On Markov measure-valued processes in a finite space
We consider stochastic flows with interaction in a finite phase space. The flows with variable generators generating evolutionary measure-valued processes are described. The influence of the interaction of particles on the entropy of the flow is analyzed.
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| Дата: | 2008 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2008
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3271 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509327976038400 |
|---|---|
| author | Ostapenko, E. V. Остапенко, Е. В. Остапенко, Е. В. |
| author_facet | Ostapenko, E. V. Остапенко, Е. В. Остапенко, Е. В. |
| author_sort | Ostapenko, E. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:49:31Z |
| description | We consider stochastic flows with interaction in a finite phase space. The flows with variable generators generating evolutionary measure-valued processes are described. The influence of the interaction of particles on the entropy of the flow is analyzed. |
| first_indexed | 2026-03-24T02:39:21Z |
| format | Article |
| fulltext |
UDK 517.21
E. V. Ostapenko (Nac. texn. un-t Ukrayn¥ „KPY”, Kyev)
O MARKOVSKYX MEROZNAÇNÁX PROCESSAX
NA KONEÇNOM PROSTRANSTVE*
Stochastic flows with the interaction on a finite phase space are considered. We describe flows with a
variable generator that give rise to evolutionary measure-valued processes. We establish how the
interaction of particles influences the entropy of the flow.
Rozhlqdagt\sq stoxastyçni potoky z vza[modi[g na skinçennomu fazovomu prostori. Opysano
taki potoky zi zminnym heneratorom, wo porodΩugt\ evolgcijni miroznaçni procesy. Vstanov-
leno, qk vza[modiq çastynok vplyva[ na entropig potoku.
V rabote [1] rassmatryvalys\ markovskye meroznaçn¥e process¥ s postoqnnoj
massoj, kotor¥e opys¥vagtsq kak perenos nekotoroj naçal\noj mer¥ stoxasty-
çeskymy potokamy. V nastoqwej stat\e opysan¥ stoxastyçeskye potoky na ko-
neçnom prostranstve s peremenn¥m heneratorom, poroΩdagwye takye proces-
s¥. V kaçestve prymera ustanovleno, kak vzaymodejstvye çastyc vlyqet na πn-
tropyg potoka.
Pryvedem obwee opredelenye meroznaçnoho processa. Pust\ ( , )� ρ — pol-
noe separabel\noe metryçeskoe prostranstvo s borelevskoj σ-alhebroj B( )� ,
� — prostranstvo veroqtnostn¥x mer na � s metrykoj slaboj sxodymosty,
( , , )Ω F P — veroqtnostnoe prostranstvo s fyl\tracyej Ft t, ≥{ }0 na nem.
Opredelenye. Markovskyj process µt t, ≥{ }0 v � otnosytel\no poto-
ka σ -alhebr Ft t, ≥{ }0 naz¥vaetsq πvolgcyonn¥m, esly dlq lgboj naçal\-
noj mer¥ µ0 ∈� suwestvuet yzmerymaq funkcyq f : Ω × 0, +∞[ ) × � → �
takaq, çto:
1) dlq lgboho t ≥ 0 suΩenye f na 0, t[ ] qvlqetsq Ft × B t0,[ ]( ) × B( )� -
yzmerym¥m;
2) dlq lgboho t ≥ 0 µt = µ0 ° ft
– (mod )1 P ;
3) dlq lgb¥x n ≥ 1, u 1, … , un ∈� sluçajn¥j process µt({ , f ut ( )1 , …
… f ut n( )), t ≥ 0} qvlqetsq markovskym otnosytel\no potoka Ft t, ≥{ }0 .
Potok Ft t, ≥{ }0 naz¥vaetsq stoxastyçeskym potokom so vzaymodejst-
vyem, sootvetstvugwym πvolgcyonnomu processu µt t, ≥{ }0 .
