Logarithmic and Linear Potentials of Signed Measures and Markov Property of Associated Gaussian Fields

Abstract

We consider the family of finite signed measures on the complex plane $\mathbb {C}$ with compact support, of finite logarithmic energy and with zero total mass. We show directly that the logarithmic potential of such a measure sits in the Beppo Levi space, namely, the extended Dirichlet space of the Sobolev space of order 1 over $\mathbb {C}$, and that the half of its Dirichlet integral equals the logarithmic energy of the measure. We then derive the (local) Markov property of the Gaussian field $\textbf {G}(\mathbb {C})$ indexed by this family of measures. Exactly analogous considerations will be made for the Beppo Levi space over the upper half plane $\mathbb {H}$ and the Cameron-Martin space over the real line $\mathbb {R}$. Some Gaussian fields appearing in recent literatures related to mathematical physics will be interpreted in terms of the present field $\textbf {G}(\mathbb {C})$.