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From the Chertkov wiki — draft for Misha

Schrödinger Bridge

The Schrödinger bridge is the minimum-relative-entropy coupling between the path law of a reference stochastic process and the set of path laws with prescribed initial and terminal marginals Schrödinger (1931); Léonard (2014). Originally posed by Erwin Schrödinger as a statistical-mechanics puzzle about the most likely evolution of a cloud of Brownian particles observed at two instants, the problem has become a unifying object in modern stochastic analysis, connecting the Monge–Kantorovich theory of optimal transport Chen, Georgiou & Pavon (2016), stochastic optimal control Kappen (2005), and score-based generative modelling De Bortoli et al. (2021).

Origin: Schrödinger 1931

In Über die Umkehrung der Naturgesetze, Schrödinger asked the following question Schrödinger (1931). Suppose a large number of independent particles diffuse according to the heat equation. At time \(t=0\) their empirical density is observed to be \(\rho_0\); at time \(t=1\), unexpectedly, the density is observed to be \(\rho_1\). What is the most likely trajectory of the cloud, conditionally on these two observations? Using a large-deviations argument avant la lettre, Schrödinger showed that the conditioned path measure is the one that minimises relative entropy (Kullback–Leibler divergence) with respect to the reference Wiener measure, subject to the two marginal constraints Léonard (2014). The resulting process is now called the Schrödinger bridge of the reference diffusion between \(\rho_0\) and \(\rho_1\).

Modern formulation

Let \(R\) denote the path law of a reference Markov process on \([0,1]\), and let \(\mathcal{P}(\rho_0,\rho_1)\) be the set of path laws with marginal \(\rho_0\) at \(t=0\) and \(\rho_1\) at \(t=1\). The Schrödinger bridge problem is

\[ \min_{P \in \mathcal{P}(\rho_0,\rho_1)} \mathrm{KL}(P \,\|\, R), \]

where \(\mathrm{KL}\) denotes relative entropy Léonard (2014). The minimiser \(P^\star\) is unique under mild regularity conditions, and its disintegration factorises as

\[ \frac{dP^\star}{dR}(\omega) = \varphi_0(\omega_0)\,\varphi_1(\omega_1), \]

for a pair of functions \((\varphi_0,\varphi_1)\) solving the Schrödinger system, a coupled pair of integral equations enforcing the two marginal constraints Léonard (2014). The existence of \((\varphi_0,\varphi_1)\) was established by Fortet, Beurling, and Jamison; a modern synthesis is given in Léonard (2014).

Doob's h-transform representation

When the reference process is a Brownian motion with generator \(\tfrac{1}{2}\Delta\), the Schrödinger bridge can be written as a Brownian motion with an additional drift given by the gradient of a logarithm. If \(\psi_t(x) = \mathbb{E}_R[\varphi_1(\omega_1) \,|\, \omega_t = x]\), then the bridge satisfies

\[ dX_t = \nabla_x \log \psi_t(X_t)\,dt + dW_t, \]

which is the Doob h-transform of the reference diffusion by the harmonic function \(\psi_t\) Chen, Georgiou & Pavon (2016). This representation, going back to Doob's 1957 work on conditioned diffusions, identifies the Schrödinger drift with the score of a forward propagation of the terminal potential, and is the bridge between the entropic formulation and the optimal-control formulation used in path-integral-diffusion and its quadratic specialisation, harmonic-path-integral-diffusion.

Connection to optimal transport

As the noise amplitude tends to zero, the Schrödinger bridge problem \(\Gamma\)-converges to the Monge–Kantorovich problem with quadratic cost, the entropic regularisation parameter playing the role of temperature Chen, Georgiou & Pavon (2016); Chen, Georgiou & Pavon (2021). In this sense the Schrödinger bridge is a stochastic regularisation of deterministic optimal transport, with the advantage that the regularised problem admits efficient iterative solvers. The most widely used is the Sinkhorn algorithm, which alternates between updating \(\varphi_0\) and \(\varphi_1\) to satisfy the two marginal constraints in turn Chen, Georgiou & Pavon (2021).

Stochastic optimal control viewpoint

The Schrödinger bridge admits a dual formulation as a stochastic optimal control problem: among all Itô diffusions

\[ dX_t = u_t(X_t)\,dt + dW_t, \qquad X_0 \sim \rho_0, \qquad \mathrm{Law}(X_1) = \rho_1, \]

the bridge is the controller that minimises the expected quadratic control cost Chen, Georgiou & Pavon (2016); Kappen (2005). This viewpoint underlies the path-integral treatments of the problem (see path-integral-diffusion) and, after a Hopf–Cole substitution, linearises the Hamilton–Jacobi–Bellman equation for the value function into a backward heat equation for \(\psi\) Kappen (2005).

Computational renaissance

While the Schrödinger system had been known since the 1940s, efficient algorithms for high-dimensional instances emerged only in the 2010s and 2020s. Three threads are now standard Chen, Georgiou & Pavon (2021):

  1. Sinkhorn-type iterations on the discretised Schrödinger system, benefitting from the matrix-scaling theory of entropic optimal transport.
  2. Stochastic optimal control solvers based on the Hopf–Cole linearisation, including Monte Carlo path-integral estimators Kappen (2005).
  3. Neural network parameterisations of the forward and backward drifts, trained by iterative proportional fitting on simulated trajectories De Bortoli et al. (2021).

A limiting case in which the iterative system collapses to a single Gaussian integral occurs when the state potential is quadratic; this is the harmonic-path-integral-diffusion regime, which is solvable in closed form.

Applications

Schrödinger bridges appear in several contemporary domains Chen, Georgiou & Pavon (2021):

Relation to path integral diffusion

When the reference process is a Brownian motion and the bridge is formulated as a stochastic optimal control problem with an additional state potential \(V_t\), the value function of the control problem satisfies a linear imaginary-time Schrödinger equation, and the optimal drift admits a Feynman–Kac path-integral representation Kappen (2005); Chernyak, Chertkov, Bierkens & Kappen (2014). This is the framework developed in path-integral-diffusion. When the terminal potential is harmonic, the path integral admits a closed-form Gaussian representation and the iterative Schrödinger system collapses to a single evaluation — see harmonic-path-integral-diffusion.

See also

References