1,721,004 research outputs found
A new approach to Poincaré-type inequalities on the Wiener space
Full text linkWe prove a new type of Poincaré inequality on abstract Wiener spaces for a family of probability measures that are absolutely continuous with respect to the reference Gaussian measure. This class of probability measures is characterized by the strong positivity (a notion introduced by Nualart and Zakai in [22]) of their Radon–Nikodym densities. In general, measures of this type do not belong to the class of log-concave measures, which are a wide class of measures satisfying the Poincaré inequality (Brascamp and Lieb [2]). Our approach is based on a pointwise identity relating Wick and ordinary products and on the notion of strong positivity which is connected to the non-negativity of Wick powers. Our technique also leads to a partial generalization of the Houdré and Kagan [11] and Houdré and Pérez-Abreu [12] Poincaré-type inequalities
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
Bayes' formula for second quantization operators
Full text linkThe Bayes’ formula provides the relationship between conditional expectations with
respect to absolutely continuous measures. The conditional expectation is in the context
of the Wiener space — an example of second quantization operator. In this note we
obtain a formula that generalizes the above-mentioned Bayes’ rule to general second
quantization operators
Translated Brownian motions and associated Wick products
No full textThe concept of Wick product is strongly related to the underlying
Brownian motion we have fixed on the probability space. Via the Girsanov’s
theorem we construct a family of new Brownian motions, obtained as
translations of the original one, and to each of them we associate a Wick
product. This produces a family of Wick products, named gamma-Wick products,
parameterized by the performed translations. We aim to describe this family of
products. We also define a new family of stochastic integrals, which are related
in a natural way to the gamma-Wick products
Computing conditional expectation of multidimensional diffusion processes
No full textWe study multidimensional diffusion processes and give an explicit representation for their conditional
expectation. Starting from the solution formula for one dimensional stochastic differential equations
found in Lanconelli and Proske [8], we compute the conditional expectation of a certain class of
multidimensional diffusions without resorting to the Markov property of the process and therefore
without requiring an explicit expression for the semi group associated to it
Standardizing densities on Gaussian spaces
No full textIn the present note we investigate the problem of standardizing random variables taking values on infinite dimensional Gaussian spaces. In particular, we focus on the transformations induced on densities by the selected standardization procedure. We discover that, under certain conditions, the Wick exponentials are the key ingredients for treating this kind of problems
Characterization Theorems for Differential Operators on White Noise Spaces
Full text linkWe characterize through their action on stochastic exponentials
the class of white noise operators which are derivations with respect to both
the point-wise and Wick products. We define the class of second order differential
operators and second order Wick differential operators and we characterize
the white noise operators belonging to both classes. We find that the
intersection of these two classes, in the first and second order cases, is identified
by a skewness condition on the coefficients of the differential operator.
Our technique relies on simple algebraic properties of commutators and on
the Gaussian structure of our white noise space. Our approach is suitable to
study differential operators of any orde
Prohorov-Type Local Limit Theorems on Abstract Wiener Spaces
No full textWe prove that the density of X1+â ̄+Xn-nE[X1]n, where Xnnâ¥1 is a sequence of independent and identically distributed random variables taking values on a abstract Wiener space, converges in L1to 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 Xnnâ¥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 Lanconelli and Stan (Bernoulli 22:2101â2112, 2016) 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 WienerâItà ́ chaos decomposition, OrnsteinâUhlenbeck semigroup and Wick product. We also verify and discuss our main assumptions on some examples arising from the applications: dimension-independent BerryâEsseen-type bounds and weak solutions of stochastic differential equations
A comparison theorem for stochastic differential equations under the Novikov condition
No full textWe consider a system of stochastic differential equations driven by a standard
n-dimensional Brownian motion where the drift function b is bounded and the diffusion
coefficient is the identity matrix.We define via a duality relation a vector Z (which depends
on b) of square integrable stochastic processes which is shown to coincide with the unique
strong solution of the previously mentioned equation. We show that the process Z is well
defined independently of the boundedness of b and that it makes sense under the more general
Novikov condition, which is known to guarantee only the existence of a weak solution.
We then prove that under this mild assumption the process Z solves in the strong sense a
related stochastic differential inequality. This fact together with an additional assumption
will provide a comparison result similar to well known theorems obtained in the presence
of strong solutions. Our framework is also suitable to treat path-dependent stochastic
differential equations and an application to the famous Tsirelson equation is presented
The Ornstein-Uhlenbeck process and a related Malliavin calculus
No full textWe consider two different Brownian motions, B and Ba; each of
them produces a Wiener-Itˆo chaos representation and therefore defines a
Malliavin derivative, D and Da, and a Skorohod integral, and a, respectively.
Our aim is to rewrite the differential operators Da and a in terms of
D and
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