Potential Theory, Replacement, and Compensation

Abstract

The potential for a transient chain N_k governed by transition matrix a is the matrix $g = \sum\limits_0^{\infty } {{a^k}}$. The corresponding potential for transient chains in continuous time is $g(t) = \int_0^{\infty } {p(t)}$ p(t) dt where p(t) is the transition matrix.^† These potentials appear as entities in the answers to many-important questions associated with a particular chain, ergocic or transient. For example, certain replacement processes N^* (t) arise from a chain N(t) when samples reaching a bad set B are replaced at a particular “replacement state”. The ergodic distribution for the replacement process involves a potential obtained from N(t) in a simple way. Directly related to the idea of replacement is a method of compensation which treats a modified process as the original process altered by the insertion of positive and negative mass at “boundary states” in such a way as to generate the modified process. In this way structural simplicity of the original process due to spatial homogeneity or independence can be retained in the treatment of the modified process. A variety of examples are given.