publications

2026

1. Existence of viscosity solutions to abstract Cauchy problems via nonlinear semigroups

   Fabian Fuchs, and Max Nendel

   Bulletin of the London Mathematical Society, 2026

   Abs DOI arXiv

   Abstract In this work, we provide conditions for nonlinear monotone semigroups on locally convex vector lattices to give rise to a generalized notion of viscosity solutions to a related nonlinear partial differential equation. The semigroup needs to satisfy a convexity estimate, so called \K\-convexity, with respect to another family of operators, defined on a potentially larger locally convex vector lattice. We then show that, under mild continuity requirements on the bounding family of operators, the semigroup yields viscosity solutions to the abstract Cauchy problem given in terms of its generator in the larger locally convex vector lattice. We apply our results to drift control problems for infinite-dimensional Lévy processes and robust optimal control problems for infinite-dimensional Ornstein-Uhlenbeck processes.

2. Projected Evolutionary Lifting and Well-Posedness of Stationary Hamilton-Jacobi-Bellman Equations in Infinite Dimensions

   Gabriele Bolli, and Fabian Fuchs

   2026

   arXiv
3. New Perspectives on Existence and Uniqueness of Viscosity Solutions

   Fabian Fuchs

   2026

   DOI
4. A comparison principle based on couplings of partial integro-differential operators

   Serena Della Corte, Fabian Fuchs, Richard C. Kraaij, and Max Nendel

   Journal de Mathématiques Pures et Appliquées, 2026

   Abs DOI

   This paper is concerned with a comparison principle for viscosity solutions to Hamilton-Jacobi (HJ), -Bellman (HJB), and -Isaacs (HJI) equations for general classes of partial integro-differential operators. Our approach contributes to the literature in three ways: (1) We cast the Crandall-Ishii Lemma into a test function framework to tackle a wide class of second-order integro-differential operators in the spirit of the classical doubling of variables method. (2) We provide a unified approach to estimate the difference of Hamiltonians by adapting the probabilistic notion of couplings to an analytic setting. (3) We strengthen the sup-norm contractivity resulting from the comparison principle to one that encodes continuity in the strict topology. We apply our theory to a variety of examples, in particular, to second-order differential operators and, more generally, generators of spatially inhomogeneous Lévy processes. Résumé en français. Cet article porte sur un principe de comparaison pour les solutions de viscosité d’équations de Hamilton-Jacobi (HJ), -Bellman (HJB) et -Isaacs (HJI), pour des classes générales d’opérateurs intégro-différentiels partiels. Notre approche contribue à la littérature des trois manières suivantes : (1) Nous reformulons le lemme de Crandall-Ishii dans un cadre de fonctions tests afin de traiter une large classe d’opérateurs intégro-différentiels du second ordre, dans l’esprit de la méthode classique du doublement des variables. (2) Nous proposons une approche unifiée afin d’estimer la différence des Hamiltoniens en adaptant la notion probabiliste de couplages à un cadre analytique. (3) Nous renforçons la contraction en norme supremum issue du principe de comparaison en une contraction encodant la continuité pour la topologie stricte. Nous appliquons notre théorie à une variété d’exemples, en particulier aux opérateurs différentiels du second ordre et, plus généralement, aux générateurs de processus de Lévy spatialement inhomogènes.

2025

1. A Strict Comparison Principle for Integro-Differential Hamilton-Jacobi-Bellman Equations on Domains with Boundary

   Serena Della Corte, Fabian Fuchs, Richard C. Kraaij, and Max Nendel

   2025

   arXiv
2. Existence of Viscosity Solutions to Abstract Cauchy Problems via Nonlinear Semigroups

   Fabian Fuchs, and Max Nendel

   2025

   arXiv