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Some properties of path measures

Christian Léonard

arXiv:1308.0217v1math.PR

TL;DR

The paper studies how disintegration, relative entropy, and conditional integration should be handled for path measures, particularly when reference measures are unbounded. It develops σ-finite and weighted conditional-expectation tools, establishes entropy regularity and additivity results, and identifies conditionability as a necessary boundary for pathwise conditioning.

Abstract

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We call any measure on a path space, a path measure. Some notions about path measures which appear naturally when solving the Schrödinger problem are presented and worked out in detail.

Introduction

The paper develops tools for path measures motivated by the Schrödinger problem, especially when reference measures are unbounded. It treats disintegration, relative entropy, and positive integration while identifying conditions needed for conditional expectations.

Aim of this article.

  • The paper addresses disintegration of unbounded measures, relative entropy relative to unbounded measures, and positive integration for Markov measures.These topics are motivated by applications to path measures and the Schrödinger problem.Source: Aim of this article · p. 1

Motivation.

  • The Schrödinger problem minimizes relative entropy over path probability measures with prescribed initial and final marginals.The reference path measure may be unbounded, as for reversible Brownian motion with Lebesgue reversing measure.Source: Motivation · p. 2
  • The (f, g)-transform uses nonnegative endpoint functions with ER(f(X0)g(X1)) = 1 and yields the prescribed endpoint marginals through conditional expectations.Its Radon–Nikodym product form is presented as a time-symmetric extension of the Doob h-transform.Source: Motivation · pp. 2–3
  • For an unbounded path measure, conditional expectations require restrictions such as conditionability, because ordinary integrability and disintegration arguments may fail.The paper introduces conditionable path measures and develops disintegration tools for this setting.Source: Aim of this article · p. 1, Motivation · p. 3, §1 · p. 4
  • Integrability of f(X0)g(X1) does not ensure separate integrability of f(X0) and g(X1), motivating positive-integration results for defining the needed conditional expectations.The positivity of f and g is used to establish a more general setting.Source: Motivation · p. 3

1. Disintegration of an unbounded path measure

  • For σ-finite R, Q admits a density with respect to R exactly when Q is absolutely continuous with respect to R and σ-finite.The proof constructs a countable Q-finite partition by intersecting an R-finite partition with density level sets.Source: §1 · pp. 3–4

An extension of the conditional expectation.

The paper extends conditional expectation from bounded to σ-finite measure settings by multiplying measures with a positive measurable weight. This preserves equivalence while enabling regular conditional probability kernels.

An extension of the conditional expectation.

  • The extended conditional expectation agrees with the original one on L1(R) whenever γRφ is bounded.The extension is defined on L1(γ(φ)R) and restricts consistently to L1(R).Source: An extension of the conditional… · pp. 4–5
  • A positive function γ can turn σ-finite push-forward measures into bounded equivalent measures, allowing bounded-measure conditional-expectation results to be applied.For two measures R and Q, γ can be chosen so γ(φ)R and γ(φ)Q are respectively equivalent to R and Q.Source: An extension of the conditional… · pp. 4–5
  • If Q ≺ R and both push-forward measures Rφ and Qφ are σ-finite, conditional expectations admit regular conditional probability kernels.The kernels map each x to probability measures conditioned on φ = x.Source: An extension of the conditional… · p. 5
  • The construction handles unbounded R and Q by applying bounded-measure results after weighting them with γ(φ).The proof uses the equivalence and boundedness supplied by the preceding lemma.Source: An extension of the conditional… · pp. 4, 6

Hilbertian conditional expectation.

For unbounded reference measures, the paper extends conditional expectation to L2 by interpreting it as an orthogonal projection rather than relying on L1 inclusion.

Hilbertian conditional expectation.

  • When R is unbounded, L2(R) need not be contained in L1(R), so conditional expectation is extended from L1(R) ∩ L2(R) to L2(R).The extension interprets the defining relation as an operation restricted to L2(R).Source: Hilbertian conditional expectation · p. 6
  • The resulting operator is the orthogonal projection onto the Hilbert subspace L2(RA).For a measurable mapping φ, the construction specializes to conditioning on the σ-field generated by φ.Source: Hilbertian conditional expectation · p. 6

Conditional expectation of path measures.

