## Osaka Journal of Mathematics

### Extensions of diffusion processes on intervals and Feller's boundary conditions

Kouji Yano

#### Abstract

For a minimal diffusion process on $(a,b)$, any possible extension of it to a standard process on $[a,b]$ is characterized by the characteristic measures of excursions away from the boundary points $a$ and $b$. The generator of the extension is proved to be characterized by Feller's boundary condition.

#### Article information

Source
Osaka J. Math., Volume 51, Number 2 (2014), 375-405.

Dates
First available in Project Euclid: 8 April 2014

https://projecteuclid.org/euclid.ojm/1396966254

Mathematical Reviews number (MathSciNet)
MR3192547

Zentralblatt MATH identifier
1311.60089

#### Citation

Yano, Kouji. Extensions of diffusion processes on intervals and Feller's boundary conditions. Osaka J. Math. 51 (2014), no. 2, 375--405. https://projecteuclid.org/euclid.ojm/1396966254

#### References

• J. Bertoin: Lévy Processes, Cambridge Tracts in Mathematics 121, Cambridge Univ. Press, Cambridge, 1996.
• R.M. Blumenthal: Excursions of Markov Processes, Probability and its Applications, Birkhäuser Boston, Boston, MA, 1992.
• R.M. Blumenthal and R.K. Getoor: Markov Processes and Potential Theory, Pure and Applied Mathematics 29, Academic Press, New York, 1968.
• W. Feller: The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. (2) 55 (1952), 468–519.
• W. Feller: Diffusion processes in one dimension, Trans. Amer. Math. Soc. 77 (1954), 1–31.
• W. Feller: Generalized second order differential operators and their lateral conditions, Illinois J. Math. 1 (1957), 459–504.
• M. Fukushima: On general boundary conditions for one-dimensional diffusions with symmetry, J. Math. Soc. Japan 66 (2014), 289–316.
• M. Hutzenthaler and J.E. Taylor: Time reversal of some stationary jump diffusion processes from population genetics, Adv. in Appl. Probab. 42 (2010), 1147–1171.
• K. Itô: Essentials of Stochastic Processes, Translations of Mathematical Monographs 231, Amer. Math. Soc., Providence, RI, 2006.
• K. Itô: Poisson point processes and their application to Markov processes, Lecture note of Mathematics Department, Kyoto University, mimeograph printing, 1969.
• K. Itô and H.P. McKean, Jr.: Brownian motions on a half line, Illinois J. Math. 7 (1963), 181–231.
• K. Itô and H.P. McKean, Jr.: Diffusion Processes and Their Sample Paths, Springer, Berlin, 1974.
• L.C.G. Rogers: Itô excursion theory via resolvents, Z. Wahrsch. Verw. Gebiete 63 (1983), 237–255, Addendum 67 (1984), 473–476.
• K. Yano: Convergence of excursion point processes and its applications to functional limit theorems of Markov processes on a half-line, Bernoulli 14 (2008), 963–987.