Abstract

We show that the stationary density fluctuations of exclusion processes with long jumps, whose rates are of the form \(c^\pm |y-x|^{-(1+\alpha )}\) where \(c\pm \) depends on the sign of \(y-x\), are given by a fractional Ornstein–Uhlenbeck process for \(\alpha \in (0,\frac{3}{2})\). When \(\alpha =\frac{3}{2}\) we show that the density fluctuations are tight, in a suitable topology, and that any limit point is an energy solution of the fractional Burgers equation, previously introduced in Gubinelli and Jara (Stoch Partial Differ Equ Anal Comput 1(2):325–350, 2013) in the finite volume setting.

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