  Sharp large deviation estimates for stochastic differential equations with
small noise, based on minimizing the Freidlin-Wentzell action functional under
appropriate boundary conditions, can be obtained by integrating certain matrix
Riccati differential equations along the large deviation minimizers or
instantons, either forward or backward in time. Previous works in this
direction often rely on the existence of isolated minimizers with positive
definite second variation. By adopting techniques from field theory and
explicitly evaluating the large deviation prefactors as functional determinant
ratios using Forman's theorem, we extend the approach to general systems where
degenerate submanifolds of minimizers exist. The key technique for this is a
boundary-type regularization of the second variation operator. This extension
is particularly relevant if the system possesses continuous symmetries that are
broken by the instantons. We find that removing the vanishing eigenvalues
associated with the zero modes is possible within the Riccati formulation and
amounts to modifying the initial or final conditions and evaluation of the
Riccati matrices. We apply our results in multiple examples including a
dynamical phase transition for the average surface height in short-time large
deviations of the one-dimensional Kardar-Parisi-Zhang equation with flat
initial profile.
