A bidirectional hitting probability for the symmetric Hunt processe

We write sigma(A) 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 lim(x ->infinity)P(x)(sigma(B) <= t) =0 implies that lim(R ->infinity)P(y)(sigma B-R <= t) = 0, for y is an element of B. Additionally, if the Hunt process is m-symmetric, then both statements are equivalent.

Automated summary

In this paper, we investigate the relation of hitting probabilities between two sets in Hunt processes, without considering spatial homogeneity. We claim that if the Hunt process satisfies the strong Feller property, then there is a certain relationship between the hitting probabilities of the two sets.