Properties of the Flows Generated by Stochastic Equations with Reflection
We consider properties of a random set $\varphi_t(\mathbb{R}_+^d)$, where $\varphi_t(x)$ is a solution of a stochastic differential equation in $\mathbb{R}_+^d$ with normal reflection on the boundary starting at the point $x$. We perform the characterization of inner and boundary points of the set...
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| Дата: | 2005 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2005
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3665 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509792619986944 |
|---|---|
| author | Pilipenko, A. Yu. Пилипенко, А. Ю. Пилипенко, А. Ю. |
| author_facet | Pilipenko, A. Yu. Пилипенко, А. Ю. Пилипенко, А. Ю. |
| author_sort | Pilipenko, A. Yu. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:01:36Z |
| description | We consider properties of a random set $\varphi_t(\mathbb{R}_+^d)$, where $\varphi_t(x)$ is a solution of a stochastic differential equation in
$\mathbb{R}_+^d$ with normal reflection on the boundary starting at the point $x$. We perform the characterization of inner and boundary points of the set $\varphi_t(\mathbb{R}_+^d)$.
We prove that the Hausdorff dimension of the boundary $\partial \varphi_t(\mathbb{R}_+^d)$ is not greater than $d - 1$. |
| first_indexed | 2026-03-24T02:46:44Z |
| format | Article |
| fulltext |
UDK 519.21
A. G. Pylypenko (Yn-t matematyky NAN Ukrayn¥, Kyev)
SVOJSTVA POTOKOV, POROÛDENNÁX
STOXASTYÇESKYMY URAVNENYQMY S OTRAÛENYEM
*
We consider properties of a random set ϕt
d
R+( ), where ϕt x( ) is a solution of a stochastic differential
equation in R+
d with normal reflection on the boundary starting at the point x. We perform the
characterization of inner and boundary points of the set ϕt
d
R+( ). We prove that the Hausdorff
dimension of the boundary ∂ϕt
d
R+( ) is not greater than d – 1.
Rozhlqdagt\sq vlastyvosti vypadkovo] mnoΩyny ϕt
d
R+( ), de ϕt x( ) — rozv’qzok stoxastyçno-
ho dyferencial\noho rivnqnnq v R+
d
z normal\nym vidbyttqm vid meΩi, wo startu[ z toçky x.
Provedeno xarakteryzacig vnutrißnix ta hranyçnyx toçok mnoΩyny ϕt
d
R+( ). Dovedeno, wo
rozmirnist\ Xausdorfa meΩi ∂ϕt
d
R+( ) ne perevywu[ d – 1.
Pust\ ϕt ( x ), t ∈ [ 0, T ], x ∈ R+
d = R
d
–
1 × [ 0, ∞ ), — reßenye sledugweho sto-
xastyçeskoho uravnenyq s normal\n¥m otraΩenyem ot hranyc¥ [1]:
d ϕt ( x ) = a0 ( ϕt ( x ) ) d t +
k
m
k t ka x dw t
=
∑ ( )
1
ϕ ( ) ( ) + n dt xξ( , ), t ∈ [ 0, T ],
(1)
ϕ0 ( x ) = x, ξ ( 0, x ) = 0, x ∈ R+
d
,
hde ak : R+
d → R
d
, k = 0, … , m, w t k mk ( ), , ,= …{ }1 — nezavysym¥e vynerov-
skye process¥, n = ( 0, … , 0, 1 ) — normal\ k hyperploskosty R
d
–
1 × { 0 }, ξ ( t,
x ) — neub¥vagwyj po t process dlq lgboho fyksyrovannoho x ∈ R+
d
, pryçem
ξ ( t, x ) =
0
01
t
xs
d ds x∫ ∈ × { }{ }−÷ ϕ ξ
( )
( , )
R
,
t. e. ξ ( t, x ) vozrastaet tol\ko v te moment¥ vremeny, kohda ϕt ( x ) ∈ R
d
–
1 × { 0 }.
PredpoloΩym, çto funkcyy ak , k = 0, … , m, udovletvorqgt uslovyg Lyp-
ßyca. NesloΩno proveryt\, çto v πtom sluçae suwestvuet edynstvennoe syl\-
noe reßenye uravnenyq (1), pryçem ( ϕt ( x ), ξ ( t, x ) ) ymeet neprer¥vnug po pare
arhumentov ( t, x ) modyfykacyg (sm., naprymer, [2]). Dalee budem sçytat\, çto
v kaçestve ( ϕ, ξ ) uΩe vzqta πta modyfykacyq.
Rassmotrym sluçajnoe mnoΩestvo ϕt
d
R+( ), kotoroe zanymagt v moment vre-
meny t obraz¥ toçek prostranstva R+
d
pry dejstvyy sluçajnoho otobraΩenyq
ϕt ( ⋅, ω ) : R+
d → R+
d
.
Cel\g stat\y qvlqetsq xarakteryzacyq naçal\n¥x znaçenyj yz R+
d
, koto-
r¥e popadagt vo vnutrennost\ yly na hranycu mnoΩestva ϕt
d
R+( ). Osnovnoj
rezul\tat soderΩytsq v sledugwyx dvux teoremax.
Teorema 1. Dlq poçty vsex ω y vsex t ∈ [ 0, T ] razmernost\ Xausdorfa
mnoΩestva ∂ϕt
d
R+( ) — hranyc¥ mnoΩestva ϕt
d
R+( ) — ne prev¥ßaet d – 1.
