Variance limite d'une marche aléatoire réversible en milieu aléatoire sur <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mi mathvariant="double-struck">Z</mml:mi></mml:math>

Depauw, Jérôme; Derrien, Jean-Marc

doi:10.1016/j.crma.2009.01.030

Cited by 17 publications

(10 citation statements)

References 7 publications

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“…) is a regular local Dirichlet form. 6 MOUSTAPHA BA AND PIERRE MATHIEU Following [8], chapter 1.5, we also define the extended domain H 1 e (I 0 ): this is the set of measurable functions f on I 0 , such that |f | < ∞ a.e and there exists aξ-Cauchy sequence…”

Section: Dirichlet Forms and Processesmentioning

confidence: 99%

A Sobolev Inequality and the Individual Invariance Principle for Diffusions in a Periodic Potential

Ba¹,

Mathieu²

2015

SIAM J. Math. Anal.

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Publication issue de la thèse de Moustapha BaInternational audienceWe consider a diffusion process in $\mathbb{R}^d$ with a generator of the form $ L:=\frac 12 e^{V(x)}div(e^{-V(x)}\nabla ) $ where $V$ is measurable and periodic. We only assume that $e^V$ and $e^{-V}$ are locally integrable. We then show that, after proper rescaling, the law of the diffusion converges to a Brownian motion for Lebesgue almost all starting points. This pointwise invariance principle was previously known under uniform ellipticity conditions (when $V$ is bounded), and was recently proved under more restrictive $L^p$ conditions on $e^V$ and $e^{-V}$. Our approach uses Dirichlet form theory to define the process, martingales and time changes and the construction of a corrector. Our main technical tool to show the sub-linear growth of the corrector is a new weighted Sobolev type inequality for integrable potentials. We heavily rely on harmonic analysis technics

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Section: Dirichlet Forms and Processesmentioning

confidence: 99%

A Sobolev Inequality and the Individual Invariance Principle for Diffusions in a Periodic Potential

Ba¹,

Mathieu²

2015

SIAM J. Math. Anal.

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“…for u = 1/c(T ω), v = (1/ ) s=1c (T s ω) and α = 1 this tends to σ −2 = [ (1/c) dμ c dμ]. The point is that, since c > 0, the convergence is still satisfied if one of these integrals is +∞ (see [3]).…”

Section: Lemma 3 For Each K ≥ 1 With F K Defined As Above We Havementioning

confidence: 96%

“…In the case when at least one of c and c −1 is not integrable, X n / √ n converges to the degenerate normal distribution. The second method is adapted from [3] and leads to the following theorem. This paper is organized as follows.…”

Section: Theorem 1 (Depauw and Derrienmentioning

confidence: 99%

“…When c is integrable but c −1 not, Derriennic and Lin have proved, in an unpublished work, the annealed central limit theorem (CLT) with null variance: lim n→∞ n −1 E ω (X 2 n ) = 0 in μmeasure, where E ω denotes the expectation relative to the randomness of the walk with the environment being fixed. For the quenched version Depauw and Derrien [3] considered a nonnegative solution f , defined on Z, of the Poisson equation (P ω − I )f = 1 and that satisfies f (0) = 0 to obtain the limit of the variance of the reversible random walk (X n ) n≥0 without using any martingale and without having any condition on function c except that c > 0. [3].)…”

Section: Introductionmentioning

confidence: 99%

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A Quenched Central Limit Theorem for Reversible Random Walks in a Random Environment on Z

Lam

2014

Journal of Applied Probability

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The main aim of this paper is to prove the quenched central limit theorem for reversible random walks in a stationary random environment on Z without having the integrability condition on the conductance and without using any martingale. The method shown here is particularly simple and was introduced by Depauw and Derrien [3]. More precisely, for a given realization ω of the environment, we consider the Poisson equation (Pω - I)g = f, and then use the pointwise ergodic theorem in [8] to treat the limit of solutions and then the central limit theorem will be established by the convergence of moments. In particular, there is an analogue to a Markov process with discrete space and the diffusion in a stationary random environment.

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“…with the convention that 1/ + ∞ = 0 (see [1], [2], and [6]). Therefore, we observe a symmetry between c and 1/c asymptotically in time.…”

Section: Remark On the Reversible Casementioning

confidence: 99%

A Symmetry Property for a Class of Random Walks in Stationary Random Environments on Z

Derrien¹,

Plantevin²

2012

J. Appl. Probab.

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Variance limite d'une marche aléatoire réversible en milieu aléatoire sur Z

Cited by 17 publications

References 7 publications

A Sobolev Inequality and the Individual Invariance Principle for Diffusions in a Periodic Potential

A Sobolev Inequality and the Individual Invariance Principle for Diffusions in a Periodic Potential

A Quenched Central Limit Theorem for Reversible Random Walks in a Random Environment on Z

A Symmetry Property for a Class of Random Walks in Stationary Random Environments on Z

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