The Experts below are selected from a list of 327 Experts worldwide ranked by ideXlab platform
Leonardo Tolomeo - One of the best experts on this subject based on the ideXlab platform.
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on the Unique Ergodicity for a class of 2 dimensional stochastic wave equations
arXiv: Analysis of PDEs, 2021Co-Authors: Justin Forlano, Leonardo TolomeoAbstract:We study the global-in-time dynamics for a stochastic semilinear wave equation with cubic defocusing nonlinearity and additive noise, posed on the $2$-dimensional torus. The noise is taken to be slightly more regular than space-time white noise. In this setting, we show existence and Uniqueness of an invariant measure for the Markov semigroup generated by the flow over an appropriately chosen Banach space. This extends a result of the second author to a situation where the invariant measure is not explicitly known.
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Unique Ergodicity for a class of stochastic hyperbolic equations with additive space time white noise
Communications in Mathematical Physics, 2020Co-Authors: Leonardo TolomeoAbstract:In this paper, we consider a certain class of second order nonlinear PDEs with damping and space-time white noise forcing, posed on the d-dimensional torus. This class includes the wave equation for $$d=1$$ and the beam equation for $$d\le 3$$. We show that the Gibbs measure is the Unique invariant measure for this system. Since the flow does not satisfy the strong Feller property, we introduce a new technique for showing Unique Ergodicity. This approach may be also useful in situations in which finite-time blowup is possible.
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Unique Ergodicity for stochastic hyperbolic equations with additive space time white noise
arXiv: Analysis of PDEs, 2018Co-Authors: Leonardo TolomeoAbstract:In this paper, we consider a certain class of second order nonlinear PDEs with damping and space-time white noise forcing, posed on the $d$-dimensional torus. This class includes the wave equation for $d=1$ and the beam equation for $d\le 3$. We show that the Gibbs measure of the equation without forcing and damping is the Unique invariant measure for the flow of this system. Since the flow does not satisfy the Strong Feller property, we introduce a new technique for showing Unique Ergodicity. This approach may be also useful in situations in which finite-time blowup is possible.
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Unique Ergodicity for a class of stochastic hyperbolic equations with additive space time white noise
arXiv: Analysis of PDEs, 2018Co-Authors: Leonardo TolomeoAbstract:In this paper, we consider a certain class of second order nonlinear PDEs with damping and space-time white noise forcing, posed on the $d$-dimensional torus. This class includes the wave equation for $d=1$ and the beam equation for $d\le 3$. We show that the Gibbs measure of the equation without forcing and damping is the Unique invariant measure for the flow of this system. Since the flow does not satisfy the Strong Feller property, we introduce a new technique for showing Unique Ergodicity. This approach may be also useful in situations in which finite-time blowup is possible.
Peter Sarnak - One of the best experts on this subject based on the ideXlab platform.
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recent progress on the quantum Unique Ergodicity conjecture
Bulletin of the American Mathematical Society, 2011Co-Authors: Peter SarnakAbstract:We report on some recent striking advances on the quantum Unique Ergodicity or “QUE” conjecture, concerning the distribution of large frequency eigenfunctions of the Laplacian on a negatively curved manifold. The account falls naturally into two categories. The first concerns the general conjecture where the tools are more or less limited to micro-local analysis and the dynamics of the geodesic flow. The second is concerned with arithmetic such manifolds where tools from number theory and ergodic theory of flows on homogeneous spaces can be combined with the general methods to resolve the basic conjecture as well as its holomorphic analogue. The recent account by Zelditch [Zel4] covers some of the same material and his and our discussion complement each other nicely, as he goes into more detail with the first category and we with the second. This note is not meant to be a survey of these topics and the discussion is not chronological . Our aim is to expose these recent developments after introducing the necessary backround which places them in their proper context. Recent Progress on QUE 2 §1. The General QUE Conjecture In Figure 1 the domains ΩE,ΩS and ΩB are an ellipse, a stadium and a “Barnett” billiard table respectively. Superimposed on these are the densities of a consecutive sequence of high frequency eigenfunctions (“states”, “modes”) of the Laplacian. That is they are solutions to 4φj + λjφj = 0 in Ω . φ ∣∣ ∂Ω = 0 (Dirichlet boundary conditions), ∫ Ω |φj| dxdy = 1 . (0) Here 4 = divgrad = ∂2 ∂x2 + ∂ 2 ∂y2 , λ1 < λ2 ≤ λ3 . . . are the eigenvalues and the eigenfunctions are normalized to have unit L-norm. The sequences are of 12 consecutive modes around the 5600 eigenvalue. They are ordered from left to right and then down and the grayscale represents the probability density |φ|, with zero white and larger values darker.
