From Schrodinger Bridge to Optimal Transport over Sub-Riemannian Manifolds

Abstract: We study the least-energy way to reshape a probability distribution when motion is constrained to a horizontal bundle, that is, optimal transport and distribution steering in sub-Riemannian geometry, motivated by density control over underactuated systems. To obtain a continuous and numerically tractable formulation, we introduce an entropic regularization by adding small noise aligned with the control directions and study the associated Schrodinger bridge problem. The resulting reference process is a degenerate diffusion on the sub-Riemannian manifold. Under bracket-generating hypotheses we obtain smooth, strictly positive transition densities and a forward--backward characterization of the optimal bridge. This leads to a practical Sinkhorn-type algorithm for the Schrodinger potentials and, as the noise level vanishes, a recovery of the deterministic sub-Riemannian optimal transport problem. We demonstrate with a numerical example.

• The paper develops an entropic regularization of sub-Riemannian distribution steering, using hypoelliptic noise and Schrödinger bridge theory to address singularities and non-uniqueness in deterministic transport.
• Its zero-noise analysis proves convergence, along subsequences, of entropic solutions to optimal sub-Riemannian transport solutions through compactness, large deviations, and Γ-convergence.
• The authors establish strong duality and a contractive Sinkhorn algorithm, then validate the framework on a Heisenberg-type system that transports a Gaussian distribution toward a ring-shaped target.

Problem setting and motivation

The paper by Adu, Elamvazhuthi, and Gharesifard (2605.11429) addresses minimum-effort distribution steering for control-affine systems with fewer controls than state dimensions. Given initial and terminal probability measures $\mu_0, \mu_f \in \mathcal{P}(\mathbb{R}^d)$ with strictly positive, compactly supported densities, and a matrix-valued map $g$ whose columns $g_1,\dots,g_m$ span a distribution $\mathcal{D} \subset T\mathbb{R}^d$ with $m < d$, the goal is to solve

$$\inf_u E\left[\int_0^{t_f} \tfrac12 \|u(t,x_t^u)\|^2\,dt\right], \qquad x_{t_f}^u \sim \mu_f,$$

subject to the deterministic dynamics $\dot{x}_t^u = g(t,x_t^u)u(t,x_t^u)$. This is an ensemble (density) control problem over the sub-Riemannian manifold $(\mathbb{R}^d, \mathcal{D}, g^{SR})$, where the metric is given via the pseudoinverse of $G = gg^\top$. The static counterpart is the Monge–Kantorovich problem with cost $\tfrac12 d_{SR}^2(x_0,x_f)$.

The authors identify the central obstruction: sub-Riemannian geometry induces non-smooth cost structure; the control-to-state map can have singularities, so minimizers may fail to be regular or unique, and even when a Monge map exists it may be discontinuous. Prior work established existence of optimal feedback controls via Young measures and, under absence-of-abnormality assumptions, regular feedback controls. The contribution here is different in kind: rather than attacking the deterministic problem directly, the authors regularize it entropically and recover the deterministic solution as a limit.

Entropic regularization as a degenerate Schrödinger bridge problem

The regularization perturbs the dynamics with noise acting only along the horizontal distribution:

$g$0

with $g$1 a standard $g$2-Brownian motion. Via Girsanov's theorem, minimizing expected quadratic control effort over this system is equivalent to the Schrödinger bridge problem (SBP): minimize the relative entropy $g$3 over path measures matching the endpoint marginals, where $g$4 is the law of the uncontrolled degenerate diffusion.

The key technical difficulty is that the reference process is genuinely degenerate: the diffusion matrix $g$5 has rank $g$6, so uniform ellipticity fails and classical parabolic theory does not apply. The paper resolves this through a structural observation that is worth emphasizing. Smoothness of the transition density follows from Hörmander's hypoellipticity theorem applied to the sum-of-squares generator $g$7 with $g$8. Strict positivity, however, is tied not to Chow–Rashevskii controllability but to the Bismut/submersion condition on the endpoint map. For a driftless diffusion, these two conditions coincide under bracket generation — this equivalence is precisely why the driftless formulation is chosen, and it yields smooth, strictly positive transition densities (Proposition 3.2). With this kernel in hand, the SBP is shown to be proper: an admissible finite-energy feedback exists, constructed explicitly via the Doob/Léonard generalized $g$9-transform, with control $g_1,\dots,g_m$0 and a finite-energy bound derived from hypoelliptic heat-kernel estimates. The authors note that Léonard's general Markovian framework treats only the elliptic Brownian case; extending the diffusion-to-OT limit to anisotropic, degenerate noise is what distinguishes this work.

Eulerian formulation and strong duality

The stochastic problem is recast in Benamou–Brenier form over the pair $g_1,\dots,g_m$1 of a space-time probability measure and its momentum, subject to the controlled Fokker–Planck equation including the Stratonovich correction drift $g_1,\dots,g_m$2. Joint strict convexity of the kinetic energy functional plus linearity of the constraints give existence and uniqueness of a measure-valued minimizer.

The first-order analysis proceeds through the dual problem with value functional $g_1,\dots,g_m$3 over functions satisfying a Hamilton–Jacobi–Bellman inequality. Assuming solvability of the forward–backward Schrödinger system

$g_1,\dots,g_m$4

strong duality holds with no gap: the dual optimizer is $g_1,\dots,g_m$5, the primal optimizer is $g_1,\dots,g_m$6 with $g_1,\dots,g_m$7, both smooth and strictly positive. The proof combines weak duality via Young's inequality, exact verification of the HJB equality using the log-transform, and smoothness inherited from the hypoelliptic kernels. A consequence is that the entire optimal ensemble transport is characterized by two scalar potentials solving linear PDEs — a structure directly exploitable numerically.

