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Convex monotone semigroups on lattices of continuous functions

Robert Denk, Michael Kupper, Max Nendel

TL;DR

This work analyzes convex monotone $C_0$-semigroups on Banach lattices and shows that the classical generator domain is typically not invariant, motivating invariant alternatives such as the monotone domain $D(A_ abla)$ and Lipschitz-type domains $D_L$ and $D_L^s$. It proves that these domains are invariant under the semigroup and establishes a uniqueness principle: the semigroup is uniquely determined by its monotone generator on $D(A_ abla)$. The abstract theory is illustrated via Hamilton–Jacobi–Bellman-type equations, including the uncertain shift semigroup and the $G$-heat equation, with precise descriptions of the symmetric Lipschitz sets (e.g., $D_L^s=W^{1, ablafty}$ for the uncertain shift and $D_L^s=W^{2, ablafty}$ for the $G$-expectation). These results provide a robust framework for analyzing nonlinear, convex, and fully nonlinear dynamics in spaces of continuous functions, with implications for viscosity and weak-solution concepts in control and uncertainty quantification.

Abstract

We consider convex monotone $C_0$-semigroups on a Banach lattice, which is assumed to be a Riesz subspace of a $σ$-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.

Convex monotone semigroups on lattices of continuous functions

TL;DR

This work analyzes convex monotone -semigroups on Banach lattices and shows that the classical generator domain is typically not invariant, motivating invariant alternatives such as the monotone domain and Lipschitz-type domains and . It proves that these domains are invariant under the semigroup and establishes a uniqueness principle: the semigroup is uniquely determined by its monotone generator on . The abstract theory is illustrated via Hamilton–Jacobi–Bellman-type equations, including the uncertain shift semigroup and the -heat equation, with precise descriptions of the symmetric Lipschitz sets (e.g., for the uncertain shift and for the -expectation). These results provide a robust framework for analyzing nonlinear, convex, and fully nonlinear dynamics in spaces of continuous functions, with implications for viscosity and weak-solution concepts in control and uncertainty quantification.

Abstract

We consider convex monotone -semigroups on a Banach lattice, which is assumed to be a Riesz subspace of a -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.

Paper Structure

This paper contains 9 sections, 16 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Setup and notation
  3. Invariant domains
  4. Uniqueness
  5. Examples
  6. The uncertain shift semigroup
  7. The $G$-expectation
  8. Some auxiliary results
  9. Directional derivatives of convex operators

Key Result

Lemma 3.4

One has $D(A)\subset D(A_\delta)\subset D_L$, and $A_\delta|_{D(A)}=A$. If the norm $\|\cdot \|$ on $X$ is $\sigma$-order continuous, then $x\in D(A_\delta)$ with $A_\delta x\in X$ implies $x\in D(A)$ and $A_\delta x=Ax$. If $X$ is $\sigma$-Dedekind complete with $\sigma$-order continuous norm, then

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • proof
  • Proposition 3.6
  • ...and 28 more