*
V¥polnena pry podderΩke Mynysterstva nauky y obrazovanyq Ukrayn¥ (proekt GP/F8/0086 ).
© A. G. PYLYPENKO, 2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8 1069
1070 A. G. PYLYPENKO
Bolee toho, dlq lgboho R > 0 mera Xausdorfa H
d
–
1 mnoΩestva
∂ϕt
d
R+( ) ∩ x x Rd∈ ≤{ }+R : koneçna.
Zameçanye 1. Lehko vydet\, çto esly mnoΩestvo ϕt
d
R+( ) soderΩyt vnut-
rennye toçky, to dim ∂ϕt
d
R+( ) ne men\ße d – 1 — razmernosty proekcyy πtoho
mnoΩestva na R
d
–
1 × { 0 }, t. e. v πtom sluçae
dim ∂ϕt
d
R+( ) = d – 1.
Zameçanye 2. V konce stat\y budet pryveden prymer, kohda s poloΩytel\-
noj veroqtnost\g ymeet mesto strohoe neravenstvo
dim ∂ϕt
d
R+( ) < d – 1.
Teorema 2. Dlq poçty vsex ω y vsex t ∈ [ 0, T ] ymeet mesto ravenstvo
sluçajn¥x mnoΩestv
∂ϕt
d
R+( ) = ϕ ∂t
d
R+( ) = ϕ τt
dx x t∈ ≤{ }+R : ( ) ,
hde τ( x ) = inf : ( )s xs
d≥ ∈ × { }{ }−0 01ϕ R — moment popadanyq reßenyq, star-
tugweho yz x, na hyperploskost\ R
d
–
1 × { 0 }.
Dokazatel\stvo teorem¥ 1. V rabote [2] b¥ly ustanovlen¥ sledugwye
svojstva ϕt ( x ).
Teorema 3. Suwestvuet mnoΩestvo Ω 0 veroqtnosty 1 takoe, çto dlq
vsex ω ∈ Ω 0 v¥polnqetsq sledugwee:
1) dlq lgb¥x x, y ∈ R+
d
, x ≠ y , y t < max ( ), ( )τ τx y{ } ymeet mesto
neravenstvo
ϕt ( x ) ≠ ϕt ( y );
2) dlq lgb¥x x ∈ R+
d y ω ∈ Ω 0 takyx, çto τ ( x, ω ) < ∞ , najdetsq y =
= y ( x, ω ) ∈ R
d
–
1 × { 0 } takoe, çto
ϕτ ( x ) ( x ) = ϕτ ( x ) ( y ),
pryçem ϕt ( x ) = ϕt ( y ) pry t ≥ τ ( x );
3) otobraΩenye ϕt : x t xd∈ <{ }+R : ( )τ → R d qvlqetsq lokal\n¥m homeo-
morfyzmom;
4) P lim inf ( )
,x
x
t T
t
d
x
→∞
∈
∈[ ]
+
= ∞
R
0
ϕ = 1.
Yz dannoj teorem¥ sleduet, çto mnoΩestvo ϕ τt x x t( ): ( ) >{ } otkr¥to, a
mnoΩestvo ϕ τt x x t( ): ( ) ≤{ } soderΩytsq v ϕ t ( R
d
–
1 × { 0 } ). Poπtomu
∂ϕt
d
R+( ) ⊂ ϕ t ( R
d
–
1 × { 0 } ), y dlq dokazatel\stva teorem¥ 1 dostatoçno usta-
novyt\, çto dlq lgboho R > 0 ymeet mesto neravenstvo (sm. p. 4 teorem¥ 3):
H x x Rd
t
d− −∈ ≤{ } × { }( )( )1 1 0ϕ R : < ∞.
Nam potrebuetsq sledugwee utverΩdenye [3].
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
SVOJSTVA POTOKOV, POROÛDENNÁX STOXASTYÇESKYMY … 1071
Teorema 4. PredpoloΩym, çto otobraΩenye f : R
n → R
m
, n ≤ m , prynad-
leΩyt sobolevskomu prostranstvu Wp, loc
1 dlq nekotoroho p > n. Pust\ U ⊂
⊂ R
n — proyzvol\noe ohranyçennoe mnoΩestvo. Tohda H
n
( f ( U ) ) < ∞, hde H
n
— n-mernaq mera Xausdorfa v R
m
.
Zameçanye 3. Yz teorem vloΩenyq Soboleva sleduet suwestvovanye nepre-
r¥vnoj modyfykacyy funkcyy f. Ymenno πta modyfykacyq rassmatryvaetsq v
teoreme 4.
Takym obrazom, dlq dokazatel\stva teorem¥ 1 dostatoçno proveryt\, çto
otobraΩenye
R
d
–
1 ∋ u → ϕt u , 0( )( ) ∈ R
d
(2)
prynadleΩyt prostranstvu
p
p
d dW
>
−
1
1 1∩ , ( , )loc R R .
Zameçanye 4. V stat\e yspol\zugtsq nekotor¥e ydey yz hl. 4 [4], hde doka-
zano, çto reßenye stoxastyçeskoho uravnenyq (bez otraΩenyq) s lypßycev¥my
koπffycyentamy v R
d
dyfferencyruemo po naçal\n¥m dann¥m poçty naver-
noe otnosytel\no mer¥ Lebeha. V çastnosty, otsgda sleduet, çto otobraΩenye
x t xd∈ <{ }+R : ( )τ ∋ x → ϕt ( x ) ∈ R
d
prynadleΩyt
p
p
d dW x t x
>
+∈ <{ }( )
1
1∩ , : ( ) ,loc R Rτ . Odnako (2) ne sleduet
neposredstvenno yz [4].
