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24
THE MARKOV CHAIN MONTE CARLO REVOLUTION
"... Abstract. The use of simulation for highdimensional intractable computations has revolutionized applied mathematics. Designing, improving and understanding the new tools leads to (and leans on) fascinating mathematics, from representation theory through microlocal analysis. 1. ..."
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Abstract. The use of simulation for highdimensional intractable computations has revolutionized applied mathematics. Designing, improving and understanding the new tools leads to (and leans on) fascinating mathematics, from representation theory through microlocal analysis. 1.
Transportationinformation inequalities for Markov processes (II): Relations . . .
, 2009
"... We continue our investigation on the transportationinformation inequalities WpI for a symmetric markov process, introduced and studied in [13]. We prove that WpI implies the usual transportation inequalities WpH, then the corresponding concentration inequalities for the invariant measure µ. We giv ..."
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Cited by 17 (2 self)
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We continue our investigation on the transportationinformation inequalities WpI for a symmetric markov process, introduced and studied in [13]. We prove that WpI implies the usual transportation inequalities WpH, then the corresponding concentration inequalities for the invariant measure µ. We give also a direct proof that the spectral gap in the space of Lipschitz functions for a diffusion process implies W1I (a result due to [13]) and a Cheeger type’s isoperimetric inequality. Finally we exhibit relations between transportationinformation inequalities and a family of functional inequalities (such as Φlog Sobolev or ΦSobolev).
Malrieu Trend to Equilibrium and Particle Approximation for a Weakly Self–Consistent VlasovFokkerPlanck Equation
, 2009
"... We consider a VlasovFokkerPlanck equation governing the evolution of the density of interacting and diffusive matter in the space of positions and velocities. We use a probabilistic interpretation to obtain convergence towards equilibrium in Wasserstein distance with an explicit exponential rate. ..."
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Cited by 9 (3 self)
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We consider a VlasovFokkerPlanck equation governing the evolution of the density of interacting and diffusive matter in the space of positions and velocities. We use a probabilistic interpretation to obtain convergence towards equilibrium in Wasserstein distance with an explicit exponential rate. We also prove a propagation of chaos property for an associated particle system, and give rates on the approximation of the solution by the particle system. Finally, a transportation inequality for the distribution of the particle system leads to quantitative deviation bounds on the approximation of the equilibrium solution of the equation by an empirical mean of the particles at given time. Introduction and main results We are interested in the long time behaviour and in a particle approximation of a distribution ft(x,v) in the space of positions x ∈ R d and velocities v ∈ R d (with d � 1) evolving according to the VlasovFokkerPlanck equation
On the long time behavior of the TCP window size process
 Stochastic Processes and their Applications 120 (2010
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Functional inequalities for heavy tails distributions and application to isoperimetry
, 2008
"... Abstract. This paper is devoted to the study of probability measures with heavy tails. Using the Lyapunov function approach we prove that such measures satisfy different kind of functional inequalities such as weak Poincaré and weak Cheeger, weighted Poincaré and weighted Cheeger inequalities and th ..."
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Abstract. This paper is devoted to the study of probability measures with heavy tails. Using the Lyapunov function approach we prove that such measures satisfy different kind of functional inequalities such as weak Poincaré and weak Cheeger, weighted Poincaré and weighted Cheeger inequalities and their dual forms. Proofs are short and we cover very large situations. For product measures onR n we obtain the optimal dimension dependence using the mass transportation method. Then we derive (optimal) isoperimetric inequalities. Finally we deal with spherically symmetric measures. We recover and improve many previous results.
Bernstein type’s concentration inequalities for symmetric Markov processes
"... Abstract. Using the method of transportationinformation inequality introduced in [28],weestablishBernsteintype’sconcentrationinequalitiesforempiricalmeans 1 ∫ t t 0 g(Xs)ds where g is a unbounded observable of the symmetric Markov process (Xt). Three approaches are proposed: functional inequalities ..."
