Transfer of absolute continuity by a flow generated by a stochastic equation with reflection
Let $\varphi_t(x),\quad x \in \mathbb{R}_+ $, be a value taken at time $t \geq 0$ by a solution of stochastic equation with normal reflection from the hyperplane starting at initial time from $x$. We characterize an absolutely continuous (with respect to the Lebesgue measure) component and a singula...
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| Datum: | 2006 |
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| Format: | Artikel |
| Sprache: | Russisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2006
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509673940058112 |
|---|---|
| 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 | 2020-03-18T19:57:46Z |
| description | Let $\varphi_t(x),\quad x \in \mathbb{R}_+ $, be a value taken at time $t \geq 0$ by a solution of stochastic equation with normal reflection from the hyperplane starting at initial time from $x$.
We characterize an absolutely continuous (with respect to the Lebesgue measure) component and a singular component of the stochastic measure-valued process $µ_t = µ ○ ϕ_t^{−1}$, which is an image of some absolutely continuous measure $\mu$ for random mapping $\varphi_t(\cdot)$.
We prove that the contraction of the Hausdorff measure $H^{d-1}$ onto a support of the singular component is $\sigma$-finite.
We also present sufficient conditions which guarantee that the singular component is absolutely continuous with respect to $H^{d-1}$. |
| first_indexed | 2026-03-24T02:44:51Z |
| format | Article |
| fulltext |
UDK 519.21
A. G. Pylypenko (Yn-t matematyky NAN Ukrayn¥, Kyev)
PERENOS ABSOLGTNOJ NEPRERÁVNOSTY POTOKOM,
POROÛDENNÁM STOXASTYÇESKYM URAVNENYEM
S OTRAÛENYEM
Let ϕt x( ), x ∈ +R , be a value taken at time t ≥ 0 by a solution of stochastic equation with normal
reflection from the hyperplane starting at initial time from x. We characterize an absolutely continuous
(with respect to the Lebesgue measure) component and a singular component of the stochastic measure-
valued process µ µ ϕt t= −� 1 , which is an image of some absolutely continuous measure µ for random
mapping ϕt ( )⋅ . We prove that the contraction of the Hausdorff measure Hd−1 onto a support of the
singular component is σ-finite. We also present sufficient conditions which guarantee that the singular
component is absolutely continuous with respect to Hd−1 .
Nexaj ϕt x( ), x ∈ +R , — znaçennq u moment çasu t ≥ 0 rozv’qzku stoxastyçnoho rivnqnnq z
normal\nym vidbyttqm vid hiperplowyny, qke startu[ v poçatkovyj moment çasu z x . U statti
oxarakteryzovano absolgtno neperervnu (vidnosno miry Lebeha) i synhulqrnu komponenty
vypadkovoho miroznaçnoho procesu µ µ ϕt t= −� 1
— obrazu deqko] absolgtno neperervno] miry
µ pry vypadkovomu vidobraΩenni ϕt ( )⋅ . Dovedeno, wo zvuΩennq miry Xausdorfa Hd−1
na
nosij synhulqrno] komponenty σ-skinçenne, a takoΩ navedeno dostatni umovy, qki harantugt\,
wo synhulqrna komponenta [ absolgtno neperervnog vidnosno Hd−1 .
Vvedenye. Pust\ ϕt x( ) — reßenye sledugweho stoxastyçeskoho uravnenyq v
R R+
−= × ∞d d 1 0[ ; ) s normal\n¥m otraΩenyem ot hyperploskosty R
d− ×1 0{ }:
d xtϕ ( ) = a x dt a x dw tt
k
m
k t k0
1
( ) ( )( ) ( ) ( )ϕ ϕ+
=
∑ + n dt xξ( , ), t ∈ [ 0, T ] ,
(0.1)
ϕ0( )x = x , ξ ( 0, x ) = 0, x d∈ +R ,
hde ak
d d: R R+ → , k = 0, … , m, udovletvorqgt uslovyg Lypßyca, { wk ( t ) , k =
= 1, … , m } — nezavysym¥e vynerovskye process¥, n = …( , , , )0 0 1 — normal\ k
hyperploskosty R
d− ×1 0{ }, ξ ( t, x ) — ne ub¥vagwyj po t process dlq lgboho
fyksyrovannoho x d∈ +R , pryçem
ξ ( t, x ) = 1
{ ( ) { }}
( , )ϕ ξ
s
dx
t
ds x∈ ×−∫ R
1 0
0
,
t. e. ξ ( t, x ) vozrastaet tol\ko v te moment¥ vremeny, kohda ϕt
dx( ) { }∈ ×−
R
1 0 .
Zameçanye 0.1. Dlq toho çtob¥ reßyt\ uravnenye (0.1), nado najty paru
processov ϕt x( ) y ξ ( t, x ), udovletvorqgwug ukazann¥m v¥ße uslovyqm.
Pry sdelann¥x predpoloΩenyqx suwestvuet y edynstvenno reßenye uravne-
nyq (0.1) (sm. [1]). Pry πtom [2] suwestvuet neprer¥vnaq po ( t, x ) modyfyka-
cyq processa ϕt x( ), kotoraq budet rassmatryvat\sq dalee.
Pust\ µ — veroqtnostnaq mera v R+
d , absolgtno neprer¥vnaq otnosytel\-
no mer¥ Lebeha λd. Rassmotrym sluçajn¥j meroznaçn¥j process µ µ ϕt t= −� 1,
hde µ ϕ� t
−1
— obraz mer¥ µ pry sluçajnom otobraΩenyy ϕ ωt( , )⋅ .
