Energy forms

Abstract

In this thesis we study energy forms. These are quadratic forms on the space of real-valued measurable $m$-a.e. determined functions $$E:L^0(m) \to [0,\infty],$$ which assign to a measurable function $f$ its energy $E(f)$. Their two defining characteristics are a contraction property and some form of continuity. The contraction property demands that for each normal contraction $C:\mathbb R \to \mathbb R$ the energy of a function $f$ satisfies $$E(C \circ f) \leq E(f).$$ This is an abstract formulation of the postulate that cutting off fluctuations of a function (which is thought to describe some physical quantity) decreases its energy. The continuity assumption that we impose on energy forms is lower semicontinuity with respect to local convergence in measure. We develop the basic theory of energy forms and then investigate their extensions and global properties.