Probability and Stochastic Analysis Seminar

Hydrodynamics for open ASEP with weak symmetry

We consider the asymmetric simple exclusion process (ASEP) on the one-dimensional lattice of size $n$. Particles can enter or exit the system from the boundares with given time-dependent rates. Those rates are regulated by a factor $n^\theta$. We investigate the hydrodynamic limit under the hyperbolic time scale in three cases: (1) $\theta > 0$ (fast boundary), (2) $\theta = 0$, the bulk dynamics performs rightward flux, but particles enter only from the right and exit only from the left (boundary against the flux), (3) $\theta < 0$ (slow boundary). In all cases, a weak symmetry (w.r.t. the hyperbolic time scale) is technically necessary. The macroscopic equation is given by Burgers equation with boundary conditions that allow typical discontinuities at boundaries (boundary layer). The boundary conditions are obtained through a grading scheme in case (1) and a vanishing current argument in case (2) and (3).

Based on the arXiv preprints 2108.09345 and 2203.15091.