Osnovnoj vopros, rassmatryvaem¥j v dannoj stat\e, zaklgçaetsq v xarakte-
ryzacyy absolgtno neprer¥vnoj y synhulqrnoj komponent processa µt . Doka-
© A. G. PYLYPENKO, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12 1663
1664 A. G. PYLYPENKO
z¥vaetsq, çto absolgtno neprer¥vnaq komponenta µt
abs
ravna
µ ϕt t
d( )R+
� — su-
Ωenyg mer¥ µt na vnutrennost\ mnoΩestva ϕt
d( )R+ , a synhulqrnaq çast\
µ µ ∂ϕt t t
d
sing =
+( )R
— suΩenyg µt na hranycu ∂ϕt
d( )R+ .
V tret\em punkte stat\y pryvodqtsq dostatoçn¥e uslovyq, harantyrugwye
absolgtnug neprer¥vnost\
µ ∂ϕt t
d( )R+
<< Hd−1. (0.2)
SuΩenye Hd−1
v R
d
na mnoΩestvo ∂ϕt
d( )R+ qvlqetsq σ-koneçnoj meroj
(sm. teoremu 1.1). Poπtomu absolgtnaq neprer¥vnost\ (0.2) ne trebuet utoçne-
nyj.
Otmetym, çto vozmoΩna sytuacyq, kohda razmernost\ Xausdorfa mnoΩestva
∂ϕt
d( )R+ men\ße d – 1 s poloΩytel\noj veroqtnost\g (sm. prymer v [3]), y vop-
ros ob absolgtnoj neprer¥vnosty (0.2) ne qvlqetsq, voobwe hovorq, tryvyal\-
n¥m.
V kaçestve yllgstracyy k poluçenn¥m rezul\tatam pryvodytsq prymer,
pokaz¥vagwyj, çto dlq lgboj absolgtno neprer¥vnoj mer¥ µ v ßare U =
= { }:x xd∈ ≤R 1 mera µ ϕ� t
−1, hde ϕt x( ) — brounovskoe dvyΩenye s otra-
Ωenyem ot hranyc¥ ßara, startugwee yz x, rasklad¥vaetsq v summu
µt =
µ ϕ
∂ϕ
� t Ut
−1
( )
+
µ ϕ
ϕ
� �t Ut
−1
( )
,
hde pervaq mera absolgtno neprer¥vna otnosytel\no λd, a vtoraq — otnosy-
tel\no Hd−1.
1. Xarakteryzacyq absolgtno neprer¥vnoj komponent¥ processa
µµ °° ϕϕt
−1
. Pust\ ϕt x( ) — reßenye stoxastyçeskoho uravnenyq s normal\n¥m ot-
raΩenyem (0.1), hde koπffycyent¥ ak udovletvorqgt uslovyg Lypßyca.
Rassmotrym sluçajn¥j meroznaçn¥j process µ µ ϕt t= −� 1, hde µ — abso-
lgtno neprer¥vnaq koneçnaq mera v R+
d .
Teorema11.1. Dlq poçty vsex ω y lgboho t ≥ 0 mera µt predstavyma v
vyde summ¥ vzaymno ortohonal\n¥x mer
µt = µ ϕ ∂ϕt t
d
t
d( ) ( )\R R+ +
+ µ ∂ϕt t
d( )R+
,
pryçem:
a) pervaq komponenta absolgtno neprer¥vna otnosytel\no d-mernoj me-
r¥ Lebeha, a vtoraq lybo synhulqrna, lybo qvlqetsq nulevoj meroj;
b) nosytel\ mer¥
µ ϕ ∂ϕ� t t
d
−
+
1
( )R
soderΩytsq v sçetnom obæedynenyy mno-
Ωestv koneçnoj ( )d − 1 -mernoj mer¥ Xausdorfa Hd−1.
Dokazatel\stvo. Ortohonal\nost\ suΩenyq mer¥ µt na neperesekagwye-
sq mnoΩestva oçevydna.
Vvedem sluçajnoe mnoΩestvo Ut( )ω = { }: ( )x t xd∈ <+R τ , hde τ( )x =
= inf : ( ) { }s xs
d≥ ∈ ×{ }−0 01ϕ R — moment pervoho popadanyq processa ϕt x( )
na hyperploskost\ R
d− ×1 0{ }.
Yz rezul\tatov [2, 4] sleduet, çto poçty navernoe dlq vsex t ≥ 0 :
1) mnoΩestva ϕt tU( ), ϕt
d
tU( \ )R+ ne peresekagtsq y ravn¥ sootvetstvenno
vnutrennosty y hranyce sluçajnoho mnoΩestva ϕt
d( )R+ , t. e. do momenta popa-
danyq na hyperploskost\ R
d− ×1 0{ } reßenye ϕt x( ) prynadleΩyt vnutrennos-
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
PERENOS ABSOLGTNOJ NEPRERÁVNOSTY POTOKOM, POROÛDENNÁM … 1665
ty sluçajnoho mnoΩestva ϕt
d( )R+ , a posle neho — hranyce;
2) ymeet mesto ravenstvo sluçajn¥x mnoΩestv
ϕt
d
tU( \ )R+ = ϕt
d( ){ }R
− ×1 0 = ∂ϕt
d( )R+ , ϕt tU( ) = ϕt
d( )R+
�;
3) dlq lgboho r > 0
H x x rd
t
d− −∈ ≤ ×1 1 0( ({ } )): { }ϕ R < ∞ ,
hde Hd−1
— mera Xausdorfa v R
d
razmernosty d − 1.
