovedeno.
NevaΩko dovesty nastupnu lemu.
Lema/3. Proces X ( ⋅ ) [ markovs\kym.
Doslidymo teper povedinku joho napivhrupy u dvovymirnomu vypadku pry t →
→ + ∞ .
Nexaj Tt
( )2
— napivhrupa dvovymirnoho vinerovoho procesu zi sklegvannqm, a
S ( t ) — napivhrupa odnovymirnoho vinerovoho procesu. Reßtu poznaçen\ zaly-
ßymo bez zmin. Nexaj ˆ ( )w ⋅ — vineriv proces, ne zaleΩnyj vid X ( ⋅ ) , i
∀ f : R
2 → R ∀ x ∈ R : ˜( )f x
df= f x x( , ).
Teorema/4. Ma[ misce zbiΩnist\
T f x x S f
x x
t t
( ) ( , ) ˜2
1 2
1 2
2
− +
→ 0, t → + ∞ .
Qkwo vykonu[t\sq umova
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4
500 T. V. MALOVIÇKO
f x x dx2( , )
R
∫ < + ∞ ,
to
T f x xt
( ) ( , )2
1 2 → 0, t → + ∞ .
Dovedennq. Ma[mo
P { τ ≤ t } = P { ˜ ( )w2 ⋅ popada[ v 0 do momentu t } =
= P0
2 1
2
ˆ ( )w
x x
t⋅ −
popada[ v do momentu =
= P0 0
2 1
2
max ˆ ( )
[ ; ]s t
w s
x x
∈
≥ −
= 2
20
2 1P ˆ ( )w t
x x≥ −
=
= P0
2 1
2
ˆ ( )w t
x x≥ −
= 1
1
2 2
2
2
2
2 1
2 1
− −
− −
−
∫πt
u
t
du
x x
x x
exp
( )/
( )/
≥
≥ 1 2
2 2
2 1− −
πt
x x
= 1 2 1− −x x
tπ
→ 1, t → + ∞ .
Tomu
lim { }
t
t
→ +∞
>P τ = 0,
i dlq dovil\no] obmeΩeno] funkci] f ∈ C ( R
2
)
Mx x tf X t
1 2, { }( ( ))÷ τ> ≤
sup ( ) { }
�
�
u
f u t
∈
>
R
2
P τ → 0, t → + ∞ .
Zi stroho markovs\ko] vlastyvosti vinerovoho procesu vyplyva[
Mx x tf X t
1 2, { }( ( ))÷ τ≤ = Mx x tf w t w t
1 2 1 1, { }( ( ), ( ))÷ τ≤ =
=
M Mx x tf w t w t
1 2 1 1, { }{ }( ( ), ( ))÷ τ τ≤ F =
Mx x t tS f w
1 2 1, { }
˜( ( ))− ≤τ ττ ÷ =
= Mx x t tS f
w
1 2
1
2, { }
˜ ˜ ( )
− ≤
τ τ
τ
÷ .
OtΩe, dlq dovil\no] obmeΩeno] funkci] f ∈ C ( R
2
)
T f x x S f
x x
t t
( ) ( , ) ˜2
1 2
1 2
2
− +
=
= M Mx x x xf X t f
w t
1 2 1 2 2
1
2, ( )/( ( )) ˜ ˆ ( )−
+ ≤
≤ M Mx x x x tf X t f X t
1 2 1 2, , { }( ( )) ( ( ))− ≤÷ τ +
+ M Mx x t t x xS f
w
f
w t w t
1 2 1 2
1 1 1
2 2 2, { } ,
˜ ˜ ( ) ˜ ( )
,
˜ ( )
− ≤
−
τ τ
τ
÷ =
=
M Mx x t x x tf X t f
w t w t
1 2 1 2
1 1
2 2, { } , { }( ( ))
˜ ( )
,
˜ ( )
÷ ÷τ τ> >+
→ 0, t → + ∞ ,
tobto
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4
VLASTYVOSTI VINEROVOHO PROCESU ZI SKLEGVANNQM 501
T f x x S f
x x
t t
( ) ( , ) ˜2
1 2
1 2
2
− +
→ 0, t → + ∞ .
