A Schrödinger bridge asks for the most likely random evolution connecting two unlikely endpoint distributions under a reference diffusion. It is a path-space entropy projection whose endpoint reduction is entropic transport and whose zero-noise limit is classical optimal transport.
The path-space problem
Definition. Let \(R^\varepsilon\) be a reference Markov law on path space \(\Omega=C([0,1];\mathbb R^d)\), typically Brownian motion with generator \(\tfrac\varepsilon2\Delta\). For prescribed endpoints \(\mu_0,\mu_1\), the Schrödinger problem is \[ \inf_{P\in\mathcal P(\Omega)} \left\{ \operatorname{KL}(P\,\|\,R^\varepsilon): P_0=\mu_0,\ P_1=\mu_1 \right\}, \] where \(P_t=(e_t)_\#P\).
The minimizer changes the reference dynamics as little as possible in relative entropy while enforcing both endpoint laws. It is not conditioned on two fixed points, but on two full marginal distributions.
Reduction to a static entropy projection
Static reduction. Let \(R^\varepsilon_{01}\) be the joint endpoint law of the reference process. Then \[ \inf_{\substack{P:P_0=\mu_0\\P_1=\mu_1}} \operatorname{KL}(P\,\|\,R^\varepsilon) = \inf_{\pi\in\Pi(\mu_0,\mu_1)} \operatorname{KL}(\pi\,\|\,R^\varepsilon_{01}). \] If \(\pi^\star\) solves the static problem, the dynamic minimizer is \[ P^\star(\mathrm{d}\omega) =\int R^\varepsilon(\mathrm{d}\omega\mid X_0=x,X_1=y) \,\pi^\star(\mathrm{d}x,\mathrm{d}y). \]
Proof. Disintegrate \(P\) and \(R^\varepsilon\) with respect to \((X_0,X_1)\). The chain rule for entropy gives \[ \operatorname{KL}(P\,\|\,R^\varepsilon) = \operatorname{KL}(P_{01}\,\|\,R^\varepsilon_{01}) +\int \operatorname{KL}\!\left( P^{xy}\,\|\,R^{\varepsilon,xy} \right) \,\mathrm{d}P_{01}(x,y). \] The second term is nonnegative and vanishes exactly when the conditional path law given endpoints is the reference bridge. Minimizing leaves only the endpoint coupling problem.
Relation to entropic optimal transport
For Brownian reference, the endpoint kernel has the form \[ r^\varepsilon_{01}(x,y)\propto \exp\!\left(-\frac{|x-y|^2}{2\varepsilon}\right) \] relative to suitable reference marginals. Expanding \(\operatorname{KL}(\pi\,\|\,R^\varepsilon_{01})\) yields, up to terms depending only on \(\mu_0,\mu_1\), \[ \frac{1}{2\varepsilon}\int|x-y|^2\,\mathrm{d}\pi +\operatorname{KL}(\pi\,\|\,\mu_0\otimes\mu_1). \] Thus the static Schrödinger bridge is quadratic entropic transport after a change of constants and parameterization.
Schrödinger potentials
Factorization theorem. Under positivity and finite-entropy assumptions, the optimal endpoint coupling has form \[ \pi^\star(\mathrm{d}x,\mathrm{d}y) =f(x)g(y)R^\varepsilon_{01}(\mathrm{d}x,\mathrm{d}y), \] where \(f,g>0\) solve \[ f(x)\int g(y)R^\varepsilon_{01}(\mathrm{d}y\mid x) =\frac{\mathrm{d}\mu_0}{\mathrm{d}R^\varepsilon_0}(x), \] \[ g(y)\int f(x)R^\varepsilon_{01}(\mathrm{d}x\mid y) =\frac{\mathrm{d}\mu_1}{\mathrm{d}R^\varepsilon_1}(y). \] The pair is unique up to reciprocal scaling.
Alternately enforcing these equations is iterative proportional fitting, the path-space version of Sinkhorn scaling.