Takym obrazom, nosytel\ mer¥
µ ϕ ∂ϕ� t t
d
−
+
1
( )R
soderΩytsq v sçetnom obæ-
edynenyy mnoΩestv koneçnoj ( )d − 1 -mernoj mer¥ Xausdorfa Hd−1
y, znaçyt,
sootvetstvugwaq mera ne moΩet b¥t\ absolgtno neprer¥vnoj, esly ona ne v¥-
roΩdena. Poπtomu dlq dokazatel\stva teorem¥J1.1 dostatoçno proveryt\ abso-
lgtnug neprer¥vnost\
µ ϕt t
d( ( ))R+
� << λd
.
Pust\ ãk — proyzvol\noe lypßycevo prodolΩenye ak na R
d , ˜ ( )ϕt x — re-
ßenye stoxastyçeskoho uravnenyq (bez otraΩenyq) s koπffycyentamy ãk .
Tohda s veroqtnost\gJ1 ymeet mesto ravenstvo ˜ ( )ϕt x = ϕt x( ) dlq vsex t ≤ τ ( x ) .
Sledovatel\no,
µ ϕt t
d( ( ))R+
� =
µ ϕU tt
� −1 =
µ ϕU tt
� ˜ −1 << µ ϕ� ˜ t
−1.
Yzvestno (sm. [5], teorema 3.3.3), çto s veroqtnost\gJ1 dlq vsex t otobraΩe-
nye ˜ ( , )ϕ ωt ⋅ qvlqetsq πlementom prostranstva Wp
d d
, ( , )loc
1
R R y suwestvuet
modyfykacyq (otnosytel\no mer¥ P × λd
) proyzvodnoj ∇ ˜ ( )ϕt x po naçal\n¥m
dann¥m (po x ), udovletvorqgwaq nekotoromu lynejnomu stoxastyçeskomu urav-
nenyg s ohranyçenn¥my koπffycyentamy. Poπtomu dlq vsex ( ω, x ) yz mno-
Ωestva polnoj ( )P × µ -mer¥ y vsex t ≥ 0 qkobyan det ˜ ( )∇ϕt x ne raven nulg.
Yz lemm¥J5.1 [6] sleduet, çto dlq poçty vsex ω y vsex t ≥ 0 ymeet mesto abso-
lgtnaq neprer¥vnost\ mer µ ϕ λ� ˜ t
d− <<1 .
TeoremaJ1.1 dokazana.
2. Xarakteryzacyq synhulqrnoj komponent¥ mer¥ µµµµ t . V teoremeJ1.1
pokazano, çto razmernost\ Xausdorfa nosytelq mer¥
µ ϕ ∂ϕ� t t
d
−
+
1
( )R
— synhu-
lqrnoj komponent¥ mer¥ µt — ne prev¥ßaet d – 1. V dannom punkte pryvo-
dqtsq dostatoçn¥e uslovyq, harantyrugwye absolgtnug neprer¥vnost\
µ ϕ ∂ϕ� t t
d
−
+
1
( )R
<< Hd
t
d
−
+
1
∂ϕ ( )R
. (2.1)
Nam ponadobytsq sledugwee abstraktnoe utverΩdenye ob absolgtnoj ne-
prer¥vnosty obraza mer¥ otnosytel\no mer¥ Xausdorfa.
Teorema12.1. Pust\ funkcyq f Ld m: R R→ ⊂ approksymatyvno dyf-
ferencyruema v λd-poçty vsex toçkax ohranyçennoho yzmerymoho mnoΩestva
U d⊂ R , hde yzmerymoe mnoΩestvo L ymeet koneçnug k-mernug meru Xaus-
dorfa, H Lk( ) < ∞ . PredpoloΩym, çto ranh matryc¥ ap
∂
∂
=
=
f x
x
i
j i m
j d
( )
,
,
1
1
, so-
stavlennoj yz approksymatyvn¥x çastn¥x proyzvodn¥x, raven k dlq poçty
vsex x U∈ , hde k d≤ .
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
1666 A. G. PYLYPENKO
Tohda
λU
d f� −1 << Hk
L , (2.2)
hde Hk
L — suΩenye mer¥ Xausdorfa na L.
Zameçanye 2.1. Esly funkcyq f ymeet sobolevskug proyzvodnug (yly
daΩe ob¥çn¥e proyzvodn¥e po napravlenyqm poçty vsgdu), to ona takΩe qvlq-
etsq poçty vsgdu approksymatyvno dyfferencyruemoj. Pry πtom sobolevskaq
y approksymatyvnaq proyzvodn¥e poçty vsgdu sovpadagt.
Zameçanye 2.2. SuΩenye mer¥ Xausdorfa Hk
L qvlqetsq koneçnoj me-
roj v otlyçye ot Hk , kotoraq ne qvlqetsq daΩe k-koneçnoj pry k d< .
Dokazatel\stvo teorem¥ analohyçno dokazatel\stvu lemm¥J5.1 y teore-
m¥J5.6 [6]. Snaçala, prymenqq teoremuJ3.1.8 [7], svedem dokazatel\stvo k slu-
çag, kohda f neprer¥vno dyfferencyruema y det
( )
,
,
∂
∂
≠
=
=
f x
x
i
j i k
j k
1
1
0 . Zatem dlq
zaverßenyq dokazatel\stva yskomoho rezul\tata nado yspol\zovat\ metod ras-
sloenyj [8] y formulu (3.2.30) [7].
Sledstvye 2.1. Pust\ µ ( ) ( )dx p x dx= — koneçnaq mera v R
d . Predpo-
loΩym, çto funkcyq f d m: R R→ µ -poçty vsgdu approksymatyvno dyf-
ferencyruema, suwestvuet mnoΩestvo L m⊂ R takoe, çto f x L( ) ∈ dlq
µ-poçty vsex x d∈R y H Lk( ) < ∞ , hde k d≤ . Dopustym, çto
rank ap
∂
∂
=
=
=
f x
x
ki
j i m
j d
( )
,
,
1
1
dlq µ-poçty vsex x. Tohda µ � f −1 << HL
k .