Rassmotrym approksymyrugwug posledovatel\nost\ processov ϕ ε
t x( )( ) , ε >
> 0, t ≥ 0, x ∈ R
d
–
1 × ( 0, ∞ ), udovletvorqgwyx sledugwym uravnenyqm:
d xtϕ ε( )( ) = a x g x dtt t0 ϕ ϕε
ε
ε( ) ( )( ) ( )( ) + ( )( ) +
k
m
k t ka x dw t
=
∑ ( )
1
ϕ ε( )( ) ( ), t ≥ 0,
ϕ ε
0
( )( )x = x,
hde gε( x ) = 0 0 2, , ,… ( )
/ψ εx
x
d
d
, ψ ∈ C
∞
( ( 0, ∞ ) ), ψ ( y ) = 0 pry y ≥ 2, ψ ( y ) = 1
pry y ≤ 1, ψ monotonno ne vozrastaet.
NesloΩno proveryt\, çto
P ∃ ≥ ∃ ∈ × ∞( ) ∈ × { }( )− −t x xd
t
d0 0 01 1
R R, : ( )( )ϕ ε = 0.
Lemma 1. Dlq lgb¥x x ∈ R
d
–
1 × ( 0, ∞ ), p > 1 y T > 0 ymeet mesto
sxodymost\
lim sup ( ) ( )
,
( )
ε
εϕ ϕ
→ + ∈[ ]
−
0 0
E
t T
t t
p
x x = 0. (3)
Zameçanye 5. Vvedenye funkcyy typa g ε y dokazatel\stvo sxodymosty
ϕ ε
t
( ) → ϕ t , ε → 0, naz¥vaetsq metodom ßtrafov (sm., naprymer, [5, 6]). M¥
pryvedem, odnako, dokazatel\stvo sootvetstvugweho utverΩdenyq, tak kak
predpoloΩenyq lemm¥ 1 neskol\ko otlyçagtsq ot uslovyj ukazann¥x rabot.
Dokazatel\stvo lemm¥ 1. DokaΩem dlq lgboho x slabug sxodymost\
raspredelenyj processov ϕ ε
⋅
( )( )x k raspredelenyg ϕ⋅( )x v prostranstve
C T d0, ,[ ]( )R .
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
1072 A. G. PYLYPENKO
Pust\ h̃ ∈ C
∞
( R ) — monotonnaq funkcyq, takaq, çto
˜( )h x = 0 pry x ≤ 1 / 2
y
˜( )h x = x pry x ≥ 1. Opredelym funkcyg hδ : R
d → R
d
, hde δ > 0,
sledugwym obrazom:
hδ ( x1 , … , xd ) = x x h
x
d
d
1 1, , , ˜…
− δ
.
Oboznaçym Br : = x x rd∈ × ∞( ) ≤{ }−
R
1 0, : .
Dokazatel\stvo sledugwej lemm¥ ymeet standartn¥j xarakter y poπtomu
ne pryvodytsq.
Lemma 2.
1) ∀ p > 1 ∀ r > 0:
sup sup sup ( ) sup ( )
, ,
( )
,x B t T
t
p
t T
t
p
r
x x
∈ ∈( ] ∈[ ] ∈[ ]
+
ε
εϕ ϕ
0 1 0 0
E E < ∞; (4)
2) ∀ x ∈ Br ∀ p > 1 ∀ δ ∈ ( 0, 1 ]:
sup sup ( )
, ,
( )
ε δ
δ
εϕ
∈ /( ] ∈[ ]
( )
0 2 0
E
t T
t
p
h x < ∞; (5)
3) ∀ x ∈ Br ∀ p > 1 ∀ δ ∈ ( 0, 1 ] ∃ c > 0 ∀ ε ∈ 0 2, δ /( ] ∀ s, t ∈ [ 0, T ]:
E h x h xt s
p
δ
ε
δ
εϕ ϕ( ) ( )( ) ( )( ) − ( ) ≤ c t s p− /2
. (6)
Pust\ x fyksyrovano. Dalee budem oboznaçat\ ϕt , ϕ ε
t
( )
vmesto ϕ t ( x ),
ϕ ε
t x( )( ) .
Yz teorem¥ 12. 2 [7] y (5), (6) sleduet, çto dlq lgboho fyksyrovannoho δ > 0
raspredelenye processov hδ
ε
ε δ
ϕ⋅( ){ } ∈ /( ]
( )
,0 2
slabo otnosytel\no kompaktno.
Poπtomu
∀ α > 0 ∃ c > 0 ∀ ε ∈ 0
2
,
δ
: sup sup
, ,
( )
ε δ
δ
εϕ
∈ /( ] ∈[ ]
( ) ≥
0 2 0
P
t T
th c ≤ α
y
∀ α > 0 ∀ β > 0 ∃ η > 0 ∀ ε ∈ 0
2
,
δ
:
P sup ( ) ( )
s t
t sh h
− <
( ) − ( ) ≥
η
δ
ε
δ
εϕ ϕ β ≤ α.
Sledovatel\no,
sup sup
, ,
( )
ε δ
εϕ δ
∈ /( ] ∈[ ]
≥ +
0 2 0
P
t T
t c ≤ α,
sup sup
,
( ) ( )
ε δ η
ε εϕ ϕ β δ
∈ /( ] − <
− ≥ +
0 2
P
s t
t s ≤ α.
