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White noise theory allows to formulate quantum white noises explicitly as elemental quantum stochastic processes. A traditional quantum stochastic differential equation of Itô type is brought into a normal-ordered white noise differential equation driven by lower powers of quantum white noises. The class of normal-ordered white noise differential equations covers quantum stochastic differential equations with highly singular noises such as higher powers or higher order derivatives of quantum white noises, which are far beyond the traditional Itô theory. For a general normal-ordered white noise differential equation unique existence of a solution is proved in the sense of white noise distribution. Its regularity properties are investigated by means of weighted Fock spaces interpolating spaces of white noise distributions and associated characterization theorems for S-transform and for operator symbols.

This article is concerned with compositions in the context of three standard quantizations in the framework of Fock spaces, namely, anti-Wick, Wick and Weyl quantizations. The first one is a composition of states also known as a Wick product and is closely related to the standard scattering identification operator encountered in Quantum Electrodynamics for issues on time dynamics (see [29, 13]). Anti-Wick quantization and Segal–Bargmann transforms are implied here for that purpose. The other compositions are for observables (operators in some specific classes) for the Wick and Weyl symbols. For the Wick and Weyl symbols of the composition of two operators, we obtain an absolutely converging series and for the Weyl symbol, the remainder terms up to any orders of the expansion are controlled, still in the Fock space framework.

For the canonical commutation relations in infinite dimensions, we offer an explicit direct construction of Weyl representations W_ϕ generated from the Fock representation by any ϕ ∈ L²(E*, μ, R) over the Q-space (E*, μ). Moreover, we obtain that, for any ϕ, ψ ∈ L²(E*, μ,R), W_ϕ+ψ and W_ϕ are unitarily equivalent, proving a conjecture posed by Robinson in Ref. 2. Our construction employs Wiener–Itô decomposition of the space L²(E*, μ, R) (respectively L²(E*, μ, C)).

We introduce the forward integral with respect to a pure jump Lévy process and prove an Itô formula for this integral. Then we use Mallivin calculus to establish a relationship between the forward integral and the Skorohod integral and apply this to obtain an Itô formula for the Skorohod integral.

Let be a Fock space and, for any non-negative integer k, let be the sum of all homogeneous chaos spaces of order at most k. For all non-negative integers m and n, the Wick product is a bounded bilinear operator from Γ (H_c)_m × Γ (H_c)_n into Γ (H_c)_{m +n} with norm greater than or equal to . In the monograph¹ S. Janson conjectured that this lower bound is exact. In this paper we prove this conjecture. In addition, we prove that a pair of nonzero vectors (ϕ, ψ)∈ Γ (H_c)_m × Γ (H_c)_n achieve this bound if and only if both vectors are multiples of the homogeneous products, i.e. ϕ =αu^⊗m, ψ =βu^⊗n, with u, v∈ H_c and α, β ∈ ℂ.

A new Gel'fand triple exp(S)₁ ⊆ (L)² ⊆ exp(S)_-1 is constructed as extension of the known Kondratiev one (S)₁ ⊆ (L)² ⊆ (S)_-1. Expansion theorems for generalized stochastic processes considered as elements of the spaces and are derived. This series expansion is used for solving a class of evolution stochastic differential equations. The Wick product is developed on the spaces exp(S)_-1, and . The series expansion of generalized stochastic processes is used for solving a class of nonlinear stochastic differential equations by means of Wick products.

We consider the forward integral with respect to fractional Brownian motion B^(H)(t) and relate this to the Wick–Itô–Skorohod integral by using the M-operator introduced by Ref. 10 and the Malliavin derivative . Using this connection we obtain a general Itô formula for the Wick–Itô–Skorohod integralswith respect to B^(H)(t), valid for .

It is known that if X is a normally distributed random variable, and ♢ and E denote the Wick product and expectation, respectively, then for any non-negative integers m and n, and any polynomial functions f and g of degrees at most m and n, respectively, the following inequality holds:

In this work we introduce a new algebra of stochastic generalized functions. The regular Hida distributions in are embedded in this algebra via their chaos expansions. As an application, we prove the existence and uniqueness of the solution of a stochastic Cauchy problem involving singularities.

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 Hölder inequality for the norms of Gaussian Wick products, reprove Nelson hypercontractivity inequality, and prove a more general inequality whose marginal cases are the Hölder and Nelson inequalities mentioned before. We will show that there is a deep connection between the Gaussian Hölder inequality and classic Hölder 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.

We derive some characteristic properties of the convolution operator acting on white noise functions and prove that the convolution product of white noise distributions coincides with their Wick product. Moreover, we show that the S-transform and the Laplace transform coincide on Fock space.

Motivated by the Schwartz space of tempered distributions and the Kondratiev space of stochastic distributions we define a wide family of nuclear spaces which are increasing unions of (duals of) Hilbert spaces , with decreasing norms ‖⋅‖_p. The elements of these spaces are functions on a free commutative monoid. We characterize those rings in this family which satisfy an inequality of the form ‖f * g‖_p ≤ A(p - q)‖f‖_q‖g‖_p for all p ≥ q + d, where _* denotes the convolution in the monoid, A(p - q) is a strictly positive number and d is a fixed natural number (in this case we obtain commutative topological ℂ-algebras). Such an inequality holds in , but not in . We give an example of such a ring which contains . We characterize invertible elements in these rings and present applications to linear system theory.

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.

