Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convex monotone semigroups and their generators with respect to $Γ$-convergence

Jonas Blessing, Robert Denk, Michael Kupper, Max Nendel

TL;DR

This work develops a Γ-convergence based infinitesimal generator framework for strongly continuous convex monotone semigroups on spaces of continuous functions, proving that the semigroup is uniquely determined by its upper Γ-generator on invariant Lipschitz sets. It combines a comparison principle with Chernoff-type approximation methods to reconstruct semigroups from infinitesimal data, and provides concrete regularization and truncation techniques to approximate the Γ-generator from smooth functions. The theory is then demonstrated in stochastic control, infinite-dimensional Wiener processes, and Wasserstein perturbations of linear transition semigroups, linking to Hamilton–Jacobi–Bellman type structures and entropy/Talagrand-type inequalities. Overall, the results offer analyticCharacterizations and practical numerical schemes for nonlinear semigroups beyond traditional resolvent-based approaches, with broad applicability to control and perturbation problems in finite and infinite dimensions.

Abstract

We study semigroups of convex monotone operators on spaces of continuous functions and their behaviour with respect to $Γ$-convergence. In contrast to the linear theory, the domain of the generator is, in general, not invariant under the semigroup. To overcome this issue, we consider different versions of invariant Lipschitz sets which turn out to be suitable domains for weaker notions of the generator. The so-called $Γ$-generator is defined as the time derivative with respect to $Γ$-convergence in the space of upper semicontinuous functions. Under suitable assumptions, we show that the $Γ$-generator uniquely characterizes the semigroup and is determined by its evaluation at smooth functions. Furthermore, we provide Chernoff approximation results for convex monotone semigroups and show that approximation schemes based on the same infinitesimal behaviour lead to the same semigroup. Our results are applied to semigroups related to stochastic optimal control problems in finite and infinite-dimensional settings as well as Wasserstein perturbations of transition semigroups.

Convex monotone semigroups and their generators with respect to $Γ$-convergence

TL;DR

This work develops a Γ-convergence based infinitesimal generator framework for strongly continuous convex monotone semigroups on spaces of continuous functions, proving that the semigroup is uniquely determined by its upper Γ-generator on invariant Lipschitz sets. It combines a comparison principle with Chernoff-type approximation methods to reconstruct semigroups from infinitesimal data, and provides concrete regularization and truncation techniques to approximate the Γ-generator from smooth functions. The theory is then demonstrated in stochastic control, infinite-dimensional Wiener processes, and Wasserstein perturbations of linear transition semigroups, linking to Hamilton–Jacobi–Bellman type structures and entropy/Talagrand-type inequalities. Overall, the results offer analyticCharacterizations and practical numerical schemes for nonlinear semigroups beyond traditional resolvent-based approaches, with broad applicability to control and perturbation problems in finite and infinite dimensions.

Abstract

We study semigroups of convex monotone operators on spaces of continuous functions and their behaviour with respect to -convergence. In contrast to the linear theory, the domain of the generator is, in general, not invariant under the semigroup. To overcome this issue, we consider different versions of invariant Lipschitz sets which turn out to be suitable domains for weaker notions of the generator. The so-called -generator is defined as the time derivative with respect to -convergence in the space of upper semicontinuous functions. Under suitable assumptions, we show that the -generator uniquely characterizes the semigroup and is determined by its evaluation at smooth functions. Furthermore, we provide Chernoff approximation results for convex monotone semigroups and show that approximation schemes based on the same infinitesimal behaviour lead to the same semigroup. Our results are applied to semigroups related to stochastic optimal control problems in finite and infinite-dimensional settings as well as Wasserstein perturbations of transition semigroups.
Paper Structure (21 sections, 37 theorems, 310 equations)

This paper contains 21 sections, 37 theorems, 310 equations.

Table of Contents

  1. Introduction
  2. Setup and notation
  3. Lipschitz sets, $\Gamma$-generators and basic continuity properties
  4. Convex monotone semigroups, comparison and uniqueness
  5. Convex monotone semigroups
  6. Comparison and uniqueness
  7. Approximation of the $\Gamma$-generator
  8. General approximation results
  9. Regularization by convolution and truncation
  10. Convolution
  11. Truncation
  12. Link to distributional derivatives
  13. Chernoff-type approximation schemes
  14. Examples
  15. Stochastic control problems
  16. ...and 6 more sections

Key Result

Lemma 2.3

Let $(I(t))_{t\geq 0}$ be a family of convex monotone operators $I(t)\colon{\rm C}_\kappa\to{\rm C}_\kappa$ with $I(s)I(t)f=I(t)I(s)f$ for all $f\in{\rm C}_\kappa$ and $s,t\geq 0$. Then,

Theorems & Definitions (88)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Remark 2.7
  • Definition 3.1
  • ...and 78 more