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Wasserstein Contraction of Stochastic Systems on Manifolds: A Differential Approach

Dongjun Wu

arXiv:2610.03264v1math.OCcs.SYeess.SY

TL;DR

The paper seeks Wasserstein contraction estimates for laws of continuous-time stochastic systems. It develops differential contraction criteria, including an orthogonal noise coupling, and derives an intrinsic condition that implies exponential W2 contraction and recovers the Ricci-curvature structure in the orthonormal-frame case.

  • Problem

    The paper seeks estimates of how the laws of a continuous-time stochastic system contract in Wasserstein distance from different initial laws.

    Source: §1 · p. 2

  • Method

    The paper constructs stochastic variational dynamics and uses an orthogonal differential coupling to formulate contraction criteria for Euclidean systems and Riemannian manifolds.

    Source: §1 · pp. 3–4

  • Results

    The intrinsic Lie-derivative criterion implies exponential W2 contraction, and in the orthonormal-frame case the criterion reduces to a familiar Ricci-curvature structure.

    Source: §1 · pp. 3–4

  • Takeaways & Limitations

    Orthogonal differential coupling can reveal contraction hidden by synchronous coupling, while recovering the familiar curvature-based condition for orthonormal frames.

    Source: §7 · p. 21

    The contraction inequality can change when the diffusion factorization changes, even though the system's probability law remains unchanged.

    Source: §4.2 · p. 15

Abstract

from arXiv · show

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.

1. Introduction.

The paper develops a differential route from stochastic variational dynamics to Wasserstein contraction, extending contraction analysis to state-dependent diffusion and Riemannian manifolds. Orthogonal differential coupling incorporates noise-rotation freedom into curvature-dependent criteria, with applications including noise-assisted stabilization on a compact manifold.

1. Introduction.

  • The framework extends prior stochastic Wasserstein contraction results by addressing state-dependent diffusion, whose derivatives enter the variational dynamics when synchronous cancellation fails.Earlier results cited in the paper either had a noise-dependent residual or assumed state-independent diffusion and drift contraction in a fixed norm.Source: §1 · pp. 1–2
  • Mean-square variational dynamics and L2 path integration turn infinitesimal contraction into Wasserstein contraction for probability laws.The construction uses differential Lyapunov functions and extends to state-dependent diffusion and time- and state-dependent contraction metrics.Source: §1 · pp. 2–3
  • The intrinsic Riemannian criterion uses Lie derivatives of the metric and implies exponential W2 contraction with respect to Riemannian distance.It is a stochastic counterpart of the deterministic condition Lf0g < 0.Source: §1 · p. 3
  • Orthogonal differential coupling modifies diffusion variation through an infinitesimal relative noise rotation, adding a coupling choice beyond synchronous coupling.The paper identifies conservatism in synchronous coupling as motivation for this additional freedom.Source: §1 · pp. 3–4
  • In the orthonormal-frame case, the curvature-dependent criterion reduces to the familiar Bakry-Émery/Ricci curvature structure for Wasserstein contraction.The coupling can be chosen so the modified diffusion variation vanishes.Source: §1 · p. 4
  • On a compact manifold, suitably shaped state-dependent diffusion yields global exponential W2 contraction and a unique invariant distribution that can be made arbitrarily concentrated around a target.The application addresses a setting where smooth deterministic feedback cannot globally stabilize the equilibrium.Source: §1 · pp. 4–5

2. Problem Formulation.

The section formulates Wasserstein contraction as a problem of finding differential conditions that ensure uniform contraction of probability laws. It sets up mean-square variational dynamics and Lyapunov functions, then identifies the corresponding Euclidean and Riemannian formulations.

