Kernels and Choquet capacities

Abstract

M. Brelot showed that the capacity corresponding to a function-kernel is a Choquet capacity, provided that the kernel satisfies the principle of equilibrium, the weak domination principle and the adjoint kernel satisfies the weak principle of equilibrium. This result is not applicable for a series of important kernels in potential theory (e.g. the fundamental solution of the heat equation, or the Kolmogorov equation), since the above principles no longer hold in this situation. New principles for function kernels guaranteeing that the capacity is a Choquet capacity are introduced and applied in the framework of balayage spaces. In particular, polar and adjoint polar sets are shown to coincide in this context.