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Dorian Le Peutrec - One of the best experts on this subject based on the ideXlab platform.
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Sharp spectral asymptotics for non-reversible metastable diffusion processes
2020Co-Authors: Dorian Le Peutrec, Laurent MichelAbstract:Let $U_h:\mathbb R^{d}\to \mathbb R^{d}$ be a smooth vector field and consider the associated overdamped Langevin equation $$dX_t=-U_h(X_t)\,dt+\sqrt{2h}\,dB_t$$ in the low temperature regime $h\rightarrow 0$. In this work, we study the spectrum of the associated diffusion $L=-h\Delta+U_h\cdot\nabla$ under the assumptions that $U_h=U_{0}+h\nu$, where the vector fields $U_{0}:\mathbb R^{d}\to \mathbb R^{d}$ and $\nu:\mathbb R^{d}\to \mathbb R^{d}$ are independent of $h\in(0,1]$, and that the dynamics admits $e^{-\frac Vh}$ as an invariant measure for some smooth Function $V:\mathbb{R}^d\rightarrow\mathbb{R}$. Assuming additionally that $V$ is a Morse Function admitting $n_0$ local minima, we prove that there exists $\epsilon>0$ such that in the limit $h\to 0$, $L$ admits exactly $n_0$ eigenvalues in the strip $\{0\leq \operatorname{Re}(z)< \epsilon\}$, which have moreover exponentially small moduli. Under a generic assumption on the potential barriers of the Morse Function $V$, we also prove that the asymptotic behaviors of these small eigenvalues are given by Eyring-Kramers type formulas.
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Bar codes of persistent cohomology and Arrhenius law for p-forms
2020Co-Authors: Dorian Le Peutrec, Francis Nier, C. ViterboAbstract:This article shows that counting or computing the small eigenvalues of the Witten Laplacian in the semi-classical limit can be done without assuming that the potential is a Morse Function as the authors did in [LNV]. In connection with persistent cohomology, we prove that the rescaled logarithms of these small eigenvalues are asymptotically determined by the lengths of the bar code of the Function f. In particular, this proves that these quantities are stable in the C 0 topology on the space of Functions. Additionally, our analysis provides a general method for computing the subexponential corrections in a large number of cases.
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Small eigenvalues of the Witten Laplacian acting on p-forms on a surface
Asymptotic Analysis, 2011Co-Authors: Dorian Le PeutrecAbstract:In this article, we are interested in the exponentially small eigenvalues of the self adjoint realization of the semiclassical Witten Laplacian $\Delta_{f,h}^{(p)}$, in the general framework of p-forms, on a connected compact Riemannian manifold without boundary. Our purpose is to notice that the knowledge of (the asymptotic formulas for) the smallest non zero eigenvalues of the self adjoint realization of $\Delta_{f,h}^{(0)}$ (acting on Functions), presented in [HeKlNi], essentially contains all the necessary information to the treatment of the case of oriented surfaces, for p-forms. The Function f is assumed to be a Morse Function on the manifold.
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Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian
Annales de la Faculté des Sciences de Toulouse. Mathématiques., 2010Co-Authors: Dorian Le PeutrecAbstract:This article follows the previous works \cite{HKN} by Helffer-Klein-Nier and \cite{HelNi1} by Helffer-Nier about the metastability in reversible diffusion processes via a Witten complex approach. Again, exponentially small eigenvalues of some self-adjoint realization of $\Delta_{f,h}^{(0)}=-h^{2}\Delta +\left|\nabla f(x)\right|^{2}-h\Delta f(x)\;,$ are considered as the small parameter $h>0$ goes to $0$. The Function $f$ is assumed to be a Morse Function on some bounded domain $\Omega$ with boundary $\partial\Omega$. Neumann type boundary conditions are considered. With these boundary conditions, some simplifications possible in the Dirichlet problem studied in \cite{HelNi1} are no more possible. A finer treatment of the three geometries involved in the boundary problem (boundary, metric, Morse Function) is carried out.
Burghelea Dan - One of the best experts on this subject based on the ideXlab platform.
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Virtually small spectral package of a Riemannian manifold
2020Co-Authors: Burghelea Dan, Lee YoonweonAbstract:For a Morse Function on a closed orientable Riemannian manifold one introduces the {\it virtually small spectral package} an analytic object consisting of a finite number of analytic quantities derived from the pair, {\it Riemannian metric, Morse Function\} which, in principle, can be calculated. One shows that they determine the {\it Torsion } of the underlying space, a parallel to the result that the dimensions of the spaces of harmonic forms calculate the {\it Euler-Poincar\'e characteristic} of the underlying space and extends the {\it Poincar\'e Duality} between harmonic forms and between Betti numbers for a closed oriented Riemannian manifold .Comment: 8 page
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Witten deformation and the spectral package of a Riemannian manifold
2020Co-Authors: Burghelea DanAbstract:The Witten deformation associated to a Morse Function on a closed Riemannian manifold, via Rellich-Kato theorem, relates analytically the spectral package of the Riemannian manifold (eigenvalues and eigenforms) to the Morse complex defined by the pair (Morse Function, Riemannian metric) coupled with the "multivariable harmonic oscillators" associated to the critical points of the Morse Function. We survey this relation and discuss some implications, including the finite subset of the spectral package referred to as the "virtually small spectral package" .Comment: 18 pages; few misprints in the previous version correcte
