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Sean P Meyn - One of the best experts on this subject based on the ideXlab platform.
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optimal kullback leibler aggregation via Spectral Theory of markov chains
IEEE Transactions on Automatic Control, 2011Co-Authors: Kun Deng, Prashant G Mehta, Sean P MeynAbstract:This paper is concerned with model reduction for complex Markov chain models. The Kullback-Leibler divergence rate is employed as a metric to measure the difference between the Markov model and its approximation. For a certain relaxation of the bi-partition model reduction problem, the solution is shown to be characterized by an associated eigenvalue problem. The form of the eigenvalue problem is closely related to the Markov Spectral Theory for model reduction. This result is the basis of a heuristic proposed for the m-ary partition problem, resulting in a practical recursive algorithm. The results are illustrated with examples.
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large deviations asymptotics and the Spectral Theory of multiplicatively regular markov processes
arXiv: Probability, 2005Co-Authors: Ioannis Kontoyiannis, Sean P MeynAbstract:We continue the investigation of the Spectral Theory and exponential asymptotics of Markov processes, following Kontoyiannis and Meyn (2003). We introduce a new family of nonlinear Lyapunov drift criteria, characterizing distinct subclasses of geometrically ergodic Markov processes in terms of inequalities for the nonlinear generator. We concentrate on the class of "multiplicatively regular" Markov processes, characterized via conditions similar to (but weaker than) those of Donsker-Varadhan. For any such process {Phi(t)} with transition kernel P on a general state space, the following are obtained. 1. Spectral Theory: For a large class of functionals F, the kernel Phat(x,dy) = e^{F(x)}P(x,dy) has a discrete spectrum in an appropriately defined Banach space. There exists a "maximal" solution to the "multiplicative Poisson equation," defined as the eigenvalue problem for Phat. Regularity properties are established for \Lambda(F) = \log(\lambda), where \lambda is the maximal eigenvalue, and for its convex dual. 2. MULTIPLICATIVE MEAN ERGODIC THEOREM: The normalized mean E_x[\exp(S_t)] of the exponential of the partial sums {S_t} of the process with respect to any one of the above functionals F, converges to the maximal eigenfunction. 3. MULTIPLICATIVE REGULARITY: The drift criterion under which our results are derived is equivalent to the existence of regeneration times with finite exponential moments for {S_t}. 4. LARGE DEVIATIONS: The sequence of empirical measures of {Phi(t)} satisfies an LDP in a topology finer than the \tau-topology. The rate function is \Lambda^* and it coincides with the Donsker-Varadhan rate function. 5. EXACTR LARGE DEVIATIONS: The partial sums {S_t} satisfy an exact LD expansion, analogous to that obtained for independent random variables.
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large deviations asymptotics and the Spectral Theory of multiplicatively regular markov processes
Electronic Journal of Probability, 2005Co-Authors: Ioannis Kontoyiannis, Sean P MeynAbstract:In this paper we continue the investigation of the Spectral Theory and exponential asymptotics of primarily discrete-time Markov processes, following Kontoyiannis and Meyn (2003). We introduce a new family of nonlinear Lyapunov drift criteria, which characterize distinct subclasses of geometrically ergodic Markov processes in terms of simple inequalities for the nonlinear generator. We concentrate primarily on the class of multiplicatively regular Markov processes, which are characterized via simple conditions similar to (but weaker than) those of Donsker-Varadhan. For any such process $\{\Phi(t)\}$ with transition kernel $P$ on a general state space $X$, the following are obtained. Spectral Theory: For a large class of (possibly unbounded) functionals $F$ on $X$, the kernel $\hat P(x,dy) = e^{F(x)} P(x,dy)$ has a discrete spectrum in an appropriately defined Banach space. It follows that there exists a "maximal," well-behaved solution to the "multiplicative Poisson equation," defined as an eigenvalue problem for $\hat P$. Multiplicative Mean Ergodic Theorem: Consider the partial sums of this process with respect to any one of the functionals $F$ considered above. The normalized mean of their moment generating function (and not the logarithm of the mean) converges to the above maximal eigenfunction exponentially fast. Multiplicative regularity: The Lyapunov drift criterion under which our results are derived is equivalent to the existence of regeneration times with finite exponential moments for the above partial sums. Large Deviations: The sequence of empirical measures of the process satisfies a large deviations principle in a topology finer that the usual tau-topology, generated by the above class of functionals. The rate function of this LDP is the convex dual of logarithm of the above maximal eigenvalue, and it is shown to coincide with the Donsker-Varadhan rate function in terms of relative entropy. Exact Large Deviations Asymptotics: The above partial sums are shown to satisfy an exact large deviations expansion, analogous to that obtained by Bahadur and Ranga Rao for independent random variables.
