101 research outputs found
Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems
We study dynamical optimal transport metrics between density matrices
associated to symmetric Dirichlet forms on finite-dimensional -algebras.
Our setting covers arbitrary skew-derivations and it provides a unified
framework that simultaneously generalizes recently constructed transport
metrics for Markov chains, Lindblad equations, and the Fermi
Ornstein--Uhlenbeck semigroup. We develop a non-nommutative differential
calculus that allows us to obtain non-commutative Ricci curvature bounds,
logarithmic Sobolev inequalities, transport-entropy inequalities, and spectral
gap estimates
Stability for a GNS inequality and the Log-HLS inequality, with application to the critical mass Keller-Segel equation
Starting from the quantitative stability result of Bianchi and Egnell for the
2-Sobolev inequality, we deduce several different stability results for a
Gagliardo-Nirenberg-Sobolev inequality in the plane. Then, exploiting the
connection between this inequality and a fast diffusion equation, we get a
quantitative stability for the Log-HLS inequality. Finally, using all these
estimates, we prove a quantitative convergence result for the critical mass
Keller-Segel system
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