Small Temperature Schrödinger Bridges

• Small temperature Schrödinger bridges are entropic regularized couplings that interpolate between probability measures, converging to deterministic optimal transport in the vanishing temperature limit.
• Their leading-order asymptotics are governed by the score function and Fisher information, unifying diffusion processes, Wasserstein gradient flows, and geometric PDEs.
• Extensions to discrete structures and manifolds enable novel approaches to transport-entropy inequalities and computational approximations via Sinkhorn iterations.

Small temperature Schrödinger bridges are a class of entropic regularized couplings and stochastic processes that interpolate between probability measures, converging in the vanishing temperature limit to deterministic optimal transport, diffusion processes, or geodesics in the underlying geometric setting. The study of their $\epsilon \to 0$ (or small-temperature) asymptotics reveals deep connections to Wasserstein gradient flows, diffusion semigroups, large deviations, transport-entropy inequalities, and their manifestations on both continuous spaces and discrete structures such as graphs.

1. Formulation of Schrödinger Bridges and the Small-Temperature Regime

A Schrödinger bridge with temperature (entropic regularization parameter) $\epsilon > 0$ is defined as the solution to a relative entropy minimization over path-space or endpoint couplings, subject to marginal constraints and a reference process (typically Brownian motion in Euclidean spaces, or more general diffusions/manifolds, or Markov chains on graphs). Concretely, in the static same-marginal case on $\mathbb{R}^d$, the bridge is given by the solution $\pi^\epsilon$ to

$$\pi^\epsilon = \underset{\pi \in \Pi(\mu,\mu)}{\arg\min}\, H(\pi\,\|\,\mu \otimes K_\epsilon)$$

where $$K_\epsilon(x, y) = (2\pi\epsilon)^{-d/2} \exp(-\|x-y\|^2/(2\epsilon))$$ is the heat kernel, and $H(\cdot\,\|\,\cdot)$ denotes Kullback–Leibler divergence. The corresponding plans and interpolating paths smooth out the classical optimal transport solution, yielding a (unique) entropic interpolation. As $\epsilon \downarrow 0$, the regularization disappears and the bridge concentrates on deterministic couplings minimizing the transport cost, e.g., the identity in the same-marginal case (Agarwal et al., 2024, Agarwal et al., 12 May 2025, Mulcahy et al., 21 Dec 2025, Nilsson et al., 4 Jun 2025, Samson, 2020).

On graphs, the temperature parameter is interpreted as a slowdown factor $\gamma \downarrow 0$ for a continuous-time Markov chain. Schrödinger bridges in this context interpolate between discrete measures, and as temperature vanishes, the bridges become supported on geodesics in the metric space of the graph, yielding the optimal $W_1$ coupling (Samson, 2020).

2. Analytical Expansions and Leading-Order Asymptotics

The small-temperature expansions of Schrödinger bridges demonstrate that, in Euclidean and manifold settings, the difference between the entropic plan (or its barycentric projection map $T_\epsilon$) and the identity is governed by the score function of the marginal density: $T_\epsilon(x) = x + \frac{\epsilon}{2} \nabla\log \mu(x) + o(\epsilon) \quad \text{in %%%%10%%%%}$ This Poisson-type expansion shows that Schrödinger bridges at small $\epsilon$ discretize the infinitesimal generator of the Langevin diffusion. More generally, for smooth test functions $\xi$,

$$\lim_{\epsilon\to 0} \frac{K_\epsilon[\xi](x) - \xi(x)}{\epsilon} = L\xi(x)$$

where $L$ is the Langevin (Fokker–Planck) generator. This generator limit underpins the convergence to continuous-time gradient flows and justifies the use of entropic optimal transport (Sinkhorn) steps in algorithms (Agarwal et al., 2024, Agarwal et al., 12 May 2025, Mulcahy et al., 21 Dec 2025).

For the relative entropy between $\pi^\epsilon$ and the stationary Langevin law $Q^\epsilon = \mathrm{Law}(X_0, X_\epsilon)$, a second-order expansion yields

$$H(\pi^\epsilon \| Q^\epsilon) = O(\epsilon^2), \quad H(Q^\epsilon \| \pi^\epsilon) = O(\epsilon^2)$$

with an explicit coefficient involving the Fisher information of $\mu$. This enables fine control of the error and underpins the convergence of discrete SB schemes to Wasserstein gradient flows such as the heat equation (Agarwal et al., 2024, Mulcahy et al., 21 Dec 2025).

On Riemannian manifolds, the small-time heat kernel expansion yields the convergence of the gradient of the Schrödinger potential to half the manifold score function: $\nabla_g \log a^T \to \frac{1}{2} \nabla_g \log \rho \quad \text{in %%%%17%%%% as %%%%18%%%%}$ This captures curvature and density corrections unique to the manifold geometry (Mulcahy et al., 21 Dec 2025).

