On Renewal Equations Appearing in Some Problems in the Theory of Generalized Diffusion Processes
We construct a Wiener process on a plane with semipermeable membrane located on a fixed circle and acting in the normal direction. The construction method takes into account the symmetry properties of both the circle and the Wiener process. For this reason, the method is reduced to the perturbation...
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| Datum: | 2005 |
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| Sprache: | Ukrainisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2005
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/3686 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509817770082304 |
|---|---|
| author | Portenko, N. I. Портенко, М. І. |
| author_facet | Portenko, N. I. Портенко, М. І. |
| author_sort | Portenko, N. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:02:00Z |
| description | We construct a Wiener process on a plane with semipermeable membrane located on a fixed circle and acting in the normal direction. The construction method takes into account the symmetry properties of both the circle and the Wiener process. For this reason, the method is reduced to the perturbation of a Bessel process by a drift coefficient that has the type of a δ-function concentrated at a point. This leads to a pair of renewal equations, using which we determine the transition probability of the radial part of the required process. |
| first_indexed | 2026-03-24T02:47:08Z |
| format | Article |
| fulltext |
UDK 519.21
M. I. Portenko (In-t matematyky NAN Ukra]ny, Ky]v)
PRO RIVNQNNQ VIDNOVLENNQ,
QKI VYNYKAGT| V DEQKYX ZADAÇAX
TEORI} UZAHAL|NENYX DYFUZIJNYX PROCESIV
We construct a Wiener process on a plane with semipermeable membrane situated on a fixed circle and
acting in the normal direction. The method of constructing takes into account the symmetry properties
both of a circle and the Wiener process. For this reason, the method is reduced to the perturbation of a
Bessel process by drift coefficient characterized as a δ-function concentrated at a point. This leads to a
pair of renewal equations which determine the transition probability of radial part of the process desired.
Pobudovano vineriv proces na plowyni z napivprozorog membranog, wo roztaßovana na fiksova-
nomu koli i di[ v normal\nomu naprqmku. Metod pobudovy vraxovu[ vlastyvosti symetri] qk ko-
la, tak i vinerovoho procesu. Tomu sprava zvodyt\sq do zburennq besselevoho procesu koefici-
[ntom perenosu, wo ma[ xarakter δ-funkci], zoseredΩeno] v toçci. Ce j pryvodyt\ do pary riv-
nqn\ vidnovlennq, z dopomohog qkyx znaxodyt\sq jmovirnist\ perexodu radial\no] çastyny ßu-
kanoho procesu.
1. Vstup. Zahal\ni metody pobudovy dyfuzijnyx procesiv, qki opysugt\ fizyç-
ne qvywe dyfuzi] v seredovywax iz roztaßovanymy na deqkyx poverxnqx membra-
namy, ne vraxovugt\ vlastyvostej symetri] qk cyx poverxon\, tak i koefici[ntiv,
wo xarakteryzugt\ dyfuzijni xarakterystyky seredovywa (dyv. [1]). Pryrodno
spodivatys\, wo vraxuvannq cyx vlastyvostej pryvodyt\ do znaçnoho sprowennq
pobudovy vidpovidnoho procesu. Metog ci[] roboty qkraz i [ pokazaty, wo take
sprowennq spravdi ma[ misce na prykladi dvovymirnoho vinerovoho procesu z
membranog, wo roztaßovana na koli i di[ po normali.
Vidomo (dyv., napryklad, [2]), wo vineriv proces na plowyni v polqrnyx ko-
ordynatax moΩna zapysaty u vyhlqdi r t t t( ), ( )θ( ) ≥0 , de r ( ⋅ ) — radial\na çasty-
na procesu, a θ ( ⋅ ) — joho cyrkulqrna çastyna. Proces r t t( )( ) ≥0 nazyva[t\sq
besselevym procesom i [ neperervnym odnoridnym procesom Markova na R+ =
= [ 0, + ∞ ) z wil\nistg jmovirnosti perexodu (wodo lebehovo] miry na R+ )
h0 ( t, ρ, r ) =
r
t
r
t
I
r
t
exp − +
{ }ρ ρ2 2
02
,
de t > 0, ρ ∈ R+ , r ∈ R+ , I0 ( ⋅ ) — tak zvana modyfikovana besseleva funkciq
nul\ovoho porqdku (dyv. [2]),
I0 ( z ) =
n
nz
n
=
∞
∑
/
0
2
2
2
( !) .
Wo stosu[t\sq cyrkulqrnoho procesu, to joho moΩna zapysaty u vyhlqdi
θ ( t ) = θ0
0
2
t
r s ds∫ −
( ) , t ≥ 0,
de θ0 0( )t t( ) ≥ — ne zaleΩnyj vid procesu r ( ⋅ ) vineriv proces na koli radiusa 1,
tobto neperervnyj odnoridnyj proces Markova na [ 0, 2 π ] (toçky 0 ta 2 π
ototoΩnggt\sq) z wil\nistg jmovirnosti perexodu (wodo lebehovo] miry na [ 0,
2 π ] )
© M. I. PORTENKO, 2005
1302 ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
PRO RIVNQNNQ VIDNOVLENNQ, QKI VYNYKAGT| V DEQKYX ZADAÇAX TEORI} … 1303
g0 ( t, θ, ζ ) =
1
2
1
2
2 2
π
ζ θ π
t t
k
k∈
∑ − − +
Z
exp ( ) ,
de t > 0, θ ∈ [ 0, 2 π ), ζ ∈ [ 0, 2 π ), a Z — mnoΩyna vsix cilyx çysel.
