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A survey of the Schr\"odinger problem and some of its connections with optimal transport

Christian Léonard

arXiv:1308.0215v1math.PRmath.FAmath.OC

TL;DR

The paper presents the Schrödinger problem as an entropy minimization over path measures with prescribed marginals, and surveys its connections with optimal transport. It relates dynamic and static solutions through endpoint couplings and reference bridges, and also proves new results.

  • Problem

    The Schrödinger problem asks for the most likely path behavior under a rare event in which the observed final profile differs from its expected profile.

    Source: §6 · p. 28

  • Method

    The paper studies entropy minimization relative to a reference path measure with prescribed endpoint marginals and relates it to a static endpoint-coupling problem.

    Source: §1 · p. 2

  • Results

    The dynamic and static Schrödinger problems have equal minimum values, and the dynamic solution shares the reference measure’s bridges while its endpoint coupling solves the static problem.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The article provides a user’s guide and literature survey on the Schrödinger problem’s connections with optimal transport, alongside new results.

    Source: §1 · p. 1

    Without assumption (vii), the optimizer’s density may lack a product form and have a more complex structure.

    Source: Summing up · p. 13

Abstract

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This article is aimed at presenting the Schr\"odinger problem and some of its connections with optimal transport. We hope that it can be used as a basic user's guide to Schr\"odinger problem. We also give a survey of the related literature. In addition, some new results are proved.

1. Introduction

The article presents the dynamic and static Schrödinger problems and their connections to optimal transport. It relates path-space solutions to endpoint couplings and surveys related literature while proving new results.

1. Introduction

  • The article aims to be a basic user’s guide to the Schrödinger problem, surveys related literature, and proves some new results.Source: §1 · p. 1
  • The setup uses reversible Brownian motion as a reference path measure and prescribes the initial and final marginals of the path law.The state space may be a complete connected Riemannian manifold without boundary; the reference measure can be unbounded on noncompact manifolds.Source: §1 · pp. 1–2
  • The dynamic Schrödinger optimizer shares the reference measure’s bridges, with its endpoint coupling solving the static problem; their minimum values are equal.Strict convexity also makes the dynamic solution unique when it exists.Source: §1 · p. 2
  • Solutions of the dynamic and static Monge–Kantorovich problems correspond when the minimizing geodesic between each endpoint pair is unique, and their values are equal.The dynamic solution is formed by placing each endpoint pair on its constant-speed geodesic.Source: §1 · p. 3

Displacement interpolations in P2(X

The section relates displacement interpolation to geodesics in Wasserstein space and describes entropic interpolation as a regular approximation to it. It also summarizes extensions and the article’s new results.

Displacement interpolations in P2(X

  • Displacement interpolation is a minimizing constant-speed geodesic in P2(X), while the entropic interpolation has analogous properties.The optimal displacement pair also satisfies a forward-backward coupled system.Source: Displacement interpolations in P2(X · pp. 3–4

The Monge-Kantorovich problem is a limit of Schr¨odinger problems.

  • The dynamic and static Monge–Kantorovich problems are Γ-limits of re-normalized Schrödinger problems; under compactness, minimizers converge, and the marginal flow converges to displacement interpolation when the transport solution is unique.Entropic interpolations are positive and regular on (0, 1)×X, unlike the limiting displacement interpolation; exact recovery of the final marginal is stated for regular densities.Source: The Monge-Kantorovich problem is a… · pp. 4–5

Extension of the framework.

  • The framework extends to Markov reference measures satisfying a large deviation principle, yielding transport problems for alternate cost functions; discrete random-walk references are also considered.Source: Extension of the framework · p. 6

New results.

  • The article’s new results include sufficient conditions for product-shaped optimizer densities, a Markov property result, and Benamou–Brenier-type formulas.The product form is also treated when the reference measure has infinite mass; for Markov references, the factors may vanish under additional requirements.Source: New results · p. 6

Outline of the paper.

  • The article proceeds from rigorous statements and optimizer structure to Markov transforms, Brownian and discrete examples, transport limits, statistical-physics motivation, and literature discussion.Source: Outline of the paper · p. 6

2. Schr¨odinger’s problem

This section sets up path measures and relative entropy, then defines dynamic and static Schrödinger problems from a reference path measure. Their minimizers are linked through endpoint marginals and conditional bridges.