V sluçae dyskretnoho prostranstva � stoxastyçeskyj potok na � udobno
ynterpretyrovat\ kak opysanye dvyΩenyq çastyc, startugwyx yz kaΩdoj toç-
ky prostranstva �. Pry πtom veroqtnostn¥e xarakterystyky poloΩenyq dvy-
Ωenyq otdel\noj çastyc¥ v buduwem zavysqt ot poloΩenyq vsex çastyc v nas-
toqwyj moment.
Pust\ � = 1{ , … , d}. Tohda prostranstvo otobraΩenyj yz � v � koneçno,
çto daet vozmoΩnost\ upravlqt\ ynfynytezymal\n¥my xarakterystykamy pro-
cessa f tt , ≥{ }0 .
Opyßem klass upravlqem¥x stoxastyçeskyx potokov f tt , ≥{ }0 na �, soot-
vetstvugwyx πvolgcyonn¥m processam µt t, ≥{ }0 .
Rassmotrym stoxastyçesky neprer¥vn¥j markovskyj process f tt , ≥{ }0 na
mnoΩestve otobraΩenyj � v � y vvedem sledugwye oboznaçenyq. Çerez i =
= (i1,…, id ) oboznaçym πlement prostranstva �
�
; s πlementom i assocyyruem
* PodderΩano hrantom GP/F13/0195.
© E. V. OSTAPENKO, 2008
1572 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
O MARKOVSKYX MEROZNAÇNÁX PROCESSAX NA KONEÇNOM PROSTRANSTVE 1573
otobraΩenye f : � → �, pry kotorom f ( )1 = i1, … , f d( ) = id . Veroqtnost\ pe-
rexoda za vremq s yz sostoqnyq j v moment vremeny t v sostoqnye i obo-
znaçymGçerez P j it t s, ( , )+ , a vektor πtyx veroqtnostej — çerez P jt t s, ( )+ =
=
P j it t s i, ( , )+ ∈( ) �� .
Kak perexodn¥e veroqtnosty stoxastyçesky neprer¥vnoho markovskoho pro-
cessa na koneçnom prostranstve, P jt t s, ( )+ udovletvorqgt nekotoromu dyffe-
rencyal\nomu uravnenyg [2]. V¥delym klass uravnenyj, reßenyq kotor¥x op-
redelqgt perexodn¥e veroqtnosty processov na �
�
, sootvetstvugwyx markov-
skym πvolgcyonn¥m processam.
Pust\ µ0 =
l l la∈∑ �
δ , al ≥ 0, l ∈�,
l la∈∑ �
= 1, — naçal\naq mera na �.
Opredelym otobraΩenye Ψ, dejstvugwee yz prostranstva otobraΩenyj � v
� v prostranstvo � sledugwym obrazom:
Ψ( )i =
l
l ia
l
∈
∑
�
δ , i ∈ �
�
.
Pust\ A — neprer¥vnoe otobraΩenye otrezka 0 1,[ ] v prostranstvo stoxas-
tyçeskyx matryc na �
�
takoe, çto πlement¥ matryc¥ Aj
i ( )⋅ dlq fyksyrovan-
noho i zavysqt tol\ko ot Ψ( )i , t.Ge. A j
i ( )⋅ = Ar
i ( )⋅ , esly Ψ( )j = Ψ( )r .
Lemma. Esly perexodn¥e veroqtnosty P jt t s, ( )+ stoxastyçesky neprer¥v-
noho markovskoho processa f tt , ≥{ }0 udovletvorqgt uravnenyg
d
ds
P jt t s, ( )+ = P j A u t s Et t s, ( ) ( ( )) –+ +( ) (1)
s naçal\n¥m uslovyem P j it t, ( , ) = 1, hde A ymeet ukazannoe svojstvo, E —
edynyçnaq matryca, to dlq lgboho k , 0 ≤ k ≤ d, process µt({ , ft ( )1 , …
… , f kt ( ))}, hde µt = µ0 ° ft( )–1
, qvlqetsq markovskym.
