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Absolutely continuous curves of order two in Wasserstein space

Definition

Fix $T > 0$ . A curve $μ : [0, T] \to P_{2} (R^{d})$ is absolutely continuous of order two, written $μ \in A C^{2} ([0, T]; P_{2})$ , if it is continuous in $W_{2}$ and there is a nonnegative Lebesgue-measurable $g$ with $\int_{0}^{T} g (t)^{2} d t < \infty$ such that $W_{2} (μ_{s}, μ_{t}) \leq \int_{s}^{t} g (r) d r$ for every $0 \leq s \leq t \leq T$ . At an interior time $t$ where the limit exists, the metric derivative is $∣ {\dot{μ}}_{t} ∣ = \lim_{h \to 0} W_{2} (μ_{t + h}, μ_{t}) / ∣ h ∣$ ; no endpoint derivative is required.

Definition references

def wasserstein two space and couplings

Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Theorem 1.1.2 and Chapter 8

Filippo Santambrogio, Optimal Transport for Applied Mathematicians, Chapter 5, §§5.1–5.3

Depends on

The Euclidean quadratic Wasserstein space

Used by

Dependency tree · two levels

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Sources

Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Theorem 1.1.2 and Chapter 8 (standard reference, not scraped)
Filippo Santambrogio, Optimal Transport for Applied Mathematicians, Chapter 5, §§5.1–5.3 (standard reference, not scraped)