On the functional equation of exit laws for lattice semigroups

Abstract

Let P = (P[t])[t>0] be a semigroup of kernels on an LCD space (X,β) such that V := ∫[∞][0] P[t]dt exists. An exit law for P is a family of positive measurable functions f = (f[t])[t>0] which verifies the following functional equation: P[s]f[t] = f[s+t], for all s,t > 0. Let R := {u ε F : u = ∫[∞][0] f[t]dt for some exit law f for P} and Im V := {Vu : u ε β}. In general we have Im V ⊂ R. If P is a lattice and submarkovian semigroup then we prove in this paper the equality Im V = R. For this purpose, we associate a semi- dynamical system to P. Moreover, we give an example of not lattice semigroup for which Im V = R.