  We consider the facilitated exclusion process, an interacting particle system
on the integer line where particles hop to one of their left or right
neighbouring site only when the other neighbouring site is occupied by a
particle. A peculiarity of this system is that, starting from the step initial
condition, the density profile develops a downward jump discontinuity around
the position of the first particle, unlike other exclusion processes such as
the asymmetric simple exclusion process (ASEP). In the weakly asymmetric
regime, we show that the field of particle positions around the jump
discontinuity converges to the solution of the multiplicative noise stochastic
heat equation (i.e. the exponential of a solution to the KPZ equation) on a
half-line subject to Dirichlet boundary condition, with initial condition given
by the derivative of a Dirac delta function. We prove this result by
reformulating the problem in terms of ASEP on a half-line with a boundary
reservoir, for which we extend known proofs of convergence to deal with
Dirichlet boundary condition and the very singular type of initial condition
that arises in our case.

Probability