Besseliv proces v R+ moΩna rozhlqdaty qk dyfuzijnyj z odynyçnym koefi-
ci[ntom dyfuzi] ta koefici[ntom perenosu α0 ( ρ ) = ( 2 ρ )
–
1
, ρ > 0. Zafiksu[mo
teper deqke çyslo a > 0 i parametr q ∈ [ – 1, 1 ]. Çerez δa ( ρ ), ρ ∈ R+ , pozna-
çymo δ-funkcig Diraka, zoseredΩenu v toçci ρ = a, tobto uzahal\nenu funk-
cig, qka di[ na probnu funkcig ϕ na R+ za pravylom 〈 δa , ϕ 〉 = ϕ ( a ). Naße
zavdannq polqha[ v pobudovi uzahal\nenoho dyfuzijnoho procesu ˜( )r t , t ≥ 0, v
R+ , u qkoho koefici[nt dyfuzi] zalyßa[t\sq odynyçnym, a koefici[nt perenosu
α ( ρ ), ρ ∈ R+ , zapysu[t\sq u vyhlqdi
α ( ρ ) = α0 ( ρ ) + q δa ( ρ ), ρ ∈ R+ .
Pobudovi c\oho procesu prysvqçeno pp. 2, 3. Qkwo proces ˜( )r ⋅ vΩe pobudovano
i qkwo vzqty ne zaleΩnyj vid n\oho variant procesu θ0( )⋅ i poklasty
˜ ( )θ t = θ0
0
2
t
r s ds∫ −
˜( ) , t ≥ 0,
to proces ˜( ), ˜ ( )r t t
t
θ( ) ≥0
i bude zadanym u polqrnyx koordynatax ßukanym pro-
cesom, tobto vinerovym procesom na plowyni, u qkoho na koli radiusa a z cent-
rom u poçatku koordynat roztaßovana membrana, qka di[ v normal\nomu naprqm-
ku i ma[ svo]m koefici[ntom prozorosti çyslo q.
Qkwo q = – 1, to çastyna c\oho procesu v kruzi Ba( )0 = x x a∈ ≤{ }R
2 : [
vinerovym procesom u c\omu kruzi z mytt[vym vidbyttqm po normali na joho me-
Ωi, tobto na koli x x a∈ ={ }R
2 : . Qkwo q = + 1, to çastyna pobudovanoho
procesu v dopovnenni do c\oho kruha [ vinerovym procesom tam iz mytt[vym vid-
byttqm po normali na tomu Ω koli, ale v protyleΩnomu naprqmku. Qkwo q = 0 ,
to membrany nema[. V usix inßyx vypadkax ma[mo dvovymirnyj vineriv proces iz
napivprozorog membranog na koli x x a∈ ={ }R
2 : .
2. Rivnqnnq vidnovlennq. Dlq t > 0, ρ ∈ R+ ta r ∈ R+ poznaçymo çerez
Q ( t, ρ, r ) poxidnu po ρ funkci] h0
Q ( t, ρ, r ) =
r
t
r
t
r I
r
t
I
r
t2
2 2
1 02
exp − +
−
ρ ρ ρ ρ
, (1)
de I1 ( ⋅ ) — modyfikovana besseleva funkciq 1-ho porqdku
I1 ( z ) =
n
nz
n n
=
∞ +
∑
+/
0
2 1
2
1!( )!
(my skorystalys\ rivnistg ′I z0( ) = I1 ( z ) ). Poklademo Hρ ( t ) = Q ( t, ρ, ρ ) dlq
t > 0, ρ > 0, tobto
Hρ ( t ) =
ρ ρ ρ ρ2
2
2
1
2
0
2
t t
I
t
I
t
exp −
−
. (2)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1304 M. I. PORTENKO
Oskil\ky I1 ( z ) < I0 ( z ) pry vsix z ∈ R
1
, to Hρ ( t ) < 0 pry vsix t > 0. Zrozumi-
lo, wo pry fiksovanomu ρ > 0 ma[mo
lim ( )
t
t H t
→+ ∞
2
ρ = – ρ
2
. (3)
Doslidyty povedinku funkcij Hρ ta Q pry t ↓ 0 moΩna z dopomohog asymp-
totyçno] formuly (dyv. [3, c. 147])
Iν ( z ) = ( ) exp ( )2 11 2 1πz z O z− −/ { } +( ) , (4)
wo spravdΩu[t\sq pry | z | → +∞, | arg z | ≤ π / 2 – α. Zvidsy pry fiksovanomu
ρ > 0 otrymu[mo asymptotyçnu formulu
Hρ ( t ) = O t− /( )1 2
, t ↓ 0. (5)
Z (3) ta (5) vyplyva[, wo pry fiksovanomu ρ > 0 isnu[ taka stala K > 0, wo pry
vsix t > 0
– Hρ ( t ) ≤ K t t− −/ ∧( )1 2 2
, (6)
de çerez x ∧ y poznaçeno menße z çysel a ∈ R
1
ta b ∈ R
1
. Ocinka (6) pokazu[,
wo funkciq Hρ pry fiksovanomu ρ > 0 [ intehrovnog na ( 0, + ∞ ). Prosti ob-
çyslennq pryvodqt\ do rezul\tatu
0
∞
∫ H t dtρ( ) = – 1. (7)
OtΩe, funkcig – Hρ ( t ), t > 0, moΩna rozhlqdaty qk wil\nist\ jmovirnisnoho
rozpodilu na ( 0, + ∞ ).