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Spectral Theory and limit theorems for geometrically ergodic markov processes
arXiv: Probability, 2002Co-Authors: Ioannis Kontoyiannis, Sean P MeynAbstract:Consider the partial sums {S_t} of a real-valued functional F(Phi(t)) of a Markov chain {Phi(t)} with values in a general state space. Assuming only that the Markov chain is geometrically ergodic and that the functional F is bounded, the following conclusions are obtained: 1. Spectral Theory: Well-behaved solutions can be constructed for the ``multiplicative Poisson equation''. 2. A ``multiplicative'' mean ergodic theorem: For all complex \alpha in a neighborhood of the origin, the normalized mean of \exp(\alpha S_t) converges exponentially fast to a solution of the multiplicative Poisson equation. 3. Edgeworth Expansions: Rates are obtained for the convergence of the distribution function of the normalized partial sums S_t to the standard Gaussian distribution. 4. Large Deviations: The partial sums are shown to satisfy a large deviations principle in a neighborhood of the mean. This result, proved under geometric ergodicity alone, cannot in general be extended to the whole real line. 5. Exact Large Deviations Asymptotics: Rates of convergence are obtained for the large deviations estimates above. Extensions of these results to continuous-time Markov processes are also given.
Rosa Senadias - One of the best experts on this subject based on the ideXlab platform.
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semi classical weights and equivariant Spectral Theory
Advances in Mathematics, 2016Co-Authors: Emily B Dryden, Victor Guillemin, Rosa SenadiasAbstract:Abstract We prove inverse Spectral results for differential operators on manifolds and orbifolds invariant under a torus action. These inverse Spectral results involve the asymptotic equivariant spectrum, which is the spectrum itself together with “very large” weights of the torus action on eigenspaces. More precisely, we show that the asymptotic equivariant spectrum of the Laplace operator of any toric metric on a generic toric orbifold determines the equivariant biholomorphism class of the orbifold; we also show that the asymptotic equivariant spectrum of a T n -invariant Schrodinger operator on R n determines its potential in some suitably convex cases. In addition, we prove that the asymptotic equivariant spectrum of an S 1 -invariant metric on S 2 determines the metric itself in many cases. Finally, we obtain an asymptotic equivariant inverse Spectral result for weighted projective spaces. As a crucial ingredient in these inverse results, we derive a surprisingly simple formula for the asymptotic equivariant trace of a family of semi-classical differential operators invariant under a torus action.
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equivariant inverse Spectral Theory and toric orbifolds
Advances in Mathematics, 2012Co-Authors: Emily B Dryden, Victor Guillemin, Rosa SenadiasAbstract:Abstract Let O 2 n be a symplectic toric orbifold with a fixed T n -action and with a toric Kahler metric g . In [10] we explored whether, when O is a manifold, the equivariant spectrum of the Laplace operator Δ g on C ∞ ( O ) determines O up to symplectomorphism. In the setting of toric orbifolds we significantly improve upon our previous results and show that a generic toric orbifold is determined by its equivariant spectrum, up to two possibilities. This involves developing the asymptotic expansion of the heat trace on an orbifold in the presence of an isometry. We also show that the equivariant spectrum determines whether the toric Kahler metric has constant scalar curvature.
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equivariant inverse Spectral Theory and toric orbifolds
arXiv: Differential Geometry, 2011Co-Authors: Emily B Dryden, Victor Guillemin, Rosa SenadiasAbstract:Let O be a symplectic toric 2n-dimensional orbifold with a fixed T^n-action and with a toric Kahler metric g. We previously explored whether, when O is a manifold, the equivariant spectrum of the Laplace operator acting on smooth functions on (O,g) determines the moment polytope of O, and hence by Delzant's theorem determines O up to symplectomorphism. In the setting of toric orbifolds we significantly improve upon our previous results and show that the moment polytope of a generic toric orbifold is determined by its equivariant spectrum, up to two possibilities and up to translation. This involves developing the asymptotic expansion of the heat trace on an orbifold in the presence of an isometry. We also show that the equivariant spectrum determines whether the toric Kahler metric has constant scalar curvature.
Hayato Chiba - One of the best experts on this subject based on the ideXlab platform.