V [4] dokazano, çto dlq poçty vsex ω y vsex t otobraΩenye ϕt( )⋅ prynad-
leΩyt prostranstvu Wp
d d
, ( , )loc
1
R R+ dlq lgboho p > 1. Yz sledstvyq 2.1 v¥te-
kaet sledugwee utverΩdenye.
Teorema12.2. PredpoloΩym, çto dlq lgboho x d∈ +R
P rank( )( ) ,∇ ≥ − ≥ϕt x d t1 0 = 1. (2.3)
Tohda dlq poçty vsex ω
P µ ϕ ∂ϕ ∂ϕ� t
d
t
d
t
dH t− −
+ +
<< ≥( )1 1 0( ) ( ),R R
= 1.
Cel\ dal\nejßej rabot¥ sostoyt v naxoΩdenyy dostatoçn¥x uslovyj, ha-
rantyrugwyx (2.3), tak kak proveryt\ yx neposredstvenno ne predstavlqetsq
vozmoΩn¥m.
V sledugwej teoreme pryvodytsq vyd stoxastyçeskoho uravnenyq, kotoromu
udovletvorqet (sobolevskaq) proyzvodnaq ∇ϕt x( ).
Teorema12.3 [9]. PredpoloΩym, çto koπffycyent¥ uravnenyq (0.1) nepre-
r¥vno dyfferencyruem¥ y ymegt ohranyçenn¥e çastn¥e proyzvodn¥e.
Dopustym, çto dlq lgboho x d∈ ×−
R
1 0{ } ymeet mesto neravenstvo
( ( )),a xk d
k
m
2
1
0
=
∑ > , (2.4)
hde ak d, — d-q koordynata funkcyy a a ak k k d
T= …( , , ), ,1 .
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
PERENOS ABSOLGTNOJ NEPRERÁVNOSTY POTOKOM, POROÛDENNÁM … 1667
Tohda sluçajnoe otobraΩenye ϕt
d d: R R+ → s veroqtnost\gJJ1 dlq vsex
t ≥ 0 prynadleΩyt prostranstvu Soboleva Wp
d d
, ( , )loc
1
R R+ , p > 1, pryçem
sobolevskaq proyzvodnaq y x xt t( ) : ( )= ∇ϕ udovletvorqet uravnenyg
dy x a x y x dtt t t( ) ( ( )) ( )= ∇ 0 ϕ +
+
k
m
k t t k ta x y x dw t P y x n x dt
=
−∑ ∇ − −
1
( ( )) ( ) ( ) ( ) ( ) ( , )ϕ 1 , (2.5)
y x0( ) = 1,
hde n x dt( , ) — toçeçnaq sluçajnaq mera takaq, çto n x t( ,{}) = 1 v tom y
tol\ko v tom sluçae, kohda ϕt x( ) prynadleΩyt hyperploskosty R
d− ×1 0{ };
1J— edynyçnaq matryca, a matryca P pij i j
d= =( ) , 1 opredelena sledugwym ob-
razom:
pij =
1 1
0
, ,
— .
i j d= ≤ −
v ostal\n¥x sluçaqx
Zameçanye 2.3. Reßenye (2.5) (dlq fyksyrovannoho x ) ponymaetsq v sle-
dugwem sm¥sle:
1) dlq lgboho momenta ostanovky τ takoho, çto ϕτ
d x( ) ≠ 0 ( ϕt
d
— d-q
koordynata ϕt ), ymeet mesto ravenstvo
y xt( ) = y x a x y x dss s
t
τ
τ
ϕ( ) ( ( )) ( )+ ∇∫ 0 +
+
k
m t
k s s ka x y x dw s
=
∑ ∫ ∇
1 τ
ϕ( ( )) ( ) ( ),
�t ∈[ , )τ τ ,
hde
�τ τ ϕ= ≥ =inf : ( ){ }s xs
d 0 ;
2) mnoΩestvo { }: ( )t xt
dϕ = 0 soderΩytsq v { }: ( )t Py xt = 0 ;
3) process ( ) ( )1 − P y xt , t ≥ 0, ymeet cádlág traektoryy;
4) process Py xt( ), t ≥ 0, ymeet neprer¥vn¥e traektoryy.
Zameçanye 2.4. Yz teorem¥J2.3 sleduet, çto dlq t < τ ( x ) proyzvodnaq
y xt( ) qvlqetsq reßenyem lynejnoho stoxastyçeskoho uravnenyq
dy x a x y x dt a x y x dw tt t t
k
m
k t t k( ) ( ( )) ( ) ( ( )) ( ) ( )= ∇ + ∇
=
∑0
1
ϕ ϕ ,
y x0( ) = 1.
Zameçanye 2.5. Sluçajnoe otobraΩenye ϕt( )⋅ moΩet ne b¥t\ neprer¥vno
dyfferencyruem¥m po x, daΩe esly vse koπffycyent¥ uravnenyq qvlqgtsq
beskoneçno dyfferencyruem¥my. Naprymer, esly ϕt x( ) — otraΩennoe brou-
novskoe dvyΩenye v [ 0, ∞ ) , startugwee yz toçky x ≥ 0 v nulevoj moment vre-
meny, to nesloΩno proveryt\ (sm. [2]), çto
ϕt x( ) =
x w t t x
t xt
+ <
≥
( ), ( ),
( ), ( ),
τ
ϕ τ0
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
1668 A. G. PYLYPENKO
hde τ( )x — moment pervoho popadanyq ϕt x( ) v nul\, w t( ) — ysxodn¥j vyne-
rovskyj process. Tohda proyzvodnaq
∇ϕt x( ) =
1
0
, ( ),
, ( ),
t x
t x
<
≥
τ
τ
ne ymeet neprer¥vnoj modyfykacyy po x dlq lgboho t > 0.
