Equilibrium measures on recurrent Markov processes
The equilibrium measures in Newtonian potential theory were interpreted by \textit{K. L. Chung} [Ann. Inst. Fourier 25(1975), No.3-4, 131-138 (1976; Zbl 0338.31001)] and others in terms of the hitting-distributions of the associated standard Brownian motion. In the present note the author treats the same problem for the case of recurrent Markov processes. The associated Green function on the given open set \(L^ c\) is supposed to be symmetric. The main results are the following: a) The equilibrium measure \(\nu^ K_ L\) of a set \(K\subset L^ c\) can be identified as \(\nu^ K_ L=\mu^ K_ L\cdot C(K,L)\) where \(\mu^ K_ L\) and C(K,L) are the concepts measure and capacity introduced earlier by \textit{T. Ueno} [Kodai Math. Sem. Rep. 12, 109-142 (1960; Zbl 0094.322)] for recurrent processes. b) Furthermore if the Green function \(g_ L(\cdot,\cdot)\) is lower semi- continuous then \(g_ L\nu^ K_ L\) could be interpreted as the distribution of the hitting-time of the process for the set \(K\) before leaving \(L^ c.\) Details of the statements are too long to be given here.
scientific article; zbMATH DE number 3934157