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Cited by 4 (1 self)
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Abstract. Using the method of transportationinformation inequality introduced in [28],weestablishBernsteintype’sconcentrationinequalitiesforempiricalmeans 1 ∫ t t 0 g(Xs)ds where g is a unbounded observable of the symmetric Markov process (Xt). Three approaches are proposed: functional inequalities approach; Lyapunov function method; and an approach through the Lipschitzian norm of the solution to the Poisson equation. Several applications and examples are studied.
Asymptotic analysis and diffusion limit of the Persistent Turning Walker Model
, 2008
"... The Persistent Turning Walker Model (PTWM) was introduced by Gautrais et al in Mathematical Biology for the modelling of fish motion. It involves a nonlinear pathwise functional of a nonelliptic hypoelliptic diffusion. This diffusion solves a kinetic FokkerPlanck equation based on an OrnsteinUhl ..."
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The Persistent Turning Walker Model (PTWM) was introduced by Gautrais et al in Mathematical Biology for the modelling of fish motion. It involves a nonlinear pathwise functional of a nonelliptic hypoelliptic diffusion. This diffusion solves a kinetic FokkerPlanck equation based on an OrnsteinUhlenbeck Gaussian process. The long time “diffusive ” behavior of this model was recently studied by Degond & Motsch using partial differential equations techniques. This model is however intrinsically probabilistic. In the present paper, we show how the long time diffusive behavior of this model can be essentially recovered and extended by using appropriate tools from stochastic analysis. The approach can be adapted to many other kinetic “probabilistic ” models. Keywords. Mathematical Biology; animal behavior; hypoelliptic diffusions; kinetic FokkerPlanck equations; Poisson equation; invariance principles; central limit theorems, Gaussian and Markov processes. AMSMSC. 82C31; 35H10; 60J60; 60F17; 92B99; 92D50; 34F05. 1
LONG TIME BEHAVIOUR AND STATIONARY REGIME OF MEMORY GRADIENT DIFFUSIONS
"... In this paper, we are interested in a diffusion process based on a gradient descent. The process is non Markov and has a memory term which is built as a weighted average of the drift term all along the past of the trajectory. For this type of diffusion, we study the long time behaviour of the proces ..."
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In this paper, we are interested in a diffusion process based on a gradient descent. The process is non Markov and has a memory term which is built as a weighted average of the drift term all along the past of the trajectory. For this type of diffusion, we study the long time behaviour of the process in terms of the memory. We exhibit some conditions for the longtime stability of the dynamical system and then provide, when stable, some convergence properties of the occupation measures and of the marginal distribution, to the associated steady regimes. When the memory is too long, we show that in general, the dynamical system has a tendency to explode, and in the particular gaussian case, we explicitly obtain the rate of divergence.
Maximum Likelihood Drift Estimation for Multiscale Diffusions
 Stochastic Processes Applications
"... We study the problem of parameter estimation using maximum likelihood for fast/slow systems of stochastic differential equations. Our aim is to shed light on the problem of model/data mismatch at small scales. We consider two classes of fast/slow problems for which a closed coarsegrained equation f ..."
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We study the problem of parameter estimation using maximum likelihood for fast/slow systems of stochastic differential equations. Our aim is to shed light on the problem of model/data mismatch at small scales. We consider two classes of fast/slow problems for which a closed coarsegrained equation for the slow variables can be rigorously derived, which we refer to as averaging and homogenization problems. We ask whether, given data from the slow variable in the fast/slow system, we can correctly estimate parameters in the drift of the coarsegrained equation for the slow variable, using maximum likelihood. We show that, whereas the maximum likelihood estimator is asymptotically unbiased for the averaging problem, for the homogenization problem maximum likelihood fails unless we subsample the data at an appropriate rate. An explicit formula for the asymptotic error in the log likelihood function is presented. Our theory is applied to two simple examples from molecular dynamics.