Higher order hydrodynamics and equilibrium fluctuations of interacting particle systems

Motivated by the recent preprint [arXiv:2004.08412] by Ayala, Carinci, and Redig, we first provide a general framework for the study of scaling limits of higher order fields. Then, by considering the same class of infinite interacting particle systems as in [arXiv:2004.08412], namely symmetric simple exclusion and inclusion processes in the d-dimensional Euclidean lattice, we prove the hydrodynamic limit, and convergence for the equilibrium fluctuations, of higher order fields. In particular, the limit fields exhibit a tensor structure. Our fluctuation result differs from that in [arXiv:2004.08412], since we consider a different notion of higher order fluctuation fields.