Alberto Lanconelli

On explicit strong solution of Itô-SDE's and the Donsker delta function of a diffusion.

We determine a new explicit representation of strong solutions of Ito-diffusions and elicit its correspondence to the general stochastic transport equation. We apply this formula to deduce an explicit Donsker delta function of a diffusion.

A HÖLDER–YOUNG–LIEB INEQUALITY FOR NORMS OF GAUSSIAN WICK PRODUCTS

An 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 Holder inequality for the norms of Gaussian Wick products, reprove Nelson hypercontractivity inequality, and prove a more general inequality whose marginal cases are the Holder and Nelson inequalities mentioned before. We will show that there is a deep connection between the Gaussian Holder inequality and classic Holder 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 Fatous lemma argument, to the infinite-dimensional framework.

Hölder-Type Inequalities for Norms of Wick Products

Various upper bounds for the 𝐿2-norm of the Wick product of two measurable functions of a random variable 𝑋, having finite moments of any order, together with a universal minimal condition, are proven. The inequalities involve the second quantization operator of a constant times the identity operator. Some conditions ensuring that the constants involved in the second quantization operators are optimal, and interesting examples satisfying these conditions are also included.

Some Norm Inequalities for Gaussian Wick Products

We provide several inequalities for the ℒ q (𝒫)-norm of the Wick product of random variables. These estimates are based on a Jensens type inequality for the Wick multiplication, which we derive via a positivity argument. As an application we study a certain type of anticipating stochastic differential equation whose solution is shown to be an element of ℒ q (𝒫) for some q ≥ 1.

Quantum white noise convolution operators with application to differential equations

Abstract In this paper we introduce a quantum white noise (QWN) convolution calculus over a nuclear algebra of operators. We use this calculus to discuss new solutions of some linear and non-linear differential equations.

WICK CALCULUS FOR THE SQUARE OF A GAUSSIAN RANDOM VARIABLE WITH APPLICATION TO YOUNG AND HYPERCONTRACTIVE INEQUALITIES

We investigate a probabilistic interpretation of the Wick product associated to the chi-square distribution in the spirit of the results obtained in Ref. 7 for the Gaussian measure. Our main theorem points out a profound difference from the previously studied Gaussian7 and Poissonian12 cases. As an application, we obtain a Young-type inequality for the Wick product associated to the chi-square distribution which contains as a particular case a known Nelson-type hypercontractivity theorem.

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

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.

On a new probabilistic representation for the solution of the heat equation

We obtain a new probabilistic representation for the solution of the heat equation in terms of a product for smooth random variables which is introduced and studied in this paper. This multiplication, expressed in terms of the Hida–Malliavin derivatives of the random variables involved, exhibits many useful properties that are to some extents opposite to some peculiar features of the Wick product.

Translated Brownian Motions and Associated Wick Products

Abstract The concept of Wick product is strongly related to the underlying Brownian motion we have fixed on the probability space. Via the Girsanovs 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 γ-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 γ-Wick products.

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

We prove a new type of Poincare 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 Poincare 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 Houdre and Kagan [11] and Houdre and Perez-Abreu [12] Poincare-type inequalities.