Entropic interpolation
The time marginals \[ \mu_t=P_t^\star \] form the entropic interpolation between \(\mu_0\) and \(\mu_1\). For a reversible Markov reference, define \[ f_t(x)=\mathbb E_R[f(X_0)\mid X_t=x], \qquad g_t(x)=\mathbb E_R[g(X_1)\mid X_t=x]. \] Then \[ \frac{\mathrm{d}\mu_t}{\mathrm{d}R_t}=f_tg_t. \] For Brownian motion, \(f_t\) and \(g_t\) evolve by backward and forward heat semigroups.
Forward, backward, and current velocities
For Brownian diffusivity \(\varepsilon\), the bridge has forward and backward drifts \[ b_t^+=\varepsilon\nabla\log g_t, \qquad b_t^-=-\varepsilon\nabla\log f_t. \] Their average \[ v_t=\frac{b_t^++b_t^-}{2} =\frac{\varepsilon}{2}\nabla\log\frac{g_t}{f_t} \] is the current velocity and satisfies \[ \partial_t\mu_t+\nabla\cdot(v_t\mu_t)=0. \] Their half-difference \[ u_t=\frac{\varepsilon}{2}\nabla\log\rho_t \] is the osmotic velocity, where \(\rho_t\) is the density relative to Lebesgue measure.
Dynamic action with Fisher information
Entropic dynamic formulation. Up to endpoint-dependent entropy constants, the Brownian Schrödinger problem is equivalent to minimizing \[ \int_0^1\int \left\{ \frac{1}{2\varepsilon}|v_t|^2 +\frac{\varepsilon}{8}|\nabla\log\rho_t|^2 \right\} \rho_t\,\mathrm{d}x\,\mathrm{d}t \] subject to the continuity equation and endpoint constraints.
Proof idea. Girsanov's theorem expresses path-space entropy through the squared controlled forward drift \(b^+\). Decompose \[ b^+=v+\frac{\varepsilon}{2}\nabla\log\rho. \] Expanding the square gives kinetic energy, Fisher information, and a cross term. The continuity equation turns the cross term into the difference of endpoint entropies, which is fixed by the constraints.
Hamilton--Jacobi--Bellman structure
The logarithmic potentials \(\phi_t=\varepsilon\log g_t\) and \(\widehat\phi_t=\varepsilon\log f_t\) satisfy viscous Hamilton--Jacobi equations. For example, \[ \partial_t\phi_t+\frac12|\nabla\phi_t|^2 +\frac{\varepsilon}{2}\Delta\phi_t=0 \] with the appropriate time orientation. As \(\varepsilon\to0\), the viscosity term disappears and the equation approaches the Hamilton--Jacobi equation of Wasserstein geodesics.
Zero-noise limit
Convergence to optimal transport. Under finite second moments and standard tightness assumptions, as \(\varepsilon\downarrow0\), the rescaled Schrödinger endpoint problem \(\Gamma\)-converges to quadratic optimal transport. Optimal endpoint couplings converge to quadratic optimal plans, and entropic interpolations converge to displacement interpolations whenever the limiting optimizer is unique.
Proof mechanism. Small-noise Brownian paths satisfy a large-deviation principle with rate \(\tfrac12\int|\dot\gamma_t|^2\,\mathrm{d}t\). Entropy minimization therefore concentrates on least-action paths. The contraction principle reduces the endpoint rate to \(\tfrac12|x-y|^2\), producing the Kantorovich cost. Exponential tightness gives compactness, and uniqueness identifies the full limit.
Gaussian bridges
For Gaussian endpoint laws and a linear Gaussian reference process, the Schrödinger bridge remains Gaussian. Its mean solves a deterministic minimum-energy interpolation, while its covariance solves coupled matrix Riccati equations. In the zero-noise limit these equations reduce to the Gaussian Wasserstein geodesic from page 12.
What the bridge adds to transport
Classical transport chooses deterministic least-action motion whenever a map exists. A Schrödinger bridge retains stochastic fluctuations around the most likely motion, remains smooth for positive noise, and naturally incorporates a reference dynamics. This makes it a genuine dynamical theory rather than merely a computational regularization.