Gaussian fields, equilibrium potentials and multiplicative chaos for Dirichlet forms

For a Dirichlet form(epsilon,F) on L-2(E;m), letG(epsilon)={X-u;u is an element of F-e} be the Gaussian field indexed by the extended Dirichlet space F-e. We first solve the equilibrium problem for a regular recurrent Dirichlet form epsilon of finding for a closed set B a probability measure mu(B) concentrated on B whose recurrent potential R-mu(B) is an element of F-e is constant q.e. on B(called a Robin constant). We next assume that E is the complex plane C and E is a regular recurrent strongly local Dirichlet form. For the closed disk (B) over bar (x,r)={z is an element of C:vertical bar z-x vertical bar <= r}, let mu(x,r) and f(x,r) be its equilibrium measure and Robin constant. Denote the Gaussian random variable X-R mu(x.r) is an element of G(epsilon) by Y-x,Y-r and let, for a given constant gamma > 0, mu(r) (A,omega)=integral(A) exp(gamma Y-x,Y-r-(1/2)gamma(2)f(x,r))dx. Under a certain condition on the growth rate of f(x,r), we prove the convergence in probability of mu(r)(A,omega) to a random measure (mu) over bar (A,omega) as r down arrow 0. The possible range of gamma to admit a non-trivial limit will then be examined in the cases that (epsilon,F) equals (1/2D(C), H-1(C)) and (a, H-1(C)), where a corresponds to the uniformly elliptic partial differential operator of divergence form.