Yz proyzvol\nosty v¥bora α , β , δ y teorem¥ 8. 2 [7] sleduet slabaq otnosy-
tel\naq kompaktnost\ raspredelenyj posledovatel\nosty ϕε
ε{ } >0
.
Pust\ εk → 0, k → ∞, — takaq posledovatel\nost\, çto ϕεk{ } slabo sxo-
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
SVOJSTVA POTOKOV, POROÛDENNÁX STOXASTYÇESKYMY … 1073
dytsq pry k → ∞. Yz ocenky (5) y lypßycevosty funkcyj a0 , … , am sleduet
takΩe slabaq otnosytel\naq kompaktnost\ raspredelenyj processov
αt
k : =
0
0
t
sa dsk∫ ( )ϕε
, βt
k : =
j
m t
j s ja dw sk
=
∑ ∫ ( )
1 0
ϕε ( ) ,
a sledovatel\no, y
ξt
k : = ϕ α βε
t t
k
t
kk n( ) − −( ), .
Bez potery obwnosty moΩno sçytat\, çto raspredelenyq processov ϕε
⋅
k
, α⋅
k
,
β⋅
k
, ξ⋅
k
slabo sxodqtsq.
Yz teorem¥ Skoroxoda (sm., naprymer, [8]) sleduet suwestvovanye edynoho
veroqtnostnoho prostranstva y posledovatel\nosty sluçajn¥x processov
ζ̃k = ˜ , ˜ , ˜ , ˜ , , , , ,ϕ α β ξ ϕ ξεk k k k k
m
k k kw w1 …( )
takoj, çto:
a) sovmestnoe raspredelenye ˜ , , ( )ϕ ξεk k( ) …( ) sovpadaet s sovmestn¥m ras-
predelenyem processov ϕ α β ξ ϕ ξε( ), , , , , , , ,k k k k
mw w1 …( ) ;
b) ζ̃k
sxodytsq (v prostranstve neprer¥vn¥x funkcyj) poçty navernoe k
nekotoromu predel\nomu processu ζ̃0 = ˜ , ˜ , ˜ , ˜ , ˜ , , ˜ , ,ϕ α β ξ ϕ ξ0 0 0 0
1
0 0w wm…( ) .
Analohyçno teoreme 1 [1] (hl. 5, § 2) moΩno proveryt\, çto
ϕ̃t
0 = x +
0
0
0
t
ta ds∫ ( )ϕ̃ +
k
m t
k s ka dw s
=
∑ ∫ ( )
1 0
0˜ ˜ ( )ϕ + n tξ̃
0
. (7)
Pry πtom ξ̃t
0
— neub¥vagwyj process kak predel neub¥vagwyx processov.
NesloΩno takΩe zametyt\, çto
0
0
0
0 1
t
s
s
d d∫ ∈ × { }{ }−÷ ˜
˜
ϕ ξ
R
= ξ̃t
0
, t ≥ 0.
Takym obrazom, para ˜ , ˜ϕ ξ0 0( ) qvlqetsq reßenyem stoxastyçeskoho uravnenyq
(7) s normal\n¥m otraΩenyem. Yz edynstvennosty reßenyq sleduet, çto ϕ̃0 =
= ϕ0
, ξ̃0 = ξ0
. Yz (4) y sxodymosty ˜ ( )ϕ εk → ϕ̃0
( = ϕ0
), ϕ ε( )k → ϕ0
pry k →
→ ∞ sleduet, çto
∀ p > 1 E sup ˜
,
( )
t T
t t
p
k
∈[ ]
−
0
0ϕ ϕε
→ 0, k → ∞,
∀ p > 1 E sup
,t T
t
k
t
p
∈[ ]
−
0
0ϕ ϕ → 0, k → ∞.
Poπtomu
E sup
,
( )
t T
t t
p
k
∈[ ]
−
0
ϕ ϕε = E sup ˜
,
( )
t T
t t
k p
k
∈[ ]
−
0
ϕ ϕε ≤
≤ 2 Ep
t T
t t
p
t
k
t
p
k−
∈[ ]
− + −( )1
0
0 0sup ˜ ˜
,
( )ϕ ϕ ϕ ϕε → 0, k → ∞.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
1074 A. G. PYLYPENKO
Yz proyzvol\nosty v¥bora posledovatel\nosty { εk } sleduet (3).
Lemma 1 dokazana.
Pust\ ak k m
( )
, ,
ε
ε{ } > =0 1
— takaq posledovatel\nost\ funkcyj, çto:
1) ak
( )ε ∈ C d d∞
+( )R R, , ε > 0, k = 0, m;
2) ak
( )ε
ravnomerno sxodytsq k ak pry ε → 0 +;
3) ak
( )ε{ } udovletvorqet uslovyg Lypßyca ravnomerno po ε > 0, k = 0, m:
∃ L ∀ x1 , x2 : a x a xk k
( ) ( )( ) ( )ε ε
1 2− ≤ L x x1 2− . (8)
Prymenqq lemm¥ 1 y 2, rassuΩdenyq, analohyçn¥e takov¥m pry dokazatel\stve
teorem¥ 8 [1] (hl. 4, § 1), y dyahonal\n¥j metod Kantora, v¥byraem takye po-
sledovatel\nosty { εk }, { δk }, çto
∀ R > 0 ∀ p > 1 ∀ ξ ∈ BR : lim sup ˆ ( ) ( )
,k t T
t
k
t
p
x x
→∞ ∈[ ]
−E
0
ϕ ϕ = 0, (9)
sup sup ˆ ( )
,x B t T
t
k p
R
x
∈ ∈[ ]
E
0
ϕ < ∞, (10)
hde ϕ̂t
k
— reßenye uravnenyq
d ˆ ( )ϕt
k x = a x g x dtk
kt
k
t
k
0
( ) ˆ ( ) ˆ ( )ε
δϕ ϕ( ) + ( )( ) +
k
m
k t
k
ka x dw tk
=
∑ ( )
1
( ) ˆ ( ) ( )ε ϕ , t ≥ 0,
ˆ ( )ϕ0
k x = x.
