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Time Evolution with and without Remote Past ( I ) : Noise Driven Automorphisms of Compact Abelian Groups

Usually the time evolution is discussed from the present to the future or from the past, precisely, from some fixed initial time in the past, to the present or to the future. But we sometimes consider the time evolution from the remote past to the remote future as in the theory of stationary stochastic processes. In the present paper we consider time evolutions governed by noise driven automorphism on compact abelian groups and give a necessary and sufficient condition for the time evolution to admit remote past. It turns out that to admit the remote past is fairly restrictive. Let G be a compact abelian group and φ be an automorphism of G. Consider the random time evolution governed by a stochastic equation