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Based on the quantum white noise theory, we formulate new types of anticipating quantum stochastic integrals by combining the Hitsuda–Skorokhod quantum stochastic integrals and the interactions between the integrands and the integrators. For our purpose, we prove various versions of analytic characterization theorems of symbols of white noise operators as more general cases. Also, several types of quantum (stochastic) gradients are formulated as continuous linear operators which are related to the quantum white noise derivatives and reflecting the interactions with the external noises. Finally, we formulate further systematical study of anticipating quantum stochastic integrals. [ABSTRACT FROM AUTHOR]

We investigate the convergence of higher order Cesàro means in Banach spaces. The main results of this paper are: (1) The proof of mean and Birkhoff-type ergodic theorems for higher order Cesàro means. (2) The existence of a one-to-one correspondence between convergent Cesàro means of different orders. (3) The proof of strong laws of large numbers for higher order sums of independent and identically distributed random elements. (4) A characterization of the ergodicity of measure preserving maps in terms of higher order mixing properties. To deal with higher order Cesàro means, one needs a notion of weighted mean more general than the one usually considered in the literature on weighted ergodic theorems. In this context, we also prove a characterization of generalized weighted means preserving Cesàro convergence. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new white noise delta function based on the Kubo–Yokoi delta function and an infinite-dimensional Brownian motion. We also give a white noise differential equation induced by the delta function through the Itô formula introducing a differential operator directed by the time derivative of the infinite-dimensional Brownian motion and an extension of the definition of the Volterra Laplacian. Moreover, we give an extension of the Itô formula for the white noise distribution of the infinite-dimensional Brownian motion. [ABSTRACT FROM AUTHOR]

We study the analytic characterization of S -transform in a general setting of white noise functionals. Then, the measurability condition of the norms generating the underlining locally convex space is a necessary and sufficient condition for the analytic characterization of the S -transform in terms of analytic and growth conditions. [ABSTRACT FROM AUTHOR]

This paper discusses the properties of the canonical quantum decomposition of the classical Pearson random variables. We show that this leads to the problem of representing the creation–annihilation–preservation (CAP) operators canonically associated to a real-valued random variable X with all moments as (normally ordered) differential operators with polynomial coefficients — a problem already studied in the literature (see references in the introduction). We deduce a formula, for the polynomial coefficients in the representation of the CAP operators of X as pseudo-differential operators, more explicit than the one existing in the literature. We give a new characterization of the Pearson distributions in terms of the Hermitianity of the associated Sturm–Liouville operators. In the second part of the paper, we introduce the notion of finite type random variable X and characterize type-2 and type-3 real-valued random variables. We prove that a necessary condition for X to be of finite type is the polynomial growth of the corresponding principal Jacobi sequence. This allows to single out three classes of random variables of infinite type and to prove that the Beta and the uniform distributions are of infinite type. [ABSTRACT FROM AUTHOR]

We revisit the analytic characterization theorem for S-transform of infinite dimensional distributions. Then we prove that the nuclearity of the space of test white noise functionals is a necessary condition for the characterization of the S-transform in terms of analytic and growth conditions. [ABSTRACT FROM AUTHOR]

We study exponential convergence rates for weighted sums of successive independent random variables in a noncommutative probability space of which the weights are in a von Neumann algebra. Then we prove a noncommutative extension of the result for the exponential convergence rate by Baum, Katz and Read. As applications, we first study a large deviation type inequality for weighted sums in a noncommutative probability space, and secondly we study exponential convergence rates for weighted free additive convolution sums of probability measures. [ABSTRACT FROM AUTHOR]

The Cesàro theorem is extended to the cases: (1) higher order Cesàro mean for sequence (discrete case); and (2) higher order, multi-dimensional and continuous Cesàro mean for functions. Also, we study the Cesàro theorem for the case of positive-order. [ABSTRACT FROM AUTHOR]

We introduce, for each a ... R+, the Brownian motion associated to the distribution derivative of order a of white noise. We prove that the generator of this Markov process is the exotic Laplacian of order 2a, given by the Cesàro mean of order 2a of the sec-ond derivatives along the elements of an orthonormal basis of a suitable Hilbert space (the Cesàro space of order 2a). In particular, for a = 1/2 one finds the usual Lévy Laplacian, but also in this case the connection with the 1/2-derivative of white noise is new. The main technical tool, used to achieve these goals, is a generalization of a result due to Accardi and Smolyanov5 extending the well-known Cesàro theorem to higher order arithmetic means. These and other estimates allow to prove existence of the heat semi-group associated to any exotic Laplacian of order ≥1/2 and to give its explicit expression in terms of infinite dimensional Fourier transform. [ABSTRACT FROM AUTHOR]

In this paper, we give a relationship between the exotic Laplacians and the Lévy Laplacians in terms of the higher-order derivatives of white noise by introducing a bijective and continuous linear operator acting on white noise functionals. Moreover, we study a relationship between exotic Laplacians, acting on higher-order singular functionals, each other in terms of the constructed operator. [ABSTRACT FROM AUTHOR]

As a general study for uniqueness of integrands in quantum martingale representation, we present a necessary and sufficient condition for uniqueness of integrands in a quantum stochastic integral. Also, several equivalent conditions to the necessary and sufficient condition are studied. [ABSTRACT FROM AUTHOR]

We revisit a CKS-space from the viewpoint of standard setup of white noise calculus and prove a general characterization theorem for white noise operators from a CKS-space into another CKS-space. The usual characterizations so far obtained for white noise distributions, white noise test functions and white noise operators are all consequences of the unified characterization theorem. [ABSTRACT FROM AUTHOR]

Duality is established for new spaces of entire functions in two infinite dimensional variables with certain growth rates determined by Young functions. These entire functions characterize the symbols of generalized Fock space operators. As an application, a proper space is found for a solution to a normal-ordered white noise differential equation having highly singular coefficients. [ABSTRACT FROM AUTHOR]

In this paper we shall discuss the existence and the uniqueness of solutions of the heat type equation and the wave type equation associated with the Lévy Laplacian acting on a domain in the space of generalized white noise functionals. [ABSTRACT FROM AUTHOR]

No abstract received. [ABSTRACT FROM AUTHOR]