Qkwo vykonu[t\sq takoΩ umova
f x x dx2( , )
R
∫ < + ∞ ,
to
S f xt
˜ ( )
2
=
R
∫ − −
f u u
t
x u
t
du
( , )
exp
( )
2 2
2 2
π
≤
≤ f u u du
t
x u
t
du2
21
2
( , ) exp
( )
R R
∫ ∫
− −
π
=
=
1
2
2
πt
f u u du( , )
R
∫
→ 0, t → + ∞ ,
i v c\omu vypadku
T f x xt
( ) ( , )2
1 2 → 0, t → + ∞ .
Teoremu dovedeno.
Rozhlqnemo deqki vlastyvosti vinerovoho potoku zi sklegvannqm, tobto pro-
cesu { X ( u, t ), u ∈ R , t ∈ [ 0; ∞ ) } v R .
Teorema/5. Nexaj X ( ⋅ , ⋅ ) — vineriv potik zi sklegvannqm. Todi
lim
( , )
u
X u t
u→ +∞ +1 ε = 0 m. n., lim
( , )u
u
X u t→ +∞ +1 ε = 0 m. n.
Dovedennq. Oskil\ky pry fiksovanomu u X ( u, ⋅ ) — vineriv proces, wo
startu[ z u, to
MX u t2( , ) = t + u2.
Poznaçymo
un
1 = n2/ε , εn
1 =
1
n
, An
1 =
X u t
u
n
( , )
1
1
+ ≥
ε ε .
Todi
P An( )1 = P X u t un n n( ) ( ),1 1 1 1≥{ }+ε ε ≤
MX u t
u
n
n n
2 1
1 2 1 2 2
( )
( ) ( )
,
ε ε+ =
t u
u
n
n n
+
+
( )
( ) ( )
1 2
1 2 1 2 2ε ε ,
t u
u
n
n n
+
+
( )
( ) ( )
1 2
1 2 1 2 2ε ε ∼
1
1 2 1 2( ) ( )ε ε
n nu
=
1
2n
, n → ∞ ,
a tomu
n
nP A
=
∞
∑
1
1( ) < + ∞ .
Za lemog Borelq – Kantelli
lim
,( )
( )n
n
n
X u t
u→ +∞ +
1
1 1 ε = 0 m. n..
Dali
∀ u > 1 ∃ ∈ +k u u uk k: ,( ]1
1
1 ,
pryçomu z monotonnosti X ( ⋅ , t ) vyplyva[, wo
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4
502 T. V. MALOVIÇKO
X u t
u
u
u
k
k
k( )
( )
,1
1 1
1 1
+
+
ε
ε
≤
X u t
u
( , )
1+ε ≤
X u t
u
u
u
k
k
k( )
( )
,+
+
+
+
+
1
1
1
1 1
1
1 1
ε
ε
.
Todi
u
u
k+ −1
1
1 =
u u
u
k+ −1
1
≤
u u
u
k k
k
+ −1
1 1
1 =
( ) / /
/
k k
k
+ −1 2 2
2
ε ε
ε → 0, u → ∞ ( k → ∞ ) ,
tobto
lim
u
ku
u→ +∞
+1
1
= 1.
Analohiçno
lim
u
ku
u→ +∞
1
= 1.
OtΩe,
lim
( , )
u
X u t
u→ +∞ +1 ε = 0 m.Bn.
Nexaj 1 + 1 / ε < δ < + ∞ . Poznaçymo
un
2 = n
δ, β =
δε ε
δ
− −1
, εn
2 =
1
2( )un
β , An
2 =
u
X u t
n
n
n
2
2 1
2
( ),
+ ≥
ε ε .
Oskil\ky δε ε− −1 > 0, to
εn
2 =
1
1nδε ε− − → 0, n → ∞ .
Vnaslidok toho, wo pry fiksovanomu u X ( u, ⋅ ) — vineriv proces, wo startu[
z u, ma[mo
P An( )2 = P
u
X u t
n
n
n
2
2 1
2
( ),
+ ≥
ε ε = P X u t
u
n
n
n
( ),
/( )
2
2
2
1 1
≤
+
ε
ε
=
=
1
2
2 2
2 2 1 1 2 2 1 1
2
π
ε ε
ε εt
e drr u t
u u
n
n n n n
− −
−( ) ( )
+ +
∫ ( ) /
/ ; /
/( ) /( )
≤
≤
2
2
2
2
1 1 2 2 1 1 2
2
π ε
εε ε
t
u u u
t
n
n
n n n
−
( ) −( )
+ +
/( )
/( )
exp
/
,
oskil\ky
un
n
2
2
1 1
ε
ε
+/( )
= ( ) /un
2 1 1− δ < un
2.
