Title: Random walks in random environment with Markov dependence on time
Author(s):

	C.Boldrighini (Dipartimento di Matematica, Università di Roma "La Sapienza", Piazzale Aldo Moro 2, 00185 Roma, Italy. Partially supported by INdAM (G.N.F.M.) and M.U.R.S.T. research founds) ,
	R.A.Minlos (Institute for Problems of Information Transmission, Russian Academy of Sciences, B. Karetnyi Per. 19, 127994, GSP-4, Moscow, Russia. Partially supported by RFBR grants 99-01-024, 97-01-00714 and CRDF research funds N RM1-2085) ,
	A.Pellegrinotti (Dipartimento di Matematica, Università di Roma Tre, Largo S. Leonardo Murialdo 1, 00146 Roma, Italy. Partially supported by INdAM (G.N.F.M.) and M.U.R.S.T. research founds)

We consider a simple model of discrete-time random walk on Ζ^ν, ν=1,2,... in a random environment independent in space and with Markov evolution in time. We focus on the application of methods based on the properties of the transfer matrix and on spectral analysis. In section 2 we give a new simple proof of the existence of invariant subspaces, with an explicit condition on the parameters. The remaining part is devoted to a review of the results obtained so far for the quenched random walk and the environment from the point of view of the random walk, with a brief discussion of the methods.

Key words: random walks, random environment, Markov chains
PACS: 05.50.Ey, 05.40.Fb