Convex monotone semigroups on lattices of continuous functions

TL;DR

This paper studies convex monotone semigroups on Banach lattices of continuous functions, proposing alternative domains for the generator to ensure invariance and establishing uniqueness of the semigroup with applications to Hamilton-Jacobi-Bellman equations.

Contribution

It introduces invariant domains like the monotone domain and Lipschitz set for convex semigroups, and proves the uniqueness of the semigroup via an extended generator.

Findings

Invariance of Lipschitz sets under the semigroup.

Extension of the generator ensures uniqueness.

Applications to Hamilton-Jacobi-Bellman equations.

Abstract

We consider convex monotone $C_0$-semigroups on a Banach lattice, which is assumed to be a Riesz subspace of a $\sigma$-Dedekind complete Banach lattice. Typical examples include the space of all bounded uniformly continuous functions and the space of all continuous functions vanishing at infinity. We show that the domain of the classical generator of a convex semigroup is typically not invariant. Therefore, we propose alternative versions for the domain, such as the monotone domain and the Lipschitz set, for which we prove invariance under the semigroup. As a main result, we obtain the uniqueness of the semigroup in terms of an extended version of the generator. The results are illustrated with several examples related to Hamilton-Jacobi-Bellman equations, including nonlinear versions of the shift semigroup and the heat equation. In particular, we determine their symmetric Lipschitz sets, which are invariant and allow to understand the generators in a weak sense.