Nonlinear semigroups and limit theorems for convex expectations

2022/10/25 by Jonas Blessing, Blessing, Jonas, Michael Kupper +1 · 5 citations
Computer Science · Mathematics · Physics and Astronomy · #60F05 #60F10 #60G50 #Advanced Banach Space Theory #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #Optimization and Variational Analysis #Primary 47H20 #Probability (math.PR) #Secondary 47J25

paper · pdf · doi:10.48550/arxiv.2210.14096

openalex publication_date 2022/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Based on the Chernoff approximation, we provide a general approximation result for convex monotone semigroups which are continuous w.r.t. the mixed topology on suitable spaces of continuous functions. Starting with a family (I(t))_{t≥ 0} of operators, the semigroup is constructed as the limit S(t)f:=lim_n→∞I((t)/(n))ⁿ f and is uniquely determined by the time derivative I'(0)f for smooth functions. We identify explicit conditions for the generating family (I(t))_{t≥ 0} that are transferred to the semigroup (S(t))_{t≥ 0} and can easily be verified in applications. Furthermore, there is a structural link between Chernoff type approximations for nonlinear semigroups and law of large numbers and central limit theorem type results for convex expectations. The framework also includes large deviation results.

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