Recovering Competing Reaction Channels in Doob’s Lagrangian Transition Path Sampling

Variational transition path sampling casts rare transitions as stochastic optimal control, minimizing an objective whose optimum is the conditioned bridge between metastable states. Committor-based initialization supplies a physically informed start. When several channels compete, two distinct failures arise. A single Gaussian path can follow only one channel, and it follows the one in which it is initialized. Intermediate metastable states within the channels cause the spectral gap that sets the convergence rate of committor initialization to collapse, while competing channels alone leave the rate of order one. A mixture control with one component per channel, each initialized from a per-channel committor, removes the representational defect. Under a channel separability condition, which excludes intermediate metastable states within a channel, we prove per-channel convergence at the within-channel relaxation rate, uniformly in the degeneracy between channels, with mixture weights that recover the transition path theory channel weights conditioned on the transit horizon. Experiments on a two-dimensional two-channel family, where committor, weights, and spectrum are exactly computable, confirm the mode collapse, its repair, and the spectral dichotomy.