2. Schr¨odinger’s problem

  • The section introduces the general framework before stating Schrödinger’s problem and discussing its main properties.Source: §2 · p. 6

Path measures.

  • A path measure contains more information than its time-marginal flow because it also encodes correlations between positions at different times.The endpoint marginal is the joint law of the path’s initial and final positions.Source: Path measures · p. 7

Why unbounded path measures.

  • Reversible Brownian motion on Rn uses Lebesgue measure as its initial distribution, so its path measure is unbounded; such reversible references can simplify computations.Source: Why unbounded path measures · pp. 7–8

Relative entropy.

  • Relative entropy is defined for a probability measure with respect to a σ-finite positive measure, with infinite entropy when absolute continuity fails.Source: Relative entropy · p. 8

Statement of Schr¨odinger’s problem.

  • The dynamic problem minimizes path-space entropy subject to prescribed endpoint marginals, while the static problem minimizes entropy of the endpoint coupling relative to the projected reference measure.The unique solutions correspond: the dynamic minimizer’s endpoint marginal solves the static problem, and the dynamic minimizer shares the reference bridges.Source: Statement of Schr¨odinger’s problem · pp. 8–9

Existence results.

The static and dynamic Schrödinger problems have a unique solution exactly when their common infimum is finite, equivalently when the prescribed marginals admit a coupling of finite relative entropy. The paper also gives sufficient integrability conditions for existence.

Existence results.

  • A unique solution to both problems exists exactly when inf (Sdyn) = inf (S) < ∞, equivalently when a coupling of the prescribed marginals has finite relative entropy with respect to R01.The equivalent coupling condition is H(πo|R01) < ∞ for some πo with marginals µ0 and µ1.Source: Existence results · p. 9
  • When R0 = R1 = m, finite marginal entropies H(µ0|m) and H(µ1|m) are necessary for either problem to have a solution.Source: Existence results · p. 9
  • Under the stated sufficient conditions, including finite marginal entropies and integrability of A+B under µ1, both problems admit a unique solution.The conditions also require nonnegative measurable functions A and B satisfying the preceding assumptions.Source: Existence results · pp. 9–10
  • If (iv) holds and ∫e^{α(A+B)}dm < ∞ for some α > 0, then condition (iii) is satisfied.Condition (iv) includes finite marginal entropies; the stated condition (v) requires m to be bounded.Source: Existence results · p. 10

The dual problem.

The Schrödinger problem has a dual formulation with equal primal and dual values, though dual attainment is not guaranteed in regular function spaces. The minimizer's density and its potentials have measurability and support qualifications that obstruct a naive product formula.

The dual problem.

  • For marginals in PB(X), the dual problem satisfies inf (S) = sup (D), but attainment in CB(X) is not guaranteed in general.When the dual is attained, the text relates equality to the entropy of the minimizer.Source: The dual problem · p. 10

Obstacle (i).

  • The density identity dbπ/dR01 = exp(ϕ⊕ψ) holds only bπ-almost everywhere, and its exceptional support set need not be a product set.The text describes the minimizer's density shape as tempting but not valid in the general case.Source: The dual problem · p. 11, Obstacle (i) · p. 11

Obstacle (ii).

  • Measurability of ϕ⊕ψ with respect to R01 does not imply that ϕ and ψ are separately R0- and R1-measurable, so the conditional expectations in (2.8) cannot generally be used.Source: Obstacle (ii) · p. 11, The dual problem · p. 11

Avoiding obstacle (i).

The paper addresses the density-form obstacle by perturbing admissible constraints into the intrinsic core, and addresses the measurability obstacle with conditions on the reference measure.

Avoiding obstacle (i).

  • For constraints in the intrinsic core, the minimizer has density exp(ϕ⊕ψ) with respect to R01, with ϕ⊕ψ jointly R01-measurable.Admissible constraints can be perturbed arbitrarily little in total variation into the intrinsic core, yielding solutions with this density form.Source: Avoiding obstacle (i) · pp. 11–12

Overcoming obstacle (ii).

