REMARKS ON EQUILIBRIUM POTENTIAL AND ENERGY

1. To M. Brelot we owe various basic principles and methods of general potential theory (see e.g. [1]). The relationship between them constitutes an important part of the development. In a recent paper [2] I established an equilibrium principle for a broad class of Markov processes by a simple probabilistic method which may be succinctly described as that of ≪ last exit ≫ In contrast, the probabilistic method of solving the Dirichlet problem, due largely to Doob, may be described as that of the ≪ first exit ≫. Now the classical method of solving the equilibrium problem, introduced by Gauss and perfected by Frostman, accrues from the minimization of a quadratic functional involving the ≪ energy ≫. It is natural to ask whether the method of last exit has anything to do with that of energy. Indeed, the first question that arises is whether the equilibrium measure obtained in [2] does minimize some kind of energy. The hypotheses made there are free from the usual duality assumptions and certainly do not require the symmetry of the (potential density) kernel, on which the classical method of energy relies heavily. Although the concept of energy has been extended to the nonsymmetric case, its utilization in a general probabilistic context appears to be still a distant goal.In this note we shall show that for a symmetric kernel, the equilibrium measure obtained by the last exit method does in fact minimize the energy. Even this little step requires much strengthening of the hypotheses in [2] to be specified below.