K-th order Hydrodynamic limits

Abstract

In this thesis, we study stochastic duality under hydrodynamic scaling in the context of interacting particles on a grid. The approach is inspired and motivated by the relation between duality and local equilibria. We identify duality relations in terms of the expectation of the density field for which the hydrodynamic limit is recovered. This is initially done both for symmetric inclusion and exclusion processes as well as for independent random walkers. We continue with the independent case and generalize to particles that also possess a, possibly scale-dependent, internal energy state. The results in this context assume generator convergence under scaling and are illustrated using run-and-tumble systems. This work also includes examples concerning instances of run-and-tumble processes that do not have convergence on a generator level. Apart from run-and-tumble processes, we examine the effect of reservoirs on the relevant duality relations and macroscopic profiles. The reservoirs are found to correspond with boundary conditions for the macroscopic profile.