Zero-noise limit: recovery of sub-Riemannian optimal transport

The main convergence result shows that as $g_1,\dots,g_m$8, the entropic solutions converge (up to subsequences) to a solution of the deterministic sub-Riemannian Benamou–Brenier problem with the continuity equation $g_1,\dots,g_m$9. The argument has three parts:

1. Compactness: the KL-based energy bound, combined with small-time heat-kernel asymptotics at the cut locus (via Barilari–Boscain–Neel), gives an $\mathcal{D} \subset T\mathbb{R}^d$0-independent energy bound; Cauchy–Schwarz then transfers tightness from $\mathcal{D} \subset T\mathbb{R}^d$1 to $\mathcal{D} \subset T\mathbb{R}^d$2, and Prokhorov's theorem yields precompactness.
2. Limit feasibility: weak limits satisfy the deterministic continuity equation.
3. Optimality: a $\mathcal{D} \subset T\mathbb{R}^d$3-convergence argument identifies the limit functional. The laws $\mathcal{D} \subset T\mathbb{R}^d$4 satisfy an LDP with good rate function equal to the sub-Riemannian energy $\mathcal{D} \subset T\mathbb{R}^d$5 (using Chiarini–Fischer for small-noise Itô processes); combining this with Léonard's machinery gives $\mathcal{D} \subset T\mathbb{R}^d$6-convergence of $\mathcal{D} \subset T\mathbb{R}^d$7 to the path-space action functional, and the recovery sequence supplies the limsup bound.

The conclusion $\mathcal{D} \subset T\mathbb{R}^d$8 means the strictly convex entropic problem acts as a selection principle among potentially many deterministic SR-optimal transports, and converts the hyperbolic continuity-equation constraint into a parabolic one — a structural change with direct numerical consequences.

Numerical method and example

The Schrödinger system is solved by a Sinkhorn/IPFP fixed-point iteration on the cone of positive functions. The update operator $\mathcal{D} \subset T\mathbb{R}^d$9 is a contraction in the Hilbert metric (following Chen–Georgiou–Pavon), so the iteration converges linearly to the unique fixed point; a continuation schedule over decreasing $m < d$0 warm-starts each stage. The authors stress a discretization subtlety absent in elliptic settings: standard Euclidean finite differences need not preserve positivity and mass under horizontal degeneracy, so their schemes discretize the generator and adjoint consistently with the sub-Riemannian geometry.

The demonstration uses the Heisenberg-type system on $m < d$1 with $m < d$2, $m < d$3, whose Lie bracket $m < d$4 satisfies Hörmander's condition everywhere. Because the vector fields are left-invariant on the Heisenberg group, the transition density admits a closed-form Fourier-integral expression, giving an explicit family of sub-Riemannian Schrödinger bridges for every $m < d$5. Steering a Gaussian blob to a ring-shaped terminal density ($m < d$6, $m < d$7) is computed for $m < d$8, showing smooth morphing of the time marginals and progressive concentration of sample trajectories onto horizontally constrained paths. Using Varadhan-type asymptotics $m < d$9 together with the explicit Heisenberg distance formula, Laplace principle yields the limiting value function $$\inf_u E\left[\int_0^{t_f} \tfrac12 \|u(t,x_t^u)\|^2\,dt\right], \qquad x_{t_f}^u \sim \mu_f,$$0 and hence the deterministic optimal feedback $$\inf_u E\left[\int_0^{t_f} \tfrac12 \|u(t,x_t^u)\|^2\,dt\right], \qquad x_{t_f}^u \sim \mu_f,$$1.

Limitations and open questions

Several restrictions bound the scope of the results. All analysis requires bounded derivatives of all orders on the $$\inf_u E\left[\int_0^{t_f} \tfrac12 \|u(t,x_t^u)\|^2\,dt\right], \qquad x_{t_f}^u \sim \mu_f,$$2 and global bracket generation on $$\inf_u E\left[\int_0^{t_f} \tfrac12 \|u(t,x_t^u)\|^2\,dt\right], \qquad x_{t_f}^u \sim \mu_f,$$3; behavior at cut loci, where the sub-Riemannian distance loses smoothness and the heat-kernel asymptotics used in the compactness step are delicate, is handled only through cited asymptotic bounds rather than analyzed intrinsically. The convergence theorem guarantees optimality of the limit only along a convergent subsequence, and uniqueness of the deterministic limit point is not established — consistent with the known non-uniqueness of sub-Riemannian Monge solutions, though the strict convexity selection mechanism suggests uniqueness would be worth investigating. The numerical example relies on an explicitly available heat kernel via group structure; how the Sinkhorn pipeline performs for generic non-left-invariant bracket-generating systems, where the kernel must be computed numerically under positivity-preserving geometric discretization, remains open. Finally, the endpoint densities are assumed compactly supported (or exponentially tailed); relaxing this to heavier tails would require refined heat-kernel integrability estimates.

Conclusion

The paper constructs a complete pipeline from entropic regularization to deterministic solution for optimal transport over sub-Riemannian manifolds: well-posedness of a degenerate Schrödinger bridge via the driftless Hörmander/Bismut equivalence, a strongly dual Eulerian characterization with smooth positive optimizers, $$\inf_u E\left[\int_0^{t_f} \tfrac12 \|u(t,x_t^u)\|^2\,dt\right], \qquad x_{t_f}^u \sim \mu_f,$$4-convergence of the zero-noise limit to the sub-Riemannian Benamou–Brenier problem, and a Hilbert-metric-contractive Sinkhorn algorithm validated on the Heisenberg group with explicit kernels. The framework mitigates the singularity and non-uniqueness pathologies of the deterministic problem while preserving its solution in the limit, and extends the entropic OT/Sinkhorn paradigm to genuinely hypoelliptic reference processes.