The paper defines conditionable and Markov path measures so conditional expectations remain meaningful for possibly unbounded path measures. A conditionable measure has σ-finite time marginals at every time.

Conditional expectation of path measures.

  • A path measure is conditionable when every time marginal Qt is σ-finite, which makes conditional expectations given any time subset well-defined.A single σ-finite marginal within a time subset suffices to establish σ-finiteness of the corresponding joint marginal.Source: Conditional expectation of path measures · p. 7
  • Even when Q(Ω) = ∞, conditioning on XT produces a probability measure and bounded conditional probabilities for measurable events.For bounded measurable b, EQ(b | XT) is also bounded and measurable.Source: Conditional expectation of path measures · p. 7
  • A path measure can fail to be conditionable when one time marginal is not σ-finite, even if the joint path measure is σ-finite.In the example, Q0 is Lebesgue and σ-finite, while Q1 is not σ-finite.Source: Conditional expectation of path measures · p. 7
  • A Markov measure is defined as a conditionable path measure whose future conditional law given the past depends only on the current state.The definition is formulated using conditional expectations with respect to path segments and Xt.Source: Conditional expectation of path measures · p. 7

2. Relative entropy with respect to an unbounded measure

The paper extends relative entropy to possibly unbounded reference measures and establishes its convexity, lower semicontinuity, measurability, and additive decomposition. These results support a path-space application to the Schrödinger problem, where endpoint optimization determines the unique solution.

Basic properties

  • For a probability reference measure, relative entropy is nonnegative, vanishes only when the two measures coincide, and is strictly convex.These properties provide uniqueness for entropy minimization.Source: §2 · pp. 7–8

Definition and extension

  • Relative entropy for an unbounded reference measure is restricted to probabilities satisfying a suitable weighted integrability condition, making the definition meaningful and independent of the admissible weight.The reference measure is σ-finite, and compatible weights yield the same entropy expression.Source: §2 · p. 8

Regularity

  • The entropy functional has an effective domain contained in the weighted probability space and is lower semicontinuous under σ(M_W(Ω), C_W(Ω)).The weighted function spaces control integrability and growth against an unbounded reference measure.Source: §2 · pp. 8–9
  • As a function of both signed measures, relative entropy is jointly convex, jointly lower semicontinuous, and therefore jointly Borel measurable.The proof represents the functional as a supremum of affine continuous functions.Source: §2 · p. 10

Additivity and measurability

  • For Polish spaces, disintegration provides measurable conditional measures, enabling the additive property of relative entropy along a measurable map.The Polish assumption addresses the measurability of the conditional kernels and the entropy integrand.Source: §2 · p. 10

Application to the Schrödinger problem

  • In the Schrödinger problem, entropy decomposes into endpoint and bridge contributions, so strict convexity yields a unique path-space optimizer whose endpoint law uniquely solves the marginal problem.The bridge is Q(·|X_0=x,X_1=y), while the endpoint law is Q_01=(X_0,X_1)#Q.Source: §2 · p. 11

Integration of nonnegative functions.

The section extends conditional-expectation identities for nonnegative functions under Markov measures without unnecessary separate integrability assumptions, including factorization across time intervals.

  • The lemma supports (f, g)-transforms of Markov processes without requiring separate integrability of the nonnegative factors.It is motivated by settings where the product is integrable but the individual factors need not be.Source: Integration of nonnegative functions · pp. 11–12

Lemma 3.1

  • For α and β supported before and after t, positivity or vanishing of ER(αβ | Xt) determines corresponding positivity or vanishing of the conditional factors.Outside an R-negligible set, a zero product implies at least one zero factor, while a positive product implies both factors are positive.Source: Integration of nonnegative functions · p. 11
  • The extended identity (3.1) holds R-a.e. with an indicator excluding infinite conditional factors, while the positivity identity holds P-a.e. rather than generally R-a.e.The distinction matters because conditional factors may be infinite even when αβ is integrable.Source: Integration of nonnegative functions · p. 12
  • ER(αβ | Xt) equals ER(α | Xt)ER(β | Xt) in the extended nonnegative sense, obtained by truncation and monotone convergence.The identity is first established through bounded truncations and increasing limits.Source: Integration of nonnegative functions · p. 12

Lemma 3.2

  • The same factorization extends to α, ζ, and β over intervals [0,s], [s,t], and [t,1], with ζ retained between the endpoint conditional expectations.For P-a.e. paths, the endpoint conditional expectations and the middle conditional product are positive and factorized.Source: Integration of nonnegative functions · p. 13

Appendix A. Conditional expectation with respect to an unbounded measure

The appendix introduces conditional expectation and disintegration for unbounded positive measures, motivated by natural unbounded path measures such as reversible Brownian motion.

