Rigorous Validation of Stochastic Transition Paths

TL;DR

This paper introduces a rigorous validation framework for stochastic transition paths, combining analytical reduction, validated numerics, and fixed-point proofs to ensure solutions are mathematically sound.

Contribution

It presents a novel method that bridges numerical approximation and rigorous analysis for stochastic systems, specifically for computing transition paths.

Findings

Successfully computed minimum-energy transition paths in test potentials.

Validated the existence of solutions using fixed-point arguments.

Demonstrated the approach on the Müller-Brown potential.

Abstract

Global dynamics in nonlinear stochastic systems is often difficult to analyze rigorously. Yet, many excellent numerical methods exist to approximate these systems. In this work, we propose a method to bridge the gap between computation and analysis by introducing rigorous validated computations for stochastic systems. The first step is to use analytic methods to reduce the stochastic problem to one solvable by a deterministic algorithm and to numerically compute a solution. Then one uses fixed-point arguments, including a combination of analytical and validated numerical estimates, to prove that the computed solution has a true solution in a suitable neighbourhood. We demonstrate our approach by computing minimum-energy transition paths via invariant manifolds and heteroclinic connections. We illustrate our method in the context of the classical M\"uller-Brown test potential.