Controlling mean exit time of stochastic dynamical systems based on quasipotential and machine learning

The paper asserts, without proof, that adding u=c∇V to the drift F=−(1/2)∇V+l rescales the quasipotential to V_c=(1−2c)V and hence the exit-time satisfies Td∼exp{(1−2c)V0/σ}; it then cancels a single prefactor b across different c to derive the update cd=c1+(σ/(2V0))ln(T1/Td) (its Eq. (3.19)) . The Hamilton–Jacobi derivation and orthogonal decomposition that justify V_c=(1−2c)V are present only as heuristic scaffolding in the paper (HJ and the decomposition are stated, but the controlled-HJ verification is not carried out) . Moreover, the paper implicitly treats the Eyring–Kramers prefactor as c-independent when eliminating b (again in Eq. (3.19)), despite earlier stating only the logarithmic-scale relation TE≈b e^{V0/σ} without characterizing b’s dependence on the drift . In general, b depends on the drift and thus on c (e.g., in gradient systems). The candidate solution supplies the missing controlled-HJ check showing V_c=(1−2c)V, and correctly qualifies the exit-time asymptotics: E[τ_c]=exp{(1−2c)V0/σ+o(1/σ)} universally, with a σ-independent but generally c-dependent prefactor under additional nondegeneracy. It also explains how to use calibration/iteration despite unknown b(c). Hence the model’s analysis is correct at the stated precision and fixes the paper’s key gaps.