Semigroups of operators that describe a Feller process on the line, which is the result of pasting together two diffusion processes
Authors:
P. P. Kononchuk and B. I. Kopytko
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 84 (2012), 87-97
MSC (2010):
Primary 60J60
DOI:
https://doi.org/10.1090/S0094-9000-2012-00868-5
Published electronically:
July 31, 2012
MathSciNet review:
2857419
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We use the method of the classical potential theory to construct the semigroup of operators that describe a Feller process on the line by pasting together two diffusion processes that satisfy a nonlocal Feller–Wentzell type condition for the pasting.
References
- H. Langer and W. Schenk, Knotting of one-dimensional Feller processes, Math. Nachr. 113 (1983), 151–161. MR 725484, DOI https://doi.org/10.1002/mana.19831130115
- William Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. (2) 55 (1952), 468–519. MR 47886, DOI https://doi.org/10.2307/1969644
- A. D. Ventcel′, Semigroups of operators that correspond to a generalized differential operator of second order, Dokl. Akad. Nauk SSSR (N.S.) 111 (1956), 269–272 (Russian). MR 0092085
- B. I. Kopytko, Pasting together of two diffusion processes on a straight line, Probabilistic methods of infinite-dimensional analysis (Russian), Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 1980, pp. 84–101, 173 (Russian). MR 623093
- M. I. Portenko, Protsesi difuzīï v seredovishchakh z membranami, Trudi Īnstitutu Matematiki Natsīonal′noï Akademīï Nauk Ukraïni [Proceedings of the Institute of Mathematics of the National Academy of Sciences of the Ukraine], vol. 10, Natsīonal′na Akademīya Nauk Ukraïni, Īnstitut Matematiki, Kiev, 1995 (Ukrainian, with Ukrainian summary). MR 1356720
- Pavlo Kononchuk, Pasting of two diffusion processes on a line with nonlocal boundary conditions, Theory Stoch. Process. 14 (2008), no. 2, 52–59. MR 2479733
- G. L. Kulīnīč, The limit distribution behavior of the solution of a stochastic diffusion equation, Teor. Verojatnost. i Primenen 12 (1967), 548–551 (Russian, with English summary). MR 0215365
- O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian). Translated from the Russian by S. Smith. MR 0241822
- L. I. Kamynin, The existence of a solution of boundary-value problems for a parabolic equation with discontinuous coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 721–744 (Russian). MR 0165245
References
- H. Langer and W. Schenk, Knotting of one-dimensional Feller processes, Math. Nachr. 118 (1983), 151–161. MR 725484 (85d:60139)
- W. Feller, The parabolic differential equations and the associated semigroups of transformations, Ann. Math. 55 (1952), 468–519. MR 0047886 (13:948a)
- A. D. Ventcel’ [Wentzell], Semigroups of operators corresponding to the generalized differential second order operator, Dokl. Akad. Nauk SSSR 111 (1956), no. 2, 269–272 (Russian) MR 0092085 (19:1060c)
- B. I. Kopytko, Pasting two diffusion processes on a line, Probabilistic methods of the infinite-dimensional analysis, Institute of Mathematics, Academy of Sciences of Ukrain. SSR, Kiev, 1980, pp. 84–101. (Russian) MR 623093 (82i:60126)
- M. I. Portenko, Diffusion Processes in Media with Membranes, Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, 1995. (Ukrainian) MR 1356720 (96k:60200)
- P. Kononchuk, Pasting of two diffusion processes on a line with nonlocal boundary conditions, Theory Stoch. Process. 14(30) (2008), no. 2, 52–59. MR 2479733 (2010c:60227)
- G. L. Kulinich, On the limit behavior of the distribution of the solution of a stochastic diffusion equation, Teor. Veroyatnost. Primenen. 12 (1967), no. 3, 548–551; English transl. in Theory Probab. Appl. 12 (1967), no. 3, 497–499. MR 0215365 (35:6206)
- O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow, 1967; English transl., American Mathematical Society, Providence, Rhode Island, 1968. MR 0241822 (39:3159b)
- L. I. Kamynin, The existence of a solution of boundary value problems for a parabolic equation with discontinuous coefficients, Izvestiya Akad. Nauk SSSR, ser. matem. 28 (1964), no. 4, 721–744. (Russian) MR 0165245 (29:2534)
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2010):
60J60
Retrieve articles in all journals
with MSC (2010):
60J60
Additional Information
P. P. Kononchuk
Affiliation:
Department of Higher Mathematics, Faculty for Mechanics and Mathematics, L’viv National Ivan Franko University, Universytets’ka Street 1, L’viv 79000, Ukraine
Email:
p.kononchuk@gmail.com
B. I. Kopytko
Affiliation:
Department of Higher Mathematics, Faculty for Mechanics and Mathematics, L’viv National Ivan Franko University, Universytets’ka Street 1, L’viv 79000, Ukraine
Email:
bohdan.kopytko@gmail.com
Keywords:
Diffusion processes,
discontinuities of trajectories,
analytic methods,
method of potential
Received by editor(s):
September 27, 2010
Published electronically:
July 31, 2012
Article copyright:
© Copyright 2012
American Mathematical Society
