Patchwork sampling of stochastic differential equations.

Area of Science:

• Computational Physics
• Statistical Mechanics
• Stochastic Processes

Background:

• Stochastic differential equations (SDEs) are crucial for modeling complex systems.
• Simulating SDEs can be challenging due to rarely visited states or slow transitions.
• Existing methods may struggle with efficiency and accuracy in such scenarios.

Purpose of the Study:

• To introduce a novel, accurate, and efficient method for sampling stationary properties of SDE solutions.
• To address the challenges posed by rarely visited regions and rare transitions in state spaces.
• To extend the concept of truncated Markov chains to non-detailed balance processes.

Main Methods:

• Partitioning the state space into ergodic patches.
• Running simulations strictly truncated to each patch.
• Calculating patch weights by counting attempted transitions between patches.
• Extending truncated Markov chains to non-detailed balance processes.

Main Results:

• The proposed method achieves accurate and efficient sampling of SDE stationary properties.
• Effective simulations are enabled even in rarely visited regions of the state space.
• The method is successfully illustrated with three distinct physical examples.

Conclusions:

• The developed method offers a significant improvement for simulating SDEs, particularly in complex systems.
• It provides a robust framework for handling systems with rare events or complex dynamics.
• The approach generalizes truncated Markov chains, broadening their applicability.