Rare events in a stochastic vegetation-water dynamical system based on machine learning

The paper’s WKB/eikonal setup and transport equation for the prefactor C(x) are consistent with standard theory and with the model’s derivation (see the Hamilton–Jacobi equation, the decomposition b = −(1/2)a∇V + l, and equation (3.3)–(3.4) for C(x) along the MPP ϕ; these are stated explicitly in the paper) . However, the paper’s prefactor formulas for the mean exit time contain dimensional and normalization errors. In the non-characteristic case (its (3.5)), the paper’s power of ε is (n−1) rather than the standard (n−1)/2, and it omits the interior Gaussian factor (√det ∇^2V(x̄)) that arises from the mass integral; the text also identifies the exit rate directly with a boundary flux without stating the required normalization of the quasi-stationary density . In the characteristic case (its (3.6)), the factor involving Hessians appears as √|det H*| divided by det ∇^2V(x̄) (no square root on the denominator), whereas the standard result is √(|det H*|/det ∇^2V(x̄)) . The model’s solution recovers the classical prefactors (non-characteristic: Laplace on ∂D vs D, giving a (2π ε)^{(n−1)/2} vs (2π ε)^{n/2} balance; characteristic: inner 1D instability giving the universal π/λ* and the Gaussian tangential √|det H*| with the interior √det ∇^2V(x̄) in the denominator), matching established literature.