Existence of Viscosity Solutions to Abstract Cauchy Problems via Nonlinear Semigroups

TL;DR

This paper establishes conditions under which nonlinear semigroups on locally convex vector lattices generate viscosity solutions to abstract Cauchy problems, with applications in infinite-dimensional control problems.

Contribution

It introduces a framework linking $K$-convexity of semigroups to viscosity solutions in a generalized setting, extending previous theories to broader infinite-dimensional contexts.

Findings

Semigroups satisfying $K$-convexity produce viscosity solutions.

Results apply to control problems involving infinite-dimensional Lévy and Ornstein-Uhlenbeck processes.

Provides new tools for analyzing nonlinear PDEs in infinite-dimensional spaces.

Abstract

In this work, we provide conditions for nonlinear monotone semigroups on locally convex vector lattices to give rise to a generalized notion of viscosity solutions to a related nonlinear partial differential equation. The semigroup needs to satisfy a convexity estimate, so called $K$-convexity, w.r.t. another family of operators, defined on a potentially larger locally convex vector lattice. We then show that, under mild continuity requirements on the bounding family of operators, the semigroup yields viscosity solutions to the abstract Cauchy problem given in terms of its generator in the larger locally convex vector lattice. We apply our results to drift control problems for infinite-dimensional L\'evy processes and robust optimal control problems for infinite-dimensional Ornstein-Uhlenbeck processes.