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Spectral theory and limit theorems for geometrically ergodic Markov processes. Part II: Empirical measures & unbounded functionals
, 2001
"... Consider the partial sums {St} of a realvalued functional F(�(t)) of a Markov chain {�(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: Spectral theory. Wellbehaved s ..."
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Cited by 96 (22 self)
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Consider the partial sums {St} of a realvalued functional F(�(t)) of a Markov chain {�(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: Spectral theory. Wellbehaved solutions fˇ can be constructed for the “multiplicative Poisson equation ” (eαF P) f ˇ = λf ˇ,wherePis the transition kernel of the Markov chain and α ∈ C is a constant. The function fˇ is an eigenfunction, with corresponding eigenvalue λ, for the kernel (eαF P) = eαF(x) P(x,dy). A “multiplicative ” mean ergodic theorem. For all complex α in a neighborhood of the origin, the normalized mean of exp(αSt) (and not the logarithm of the mean) converges to fˇ exponentially fast, where fˇ is a solution of the multiplicative Poisson equation. Edgeworth expansions. Rates are obtained for the convergence of the distribution function of the normalized partial sums St to the standard Gaussian distribution. The first term in this expansion is of order (1 / √ t) and it depends on the initial condition of the Markov chain through the solution ̂ F of the associated Poisson equation (and not the solution fˇ of the multiplicative Poisson equation). 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. Exact large deviations asymptotics. Rates of convergence are obtained for the large deviations estimates above. The polynomial preexponent is of order (1 / √ t) and its coefficient depends on the initial condition of the Markov chain through the solution fˇ of the multiplicative Poisson equation. Extensions of these results to continuoustime Markov processes are also given. 1. Introduction. Consider a Markov process � ={�(t): t ∈ T
Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré
 J. Func. Anal
, 1996
"... Abstract. We study the relationship between two classical approaches for quantitative ergodic properties: the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one based on functional inequalities (of Poincaré type). We show that they can be linked through new ..."
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Cited by 64 (24 self)
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Abstract. We study the relationship between two classical approaches for quantitative ergodic properties: the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one based on functional inequalities (of Poincaré type). We show that they can be linked through new inequalities (LyapunovPoincaré inequalities). Explicit examples for diffusion processes are studied, improving some results in the literature. The example of the kinetic FokkerPlanck equation recently studied by HérauNier, HelfferNier and Villani is in particular discussed in the final section.
Large deviations asymptotics and the spectral theory of multiplicatively regular Markov processes
 Electron. J. Probab
"... In this paper we continue the investigation of the spectral theory and exponential asymptotics of primarily discretetime Markov processes, following Kontoyiannis and Meyn [32]. We introduce a new family of nonlinear Lyapunov drift criteria, which characterize distinct subclasses of geometrically er ..."
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Cited by 51 (10 self)
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In this paper we continue the investigation of the spectral theory and exponential asymptotics of primarily discretetime Markov processes, following Kontoyiannis and Meyn [32]. 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 DonskerVaradhan. For any such process Φ = {Φ(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: X → C, the kernel ̂ 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 ” solution (λ, ˇ f) to the multiplicative Poisson equation, defined as the eigenvalue problem ̂ P ˇ f = λ ˇ f. The functional Λ(F) = log(λ) is convex, smooth, and its convex dual Λ ∗ is convex, with compact sublevel sets.
A FlemingViot Particle Representation Of Dirichlet Laplacian
 Comm. Math. Phys
"... : We consider a model with a large number N of particles which move according to independent Brownian motions. A particle which leaves a domain D is killed; at the same time, a di#erent particle splits into two particles. For large N , the particle distribution density converges to the normalized he ..."
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Cited by 42 (7 self)
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: We consider a model with a large number N of particles which move according to independent Brownian motions. A particle which leaves a domain D is killed; at the same time, a di#erent particle splits into two particles. For large N , the particle distribution density converges to the normalized heat equation solution in D with Dirichlet boundary conditions. The stationary distributions converge as N ## to the first eigenfunction of the Laplacian in D with the same boundary conditions. 1. Introduction. Our article is closely related to a model studied by Burdzy, Ho#lyst, Ingerman and March (1996) using heuristic and numerical methods. Although we are far from proving conjectures stated in that article, the present paper seems to lay solid theoretical foundations for further research in this direction. The model is related to many known ideas in probability and physicswe review them in the Appendix (Section 3). We present the model and state our main results in this section. Secti...
Probabilistic Approach for Granular Media Equations in the Non Uniformly Convex Case
, 2007
"... We use here a particle system to prove a convergence result as well as a deviation inequality for solutions of granular media equation when the confinement potential and the interaction potential are no more uniformly convex. Proof is straightforward, simplifying deeply proofs of CarrilloMcCannVil ..."
