Dynamic Schrödinger Bridge

• Dynamic Schrödinger Bridge is a stochastic process that optimally transports initial to final probability distributions by minimizing the path-space KL divergence relative to a reference diffusion.
• It employs controlled diffusion and reaction–diffusion PDEs, using forward-backward systems to compute optimal drifts and ensure a Markov process structure.
• Numerical methods like dynamic Sinkhorn recursion and closed-form kernels enable efficient computation and extend the framework to nonlinear drifts, jump processes, and multi-marginal constraints.

A dynamic Schrödinger bridge is a stochastic process that optimally transports a given initial probability distribution to a specified final distribution over a fixed time interval, by minimizing a path-space Kullback–Leibler (KL) divergence relative to a specified reference (often Brownian) process. This entropic regularization of the classic optimal transport problem yields a time-evolving interpolation—called the entropic interpolation—between the endpoint distributions and is realized by a controlled diffusion or, more generally, by a Markov process whose drift and, in extensions, jump rates, are computed by a variational principle. The theory provides a dynamical generalization of optimal mass transport with rich structure and exact solvability in a number of cases. Dynamic Schrödinger bridges underpin contemporary research in generative modeling, stochastic control, and nonequilibrium statistical mechanics.

1. Stochastic Control Formulation

In the classical dynamic Schrödinger Bridge, one seeks a law $P^*$ on path space $C([0,T],\mathbb{R}^n)$ that matches prescribed initial and final marginals $\rho_0$ and $\rho_1$ and minimizes the relative entropy to a reference process $Q$ (often a Wiener measure) over $[0,T]$: $$P^* = \arg\min_{P:\, P_{X_0}=\rho_0,\, P_{X_T}=\rho_1} \mathrm{KL}(P\|Q)$$ Equivalently, for the reference SDE $dX_t = \sqrt{2}\,dW_t$ with $X_0\sim\rho_0$, the bridge can be formulated as a stochastic optimal control problem: $$\inf_{u} E \left[ \int_0^{T} \frac{1}{2}\|u_t\|^2\,dt \right]$$ subject to $$dX_t = u_t\,dt + \sqrt{2}\,dW_t,\ X_0\sim\rho_0,\ X_T\sim\rho_1$$ (Teter et al., 2024).

With state-regularization, a quadratic cost term is introduced: $$\inf_u E \left[ \int_0^T \left( \frac{1}{2}\|u_t\|^2 + \frac{\kappa}{2}\|X_t\|^2 \right) dt \right]$$ which incentivizes trajectories to remain close to the origin or another nominal state (Teter et al., 2024).

The optimal solution is always Markov. For diffusion processes, the optimal drift is characterized by a factorization $\rho(x,t) = \hat\varphi(x,t)\varphi(x,t)$, where $(\hat\varphi,\varphi)$ solve a system of forward and backward PDEs ("Schrödinger system").

2. Reaction–Diffusion PDEs: The Bridge System

The optimal controlled process admits a drift $u^*(t,x)=\nabla_x\log\varphi(t,x)$, where $\varphi$ solves the backward PDE: $\partial_t\varphi = \Delta\varphi - q(x)\varphi$ with $q(x) = \frac{1}{2}\kappa\|x\|^2$ for quadratic state cost (Teter et al., 2024, Teter et al., 2024). The forward factor $\hat\varphi$ solves a time-reversed PDE: $$\partial_t\hat\varphi = \Delta\hat\varphi - q(x)\hat\varphi$$ For $q\equiv0$, these are heat equations; for quadratic $q$, they become non-self-adjoint reaction–diffusion equations, featuring mass creation/killing at a state-dependent rate (Teter et al., 2024, Teter et al., 2024).

The corresponding Fokker–Planck/Hamilton–Jacobi–Bellman system reads: $$\partial_t\psi + \frac{1}{2}|\nabla\psi|^2 + \Delta\psi = q(x), \quad \partial_t\rho + \nabla\cdot(\rho\nabla\psi) = \Delta\rho$$ where $\psi=\log\varphi$.

3. Closed-Form Markov Kernel and Exact Solvability

Exact solvability is achieved if the Green's function (transition kernel) for the uncontrolled PDE is in closed form. For $q(x) = \frac{1}{2}x^\top Q x$, with $Q \succcurlyeq 0$, this kernel can be constructed: $$\kappa_{++}(t_0,x;t,y) = \frac{(\det M)^{1/4}}{(2\pi)^{n/2}\sqrt{\det\sinh(2\Delta t\sqrt{D})}} \exp\left\{-\frac{1}{2}([Vx;Vy]^\top M [Vx;Vy])\right\}$$ where $Q/2=V^\top D V$ and $M$ is a block-diagonal matrix encoding the hyperbolic structure from the quadratic reaction term (Teter et al., 2024, Teter et al., 2024).