  • If R01 ≪ R0⊗R1, the potentials ϕ and ψ can be chosen respectively µ0- and µ1-measurable.The text also gives a separate sufficient condition: every R01-measurable function is R0⊗R1-measurable.Source: Overcoming obstacle (ii) · p. 12

Summing up.

Under the stated reference-measure and marginal conditions, the Schrödinger system characterizes the unique dynamic solution. The paper notes that a positivity condition affects whether its density has a product form and that the Markov case admits a more efficient theorem.

Summing up.

  • Theorem 2.8 assumes common endpoint reference measure m, bounds on R01, an integrability condition, a relation between R01 and m⊗2, and internal constraints on the prescribed marginals.The constraints are internal, for example, when m is a probability measure and both marginals dominate εm for some ε > 0.Source: Summing up · p. 12
  • Without assumption (vii), the density dbπ/dR01 may lack product form and have a complex structure.The text states that (vii) ensures dbπ/dR01 > 0.Source: Summing up · p. 13
  • Under Theorem 2.8's hypotheses, the dynamic problem has a unique solution represented using positive measurable functions f0 and g1 that solve the Schrödinger system.The system matches conditional expectations under R to the marginal densities with respect to m.Source: Summing up · p. 13

The special case where R is Markov.

Markov property.

When the reference path measure is Markov, the dynamic Schrödinger problem’s solution is also Markov. The proof uses conditional independence of past and future given the present state.

Markov property.

  • The reference measure’s time marginals must be σ-finite so conditional versions such as R(· | Xt) can be defined.The stated assumption applies at any time marginal.Source: Markov property · p. 13
  • Under the reference measure, past and future are conditionally independent given the state at time t.The section identifies this time-symmetric conditional-independence property with the usual forward Markov property.Source: Markov property · pp. 13–14
  • A solution of the dynamic Schrödinger problem inherits the Markov property from a Markov reference measure.The proof rules out a non-Markov solution by constructing a competing measure with lower relative entropy.Source: Markov property · pp. 13–14

Reversibility.

For a reversible Markov reference measure, Theorem 2.12 gives conditions under which the dynamic Schrödinger solution has an endpoint-product form. The theorem also allows the endpoint functions to vanish under its assumptions.

Reversibility.

  • Reversibility means invariance under time reversal; it implies that every time marginal equals the reversing measure.The measure is therefore stationary in time.Source: Reversibility · p. 14
  • The theorem assumes a Markov reference measure and additional requirements, including positive mass on a measurable state subset at an intermediate time.For the stated properties, a reversible Markov measure with a regenerative set is sufficient.Source: Reversibility · pp. 14–15
  • Theorem 2.12 characterizes when the dynamic solution is a Markov measure weighted by nonnegative functions of its initial and terminal states.Conversely, a transform of that form whose functions solve the Schrödinger system is the unique dynamic solution.Source: Reversibility · p. 15
  • Unlike Theorem 2.8, Theorem 2.12 does not require internal constraints, and its endpoint functions may vanish on sets of positive reference marginal measure.The functions are nonnegative rather than necessarily positive.Source: Reversibility · p. 15

3. (f, g)-transform of a Markov measure

The article introduces endpoint-weighted transforms of Markov measures, characterizes when they solve the Schrödinger problems, and derives their time marginals and dynamics. Under reversibility, their potentials and generators have forward and backward descriptions.

3. (f, g)-transform of a Markov measure

  • We call f0(X0)g1(X1)R, for a Markov measure R, an (f, g)-transform; it is a time-symmetric version of Doob’s usual h-transform.Source: §3 · p. 16

(f, g)-transform of a reversible Markov measure.

  • Under Theorem 3.3’s stated integrability condition, the transform and its endpoint law uniquely solve the dynamic and static Schrödinger problems for constraints satisfying the Schrödinger system.The reference measure is assumed Markov and reversible.Source: (f, g)-transform of a reversible… · p. 16
  • A dynamic Schrödinger solution need not be an (f, g)-transform when its support is not a rectangle.The cited discussion points to [FG97, §2] and [L´eo01b, §5] for this case.Source: (f, g)-transform of a reversible… · p. 17
  • For an (f, g)-transform, the time marginal is given by the product of the conditional expectations of the endpoint weights given the state at time t.These conditional expectations are positive and finite P-almost everywhere, but not necessarily R-almost everywhere.Source: (f, g)-transform of a reversible… · p. 17

The dynamics of the (f, g)-transform.

