Clark-Ocone formulas and Poincare inequalities on the discrete cube

We establish Poincare inequalities for the continuous time random walk on t he cube {-1, +1}(d). A first method is based on the study of cylindrical fu nctionals. A Poincare inequality is proved for these functionals and extend ed to arbitrary functionals. A second method is based on martingale represe ntation formulas. A whole family of Clark-Ocone formulas is then available, which leads to the corresponding family of Poincare inequalities. These va rious inequalities are compared through examples. We also show that the cyl indrical method extends to some asymmetric continous time random walks on { -1, +1}(d). (C) 2001 Editions scientifiques et medicales Elsevier SAS.