This explicit kernel allows for numerically stable and mesh-free forward/backward propagation of Schrödinger factors by simple integral transforms, bypassing PDE solvers (Teter et al., 2024).

4. Limiting Cases and Connections to Quantum Propagators

Dynamic Schrödinger bridge theory recovers and interpolates between a variety of models:

• In the limit $\kappa\to0$, the kernel recovers the classical heat kernel and thus the standard entropy-regularized Schrödinger bridge (Teter et al., 2024).
• For $Q/2=I$, the kernel reduces to the Mehler kernel (isotropic quantum harmonic oscillator propagator).
• The explicit kernel connects formally to quantum mechanics via Wick rotation $\Delta\to i\partial_\tau$. Thus, reaction–diffusion bridges with quadratic costs generalize both classical diffusions and exactly solvable quantum systems (Teter et al., 2024, Teter et al., 2024).

5. Algorithmic Realization: Dynamic Sinkhorn Recursion

The Schrödinger bridge system can be solved numerically for arbitrary endpoint distributions (of finite second moments) using dynamic Sinkhorn recursion:

1. Initialize $\hat\varphi_0^{(k)}$ as a positive guess.
2. Forward propagate: $$\hat\varphi^{(k)}(T, x) = \int \kappa(0, x; T, y)\hat\varphi_0^{(k)}(x) dx$$ Then set $$\varphi_T^{(k)} = \rho_1 / \hat\varphi^{(k)}(T, \cdot)$$.
3. Backward propagate: $$\varphi^{(k)}(0, x) = \int \kappa(0, x; T, y) \varphi_T^{(k)}(y) dy$$ Then update $$\hat\varphi_0^{(k+1)} = \rho_0 / \varphi^{(k)}(0, \cdot)$$.
4. Repeat until convergence.

This procedure is guaranteed to converge geometrically due to the Hilbert-metric contraction of the kernel map (Teter et al., 2024).

The resulting factors $\hat\varphi(t, x)$, $\varphi(t, x)$ can be recomputed for intermediate times via the same forward/backward integral transforms, yielding for any $t$, $\rho(t, x) = \hat\varphi(t, x) \varphi(t, x)$ and $u^*(t, x) = \nabla\log\varphi(t, x)$.

6. Extensions: Nonlinear Drift, Jumps, and Multi-Marginal Constraints

Dynamic Schrödinger bridges extend naturally to:

• Nonlinear drift backgrounds, where the uncontrolled process is not pure diffusion but has deterministic drift $b(x)$ or higher-order terms. In such cases, the Schrödinger system becomes forward-backward Kolmogorov PDEs potentially reducible to initial value problems under certain conditions (Caluya et al., 2019).
• Jump diffusions and regime-switching Markov processes, relevant for discontinuous and non-Gaussian stochastic systems. The bridge remains Markov in path space, and its SDE/PIDE can be characterized dynamically (Zlotchevski et al., 8 Nov 2025, Marco et al., 23 Feb 2026).
• Multi-marginal constraints, where the bridge is required to pass through a sequence of prescribed marginal distributions at intermediate times, admitting a construction via gluing of local bridges and ensuring global Markovianity (Park et al., 18 Oct 2025).
• Integral or ensemble path constraints (Maximum Caliber framework), enabling path-wise conditioning and inference of time-dependent potential landscapes (Miangolarra et al., 2024).

7. Applications and Impact in Modern Research

Dynamic Schrödinger bridges are foundational in modern computational optimal transport, generative diffusion modeling, and stochastic control. Mesh-free and scalable numerics (dynamic Sinkhorn, score-based methods, continuous normalizing flows) enable high-dimensional and sample-based modeling across physics (e.g., quantum analogies, molecular dynamics), astrophysics (inverse problems in star formation), and data science (distribution matching, image translation, time series synthesis) (Teter et al., 2024, Pavon et al., 2018, Zhu et al., 9 Jun 2025, Jing et al., 22 Mar 2025, Bortoli et al., 2024, Marco et al., 23 Feb 2026).

Adding a quadratic state cost transforms the classical bridge into a reaction–diffusion process with explicit kernel, suitable for controlled sampling, high-dimensional generative modeling, and biasing towards desired state regions or probability landscapes.

Closed-form solvability ensures robustness and tractability for both analysis and implementation—an essential property for both theoretical development and empirical validation in high-dimensional systems (Teter et al., 2024, Teter et al., 2024).