Speaker: Mario Ayala Valenzuela TU Munich
Abstract: In this talk, our focus lies in establishing scaling limits for the symmetric inclusion process. Specifically, we aim to derive hydrodynamic limits and investigate non- equilibrium fluctuations in the case where particles can interact with other particles arbitrarily far apart, allowing for so-called long-jumps. Results of this type are well-established for the long-jumps exclusion process. For this model the hydrodynamic limit is formulated in terms of a Cauchy problem associated with an infinitesimal generator of jump type. The scaling limit of density fluctuations, instead, corresponds to a fractional generalized Ornstein–Uhlenbeck process. Despite the fundamental difference in their nature, the inclusion process exhibits similarities to the exclusion process in the context of scaling limits. Both processes share a similar structure concerning ”short-jump” hydrodynamics and equilibrium fluctuations. In this work, we corroborate that these structural similarities persist even in the long-jump setting. Firstly, we establish that, under appropriate rescaling, the hydrodynamic equation of the long-jump version coincides with that of the exclusion process, sharing the same underlying random walker. Moreover, our main contribution lies in the establishment of non-equilibrium fluctuations. We find that the density fluctuation field, starting from a class of appropriate non-equilibrium measures, converges to a time-dependent generalized Ornstein–Uhlenbeck process. Notably, the characteristics of this process can be verified to coincide, modulo a constant, and when simplified to stationarity, with those found for the exclusion process with long-jumps. Work in progress with Johannes Zimmer.