Antishocks in the ASEP with open boundaries conditioned on low current

We study the time evolution of the ASEP on a finite lattice with L sites and open boundaries, conditioned on an atypically low current up to a finite time t. By an exact computation, we show that for a one-parameter family of boundary densities and a special value of the conditioned current, an initial product measure with an antishock at site k evolves into a convex combination of such antishocks at sites k'. The weights p(k', t vertical bar k, 0) are shown to be the transition probabilities of simple biased random walk with reflecting boundaries. We compute explicitly these transition rates. Our result implies that the antishock remains microscopically stable under the locally conditioned dynamics.