Ergodic Decomposition of Dirichlet Forms via Direct Integrals and Applications

Abstract

We study superpositions and direct integrals of quadratic and Dirichlet forms. We show that each quasi-regular Dirichlet space over a probability space admits a unique representation as a direct integral of irreducible Dirichlet spaces, quasi-regular for the same underlying topology. The same holds for each quasi-regular strongly local Dirichlet space over a metrizable Luzin, Radon measure space, and admitting carr\'e du champ operator. In this case, the representation is only projectively unique.