Convex Semigroups on Banach Lattices

Robert Denk (), Michael Kupper and Max Nendel
Michael Kupper: Center for Mathematical Economics, Bielefeld University
Max Nendel: Center for Mathematical Economics, Bielefeld University

Abstract: In this paper, we investigate convex semigroups on Banach lattices. First, we consider the case, where the Banach lattice is $\sigma$-Dedekind complete and satisfies a monotone convergence property, having L$^p$--spaces in mind as a typical application. Second, we consider monotone convex semigroups on a Banach lattice, which is a Riesz subspace of a $\sigma$-Dedekind complete Banach lattice, where we consider the space of bounded uniformly continuous functions as a typical example. In both cases, we prove the invariance of a suitable domain for the generator under the semigroup. As a consequence, we obtain the uniqueness of the semigroup in terms of the generator. The results are discussed in several examples such as semilinear heat equations (g-expectation), nonlinear integro-differential equations (uncertain compound Poisson processes), fully nonlinear partial differential equations (uncertain shift semigroup and G-expectation).

Keywords: Convex semigroup; nonlinear Cauchy problem; fully nonlinear PDE; well-posedness and uniqueness; Hamilton-Jacobi-Bellman equations (search for similar items in EconPapers)
Pages: 36
Date: 2019-09-10
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