‰ªŽR‘åŠwŠÂ‹«—��HŠw•” Acta Medica Okayama 1341-9099 1 1 1996 Random walks and isotropic Markov chains on homogeneous spaces 21 26 EN Akihito Hora Let G be a topological group acting on S transitively from the left with a compact stabilizer K. We show that every isotropic (i.e. spatially homogeneous w.r.t. the G-actions) Markov chain on S can be lifted to a right random walk on G and give a one-to-one correspondence between the isotropic Markov chains on S and the totality of sequences of probabilities (ƒË,ƒÊ1,ƒÊ2,¥¥¥) where ƒË is a probability on G/K and each ƒÊn is that on K�_G/K. No potential conflict of interest relevant to this article was reported.

‰ªŽR‘åŠwŠÂ‹«—��HŠw•” Acta Medica Okayama 1341-9099 2 1 1997 A Critical Phenomenon Appearing in the Process of Patricle Diffusion in Classical Statistical Mechanics 1 8 EN Akihito Hora An aspect of the cut-off phenomenon is reviewed. It is a sort of critical phenomenon observed in a wide variety of diffusion. We introduce a framework in which such a phenomenon can be understood and analyzed in a rigorous manner. We illustrate the mechanism with some concrete models. No potential conflict of interest relevant to this article was reported.