Strong Convergence for Split-Step Methods in Stochastic Jump Kinetics

Mesoscopic models in the reaction-diffusion framework have gained recognition as a viable approach to describing chemical processes in cell biology. The resulting computational problem is a continuous-time Markov chain on a discrete and typically very large state space. Due to the many temporal and spatial scales involved many different types of computationally more effective multiscale models have been proposed, typically coupling different types of descriptions within the Markov chain framework. In this work we look at the strong convergence properties of the basic first order Strang, or Lie–Trotter, split-step method, which is formed by decoupling the dynamics in finite time steps. Thanks to its simplicity and flexibility, this approach has been tried in many different combinations. We develop explicit sufficient conditions for pathwise well-posedness and convergence of the method, including error estimates, and we illustrate our findings with numerical examples. In doing so, we also suggest a certain partition of unity representation for the split-step method, which in turn implies a concrete simulation algorithm under which trajectories may be compared in a pathwise sense.