Dokazatel\stvo. Dlq matryçnoznaçnoj funkcyy B t( ), t ≥ 0, oboznaçym
çerez Ωt
t s B+ ( ) matrycant uravnenyq ẋ = x B t( ) , x t( ) = x0. Sohlasno [3], matry-
cant uravnenyq (1) ymeet vyd Ωt
t s A E+ ( – ) = Ωt
t s A+ ( ) e s–
, a matrycant Ωt
t s A+ ( )
predstavym v vyde
Ωt
t s A+ ( ) = E +
t
t s
A d
+
∫ ( )τ τ +
t
t s
t
t
A A d d
+ +
∫ ∫( ) ( )τ τ τ τ
τ
1 1 + … .
Otsgda sleduet, çto matryca Ωt
t s A+( ( ) – E) ymeet ukazannoe dlq matryc¥ A
svojstvo, a dlq P jt t s, ( )+ poluçaem predstavlenye P jt t s, ( )+ = P jt t, ( ) Ωt
t s A+( ( ) –
E) e s– + P jt t, ( ) e s–
, hde pervoe slahaemoe zavysyt tol\ko ot Ψ( )j .
Zametym, çto µt = Ψ( )ft . Tohda dlq proyzvol\n¥x t, s ≥ 0 y ∆ ∈� ,
i1 ∈� , … , ik ∈�
P µ µt s t s t s k r r rf i f k i f f k r t+ + +∈ = … = … ≤{ }∆, ( ) , , ( ) / , ( ), , ( ),1 11 =
= M I f I f f k f f f k r tt s i i t s t s r r rk∆ Ψ Ψ( ) ( ), , ( ) / ( ), ( ), , ( ),( , , )+ … + +( ) …( ) … ≤{ }1
1 1 =
=
i i
t s i i i i t s t s r
k d
k k d
I f I f f d f
+
+
…
+ … … + +∑ …( ){
1
1 1 1
1
, ,
( ) ( , , , , , )– ( ) ( ), , ( ) / ( )M Ψ ∆ Ψ ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
1574 E. V. OSTAPENKO
f f k r tr r( ), , ( ),1 … ≤ } =
i i
t s i t s r
k d
I f I f f
+ …
+ +∑ {
1
1
, ,
( )– ( ) ( )/ ( )M Ψ ∆ Ψ ,
f f k r tr r( ), , ( ),1 … ≤ } .
Voz\mem uslovnoe matematyçeskoe oΩydanye po bolee ßyrokoj σ-alhebre, po-
roΩdennoj processom f r tr , ≤{ }:
M I f I f f f f k r tt s i t s r r rΨ ∆ Ψ– ( )
( ) ( )/ ( ), ( ), , ( ),1 1+ + … ≤{ } =
= M M I i I f f r t fi t s r rΨ ∆ Ψ– ( )
( ) ( )/ , / ( )1 + ≤{ }{ , f f k r tr r( ), , ( ),1 … ≤ } =
= M I i P f i f f f k r tt t s t r r rΨ ∆ Ψ– ( ) ,( ) ( , )/ ( ), ( ), , ( ),1 1+ … ≤{ } .
Vospol\zuemsq predstavlenyem P f it t s t, ( , )+ :
P µ µt s t s t s k r r rf i f k i f f k r t+ + +∈ = … = … ≤{ }∆, ( ) , , ( ) / , ( ), , ( ),1 11 =
=
i i
t
t s
f
i s
r
k d
t
I i A E e f
+ …
+∑ ( ){
1
1
, ,
( )
–
– ( ) ( ) – / ( )M Ψ ∆ Ω Ψ , f f k r tr r( ), , ( ),1 … ≤ } +
+
i i
s
i t r
k d
e I i I f f
+ …
{ }∑ {
1
1
, ,
–
( )– ( ) ( )/ ( )M Ψ ∆ Ψ , f f k r tr r( ), , ( ),1 … ≤ } =
=
i i
t
t s
f
i s
t
k d
t
I i A E e f
+ …
+∑ ( ){ }
1
1
, ,
( )
–
– ( ) ( ) – / ( )M Ψ ∆ Ω Ψ +
+ e I i I f I f f k fs
i
i t i i t t rk
–
( ) ( , , )– ( ) ( ) ( ), , ( ) ( )/M
∈
…∑
…( )
��
Ψ ∆ Ψ1 1
1 ,
f f k r tr r( ), , ( ),1 … ≤
=
=
i i
t
t s
f
i s
t
k d
t
I i A E e f
+ …
+∑ ( ){ }
1
1
, ,
( )
–
– ( ) ( ) – / ( )M Ψ ∆ Ω Ψ +
+ e I f I f f k fs
t i i t t tk
–
( ) ( , , )– ( ) ( ), , ( ) / ( )M{ …( )…Ψ ∆ Ψ1 1
1 , f f kt t( ), , ( )1 … }.
