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Convergence of infinitesimal generators and stability of convex monotone semigroups

Jonas Blessing, Michael Kupper, Max Nendel

TL;DR

The paper develops an abstract stability framework for sequences of convex monotone semigroups on weighted function spaces, showing that convergence of their generators in the mixed topology yields convergence of the semigroups themselves. It replaces viscosity-based arguments with a comparison principle based on the Γ-generator on Lipschitz sets, which ensures uniqueness of the limit and subsequence independence. The results accommodate discretizations in time and space, including Chernoff-type schemes, and unify several applications such as Euler/Yosida approximations for upper envelopes, vanishing-viscosity limits for convex HJB equations, large deviations for randomized Euler schemes, and discretizations of stochastic control problems under model uncertainty. This provides a robust, operator-theoretic route to both convergence analysis and numerical approximation for nonlinear, convex semigroups arising in control, finance, and stochastic processes. The framework clarifies how limit operators govern the evolution of value functions and related objects even in non-smooth or uncertain settings, with potential for wide applicability in nonlinear PDEs and stochastic analysis.

Abstract

Based on the convergence of their infinitesimal generators in the mixed topology, we provide a stability result for strongly continuous convex monotone semigroups on spaces of continuous functions. In contrast to previous results, we do not rely on the theory of viscosity solutions but use a recent comparison principle which uniquely determines the semigroup via its $Γ$-generator defined on the Lipschitz set and therefore resembles the classical analogue from the linear case. The framework also allows for discretizations both in time and space and covers a variety of applications. This includes Euler schemes and Yosida-type approximations for upper envelopes of families of linear semigroups, stability results and finite-difference schemes for convex HJB equations, Freidlin-Wentzell-type results and Markov chain approximations for a class of stochastic optimal control problems and continuous-time Markov processes with uncertain transition probabilities.

Convergence of infinitesimal generators and stability of convex monotone semigroups

TL;DR

The paper develops an abstract stability framework for sequences of convex monotone semigroups on weighted function spaces, showing that convergence of their generators in the mixed topology yields convergence of the semigroups themselves. It replaces viscosity-based arguments with a comparison principle based on the Γ-generator on Lipschitz sets, which ensures uniqueness of the limit and subsequence independence. The results accommodate discretizations in time and space, including Chernoff-type schemes, and unify several applications such as Euler/Yosida approximations for upper envelopes, vanishing-viscosity limits for convex HJB equations, large deviations for randomized Euler schemes, and discretizations of stochastic control problems under model uncertainty. This provides a robust, operator-theoretic route to both convergence analysis and numerical approximation for nonlinear, convex semigroups arising in control, finance, and stochastic processes. The framework clarifies how limit operators govern the evolution of value functions and related objects even in non-smooth or uncertain settings, with potential for wide applicability in nonlinear PDEs and stochastic analysis.

Abstract

Based on the convergence of their infinitesimal generators in the mixed topology, we provide a stability result for strongly continuous convex monotone semigroups on spaces of continuous functions. In contrast to previous results, we do not rely on the theory of viscosity solutions but use a recent comparison principle which uniquely determines the semigroup via its -generator defined on the Lipschitz set and therefore resembles the classical analogue from the linear case. The framework also allows for discretizations both in time and space and covers a variety of applications. This includes Euler schemes and Yosida-type approximations for upper envelopes of families of linear semigroups, stability results and finite-difference schemes for convex HJB equations, Freidlin-Wentzell-type results and Markov chain approximations for a class of stochastic optimal control problems and continuous-time Markov processes with uncertain transition probabilities.
Paper Structure (14 sections, 25 theorems, 193 equations)

This paper contains 14 sections, 25 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Setup and main results
  3. Basic notation and the mixed topology
  4. Stability of convex monotone semigroups
  5. Uniqueness of the limit semigroup
  6. Equicontinuity w.r.t. the mixed topology
  7. Examples
  8. Euler formula and Yosida approximation for upper envelopes
  9. Vanishing viscosity and stability of convex HJB equations
  10. Large deviations for randomized Euler schemes
  11. Discretization of stochastic optimal control problems
  12. A version of Arzéla Ascoli's theorem
  13. Basic convexity estimates
  14. Continuity from above

Key Result

Theorem 2.5

Let $(S_n)_{n\in\mathbb{N}}$ be a sequence satisfying Assumption ass:Sn. Then, there exist a strongly continuous convex monotone semigroup $(S(t))_{t\geq 0}$ on ${\rm C}_\kappa(\mathbb{R}^d)$ with generator $A\colon D(A)\to{\rm C}_\kappa(X)$ and a subsequence $(n_l)_{l\in\mathbb{N}}\subset\mathbb{N} where $t_n\in\mathcal{T}_n$ with $t_n\to t$ and $f_n\in{\rm C}_\kappa(X_n)$ with $f_n\to f$ are arb

Theorems & Definitions (53)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.4
  • Theorem 2.5
  • proof
  • Theorem 2.7
  • proof
  • Theorem 2.9
  • proof
  • Theorem 2.10
  • ...and 43 more