A free probability analogue of the Wasserstein metric on the trace-state space

By Philippe Biane, D. Voiculescu. 2000

115 citations recorded by the source corpus.

Abstract. We define a free probability analogue of the Wasserstein metric, which extends the classical one. In dimension one, we prove that the square of the Wasserstein distance to the semi-circle distribution is majorized by a modified free entropy quantity.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/199506905c85c2723df9c362a4560bcdd97497d5

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In Applied Mathematics

In Computational Theory and Mathematics

In Statistics and Probability

Citations in the corpus

A free stochastic partial differential equation (2010, background): “,Xn(t) have Lipschitz conjugate variable (in the above sense for all t > 0, which is by no means a trivial assumption) then the argument of [4], variant of [29], gives the corresponding free Talagrand transportation cost inequality: dW ( (X1, .”
An Analog of the 2-Wasserstein Metric in Non-Commutative Probability Under Which the Fermionic Fokker–Planck Equation is Gradient Flow for the Entropy (2012, methodology): “Based on this idea, an analog of the Wasserstein metric has been defined by Biane and Voiculescu in the setting of free probability [4].”
Noncommutative martingale deviation and Poincaré type inequalities with applications (2012, background): “The Wasserstein distance has been extensively studied in the noncommutative setting; see[7,44,52].”
Transport Inequalities. A Survey (2010, background): “In [15], Biane and Voiculescu have obtained the following transport inequality for the semicircular distribution σ (61) T2(ν, σ) ≤ 2Σ̃(ν|σ), which holds for all ν ∈ P(R) with compact support (see [15, Theorem 2.”