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

Abstract

Let p( t, x, y) be a symmetric transition density with respect to a σ-finite measure m on ( E, E ), g( x, y)=∫ p( t, x, y)d t, and M={σ-finite measures μ⩾0:∫g(x,y)μ( dx)μ( dy)<∞} . There exists a Gaussian random field Φ={ϕ μ:μϵ M} with mean 0 and covariance Eϕ μϕ ν=∫g(x,y)μ( dx)ν( dy) . Letting F(B)=σ{ϕ μ:μ(B c)=0} we consider necessary and sufficient conditions for the Markov property (MP) on sets B, C: F ( B), F ( C) c.i. given F ( B ∩ C). Of crucial importance is the following, proved by Dynkin: E{ϕ μ∣ F(B)}=ϕ μB , where μ B is the hitting distribution of the process corresponding to p, m with initial law μ. Another important fact is that ϕ μ = ϕ ν iff μ, ν 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 xϵ E, T B ∩ C= T B + T C ∮ θ T B = T C + T B ∮ θ T C a.s. P x where, for D ϵ E , T D 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∪ C= E and B, C are finely closed.