Strong convergence for split-step methods in stochastic jump kinetics

TL;DR

This paper analyzes the strong convergence of split-step methods for stochastic jump kinetics, providing explicit conditions, error estimates, and a new simulation algorithm for path-wise comparison in reaction-diffusion models.

Contribution

It establishes convergence conditions and error bounds for the split-step method in stochastic jump processes, introducing a partition of unity approach for improved simulation.

Findings

Explicit convergence conditions and error estimates are derived.

A new partition of unity representation enhances simulation accuracy.

Numerical examples demonstrate the theoretical results.

Abstract

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 path-wise 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 path-wise sense.