1,720,965 research outputs found
A note on a local limit theorem for Wiener space valued random variables
Full text linkWe prove a local limit theorem, i.e. a central limit theorem for densities, for a sequence of independent and identically distributed random variables taking values on an abstract Wiener space; the common law of those random variables is assumed to be absolutely continuous with respect to the reference Gaussian measure. We begin by showing that the key roles of scaling operator and convolution product in this infinite dimensional Gaussian framework are played by the Ornstein-Uhlenbeck semigroup and Wick product, respectively. We proceed by establishing a necessary condition on the density of the random variables for the
local limit theorem to hold true. We then reverse the implication and prove under an additional assumption the desired L1-convergence of the density of X1+···+Xn/sqrt{n}. We close the paper comparing our result with certain Berry-Esseen bounds for multidimensional central limit theorems
An extension of the Beckner’s type Poincaré inequality to convolution measures on abstract Wiener spaces
No full textWe 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
Holder-Young-Lieb inequalities for norms of Gaussian Wick products
No full textAn important connection between the finite dimensional Gaussian Wick products and
Lebesgue convolution products will be proven first. Then this connection will be used
to prove an important H ̈older inequality for the norms of Gaussian Wick products,
reprove Nelson hypercontractivity inequality, and prove a more general inequality whose
marginal cases are the H ̈older and Nelson inequalities mentioned before. We will show
that there is a deep connection between the Gaussian H ̈older inequality and classic
Ho ̈lder inequality, between the Nelson hypercontractivity and classic Young inequality
with the sharp constant, and between the third more general inequality and an extension
by Lieb of the Young inequality with the best constant. Since the Gaussian probability
measure exists even in the infinite dimensional case, the above three inequalities can be
extended, via a classic Fatou’s lemma argument, to the infinite dimensional framework
An Ito formula for a family of stochastic integrals and related Wong-Zakai type theorems
No full textThe aim of this paper is to generalize two important results known for the Stratonovich and Itˆo integrals
to any stochastic integral obtained as limit of Riemann sums with arbitrary evaluating point: the ordinary
chain rule for certain nonlinear functions of the Brownian motion and the Wong–Zakai approximation
theorem. To this scope we begin by introducing a new family of products for smooth random variables
which reduces for specific choices of a parameter to the pointwise and to the Wick products. We show
that each product in that family is related in a natural way to a precise choice of the evaluating point in the
above mentioned Riemann sums and hence to a certain notion of stochastic integral. Our chain rule relies on
a new probabilistic representation for the solution of the heat equation while the Wong–Zakai type theorem
follows from a reduction method for quasi-linear SDEs together with a formula of Gjessing’s type
A sharp interpolation between the Hölder and Gaussian Young inequalities
Full text linkWe prove a very general sharp inequality of the Hölder-Young-type for functions defined on infinite dimensional Gaussian spaces. We begin by considering a family of commutative products for functions which interpolates between the pointwise and Wick products; this family arises naturally in the context of stochastic differential equations, through Wong-Zakai-type approximation theorems, and plays a key role in some generalizations of the Beckner-type Poincaré inequality. We then obtain a crucial integral representation for that family of products which is employed, together with a generalization of the classic Young inequality due to Lieb, to prove our main theorem. We stress that our main inequality contains as particular cases the Hölder inequality and Nelson's hyper-contractive estimate, thus providing a unified framework for two fundamental results of the Gaussian analysis
A sharp interpolation between the Hölder and Gaussian Young inequalities
No full textWe prove a very general sharp inequality of the Hölder-Young-type for functions defined on infinite dimensional Gaussian spaces. We begin by considering a family of commutative products for functions which interpolates between the pointwise and Wick products; this family arises naturally in the context of stochastic differential equations, through Wong-Zakai-type approximation theorems, and plays a key role in some generalizations of the Beckner-type Poincaré inequality. We then obtain a crucial integral representation for that family of products which is employed, together with a generalization of the classic Young inequality due to Lieb, to prove our main theorem. We stress that our main inequality contains as particular cases the Hölder inequality and Nelson's hyper-contractive estimate, thus providing a unified framework for two fundamental results of the Gaussian analysis
An extension of the Beckner’s type Poincaré inequality to convolution measures on abstract Wiener spaces
No full textWe 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
An Ito formula for a family of stochastic integrals and related Wong-Zakai type theorems
No full textThe aim of this paper is to generalize two important results known for the Stratonovich and Itˆo integrals
to any stochastic integral obtained as limit of Riemann sums with arbitrary evaluating point: the ordinary
chain rule for certain nonlinear functions of the Brownian motion and the Wong–Zakai approximation
theorem. To this scope we begin by introducing a new family of products for smooth random variables
which reduces for specific choices of a parameter to the pointwise and to the Wick products. We show
that each product in that family is related in a natural way to a precise choice of the evaluating point in the
above mentioned Riemann sums and hence to a certain notion of stochastic integral. Our chain rule relies on
a new probabilistic representation for the solution of the heat equation while the Wong–Zakai type theorem
follows from a reduction method for quasi-linear SDEs together with a formula of Gjessing’s type
Holder-Young-Lieb inequalities for norms of Gaussian Wick products
No full textAn important connection between the finite dimensional Gaussian Wick products and
Lebesgue convolution products will be proven first. Then this connection will be used
to prove an important H ̈older inequality for the norms of Gaussian Wick products,
reprove Nelson hypercontractivity inequality, and prove a more general inequality whose
marginal cases are the H ̈older and Nelson inequalities mentioned before. We will show
that there is a deep connection between the Gaussian H ̈older inequality and classic
Ho ̈lder inequality, between the Nelson hypercontractivity and classic Young inequality
with the sharp constant, and between the third more general inequality and an extension
by Lieb of the Young inequality with the best constant. Since the Gaussian probability
measure exists even in the infinite dimensional case, the above three inequalities can be
extended, via a classic Fatou’s lemma argument, to the infinite dimensional framework
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