Z asymptotyçno] formuly (4) vyplyva[, krim toho, wo pry ρ ≠ r qdro Q
moΩna zobrazyty u vyhlqdi
Q ( t, ρ, r ) =
r r
t t
r
ρ
ρ
π
ρ− − −
2
1
23
2exp ( ) + R ( t, ρ, r ), (8)
de qdro R pry t ↓ 0 zadovol\nq[ spivvidnoßennq
R ( t, ρ, r ) = exp ( )− −
( )− /1
2
2 1 2
t
r O tρ , (9)
pryçomu dlq koΩnoho ε > 0 isnugt\ taki δ > 0 ta Kε , wo pry ρ ∈ [ ε, ε
–
1
],
r ∈ [ ε, ε
–
1
] (vvaΩa[mo ε < 1) ta t < δ vykonu[t\sq nerivnist\
| R ( t, ρ, r ) | ≤ Kε t
–
1
/
2
. (10)
Na pidstavi cyx faktiv moΩna dovesty take tverdΩennq.
Lema 1. Nexaj v( )t t( ) >0 — neperervna funkciq, dlq qko] pry deqkomu T > 0
(a otΩe, i pry vsix T < +∞ )
0
T
t dt∫ v( ) < + ∞.
Todi:
1) pry t > 0 ta ρ > 0 ma[ misce spivvidnoßennq
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
PRO RIVNQNNQ VIDNOVLENNQ, QKI VYNYKAGT| V DEQKYX ZADAÇAX TEORI} … 1305
lim ( ) ( , , )
r
t
Q t r d
→ ± ∫ −
ρ
τ τ ρ τ
0
v = ± v ( t ) +
0
t
H t d∫ −v( ) ( )τ τ τρ ;
2) pry t > 0 ta r > 0 ma[ misce spivvidnoßennq
lim ( ) ( , , )
ρ
τ τ ρ τ
→ ± ∫ −
r
t
Q t r d
0
v = ∓ v ( t ) +
0
t
rH t d∫ −v( ) ( )τ τ τ .
Dovedennq. Z formul (8) ta (9) vyplyva[, wo intehral
0
t
Q t r d∫ −v( ) ( , , )τ τ ρ τ
isnu[ pry t > 0 ta ρ ≠ r ( ρ > 0, r > 0 ), a zi spivvidnoßennq (5) — wo intehral
0
t
H t d∫ −v( ) ( )τ τ τρ
isnu[ pry t > 0 ta ρ > 0. Dali, rivnosti
lim ( )
( )
exp
( )
( )ρ
τ
ρ
ρ
π τ
ρ
τ
τ
→ ± ∫ −
−
− −
−
r
t
r r
t
r
t
d
0
3
2
2 2
v = ∓ v ( t )
ta
lim ( )
( )
exp
( )
( )r
t
r r
t
r
t
d
→ ± ∫ −
−
− −
−
ρ
τ
ρ
ρ
π τ
ρ
τ
τ
0
3
2
2 2
v = ± v ( t )
vstanovlggt\sq elementarno. OtΩe, zalyßa[t\sq obçyslyty hranyci pry ρ →
→ r ± ta pry r → ρ ± vyrazu
0
t
R t r d∫ −v( ) ( , , )τ τ ρ τ .
Qkwo t > 0 fiksovane, to dlq dovil\noho dodatnoho δ ′ < t ma[mo
lim ( ) ( , , )
ρ
δ
τ τ ρ τ
→ ±
− ′
∫ −
r
t
R t r d
0
v =
0
t
rH t d
− ′
∫ −
δ
τ τ τv( ) ( )
ta
lim ( ) ( , , )
r
t
R t r d
→ ±
− ′
∫ −
ρ
δ
τ τ ρ τ
0
v =
0
t
H t d
− ′
∫ −
δ
ρτ τ τv( ) ( ) .
