Analysis and Geometry of Markov Diffusion Operators

By D. Bakry, Ivan Gentil, Michel Ledoux. 2013

1,715 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/044f8706a3b078cfe9c5d1448e42293b33ccf26a

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Citations in the corpus

The Li-Yau inequality and applications under a curvature-dimension condition (2014, methodology): “(ii) For any positive function f ∈ A and t > 0 , then X < 1 + π 2 ρ 2 t 2 and (iii) For any positive function f ∈ A and t > 0 , then X 1 + π 2 ρ 2 t 2 and As is [1], inequality (2.3) may be called a local logarithmic Sobolev inequality and (2.2) a local reverse logarithmic Sobolev inequality.”
SVGD as a kernelized Wasserstein gradient flow of the chi-squared divergence (2020, background): “More precisely, the starting point for our approach is the following integration-by-parts formula, which is a crucial component of the theory of Markov semigroups [BGL14]:”
Logarithmic bounds for isoperimetry and slices of convex sets (2023, background): “(7) The name of this inequality stems from an analogy to investigations by Lichnerowicz in Riemannian geometry, see [2, 35].”
Poincaré and Brunn-Minkowski inequalities on the boundary of weighted Riemannian manifolds (2017, background): “, also [5, 6, 20, 21, 38, 51, 57, 64, 77, 78] and the references therein), especially since Perelman’s work on the Poincaré Conjecture [62], and the extension of the Curvature-Dimension condition to the metric-measure space setting by Lott-Sturm-Villani [46, 72].”
Stability Results for Logarithmic Sobolev and Gagliardo–Nirenberg Inequalities (2014, methodology): “Much more can also be found in Bakry et al. [2014]. Not all Gagliardo-Nirenberg-Sobolev inequalities are covered by our remarks and we shall refer to Carlen et al.”
A formula for the time derivative of the entropic cost and applications (2019, background): “For its proof as well as for all the regularizing properties of Pt that will be used throughout the paper, we address once more the reader to [5].”
Nonsmooth differential geometry - An approach tailored for spaces with Ricci curvature bounded from below (2014, methodology): “In the rest of this introduction we will recall some regularity results due to Savaré [52] which have been obtained by proper generalization and adaptation of the arguments proposed by Bakry [17] (see also the recent survey [18]).”
Gradient estimates for heat kernels and harmonic functions (2017, background): “(BE) |∇Ht f (x)|(2) ≤ Ht(|∇ f |(2))(x), which, on manifolds, is known to be equivalent to non-negativity of Ricci curvature; see [10] and also [8, 9, 111].”
Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis (2017, methodology): “Our analysis draws heavily on the theory of optimal transportation (Villani, 2003) and on the analysis of Markov di ﬀ usion operators (Bakry et al., 2014) (the necessary background on Markov semigroups and functional inequalities is given in Appendix A).”
Semi Log-Concave Markov Diffusions (2013, background): “For further informations and more, see the forthcoming book [5].”
Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions (2015, background): “97-100 in [2]), with eigenfunctions uk orthonormal in L2(μ).”
Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces (2015, neutral): “[34, 35, 36, 12, 9, 14]).”