Dali
P An( )2 ≤
2
2
2 1 1
2 1 1 2 2
π
δ
δ
t
u
u u
tn
n n( )
( )/
/
exp−
−
−
−( )
=
2 1 1
2
1
2 2
π
δ
δ
t
n
n n
t
− − −
exp
( / )
.
ZvaΩagçy na te, wo pry dostatn\o velykyx znaçennqx x
e x− <
1
x
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4
VLASTYVOSTI VINEROVOHO PROCESU ZI SKLEGVANNQM 503
pry dostatn\o velykyx n otrymu[mo
P An( )2 ≤
2 2
1
1
2
2
π
δ
δt
n
t
n
n
n
−
−
∼
2 2 1
1
t
nπ δ+ , n → ∞ .
Tomu
n=
∞
∑
1
P An( )2 < + ∞ .
Za lemog Borelq – Kantelli
lim
,( )n
n
n
u
X u t→∞ +
2
2 1 ε = 0 m. n.,
∀ u > 1 ∃ ∈ +m u u um m: ,( ]2
1
2 ,
pryçomu z monotonnosti X ( ⋅ , t ) vyplyva[, wo
( )
( )
/( ) /( )
,
u
X u t
u
u
m
m m
+
+
+ +
+
1
2 1 1
1
2
1
2
1 1ε ε
≤
u
X u t
1 1/( )
( , )
+ε
≤
( )
( )
/( ) /( )
,
u
X u t
u
u
m
m m
2 1 1
2 2
1 1+ +
ε ε
,
u
um
2 1− =
u u
u
m
m
− 2
2 ≤
u u
u
m m
m
+ −1
2 2
2 =
( )m m
m
+ −1 δ δ
δ → 0, u → ∞ ( m → ∞ ) ,
tobto
lim
u
m
u
u→ +∞ 2 = 1.
Analohiçno
lim
u
m
u
u→ +∞ +1
2 = 1.
OtΩe,
lim
( , )
/( )
n
u
X u t→∞
+1 1 ε
= 0 m. n.,
tobto
lim
( , )n
u
X u t→∞ +1 ε = 0 m. n.
Teoremu dovedeno.
Lema/4. Nexaj X ( ⋅ , ⋅ ) — vineriv potik zi sklegvannqm. Todi dlq dovil\noho
t i dovil\noho vidrizka [ a; b ] z imovirnistg 1 isnu[ vidrizok [ α; β ] ⊂ [ a; b ]
( α < β ) takyj, wo
X ( α, t ) = X ( β, t ) .
Dovedennq. PokaΩemo spoçatku, wo
A = ∃ ⊂ ={ }[ , ] [ ; ] : ( , ) ( , )α β α βa b X t X t
[ vypadkovog podi[g, tobto A ∈ F . Dlq c\oho poznaçymo
B =
r a b r r b
X r t X r t
1 2 1
1 2
∈ ∈
={ }
Q Q∩ ∩
∪ ∩
[ ; ) ( ; ]
( , ) ( , ) =
= ∃ ∈ ≤ < ≤ ={ }r r a r r b X r t X r t1 2 1 2 1 2, : , ( , ) ( , )Q .
Todi B ∈ F , oskil\ky
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4
504 T. V. MALOVIÇKO
∀r r1 2, : X r t X r t( , ) ( , )1 2={ } ∈ F .
Oçevydno, wo B ⊂ A .
Dali,
∀ ∈ω A ∃α β0 0, : a b≤ < ≤α β0 0 , X t( , , )ω α0 = X t( , , )ω β0 .
Oskil\ky α0 < β0 , to
∃ ∈˜ , ˜r r1 2 Q : α β0 1 2 0≤ < ≤˜ ˜r r .
Z monotonnosti X ( ⋅ , t ) vyplyva[
X r t( , ˜ , )ω 1 = X r t( , ˜ , )ω 2 .
Takym çynom, ω ∈ B, a tomu A ⊂ B .
OtΩe, A = B ∈ F .