P. Gutiérrez - One of the best experts on this subject based on the ideXlab platform.
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Splitting Potential and the Poincaré-Melnikov Method for Whiskered Tori in Hamiltonian Systems
Journal of Nonlinear Science, 2000Co-Authors: A. Delshams, P. GutiérrezAbstract:We deal with a perturbation of a hyperbolic integrable Hamiltonian system with n+1 degrees of freedom. The integrable system is assumed to have n -dimensional hyperbolic invariant tori with coincident whiskers (separatrices). Following Eliasson, we use a geometric approach closely related to the Lagrangian properties of the whiskers, to show that the splitting distance between the perturbed stable and unstable whiskers is the gradient of a periodic scalar Function of n phases, which we call splitting potential. This geometric approach works for both the singular (or weakly hyperbolic) case and the regular (or strongly hyperbolic) case, and provides the existence of at least n+1 homoclinic intersections between the perturbed whiskers. In the regular case, we also obtain a first-order approximation for the splitting potential, that we call Melnikov potential. Its gradient, the (vector) Melnikov Function, provides a first-order approximation for the splitting distance. Then the nondegenerate critical points of the Melnikov potential give rise to transverse homoclinic intersections between the whiskers. Generically, when the Melnikov potential is a Morse Function, there exist at least 2 ^ n critical points. The first-order approximation relies on the n -dimensional Poincaré-Melnikov method, to which an important part of the paper is devoted. We develop the method in a general setting, giving the Melnikov potential and the Melnikov Function in terms of absolutely convergent integrals, which take into account the phase drift along the separatrix and the first-order deformation of the perturbed hyperbolic tori. We provide formulas useful in several cases, and carry out explicit computations that show that the Melnikov potential is a Morse Function, in different kinds of examples.
A. Delshams - One of the best experts on this subject based on the ideXlab platform.
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Splitting Potential and the Poincaré-Melnikov Method for Whiskered Tori in Hamiltonian Systems
Journal of Nonlinear Science, 2000Co-Authors: A. Delshams, P. GutiérrezAbstract:We deal with a perturbation of a hyperbolic integrable Hamiltonian system with n+1 degrees of freedom. The integrable system is assumed to have n -dimensional hyperbolic invariant tori with coincident whiskers (separatrices). Following Eliasson, we use a geometric approach closely related to the Lagrangian properties of the whiskers, to show that the splitting distance between the perturbed stable and unstable whiskers is the gradient of a periodic scalar Function of n phases, which we call splitting potential. This geometric approach works for both the singular (or weakly hyperbolic) case and the regular (or strongly hyperbolic) case, and provides the existence of at least n+1 homoclinic intersections between the perturbed whiskers. In the regular case, we also obtain a first-order approximation for the splitting potential, that we call Melnikov potential. Its gradient, the (vector) Melnikov Function, provides a first-order approximation for the splitting distance. Then the nondegenerate critical points of the Melnikov potential give rise to transverse homoclinic intersections between the whiskers. Generically, when the Melnikov potential is a Morse Function, there exist at least 2 ^ n critical points. The first-order approximation relies on the n -dimensional Poincaré-Melnikov method, to which an important part of the paper is devoted. We develop the method in a general setting, giving the Melnikov potential and the Melnikov Function in terms of absolutely convergent integrals, which take into account the phase drift along the separatrix and the first-order deformation of the perturbed hyperbolic tori. We provide formulas useful in several cases, and carry out explicit computations that show that the Melnikov potential is a Morse Function, in different kinds of examples.
Valerio Pascucci - One of the best experts on this subject based on the ideXlab platform.
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Loops in Reeb Graphs of 2-Manifolds
Discrete and Computational Geometry, 2004Co-Authors: Kree Cole-mclaughlin, Herbert Edelsbrunner, John Harer, Vijay Natarajan, Valerio PascucciAbstract:Given a Morse Function f over a 2-manifold with or without boundary, the Reeb graph is obtained by contracting the connected components of the level sets to points. We prove tight upper and lower bounds on the number of loops in the Reeb graph that depend on the genus, the number of boundary components, and whether or not the 2-manifold is orientable. We also give an algorithm that constructs the Reeb graph in time O(n log n), where n is the number of edges in the triangulation used to represent the 2-manifold and the Morse Function.
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Symposium on Computational Geometry - Loops in reeb graphs of 2-manifolds
Proceedings of the nineteenth conference on Computational geometry - SCG '03, 2003Co-Authors: Kree Cole-mclaughlin, Herbert Edelsbrunner, John Harer, Vijay Natarajan, Valerio PascucciAbstract:Given a Morse Function f over a 2-manifold with or without boundary, the Reeb graph is obtained by contracting the connected components of the level sets to points. We prove tight upper and lower bounds on the number of loops in the Reeb graph that depend on the genus, the number of boundary components, and whether or not the 2-manifold is orientable. We also give an algorithm that constructs the Reeb graph in time O(nlogn), where n is the number of edges in the triangulation used to represent the 2-manifold and the Morse Function.
