  We present a new algorithm, the efficient jet marching method (EJM), for
computing the quasipotential and its gradient for two-dimensional SDEs. The
quasipotential is a potential-like function for nongradient SDEs that gives
asymptotic estimates for the invariant probability measure, expected escape
times from basins of attractors, and maximum likelihood escape paths. The
quasipotential is a solution to an optimal control problem with an anisotropic
cost function which can be solved for numerically via Dijkstra-like
label-setting methods. Previous Dijkstra-like quasipotential solvers have
displayed in general 1st order accuracy in the mesh spacing. However, by
utilizing higher order interpolations of the quasipotential as well as more
accurate approximations of the minimum action paths (MAPs), EJM achieves
second-order accuracy for the quasipotential and nearly second-order for its
gradient. Moreover, by using targeted search neighborhoods for the fastest
characteristics following the ideas of Mirebeau, EJM also enjoys a reduction in
computation time. This highly accurate solver enables us to compute the
prefactor for the WKB approximation for the invariant probability measure and
the Bouchet-Reygner sharp estimate for the expected escape time for the
Maier-Stein SDE. Our codes are available on GitHub.