Podilymo vidrizok [ a; b ] na n rivnyx vidrizkiv toçkamy uk , de a = u0 <
< u1 < … < un = b. Dlq dovil\nyx n ta k ≤ n
P ( A ) ≥ P X u t X u tk k( , ) ( , )− ={ }1 =
= P iw w tu uk k−
⋅ ⋅{ }1
( ) ( ) zustrilysq do momentu =
= P
w w
t
u uk k
( ) ( )⋅ − ⋅
−1
2
0popada[ v do momentu =
=
P ˜ ( )w
u u
tk k⋅ −
−
popada[ v do momentu
1
2
=
= P max ˜ ( )
[ ; ]s t
k kw s
u u
∈
−≥ −
0
1
2
= 2
2
1P ˜ ( )w t
u uk k≥ −
− =
= P ˜ ( )w t
u uk k≥ −
−1
2
= 1
1
2
2
1 1
2
2 2
− −
− − −
− −
∫πt
e duu t
u u u uk k k k
/
;
≥
≥ 1 1− − −u u
t
k k
π
= 1 − −b a
t nπ
,
de wuk −
⋅
1
( ) ta wuk
( )⋅ — nezaleΩni vinerovi procesy, wo startugt\ z toçok uk−1
ta uk vidpovidno, a ˜ ( )w ⋅ — ne zaleΩnyj vid nyx vineriv proces, qkyj startu[ z 0.
Oskil\ky
lim
n
b a
t n→∞
− −
1
π
= 1,
to P ( A ) = 1.
Lemu dovedeno.
1. Dorogovtsev A. A., Kotelenez P. Stochastic flows with interaction and random measures. –
Dordrecht: Kluwer Acad. Publ., 2004.
2. D¥nkyn E. B., Gßkevyç A. A. Teorem¥ y zadaçy o processax Markova. – M.: Nauka, 1967. –
232Bs.
3. Yvanov V. V. Topolohyçeskaq stepen\ // Matematyka sehodnq. – Kyev: V¥wa ßk., 1987. –
231Bs.
OderΩano 28.10.2004
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4
|
| id | umjimathkievua-article-3469 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:43:09Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/db/f34007849c49da2a1b2aec4ef35882db.pdf |
| spelling | umjimathkievua-article-34692020-03-18T19:55:25Z Properties of a wiener process with coalescence Властивості вінерового процесу зі склеюванням Malovichko, T. V. Маловичко, Т. В. A Wiener process with coalescence and its analog are discussed. We prove the existence of an initial distribution with preset final probabilities for this analog and investigate the problem of the existence of such distributions concentrated at a single point or absolutely continuous with respect to the Lebesgue measure. The behavior of a semigroup of a Wiener process with coalescence in the two-dimensional case and properties of a Wiener flow with coalescence are studied. Розглянуто вінерів процес зі склеюванням i його аналог. Доведено існування початкового розподілу із заданими фінальними ймовірностями для останнього процесу та досліджено існування таких розподілів, сконцентрованих в одній точці або абсолютно неперервних відносно міри Лебега. Вивчаються поведінка напівгрупи вінерового процесу зі склеюванням у двовимірному випадку та властивості вінерового потоку зі склеюванням. Institute of Mathematics, NAS of Ukraine 2006-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3469 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 4 (2006); 489–504 Український математичний журнал; Том 58 № 4 (2006); 489–504 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3469/3675 https://umj.imath.kiev.ua/index.php/umj/article/view/3469/3676 Copyright (c) 2006 Malovichko T. V. |
| spellingShingle | Malovichko, T. V. Маловичко, Т. В. Properties of a wiener process with coalescence |
| title | Properties of a wiener process with coalescence |
| title_alt | Властивості вінерового процесу зі склеюванням |
| title_full | Properties of a wiener process with coalescence |
| title_fullStr | Properties of a wiener process with coalescence |
| title_full_unstemmed | Properties of a wiener process with coalescence |
| title_short | Properties of a wiener process with coalescence |
| title_sort | properties of a wiener process with coalescence |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3469 |
| work_keys_str_mv | AT malovichkotv propertiesofawienerprocesswithcoalescence AT malovičkotv propertiesofawienerprocesswithcoalescence AT malovichkotv vlastivostívínerovogoprocesuzískleûvannâm AT malovičkotv vlastivostívínerovogoprocesuzískleûvannâm |