Sluçajn¥j process ˆ ( )ϕt
k x ymeet modyfykacyg, neprer¥vno dyfferency-
ruemug po x [9] (dalee rassmatryvaem tol\ko πtu modyfykacyg), pry πtom
proyzvodnaq y xt
k ( ) : =
∂ϕ
∂
ˆ ( )t
k x
x
udovletvorqet uravnenyg
d y xt
k ( ) = ∇ ( ) + ∇ ( )( )a x g x y x dtt
k
t
k
t
k
k0
( ) ˆ ( ) ˆ ( ) ( )ε
δϕ ϕ +
+
k
m
k t
k
t
k
ka x y x dw t
=
∑ ∇ ( )
1
( ) ˆ ( ) ( ) ( )ε ϕ ,
y xk
0 ( ) = ÷,
hde ÷ — edynyçnaq ( d × d )-matryca.
Oboznaçym çerez 〈 ⋅, ⋅ 〉 y || ||⋅ sootvetstvenno skalqrnoe proyzvedenye y nor-
mu Hyl\berta – Ímydta na prostranstve ( d × d )-matryc. Yz toho, çto ψ ′ ( x ) ≤ 0
pry x > 0 (sm. opredelenye funkcyy g ), sleduet, çto
∇g x y yδ ( ) , ≤ 0 (11)
dlq lgboho x ∈ R+
d
y lgboj matryc¥ y. Yspol\zuq (11), nesloΩno proveryt\,
çto
∀ p > 1 ∃ c = c ( p, L ) ∀ x ∈ R+
d
∀ t ∈ [ 0, T ] ∀ ε > 0:
E sup ( )
,s t
s
k p
y x
∈[ ]0
≤ d p /2 + c y x dz
t
z s
z
k p
0 0
∫
∈[ ]
E sup ( )
,
,
hde L — konstanta Lypßyca yz (8).
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
SVOJSTVA POTOKOV, POROÛDENNÁX STOXASTYÇESKYMY … 1075
Sledovatel\no,
sup sup sup ˆ ( )
,ε
ϕ
> ∈ ∈[ ]+
∇
0 0x t T
t
k p
d
x
R
E < ∞. (12)
Nam ponadobytsq sledugwaq texnyçeskaq lemma.
Lemma 3. Pust\ U ⊂ R
k — ohranyçennoe otkr¥toe mnoΩestvo. Dopus-
tym, çto posledovatel\nost\ yzmerym¥x sluçajn¥x polej ξt
n
n
x( ){ } ≥1
, x ∈ U,
t ∈ [ 0, T ], udovletvorqet uslovyqm:
1) dlq lgboho n ≥ 1 sluçajn¥j process ξt
n( )⋅ , t ∈ [ 0, T ], prynymaet zna-
çenyq v W Up
d1( , )R y ymeet neprer¥vn¥e traektoryy (v Wp
1);
2) sup sup
,n t T
t
n
Wp≥ ∈[ ]1 0
1E ξ > ∞;
3) suwestvuet sluçajnoe pole ξt ( x ), t ∈ [ 0, T ], x ∈ U, takoe, çto
lim sup ( ) ( )
,n t T U
t
n
t
p
x x d x
→∞ ∈[ ]
∫ −E
0
ξ ξ = 0.
Tohda process ξt prynymaet znaçenyq v W Up
d1( , )R , pryçem sup
,t T
t Wp∈[ ]0
1ξ
qvlqetsq yzmerym¥m otnosytel\no popolnenyq ysxodnoj σ-alhebr¥ mnoΩest-
vamy nulevoj mer¥ y
E sup
,t T
t Wp∈[ ]0
1ξ ≤ sup sup
,n t T
t
n
Wp
E
∈[ ]0
1ξ .
Dokazatel\stvo. Zametym, çto posledovatel\nost\ ∇ ⋅ ⋅{ } ≥
ξn
n
( )
1
ohrany-
çena v prostranstve L p ( Ω × U × [ 0, T ], d P × d x × d t ) y, sledovatel\no, slabo
kompaktna. Sohlasno teoreme Banaxa – Saksa [10], suwestvuet posledovatel\-
nost\ v¥pukl¥x lynejn¥x kombynacyj η
n =
j
n
n j
jc=∑ 1 , ξ takaq, çto ∇ η
n
sxo-
dytsq v Lp k nekotoromu sluçajnomu πlementu g. Bez potery obwnosty moΩno
sçytat\, çto dlq poçty vsex t ∈ [ 0, T ] (otnosytel\no mer¥ Lebeha) y poçty vsex
ω ∈ Ω ymeet mesto sxodymost\
η ωt
n( , )⋅ → ξ ωt ( , )⋅ v Lp( U, R
d
),
∇ η ωt
n( , )⋅ → gt ( , )⋅ ω v Lp( U, R
d
×
k
).
