Convergence of infinitesimal generators and stability of convex monotone semigroups

Jonas Blessing; Michael Kupper; Max Nendel

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arXiv · 2305.18981

Convergence of infinitesimal generators and stability of convex monotone semigroups

Jonas Blessing·Michael Kupper·Max Nendel

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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.

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Jonas Blessing, Michael Kupper, Max Nendel. 2023-05-30. Convergence of infinitesimal generators and stability of convex monotone semigroups. https://arxiv.org/abs/2305.18981

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