Itô’s stochastic calculus and Heisenberg commutation relations

Stochastic calculus and stochastic differential equations for Brownian motion were introduced by K. Itô in order to give a pathwise construction of diffusion processes. This calculus has deep connections with objects such as the Fock space and the Heisenberg canonical commutation relations, which have a central role in quantum physics. We review these connections, and give a brief introduction to the noncommutative extension of Itô’s stochastic integration due to Hudson and Parthasarathy. Then we apply this scheme to show how finite Markov chains can be constructed by solving stochastic differential equations, similar to diffusion equations, on the Fock space.     

Infinite-Dimensional Analysis and Analytic Number Theory

We consider arithmetical aspects of analysis on Fock spaces (Boson Fock space, Fermion Fock space, and Boson–Fermion Fock space) with applications to analytic number theory.     

Dixmier trace and the Fock space

We give criteria for products of Toeplitz and Hankel operators on the Fock (Segal–Bargmann) space to belong to the Dixmier class, and compute their Dixmier trace. Along the road, analogous results for the Weyl pseudodifferential operators are also obtained.     

On the Fock space of metaanalytic functions

We develop the theory on the Fock space of metaanalytic functions, a generalization of some recent results on the Fock space of polyanalytic functions. We show that the metaanalytic Bargmann transform is a unitary mapping between vector-valued Hilbert spaces and metaanalytic Fock spaces. A reproducing kernel of the metaanalytic Fock space is derived in an explicit form. Furthermore, we establish a complete characterization of all lattice sampling and interpolating sequence for the Fock space of metaanalytic functions.     

On the spectral analysis of quantum field Hamiltonians

We define C^∗-algebras on a Fock space such that the Hamiltonians of quantum field models with positive mass are affiliated to them. We describe the quotient of such algebras with respect to the ideal of compact operators and deduce consequences in the spectral theory of these Hamiltonians: we compute their essential spectrum and give a systematic procedure for proving the Mourre estimate.     

Variationally trivial Lagrangians and locally variational differential equations of arbitrary order

We continue our investigation of the Lagrangian formalism on jet bundle extensions using Fock space methods. We are able to provide the most general form of a variationally trivial Lagrangian of arbitrary order and we also give a generic expression for the most general locally variational differential equation. As anticipated in the literature, these expressions involve some special combinations of the highest order derivatives, called hyper-Jacobians.     

Brownian bridge in quantum probability

The Classical Brownian Bridge is constructed in Symmetric Fock space over an appropriate base Hilbert space. While the representation of the classical Ito-Wiener integral with respect to the increments of the Brownian bridge implements the unitary isomorphism between the Fock space and the (classical) L² space of the Brownian bridge (as is the case with the standard Brownian motion (SBM)), the quantum Ito-integrals with respect to the associated creation and annihilation bridge processes give different left-and right-integrals. This essentially displays the feature that the Brownian Bridge is not a process of independent increments.     

Characterization of Weyl operator in terms of Mehler–Fock transform

In this paper, we define the windowed-Mehler–Fock transform and introduce the corresponding Weyl transform. Further, we examine the boundedness of windowed-Mehler–Fock transform in Lebesgue space and establish some of its fundamental properties. Also, we give the criteria of boundedness and compactness of Weyl transform in Lebesgue space.     

Fock空间上径向函数诱导的Toeplitz算子

本文讨论了Fock空间上以径向函数和拟齐次函数为符号的Toeplitz算子的代数性质,给出了两个以径向函数为符号的Toeplitz算子的积仍为Toeplitz算子的充分必要条件,并且研究了以拟齐次函数为符号的Toeplitz算子的交换性.     

Hilbert Module Realization of the Square of White Noise and Finite Difference Algebras

We develop an approach to the representations theory of the algebra of the square of white noise based on the construction of Hilbert modules. We find the unique Fock representation and show that the representation space is the usual symmetric Fock space. Although we started with one degree of freedom we end up with countably many degrees of freedom. Surprisingly, our representation turns out to have a close relation to Feinsilver's finite difference algebra. In fact, there exists a holomorphic image of the finite difference algebra in the algebra of square of white noise. Our representation restricted to this image is the Boukas representation on the finite difference Fock space. Thus we extend the Boukas representation to a bigger algebra, which is generated by creators, annihilators, and number operators.     

A unified view of some vertex operator constructions

We present a general vertex operator construction based on the Fock space for affine Lie algebras of typeA. This construction allows us to give a unified treatment for both the homogeneous and principle realizations of the affine Lie algebras as well as for some extended affine Lie algebras coordinatized by certain quantum tori.     

Second order freeness and fluctuations of random matrices: I. Gaussian and Wishart matrices and cyclic Fock spaces

We extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations. We introduce the concept of “second order freeness” and interpret the global fluctuations of Gaussian and Wishart random matrices by a general limit theorem for second order freeness. By introducing cyclic Fock space, we also give an operator algebraic model for the fluctuations of our random matrices in terms of the usual creation, annihilation, and preservation operators. We show that orthogonal families of Gaussian and Wishart random matrices are asymptotically free of second order.     

Martingale Representation of Functionals of Lévy Processes

Abstract

The main focus of the paper is a Clark–Ocone–Haussman formula for Lévy processes. First a difference operator is defined via the Fock space representation of L ²(P), then from this definition a Clark–Ocone–Haussman type formula is derived. We also derive some explicit chaos expansions for some common functionals. Later we prove that the difference operator defined via the Fock space representation and the difference operator defined by Picard [Picard, J. Formules de dualitésur l'espace de Poisson. Ann. Inst. Henri Poincaré 1996, 32 (4), 509–548] are equal. Finally, we give an example of how the Clark–Ocone–Haussman formula can be used to solve a hedging problem in a financial market modelled by a Lévy process.     

Fock空间上的乘法算子

Fock空间是由整函数组成的具有再生核的Hilbert空间.Fock空间上的乘法算子的定义域不是整个Fock空间,它在Fock空间上是稠定的.研究了Fock空间上的乘法算子的性质,对其值域进行了刻画,并得出了乘法算子作用在Fock空间的拟不变子空间上值域余一维的充要条件.     

A Note on Invertible Weighted Composition Operators on the Fock Space of C~N

This paper shows that a bounded invertible weighted composition operator on the Fock space of C~N is nonzero multiples of a unitary operator, which is an addition to the recent result on invertible weighted composition operator on the Fock space of C~N and an extension to the corresponding result on the Fock space of C.     

Stochastic integral of Hitsuda–Skorokhod type on the extended Fock space

We review some recent results related to stochastic integrals of the Hitsuda–Skorokhod type acting on the extended Fock space and its riggings.     

广义Segal-Bargmann空间上无界Toeplitz算子的交换

本文给出了复平面C上广义Fock空间中两个Toeplitz算子T_u和T_v的性质.假设u是一个径向函数,两算子是可交换的.在一定的增长条件之下,我们证明出u也是一个径向函数.最后还构造了一个具有本性无界符号的S_p紧,Toeplitz算子.     

Weighted composition operator on the Fock space

We characterize the boundedness and compactness of a weighted composition operator on the Fock space. Our results use a certain integral transform. We also estimate the essential norm of a weighted compositon operator. The result could be extended to the higher-dimensional case.