In this talk, our focus lies in establishing scaling limits for the symmetric inclusion process, introduced in [2]. 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.

Hydrodynamic limits in the context of long-jumps for the exclusion process were first es- tablished by [4], who formulated a Cauchy problem associated with an infinitesimal generator of jump type. This is in contrast to the situation with short-jumps, where the Cauchy problem is known to be associated with a generator of diffusive type. Subsequently, [3] investigated fluctuations from the hydrodynamic limit for the exclusion process, considering initializations from Bernoulli product measures. They demonstrated that the scaling limit of density fluctuations corresponds to a fractional generalized Ornstein–Uhlenbeck process.

Despite the fundamental difference in their nature, the inclusion process exhibits similar- ities to the exclusion process in the context of scaling limits. Both processes share a similar structure concerning ”short-jump” hydrodynamics and equilibrium fluctuations [1]. In this paper, 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 appropri- ate 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 in [3] for the exclusion process with long-jumps.

Work in progress with Johannes Zimmer.

References [1] Mario Ayala, Gioia Carinci, and Frank Redig. Higher order fluctuation fields and orthogonal duality polynomials. Electronic Journal of Probability, 26:1–35, 2021. [2] Cristian Giardina, Jorge Kurchan, and Frank Redig. Duality and exact correlations for a model of heat conduction. Journal of mathematical physics, 48(3):033301, 2007. [3] Patricia Goncalves and Milton Jara. Density fluctuations for exclusion processes with long jumps. Probability Theory and Related Fields, 170:311–362, 2018. [4] Milton Jara. Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps. Communications on pure and applied mathematics, 62(2):198–214, 2009.