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a Spectral Theory of linear operators on rigged hilbert spaces under analyticity conditions
Advances in Mathematics, 2015Co-Authors: Hayato ChibaAbstract:Abstract A Spectral Theory of linear operators on rigged Hilbert spaces (Gelfand triplets) is developed under the assumptions that a linear operator T on a Hilbert space H is a perturbation of a selfadjoint operator, and the Spectral measure of the selfadjoint operator has an analytic continuation near the real axis in some sense. It is shown that there exists a dense subspace X of H such that the resolvent ( λ − T ) − 1 ϕ of the operator T has an analytic continuation from the lower half plane to the upper half plane as an X ′ -valued holomorphic function for any ϕ ∈ X , even when T has a continuous spectrum on R, where X ′ is a dual space of X. The rigged Hilbert space consists of three spaces X ⊂ H ⊂ X ′ . A generalized eigenvalue and a generalized eigenfunction in X ′ are defined by using the analytic continuation of the resolvent as an operator from X into X ′ . Other basic tools of the usual Spectral Theory, such as a spectrum, resolvent, Riesz projection and semigroup are also studied in terms of a rigged Hilbert space. They prove to have the same properties as those of the usual Spectral Theory. The results are applied to estimate asymptotic behavior of solutions of evolution equations.
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a Spectral Theory of linear operators on rigged hilbert spaces under analyticity conditions
arXiv: Spectral Theory, 2011Co-Authors: Hayato ChibaAbstract:A Spectral Theory of linear operators on rigged Hilbert spaces (Gelfand triplets) is developed under the assumptions that a linear operator $T$ on a Hilbert space $\mathcal{H}$ is a perturbation of a selfadjoint operator, and the Spectral measure of the selfadjoint operator has an analytic continuation near the real axis in some sense. It is shown that there exists a dense subspace $X$ of $\mathcal{H}$ such that the resolvent $(\lambda -T)^{-1}\phi$ of the operator $T$ has an analytic continuation from the lower half plane to the upper half plane as an $X'$-valued holomorphic function for any $\phi \in X$, even when $T$ has a continuous spectrum on $\mathbf{R}$, where $X'$ is a dual space of $X$. The rigged Hilbert space consists of three spaces $X \subset \mathcal{H} \subset X'$. A generalized eigenvalue and a generalized eigenfunction in $X'$ are defined by using the analytic continuation of the resolvent as an operator from $X$ into $X'$. Other basic tools of the usual Spectral Theory, such as a spectrum, resolvent, Riesz projection and semigroup are also studied in terms of a rigged Hilbert space. They prove to have the same properties as those of the usual Spectral Theory. The results are applied to estimate asymptotic behavior of solutions of evolution equations.
Tomohiro Sasamoto - One of the best experts on this subject based on the ideXlab platform.
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Spectral Theory for interacting particle systems solvable by coordinate bethe ansatz
Communications in Mathematical Physics, 2015Co-Authors: Alexei Borodin, Ivan Corwin, Leonid Petrov, Tomohiro SasamotoAbstract:We develop Spectral Theory for the q-Hahn stochastic particle system introduced recently by Povolotsky. That is, we establish a Plancherel type isomorphism result that implies completeness and biorthogonality statements for the Bethe ansatz eigenfunctions of the system. Owing to a Markov duality with the q-Hahn TASEP (a discrete-time generalization of TASEP with particles’ jump distribution being the orthogonality weight for the classical q-Hahn orthogonal polynomials), we write down moment formulas that characterize the fixed time distribution of the q-Hahn TASEP with general initial data. The Bethe ansatz eigenfunctions of the q-Hahn system degenerate into eigenfunctions of other (not necessarily stochastic) interacting particle systems solvable by the coordinate Bethe ansatz. This includes the ASEP, the (asymmetric) six-vertex model, and the Heisenberg XXZ spin chain (all models are on the infinite lattice). In this way, each of the latter systems possesses a Spectral Theory, too. In particular, biorthogonality of the ASEP eigenfunctions, which follows from the corresponding q-Hahn statement, implies symmetrization identities of Tracy and Widom (for ASEP with either step or step Bernoulli initial configuration) as corollaries. Another degeneration takes the q-Hahn system to the q-Boson particle system (dual to q-TASEP) studied in detail in our previous paper (2013). Thus, at the Spectral Theory level we unify two discrete-space regularizations of the Kardar–Parisi–Zhang equation/stochastic heat equation, namely, q-TASEP and ASEP.
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Spectral Theory for the boson particle system
Compositio Mathematica, 2015Co-Authors: Alexei Borodin, Ivan Corwin, Leonid Petrov, Tomohiro SasamotoAbstract:We develop Spectral Theory for the generator of the -Boson results to a discrete delta Bose gas considered previously by van Diejen, as well as to another discrete delta Bose gas that describes the evolution of moments of the semi-discrete stochastic heat equation (or equivalently, the O’Connell–Yor semi-discrete directed polymer partition function). A further limit takes us to the delta Bose gas which arises in studying moments of the stochastic heat equation/Kardar–Parisi–Zhang equation.
