Logarithmic and linear potentials of signed measures and Markov property of associated Gaussian fields (Q667727)

Let $D(f,g):= \int_{\mathbb C} \nabla f(x)\nabla g(x)\,dx$ be the classical Dirichlet form on $L^2(\mathbb C) = L^2(\mathbb C,dx)$ with domain $H^1(\mathbb C)=\left\{f\in L^2(\mathbb C) : |\nabla f|\in L^2(\mathbb C)\right\}$. The corresponding extended Dirichlet space is known to coincide with the Beppo-Levi space $\mathrm{BL}(\mathbb C)=\left\{f\in L^2_{\text{loc}}(\mathbb C) : |\nabla f|\in L^2(\mathbb C)\right\}$. The paper starts with an elementary (direct) proof that for any finite signed measure $\mu$ on $\mathbb C$ with compact support and $\mu(\mathbb C)=0$ the logarithmic potential $U\mu$ is in $\mathrm{BL}(\mathbb C)$; moreover $U\mu$ is $D_1$-quasi continuous and $\frac 12 D(U\mu,u)=\int \widetilde u\,d\mu$ for any quasi-continuous modification $\widetilde u$ of $u\in \mathrm{BL}(\mathbb C)$. Moreover, $I(\mu,\nu) = \int U\nu\,d\mu$ is an inner product on the space of finite signed measures on $\mathbb C$ with compact support and $\mu(\mathbb C)=0$. The latter observation allows one to construct a mean-zero Gaussian random field indexed by such measures whose covariance function is $I(\mu,\nu)$. It is shown that this Markov field has the local Markov property. In the second part of the paper these results are extended to the upper complex half-plane (again with the Beppo-Levi space thereon) and the Dirichlet form obtained by a reflecting Brownian motion and to the setting of linear potentials on the real line (with the Cameron-Martin space). Finally, several examples of Gaussian fields and processes are constructed as subfields of the above Gaussian fields.