Dalee v stat\e predpolahaetsq, çto uslovyq teorem¥J2.3 v¥polnqgtsq.
Rassmotrym vopros o spravedlyvosty ravenstva (2.3). Pust\ x d∈ +R fyksy-
rovan. Dalee budem oboznaçat\ ϕt , yt vmesto ϕt x( ), y xt( ).
Pust\ T > 0. Vvedem moment¥ ostanovky
τ ϕ0 0 0( ) inf :{ }c t Tt
d= ≥ = ∧ ,
σ τ ϕn n t
dc t c c T( ) inf ( ) :{ }= ≥ = ∧ ,
τ σ ϕn n t
dc t c T+ = ≥ = ∧1 0( ) inf ( ) :{ } ,
hde c > 0.
NesloΩno zametyt\, çto dlq lgboho t > 0 s veroqtnost\gJJ1 suwestvuet
takoe koneçnoe n, çto t c cn n∈ +[ ( ), ( ))σ τ 1 .
Yz dokazatel\stva teorem¥J1 [9] sleduet, çto
∀ > ∀ >T p0 1: E sup
[ , ]t T
t t
cy y
∈
− →
0
0, c → +0 , (2.6)
hde process yt
c
opredelen sledugwym obrazom:
yc
0 = 1 ,
y y a y ds a y dw st
c
c
c
c
t
s s
c
k
m
c
t
k s s
c
kn
n n
= + ∇ + ∇∫ ∑ ∫
=
σ
σ σ
ϕ ϕ( )
( ) ( )
( ) ( ) ( )0
1
pry t c cn n∈ +[ ( ), ( ))σ τ 1 ,
y Pyt
c
c
c
n
= −τ ( ) , t c cn n∈[ ( ), ( )]τ σ .
Oboznaçym çerez Us t, , s ≤ t , reßenye lynejnoho stoxastyçeskoho uravnenyq
dU a U dt a U dw tst t st
k
m
k t st k= ∇ + ∇
=
∑0
1
( ) ( ) ( )ϕ ϕ ,
(2.7)
Uss = 1 .
Lehko vydet\, çto
y U PU PUt
c
c t
k
n
c c cn k k
=
=
−
∏σ σ τ τ( ), ( ), ( ) , ( )( )
1
1
0 0
=
= U P PU P PU
n k kc t
k
n
c c cσ σ τ τ( ), ( ), ( ) , ( )( )
=
−
∏
1
1
0 0
. (2.8)
Nam ponadobytsq sledugwaq lemma o nev¥roΩdennosty predela proyzvede-
nyj matryc.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
PERENOS ABSOLGTNOJ NEPRERÁVNOSTY POTOKOM, POROÛDENNÁM … 1669
Lemma 2.1. Pust\ { }, , , ( )A k N nn k = 1 , n ≥ 1, — posledovatel\nost\ se-
ryj sluçajn¥x matryc razmera d × d . Dopustym, çto:
1) dlq lgboho k suwestvugt predel¥
Ak : = lim ,
n
n kA
→∞
poçty navernoe,
lim lim max
( )
,
m n m k N n
n kA
→∞ →∞ ≤ ≤
= 0 poçty navernoe;
2) suwestvugt perestanovky σn çysel 1, , ( )… N n takye, çto ymeet mes-
to predel po veroqtnosty
B : = P lim
( )
, ( )( )
n k
N n
n kA
n→∞ =
∏ +
1
1 σ ;
3) suwestvuet y koneçen predel po veroqtnosty
α : = P lim
( )
,
n k
N n
n kA
→∞ =
∑
1
;
4) K : = sup
( )
,
n k
N n
n kAE
=
∏
1
2
< ∞ ;
5) dlq lgboho k ≥ 1
det ( )1 + Ak ≠ 0 poçty navernoe.
Tohda det B ≠ 0 poçty navernoe.
Dokazatel\stvo. Determynant matryc¥ qvlqetsq neprer¥vnoj funkcyej
(ot matryc¥), poπtomu
det B = P lim det
( )
,( )
n k
N n
n kA
→∞ =
∏ +
1
1 ,
y, znaçyt, det B ≠ 0 poçty navernoe, esly y tol\ko esly
P lim ln det
( )
,( )
n k
N n
n kA
→∞ =
∑ +( )
1
1 ≠ – ∞ poçty navernoe,
hde sçytaetsq, çto ln 0 = – ∞ .
Zametym, çto dlq poçty vsex ω pry k k≥ 0( )ω , n n≥ 0( )ω ymeet mesto ne-
ravenstvo An k, /< 1 2 y, sledovatel\no,
ln det( ),1 +( )An k = tr A f An k n k n k, , ,+ ( )2
,
hde f x c xn k, ( ) ≤ , x ≤ 1, c = const .
Tohda
ln det B( ) = lim
( )
,
( )
, ,
n k
N n
n k
k k
N n
n k n k nA f A
→∞ = =
∑ ∑+ ( ) +
1
2
0
tr ζ ,
hde
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
1670 A. G. PYLYPENKO
ζn = ln det( ), ,
k
k
n k
k
k
n kA A
=
−
=
−
∏ ∑+
−
1
1
1
10 0
1 tr
sxodytsq pry n → ∞ k nekotoroj sobstvennoj sluçajnoj velyçyne ζ . Poπto-
mu
E ln det B( ) − −α ζ =
= E lim ln det
( )
,
n k
N n
n k nB A
→∞ =
( ) − −∑
1
tr ζ =
= E lim
( )
, ,
n k k
N n
n k n kf A
→∞ =
∑ ( )
0
2
≤
≤ E lim
( )
,
n k
N n
n kA
→∞ =
∑
1
2
≤ K < ∞ .
