A bidirectional hitting probability for the symmetric Hunt processes

Abstract

We write σA the first hitting time of set A for the Hunt processes. Let B and BR be compact sets, where BR states far away from B. We assume that the Hunt process is irreducible and conservative and satisfies the Feller property. We consider a relation of the hitting probability of B from BR with the hitting probability of BR from B, without the spatial homogeneity. Our claim is that if the Hunt process satisfies the strong Feller property, then limx→∞Px(σB≤t)=0 implies that limR→∞Py(σBR≤t)=0, for y∈B. Additionally, if the Hunt process is m-symmetric, then both statements are equivalent.