Learning forecasts of rare stratospheric transitions from short simulations

The paper’s Appendix B derives (i) the Feynman–Kac boundary value problem for F defined in Eq. (B6) and its PDE/BC form (B13) via an Itô/martingale argument, (ii) the finite-time “stopped” operator identity (T^Δt_{D^c} − 1)F = I^Δt_{D^c}[VF − H] (B19), (iii) specializations to the committor and MFPT in the main text (13)–(15), and (iv) the homogenized Petrov–Galerkin system (B20)–(B22), together with Monte Carlo inner products (B23)–(B28) . The candidate solution reproduces items (i)–(iii) correctly and matches the operator/Galerkin framing, but it introduces a sign error in the RHS of the homogenized finite-time equation and hence the Galerkin right-hand side: the paper’s (B21)–(B22) gives −(T−1)ĝ − I[H − Vĝ], whereas the candidate writes −(T−I)ĝ + I[H − Vĝ], i.e., the integral term has the wrong sign. This propagates to their Monte Carlo estimator for b, where the integral should enter with a minus sign, not a plus .