Non-bilinear Dirichlet Functionals: Markovianity, locality, invariance
Giovanni Brigati, Lorenzo Dello Schiavo
TL;DR
This paper develops a comprehensive framework for non-bilinear Dirichlet functionals on Hilbert spaces, unifying nonlinear semigroups, resolvents, generators, and energy functionals (TAJE-operators) and providing complete Markovianity characterizations across these relations. It introduces strong notions of locality and locality-type contractions, together with a detailed invariance theory including double invariance and invariance under projections, resolvents, and semigroups. Central to the work are equivalence theorems linking energy functionals to their cyclically monotone generators via cyclical monotonicity and Painlevé–Kuratowski graph convergence, ensuring constructive correspondences between the analytic and variational perspectives. The results extend the classical bilinear Dirichlet form theory to a broad nonlinear setting, with implications for nonlinear Markov processes, potential theory, and nonlinear evolution PDEs, and outline clear paths to extend beyond Hilbert spaces to Banach lattices. Overall, the framework provides a robust, general toolkit for studying nonlinear Dirichlet functionals with well-posed notions of Markovianity, locality, and invariance.
Abstract
We present in a unified setting the foundations for a theory of non-bilinear Dirichlet functionals on Hilbert spaces. We prove known and new equivalences between non-linear semigroups, non-linear resolvents, non-linear generators, and their energy functionals, including a complete characterization of Markovianity. We introduce and characterize strong notions of invariance for general lower semicontinuous convex functionals, and notions of locality and strong locality for non-bilinear Dirichlet functionals. Contrary to many partial results in the literature, these characterizations are complete and correctly extend the analogous assertions for the bilinear case in full generality.