A coupled backward stochastic differential equation (BSDE) framework for metastable transition precursors in a stochastically forced Duffing oscillator

This work develops a coupled backward stochastic differential equation framework for metastable transition precursors in a stochastically forced Duffing oscillator. The nonlinear oscillator evolves forward under additive noise, while two coupled backward fields are introduced to encode risk accumulation and directional transition tendency. The first field is designed to reflect future amplitude-related excursions in phase space, and the second field captures sign-sensitive transition bias between metastable wells. In a Markovian setting, the coupled forward-backward system yields a probabilistic representation of a semilinear partial differential equation system on phase space through a nonlinear Feynman-Kac relation. The formulation is tailored to nonlinear multistable dynamics and emphasizes interpretable precursor signatures before noise-induced inter-well switching. A numerical strategy based on time discretization, conditional expectation regression, and Picard iteration is presented in a directly implementable form. The proposed framework provides a compact connection between stochastic nonlinear dynamics and coupled backward equations, and it is readily extendable to excitable oscillators, multistable networks, and related stochastic systems.