Abstract
On a finite random walk {X,p(x,y)}, an averaging operator A is defined by Au(x)=∑p(x,y)u(y) and its perturbation is Aφu(x)=Au(x)−φ(x)u(x), where φ(x) is a real-valued function on the states of X. The properties of the solutions and the supersolutions of Aφu(x)=0 are studied which fall into three categories depending on the greatest eigenvalue (a term made precise) of the non-symmetric matrix representing Aφ. Relative to the operator Aφ, the Dirichlet-Poisson solution, the Green function, the Equilibrium Principle and the Condenser problem are investigated.
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