## Journal of the Mathematical Society of Japan

### Functional limit theorems for processes pieced together from excursions

Kouji YANO

#### Abstract

A notion of convergence of excursion measures is introduced. It is proved that convergence of excursion measures implies convergence in law of the processes pieced together from excursions. This result is applied to obtain homogenization theorems of jumping-in extensions for positive self-similar Markov processes, for Walsh diffusions and for the Brownian motion on the Sierpiński gasket.

#### Article information

Source
J. Math. Soc. Japan, Volume 67, Number 4 (2015), 1859-1890.

Dates
First available in Project Euclid: 27 October 2015

https://projecteuclid.org/euclid.jmsj/1445951170

Digital Object Identifier
doi:10.2969/jmsj/06741859

Mathematical Reviews number (MathSciNet)
MR3417517

Zentralblatt MATH identifier
1344.60035

Subjects
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60G18: Self-similar processes

#### Citation

YANO, Kouji. Functional limit theorems for processes pieced together from excursions. J. Math. Soc. Japan 67 (2015), no. 4, 1859--1890. doi:10.2969/jmsj/06741859. https://projecteuclid.org/euclid.jmsj/1445951170

#### References

• M. Barlow, J. Pitman and M. Yor, On Walsh's Brownian motions, In Séminaire de Probabilités, XXIII, Lecture Notes in Math., 1372, pp.,275–293. Springer, Berlin, 1989.
• M. T. Barlow and E. A. Perkins, Brownian motion on the Sierpiński gasket, Probab. Theory Related Fields, 79 (1988), 543–623.
• R. G. Bartle, An extension of Egorov's theorem, Amer. Math. Monthly, 87 (1980), 628–633.
• R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, Pure Appl. Math., 29, Academic Press, New York, 1968.
• C. Donati-Martin, B. Roynette, P. Vallois and M. Yor, On constants related to the choice of the local time at 0, and the corresponding Itô measure for Bessel processes with dimension $d=2(1-\alpha)$, $0<\alpha<1$, Studia Sci. Math. Hungar., 45 (2008), 207–221.
• W. Feller, Generalized second order differential operators and their lateral conditions, Illinois J. Math., 1 (1957), 459–504.
• P. J. Fitzsimmons, On the existence of recurrent extensions of self-similar Markov processes, Electron. Comm. Probab., 11 (2006), 230–241.
• P. J. Fitzsimmons and K. Yano, Time change approach to generalized excursion measures, and its application to limit theorems, J. Theoret. Probab., 21 (2008), 246–265.
• 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.
• N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, North-Holland Mathematical Library, 24, North-Holland Publishing Co., Amsterdam, second edition, 1989.
• K. Itô, Poisson point processes and their application to Markov processes, Lecture note of Mathematics Department, Kyoto University, 1969, Available at http://mathsoc.jp/publication/ItoArchive/.
• K. Itô, Poisson point processes attached to Markov processes, In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol.,III: Probability theory, Univ. California Press, Berkeley, Calif., 1972, pp.,225–239.
• K. Itô and H. P. McKean, Jr., Brownian motions on a half line, Illinois J. Math., 7 (1963), 181–231.
• A. E. Kyprianou, Introductory lectures on fluctuations of Lévy processes with applications, Universitext, Springer-Verlag, Berlin, 2006.
• A. Lambert and F. Simatos, The weak convergence of regenerative processes using some excursion path decompositions, Ann. Inst. H. Poincaré Probab. Stat., 50 (2014), 492–511.
• J. Lamperti, Semi-stable Markov processes, I, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 22 (1972), 205–225.
• V. Rivero, Recurrent extensions of self-similar Markov processes and Cramér's condition, Bernoulli, 11 (2005), 471–509.
• V. Rivero, Recurrent extensions of self-similar Markov processes and Cramér's condition, II, Bernoulli, 13 (2007), 1053–1070.
• T. S. Salisbury, On the Itô excursion process, Probab. Theory Related Fields, 73 (1986), 319–350.
• J. B. Walsh, A diffusion with a discontinuous local time, In Temps locaux, Astérisque, 52–53, Société Mathématique de France, Paris, 1978, pp.,37–45.
• W. Whitt, Some useful functions for functional limit theorems, Math. Oper. Res., 5 (1980), 67–85.
• 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.
• K. Yano, Extensions of diffusion processes on intervals and Feller's boundary conditions, Osaka J. Math., 51 (2014), 375–405.