3. Large Deviations, Convergence, and Optimal Transport Limits

The vanishing temperature limit connects SBs to deterministic optimal transport via large deviation principles (LDPs). For a family of entropic couplings $\{\pi^\epsilon\}_\epsilon$ on convex domains with reflected Brownian motion or more general reference diffusions, a sharp LDP establishes exponential concentration of $\pi^\epsilon$ around the unique optimal transport plan $\pi$ solving

$$\min_{\pi \in \Pi(\mu_0, \mu_1)} \int c(x, y) \, d\pi(x, y)$$

with $c(x, y)$ typically the potential limit of the scaled cost $c_\epsilon$ (Nilsson et al., 4 Jun 2025). The LDP rate function

$I(x, y) = c(x, y) - [-\psi(x) + \psi^c(y)] \geq 0$

(where $\psi$ is a Kantorovich potential) quantifies how fast probability decays for deviations from the OT plan, with rate $-\inf_{(x, y) \in A} I(x, y)/\epsilon$ for measurable $A$ disjoint from the optimal coupling's support.

These results extend to reflected SDEs, accommodating hard boundaries and providing a theoretical foundation for generative modeling schemes constrained by domain geometry (Nilsson et al., 4 Jun 2025).

4. Discrete Structures and Graph Extensions

On graphs, zero-temperature SBs recover geodesic interpolations and $W_1$-optimal couplings. The discrete setup uses jump Markov chains with a slowdown parameter $\gamma$, with transitions weighted by $\gamma^{d(x, y)}$. As $\gamma \downarrow 0$, the bridges converge to constant-speed $W_1$-geodesics and the associated endpoint couplings solve the $W_1$-optimal transport problem. Explicit binomial- or combinatorial-weighted formulas arise for transitions and interpolations on the hypercube, integer lattice, circle, and Bernoulli-Laplace slice (Samson, 2020).

The SB framework in the discrete setting underpins a systematic approach to entropic curvature, via displacement convexity of entropy, yielding explicit $W_1$- and $T_2$-entropic curvature bounds and new transport-entropy inequalities. Notably, sharp $W_2$--$W_1$ inequalities (e.g., on the hypercube and Bernoulli-Laplace) emerge that are not attainable by classical induction arguments (Samson, 2020).

5. Particle-based and Computational Approximations

In high-dimensional numerical applications, small-$\epsilon$ Schrödinger bridges are implemented by entropic optimal transport via Sinkhorn iterations. For empirical measures, the entropic coupling is represented as a matrix of weights $\Gamma^{*}$ solving

$$\min_{\Gamma \ge 0} \sum_{i, j} \Gamma_{ij} \left( \frac{\|x_i-x_j\|^2}{2\epsilon} + \log \Gamma_{ij} \right)$$

subject to prescribed marginals. The barycentric projection furnishes a map updating particles in an explicit scheme. As $\epsilon \downarrow 0$, these maps inherit the same leading error as their continuum counterparts, ensuring convergence of empirical SB flows to gradient flows (e.g., the heat flow in Wasserstein space) (Agarwal et al., 2024).

6. Implications for Information Geometry, Machine Learning, and Inequalities

The small-temperature regime reveals that Schrödinger bridge dynamics approximate infinitesimal steps along Wasserstein gradient flow, thus connecting to diffusion and score-based modeling. In modern machine learning, this connection has been leveraged to rigorously justify algorithmic practices: Sinkhorn-enforced doubly-stochastic self-attention in transformers converges, in the small-$\epsilon$ regime, to heat-flow updates. More generally, deep architectures stacking SB/entropic-OT layers approximate continuous-time diffusions and geometric PDEs (Agarwal et al., 2024).

In the discrete context, the SB perspective unifies and sharpens discrete curvature, transport-entropy, and concentration inequalities: for example, zero-temperature SBs provide a functional-analytic route to Prékopa–Leindler, Talagrand, and Csiszár-Pinsker inequalities, as well as new weak transport and modified log–Sobolev inequalities on graphs and slices (Samson, 2020).

Table: Key Small-Temperature Schrödinger Bridge Expansions

Setting	Leading Asymptotics	OT/Gradient-Flow Limit
$\mathbb{R}^d$ (same marg)	$$T_\epsilon(x) = x + \frac{\epsilon}{2} \nabla\log \mu(x) + o(\epsilon)$$	Heat equation, identity map
Riemannian manifold	$$\nabla_g \log a^T \to \frac{1}{2} \nabla_g \log \rho$$	Generator of reversible diffusion
Graph (discrete)	SB $\to$ $W_1$ geodesic coupling	Deterministic OT on graphs
Bounded domain (reflected SDE)	$\pi^\epsilon \to$ c-optimal plan exponentially fast	OT with hard constraints

7. Summary and Outlook

Small-temperature Schrödinger bridges provide a rigorous, unified perspective on the interplay between entropic optimal transport, diffusion processes, Wasserstein geometry, and functional inequalities. Their leading-order expansions are governed by the score function and Fisher information, enabling both theoretical analysis and particle-based numerical schemes. The framework integrates continuous and discrete geometries, with applications ranging from gradient flows, generative modeling, and self-attention mechanisms, to discrete curvature and concentration in graphs (Agarwal et al., 2024, Agarwal et al., 12 May 2025, Mulcahy et al., 21 Dec 2025, Nilsson et al., 4 Jun 2025, Samson, 2020).