Markov processes and martin boundaries part I

Abstract

This condition determines G(x, y; X) uniquely up to a set of (m) measure 0 for each x. The crucial point is to eliminate arbitrariness of measure 0. We will denote by G,(x, A; X) the resolvent kernel of X and assume the existence of the co-resolvent kernel ,(x, A;X) defined by the relation fB G,(x, A; X)n (dx) f ,(y, B; X)m (dy), where a is any positive number and A and B are any Borel sets of S with compact closure. The definition of the potential kernel involves the co- resolvent kernel as well as the resolvent kernel and the measure m. G(x, y; X) is said to be a potential kernel (of function type) if it is excessive in x for. each y and co-excessive in y for each x and if (1.2) and (0(y, A; X) f G(x, y; X)m (dx) are satisfied, where 0(y, A; X) lim,0 ,(y, A; X). Such a kernel, if it exists, is unique (with arbitrariness of (m) measure 0 eliminated). An obvious necessary condition for the existence of such kernel G(x, y; X) is that Go(x, A; X) and (0(x, A; X) are absolutely continuous with respect to re(A) for each x. The first key result (Theorem 1) is that this condition is also sufficient.