On σ-Finite Measures Related to the Martin Boundary of Recurrent Markov Chains
In our monograph with Roynette and Yor (Najnudel et al., A Global View of Brownian Penalisations, MSJ Memoirs, vol. 19, Mathematical Society of Japan, Tokyo, 2009), we construct a σ-finite measure related to penalisations of different stochastic processes, including the Brownian motion in dimension 1 or 2, and a large class of linear diffusions. In the last chapter of the monograph, we define similar measures from recurrent Markov chains satisfying some technical conditions. In the present paper, we give a classification of these measures, in function of the minimal Martin boundary of the Markov chain considered at the beginning. We apply this classification to the examples considered at the end of Najnudel et al. (A Global View of Brownian Penalisations, MSJ Memoirs, vol. 19, Mathematical Society of Japan, Tokyo, 2009).
KeywordsMarkov Chain Multiplicative Constant Finite Measure Simple Random Walk Martin Boundary
I would like to thank Ph. Biane and Ph. Bougerol for the discussion we had on the possibility of a link between the Martin boundary and the \(\sigma\)-finite measures studied here.
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