Sledovatel\no,
P µt s t s t s kf i f k i+ + +∈ = … ={ ∆, ( ) , , ( )1 1 / µ r r rf f k r t, ( ), , ( ),1 … ≤ } =
= P µt s t s t s kf i f k i+ + +∈ = … ={ ∆, ( ) , , ( )1 1 / µt t tf f k, ( ), , ( )1 … }.
Lemma dokazana.
Takym obrazom, sluçajn¥j process µ t , t ≥ 0 qvlqetsq markovskym πvolg-
cyonn¥m processom v �, otnosytel\no stoxastyçeskoho potoka f t{ , t ∈
∈ 0, T[ ]}.
Rassmotrym sluçaj dyskretnoho vremeny k k, ≥{ }0 y cep\ Markova f k{ ,
k ≥ }0 s matrycej perexodn¥x veroqtnostej A u( ) = u Q + ( – )1 u P , hde u ∈
∈ 0 1,[ ] ( dlq processa s neprer¥vn¥m vremenem moΩno rassmotret\ vloΩennug
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
O MARKOVSKYX MEROZNAÇNÁX PROCESSAX NA KONEÇNOM PROSTRANSTVE 1575
cep\ Markova f kt
u
k
, ≥{ }0 , hde t kk, ≥{ }0 — moment¥ skaçkov processa ft
u{ ,
t ≥ })0 .
Opyßem dvyΩenyq çastyc, sootvetstvugwye matrycam P y Q . Pust\ pere-
xod kaΩdoj çastyc¥ v lgboe sostoqnye ravnoveroqten y raven 1 / d. Tohda esly
vse çastyc¥ dvyΩutsq nezavysymo, to matryca perexodn¥x veroqtnostej ymeet
vyd P =
1/dd
j
i( ) ∈
∈
�
�
�
�
, a esly vstretyvßyesq çastyc¥ prodolΩagt dvyhat\sq
vmeste, to Q =
qj
i
j
i( ) ∈
∈
�
�
�
�
, hde qj
i = 0, esly suwestvugt m, n takye, çto j n =
= j m , no i n ≠ i m ; v protyvnom sluçae qj
i = 1/ds
, hde S = j 1{ , … , j d} — ko-
lyçestvo razn¥x koordynat vektora j.
Pust\ naçal\naq mera µ0 =
1
d l l∈∑ �
δ . Rassmotrym πntropyg mer¥ µk
u =
= µ0 ° f k
u( )–1
:
H k = – ( ) ln ( )
l
k
u
k
ul l
∈
∑
�
µ µ .
Oboznaçym çerez H i( ) znaçenye πntropyy v sluçae, kohda f k
u = i, çerez πu
ynvaryantnoe raspredelenye v �
�
dlq matryc¥ A u( ). Poskol\ku Pj
i = 1/d d ≠
≠ 0, j ∈��
, i ∈��
, takoe raspredelenye edynstvenno. Çerez Hu oboznaçym
sootvetstvugwee πu srednee znaçenye πntropyy
H u =
i
uH i i
∈
∑
��
( ) ( )π .
Teorema. Esly u1 < u2, to H u1
> H u 2
.
Dokazatel\stvo. Pust\ νk k; ≥{ }1 — posledovatel\nost\ nezavysym¥x
odynakovo raspredelenn¥x sluçajn¥x velyçyn:
νk =
0
1
s veroqtnost\g ( – ),
s veroqtnost\g
1 u
u.
