Journal of the Mathematical Society of Japan

Conditional distributions which do not satisfy the Chapman-Kolmogorov equation

Masaru IIZUKA, Miyuki MAENO, and Matsuyo TOMISAKI

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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

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

First available in Project Euclid: 10 December 2007

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Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]

generalized diffusion process boundary condition Chapman-Kolmogorov equation population genetics conditional diffusion process


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.

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