Potential theory for recurrent symmetric infinitely divisible processes

Abstract

Let X be a locally compact, second countable Abelian group. Let ϕ(t), t≥0, be an irreducible, recurrent, symmetric infinitely divisible Hunt process on X such that for t>0, ϕ(t)-ϕ(0) has a bounded continuous density p(t,·) with respect to Haar measure on X. Potential theory is developed for the kernel k = ∫ ¹₀ p(t,·)dt · ∫ ^∞₁ (p(t,·)−p(t,0))dt. In particular, balayage and equilibrium problems corresponding to an arbitrary relatively compact Borel set are formulated and solved and the solutions are characterized in terms of energy. Logarithmic potential theory is included as the special case corresponding to planar Brownian motion.