  • The transform’s forward and backward generators are expressed using the evolving functions ft and gt and the reference process’s generator.The stated generator formulas apply under hypotheses on the reference measure and a class of regular functions.Source: The dynamics of the (f, g)-transform · p. 18
  • With sufficient regularity, ft and gt solve forward and backward parabolic equations, while their logarithms give Schrödinger potentials satisfying corresponding Hamilton-Jacobi equations.The Schrödinger potentials are introduced by analogy with Kantorovich potentials.Source: The dynamics of the (f, g)-transform · p. 19

4. Standard examples

The examples apply the general results to reversible Brownian motion and reversible random walks on graphs. In the Brownian setting, the Schrödinger solution also solves a minimal-action problem with a uniquely specified velocity field.

4. Standard examples

  • The section applies the general results to reversible Brownian motion and reversible random walks on graphs.It derives the corresponding dynamics and minimal-action formulations.Source: §4 · p. 19

Reversible Brownian motion.

  • For reversible Brownian motion, the transform’s forward and backward drifts are the gradients of its Schrödinger potentials.The heat semigroup’s smoothing permits these gradients to be defined at the stated times.Source: Reversible Brownian motion · p. 20

Back to Schr¨odinger problem.

  • Under Theorem 3.3’s assumptions, the Brownian transform solves the dynamic Schrödinger problem, with its forward and backward generators given by the potential-based formulas.The solution is an (f, g)-transform of the reference measure.Source: Back to Schr¨odinger problem · p. 20

Minimal action.

  • For Brownian reference motion and finite-entropy endpoint marginals in P2(R^n), the unique minimal-action solution uses the dynamic Schrödinger path and velocity ∇ψ.ψ is the unique classical solution of the stated Hamilton-Jacobi-Bellman equation.Source: Minimal action · p. 21

Reversible random walk on a graph.

  • On a reversible graph, the (f, g)-transform has forward and backward jump generators associated with its time-dependent jump measures.The evolving functions are obtained from the graph heat semigroup.Source: Reversible random walk on a graph · p. 23
  • For the graph minimal-action problem, the unique solution uses the marginal path and a flux determined by ψ, which is the unique classical solution of the stated Hamilton-Jacobi-Bellman equation.The theorem states the result for endpoint marginals with finite static Schrödinger infimum.Source: Reversible random walk on a graph · pp. 23–24

5. Slowing down

Slowing the reference process yields Schrödinger problems whose limits recover optimal transport. Under the stated Γ-convergence results, minimizers converge to optimal transport solutions, with a unique selected limit in the graph setting.

5. Slowing down

  • Slowing the reference process provides a route from Schrödinger problems to optimal transport, with rigorous treatment stated for random walks on graphs.The section presents the general idea heuristically and refers to prior work for the diffusion case.Source: §5 · pp. 24, 26
  • The convergence argument uses a large deviation principle for the slowed process and pointwise convergence of dual objectives to identify the Kantorovich dual problem.The resulting variational convergence is described as Γ-convergence.Source: §5 · pp. 25–26
  • For reversible Brownian motion on R^n, the limiting cost is |y − x|^2/2, and the Schrödinger problem converges to quadratic Monge-Kantorovich transport.The bridges converge to constant-speed geodesic paths; when the dynamic transport solution is unique, the path measures converge to it.Source: §5 · p. 26

Γ-convergence.

  • The slowed Schrödinger problems Γ-converge to dynamic and static Monge-Kantorovich problems, and their optimal values converge.For the random walk on a graph, the limiting cost is graph distance.Source: §5 · p. 26, Γ-convergence · p. 27
  • In the graph setting, the Schrödinger minimizers converge to unique limits, selecting one solution among the infinitely many dynamic Monge-Kantorovich solutions.The selected dynamic limit is characterized as the unique solution of an auxiliary entropy minimization problem.Source: Γ-convergence · pp. 27–28

6. The statistical physics motivation of Schr¨odinger’s problem

The article presents Schrödinger’s question as finding the most likely evolution of independent particles conditioned on an atypical final profile. It relates that conditional law to entropy minimization and connects entropic interpolations with curvature-related ideas.