Otsgda sleduet koneçnost\ ln det B( ) poçty navernoe, t. e. det B ≠ 0 poçty
navernoe, çto y trebovalos\ dokazat\.
Predstavym sluçajnoe mnoΩestvo A s t s
d= ∈ >{ }[ , ] :0 0ϕ v vyde obæedyne-
nyq neperesekagwyxsq sluçajn¥x yntervalov
A = [ , ) ( , ] ( , )α β α β α β0 0 1 1
2
∪ ∪ ∪ k k
k=
∞
,
hde α0 0= , β1 = T .
Teorema12.4. Dopustym, çto v¥polnqgtsq uslovyq teorem¥J2.3 y dlq
lgboho k ≥ 0
P( ( ( ) ) ),rank PU x P d
k kα β = − 1 = 1.
Tohda P( ( ) , [ , ])rank y x d s ts ≥ − ∈1 0 = 1.
Dokazatel\stvo. Zametym, çto funkcyq rank ys , s ≥ 0, nevozrastagwaq,
poπtomu dlq dokazatel\stva teorem¥ dostatoçno proveryt\, çto rank yt ≥ d – 1
poçty navernoe. TakΩe bez potery obwnosty moΩno sçytat\, çto A sostoyt yz
beskoneçnoho çysla yntervalov.
Dlq matryc¥ B razmerom d × d çerez B̃ budem oboznaçat\ matrycu, sosto-
qwug yz perv¥x d – 1 strok y d – 1 stolbcov matryc¥ B.
Yz (2.6), (2.8) sleduet
˜ ˜ ˜ ˜
( ), ( ), ( ) , ( )y U U Ut
c
c t
k
n
c c cn k k
=
=
−
∏σ σ τ τ
1
1
0 0
→ ỹt , c → 0 + . (2.9)
Vospol\zuemsq lemmojJ2.1. PoloΩym
A Uk k k
: ˜
,= −α β 1 , A Un k n nj jk k
, ( / ), ( / ): ˜= −σ τ1 1 1 ,
hde ynterval ( / / )( ), ( )σ τj jk k
n n1 1 soderΩytsq v ( , )α βk k .
UslovyeJ1 lemm¥ v¥polnqetsq, tak kak Us t, , s ≤ t , neprer¥vno po ( s, t )
[10]; uslovye 2 v¥polnqetsq vsledstvye (2.9), a uslovye 5 — v sylu predpolo-
Ωenyj teorem¥.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
PERENOS ABSOLGTNOJ NEPRERÁVNOSTY POTOKOM, POROÛDENNÁM … 1671
Dlq proverky uslovyq 3 lemm¥ dostatoçno ubedyt\sq v suwestvovanyy pre-
dela po veroqtnosty posledovatel\nosty
k c t
c t
s c t s
j
m
c t
c t
k s c t s k
k
k
k
k
k
k
a U ds a U dw s∑ ∫ ∑ ∫
∧
∧
∧
= ∧
∧
∧∇ + ∇
σ
τ
σ
σ
τ
σϕ ϕ
( )
( )
( ) ,
( )
( )
( ) ,( ) ( ) ( )0
1
pry c → 0 + .
Zametym, çto pervoe slahaemoe sxodytsq dlq poçty vsex ω k v¥raΩenyg
0
0
t
k
s sk k k
s a U ds∫ ∑ ∇1( , ) ,( ) ( )α β αϕ
po teoreme Lebeha o maΩoryruemoj sxodymosty.
Posledovatel\nost\ vtor¥x slahaem¥x fundamental\na v L2( , , )Ω F P .
Dejstvytel\no,
E
σ
τ
σϕ
p
p
p
c t
c t
j s c t s k
j
m
p
a U dw s
( )
( )
( ) ,( ) ( )
1
1
1
1 ∧
∧
∧
=
∫∑∑ ∇ –
–
σ
τ
σϕ
l
l
l
c t
c t
j s c t s k
j
m
l
a U dw s
( )
( )
( ) ,( ) ( )
2
2
2
1
2
∧
∧
∧
=
∫∑∑ ∇ ≤
j
m
x
ja x
=
∑ ∇
1
sup ( )
2
×
× E
0
2
1 1 1 2 2 2
t
p
c t s c t c t
l
c t s c t c tU s U s ds
p p p l l l∫ ∑ ∑∧ ∧ ∧ ∧ ∧ ∧−σ σ τ σ σ τ( ) , [ ( ) , ( ) ] ( ) , [ ( ) , ( ) ]( ) ( )1 1 → 0
pry c c1 2 0, → + po teoreme Lebeha o maΩoryruemoj sxodymosty, tak kak [10]
E sup
0
2
≤ ≤ ≤s t t
stU < ∞ .
Analohyçn¥m obrazom proverqetsq uslovye 4 lemm¥J2.1. Tem sam¥m teore-
maJ2.4 dokazana.
Sledstvye 2.2. PredpoloΩym, çto dlq lgb¥x x y d, ( , )∈ × ∞−
R
1 0 , t ≥ 0
y ( d × d ) -matryc¥ V, rankV d= − 1, uslovnaq veroqtnost\
P{ }, ( , )( ) / ( )rank PU x V d x yt t x tβ ϕ< − =1 = 0, (2.10)
hde U xs t, ( ) — reßenye (2.7) s ϕ ϕt t x= ( ),
β ϕ( , ) inf : ( ) { }{ }t x z t xz
d= ≥ ∈ ×−
R
1 0 .
Tohda spravedlyvo (2.3).
