We generalize the Beckner’s type Poincaré inequality (Beckner, W. Proc. Amer. Math. Soc. (1989) 105:397–400) to a large class of probability measures on an abstract Wiener space of the form μ⋆ν, where μ is the reference Gaussian measure and ν is a probability measure satisfying a certain integrability condition. As the Beckner inequality interpolates between the Poincaré and logarithmic Sobolev inequalities, we utilize a family of products for functions which interpolates between the usual point-wise multiplication and the Wick product. Our approach is based on the positivity of a quadratic form involving Wick powers and integration with respect to those convolution measures. In addition, we prove that in the finite-dimensional case the class of densities of convolutions measures satisfies a point-wise covariance inequality.
Da Pelo Paolo, Lanconelli Alberto, Stan Aurel I. (2016). An extension of the Beckner’s type Poincaré inequality to convolution measures on abstract Wiener spaces. STOCHASTIC ANALYSIS AND APPLICATIONS, 34(1), 47-64 [10.1080/07362994.2015.1099443].
An extension of the Beckner’s type Poincaré inequality to convolution measures on abstract Wiener spaces
Lanconelli Alberto
Investigation
;
2016
Abstract
We generalize the Beckner’s type Poincaré inequality (Beckner, W. Proc. Amer. Math. Soc. (1989) 105:397–400) to a large class of probability measures on an abstract Wiener space of the form μ⋆ν, where μ is the reference Gaussian measure and ν is a probability measure satisfying a certain integrability condition. As the Beckner inequality interpolates between the Poincaré and logarithmic Sobolev inequalities, we utilize a family of products for functions which interpolates between the usual point-wise multiplication and the Wick product. Our approach is based on the positivity of a quadratic form involving Wick powers and integration with respect to those convolution measures. In addition, we prove that in the finite-dimensional case the class of densities of convolutions measures satisfies a point-wise covariance inequality.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.