Pry fyksyrovannoj posledovatel\nosty νk k; ≥{ }1 vse N razbyvagtsq na yn-
terval¥, sostoqwye yz nulej y edynyc. Rassmotrym neodnorodnug cep\ Marko-
va g kk, ≥{ }1 s matrycamy perexodn¥x veroqtnostej na kaΩdom ßahe P, esly
νk = 0, y Q, esly νk = 1, g0 = e — toΩdestvennoe otobraΩenye. Tohda
M g i g ik
k
ν P 1
1= … ={ }, , = P f i f ik
k
1
1= … ={ }, , , i 1,…, i k
∈ �
�.
Poπtomu yzuçym povedenye πntropyy vdol\ g kk, ≥{ }1 .
Oboznaçym çerez πk raspredelenye gk . Naçal\noe raspredelenye qvlqet-
sq ravnomern¥m:
π0( )j =
1
d d , j ∈�� .
Na yntervalax, sostoqwyx yz nulej raspredelenye ne menqetsq, tak kak πk =
= πk –1 P = π0 dlq lgboho πk –1, a sledovatel\no, ne menqetsq y Mg e0 = H k =
= –ln /1 d = H0.
Na yntervalax, sostoqwyx yz edynyc, matematyçeskye oΩydanye H k ub¥va-
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
1576 E. V. OSTAPENKO
et. Dejstvytel\no, pust\ νk = νk +1 = 1, tohda
M Hg e k0 1= + =
j i
j
i
kH i q j∑ ∑ ( ) ( )π .
Poskol\ku qj
i = 0, esly H j( ) < H i( ) , to
i j
iH i q∑ ( ) < H j( ) . Sledovatel\no,
Mg e0 = H k +1 <
j
H j∑ ( ) πk j( ) = Mg e0 = H k .
Oboznaçym çerez Il
0 y I l
1 l-e ynterval¥, sostoqwye yz nulej y edynyc soot-
vetstvenno. Rassmotrym sledugwye sluçajn¥e velyçyn¥: τ0
l
— dlyna Il
0 , τ1
l
— dlyna I l
1 , ξl =
k I g el M∈ =∑
0 0
H k , ηl =
k I g el M∈ =∑
1 0
H k .
V sylu svojstv posledovatel\nosty νk k, ≥{ }1 sluçajn¥e velyçyn¥ τ0
l({ +
+ τ1
l ) , l ≥ }1 , ξl l, ≥{ }1 y ηl l, ≥{ }1 qvlqgtsq nezavysym¥my odynakovo raspre-
delenn¥my.
Pust\
l0 = M τ0
l =
k
ku u k
≥
∑ ( )
1
11 – – =
1
u
y
l1 = Mτ1
l =
k
ku u k
≥
∑ ( )
1
11 – – =
1
1 – u
— srednye dlyn¥ yntervalov. V sylu zakona bol\ßyx çysel poçty vsgdu spra-
vedlyv¥ sootnoßenyq
1
1
0 1n l
n
l l
=
∑ +( )τ τ → ( )l l0 1+ , n → ∞,
1
1n l
n
l
=
∑ξ → l H0 0 , n → ∞,
1
1n l
n
l
=
∑η → Mη1, n → ∞.
Dlq N ≥ 1 sçytaem n takym, çto
N =
l
n
l l
=
∑ +( )
1
0 1τ τ + τ,
hde 0 ≤ τ < τ0
1n + + τ1
1n +
. Tohda poçty vsgdu
N
n
= 1
1
0 1n l
n
l l
=
∑ +( )τ τ + τ
n
→ ( )l l0 1+ , n → ∞.
Poskol\ku
k
N
g e kM H
=
=∑
1
0
=
l
n
l
=
∑
1
ξ +
l
n
l
=
∑
1
η +
k N
N
g e kM H
= +
=∑
– τ 1
0
,
y
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
O MARKOVSKYX MEROZNAÇNÁX PROCESSAX NA KONEÇNOM PROSTRANSTVE 1577
0 ≤
k N
N
g e kM H
= +
=∑
– τ 1
0
≤ τH0 ,
to
1
1
0N
M H
k
N
g e k
=
=∑ →
l H
l l
0 0 1
0 1
+
+
Mη
, n → ∞.