2. Problem Formulation.

  • The objective is an exponential Wp contraction estimate for arbitrary initial probability laws, with constants independent of those laws.The paper treats W1 and W2 as particular cases of the broader Wp problem.Source: §2 · p. 5
  • The differential approach propagates an infinitesimal displacement through a stochastic variational equation and integrates its decay along a curve of initial conditions.Mean-square differentiation supplies the weaker derivative notion used in the stochastic setting.Source: §2 · pp. 5–6
  • A stochastic Finsler-Lyapunov function underlies this formulation, which includes p = 1, leading to W1 theory, as well as the quadratic case p = 2, relevant to W2.The quadratic structure is the stochastic analogue of a Riemannian contraction metric.Source: §2 · p. 5
  • On a Riemannian manifold, the metric’s squared tangent-vector length fixes the differential Lyapunov function, and the problem includes possible coupling choices for the driving noises.The manifold formulation asks for conditions on drift, diffusion fields, and metric that guarantee W2 contraction.Source: §2 · p. 6

3. Wasserstein Contraction in Euclidean Spaces .

The section constructs mean-square variational dynamics and uses differential Lyapunov conditions to derive Wasserstein contraction. It also gives a stochastic matrix-measure criterion for W1 contraction.

3.1. Mean-square variational dynamics.

  • Mean-square differentiation yields stochastic variational dynamics, and L2 path integration turns infinitesimal estimates into incremental estimates.The derivative is characterized by the stochastic analogue of the classical variational equation.Source: §3.1 · p. 7

3.2. Wasserstein Contraction in Euclidean Spaces.

  • The passage states that system (1.3) is Wp contracting; path integration estimation holds in the L2 sense and leads to incremental estimations.The states start from an optimal initial coupling and evolve with the same Brownian motion.Source: §3.1 · p. 7, §3.2 · pp. 8–9
  • For linear systems, W2 contraction holds if and only if positive definite matrices P and Q satisfy the stated condition.The sufficiency direction follows by choosing V(t, x, y) = y⊤Py.Source: §3.2 · p. 10

3.3. Stochastic Matrix Measure and W1 contraction.

  • A stochastic matrix-measure bound gives a sufficient condition for W1 contraction, arising naturally from the Finsler-Lyapunov choice V(t, x, y) = |y|∗.The resulting expression is pointwise in the drift, diffusion matrices, and derivatives of the norm.Source: §3.3 · pp. 10–11
  • For scalar systems with the Euclidean norm, W1 contraction follows when the deterministic drift system is contracting, a property the passage distinguishes from W2 contraction.The passage presents this as a phenomenon that does not hold for W2 contraction.Source: §3.3 · p. 11

4. Wasserstein Contraction on Riemannian Manifolds.

On Riemannian manifolds, stochastic prolongation on the tangent bundle supplies infinitesimal dynamics, and an intrinsic criterion yields W2 contraction under its stated condition.

4.1. Stochastic prolongation on the tangent bundle.

  • Complete lifts of the vector fields produce tangent-bundle stochastic prolongation whose second component is the local stochastic variational dynamics.The manifold construction assumes geodesic completeness and the stated regularity of the vector fields.Source: §4.1 · pp. 11–12, §4 · p. 11
  • The prolonged dynamics represent mean-square variation, and path integration along curves provides the manifold analogue of the Euclidean incremental estimate.The construction is extended through relatively compact charts, with nonexplosion allowing localization times to tend to infinity.Source: §4.1 · pp. 12–13

4.2. An intrinsic contraction criterion .

  • Under Assumption 2, the intrinsic criterion in Theorem 2 implies W2 contraction on the manifold.The criterion is stated for a positive contraction rate.Source: §4.2 · p. 13
  • For Brownian motion on the sphere, the synchronous criterion gives contraction rate (n−3)/2 when n ≥ 4, while the passage calls this estimate conservative.An appropriate differential coupling is said to recover the sharper rate (n−2)/2.Source: §4.2 · p. 14
  • For the stated SO(3) system with a bi-invariant metric and Killing fields, Theorem 2 cannot certify any strictly positive W2 contraction rate.The methods are described as restrictive because changing the diffusion factorization can change the criterion without changing the law.Source: §4.2 · p. 15

5. W2 Contraction Beyond Synchronous Coupling.

The section extends Wasserstein contraction beyond synchronous coupling by treating relative noise rotation as an additional design freedom. Its manifold criterion exposes curvature’s role and recovers Ricci-based contraction in the orthonormal-frame case.

