  Based on the Heisenberg-picture analog of the master equation, we develop a
method for computing the exact time dependence of noise-averaged observables
for general noninteracting fermionic systems with noisy fluctuations. Upon
noise averaging, these fluctuations generate effective interactions, limiting
analytical approaches. While the short-time dynamics can be studied with
Langevin-type numerical simulations, the long-time limit is not amenable to
such simulations. Our results provide access to this long-time limit. As a
simple example, we examine the fate of the fractional charge in cold-atom
emulations of polyacetylene after stochastic driving. We find that in a quantum
quench to a fluctuating hopping Hamiltonian, the fractional charge remains
robust for hopping between different sublattices, while it becomes unstable in
the presence of noisy hopping on the same sublattice.
