Hunt’s hypothesis (H) and triangle property of the Green function

Let XX be a locally compact abelian group with countable base and let WW be a convex cone of positive numerical functions on XX which is invariant under the group action and such that (X,W)(X,W) is a balayage space or (equivalently, if 1∈W1∈W) such that WW is the set of excessive functions of a Hunt process on  XX, WW separates points, every function in WW is the supremum of its continuous minorants in WW, and there exist strictly positive continuous u,v∈Wu,v∈W such that u/v→0u/v→0 at infinity.Assuming that there is a Green function G>0G>0 for XX which locally satisfies the triangle inequality G(x,z)∧G(y,z)≤CG(x,y)G(x,z)∧G(y,z)≤CG(x,y) (true for many Lévy processes), it is shown that Hunt’s hypothesis (H) holds, that is, every semipolar set is polar.