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

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **84** (2011).

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

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.

**1.**H. Langer and W. Schenk,*Knotting of one-dimensional Feller processes*, Math. Nachr.**118**(1983), 151-161. MR**725484 (85d:60139)****2.**W. Feller,*The parabolic differential equations and the associated semigroups of transformations*, Ann. Math.**55**(1952), 468-519. MR**0047886 (13:948a)****3.**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)****4.**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)****5.**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)****6.**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)****7.**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)****8.**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)****9.**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)**

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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

DOI:
https://doi.org/10.1090/S0094-9000-2012-00868-5

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