Overview
This study investigates the long-time behavior of Dynamical Orthogonal (DO) approximations applied to Stochastic Differential Equations (SDEs). The DO formulation represents the solution of an SDE using a low-rank decomposition. This approach results in a coupled system that includes an evolution equation on the Stiefel manifold and an associated reduced stochastic process.
Approach
The research first establishes the well-posedness of the strong DO system. Following this, it derives quantitative error estimates. These estimates compare the original SDE with its low-rank approximation, utilizing the Wasserstein distance as the metric for comparison. The core of the methodology focuses on analyzing invariant probability measures within the DO dynamics.
To prove the existence of an invariant probability measure for the strong DO system, the researchers employed a combination of techniques. These techniques include uniform moment estimates, a Krylov–Bogoliubov argument applied to an associated frozen system, and a Kakutani-Fan-Glicksberg fixed-point theorem. These methods were used in conjunction to address the self-consistent dynamics inherent in the DO system.
The conditions under which these proofs hold require specific properties of the coefficients guiding the SDE. These properties include suitable dissipativity, Lipschitz continuity, and non-degeneracy assumptions.
Findings
- The strong DO system's well-posedness was established.
- Quantitative error estimates were derived between the original SDE and its low-rank approximation in the Wasserstein distance.
- Under suitable dissipativity, Lipschitz continuity, and non-degeneracy assumptions on the coefficients, the existence of an invariant probability measure for the strong DO system was proven.
- The proof for the invariant probability measure involved uniform moment estimates, a Krylov–Bogoliubov argument for an associated frozen system, and a Kakutani-Fan-Glicksberg fixed-point theorem to recover the self-consistent dynamics.
- It was shown that the induced low-rank process also admits an invariant probability measure.
- The structure of invariant measures was discussed through several illustrative examples.
Why This Matters
These findings provide a rigorous foundation for employing dynamical low-rank approximations to approximate the long-time statistical properties of stochastic dynamical systems. This contributes to the theoretical understanding and validation of such approximation techniques in relevant applications.