On Dynkin's Markov property of random fields associated with symmetric processes

Let p(t, x, y) be a symmetric transition density with respect to a [sigma]-finite measure m on (E, ), g(x,y)=[integral operator]p(t,x,y)dt, and . There exists a Gaussian random field with mean 0 and covariance E[phi][mu][phi][nu]=[integral operator]g(x,y)[mu](dx)[nu](dy). Letting we consider necessary and sufficient conditions for the Markov property (MP) on sets B, C: (B), (C) c.i. given (B [intersection] C). Of crucial importance is the following, proved by Dynkin: , where [mu]B is the hitting distribution of the process corresponding to p, m with initial law [mu]. Another important fact is that [phi][mu]=[phi][nu] iff [mu], [nu] have the same potential. Putting these together with an additional transience assumption, we present a potential theoretic proof of the following necessary and sufficient condition for (MP) on sets B, C: For every x[epsilon]E, TB[intersection]C=TB+TC[contour integral operator] [theta]TB=TC+TB[contour integral operator][theta]TC a.s. Px where, for D [epsilon] , TD is the hitting time of D for the process associated with p, m. This implies a necessary condition proved by Dynkin in a recent preprint for the case where B[union or logical sum]C=E and B, C are finely closed.