  • Conditional expectation theory is needed for natural unbounded path measures, including reversible Brownian motion on Rn.The appendix treats this from the measure-theoretic perspective of disintegrating unbounded positive measures.Source: Appendix A · p. 13
  • The Radon-Nikodym theorem is presented as the keystone for conditioning, with σ-finiteness made explicit in the recalled proof.The setup defines absolute continuity, concentration, and mutual singularity for positive measures.Source: The role of σ-finiteness in… · p. 13

The role of σ-finiteness in Radon-Nikodym theorem.

The section shows that σ-finiteness enables Radon-Nikodym representations for positive measures, whereas without it absolute continuity need not yield a density.

  • For bounded measures, Lebesgue decomposition separates P uniquely into an R-absolutely-continuous part and an R-singular part.The absolutely continuous component has an L1(R) density.Source: The role of σ-finiteness in… · p. 13
  • The proof uses a Hilbert-space representation whose continuity depends crucially on boundedness, although σ-finite extensions are obtained by partitioning.The σ-finite proof applies the bounded theorem on a countable partition and recollects the pieces.Source: The role of σ-finiteness in… · pp. 13–15
  • σ-finite measures can lose σ-finiteness under push-forward, as illustrated by projections and push-forwards of Lebesgue measure.The examples show infinite mass assigned to non-negligible sets after push-forward.Source: The role of σ-finiteness in… · p. 14
  • σ-finite positive measures P ≪ R admit a unique measurable density θ with P(A)=∫A θ dR, and P is bounded exactly when θ∈L1(R).The density need only be locally integrable when the measures are unbounded.Source: The role of σ-finiteness in… · pp. 13–15
  • Without σ-finiteness, absolute continuity alone may fail to produce a Radon-Nikodym density.Counting measure on [0,1] dominates Lebesgue measure, but no measurable density represents Lebesgue measure with respect to it.Source: The role of σ-finiteness in… · p. 15

Conditional expectation with respect to a positive measure.

The section defines conditional expectation for positive measures under σ-finiteness, establishes its basic L1 properties, and identifies conditions for regular conditional probability kernels.

  • Conditional expectation ER(f | A) is the unique R-a.e. function that is integrable, A-measurable, and satisfies the defining integral identity when RA is σ-finite.The σ-finiteness of the restricted measure is essential to the definition.Source: Conditional expectation with respect… · p. 15
  • The operator ER(· | A): L1(R)→L1(RA) is a contraction and fixes every A-measurable integrable function.Its operator norm is therefore controlled by the usual L1 norms.Source: Conditional expectation with respect… · p. 15
  • For a bounded measure R, ER(· | A) maps nonnegative f to nonnegative values and maps 1 to 1, both RA-a.e.; moreover, limn→∞ER(fn | A) = 0 RA-a.e.Source: Conditional expectation with respect… · p. 16
  • A regular conditional probability kernel cannot be assumed solely from pointwise conditional-expectation identities because the relevant negligible sets vary with the tested functions and sequences.The section notes that a general kernel requires restrictions on the measurable structure.Source: Conditional expectation with respect… · p. 16
  • The σ-field generated by a measurable map φ is φ−1(B), and every measurable function with respect to it can be expressed as a measurable function of φ.This supports notation for conditional expectation given φ.Source: Conditional expectation with respect… · p. 17
  • For bounded R and measurable φ into a Polish space, a regular conditional probability kernel exists and yields a disintegration formula.The kernel represents ER(· | φ) through probability measures indexed by φ's value.Source: Conditional expectation with respect… · p. 17

Radon-Nikodym derivative and conditioning.

The section introduces a positive measure P absolutely continuous with respect to R and develops related conditioning identities under boundedness assumptions.

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