6. The statistical physics motivation of Schr¨odinger’s problem

  • Schrödinger’s question asks for the most likely dynamics when many independent particles end in a profile far from the expected one, a rare event.The system is represented by its empirical path measure.Source: §6 · p. 28

Solving the problem without getting into details.

  • Sanov’s theorem links empirical-measure probabilities to relative entropy, and conditioning on the endpoint profile leads to the entropy-minimizing path law.The article identifies the rate function as relative entropy and describes the resulting minimization as a Boltzmann-Gibbs conditioning principle.Source: Solving the problem without getting… · pp. 29–31
  • Conditioned on an atypical final empirical profile, the particle system’s limiting law is the unique path measure minimizing relative entropy subject to the prescribed endpoint marginals.The conditional convergence is stated to be exponentially fast, yielding an almost-sure version through the Borel-Cantelli lemma.Source: Solving the problem without getting… · pp. 30–31

The lazy gas experiment.

  • For displacement interpolations, entropy convexity is connected to Ricci lower bounds, while the article frames entropic interpolation as a more realistic variant of the lazy-gas experiment.The cited connection to Ricci lower bounds is attributed to Sturm and von Renesse.Source: The lazy gas experiment · p. 31
  • The Schrödinger problem suggests replacing displacement interpolations with entropic interpolations in the lazy-gas thought experiment.The article reports a rigorous entropic-interpolation formula and notes possible connections of related formulas to graph curvature.Source: The lazy gas experiment · pp. 31–32

7. A short history of Schr¨odinger’s problem and related literature

The section traces Schrödinger’s entropy-minimization problem from its probabilistic origins through later work on reciprocal processes, stochastic mechanics, and optimal transport, ending with technical results on entropy for unbounded measures.

7. A short history of Schr¨odinger’s problem and related literature

  • Schrödinger’s entropy minimization gives a classical analogue of quantum wave-function propagation and Born’s formula, motivated by the most-probable evolution of an observed deviation from equilibrium.The article notes that Schrödinger did not explicitly formulate the dynamic and static problems; Föllmer later did so.Source: §7 · p. 33, F¨ollmer’s contribution · p. 34

Reciprocal processes, 1932.

  • Bernstein introduced reciprocal processes as time-reversal-symmetric correlations less restrictive than the Markov property; Jamison and Föllmer later connected Schrödinger solutions to Markov processes and h-transforms.Kolmogorov also cited Schrödinger’s 1931 paper as motivation for his work on Markov processes and time-reversal.Source: Reciprocal processes, 1932 · p. 34, Time-reversal, 1936 · p. 34

Schr¨odinger system, 1940.

  • Fortet, Beurling, and Jamison progressively developed criteria and solutions for Schrödinger’s system, with Jamison obtaining its complete solution.Source: Schr¨odinger system, 1940 · p. 34

Stochastic deformations of mechanics.

  • Euclidean quantum mechanics develops rigorous stochastic analogues of quantum-mechanical ideas, using reciprocal processes and, in important cases, Markov processes.The article notes that Feynman’s integral is not mathematically rigorous, while stochastic calculus applied in its place suggests results about diffusion processes.Source: Stochastic deformations of mechanics · p. 35

An interesting problem.

  • The article proposes investigating a Noether-type theorem for dynamical optimal transport, asking which quantities remain conserved under potentials with symmetries.For the kinetic action, the displacement interpolation has constant speed; the corresponding conservation question for a potential-dependent action remains open.Source: An interesting problem · p. 35

Stochastic optimal control.

  • Mikami’s stochastic-control formulation connects the Brownian Schrödinger problem with dynamical optimal transport, with convergence to the latter established for quadratic costs but unclear otherwise.Mikami and Thieullen also proved a Kantorovich-type dual equality for the control formulation.Source: Stochastic optimal control · p. 36

Penalized Monge-Kantorovich problem.

Annexe A. Relative entropy with respect to an unbounded measure

  • The appendix defines relative entropy for unbounded reference measures and establishes its convexity and strict convexity under stated integrability and topological conditions.Its variational representation yields lower semicontinuity in the specified weak topology.Source: Annexe A. Relative entropy with… · pp. 37–38

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