Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians

By Bernard Helffer, Francis Nier. 2005

354 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a30aba8309f658a09ab08f6d16918b9a654812aa

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Citations in the corpus

Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré (2007, background): “3) and can be extended to all dimensions using some isometric correspondence between Fokker-Planck and Schrödinger equations, namely |∇F |2(x) − ∆F (x) ≥ b > 0 for all |x| large enough (for a detailed discussion of the spectral theory of these operators see [17] in particular Proposition 3.”
Couplings and quantitative contraction rates for Langevin dynamics (2017, background): “Several analytic approaches to convergence to equilibrium for kinetic Fokker-Planck equations have been proposed during the last 15 years [10, 17, 27, 25, 26, 40, 42, 11, 7, 33, 19, 20, 3].”
Metastability: A Potential-Theoretic Approach (2016, background): “Very recently, there has been renewed progress on this issue in the case of the reversible diffusions that we will comment on below [18], [19], [20].”
Precise Arrhenius Law for p-forms: The Witten Laplacian and Morse–Barannikov Complex (2011, methodology): “For a general p, the global quasimodes of the form χU0(x)e − f(x)−f(U0) h used for the case p = 0 in [19–21,33–35] and which propagate the information through weakly resonant quantum wells, are missing.”
Kramers' law: Validity, derivations and generalisations (2011, background): “which follow from d = 0, then allow to infer more precise information on the spectrum of ∆ V,ε, and hence of the generator L of the diffusion [39].”
Short and long time behavior of the Fokker-Planck equation in a confining potential and applications (2005, methodology): “which proof is essentially contained in [15]-[ 14 ] about the parabolic (smoothing) properties of operator K in B 2 .”
Spectral analysis of semigroups and growth-fragmentation equations (2013, background): “evolution dynamics and then provides a spectral gap between the ﬁrst eigenvalue and the remainder of the spectrum. We refer the interested reader to [104] for a pedagogical introduction as well as to [56, 53, 88, 54, 55, 105, 29, 28] for some of the original articles. In the seminal work [87], C. Mo”
Hypocoercive Diffusion Operators (2006, methodology): “Hypoellipticity and hypocoercivity are often found together, and have been actually studied together, by refined hypoelliptic techniques [6], [7], [15], [16], [19], and sometimes by probabilistic methods [8], [22], [23].”
A Gentle Stochastic Thermostat for Molecular Dynamics (2009, neutral): “The main application of the Hörmander’s condition is Hörmander’s theorem [9, 12, 23].”
Quantization on Nilpotent Lie Groups (2016, methodology): “We refer the interested reader on this subject to the lecture notes by Helffer and Nier [HN05].”
Exponential Stability of Slowly Decaying Solutions to the Kinetic-Fokker-Planck Equation (2014, result): “Such a result has been proved in [20], see also [16, 15, 13, 8, 9].”
From resolvent bounds to semigroup bounds (2010, methodology): “Actually, using a form of semi-classical sub-ellipticity (closely related in spirit to the one established in [15] and further studied in [12]) it was also shown that this estimate holds in a larger parabolic neighborhood of iR away from the disc D(0, C̃h), and using this stronger result and a conto”
Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations (2014, methodology): “,v) ∈ R2n. (7.27) is then known as the kinetic Fokker-Planck equation with conﬁnement potential V. It has been extensively studied due its importance in mathematical physics. We refer for instance to [29, 37, 50, 56]. This equation is the Kolmogorov-Fokker-Planck equation associated to the stochasti”