Sledovatel\no, dlq πtyx t, ω
ξ ωt ( , )⋅ ∈ W Up
d1( , )R , ∇ ξ ωt ( , )⋅ = gt ( , )⋅ ω
y
ξ ωt Wp
( , )⋅ 1 = lim ( , )
n
t
n
Wp→∞
⋅η ω 1 .
Pust\ Θ ⊂ [ 0, T ], Ω 0 ⊂ Ω — sootvetstvugwye mnoΩestva polnoj mer¥, Θ̃ ⊂
⊂ Θ — sçetnoe plotnoe podmnoΩestvo. Tohda sohlasno lemme Fatu
Esup
˜t
t Wp
∈Θ
ξ 1 ≤ lim sup
,n t T
t
n
Wp→∞ ∈[ ]
E
0
1η ≤ sup sup
,n t T
t
n
Wp≥ ∈[ ]1 0
1E ξ .
Poskol\ku mnoΩestvo Θ̃ plotno v [ 0, T ] y process ξt , t ∈ [ 0, T ], ymeet
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
1076 A. G. PYLYPENKO
neprer¥vn¥e traektoryy v Lp (sm. uslovyq 1, 3 lemm¥), s pomow\g rassuΩde-
nyj, analohyçn¥x pred¥duwym, lehko ustanovyt\, çto dlq vsex ω yz nekotoro-
ho mnoΩestva Ω 1 , P ( Ω 1 ) = 1, y vsex t ∈ [ 0, T ] ymeet mesto vklgçenye
ξ ωt ( , )⋅ ∈ W Up
d1( , )R , pryçem
ξ ωt Wp
( , )⋅ 1 ≤ sup ( , )
˜s
s Wp
∈
⋅
Θ
ξ ω 1 .
Otsgda sleduet ravenstvo sup ˜t t Wp∈Θ
ξ 1 = sup ,t T t Wp∈[ ]0 1ξ p. n., y, sledova-
tel\no, dokazatel\stvo lemm¥ 3.
Pust\ R > 0, p > 1. Dlq x = ( x1 , … , xd ) ∈ R
d
oboznaçym x̃ : = ( x1 , … , xd – 1 ).
Pust\ UR = ˜ : ˜x x Rd∈ <{ }−
R
1
. Yz (9), (10), (12) y lemm¥ 3 sleduet, çto dlq
lgboho fyksyrovannoho xd > 0 otobraΩenye
R
d
–
1 ⊃ UR ∋ x̃ � ϕt dx x˜,( ) ∈ R
d
qvlqetsq πlementom sobolevskoho prostranstva W Up R
d1( , )R dlq poçty vsex ω
y vsex t ∈ [ 0, T ], pryçem
sup sup ( , )
, ,
( , )
x t T
t d W U
d
p R
dx
∈( ] ∈[ ]
⋅
0 1 0
1E ϕ
R
< ∞.
Kak uΩe upomynalos\ v naçale stat\y, ϕt ( x ) s veroqtnost\g 1 neprer¥ven po
x, poπtomu yz (4) ymeem sxodymost\
U
t d t
p
R
x x x dx∫ −ϕ ϕ( ˜, ) ( ˜, ) ˜0 → 0, xd → 0 +.
Prymenqq lemmu 3 ewe raz, poluçaem
∀ R > 0 : E sup ( , )
,
( , )
t T
t W Up R
d
∈[ ]
⋅
0
0 1ϕ
R
< ∞.
Yspol\zuq teorem¥ 3, 4, poluçaem dokazatel\stvo teorem¥ 1.
Zameçanye 6. Analohyçno dokazatel\stvu teorem¥ 1 nesloΩno proveryt\,
çto otobraΩenye R+
d ∋ x � ϕ t ( x ) ∈ R
d
prynadleΩyt prostranstvam Soboleva
Wp
d1
R+( ), p > 1, dlq vsex t ≥ 0 s veroqtnost\g 1.
Dokazatel\stvo teorem¥ 2. ProdolΩym funkcyy ak : R+
d → R
d
, k =
= 0, … , m, do lypßycev¥x otobraΩenyj, opredelenn¥x na vsem R
d
:
˜ ( , , , )a x x xk d d1 1… − : =
a x x x
a x x x
k d d
k d d
( , , ), ,
( , , , ), .
1
1 1
0
0 0
… ≥
… <
−
Rassmotrym sledugwee stoxastyçeskoe uravnenye v R
d
:
d xt˜ ( )ϕ = ˜ ˜ ( )a x dtt0 ϕ( ) +
k
m
k t ka x dw t
=
∑ ( )
1
˜ ˜ ( ) ( )ϕ , t ≥ s,
(13)
˜ ( )ϕ0 x = x, x ∈ R
d
.
Sluçajn¥j process ϕ̃ ymeet modyfykacyg [9], neprer¥vnug po ( t, x ), pryçem
dlq lgb¥x ω ∈ Ω, t ≥ 0:
otobraΩenye ϕ̃t : R
d → R
d qvlqetsq homeomorfyzmom. (14)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
SVOJSTVA POTOKOV, POROÛDENNÁX STOXASTYÇESKYMY … 1077
∏ta modyfykacyq y budet rassmatryvat\sq dalee.