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estimates for rankin selberg l functions and quantum Unique Ergodicity
Journal of Functional Analysis, 2001Co-Authors: Peter SarnakAbstract:Abstract Subconvex bounds in the weight aspect for Rankin–Selberg L -functions associated to two cusp forms are established. These bounds are applied to prove the equidistribution of mass conjecture for CM forms.
Tolomeo Leonardo - One of the best experts on this subject based on the ideXlab platform.
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On the Unique Ergodicity for a class of 2 dimensional stochastic wave equations
2021Co-Authors: Forlano Justin, Tolomeo LeonardoAbstract:We study the global-in-time dynamics for a stochastic semilinear wave equation with cubic defocusing nonlinearity and additive noise, posed on the $2$-dimensional torus. The noise is taken to be slightly more regular than space-time white noise. In this setting, we show existence and Uniqueness of an invariant measure for the Markov semigroup generated by the flow over an appropriately chosen Banach space. This extends a result of the second author to a situation where the invariant measure is not explicitly known.Comment: 42 page
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Unique Ergodicity for a class of stochastic hyperbolic equations with additive space-time white noise
'Springer Science and Business Media LLC', 2020Co-Authors: Tolomeo LeonardoAbstract:In this paper, we consider a certain class of second order nonlinear PDEs with damping and space-time white noise forcing, posed on the $d$-dimensional torus. This class includes the wave equation for $d=1$ and the beam equation for $d\le 3$. We show that the Gibbs measure of the equation without forcing and damping is the Unique invariant measure for the flow of this system. Since the flow does not satisfy the Strong Feller property, we introduce a new technique for showing Unique Ergodicity. This approach may be also useful in situations in which finite-time blowup is possible.Comment: The title has being changed from "Unique Ergodicity for stochastic hyperbolic equations with additive space-time white noise" to "Unique Ergodicity for a class of stochastic hyperbolic equations with additive space-time white noise", many typo corrections, some minor corrections in the proof
Nessim Sibony - One of the best experts on this subject based on the ideXlab platform.
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Unique Ergodicity for foliations on compact kahler surfaces
arXiv: Complex Variables, 2018Co-Authors: Tiencuong Dinh, Vietanh Nguyen, Nessim SibonyAbstract:Let \Fc be a holomorphic foliation by Riemann surfaces on a compact K\"ahler surface X. Assume it is generic in the sense that all the singularities are hyperbolic and that the foliation admits no directed positive closed (1,1)-current. Then there exists a Unique (up to a multiplicative constant) positive \ddc-closed (1,1)-current directed by \Fc. This is a very strong ergodic property of \Fc. Our proof uses an extension of the theory of densities to a class of non-\ddc-closed currents. A complete description of the cone of directed positive \ddc-closed (1,1)-currents is also given when \Fc admits directed positive closed currents.
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Unique Ergodicity for foliations in mathbb p 2 p 2 with an invariant curve
Inventiones Mathematicae, 2018Co-Authors: Tiencuong Dinh, Nessim SibonyAbstract:Consider a foliation in the projective plane admitting a projective line as the Unique invariant algebraic curve. Assume that the foliation is generic in the sense that its singular points are hyperbolic. We show that there is a Unique positive $${dd^c}$$ -closed (1, 1)-current of mass 1 which is directed by the foliation and this is the current of integration on the invariant line. A Unique Ergodicity theorem for the distribution of leaves follows: for any leaf L, appropriate averages of L converge to the current of integration on the invariant line. The result uses an extension of our theory of densities for currents. Foliations on compact Kahler surface with one or more invariant curves are also considered.
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Unique Ergodicity for foliations in p 2 with an invariant curve
arXiv: Complex Variables, 2015Co-Authors: Tiencuong Dinh, Nessim SibonyAbstract:Consider a foliation in the projective plane admitting a projective line as the Unique invariant algebraic curve. Assume that the foliation is generic in the sense that its singular points are hyperbolic. We show that there is a Unique positive ddc-closed (1,1)-current of mass 1 which is directed by the foliation and this is the current of integration on the invariant line. A Unique Ergodicity theorem for the distribution of leaves follows: for any leaf L, appropriate averages of L converge to the current of integration on the invariant line. This property is surprising because for most of such foliations the leaves (except the invariant line) are dense in the projective plane. So one could expect that they spend a significant amount of hyperbolic time in every open set and that there should be a fat ddc-closed non-closed current with support equal to the projective plane. The proof uses an extension of our theory of densities for currents. Foliations on compact Kaehler surfaces with one or several invariant curves are also considered.
