Nonlinear semigroups and limit theorems for convex expectations

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 \((I(t))_{t\geq 0}\) of operators, the semigroup is constructed as the limit \(S(t)f:=\lim_{n\to\infty}I(\frac{t}{n})^n f\) and is uniquely determined by the time derivative \(I'(0)f\) for smooth functions. We identify explicit conditions for the generating family \((I(t))_{t\geq 0}\) that are transferred to the semigroup \((S(t))_{t\geq 0}\) 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.