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Cited by 35 (9 self)
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We use here a particle system to prove a convergence result as well as a deviation inequality for solutions of granular media equation when the confinement potential and the interaction potential are no more uniformly convex. Proof is straightforward, simplifying deeply proofs of CarrilloMcCannVillani [CMCV03, CMCV06] and completing results of Malrieu [Mal03] in the uniformly convex case. It relies on an uniform propagation of chaos property and a direct control in Wasserstein distance of solutions starting with different initial measures. The deviation inequality is obtained via a T1 transportation cost inequality replacing the logarithmic Sobolev inequality which is no more clearly dimension free.
Biomolecular Conformations can be Identified as Metastable Sets of Molecular Dynamics
"... This article summarizes the present state of the transfer operator approach to biomolecular conformations with special emphasis on the conceptual and mathematical foundations. ..."
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Cited by 34 (3 self)
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This article summarizes the present state of the transfer operator approach to biomolecular conformations with special emphasis on the conceptual and mathematical foundations.
Phase transitions and metastability in Markovian and molecular systems
, 2002
"... Diffusion models arising in analysis of large biochemical models and other complex systems are typically far too complex for exact solution, or even meaningful simulation. The purpose of this paper is to develop foundations for model reduction, and new modeling techniques for diffusion models. These ..."
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Cited by 28 (12 self)
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Diffusion models arising in analysis of large biochemical models and other complex systems are typically far too complex for exact solution, or even meaningful simulation. The purpose of this paper is to develop foundations for model reduction, and new modeling techniques for diffusion models. These foundations are all based upon recent spectral theory of Markov processes. The main assumption imposed is Vuniform ergodicity of the process. This is equivalent to any common formulation of exponential ergodicity, and is known to be far weaker than the DonskerVaradahn conditions in large deviations theory. Under this assumption it is shown that the associated semigroup admits a spectral gap in a weighted L∞norm, and real eigenfunctions provide a decomposition of the state space into ‘almost’absorbing subsets. It is shown that the process mixes rapidly in each of these subsets prior to exiting, and that the conditional distributions of exit times are approximately exponential. These results represent a significant expansion of the classical Wentzell–Freidlin theory. In particular, the results require no special structure beyond geometric ergodicity; reversibility is not assumed; and meaningful conclusions can be drawn even for models with significant variability.
CONVERGENCE OF NUMERICAL TIMEAVERAGING AND STATIONARY MEASURES VIA POISSON EQUATIONS
, 908
"... Abstract. Numerical approximation of the long time behavior of a stochastic differential equation (SDE) is considered. Error estimates for timeaveraging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the SDE. The error analysi ..."
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Cited by 19 (2 self)
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Abstract. Numerical approximation of the long time behavior of a stochastic differential equation (SDE) is considered. Error estimates for timeaveraging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the SDE. The error analysis is based on using an associated Poisson equation for the underlying SDE. The main advantage of this approach is its simplicity and universality. It works equally well for a range of explicit and implicit schemes including those with simple simulation of random variables, and for general hypoelliptic SDEs. An analogy between this approach and Stein’s method is indicated. Some practical implications of the results are discussed. AMS 000 subject classification. Primary 65C30; secondary 60H35, 37H10, 60H10.
Langevin diffusions and MetropolisHastings algorithms
 IN APPLIED PROBABILITY
, 2003
"... We consider a class of Langevin diffusions with statedependent volatility. The volatility of the diffusion is chosen so as to make the stationary distribution of the diffusion with respect to its natural clock, a heated version of the stationary density of interest. The motivation behind this cons ..."
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Cited by 17 (0 self)
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We consider a class of Langevin diffusions with statedependent volatility. The volatility of the diffusion is chosen so as to make the stationary distribution of the diffusion with respect to its natural clock, a heated version of the stationary density of interest. The motivation behind this construction is the desire to construct uniformly ergodic diffusions with required stationary densities. Discrete time algorithms constructed by Hastings accept reject mechanisms are constructed from discretisations of the algorithms, and the properties of these algorithms are investigated.
Workload Models for Stochastic Networks: Value Functions and Performance Evaluation
, 2005
"... This paper concerns control and performance evaluation for stochastic network models. Structural properties of value functions are developed for controlled Brownian motion (CBM) and deterministic (fluid) workloadmodels, leading to the following conclusions: Outside of a nullset of network paramete ..."
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Cited by 16 (9 self)
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This paper concerns control and performance evaluation for stochastic network models. Structural properties of value functions are developed for controlled Brownian motion (CBM) and deterministic (fluid) workloadmodels, leading to the following conclusions: Outside of a nullset of network parameters, (i) The fluid valuefunction is a smooth function of the initial state. Under further minor conditions, the fluid valuefunction satisfies the derivative boundary conditions that are required to ensure it is in the domain of the extended generator for the CBM model. Exponential ergodicity of the CBM model is demonstrated as one consequence. (ii) The fluid valuefunction provides a shadow function for use in simulation variance reduction for the stochastic model. The resulting simulator satisfies an exact large deviation principle, while a standard simulation algorithm does not satisfy any such bound. (iii) The fluid valuefunction provides upper and lower bounds on performance for the CBM model. This follows from an extension of recent linear programming approaches to performance evaluation.