Markov processes and martin boundaries part I

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.Sections 6-8 are devoted to a potential theory based on the kernel G(x, y; X) under hypothesis (B) involving some regularity properties of the co-resolvent kernel.But since the potential theory itself is not our purpose, we will only give results enough to cover the application to Martin boundaries.A new phenomenon in the potential theory based on G(x, y; X) is that G(x, y; X) may not be a potential as a function of x for some y.We will denote by Sp the set of y's for which G(., y; X) is a potential.Conditions for S Sp are studied in Section 13.In the Brownian motion case, the resolvent kernel of Z is also the unique co-resolvent kernel (relative to the Lebesgue measure of R) satisfying hypothesis (B), and the corresponding potential kernel turns out to be the Newtonian potential kernel.The Martin boundary is introduced in Section 9. Let r be a measure de- fined over S such that f r(dx)G(x, y; X) is continuous in y, taking values in (0, oo ].The (generalized) Martin potential kernel associated with G(x, y; X) and with r is defined by (x, y; X) G(x, y; X) if f r (dx)