Yz edynstvennosty reßenyj (1) y (13) y neprer¥vnosty po ( s, t, x ) sleduet
suwestvovanye mnoΩestva Ω 0 polnoj mer¥ takoho, çto
∀ t ≥ 0 ∀ ω ∈ Ω 0 ∀ x ∈ R+
d , τ ( x ) > t : ϕt ( x, ω ) = ˜ ( , )ϕ ωt x , (15)
hde τ ( x ) = inf : ( )s xs
d≥ ∈ × { }{ }−0 01ϕ R .
Pust\ Ω 1 — mnoΩestvo polnoj mer¥ yz teorem¥ 1 takoe, çto
∀ ω ∈ Ω 1 ∀ t ≥ 0 : dim ,ϕ ωt
d
R
− × { }( )1 0 ≤ d – 1.
Kak b¥lo otmeçeno v naçale dokazatel\stva teorem¥ 1, hranyca ∂ ( )+ϕt
d
R
soderΩytsq v mnoΩestve ϕt
d
R
− × { }( )1 0 . PredpoloΩym protyvnoe k utverΩ-
denyg teorem¥ 2. Tohda najdutsq ω ∈ Ω 0 ∩ Ω 1 y x ∈ R
d− × { }1 0 takye, çto
toçka ϕt ( x, ω ) qvlqetsq vnutrennej toçkoj mnoΩestva ϕ ωt
d
R+( ), .
Pust\ U ⊂ ϕt
d
R+( ) — otkr¥toe mnoΩestvo, soderΩawee ϕt ( x ) (ω sejças
fyksyrovano). Yz (14) sleduet, çto V = ˜ ( )ϕt U−1
takΩe otkr¥to, pryçem
mnoΩestvo A : = U ∩ ˜ ;ϕt
dV R∩ − × −∞( )( )( )1 0 otkr¥to y ne pusto.
Yz teorem¥ 3 y (15) sleduet, çto dlq lgboho y ∈ R+
d
ymeet mesto, po
krajnej mere, odno yz utverΩdenyj
ϕt ( y ) = ϕ̃t y( ) yly ϕt ( y ) ∈ ϕt
d
R
− × { }( )1 0 .
No ϕ̃t
d
R+( ) ∩ ˜ ;ϕt
dR − × −∞( )( )1 0 = ∅ , poπtomu A ⊂ ϕt
d
R
− × { }( )1 0 , çto ne-
vozmoΩno, tak kak dim A = d > d – 1 ≥ dim ϕt
d
R
− × { }( )1 0 .
Poluçennoe protyvoreçye dokaz¥vaet teoremu 2.
V sledugwem prymere pryvedeno uravnenye, dlq kotoroho s poloΩytel\noj
veroqtnost\g v¥polnqetsq neravenstvo dim ϕt
d
R+( ) < d – 1 (sr. s teoremoj 1 y
zameçanyem 1).
Prymer. Pust\ d = 3, S = x x∈ ={ }+R
3 1: . V¥berem mnoΩestvo D ⊂ R
2
y byekcyg g : S → D tak, çto:
1) D — ohranyçennoe v¥pukloe mnoΩestvo s hladkoj hranycej;
2) hranyca ∂ D soderΩyt dva perpendykulqrn¥x otrezka;
3) otobraΩenyq g y g
–
1
dyfferencyruem¥ beskoneçnoe çyslo raz y yme-
gt ohranyçenn¥e proyzvodn¥e lgboho porqdka.
Pust\ ψt ( y ), y ∈ D, — brounovskoe dvyΩenye v D, startugwee yz toçky y
s normal\n¥m otraΩenyem ot hranyc¥:
ψt ( y ) = y + wt +
0
t
s y ds y∫ ( )ν ψ η( ) ( , ),
hde ν — vnutrennqq normal\ k ∂ D, η ( t, y ) ne ub¥vaet po t pry
fyksyrovannom y,
η ( 0, y ) = 0,
0
t
y Ds
ds y∫ ∈∂{ }÷ ψ η( ) ( , ) = η ( t, y ).
Rassmotrym process
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
1078 A. G. PYLYPENKO
ψt ( x ) = x g g
x
xt
−
1 ψ . (16)
Zametym, çto
∀ r > 0 ϕt ( x ) = r
x
rtϕ
, ϕt x( ) = x ,
(17)
ψ t g
x
x
= g
x
x
t
t
ϕ
ϕ
( )
( )
.
Prymenqq formulu Yto k predstavlenyg (16), a zatem yspol\zuq (17),
zameçaem, çto ϕt ( x ) udovletvorqet uravnenyg
d ϕt ( x ) = h x dwt t1 ϕ ( )( ) + h x dtt2 ϕ ( )( ) + n x dtξ( , ),
ϕ0 ( x ) = x, ξ ( x, 0 ) = 0, ξ ( x, t ) =
0
t
xs
d x ds∫ ∈∂{ }+
÷ ϕ ξ
( )
( , )
R
,
hde h1 , h2 — nekotor¥e lypßycev¥ funkcyy.
Yzvestno, çto esly oblast\ D udovletvorqet sformulyrovann¥m dlq nee
uslovyqm, to dlq lgboho t > 0 s poloΩytel\noj veroqtnost\g ymeet mesto
ravenstvo [11]
∀ y1 , y2 ∈ D : ψt ( y1 ) = ψt ( y2 ),
t. e. s poloΩytel\noj veroqtnost\g sluçajnoe otobraΩenye ψt perevodyt vse
toçky oblasty D v odnu. Sledovatel\no, dlq sootvetstvugwyx ysxodov
mnoΩestvo ϕt
d
R+( ) qvlqetsq poluprqmoj y ymeet razmernost\ 1.