We investigate the properties of the Wick square of Gaussian white noises through a new method to perform nonlinear operations on Hida distributions. This method lays in between the Wick product interpretation and the usual definition of nonlinear functions. We prove an Itô-type formula and solve stochastic differential equations driven by the renormalized square of the Gaussian white noise. Our approach works with standard assumptions on the coefficients of the equations, global Lipschitz continuity, and produces existence and uniqueness results in the space where the noise lives. The linear case is studied in details and positivity of the solution is proved.

An integral representation of the Wick product for Gamma distributed random variables, with mean greater than $1/2$, is presented first. We use this integral representation to prove a Hölder inequality for norms of Gamma Wick products.

In this paper, we unify techniques of Pascal white noise analysis and harmonic analysis on configuration spaces establishing relations between the main structures of both ones. Fix a Random measure $\sigma$ on a Riemannian manifold $X$, we construct on the space of finite compound configuration space ${\mathrm{\Omega }}_0$ the so-called Lebesgue–Pascal measure ${\lambda }_{\widehat{\sigma }}$ and as a consequence we obtain the Pascal measure ${\mu }_{\widehat{\sigma }}$ on the compound configuration space $\mathrm{\Omega }$. Next, the natural realization of the symmetric Fock space over $L^2(\widehat{\sigma })$ as the space $L^2({\lambda }_{\widehat{\sigma }})$ leads to the unitary isomorphism ${{𝒥}}_{\lambda \mu }$ between the space $L^2({\lambda }_{\widehat{\sigma }})$ and $L^2({\mu }_{\widehat{\sigma }})$. Finally, in the first application we study some algebraic products, namely, the Borchers product on the Fock space, the Wick product on the Pascal space, and the ⋆-convolution on the Lebesgue–Pascal space and we prove that the Pascal white noise analysis and harmonic analysis are related through an equality of operators involving ${{𝒥}}_{\lambda \mu }$. The second application is devoted to solve the implementation problem.

Based on the distributions space on ${{𝒮}}_{ℂ}^{\prime }$ (denoted by ${{ℱ}}_{𝜃}^{\ast }({{{𝒮}}^{\prime }}_{ℂ})$) which is the topological dual space of the space of entire functions with exponential growth of order $𝜃$ and of minimal type, we introduce a new type of differential equations using the Wick derivation operator and the Wick product of elements in ${{ℱ}}_{𝜃}^{\ast }({{{𝒮}}^{\prime }}_{ℂ})$. These equations are called generalized Bernoulli Wick differential equations which are the analogue of the classical Bernoulli differential equations. We solve these generalized Wick differential equations. The present method is exemplified by several examples.

We consider the (unique) mild solution $u(t,x)$ of a one-dimensional stochastic heat equation on $[0,T]\times ℝ$ driven by time-homogeneous white noise in the Wick–Skorokhod sense. The main result of this paper is the computation of the spatial derivative of $u(t,x)$, denoted by ${\partial }_{x}u(t,x)$, and its representation as a Feynman–Kac type closed form. The chaos expansion of ${\partial }_{x}u(t,x)$ makes it possible to find its (optimal) Hölder regularity especially in space.

This paper identifies certain interesting mathematical problems of stochastic quantization type in the modeling of Laser propagation through turbulent media. In some of the typical physical contexts, the problem reduces to stochastic Schrödinger equation with space–time white noise of Gaussian or Poisson or Lévy type. We identify their mathematical resolution via stochastic quantization. Nonlinear phenomena such as Kerr effect can be modeled by a stochastic nonlinear Schrödinger equation in the focusing case with space–time white noise. A treatment of stochastic transport equation, the Korteweg–De Vries equation as well as a number of other nonlinear wave equations with space–time white noise is also given. The main technique is the S-transform (we will actually use the closely related Hermite transform) which converts the stochastic partial differential equation (PDE) with space–time white noise to a deterministic PDE defined on the Hida–Kondratiev white noise distribution space. We then utilize the inverse S-transform/Hermite transform known as the characterization theorem combined with the infinite-dimensional implicit function theorem for analytic maps to establish local existence and uniqueness theorems for path-wise solutions of this class of problems. The particular focus of this paper on singular white noise distributions is motivated by practical situations where the refractive index fluctuations in the propagation medium in space and time are intense due to turbulence, ionospheric plasma turbulence, marine-layer fluctuations, etc. Since a large class of PDEs, that arise in nonlinear wave propagation, have polynomial-type nonlinearities, white noise distribution theory is an effective tool in studying these problems subject to different types of white noises.

In a recent paper, Lanconelli and Scorolli¹⁰ extended to the multidimensional case a Wong–Zakai-type approximation for Itô stochastic differential equations (SDEs) proposed by Øksendal and Hu⁷. The aim of this paper is to extend the latter result to system of SDEs of Itô type driven by fractional Brownian motion (fBm) like those considered by Hu⁶. This extension is not trivial since the covariance structure of the fBm precludes us from using the same approach as that used by Lanconelli and Scorolli. Instead we employ a truncated Cameron–Martin expansion as the approximation for the fBm. We are naturally led to the investigation of a semilinear hyperbolic system of evolution equations in several space variables that we utilize for constructing a solution of the Wong–Zakai approximated systems. We show that the law of each element of the approximating sequence solves in the sense of distribution a Fokker–Planck equation and that the sequence converges to the solution of the Itô equation, as the number of terms in the expansion goes to infinite.