Abstract
In this paper, we investigate convex semigroups on Banach lattices with order continuous norm, having L^p-spaces in mind as a typical application. We show that the basic results from linear C_0-semigroup theory extend to the convex case. We prove that the generator of a convex C_0-semigroup is closed and uniquely determines the semigroup whenever the domain is dense. Moreover, the domain of the generator is invariant under the semigroup, a result that leads to the well-posedness of the related Cauchy problem. In a last step, we provide conditions for the existence and strong continuity of semigroup envelopes for families of C_0-semigroups. The results are discussed in several examples such as semilinear heat equations and nonlinear integro-differential equations.
Highlights
Decision-making in a dynamic random environment naturally leads to so-called stochastic optimal control problems
The dynamic programming principle typically leads to convex partial differential equations, so-called Hamilton–Jacobi–Bellman (HJB) equations, where, intuitively speaking, the convexity comes from optimizing among a certain class of Markov processes, each one linked to a linear PDE via its infinitesimal generator
One classical approach to treat nonlinear partial differential equations uses the theory of maximal monotone or m-accretive operators; see, e.g., Barbu [2], Bénilan-Crandall [5], Brézis [6], Evans [10], Kato [14], and the references therein
Summary
Decision-making in a dynamic random environment naturally leads to so-called stochastic optimal control problems. We extend classical results from semigroup theory regarding uniqueness of the semigroup in terms of the generator, space and time regularity of solutions in terms of initial data, more precisely, invariance of the domain under the semigroup, and classical well-posedness of related Cauchy problems to the convex case. We provide invariance of the domain, uniqueness of the semigroup in terms of the generator and classical well-posedness of the related Cauchy problem. 5, we provide conditions for the existence and strong continuity of the semigroup envelope for families (Sλ)λ∈ of linear convolution semigroups on L p(μ), and relate the generator of the semigroup envelope to supλ∈ Aλ, i.e., the smallest upper bound of the generators ( Aλ)λ∈ of (Sλ)λ∈. We collect some additional results on bounded convex operators on general Banach lattices including a version of the uniform boundedness principle for convex operators
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