Dokazatel\stvo. Pust\
α ϕ( , ) sup [ ; ] : ( ) { }{ }t x s t xs
d= ∈ ∈ ×−0 01
R y α( , )t x = 0 ,
esly sootvetstvugwee mnoΩestvo pusto, t. e. α β( , ), ( , )t x t x( ) — maksymal\n¥j
ynterval, na kotorom ϕt
d x( ) ≠ 0.
Zametym, çto PU Pt x t xα β( , ), ( , ) = PU U Pt t x t x t, ( , ) ( , ),β α , pryçem U Pt x tα( , ),
qvlqetsq Ft-yzmerym¥m. Pust\ ρt dy dV( , ) — raspredelenye par¥ processov
ϕ αt t x tx U P( ), ( , ),( ). Tohda
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
1672 A. G. PYLYPENKO
P( )( , ), ( , )rank PU P dt x t xα β < − 1 =
= P( ), ( , ) ( , ),rank PU U P dt t x t x tβ α < − 1 =
= ∫ < − = =P( / ( ) , ) ( , ), ( , ) ( ( , ), )rank PU V d x y U V dy dVt t x t t x t tβ αϕ ρ1 = 0.
Dlq zaverßenyq dokazatel\stva ostaetsq zametyt\, çto P( ( , ),rank U Pt x tα =
= d − =1 1) , tak kak s veroqtnost\gJJ1 matryca Us t, nev¥roΩdena dlq vsex s,
t, s ≤ t, y prymenyt\ teoremuJ2.4.
Prymer12.1. Uslovyq sledstvyqJ2.2 moΩno ynterpretyrovat\ kak uslovyq
nepopadanyq processa Yto ( ), det ˜ϕt
d
stU , t ≥ s , v toçku (0; 0). Nekotor¥e dos-
tatoçn¥e uslovyq dlq πtoho ymegt dovol\no prostoj vyd.
Rassmotrym dvumernug sytuacyg. V πtom sluçae process det Ũst raven Ust
11
— πlementu pervoho stolbca y pervoj stroky matryc¥ Ust . Para ( ),ϕt stU2 11
udovletvorqet sledugwemu sootnoßenyg:
d a a dw tt t k t k
k
m
ϕ ϕ ϕ2
0
2 2
1
= +
=
∑( ) ( ) ( ), t s z s xz∈ ≥ =[ { }); inf ( )ϕ2 0 ,
dU a U a U dtst t st t st
11
0
1
1
11
0
1
2
12= ′ + ′(( ) ( ) )( ) ( )ϕ ϕ +
+
k
m
k t st k t st ka U a U dw t
=
∑ ′ + ′
1
1
1
11 1
2
12(( ) ( ) )( ) ( ) ( )ϕ ϕ .
Vse process¥ pod dyfferencyalom qvlqgtsq neprer¥vn¥my. Poπtomu dos-
tatoçn¥m uslovyem nepopadanyq ( ),ϕt stU11
v (0; 0) qvlqetsq, naprymer, nev¥-
roΩdennost\ vo vsex toçkax dyffuzyonnoj xarakterystyky. Poskol\ku mat-
ryca Ust nev¥roΩdena s veroqtnost\gJJ1 dlq vsex s, t, vektor ( ),U Ust st
11 12
ne-
nulevoj, y takym dostatoçn¥m uslovyem qvlqetsq, naprymer, sledugwee:
dlq lgb¥x x ∈ +R
2 , y ∈R
2 , y ≠ 0 , vektor¥ ( )( )a xk k m
2
1≤ ≤ y ( )( )∇ ≤ ≤y k k ma x1
1
lynejno nezavysym¥.
Prymer12.2. Pust\ ϕt x( ) — brounovskoe dvyΩenye s otraΩenyem v edynyç-
nom ßare prostranstva R
d , µ — absolgtno neprer¥vnaq mera v { }x ≤ 1 ,
µ µ ϕt t= −� 1.
V¥polnyv dostatoçno hladkye lokal\n¥e zamen¥ peremenn¥x, nesloΩno
rasprostranyt\ rezul\tat¥ stat\y na dann¥j prymer.
V dannom sluçae Us t, = 1 y uslovye, kotoroe sootvetstvuet (2.10), prynyma-
et sledugwyj vyd:
dlq lgboho x, x < 1, y matryc¥JJV, rank V d= − 1,
P{ ( ) }( ( , ))rank P t x V dtϕ β < − =1 0, (2.11)
hde P ( y ) — proektor na kasatel\nug ploskost\ sfer¥ { }x = 1 v toçke y.
Oçevydno, (2.11) ystynno.
Takym obrazom, µt predstavyma v vyde summ¥ absolgtno neprer¥vnoj
µt
abs : = µ ϕ ϕt x xt t( ) ( )\≤ ∂ ≤1 1 y synhulqrnoj µ µ ϕt t xt
sing : ({ })= ∂ ≤ 1 kompo-
nent. Pry πtom H xd
t
− ∂ ≤ < ∞1 1( ({ }))ϕ y µt
sing
absolgtno neprer¥vna ot-
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
PERENOS ABSOLGTNOJ NEPRERÁVNOSTY POTOKOM, POROÛDENNÁM … 1673
nosytel\no suΩenyq mer¥ Xausdorfa Hd−1
na mnoΩestvo ∂ ≤ϕt x({ })1 .
Otmetym takΩe, çto esly t s w s> ≥inf : ( ){ }2 , to dlq vsex x moment
τ( )x popadanyq reßenyq, startugweho yz x, na hranycu ßara ne prev¥ßaet t.