Preobrazuem v¥raΩenye
l H
l l
0 0 1
0 1
+ Mη
+
:
l H
l l
0 0 1
0 1
+
+
Mη
= ( – )1 0u H + u u M H
k
g e k( – )1
1
1
1
0
M
=
=∑
τ
=
= ( – )1 0u H + u u n M H
n k
n
g e k( – )1
1
1
1
1
0
≥ =
=∑ ∑={ }P τ =
= ( – )1 0u H + u u M H u
k
g e k
k( – ) –1
1
1
0
≥
=∑ ( ) =
= ( – )1 0u H +
k
g e k
kM H u
≥
=∑ ( )
1
0
–
k
g e k
kM H u
≥
=
+∑ ( )
1
1
0
=
= H0 +
k
k
g e k g e ku M H M H
≥
= =∑ ( )
1
10 0
– – .
Vsledstvye toho, çto na yntervale, sostoqwem yz edynyc, Mg e0 = Hk < Mg e0 = ×
× Hk –1, poluçennoe v¥raΩenye ub¥vaet s rostom u.
S druhoj storon¥, sohlasno yndyvydual\noj πrhodyçeskoj teoreme, dlq po-
sledovatel\nosty sluçajn¥x velyçyn H kk , ≥{ }1 poçty vsgdu
1
1N
H
k
N
k
=
∑ → Hu, N → ∞,
otkuda
1
1
0N
M H
k
N
g e k
=
=∑ → Hu, N → ∞.
Sledovatel\no, Hu =
l H
l l
0 0
0 1+
+
Mη1
0 1l l+
ub¥vaet s rostom u.
Teorema dokazana.
1. Dorogovtsev A. A. Stochastic flows with interaction and measure-valued processes // Int. J. Math.
and Math. Statist. – 2003. – # 63. – P. 3963 – 3977.
2. Skoroxod A. V. Lekci] z teori] vypadkovyx procesiv. – Ky]v: Lybid\, 1990. – 168 s.
3. Hantmaxer F. R. Teoryq matryc. – M.: Nauka, 1967. – 576 s.
Poluçeno 18.06.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
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| id | umjimathkievua-article-3271 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:39:21Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/77/66cde862d764b2e72b5ec75b4cb96e77.pdf |
| spelling | umjimathkievua-article-32712020-03-18T19:49:31Z On Markov measure-valued processes in a finite space О марковских мерозначных процессах на конечном пространстве Ostapenko, E. V. Остапенко, Е. В. Остапенко, Е. В. We consider stochastic flows with interaction in a finite phase space. The flows with variable generators generating evolutionary measure-valued processes are described. The influence of the interaction of particles on the entropy of the flow is analyzed. Розглядаються стохастичні потоки з взаємодією на скінченному фазовому просторі. Описано такі потоки зі змінним генератором, що породжують еволюційні мірозначні процеси. Встановлено, як взаємодія частинок впливає на ентропію потоку. Institute of Mathematics, NAS of Ukraine 2008-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3271 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 11 (2008); 1572–1577 Український математичний журнал; Том 60 № 11 (2008); 1572–1577 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3271/3288 https://umj.imath.kiev.ua/index.php/umj/article/view/3271/3289 Copyright (c) 2008 Ostapenko E. V. |
| spellingShingle | Ostapenko, E. V. Остапенко, Е. В. Остапенко, Е. В. On Markov measure-valued processes in a finite space |
| title | On Markov measure-valued processes in a finite space |
| title_alt | О марковских мерозначных процессах на конечном пространстве |
| title_full | On Markov measure-valued processes in a finite space |
| title_fullStr | On Markov measure-valued processes in a finite space |
| title_full_unstemmed | On Markov measure-valued processes in a finite space |
| title_short | On Markov measure-valued processes in a finite space |
| title_sort | on markov measure-valued processes in a finite space |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3271 |
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