Hydrodynamic limits and non-equilibrium fluctuations for the Symmetric Inclusion Process with long jumpsMI 02.08.011 (Boltzmannstr. 3, 85748 Garching)

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.