Tomu lemu bude dovedeno, qkwo pokaΩemo, wo rivnomirno wodo ρ ta r v de-
qkomu promiΩku, vidokremlenomu vid nulq, vykonu[t\sq spivvidnoßennq
lim ( ) ( , , )
′↓
− ′
∫ −
δ
δ
τ τ ρ
0
t
t
R t r drv = 0. (11)
Zafiksu[mo deqke ε > 0. Todi (dyv. (10)) isnugt\ δ > 0 ta stala Kε > 0, dlq
qkyx
| R ( t – τ, ρ, r ) | ≤ Kε ( t – τ )
–
1
/
2
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1306 M. I. PORTENKO
pry τ ∈ ( ) ,t t− ∨[ ]δ 0 rivnomirno wodo ρ ∈ [ ε, ε
–
1
] ta r ∈ [ ε, ε
–
1
]. Tomu pry
δ ′ < δ (vvaΩa[mo, wo δ ′ < t ) ma[mo
t
t
R t r d
− ′
∫ −
δ
τ τ ρ τv( ) ( , , ) ≤
2 K
t t
ε
δ τ
δ τ′
− ′≤ ≤
max ( )v ,
zvidky i vyplyva[ (11).
Lemu dovedeno.
Zafiksu[mo teper çysla a > 0 ta q ∈ [ – 1, 1 ] i v oblasti t > 0 rozhlqnemo
intehral\ne rivnqnnq
U ( t ) = f ( t ) + q U H t d
t
a
0
∫ −( ) ( )τ τ τ, (12)
v qkomu f t t( )( ) >0 — zadana neperervna funkciq, a U t t( )( ) >0 — nevidoma funk-
ciq. Vykorystavßy zvyçajne poznaçennq dlq zhortky dvox funkcij α ta β na
( 0, + ∞ )
α * β ( t ) =
0
t
t d∫ −α τ β τ τ( ) ( ) , t > 0,
moΩemo perepysaty rivnqnnq (12) u vyhlqdi
U ( t ) = f ( t ) + q U * Ha ( t ). (13)
Ce rivnqnnq nazyva[t\sq rivnqnnqm vidnovlennq. Nastupni mirkuvannq [ vido-
mymy v teori] takyx rivnqn\ (dyv., napryklad, [4]).
Poklademo H ta
( )( )1 = Ha ( t ) ta H ta
n( )( )+1 = Ha * H ta
n( )( ) pry t > 0 ta n ≥ 1.
Todi, qk vyplyva[ z (6), lokal\no rivnomirno wodo t > 0 zbiha[t\sq rqd
Ga ( t ) =
n
n
a
nq H t
=
∞
∑
1
( ) (14)
i pry c\omu dlq koΩnoho T > 0 isnu[ stala NT taka, wo pry t ∈ ( 0, T ] vykonu-
[t\sq nerivnist\
| Ga ( t ) | ≤ KT t
–
1
/
2
. (15)
Funkciq Ga [ rozv’qzkom rivnqnnq
Ga ( t ) = q Ha ( t ) + q Ga * Ha ( t ), t > 0. (16)
Teper rozv’qzok rivnqnnq (13) moΩna zapysaty u vyhlqdi
U ( t ) = f ( t ) + f * Ga ( t ), t > 0. (17)
Vid funkci] f pry c\omu dosyt\ vymahaty, wob vona bula neperervnog pry t > 0
i wob pry deqkomu T > 0 (a otΩe, i pry koΩnomu T < + ∞ ) vykonuvalas\
nerivnist\
0
t
f t dt∫ ( ) < ∞.
Za takyx umov i rozv’qzok (17) rivnqnnq (13) bude maty tu Ω vlastyvist\. Bil\ß
toho, takyj rozv’qzok bude [dynym.
Rozhlqnemo teper intehral\ni rivnqnnq
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
PRO RIVNQNNQ VIDNOVLENNQ, QKI VYNYKAGT| V DEQKYX ZADAÇAX TEORI} … 1307
V ( t, ρ ) = h0 ( t, ρ, a ) + q V H t d
t
a
0
∫ −( , ) ( )τ ρ τ τ , (18)
˜ ( , )V t r = Q ( t, a, r ) + q V r H t d
t
a
0
∫ −˜ ( , ) ( )τ τ τ. (19)
Zminna ρ ≥ 0 v perßomu rivnqnni ta r ≥ 0 u druhomu vidihragt\ rol\ paramet-
riv (nahadu[mo, wo a > 0 ta q ∈ [ – 1, 1 ] — fiksovani parametry).
Qk vyplyva[ z formuly (4), funkciq h0 ( t, ρ, a ) pry fiksovanomu ρ > 0 dlq
malyx t zadovol\nq[ nerivnist\
h0 ( t, ρ, a ) ≤ ′ − −
− /K
a
t
t
a
ρ
π ρ( ) exp ( )2
1
2
1 2 2
z deqkog stalog K ′ > 0. Krim toho, ce neperervna za sukupnistg zminnyx t > 0
ta ρ > 0 funkciq. Tomu pry koΩnomu T > 0 ta fiksovanomu ρ > 0 ma[mo
0
0
T
h t a dt∫ ( , , )ρ < ∞.
