Adaptive Fast-Slow Operator Splitting for Multiscale Biochemical Stochastic Dynamics

Stochastic reaction networks governed by Chemical Langevin Equations (CLE) exhibit pronounced multiscale dynamics spanning fast molecular reactions, intermediate transport, and slow cellular regulation, posing significant challenges for efficient and accurate simulation. Although operator splitting naturally decouples fast and slow subsystems, a rigorous error characterization for CLE splitting schemes has been lacking. We propose a modular operator-splitting framework with adaptive discretization that enables reliable and efficient simulation across fast-slow dynamics with explicit control of discretization error. Using stochastic logarithmic representations, we present a complete error analysis of the fast-slow Lie-Trotter splitting method, decomposing the one-step error into stochastic flow truncation error, commutator errors due to subsystem noncommutativity, and numerical discretization errors from fast and slow integrations. Guided by this analysis, we develop a proportional-integral (PI) adaptive controller that jointly selects macro time steps and fast microsteps, achieving substantial efficiency gains while maintaining accuracy.