Absolutely continuous curves of laws of random curves #
Let η be a law on the path space C([a, b], X) of a separable metric space, concentrated on
absolutely continuous paths, and let η_t be its law at time t, the pushforward of η along
the evaluation γ ↦ γ t. For an exponent 1 ≤ p < ∞, this file shows that the curve t ↦ η_t
moves in the p-Wasserstein distance no faster than the Lᵖ(η) average of the speeds of the
paths:
W_p(η_s, η_u) ≤ ∫_s^u ‖|γ'|(r)‖_{Lᵖ(η)} dr.
Indeed γ ↦ (γ s, γ u) pushes η to a coupling of η_s and η_u, each path covers the distance
d(γ s, γ u) ≤ ∫_s^u |γ'|(r) dr, and Minkowski's integral inequality
(TauCeti.lintegral_Lp_lintegral_le) moves the Lᵖ(η) seminorm inside the time integral.
When moreover the expected p-action ∫ A_p(γ) dη is finite, the averaged speed
r ↦ ‖|γ'|(r)‖_{Lᵖ(η)} is integrable, so a curve μ in the finite-moment Wasserstein space
P_p(X) with μ t = η_t is absolutely continuous, and since its metric derivative is the smallest
integrable density bounding its distances,
|μ'|(t) ^ p ≤ ∫ |γ'|(t) ^ p dη(γ) for almost every t, hence A_p(μ) ≤ ∫ A_p(γ) dη(γ).
Every representation of an absolutely continuous curve of laws as the time marginals of a law on
paths therefore averages the speeds of the paths to at least the metric speed of the curve. For
1 < p < ∞, the superposition principle of Lisini constructs a representation attaining
equality, which the inequality proved here shows to be minimal among all representations. The
finite moments needed to place the laws η_t in P_p(X) propagate from a single time
(TauCeti.hasFiniteMoment_map_IccExtend).
Main results #
TauCeti.wassersteinEDist_map_IccExtend_le_lintegral_eLpNorm: the Wasserstein distance between two time marginals of a law on absolutely continuous paths is at most the time integral of theLᵖnorms of the metric derivatives of the paths.TauCeti.hasFiniteMoment_map_IccExtend: with finite expected action, the time marginals all have finitep-moment once one of them does.TauCeti.absolutelyContinuousOnInterval_of_coe_eq_map_IccExtend: a curve inP_p(X)traced by the time marginals of a law on absolutely continuous paths of finite expected action is absolutely continuous.TauCeti.ae_metricDerivative_rpow_le_lintegral_of_coe_eq_map_IccExtend: its metric derivative satisfies|μ'|(t) ^ p ≤ ∫ |γ'|(t) ^ p dηat almost every time.TauCeti.curveAction_le_lintegral_of_coe_eq_map_IccExtend: itsp-action is at most the expectedp-action of the paths.TauCeti.ae_metricDerivative_rpow_eq_lintegral_of_coe_eq_map_IccExtend: when the two actions are equal and finite,|μ'|(t) ^ p = ∫ |γ'|(t) ^ p dηat almost every time.
References #
- S. Lisini, Characterization of absolutely continuous curves in Wasserstein spaces, Calc. Var. Partial Differential Equations 28 (2007), 85--120.
- L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed., Birkhäuser 2008, §8.2, for the Euclidean case.
Time marginals of a law on paths move no faster than the averaged speed. Let η be a
σ-finite measure on C([a, b], X) concentrated on paths that are absolutely continuous on
[a, b], and let 1 ≤ p < ∞. For times s, u ∈ [a, b], the p-Wasserstein distance between the
laws of γ s and γ u is at most the integral over the times between s and u of the Lᵖ(η)
norm of the metric derivative of the paths.
Finite moments propagate along a law on paths of finite expected action. Let η be a
probability measure on C([a, b], X) concentrated on paths absolutely continuous on [a, b]
with finite expected p-action, for 1 ≤ p < ∞. If the law of γ t₀ has finite p-moment for
one time t₀ ∈ [a, b], then so does the law of γ t for every time t.
Time marginals of a law on paths form an absolutely continuous curve. Let 1 ≤ p < ∞ and
let η be a measure on C([a, b], X) concentrated on paths absolutely continuous on [a, b],
with finite expected p-action. A curve μ in the finite-moment Wasserstein space P_p(X) whose
value at each time t ∈ [a, b] is the law of γ t under η is absolutely continuous on
[a, b].
The metric speed of the time marginals is at most the averaged speed of the paths. Let
1 ≤ p < ∞ and let η be a measure on C([a, b], X) concentrated on paths absolutely continuous
on [a, b], with finite expected p-action. If a curve μ in P_p(X) takes at each time
t ∈ [a, b] the law of γ t under η as its value, then at almost every t ∈ [a, b] its metric
derivative satisfies |μ'|(t) ^ p ≤ ∫ |γ'|(t) ^ p dη(γ).
The action of the time marginals is at most the expected action of the paths. Let
1 ≤ p < ∞ and let η be a measure on C([a, b], X) concentrated on paths absolutely continuous
on [a, b]. If a curve μ in P_p(X) takes at each time t ∈ [a, b] the law of γ t under η
as its value, then the p-action of μ over [a, b] is at most the expected p-action
∫ A_p(γ) dη(γ) of the paths.
A representation of minimal expected action has the metric speed of the curve. Let
1 ≤ p < ∞ and let η be a measure on C([a, b], X) concentrated on paths absolutely continuous
on [a, b], whose time marginals trace a curve μ in P_p(X) on [a, b]. If the expected
p-action of the paths is finite and equal to the p-action of μ, then at almost every
t ∈ [a, b] the metric derivative of μ is the averaged speed of the paths:
|μ'|(t) ^ p = ∫ |γ'|(t) ^ p dη(γ).