Symplectic techniques for stochastic differential equations on reductive Lie groups with applications to Langevin diffusions

Abstract: We show how Langevin diffusions can be interpreted in the context of stochastic Hamiltonian systems with structure-preserving noise and dissipation on reductive Lie groups. Reductive Lie groups provide the setting in which the Lie group structure is compatible with Riemannian structures, via the existence of bi-invariant metrics. This structure allows for the explicit construction of Riemannian Brownian motion via symplectic techniques, which permits the study of Langevin diffusions with noise in the position coordinate as well as Langevin diffusions with noise in both momentum and position.

• The paper introduces a framework for efficient sampling of SDEs on reductive Lie groups using Riemannian Brownian motion and symplectic techniques.
• It derives Langevin equations as stochastic Hamiltonian systems with geometric dissipation, ensuring that the invariant measure is the Gibbs measure.
• The work lays a foundation for structure-preserving numerical schemes applicable to momentum, position, and symplectic Langevin dynamics.

Symplectic Techniques for Stochastic Differential Equations on Reductive Lie Groups

This paper introduces a framework for Langevin diffusions on reductive Lie groups using stochastic Hamiltonian systems with structure-preserving noise and dissipation. The authors leverage the unique properties of reductive Lie groups, particularly the existence of bi-invariant metrics, to construct Riemannian Brownian motion (RBM) explicitly, facilitating the study of Langevin diffusions with noise in both position and momentum coordinates.

Main Contributions

The paper makes two primary contributions:

• Efficient Sampling of SDEs on Reductive Lie Groups: The authors demonstrate that RBM, and more generally, solutions to SDEs on reductive Lie groups, can be sampled efficiently in a pathwise strong sense, avoiding the limitations of geodesic random walks.
• Symplectic Derivation of Langevin Equations: Using symplectic techniques, the paper derives families of Langevin equations as special cases of stochastic Hamiltonian systems with geometric dissipation on reductive Lie groups, proving that their ergodic invariant measure is the Gibbs measure.

Key Concepts and Techniques

Reductive Lie Groups

The paper focuses on reductive Lie groups, which possess a compatible Lie group structure and Riemannian structure due to the existence of bi-invariant metrics. Examples include $GL(n)$, $U(n)$, $SU(n)$, $Sp(n)$, and $SO(n)$.

Riemannian Brownian Motion

The authors construct RBM on reductive Lie groups using symplectic techniques, enabling the study of Langevin diffusions with noise in position and momentum.

Stochastic Hamiltonian Systems

The paper formulates Langevin diffusions as stochastic Hamiltonian systems with geometric dissipation on reductive Lie groups. This approach ensures that the invariant measure is always a Gibbs measure and motivates the use of structure-preserving numerical integrators.

Double-Bracket Dissipation

The authors introduce double-bracket dissipation, a geometry-preserving dissipation, to cancel the effect of noise on the invariant measure, ensuring that the Gibbs measure remains the invariant measure for the SDEs.

Implementation Details

Eells-Elworthy-Malliavin Construction

The paper utilizes the Eells-Elworthy-Malliavin construction for RBM, simplifying it for compact Lie groups with bi-invariant metrics. The SDE describing RBM on a compact Lie group is given by:

$$\boldsymbol{d}g = (dL_{g})_e(X_i\circ\boldsymbol{d}W_t^i) = (dL_g)_e(\circ\boldsymbol{d}B_t)$$

which can be solved analytically by means of the Lie exponential for the initial data $g(0) = e\in G$:

$g_t = \exp B_t$

Stochastic Geometric Mechanics

The paper employs Malliavin's transfer principle to lift deterministic geometric mechanics to stochastic geometric mechanics, formulating stochastic mechanics as:

$$\boldsymbol{d}S = X_{\mathcal{H}_\alpha}(S)\circ \boldsymbol{d}Z_t^\alpha$$

where $S_t:\mathbb{R}\to M$ describes the state of the system, $M$ is a symplectic manifold, $\mathcal{H}_\alpha$ are Hamiltonians, and $Z_t^\alpha$ are semimartingales.

Invariant Measure

The authors show that the stochastic symplectic dynamics with double-bracket dissipation has the Gibbs measure as its invariant measure:

$$\boldsymbol{d}S_t = \left(X_{\mathcal{H}_0}(S_t) + \frac{\beta}{2}\{\mathcal{H}_0,\mathcal{H}_i\}X_{\mathcal{H}_i}(S_t) + \frac{1}{2}X_{\mathcal{H}_i}X_{\mathcal{H}_i}(S_t)\right)\boldsymbol{d}t + X_{\mathcal{H}_i}(S_t)\boldsymbol{d}W_t^i$$

Examples

The paper provides examples of momentum Langevin, position Langevin, and symplectic Langevin dynamics on reductive Lie groups and Euclidean space, demonstrating the applicability of the framework.

Implications and Future Directions

This research provides a foundation for developing provably effective and efficient numerical schemes for sampling measures on reductive Lie groups. The Hamiltonian framework enables the use of structure-preserving numerical integrators for long-time accuracy and stability.

Future work will focus on error analysis, numerical implementation of SDEs and Langevin diffusions, extension to reductive homogeneous spaces, and incorporation of curved geometries with negative curvature using bi-invariant pseudo-metrics.

Conclusion

The paper offers a comprehensive approach to Langevin diffusions on reductive Lie groups, combining symplectic techniques, stochastic Hamiltonian systems, and geometric dissipation. The explicit construction of RBM and the preservation of the Gibbs measure as the invariant measure make this framework a valuable tool for sampling and simulation in various applications.