On the Poincaré inequality for infinitely divisible measures
Let \(S\) be a locally compact separable metric space and let \(M\) denote the set of all nonnegative Radon measures on \(S\), with the \(\sigma \)-algebra generated by the vague topology. With each infinitely divisible measure \(m\) on \(M\), one associates canonically a nonnegative measure \(\Pi \) on \(M\setminus \{0\}\) such that \[ -\log \int e^{-} m(d\mu ) = +\int 1 - e^{-} \Pi (d\mu ) \] A one parameter convolution semigroup \((m_{t})_{t\geq 0}\) can be associated. A gradient form \[ {\mathcal E}_{m} (F,G) := \int _ {M} \left < \mu , \Gamma \left ( \frac{\partial F}{\partial \delta } (\mu ), \frac{\partial G}{\partial \delta } (\mu )\right ) \right > m(d\mu ) \] is well-defined. The fact that \({\mathcal E}_m\) satisfies a Poincaré inequality means that: \[ \int \left (F(\mu ) - _{m} \right )^{2} m(d\mu ) \leq \kappa {\mathcal E}_{m} (F, F) \] The main result of the paper gives a necessary and sufficient condition for Poincaré inequality to hold for bilinear forms of gradient type on \(L^ {2}\), with respect to an infinitely divisible measure. As an application, a strong law of large numbers for time-inhomogeneous one-dimensional subordinators is proved.
scientific article; zbMATH DE number 2174518