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Unique Ergodicity of harmonic currents on singular foliations of mathbb p 2
Geometric and Functional Analysis, 2010Co-Authors: John Erik Fornaess, Nessim SibonyAbstract:Let \({\mathcal{F}}\) be a holomorphic foliation of \({\mathbb{P}^2}\) by Riemann surfaces. Assume all the singular points of \({\mathcal{F}}\) are hyperbolic. If \({\mathcal{F}}\) has no algebraic leaf, then there is a Unique positive harmonic (1, 1) current T of mass one, directed by \({\mathcal{F}}\). This implies strong ergodic properties for the foliation \({\mathcal{F}}\). We also study the harmonic flow associated to the current T.
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Unique Ergodicity of harmonic currents on singular foliations of p2
arXiv: Dynamical Systems, 2006Co-Authors: John Erik Fornaess, Nessim SibonyAbstract:Let F be a holomorphic foliation of P^2 by Riemann surfaces. Assume all the singular points of F are hyperbolic. If F has no algebraic leaf, then there is a Unique positive harmonic $(1,1)$ current $T$ of mass one, directed by F. This implies strong ergodic properties for the foliation. We also study the harmonic flow associated to the current $T.$
Lior Silberman - One of the best experts on this subject based on the ideXlab platform.
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entropy bounds and quantum Unique Ergodicity for hecke eigenfunctions on division algebras
arXiv: Number Theory, 2016Co-Authors: Lior Silberman, Akshay VenkateshAbstract:We prove the arithmetic quantum Unique Ergodicity (AQUE) conjecture for non-degenerate sequences of Hecke eigenfunctions on quotients $\Gamma \backslash G/K$, where $G\simeq\mathrm{PGL}_{d}(\mathbb{R})$, $K$ is a maximal compact subgroup of $G$ and $\Gamma
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quantum Unique Ergodicity on locally symmetric spaces the degenerate lift
Canadian Mathematical Bulletin, 2015Co-Authors: Lior SilbermanAbstract:Given a measure μ∞ on a locally symmetric space Y = Γ\G/K, obtained as a weak-* limit of probability measures associated to eigenfunctions of the ring of invariant differential operators, we construct a measure μ∞ on the homogeneous space X = Γ\G which lifts μ∞ and which is invariant by a connected subgroup A1 ⊂ A of positive dimension, where G = NAK is an Iwasawa decomposition. If the functions are, in addition, eigenfunctions of the Hecke operators, then μ∞ is also the limit of measures associated to Hecke eigenfunctions on X. This generalizes results of the author with A. Venkatesh in the case where the spectral parameters stay away from the walls of the Weyl chamber.
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quantum Unique Ergodicity on locally symmetric spaces the degenerate lift
arXiv: Representation Theory, 2011Co-Authors: Lior SilbermanAbstract:Given a measure $\bar\mu$ on a locally symmetric space $Y=\Gamma\backslash G/K$, obtained as a weak-{*} limit of probability measures associated to eigenfunctions of the ring of invariant differential operators, we construct a measure $\mu$ on the homogeneous space $X=\Gamma\backslash G$ which lifts $\bar\mu$ and which is invariant by a connected subgroup $A_{1}\subset A$ of positive dimension, where $G=NAK$ is an Iwasawa decomposition. If the functions are, in addition, eigenfunctions of the Hecke operators, then $\mu$ is also the limit of measures associated to Hecke eigenfunctions on $X$. This generalizes previous results of the author and A.\ Venkatesh to the case of "degenerate" limiting spectral parameters.
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on quantum Unique Ergodicity for locally symmetric spaces
GAFA Geometric And Functional Analysis, 2007Co-Authors: Lior Silberman, Akshay VenkateshAbstract:We construct an equivariant microlocal lift for locally symmetric spaces. In other words, we demonstrate how to lift, in a semi-canonical fashion, limits of eigenfunction measures on locally symmetric spaces to Cartan-invariant measures on an appropriate bundle. The construction uses elementary features of the representation theory of semisimple real Lie groups, and can be considered a generalization of Zelditch’s results from the upper half-plane to all locally symmetric spaces of noncompact type. This will be applied in a sequel to settle a version of the quantum Unique Ergodicity problem on certain locally symmetric spaces.
