On the Poincaré Inequality for Infinitely Divisible Measures

We completely characterize the Poincaré inequality for bilinear forms of gradient type defined on L2-spaces w.r.t. infinitely divisible measures m in terms of the canonical measure Π associated with m. The characterization is based on an elementary algebraic observation concerning certain quadratic forms associated with m and Π, which is of its own interest (see Lemma 3.4). Examples include canonical Dirichlet forms on configuration spaces and Dirichlet forms associated to continuous state branching processes. As an application, a strong law of large numbers for time-inhomogeneous one-dimensional subordinators is obtained.