5. W2 Contraction Beyond Synchronous Coupling.

  • Orthogonal differential coupling adds an infinitesimal relative rotation of the driving Brownian motions, with the differential rotation—not the finite coupling—entering the contraction condition.The synchronous result is recovered by setting the rotation to zero.Source: §5 · p. 15, §5.1 · p. 16

5.1. Orthogonal differential coupling in Euclidean space.

  • A differential Lyapunov criterion using a linear map from tangent directions to skew-symmetric matrices is sufficient for W2 contraction in Euclidean space.The theorem assumes the paper’s Assumption 1 and regularity and positivity conditions on the Lyapunov function.Source: §5.1 · p. 16
  • For geometric Brownian motion, the coupling contribution vanishes at x = 0, so it cannot weaken the synchronous condition in that example.The paper notes that additional coupling freedom does not necessarily improve every contraction condition.Source: §5.1 · p. 17

5.2. Orthogonal coupling on Riemannian manifolds.

  • On a Riemannian manifold, directly rotating diffusion fields in Stratonovich form can add Itô drift terms and fail to preserve the marginal law; the local-coordinate construction yields a W2 contraction criterion.The theorem uses a tangent-space map into skew-symmetric matrices.Source: §5.2 · p. 17

5.3. Ricci curvature effect and examples.

  • For an orthonormal diffusion frame, the coupling removes frame variation so only curvature remains, recovering the classical Ricci/Bakry-Émery structure.The paper identifies this as a special case of its differential contraction condition.Source: §5.2 · p. 18, §5.3 · p. 18
  • On S^(n−1), orthogonal coupling improves the synchronous estimate from (n − 3)/2 to (n − 2)/2 and also recovers contraction on S2.The result states contraction for the sphere when n ≥ 3.Source: §5.3 · p. 19
  • On SO(3), the example has W2 contraction rate 1/4, whereas synchronous coupling certifies no positive rate.Source: §5.3 · p. 19
  • For diffusion on positive definite matrices with the affine-invariant metric, the stated system is W2 contracting when k > n/8.Source: §5.3 · p. 20

6. Case Study: Noise-assisted Global Stabilization on a Compact Manifold.

On the sphere, smooth deterministic feedback toward a target leaves its antipode as another equilibrium, preventing global stabilization. The paper adds state-dependent noise to obtain global W2 contraction while making the invariant distribution arbitrarily concentrated around the target without losing a positive contraction rate.

6. Case Study: Noise-assisted Global Stabilization on a Compact Manifold.

  • The deterministic feedback u = kP_xx⋆ leaves −x⋆ as an equilibrium, so it does not globally stabilize x⋆.Source: §6 · p. 20
  • The proposed state-dependent diffusion is small near the target and large near the antipodal point, where the deterministic obstruction occurs.The paper describes the noise as weak near the desired state and strong near the undesired equilibrium.Source: §6 · pp. 20–21
  • For each ε > 0, the stochastic system is globally W2 contracting and admits a unique invariant probability measure.Source: §6 · p. 21
  • The invariant distribution can be made arbitrarily concentrated around x⋆ while preserving a strictly positive global contraction rate.The parameter k sets the convergence time scale, while ε determines the noise remaining near the target.Source: §6 · p. 21

7. Conclusion.

The paper’s differential approach yields Euclidean W1 and W2 contraction criteria and a manifold criterion that recovers curvature-based contraction in the orthonormal-frame case. Its examples show that orthogonal noise coupling can reveal contraction hidden by synchronous coupling, and that noise-assisted feedback moves the probability distribution from an undesired equilibrium toward the target.

