Journal ArticleDOI
The variational formulation of the Fokker-Planck equation
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The Fokker-Planck equation as mentioned in this paper describes the evolution of the probability density for a stochastic process associated with an Ito Stochastic Differential Equation.Abstract:
The Fokker--Planck equation, or forward Kolmogorov equation, describes the evolution of the probability density for a stochastic process associated with an Ito stochastic differential equation. It ...read more
Citations
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Book
Optimal Transport: Old and New
TL;DR: In this paper, the authors provide a detailed description of the basic properties of optimal transport, including cyclical monotonicity and Kantorovich duality, and three examples of coupling techniques.
Book
Gradient Flows: In Metric Spaces and in the Space of Probability Measures
TL;DR: In this article, Gradient flows and curves of Maximal slopes of the Wasserstein distance along geodesics are used to measure the optimal transportation problem in the space of probability measures.
Journal ArticleDOI
The geometry of dissipative evolution equations: the porous medium equation
TL;DR: In this paper, the authors show that the porous medium equation has a gradient flow structure which is both physically and mathematically natural, and they use the intuition and the calculus of Riemannian geometry to quantify this asymptotic behavior.
Journal ArticleDOI
On the geometry of metric measure spaces. II
TL;DR: In this article, a curvature-dimension condition CD(K, N) for metric measure spaces is introduced, which is more restrictive than the curvature bound for Riemannian manifolds.
Journal ArticleDOI
Ricci curvature for metric-measure spaces via optimal transport
John Lott,Cédric Villani +1 more
TL;DR: In this paper, a notion of a length space X having nonnegative N-Ricci curvature, for N 2 [1;1], or having 1-RICci curvatures bounded below by K, for K2 R, was given.
References
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Journal ArticleDOI
Brownian motion in a field of force and the diffusion model of chemical reactions
TL;DR: In this article, a particle which is caught in a potential hole and which, through the shuttling action of Brownian motion, can escape over a potential barrier yields a suitable model for elucidating the applicability of the transition state method for calculating the rate of chemical reactions.
Book
Probability and Measure
TL;DR: In this paper, the convergence of distributions is considered in the context of conditional probability, i.e., random variables and expected values, and the probability of a given distribution converging to a certain value.
MonographDOI
Linear and Quasi-linear Equations of Parabolic Type
TL;DR: In this article, the authors introduce a system of linear and quasi-linear equations with principal part in divergence (PCI) in the form of systems of linear, quasilinear and general systems.