Tightness and closedness of laws of curves with bounded expected action #
Let 1 < p. By the Hölder estimate
d(γ s, γ t) ^ p ≤ |t - s| ^ (p - 1) * A_p(γ)
(AbsolutelyContinuousOnInterval.edist_rpow_le_mul_curveAction), the paths in C([a, b], X) that
are absolutely continuous with p-action A_p(γ) ≤ M < ∞ are uniformly equicontinuous, with the
common modulus δ ↦ δ ^ (1 - 1 / p) * M ^ (1 / p).
Consider a set S of measures on the path space whose members are concentrated on absolutely
continuous paths. If the expected actions are uniformly bounded, ∫ A_p dP ≤ C < ∞ for P ∈ S,
then Markov's inequality P {A_p ≥ M} ≤ C / M makes the action uniformly small in probability, and
the equicontinuity above turns this into the uniform modulus of continuity in probability of the
tightness criterion TauCeti.isTightMeasureSet_of_map_eval_of_continuity_modulus. If moreover the
time marginals of S at a dense set of times are tight and the pseudometric space X is
complete, then S is tight.
This is the compactness step for laws of random curves of finite action, for instance in the
superposition principle for absolutely continuous curves of probability measures. The bound on the
expected action also passes to weak limits: the probability measures concentrated on absolutely
continuous paths with ∫ A_p dP ≤ C < ∞ form a weakly closed set, because the action extended by
∞ off the absolutely continuous paths is lower semicontinuous on C([a, b], X).
The action bound supplies only the modulus of continuity. It does not control where the curves
are, so on a carrier that is not proper the tightness of the time marginals remains a separate
hypothesis: the Dirac masses at the curves γₙ(t) = t • eₙ in ℓ² have p-action at most 1 for
every p ≥ 0, yet they do not form a tight set
(TauCeti.exists_lintegral_curveAction_le_one_not_isTightMeasureSet). Both restrictions on the
curves are needed for the modulus: for p = 1 the action bounds only the length, which allows
arbitrarily steep curves, and without absolute continuity the action does not bound the
displacement at all, since a continuous curve can have metric derivative zero almost everywhere
and still move, like the Cantor function.
Main results #
TauCeti.measure_continuity_modulus_le_lintegral_curveAction: the explicit Hölder–Markov bound on oscillation probabilities, valid already for1 ≤ p.TauCeti.uniformEquicontinuous_of_absolutelyContinuousOnInterval_of_curveAction_le: absolutely continuous paths ofp-action at mostM < ∞are uniformly equicontinuous, for1 < p.TauCeti.exists_continuity_modulus_le_of_absolutelyContinuousOnInterval_of_curveAction: if the paths are almost surely absolutely continuous and the action is uniformly small in probability, the laws satisfy a uniform modulus of continuity in probability.exists_continuity_modulus_le_of_absolutelyContinuousOnInterval_of_lintegral_curveAction_le: the same under a uniform bound on the expected action.isTightMeasureSet_of_map_eval_of_absolutelyContinuousOnInterval_of_lintegral_curveAction_le: tight time marginals, almost surely absolutely continuous paths and a uniformly bounded expected action give tightness.TauCeti.isClosed_setOfPred_ae_absolutelyContinuousOnInterval_and_lintegral_le: the laws concentrated on absolutely continuous paths with expected action at mostC < ∞form a weakly closed set, for1 < p.TauCeti.exists_lintegral_curveAction_le_one_not_isTightMeasureSet: the bound on the expected action alone does not give tightness.
References #
- S. Lisini, Characterization of absolutely continuous curves in Wasserstein spaces, Calc. Var. Partial Differential Equations 28 (2007), 85--120.
- P. Billingsley, Convergence of Probability Measures, 2nd ed., Wiley 1999, Theorem 7.3.
Hölder–Markov estimate. Let P be a measure on C([a, b], X) concentrated on curves
absolutely continuous on [a, b], and 1 ≤ p. For η > 0 and every δ, the curves with two
times closer than δ at which their values are at least η apart have P-measure at most
δ ^ (p - 1) * (∫ A_p dP) / η ^ p, where A_p is the p-action over [a, b].
Paths of bounded action are uniformly equicontinuous. For 1 < p and M < ∞, the paths
γ ∈ C([a, b], X) that are absolutely continuous with p-action at most M form a uniformly
equicontinuous family: by the Hölder estimate, d(γ s, γ t) ^ p ≤ |t - s| ^ (p - 1) * M.
An action small in probability gives a modulus of continuity in probability. Let 1 < p
and let S be a set of measures on C([a, b], X), each concentrated on absolutely continuous
paths, whose p-action is uniformly small in probability: for every ε > 0 there is M < ∞
with P {γ | M ≤ A_p(γ)} ≤ ε for all P ∈ S. Then for all η, ε > 0 there is δ > 0 such that,
for every P ∈ S, the paths with two times closer than δ at which their values are at least
η apart have P-measure at most ε.
A bounded expected action gives a modulus of continuity in probability. Let 1 < p and
let S be a set of measures on C([a, b], X), each concentrated on absolutely continuous paths,
with uniformly bounded expected p-action ∫ A_p dP ≤ C < ∞. Then for all η, ε > 0 there is
δ > 0 such that, for every P ∈ S, paths with two times closer than δ at which their values
are at least η apart have P-measure at most ε.
Tightness of laws of curves with bounded expected action. Let X be a complete pseudometric
space, 1 < p, and D ⊆ [a, b] dense. A set S of measures on C([a, b], X) is tight if the time
marginals at the times of D are tight and its members are concentrated on absolutely continuous
paths with uniformly bounded expected p-action ∫ A_p dP ≤ C < ∞.
Closedness of the laws of bounded expected action. For 1 < p and C ≠ ∞, the
probability measures on C([a, b], X) that are concentrated on paths absolutely continuous on
[a, b] and have expected p-action at most C form a closed set for the weak topology.
A bounded expected action does not give tightness. The Dirac masses at the curves
γₙ(t) = t • eₙ, t ∈ [0, 1], along the standard basis vectors eₙ of ℓ² are concentrated on
absolutely continuous paths starting at zero, of p-action at most 1 for every p ≥ 0, but they
do not form a tight set of measures on C([0, 1], ℓ²). The hypothesis on the time marginals of
isTightMeasureSet_of_map_eval_of_absolutelyContinuousOnInterval_of_lintegral_curveAction_le
therefore cannot be dropped.