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Spectral Theory for the q boson particle system
Compositio Mathematica, 2015Co-Authors: Alexei Borodin, Ivan Corwin, Leonid Petrov, Tomohiro SasamotoAbstract:We develop Spectral Theory for the generator of the $q$ -Boson (stochastic) particle system. Our central result is a Plancherel type isomorphism theorem for this system. This theorem has various implications. It proves the completeness of the Bethe ansatz for the $q$ -Boson generator and consequently enables us to solve the Kolmogorov forward and backward equations for general initial data. Owing to a Markov duality with $q$ -TASEP ( $q$ -deformed totally asymmetric simple exclusion process), this leads to moment formulas which characterize the fixed time distribution of $q$ -TASEP started from general initial conditions. The theorem also implies the biorthogonality of the left and right eigenfunctions. We consider limits of our $q$ -Boson results to a discrete delta Bose gas considered previously by van Diejen, as well as to another discrete delta Bose gas that describes the evolution of moments of the semi-discrete stochastic heat equation (or equivalently, the O’Connell–Yor semi-discrete directed polymer partition function). A further limit takes us to the delta Bose gas which arises in studying moments of the stochastic heat equation/Kardar–Parisi–Zhang equation.
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Spectral Theory for the q boson particle system
arXiv: Mathematical Physics, 2013Co-Authors: Alexei Borodin, Ivan Corwin, Leonid Petrov, Tomohiro SasamotoAbstract:We develop Spectral Theory for the generator of the q-Boson (stochastic) particle system. Our central result is a Plancherel type isomorphism theorem for this system. This theorem has various implications. It proves the completeness of the Bethe ansatz for the q-Boson generator and consequently enables us to solve the Kolmogorov forward and backward equations for general initial data. Owing to a Markov duality with q-TASEP, this leads to moment formulas which characterize the fixed time distribution of q-TASEP started from general initial conditions. The theorem also implies the biorthogonality of the left and right eigenfunctions. We consider limits of our q-Boson results to a discrete delta Bose gas considered previously by van Diejen, as well as to another discrete delta Bose gas that describes the evolution of moments of the semi-discrete stochastic heat equation (or equivalently, the O'Connell-Yor semi-discrete directed polymer partition function). A further limit takes us to the delta Bose gas which arises in studying moments of the stochastic heat equation / Kardar-Parisi-Zhang equation.
Barry Simon - One of the best experts on this subject based on the ideXlab platform.
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a new approach to inverse Spectral Theory
2016Co-Authors: Barry SimonAbstract:We present a new approach (distinct from Gel'fand-Levitan) to the theorem of Borg-Marchenko that the m-function (equivalently, Spectral measure) for a finite interval or half-line Schr6dinger operator determines the potential. Our approach is an analog of the continued fraction approach for the moment problem. We prove there is a representation for the m-function m(-n2) = -/ - 0bA(a)e-2ada + O(e-(2b-6)). A on [0,a] is a function of q on [0, a] and vice-versa. A key role is played by a differential equation that A obeys after allowing x-dependence: aA _A o a
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Natural boundaries and Spectral Theory
Advances in Mathematics, 2011Co-Authors: Jonathan Breuer, Barry SimonAbstract:We present and exploit an analogy between lack of absolutely continuous spectrum for Schrodinger operators and nat- ural boundaries for power series. Among our new results are gen- eralizations of Hecke's example and natural boundary examples for random power series where independence is not assumed.
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Natural Boundaries and Spectral Theory
arXiv: Complex Variables, 2010Co-Authors: Jonathan Breuer, Barry SimonAbstract:We present and exploit an analogy between lack of absolutely continuous spectrum for Schroedinger operators and natural boundaries for power series. Among our new results are generalizations of Hecke's example and natural boundary examples for random power series where independence is not assumed.
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equilibrium measures and capacities in Spectral Theory
arXiv: Spectral Theory, 2007Co-Authors: Barry SimonAbstract:This is a comprehensive review of the uses of potential Theory in studying the Spectral Theory of orthogonal polynomials. Much of the article focuses on the Stahl-Totik Theory of regular measures, especially the case of OPRL and OPUC. Links are made to the study of ergodic Schrodinger operators where one of our new results implies that, in complete generality, the Spectral measure is supported on a set of zero Hausdorff dimension (indeed, of capacity zero) in the region of strictly positive Lyapunov exponent. There are many examples and some new conjectures and indications of new research directions. Included are appendices on potential Theory and on Fekete-Szego Theory.