Ce dozvolq[ zapysaty rozv’qzok rivnqnnq (18) u vyhlqdi (17), tobto
V ( t, ρ ) = h0 ( t, ρ, a ) +
0
0
t
ah a G t d∫ −( , , ) ( )τ ρ τ τ . (20)
Oçevydno, funkciq V [ neperervnog za sukupnistg zminnyx t > 0 ta ρ > 0.
Dali, qk vyplyva[ z formul (8), (9), pry r ≠ a funkciq Q ( t, a, r ) zalyßa-
[t\sq obmeΩenog pry t ↓ 0. Krim toho, vona [ neperervnog po t > 0 (pry r ≠ a ),
i tomu
0
T
Q t a r dt∫ ( , , ) < ∞
pry vsix T < + ∞. Vykorystavßy znovu formulu (17), moΩemo zapysaty pry r ≠
≠ a rozv’qzok rivnqnnq (19) u vyhlqdi
˜ ( , )V t r = Q ( t, a, r ) +
0
t
aQ a r G t d∫ −( , , ) ( )τ τ τ . (21)
Oçevydno, cq funkciq [ neperervnog po t > 0 ta r > 0 pry r ≠ a.
Qkwo r = a, to rivnqnnq (19) nabyra[ vyhlqdu
˜ ( , )V t a = Ha ( t ) + q V a H t d
t
a
0
∫ −˜ ( , ) ( )τ τ τ. (22)
Z rivnqnnq (16) vyplyva[, wo rozv’qzkom rivnqnnq (22) [ funkciq
˜ ( , )V t a =
1
q
G ta( ), t > 0. (23)
Ce tak zvane prqme znaçennq funkci]
˜ ( , )V t r pry r = a. Zastosuvavßy lemu 1
do intehrala u pravij çastyni (21), otryma[mo hranyçni znaçennq funkci] Ṽ
pry r → a ±:
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1308 M. I. PORTENKO
˜ ( , )V t a ± = Ha ( t ) ± Ga ( t ) +
0
t
a aH G t d∫ −( ) ( )τ τ τ .
Vzqvßy do uvahy (16), distanemo
˜ ( , )V t a ± =
1
1
q
G ta±
( ). (24)
OtΩe, funkciq
˜ ( , )V t r ma[ rozryv, qk funkciq arhumentu r, u toçci r = a.
Teper dlq t > 0, ρ ≥ 0, r ≥ 0 poklademo
h ( t, ρ, r ) = h0 ( t, ρ, r ) + q V Q t a r d
t
0
∫ −( , ) ( , , )τ ρ τ τ , (25)
˜( , , )h t rρ = h0 ( t, ρ, r ) + q h a V t r d
t
0
0∫ −( , , ) ˜ ( , )τ ρ τ τ . (26)
(Nahadu[mo, wo parametry a > 0 ta q ∈ [ – 1, 1 ] tut, qk i vywe, [ fiksovanymy.)
Z formul (20), (21) ta (23) vyplyva[, wo intehraly u pravyx çastynax (25) ta (26)
isnugt\. Bil\ß toho, nevaΩko baçyty, wo funkci] h ta h̃ [ neperervnymy za
sukupnistg zminnyx ( t, ρ, r ) v oblasti t > 0, ρ ≥ 0, r ≥ 0 pry r ≠ a.
Na pidstavi lemy 1 ta navedenyx vywe formul dlq funkcij V i Ṽ nevaΩko
obçyslyty prqmi ta hranyçni znaçennq funkcij h ta h̃ v toçci r = a. Navede-
mo rezul\taty cyx obçyslen\:
h ( t, ρ, a ) = ˜( , , )h t aρ = V ( t, ρ ), t > 0, ρ ≥ 0,
h ( t, ρ, a ± ) = ˜( , , )h t aρ ± = ( 1 ± q ) V ( t, ρ ), t > 0, ρ ≥ 0.
Qk baçymo, ci znaçennq dlq funkcij h ta h̃ zbihagt\sq miΩ sobog. Vyqvlq-
[t\sq, cej zbih ne [ vypadkovym. My zaraz pobaçymo, wo formuly (25) ta (26)
vyznaçagt\ odnu j tu Ω funkcig, tobto h ( t, ρ, r ) = ˜( , , )h t rρ .
Spravdi, vykorystovugçy formuly (25), (20), (21) ta (26), moΩemo zapysaty
h ( t, ρ, r ) = h0 ( t, ρ, r ) + q V Q a r t( , ) ( , , ) ( )⋅ ∗ ⋅( )ρ =
= h0 ( t, ρ, r ) + q h a h a G Q a r ta0 0( , , ) ( , , ) ( , , )( )⋅ + ⋅ ∗[ ]( ) ∗ ⋅ρ ρ =
= h0 ( t, ρ, r ) + q h a Q a r t0( , , ) ( , , ) ( )⋅ ∗ ⋅( )ρ + q h a G Q a r ta0( , , ) ( , , ) ( )⋅ ∗ ∗ ⋅( )( )ρ =
= h0 ( t, ρ, r ) + q h a Q a r t0( , , ) ( , , ) ( )⋅ ∗ ⋅( )ρ +
+ q h a V r Q a r t0( , , ) ˜ ( , ) ( , , ) ( )⋅ ∗ ⋅ − ⋅[ ]( )ρ =
= h0 ( t, ρ, r ) + q h a V r t0( , , ) ˜ ( , ) ( )⋅ ∗ ⋅( )ρ = ˜( , , )h t rρ ,
wo j potribno bulo dovesty.
