Avoidable collections of balls and minimal thinness for jump processes

Abstract: There are various concepts of small sets with respect to a Markov process. In this talk I will focus on sets near the boundary of the state space of the underlying process that are small in the sense that the (conditioned) process has positive probability of avoiding the set. The standard notion that describes the smallness of a set at a boundary point is minimal thinness. I will present several criteria for minimal thinness with respect to some classes of jump Markov processes. A closely related problem is avoidability of a countable collection of closed balls. I will show how this problem is connected to minimal thinness and give necessary and sufficient conditions for avoidability of collections of balls for two classes of jump Markov processes.