Coupling rare event algorithms with data-based learned committor functions using the analogue Markov chain

The paper proves/uses the standard characterization of the committor for the analogue Markov chain via the absorbing modification G̃ and the fixed-point equation G̃ q = q with boundary values q_{i_A}=0, q_{i_B}=1, under an ergodicity assumption on G; it further argues that the 1-eigenspace of G̃ is two-dimensional and proposes computing q as a linear combination of two leading eigenvectors, with α,β fixed by the boundary conditions . It also notes an a posteriori check that valid solutions must satisfy 0 ≤ q_i ≤ 1 and that violations indicate non-ergodic realizations to be discarded . The candidate solution derives the same characterization but proves uniqueness by block elimination on the transient block (I−P), establishes ρ(P)<1 via a uniform hitting argument and Neumann series, shows 0 ≤ q ≤ 1 via a maximum-principle style argument, and recommends solving the sparse linear system (I−P) q_T = s numerically (with an equivalent spectral/eigenspace view). These are mathematically standard and correct, and align with the paper’s algorithmic guidance on K, distance choice, and dataset construction (including excluding the final index so T_{n,k}+1 stays in range) , . Hence, both are correct; the paper emphasizes a spectral computation route, whereas the model emphasizes linear-system/M-matrix arguments and diagnostics.