3. Vlastyvosti funkci] h ( t, ρρρρ , r ). Dovedemo teper, wo funkciq h
zadovol\nq[ rivnqnnq Kolmohorova – Çepmena
h ( t + s, ρ, r ) =
0
∞
∫ h t h s r d( , , ) ( , , )ρ ζ ζ ζ (27)
pry vsix s > 0, t > 0, ρ ≥ 0, r ≥ 0. Spoçatku dovedemo take tverdΩennq.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
PRO RIVNQNNQ VIDNOVLENNQ, QKI VYNYKAGT| V DEQKYX ZADAÇAX TEORI} … 1309
Lema 2. Pry θ > 0 ta τ > 0 ma[ misce spivvidnoßennq
Ga ( θ + τ ) = q V V d
0
∞
∫ ˜ ( , ) ( , )θ ξ τ ξ ξ . (28)
Dovedennq. Ma[mo (dyv. (17))
Ga ( θ + τ ) = q Ha ( θ + τ ) + q G H da a
0
θ τ
σ θ τ σ σ
+
∫ + −( ) ( ) =
= q Q a G Q a d h a da
0 0
0
∞
∫ ∫+ −
( , , ) ( ) ( , , ) ( , , )θ ξ σ θ σ ξ σ τ ξ ξ
θ
+
+ q G H da a
θ
θ τ
σ θ τ σ σ
+
∫ + −( ) ( ) =
= q V h a d
0
0
∞
∫ ˜ ( , ) ( , , )θ ξ τ ξ ξ + q G H da a
0
τ
θ σ τ σ σ∫ + −( ) ( ) , (29)
de vykorystano tu obstavynu, wo funkciq h0 zadovol\nq[ rivnqnnq Kolmoho-
rova – Çepmena. Poperednq vykladka pokazu[, wo funkciq uθ ( τ ) = Ga ( θ + τ ) [
rozv’qzkom rivnqnnq (v oblasti τ > 0, θ — fiksovane)
uθ ( τ ) = fθ ( τ ) + q u H da
0
τ
θ σ τ σ σ∫ −( ) ( ) , (30)
de
fθ ( τ ) = q V h a d
0
0
∞
∫ ˜ ( , ) ( , , )θ ξ τ ξ ξ.
Rivnqnnq (30) — ce çastynnyj vypadok rivnqnnq (13). Zhidno z formulog (17)
joho rozv’qzok moΩna zapysaty u vyhlqdi
uθ ( τ ) = fθ ( τ ) +
0
τ
θσ τ σ σ∫ −G f da( ) ( ) ,
abo inakße
Ga ( θ + τ ) = q V h a d
0
0
∞
∫ ˜ ( , ) ( , , )θ ξ τ ξ ξ +
+ q G d V h a da
0 0
0
τ
σ σ θ ξ τ σ ξ ξ∫ ∫
∞
−( ) ˜ ( , ) ( , , ) =
= q V h a q h a G d da
0
0
0
0
∞
∫ ∫+ −
˜ ( , ) ( , , ) ( , , ) ( )θ ξ τ ξ τ σ ξ σ σ ξ
τ
=
= q V V d
0
∞
∫ ˜ ( , ) ( , )θ ξ τ ξ ξ ,
wo j potribno bulo dovesty.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1310 M. I. PORTENKO
Teper dovedemo spravedlyvist\ spivvidnoßennq (27). Ma[mo
h ( t + s, ρ, r ) = h0 ( t + s, ρ, r ) + q V Q t s a r d
t s
0
+
∫ + −( , ) ( , , )τ ρ τ τ =
=
0
0
0
0
∞
∫ ∫+ −
h t q V Q t a d h s r d
t
( , , ) ( , ) ( , , ) ( , , )ρ ξ τ ρ τ ξ τ ξ ξ +
+ q V Q t s a r d
t
t s+
∫ + −( , ) ( , , )τ ρ τ τ =
0
0
∞
∫ h t h s r d( , , ) ( , , )ρ ξ ξ ξ +
+ q V t Q s a r d
s
0
∫ + −( , ) ( , , )τ ρ τ τ.
