Variance limite d'une marche aléatoire réversible en milieu aléatoire sur Z (Limit of the Variance of a Reversible Random Walk in Random Medium on Z)

2009/02/03 by Jérôme Depauw, Depauw, Jérôme, Jean-Marc Derrien +1
Mathematics · Physics and Astronomy · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR

paper · pdf · doi:10.48550/arxiv.0902.0584

arxiv created 2009/02/03 · openalex publication_date 2009/02/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Central Limit Theorem for the random walk on a stationary random network of conductances has been studied by several authors. In one dimension, when conductances and resistances are integrable, and following a method of martingale introduced by S. Kozlov (1985), we can prove the Quenched Central Limit Theorem. In that case the variance of the limit law is not null. When resistances are not integrable, the Annealed Central Limit Theorem with null variance was established by Y. Derriennic and M. Lin (personal communication). The quenched version of this last theorem is proved here, by using a very simple method. The similar problem for the continuous diffusion is then considered. Finally our method allows us to prove an inequality for the quadratic mean of a diffusion (X_t)_t at all time t.

Related

• A quantitative central limit theorem for the random walk among random\n conductances
• The quenched central limit theorem for a model of random walk in random\n environment
• A conditional quenched CLT for random walks among random conductances on ℤ^d
• Quenched Central Limit Theorem for Random Walks in Doubly Stochastic\n Random Environment
• A Quenched Functional Central Limit Theorem for Random Walks in Random Environments under (T)_γ