## Journal of the Mathematical Society of Japan

### Conditional distributions which do not satisfy the Chapman-Kolmogorov equation

#### Abstract

We consider one-dimensional generalized diffusion processes (ODGDPs for brief), where both boundary points are accessible or asymptotically accessible. For such ODGDPs we consider stochastic processes induced by conditioning on hitting or asymptotical hitting the right boundary point before hitting or asymptotical hitting the left boundary point. The induced stochastic processes are again ODGDPs when the right boundary point is either accessible with the absorbing boundary condition or asymptotically accessible. However the probability distributions of the induced stochastic processes do not satisfy the Chapman-Kolmogorov equation when the right boundary point is accessible with the reflecting or elastic boundary condition.

#### Article information

Source
J. Math. Soc. Japan, Volume 59, Number 4 (2007), 971-983.

Dates
First available in Project Euclid: 10 December 2007

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1197320622

Digital Object Identifier
doi:10.2969/jmsj/05940971

Mathematical Reviews number (MathSciNet)
MR2370000

Zentralblatt MATH identifier
1135.60050

#### Citation

IIZUKA, Masaru; MAENO, Miyuki; TOMISAKI, Matsuyo. Conditional distributions which do not satisfy the Chapman-Kolmogorov equation. J. Math. Soc. Japan 59 (2007), no. 4, 971--983. doi:10.2969/jmsj/05940971. https://projecteuclid.org/euclid.jmsj/1197320622

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