@article{10613,
  abstract     = {Motivated by the recent preprint [\emph{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 [\emph{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 [\emph{arXiv:2004.08412}], since we considered-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 [\emph{arXiv:2004.08412}], since we consider a different notion of higher-order fluctuation fields.},
  author       = {Chen, Joe P. and Sau, Federico},
  issn         = {1024-2953},
  journal      = {Markov Processes And Related Fields},
  keywords     = {interacting particle systems, higher-order fields, hydrodynamic limit, equilibrium fluctuations, duality},
  number       = {3},
  pages        = {339--380},
  publisher    = {Polymat Publishing},
  title        = {{Higher-order hydrodynamics and equilibrium fluctuations of interacting particle systems}},
  volume       = {27},
  year         = {2021},
}

@article{1317,
  abstract     = {We analyze the behaviour of free boundaries in thin-film flow in the regime of strong slippage n∈[1,2) and in the regime of very weak slippage n∈,3) qualitatively and quantitatively. In the regime of strong slippage, we construct initial data which are bounded from above by the steady state but for which nevertheless instantaneous forward motion of the free boundary occurs. This shows that the initial behaviour of the free boundary is not determined just by the growth of the initial data at the free boundary. Note that this is a new phenomenon for degenerate parabolic equations which is specific for higher-order equations. Furthermore, this result resolves a controversy in the literature over optimality of sufficient conditions for the occurrence of a waiting time phenomenon. In contrast, in the regime of very weak slippage we derive lower bounds on free boundary propagation which are optimal in the sense that they coincide up to a constant factor with the known upper bounds. In particular, in this regime the growth of the initial data at the free boundary fully determines the initial behaviour of the interface.},
  author       = {Fischer, Julian L},
  issn         = {1873-1430},
  journal      = {Annales de l'Institut Henri Poincare (C) Non Linear Analysis},
  keywords     = {Thin-film equation, Free boundary, Waiting time, Qualitative behaviour, Higher-order parabolic equation, Degenerate parabolic equation},
  number       = {5},
  pages        = {1301 -- 1327},
  publisher    = {EMS Press},
  title        = {{Behaviour of free boundaries in thin-film flow: The regime of strong slippage and the regime of very weak slippage}},
  doi          = {10.1016/j.anihpc.2015.05.001},
  volume       = {33},
  year         = {2016},
}

@article{1318,
  abstract     = {We develop a large-scale regularity theory of higher order for divergence-form elliptic equations with heterogeneous coefficient fields a in the context of stochastic homogenization. The large-scale regularity of a-harmonic functions is encoded by Liouville principles: The space of a-harmonic functions that grow at most like a polynomial of degree k has the same dimension as in the constant-coefficient case. This result can be seen as the qualitative side of a large-scale Ck,α-regularity theory, which in the present work is developed in the form of a corresponding Ck,α-“excess decay” estimate: For a given a-harmonic function u on a ball BR, its energy distance on some ball Br to the above space of a-harmonic functions that grow at most like a polynomial of degree k has the natural decay in the radius r above some minimal radius r0. Though motivated by stochastic homogenization, the contribution of this paper is of purely deterministic nature: We work under the assumption that for the given realization a of the coefficient field, the couple (φ, σ) of scalar and vector potentials of the harmonic coordinates, where φ is the usual corrector, grows sublinearly in a mildly quantified way. We then construct “kth-order correctors” and thereby the space of a-harmonic functions that grow at most like a polynomial of degree k, establish the above excess decay, and then the corresponding Liouville principle.},
  author       = {Fischer, Julian L and Otto, Felix},
  issn         = {1532-4133},
  journal      = {Communications in Partial Differential Equations},
  keywords     = {Ck, α regularity, higher-ordercorrectors, Liouville principle, random elliptic operator, regularity theory, stochastic homogenization},
  number       = {7},
  pages        = {1108 -- 1148},
  publisher    = {Taylor & Francis},
  title        = {{A higher-order large scale regularity theory for random elliptic operators}},
  doi          = {10.1080/03605302.2016.1179318},
  volume       = {41},
  year         = {2016},
}