Obçyslymo V ( t + τ, ρ ). Vykorystovugçy formulu (21) ta lemu 2, otrymu[mo
V ( t + τ, ρ ) = h0 ( t + τ, ρ, a ) +
0
0
t
ah a G t d
+
∫ + −
τ
θ ρ τ θ θ( , , ) ( ) =
=
0
0 0
0
0
∞
∫ ∫+ −
h t h a h a G d da( , , ) ( , , ) ( , , ) ( )ρ ξ τ ξ τ θ ξ θ θ ξ
τ
+
+
τ
τ
τ θ ρ θ θ
+
∫ + −
t
ah t a G d0( , , ) ( ) =
0
0
∞
∫ h t V d( , , ) ( , )ρ ξ τ ξ ξ +
+
0
0
t
ah t a G d∫ − +( , , ) ( )θ ρ θ τ θ =
0
0
∞
∫ h t V d( , , ) ( , )ρ ξ τ ξ ξ +
+ q h t a V V d d
t
0
0
0
∫ ∫−
∞
( , , ) ˜ ( , ) ( , )θ ρ θ ξ τ ξ ξ θ =
=
0
0
0
0
∞
∫ ∫+ −
h t q h t a V d V d
t
( , , ) ( , , ) ˜ ( , ) ( , )ρ ξ θ ρ θ ξ θ τ ξ ξ =
0
∞
∫ h t V d( , , ) ( , )ρ ξ τ ξ ξ .
OtΩe,
V ( t + τ, ρ ) =
0
∞
∫ h t V d( , , ) ( , )ρ ξ τ ξ ξ .
Pidstavyvßy cej vyraz u znajdenu vywe formulu dlq h ( t + s, ρ, r ), distanemo
h ( t + s, ρ, r ) =
0
0
∞
∫ h t h s r d( , , ) ( , , )ρ ξ ξ ξ +
+ q Q s a r h t V d d
s
0 0
∫ ∫−
∞
( , , ) ( , , ) ( , )τ ρ ξ τ ξ ξ τ =
=
0
0
0
∞
∫ ∫+ −
h t h s r q Q s a r V d d
s
( , , ) ( , , ) ( , , ) ( , )ρ ξ ξ τ τ ξ τ ξ =
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
PRO RIVNQNNQ VIDNOVLENNQ, QKI VYNYKAGT| V DEQKYX ZADAÇAX TEORI} … 1311
=
0
∞
∫ h t h s r d( , , ) ( , , )ρ ξ ξ ξ .
Tym samym rivnist\ (27) vstanovleno.
Oçevydno,
0
∞
∫ h t r dr( , , )ρ = 1
pry vsix t > 0, ρ ≥ 0. OtΩe, wob dovesty, wo funkciq h [ wil\nistg jmovir-
nosti perexodu deqkoho procesu, zalyßylos\ perekonatysq v tomu, wo vona na-
buva[ lyße nevid’[mnyx znaçen\.
Z ci[g metog zauvaΩymo, wo v oblasti ( , ): , ,t t aρ ρ ρ> ≥ ≠{ }0 0 funkciq
h zadovol\nq[ rivnqnnq
∂
∂
h
t
=
1
2
2
2
∂
∂ρ
h
+
1
2ρ
∂
∂ρ
h
.
Tomu qkby dlq deqkoho r > 0 ta T > 0 my maly
inf ( , , )
, ,t T
h t r
∈( ] ≥0 0ρ
ρ = γ < 0,
to isnuvalo b take t0 ∈ ( 0, T ], wo γ = h ( t0 , a, r ).
Z lemy 1 ta z (26) znaxodymo
∂ ρ
∂ρ ρ
h t r
a
( , , )
= ±
=
∂ ρ
∂ρ ρ
h t r
a
0( , , )
=
∓ q ˜ ( , )V t r +
+ q H V t r d
t
a
0
∫ −( ) ˜ ( , )τ τ τ = ( ) ˜ ( , )1 ∓ q V t r .
OtΩe, v toçci t0 ma[mo
∂ ρ
∂ρ ρ
h t r
a
( , , )0
= ±
= ( ) ˜ ( , )1 0∓ q V t r . (31)
Zvidsy vyplyva[, wo poxidni
∂ ρ
∂ρ ρ
h t r
a
( , , )0
= +
ta
∂ ρ
∂ρ ρ
h t r
a
( , , )0
= −
odnoçasno abo nevid’[mni, abo Ω nedodatni. Ale, qk ce vyplyva[ z teoremy 14 [5,
c. 69], u toçci strohoho minimumu povynno buty
∂ ρ
∂ρ ρ
h t r
a
( , , )0
= −
< 0,
∂ ρ
∂ρ ρ
h t r
a
( , , )0
= +
> 0.
Takym çynom, distaly supereçnist\ iz rivnostqmy (31). OtΩe, prypuwennq,
wo γ < 0, [ nevirnym, wo j potribno bulo dovesty.
Zalyßa[mo çytaçevi perevirku rivnostej:
lim sup ( , , )
:
t
r r
t
h t r dr
↓ ≥ − >{ }
∫0 0
1
ρ ρ ε
ρ = 0, (32)
qkym by ne bulo ε > 0;
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1312 M. I. PORTENKO
lim ( ) ( , , )
t t
r h t r dr
↓
∞
∫ −
0
0
1 ρ ρ = α ( ρ ), (33)
qkym by ne bulo ρ (funkcig α ( ⋅ ) vyznaçeno u vstupi);
lim ( ) ( , , )
t t
r h t r dr
↓
∞
∫ −
0
0
21 ρ ρ = 1 (34)
pry vsix ρ.
