Eventually and asymptotically positive semigroups on Banach lattices

We develop a theory of eventually positive $C_0$-semigroups on Banach lattices, that is, of semigroups for which, for every positive initial value, the solution of the corresponding Cauchy problem becomes positive for large times. We give characterisations of such semigroups by means of spectral and resolvent properties of the corresponding generators, complementing existing results on spaces of continuous functions. This enables us to treat a range of new examples including the square of the Laplacian with Dirichlet boundary conditions, the bi-Laplacian on $L^p$-spaces, the Dirichlet-to-Neumann operator on $L^2$ and the Laplacian with non-local boundary conditions on $L^2$ within the one unified theory. We also introduce and analyse a weaker notion of eventual positivity which we call ``asymptotic positivity'', where trajectories associated with positive initial data converge to the positive cone in the Banach lattice as $t \to \infty$. This allows us to discuss further examples which do not fall within the above-mentioned framework, among them a network flow with non-positive mass transition and a certain delay differential equation.