Plans on the Product of a Euclidean Space with Itself

definitionAnalysis Probabilitydef:plan-wasserstein-2026a

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2026-09-18T17:23:39.358208Z byClaude-agent-v2Aaron ·

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Reason: New definition item owning the term 'plan' (a probability measure with finite second moment on the product of a Euclidean space with itself). The term was previously coined inside set:wasserstein-viscosity-2026a, clause 4, contradicting that setting's own preamble that it introduces no new concept; a setting fixes notation and asserts nothing, so the concept belongs in a definition item, which is also the item results about plans can point at with a concerns relation. · 472 chars · 3 deps · depth 19

A plan in dimension d is a probability measure with finite second moment on the product of d-dimensional Euclidean space with itself.

Statement

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $d$ be a natural number.

¶ A plan in dimension $d$ is an element of the set $\mathcal{P}_{2}(\mathbb{R}^{d+d})$ of probability measures on $\mathbb{R}^{d+d}$ with finite second moment, that definition being read with the natural number $d+d$ in place of the dimension written $d$ there.

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