Convergence of infinitesimal generators and stability of convex monotone semigroups

TL;DR

This paper establishes a stability framework for convex monotone semigroups using infinitesimal generator convergence, avoiding viscosity solutions and enabling various discretization methods.

Contribution

It introduces a novel stability result based on generator convergence, applicable to a wide range of semigroups and discretization schemes, without relying on viscosity solutions.

Findings

Provides a stability result for convex monotone semigroups via generator convergence.

Enables discretizations like Euler schemes and finite differences for HJB equations.

Covers applications in stochastic control, Markov processes, and approximation schemes.

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 $\Gamma$-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.