On h-Transforms of One-Dimensional Diffusions Stopped upon Hitting Zero
For a one-dimensional diffusion on an interval for which 0 is the regular-reflecting left boundary, three kinds of conditionings to avoid zero are studied. The limit processes are h-transforms of the process stopped upon hitting zero, where h’s are the ground state, the scale function, and the renormalized zero-resolvent. Several properties of the h-transforms are investigated.
KeywordsBrownian Motion Resolvent Operator Generalize Diffusion Natural Scale Bessel Process
The authors are thankful to Prof. Masatoshi Fukushima for drawing their attention to the paper . They also thank Prof. Matsuyo Tomisaki and Dr. Christophe Profeta for their valuable comments.
The research of the first author, Kouji Yano, was supported by KAKENHI (26800058) and partially by KAKENHI (24540390). The research of the second author, Yuko Yano, was supported by KAKENHI (23740073).
- 2.R.M. Blumenthal, R.K. Getoor, Markov Processes and Potential Theory. Pure and Applied Mathematics, vol. 29 (Academic, New York, 1968)Google Scholar
- 5.K.L. Chung, J.B. Walsh, Markov Processes, Brownian Motion, and Time Symmetry, 2nd edn. Grundlehren der Mathematischen Wissenschaften, vol. 249 (Springer, New York, 2005)Google Scholar
- 6.R.A. Doney, Fluctuation Theory for Lévy Processes. Lecture Notes in Mathematics, vol. 1897 (Springer, Berlin, 2007) [Lectures from the 35th Summer School on Probability Theory held in Saint-Flour, 6–23 July 2005, Edited and with a foreword by Jean Picard]Google Scholar
- 7.W. Feller, On second order differential operators. Ann. Math. (2) 61, 90–105 (1955)Google Scholar
- 9.P.J. Fitzsimmons, R.K. Getoor, Smooth measures and continuous additive functionals of right Markov processes, in Itô’s Stochastic Calculus and Probability Theory (Springer, Tokyo, 1996), pp. 31–49Google Scholar
- 11.N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland Mathematical Library, vol. 24 (North-Holland, Amsterdam, 1989)Google Scholar
- 12.K. Itô, Essentials of Stochastic Processes. Translations of Mathematical Monographs, vol. 231 (American Mathematical Society, Providence, 2006) [Translated from the 1957 Japanese original by Yuji Ito]Google Scholar
- 16.M. Maeno, One-dimensional h-path generalized diffusion processes, in Annual Reports of Graduate School of Humanities and Sciences, Nara Women’s University, vol. 21 (2006), pp. 167–185Google Scholar
- 18.H.P. McKean Jr., Excursions of a non-singular diffusion. Z. Wahrsch. Verw. Gebiete 1, 230–239 (1962/1963)Google Scholar
- 20.C. Profeta, Penalizing null recurrent diffusions. Electron. J. Probab. 17(69), 23 (2012)Google Scholar
- 21.D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, 3rd edn. Grundlehren der Mathematischen Wissenschaften, vol. 293 (Springer, Berlin, 1999)Google Scholar
- 22.L.C.G. Rogers, Itô excursion theory via resolvents. Z. Wahrsch. Verw. Gebiete 63(2), 237–255 (1983); Addendum 67(4), 473–476 (1984)Google Scholar
- 25.P. Salminen, M. Yor, Tanaka formula for symmetric Lévy processes, in Séminaire de Probabilités XL. Lecture Notes in Mathematics, vol. 1899 (Springer, Berlin, 2007), pp. 265–285Google Scholar
- 26.T. Takemura, State of boundaries for harmonic transforms of one-dimensional generalized diffusion processes, in Annual Reports of Graduate School of Humanities and Sciences, Nara Women’s University, vol. 25 (2010), pp. 285–294Google Scholar
- 28.S. Watanabe, On time inversion of one-dimensional diffusion processes. Z. Wahrsch. Verw. Gebiete 31, 115–124 (1974/1975)Google Scholar
- 31.K. Yano, Two kinds of conditionings for stable Lévy processes, in Probabilistic Approach to Geometry. Advanced Studies in Pure Mathematics, vol. 57 (Mathematical Society of Japan, Tokyo, 2010), pp. 493–503Google Scholar
- 32.K. Yano, On harmonic function for the killed process upon hitting zero of asymmetric Lévy processes. J. Math. Ind. 5A, 17–24 (2013)Google Scholar