Random walks and isotropic Markov chains on homogeneous spaces

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 (ν,μ1,μ2,･･･) where ν is a probability on G/K and each μn is that on K＼G/K.