Martin Capacity For Markov Chains And Random Walks In Varying Dimensions

this paper is organized as follows. Martin capacity for Markov chains is the focus of Section 2. Several examples are given, including an interesting relation between simple random walk in three dimensions and the time-space chain arising from simple random walk in the plane. Section 3 shows how to derive Lyons' percolation Theorem from the general capacity estimate for Markov chains (Theorem 2.2.) In Section 4 we give the easy proofs of Proposition 1.1 and related results concerning Brownian motion. Random walks in varying dimension are analyzed in Section 5. This section is written so it can be read independently of the rest of the paper. However, it is connected to the previous sections both in the methods of proof and in that in both settings the number of returns to the origin can be almost surely finite but have infinite expectation. Finally, Section 6 contains the examples of transient walks which interlace one-dimensional and two-dimensional steps. 2 Polarity for Markov chains First we recall some potential theory notions. Definition 2.1 Let be a set and B a oe-field of subsets of . Given a measurable functio