A new approach to Poincaré-type inequalities on the Wiener space

Lanconelli, Alberto

doi:10.1142/s0219493716500027

Cited by 4 publications

(3 citation statements)

References 26 publications

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“…This last inequality, obtained in [13], is weaker than the classic Poincaré inequality for the measure ρ since in general we have…”

Section: Introductionmentioning

confidence: 83%

“…The aim of the present paper is to show that the Beckner's inequality (1.1) can be generalized in a natural way to convolution measures on abstract Wiener spaces. This generalization passes through the use of the products • α defined in (1.3) and contains as a particular case the Poincaré-type inequality obtained in [13]. More precisely, we will prove the following inequality:…”

Section: Introductionmentioning

confidence: 93%

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An extension of the Beckner's type Poincaré inequality to convolution measures on abstract Wiener spaces

Pelo¹,

Lanconelli²,

Stan³

2014

Preprint

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We generalize the Beckner's type Poincaré inequality [3] 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. Our dimension-independent results are compared with some very recent findings in the literature. In addition, we prove that in the finite dimensional case the class of densities of convolutions measures satisfies a point-wise covariance inequality.

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“…This last inequality, obtained in [13], is weaker than the classic Poincaré inequality for the measure ρ since in general we have…”

Section: Introductionmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 93%

An extension of the Beckner's type Poincaré inequality to convolution measures on abstract Wiener spaces

Pelo¹,

Lanconelli²,

Stan³

2014

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Assume that X = Z + Y where Z and Y are independent, the law of Z is µ and the law of Y , say ν, is supported on H. Then, the law of X is given by µ ⋆ ν where ⋆ denotes the convolution of probability measures. This class of probability measures has an important role in the applications being a Gaussian (white noise) perturbation of a probability measure on the Hilbert space H. Poincaré-type inequalities with respect to this class of measures have been investigated in [14] and [8]. Observe that the measure µ ⋆ ν is absolutely continuous with respect to µ with a density given by…”

Section: Convolution Measuresmentioning

confidence: 99%

Prohorov-Type Local Limit Theorems on Abstract Wiener Spaces

Lanconelli

2018

Mediterr. J. Math.

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We prove that the density of X 1 +···+Xn−nE[X 1 ] √ n , where {X n } n≥1 is a sequence of independent and identically distributed random variables taking values on a abstract Wiener space, converges in L 1 to the density of a certain Gaussian measure which is absolutely continuous with respect to the reference Wiener measure. The crucial feature in our investigation is that we do not require the covariance structure of {X n } n≥1 to coincide with the one of the Wiener measure. This produces a non trivial (different from the constant function one) limiting object which reflects the different covariance structures involved. The present paper generalizes the results proved in [18] and deepens the connection between local limit theorems on (infinite dimensional) Gaussian spaces and some key tools from the Analysis on the Wiener space, like the Wiener-Itô chaos decomposition, Ornstein-Uhlenbeck semigroup and Wick product. We also verify and discuss our main assumptions on some examples arising from the applications: dimension independent Berry-Esseen-type bounds and weak solutions of stochastic differential equations.Assumption 1.1 The law of the X n 's is absolutely continuous with respect to µ with a density f belonging to L 2 (W, µ) Assumption 1.1 has several important implications. First of all, it yields the finiteness of all the moments of the scalar random variable X n , ϕ for ϕ ∈ W * . In fact, for any m ∈ N and ϕ ∈ W * a simple application of the Cauchy-Schwartz inequality gives

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An extension of the Beckner’s type Poincaré inequality to convolution measures on abstract Wiener spaces

Pelo

Lanconelli

Stan

2015

Stochastic Analysis and Applications

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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

show abstract

A new approach to Poincaré-type inequalities on the Wiener space

Cited by 4 publications

References 26 publications

An extension of the Beckner's type Poincaré inequality to convolution measures on abstract Wiener spaces

An extension of the Beckner's type Poincaré inequality to convolution measures on abstract Wiener spaces

Prohorov-Type Local Limit Theorems on Abstract Wiener Spaces

An extension of the Beckner’s type Poincaré inequality to convolution measures on abstract Wiener spaces

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