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2020 Functional inequalities for weighted Gamma distribution on the space of finite measures
Feng-Yu Wang
Electron. J. Probab. 25: 1-27 (2020). DOI: 10.1214/20-EJP426


Let $\mathbb{M} $ be the space of finite measures on a locally compact Polish space, and let $\mathcal{G} $ be the Gamma distribution on $\mathbb{M} $ with intensity measure $\nu \in \mathbb{M} $. Let $\nabla ^{ext}$ be the extrinsic derivative with tangent bundle $T\mathbb{M} = \cup _{\eta \in \mathbb{M} } L^{2}(\eta )$, and let $\mathcal{A} : T\mathbb{M} \rightarrow T\mathbb{M} $ be measurable such that $\mathcal{A} _{\eta }$ is a positive definite linear operator on $L^{2}(\eta )$ for every $\eta \in \mathbb{M} $. Moreover, for a measurable function $V$ on $\mathbb{M} $, let ${\mathrm{{d}} }{\mathcal{G} }^{V}= {\mathrm{{e}} }^{V}{\mathrm{{d}} }{\mathcal{G} }$. We investigate the Poincaré, weak Poincaré and super Poincaré inequalities for the Dirichlet form \[ \mathcal{E} _{\mathcal{A} ,V}(F,G):= \int _{\mathbb{M} }\langle \mathcal{A} _{\eta }\nabla ^{ext}F(\eta ), \nabla ^{ext}G(\eta )\rangle _{L^{2}(\eta )}\, {\mathrm{{d}} }{\mathcal{G} }^{V}(\eta ), \] which characterize various properties of the associated Markov semigroup. The main results are extended to the space of finite signed measures.


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Feng-Yu Wang. "Functional inequalities for weighted Gamma distribution on the space of finite measures." Electron. J. Probab. 25 1 - 27, 2020.


Received: 9 February 2019; Accepted: 31 January 2020; Published: 2020
First available in Project Euclid: 6 February 2020

MathSciNet: MR4073680
Digital Object Identifier: 10.1214/20-EJP426

Primary: 60G45 , 60G57 , 60H99

Keywords: extrinsic derivative , Poincaré inequality , Super Poincaré inequality , Weak Poincaré inequality , weighted Gamma distribution


Vol.25 • 2020
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