Wasserstein Contraction of Stochastic Systems on Manifolds: A Differential Approach
Abstract
Contraction analysis of dynamical systems turns a global convergence problem into a local one. If the variational system is stable in a uniform sense, then all trajectories of the original system converge toward each other. This paper develops a stochastic analogue for continuous-time stochastic systems, where convergence is measured by Wasserstein distance between probability laws. The key step is to employ mean-square differentiation to construct stochastic variational dynamics. Its uniform L2 stability can then be integrated along curves of initial conditions, yielding Wasserstein contraction of the associated probability laws. This yields criteria formulated in terms of differential Lyapunov functions, and on manifolds, it leads to intrinsic Riemannian formulations. A distinctive feature of the stochastic setting is that the coupling of the noises becomes a design freedom. To exploit this freedom, we introduce an orthogonal differential coupling that extends the framework beyond synchronous coupling. The resulting criterion explicitly exposes how the curvature of the state space enters the contraction mechanism and, in the orthonormal-frame case, recovers the classical Ricci-curvature structure. Finally, we apply the framework to stochastic control on a compact manifold. By introducing stochastic control input, we overcome a deterministic obstruction to global stabilization. That is, a system that cannot be globally stabilized through smooth time-invariant deterministic feedback is rendered globally contracting in W2, and its steady state distribution can be made arbitrarily concentrated around a prescribed target equilibrium.
Related
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- Local contraction of stochastic affine recursions of ℝd
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BibTeX
@misc{wu2026wasserstein,
title = {{Wasserstein Contraction of Stochastic Systems on Manifolds: A Differential Approach}},
author = {Dongjun Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03264},
eprint = {2610.03264},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03264},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Wasserstein Contraction of Stochastic Systems on Manifolds: A Differential Approach
AU - Dongjun Wu
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03264
UR - https://arxiv.org/abs/2610.03264
AB - Contraction analysis of dynamical systems turns a global convergence problem into a local one. If the variational system is stable in a uniform sense, then all trajectories of the original system converge toward each other. This paper develops a stochastic analogue for continuous-time stochastic systems, where convergence is measured by Wasserstein distance between probability laws. The key step is to employ mean-square differentiation to construct stochastic variational dynamics. Its uniform L_2 stability can then be integrated along curves of initial conditions, yielding Wasserstein contraction of the associated probability laws. This yields criteria formulated in terms of differential Lyapunov functions, and on manifolds, it leads to intrinsic Riemannian formulations. A distinctive feature of the stochastic setting is that the coupling of the noises becomes a design freedom. To exploit this freedom, we introduce an orthogonal differential coupling that extends the framework beyond synchronous coupling. The resulting criterion explicitly exposes how the curvature of the state space enters the contraction mechanism and, in the orthonormal-frame case, recovers the classical Ricci-curvature structure. Finally, we apply the framework to stochastic control on a compact manifold. By introducing stochastic control input, we overcome a deterministic obstruction to global stabilization. That is, a system that cannot be globally stabilized through smooth time-invariant deterministic feedback is rendered globally contracting in W_2, and its steady state distribution can be made arbitrarily concentrated around a prescribed target equilibrium.
ER -