  For many non-equilibrium dynamics driven by small noise, in physics,
chemistry, biology, or economy, rare events do matter. Large deviation theory
then explains that the leading order term of the main statistical quantities
have an exponential behavior. The exponential rate is often obtained as the
infimum of an action, which is minimized along an instanton. In this paper, we
consider the computation of the next order sub-exponential prefactors, which
are crucial for a large number of applications. Following a path integral
approach, we derive the dynamics of the Gaussian fluctuations around the
instanton and compute from it the sub-exponential prefactors. As might be
expected, the formalism leads to the computation of functional determinants and
matrix Riccati equations. By contrast with the cases of equilibrium dynamics
with detailed balance or generalized detailed balance, we stress the specific
non locality of the solutions of the Riccati equation: the prefactors depend on
fluctuations all along the instanton and not just at its starting and ending
points. We explain how to numerically compute the prefactors. The case of
statistically stationary quantities requires considerations of non trivial
initial conditions for the matrix Riccati equation.