ZauvaΩymo, wo oskil\ky funkciq α ( ⋅ ) mistyt\ δ-funkcig, to rivnist\ (33)
slid rozumity v tomu sensi, wo intehraly vid livo] ta pravo] çastyn, pomnoΩenyx
na probnu funkcig ϕ ( ρ ), ρ ≥ 0, zbihagt\sq miΩ sobog.
Rivnosti (32) – (34) svidçat\ pro te, wo isnu[ neperervnyj proces Markova
˜( )r t , t ≥ 0, na [ 0, + ∞ ), qkyj [ tam uzahal\nenym dyfuzijnym z odynyçnym
koefici[ntom dyfuzi] ta koefici[ntom perenosu α ( ρ ), ρ ≥ 0.
1. Portenko M. I. Procesy dyfuzi] v seredovywax z membranamy // Pr. In-tu matematyky
NANRUkra]ny. – 1995. – 10. – 200 s.
2. Yto K., Makkyn H. Dyffuzyonn¥e process¥ y yx traektoryy. – M.: Myr, 1968. – 394 s.
3. Kuznecov D. S. Specyal\n¥e funkcyy. – M.: V¥sß. ßk., 1965. – 424 s.
4. Feller V. Vvedenye v teoryg veroqtnostej y ee pryloΩenyq: V 2 t. – M.: Myr, 1984. – T. 2.
– 752 s.
4. Frydman A. Uravnenyq s çastn¥my proyzvodn¥my parabolyçeskoho typa. – M.: Myr, 1968.
– 424 s.
OderΩano 17.06.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
|
| id | umjimathkievua-article-3686 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:47:08Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a2/622dff78f536ebf1347dc796ff759aa2.pdf |
| spelling | umjimathkievua-article-36862020-03-18T20:02:00Z On Renewal Equations Appearing in Some Problems in the Theory of Generalized Diffusion Processes Про рівняння відновлення, які виникають в деяких задачах теорії узагальнених дифузійних процесів Portenko, N. I. Портенко, М. І. We construct a Wiener process on a plane with semipermeable membrane located on a fixed circle and acting in the normal direction. The construction method takes into account the symmetry properties of both the circle and the Wiener process. For this reason, the method is reduced to the perturbation of a Bessel process by a drift coefficient that has the type of a δ-function concentrated at a point. This leads to a pair of renewal equations, using which we determine the transition probability of the radial part of the required process. Побудовано вінерів процес на площині з напівпрозорою мембраною, що розташована на фіксованому колі і діє в нормальному напрямку. Метод побудови враховує властивості симетрії як кола, так і вінерового процесу. Тому справа зводиться до збурення бесселевого процесу коефіцієнтом переносу, що має характер δ-функції, зосередженої в точці. Це й приводить до пари рівнянь відновлення, з допомогою яких знаходиться ймовірність переходу радіальної частини шуканого процесу. Institute of Mathematics, NAS of Ukraine 2005-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3686 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 9 (2005); 1302–1312 Український математичний журнал; Том 57 № 9 (2005); 1302–1312 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3686/4098 https://umj.imath.kiev.ua/index.php/umj/article/view/3686/4099 Copyright (c) 2005 Portenko N. I. |
| spellingShingle | Portenko, N. I. Портенко, М. І. On Renewal Equations Appearing in Some Problems in the Theory of Generalized Diffusion Processes |
| title | On Renewal Equations Appearing in Some Problems in the Theory of Generalized Diffusion Processes |
| title_alt | Про рівняння відновлення, які виникають в деяких задачах теорії узагальнених дифузійних процесів |
| title_full | On Renewal Equations Appearing in Some Problems in the Theory of Generalized Diffusion Processes |
| title_fullStr | On Renewal Equations Appearing in Some Problems in the Theory of Generalized Diffusion Processes |
| title_full_unstemmed | On Renewal Equations Appearing in Some Problems in the Theory of Generalized Diffusion Processes |
| title_short | On Renewal Equations Appearing in Some Problems in the Theory of Generalized Diffusion Processes |
| title_sort | on renewal equations appearing in some problems in the theory of generalized diffusion processes |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3686 |
| work_keys_str_mv | AT portenkoni onrenewalequationsappearinginsomeproblemsinthetheoryofgeneralizeddiffusionprocesses AT portenkomí onrenewalequationsappearinginsomeproblemsinthetheoryofgeneralizeddiffusionprocesses AT portenkoni prorívnânnâvídnovlennââkívinikaûtʹvdeâkihzadačahteorííuzagalʹnenihdifuzíjnihprocesív AT portenkomí prorívnânnâvídnovlennââkívinikaûtʹvdeâkihzadačahteorííuzagalʹnenihdifuzíjnihprocesív |