On Differential Geometry and Homogeneous Spaces

* Work supported by the Office of Ordnance Research, United States Army, Contract No. DA-36-034-ORD-1296. 1 W. Feller, "The General Diffusion Operator and Positivity-preserving Semi-groups in One Dimension," Ann. Math., 60, 417-436, 1954; "The Parabolic Differential Equations and the Associated Semni-groups of Transformations," ibid., 55, 468-519, 1952; J. L. Doob, Stochastic Processes (New York: John Wiley & Sons, 1953). 2 Given such a measure m(dx), we make the following definition: Bv = w means that the onesided derivatives v+ and vexist on (0, 1), that v+ = von K R, and that v+(b) v-(a) = 2 w(ri)mi (a < b), the summation being taken over (i: ri e [a, b]). 3P. Levy, "Syst6mes markoviens et stationnaires: Cas denombrable," Ann. sci. &cole norm. super., 68, 327-381, 1951. 4We note that this contradicts P. Levy's statement (ibid., p. 375) that a chain must have stable states. 6 Compare W. Feller, "The Parabolic Differential Equations and the Associated Semi-groups of Transformations," Ann. Math., 55, 468-519, 1952. 6 This is a special case of a theorem due to J. Doob, "Topics in the Theory of Markov Chains," Trans. Am. Math. Soc., 52, 37-64, 1942. 7D. Ray, "Stationary Markov Processes with Continuous Paths," to appear in Trans. Am. Math. Soc. (1956). 8 P. Levy, op. cit., p. 373. His argument contains a slip, but it is easy to correct.