7. Conclusion.

  • Euclidean criteria based on general Finsler-Lyapunov functions yield both W1 and W2 contraction results.The criteria include time- and state-dependent contraction metrics.Source: §7 · p. 21
  • Orthogonal differential coupling incorporates noise-coupling freedom into variational dynamics and recovers curvature-based contraction on manifolds in the orthonormal-frame case.The sphere and Lie-group examples illustrate contraction hidden by synchronous coupling.Source: §7 · p. 21
  • From −x⋆, noise-assisted trajectories escape the undesired deterministic equilibrium and the distribution concentrates near x⋆, while the deterministic trajectory remains at x⊤.The ensemble mean approaches the stationary mean associated with πε.Source: Figure 6.1 · p. 22, Figure 6.2 · p. 22, §7 · p. 22
  • Shaping diffusion as a control input lets noise modify the global behavior of the probability law.The paper presents this as a consequence of its stochastic-control example.Source: §7 · p. 22
  • Using Wasserstein contraction for observer design remains an open direction, with non-synchronous noise needing consideration.The conclusion identifies observer design as one of several directions left open.Source: §7 · p. 22

A.1. Some Riemannian geometry.

This appendix section introduces Riemannian geometry objects used in the paper, including curvature, the metric Lie derivative, and gradient and Hessian notation.

A.1. Some Riemannian geometry.

  • The curvature tensor is defined through the metric pairing of R(X,Y)Z with W.The passage specifies R(X,Y,Z,W) = g(R(X,Y)Z,W).Source: §A.1 · p. 23
  • The Ricci tensor is introduced using an orthonormal basis.The supplied passage states the basis condition but does not provide the defining expression.Source: §A.1 · p. 23
  • The metric Lie derivative is defined, and X is called a Killing field when LXg = 0.The definition is given for vectors u and v in TxM.Source: §A.1 · p. 23
  • The gradient of a scalar function is defined by metric duality, and Hess φ is introduced as its Hessian.The gradient satisfies g(∇φ,v) = dφ(v) for v in TM.Source: §A.1 · p. 23

A.2. A Lemma on Metric Derivative.

Lemma 5 is stated for a continuous curve in a metric space and a continuous map from its parameter interval into that space.

A.2. A Lemma on Metric Derivative.

  • Lemma 5 assumes a metric space (X,d) and a continuous curve γ:[0,1]→X.The passage also introduces a continuous map v:[0,1]→X.Source: §A.2 · p. 23

A.3. Proof of Proposition 1.

The proof uses continuous-dependence estimates and linear-SDE bounds to establish convergence of the variational approximation and continuity of the variational solution in its initial conditions.

A.3. Proof of Proposition 1.

  • Bounded coefficients on [0,T] and standard linear-SDE estimates support the variational-equation convergence argument.The estimates cited include Burkholder–Davis–Gundy and Grönwall bounds.Source: §A.3 · p. 24
  • Continuous dependence of the original solutions and coefficient derivatives, together with uniform integrability and Vitali’s theorem, yields (3.5).The passage also invokes dominated convergence in time.Source: §A.3 · p. 24
  • Continuous-dependence estimates for two variational-equation solutions establish continuity of (x,y)↦Y^(x,y) in L2(Ω;C([0,T];R^n)).The proof also uses continuous dependence of X^x on x.Source: §A.3 · p. 24

A.4. Proof of Theorem 4.

The proof globalizes a local differential estimate by switching coordinate charts while preserving the metric quantity, then removes compact stopping through nonexplosion. Its sphere calculation uses orthogonal differential coupling in the contraction argument.

A.4. Proof of Theorem 4.

  • The local metric-energy estimate is ΩVα ≤ -2λVα, which is concatenated across coordinate intervals to obtain the stopped estimate.Nonexplosion and Fatou’s lemma then pass from compactly stopped paths to the unstopped estimate.Source: §A.4 · pp. 27–29
  • Chart transitions preserve the relevant metric quantity, so switching coordinates introduces no chart-dependent multiplicative constants.The proof also rules out infinitely many chart switches in finite time using the finite cover, positive separation, and path continuity.Source: §A.4 · p. 28

A.5. Proof of Proposition 3.

  • On S2, the orthogonal differential coupling cancels the second term in the contraction calculation, and the contraction estimate follows from Theorem 4.The sphere’s Ricci curvature is Ric = g.Source: §A.5 · pp. 29–30
  • To establish invariance of πϵ = ρϵdvol(x), the proof reduces the claim to L*ρϵ = 0 and uses azimuthal coordinates on the sphere.In these coordinates, dg(x, x⋆) = θ and dvol = sin θdθdϕ.Source: §A.5 · p. 31
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