1. Hyxman Y. Y., Skoroxod A. V. Stoxastyçeskye dyfferencyal\n¥e uravnenyq y yx prylo-
Ωenyq. – Kyev: Nauk. dumka, 1982. – 612 s.
2. Pilipenko A. Yu. Flows generated by stochastic equations with reflection // Random Oper. and
Stochast. Equat. – 2004. – 12, # 4. – P. 389 – 396.
3. Calderon A. P. On the differentiability of absolutely continuous functions // Riv. mat. Univ.
Parma. – 1951. – 2. – P. 203 – 213.
4. Bouleau N., Hirsch F. Dirichlet forms and analysis on Wiener space // Stud. Math. – Berlin: Walter
de Gruyter, 1991. – 14. – x + 325 p.
5. Lions P. L., Menaldi J.-L., Sznitman A.-S. Construction de processus de diffusion re’fle’chis par
pe’nalisation du domaine // C. r. Acad. sci. Math. – 1981. – 292, # 11. – P. 559 – 562.
6. Menaldi J.-L. Stochastic variational inequality for reflected diffusion // Indiana Univ. Math. J. –
1983. – 2, # 5. – P. 733 – 744.
7. Byllynhsly P. Sxodymost\ veroqtnostn¥x mer. – M.: Nauka, 1977. – 351 s.
8. Kallenberg O. Foundations of modern probability. – 2nd ed. – New York: Springer, 2002. – 638 p.
9. Kunita H. Stochastic flows and stochastic differential equations // Cambridge Stud. Adv. Math. –
1990. – 346 p.
10. Danford N., Ívarc DΩ. T. Lynejn¥e operator¥, obwaq teoryq. – M.: Yzd-vo ynostr. lyt.,
1962. – 895 s.
11. Cranston M., Le Jan Y. Noncoalescence for the Skorohod equation in a convex domain of R
2
//
Probab. Theory Relat. Fields. – 1990. – 87, # 2. – P. 241 – 252.
Poluçeno 24.11.2004
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
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| id | umjimathkievua-article-3665 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:46:44Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/2d/ef667e2467ec2813929f214223dbd02d.pdf |
| spelling | umjimathkievua-article-36652020-03-18T20:01:36Z Properties of the Flows Generated by Stochastic Equations with Reflection Свойства потоков, порожденных стохастическими уравнениями с отражением Pilipenko, A. Yu. Пилипенко, А. Ю. Пилипенко, А. Ю. We consider properties of a random set $\varphi_t(\mathbb{R}_+^d)$, where $\varphi_t(x)$ is a solution of a stochastic differential equation in $\mathbb{R}_+^d$ with normal reflection on the boundary starting at the point $x$. We perform the characterization of inner and boundary points of the set $\varphi_t(\mathbb{R}_+^d)$. We prove that the Hausdorff dimension of the boundary $\partial \varphi_t(\mathbb{R}_+^d)$ is not greater than $d - 1$. Розглядаються властивості випадкової множини $\varphi_t(\mathbb{R}_+^d)$, де $\varphi_t(x)$ — розв'язок стохастичного диференціального рівняння в $\mathbb{R}_+^d$ з нормальним відбиттям від межі, що стартує з точки $x$. Проведено характеризацію внутрішніх та граничних точок множини $\varphi_t(\mathbb{R}_+^d)$. Доведено, що розмірність Хаусдорфа межі $\partial \varphi_t(\mathbb{R}_+^d)$ не перевищує $d - 1$. Institute of Mathematics, NAS of Ukraine 2005-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3665 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 8 (2005); 1069 – 1078 Український математичний журнал; Том 57 № 8 (2005); 1069 – 1078 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3665/4059 https://umj.imath.kiev.ua/index.php/umj/article/view/3665/4060 Copyright (c) 2005 Pilipenko A. Yu. |
| spellingShingle | Pilipenko, A. Yu. Пилипенко, А. Ю. Пилипенко, А. Ю. Properties of the Flows Generated by Stochastic Equations with Reflection |
| title | Properties of the Flows Generated by Stochastic Equations with Reflection |
| title_alt | Свойства потоков, порожденных стохастическими уравнениями с отражением |
| title_full | Properties of the Flows Generated by Stochastic Equations with Reflection |
| title_fullStr | Properties of the Flows Generated by Stochastic Equations with Reflection |
| title_full_unstemmed | Properties of the Flows Generated by Stochastic Equations with Reflection |
| title_short | Properties of the Flows Generated by Stochastic Equations with Reflection |
| title_sort | properties of the flows generated by stochastic equations with reflection |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3665 |
| work_keys_str_mv | AT pilipenkoayu propertiesoftheflowsgeneratedbystochasticequationswithreflection AT pilipenkoaû propertiesoftheflowsgeneratedbystochasticequationswithreflection AT pilipenkoaû propertiesoftheflowsgeneratedbystochasticequationswithreflection AT pilipenkoayu svojstvapotokovporoždennyhstohastičeskimiuravneniâmisotraženiem AT pilipenkoaû svojstvapotokovporoždennyhstohastičeskimiuravneniâmisotraženiem AT pilipenkoaû svojstvapotokovporoždennyhstohastičeskimiuravneniâmisotraženiem |