Poπtomu ∂ ≤ = ≤ϕ ϕt tx x({ }) ({ })1 1 (sootvetstvugwyj fakt upomynalsq v
dokazatel\stve teorem¥J1.1, odnako moΩet b¥t\ perenesen y na sluçaj ohrany-
çennoj oblasty s hladkoj hranycej).
Sledovatel\no, dlq takyx t absolgtno neprer¥vnaq komponenta qvlqetsq
nulevoj meroj.
1. Tanaka H. Stochastic differential equations with reflecting boundary condition in convex regions //
Hiroshima Math. J. – 1979. – 9, # 1. – P. 163 – 177.
2. Pilipenko A. Yu. Flows generated by stochastic equations with reflection // Random Oper. and Sto-
chast. Equat. – 2004. – 12, # 4. – P. 389 – 396.
3. Pylypenko A. G. Stoxastyçeskye potoky s otraΩenyem // Dopov. NAN Ukra]ny. – 2005. –
# 10. – S. 23 – 29.
4. Pylypenko A. G. Svojstva potokov, poroΩdenn¥x stoxastyçeskymy uravnenyqmy s otra-
Ωenyem // Ukr. mat. Ωurn. – 2005. – 57, # 8. – S.J1069J–J1078.
5. Bouleau N., Hirsch F. Dirichlet forms and analysis on Wiener space // Stud. Math. – Berlin: Walter
de Gruyter & Co, 1991. – 14. – X + 325 p.
6. Bohaçev V. Y., Smolqnov O. H. Analytyçeskye svojstva beskoneçnomern¥x raspredelenyj //
Uspexy mat. nauk. – 1990. – 45, v¥p. 3(273). – S. 3 – 83.
7. Federer H. Heometryçeskaq teoryq mer¥: Per. s anhl. – M.: Nauka, 1987. – 760 s.
8. Dav¥dov G. A., Lyfßyc M. A. Metod rassloenyj v nekotor¥x veroqtnostn¥x zadaçax //
Ytohy nauky y texnyky. Teoryq veroqtnostej, mat. statystyka, teor. kybernetyka / VYNY-
TY. – 1984. – 22. – S. 61 – 157.
9. Pylypenko A. G. Pro uzahal\nenu dyferencijovanist\ za poçatkovymy danymy potoku, po-
rodΩenoho stoxastyçnym rivnqnnqm z vidbyttqm // Teoryq veroqtnostej y mat. statystyka. –
2006. – 75. – S. 136 – 148.
10. Kunita H. Stochastic flows and stochastic differential equations // Cambridge Stud. Adv. Math. –
1990. – 24. – 346 p.
Poluçeno 15.03.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
|
| id | umjimathkievua-article-3562 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:44:51Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a9/b8e0880ce2e76454d48faf20da8917a9.pdf |
| spelling | umjimathkievua-article-35622020-03-18T19:57:46Z Transfer of absolute continuity by a flow generated by a stochastic equation with reflection Перенос абсолютной непрерывности потоком, порожденным стохастическим уравнением с отражением Pilipenko, A. Yu. Пилипенко, А. Ю. Пилипенко, А. Ю. Let $\varphi_t(x),\quad x \in \mathbb{R}_+ $, be a value taken at time $t \geq 0$ by a solution of stochastic equation with normal reflection from the hyperplane starting at initial time from $x$. We characterize an absolutely continuous (with respect to the Lebesgue measure) component and a singular component of the stochastic measure-valued process $µ_t = µ ○ ϕ_t^{−1}$, which is an image of some absolutely continuous measure $\mu$ for random mapping $\varphi_t(\cdot)$. We prove that the contraction of the Hausdorff measure $H^{d-1}$ onto a support of the singular component is $\sigma$-finite. We also present sufficient conditions which guarantee that the singular component is absolutely continuous with respect to $H^{d-1}$. Нехай $\varphi_t(x),\quad x \in \mathbb{R}_+ $ — значення у момент часу $t \geq 0$ розв'язку стохастичного рівняння з нормальним відбиттям від гіперплощини, яке стартує в початковий момент часу з $x$. У статті охарактеризовано абсолютно неперервну (відносно міри Лебега) і сингулярну компоненти випадкового мірозначного процесу $\mu_t = \mu \circ \varphi_t^{-1}$ — образу деякої абсолютно неперервної міри $\mu$ при випадковому відображенні $\varphi_t(\cdot)$. Доведено, що звуження міри Хаусдорфа $H^{d-1}$ на носій сингулярної компоненти $\sigma$-скінченне, а також наведено достатні умови, які гарантують, що сингулярна компонента є абсолютно неперервною відносно $H^{d-1}$. Institute of Mathematics, NAS of Ukraine 2006-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3562 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 12 (2006); 1663–1673 Український математичний журнал; Том 58 № 12 (2006); 1663–1673 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3562/3858 https://umj.imath.kiev.ua/index.php/umj/article/view/3562/3859 Copyright (c) 2006 Pilipenko A. Yu. |
| spellingShingle | Pilipenko, A. Yu. Пилипенко, А. Ю. Пилипенко, А. Ю. Transfer of absolute continuity by a flow generated by a stochastic equation with reflection |
| title | Transfer of absolute continuity by a flow generated by a stochastic equation with reflection |
| title_alt | Перенос абсолютной непрерывности потоком, порожденным стохастическим уравнением с отражением |
| title_full | Transfer of absolute continuity by a flow generated by a stochastic equation with reflection |
| title_fullStr | Transfer of absolute continuity by a flow generated by a stochastic equation with reflection |
| title_full_unstemmed | Transfer of absolute continuity by a flow generated by a stochastic equation with reflection |
| title_short | Transfer of absolute continuity by a flow generated by a stochastic equation with reflection |
| title_sort | transfer of absolute continuity by a flow generated by a stochastic equation with reflection |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3562 |
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