arXiv subjects

Caishi Wang

Publications and source records attributed to Caishi Wang.

8 recordsLinked to original sources

arXiv2023

Stationary Measures of Space-Inhomogeneous Three-State Quantum Walks on Line: Revisited

Of a quantum walk, its stationary measures play an important role in understanding its evolution behavior. In this paper we investigate stationary measures of two models of space-inhomogeneous three-state quantum walk on the line. By using the method of reduced matrix, we find out stationary measures of the two models under some mild conditions. Our results generalize the corresponding ones existing in the literature.

quant-ph↗

arXiv2022

Generalized weighted number operators on functionals of discrete-time normal martingales

Let $M$ be a discrete-time normal martingale that has the chaotic representation property. Then, from the space of square integrable functionals of $M$, one can construct generalized functionals of $M$. In this paper, by using a type of weights, we introduce a class of continuous linear operators acting on generalized functionals of $M$, which we call generalized weighted number (GWN) operators. We prove that GWN operators can be represented in terms of generalized annihilation and creation operators (acting on generalized functionals of $M$). We also examine commutation relations between a GWN operator and a generalized annihilation (or creation) operator, and obtain several formulas expressing such commutation relations.

math.PR↗

arXiv2021

Spectral Integrals of Bernoulli Generalized Functionals

Let $\mathcal{S}\subset \mathcal{L}^2 \subset \mathcal{S}^*$ be the Gel'fand triple over the Bernoulli space, where elements of $\mathcal{S}^*$ are called Bernoulli generalized functionals. In this paper, we define integrals of Bernoulli generalized functionals with respect to a spectral measure (projection operator-valued measure) in the framework of $\mathcal{S}\subset \mathcal{L}^2 \subset \mathcal{S}^*$, and examine their fundamental properties. New notions are introduced, several results are obtained and examples are also shown.

math.FA↗

arXiv2019

Weighted Number Operators on Bernoulli Functionals and Quantum Exclusion Semigroups

Quantum Bernoulli noises (QBN, for short) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation (CAR) in equal-time. In this paper, by using QBN, we first introduce a class of self-adjoint operators acting on Bernoulli functionals, which we call the weighted number operators. We then make clear spectral decompositions of these operators, and establish their commutation relations with the annihilation as well as the creation operators. We also obtain a necessary and sufficient condition for a weighted number operator to be bounded. Finally, as application of the above results, we construct a class of quantum Markov semigroups associated with the weighted number operators, which belong to the category of quantum exclusion semigroups. Some basic properties are shown of these quantum Markov semigroups, and examples are also given.

math.FA↗

arXiv2019

Clark-Ocone Formula for Generalized Functionals of Discrete-Time Normal Noises

The Clark-Ocone formula in the theory of discrete-time chaotic calculus holds only for square integrable functionals of discrete-time normal noises. In this paper, we aim at extending this formula to generalized functionals of discrete-time normal noises. Let $Z$ be a discrete-time normal noise that has the chaotic representation property. We first prove a result concerning the regularity of generalized functionals of $Z$. Then, we use the Fock transform to define some fundamental operators on generalized functionals of $Z$, and apply the above mentioned regularity result to prove the continuity of these operators. Finally, we establish the Clark-Ocone formula for generalized functionals of $Z$, and show its application results, which include the covariant identity result and the variant upper bound result for generalized functionals of $Z$.

math.PR↗

arXiv2017

Dirichlet Forms Constructed from Annihilation Operators on Bernoulli Functionals

The annihilation operators on Bernoulli functionals (Bernoulli annihilators, for short) and their adjoint operators satisfy a canonical anti-commutation relation (CAR) in equal-time. As a mathematical structure, Dirichlet forms play an important role in many fields in mathematical physics. In this paper, we apply the Bernoulli annihilators to constructing Dirichlet forms on Bernoulli functionals. Let $w$ be a nonnegative function on $\mathbb{N}$. By using the Bernoulli annihilators, we first define in a dense subspace of the $L^2$-space of Bernoulli functionals a positive, symmetric bilinear form $\mathcal{E}_w$ associated with $w$. And then we prove that $\mathcal{E}_w$ is closed and has the contraction property, hence it is a Dirichlet form. Finally, we consider an interesting semigroup of operators associated with $w$ on the $L^2$-space of Bernoulli functionals, which we call the $w$-Ornstein-Uhlenbeck semigroup, and by using the Dirichlet form $\mathcal{E}_w$ we show that the $w$-Ornstein-Uhlenbeck semigroup is a Markov semigroup.

math.PR↗

arXiv2015

Convergence Theorems for Generalized Functional Sequences of Discrete-Time Normal Martingales

The Fock transform recently introduced by the authors in a previous paper is applied to investigate convergence of generalized functional sequences of a discrete-time normal martingale $M$. A necessary and sufficient condition in terms of the Fock transform is obtained for such a sequence to be strong convergent. A type of generalized martingales associated with $M$ are introduced and their convergence theorems are established. Some applications are also shown.

math.PR↗

arXiv2015

Characterization Theorems for Generalized Functionals of Discrete-Time Normal Martingale

In this paper, we aim at characterizing generalized functionals of discrete-time normal martingales. Let $M=(M_n)_{n\in \mathbb{N}}$ be a discrete-time normal martingale that has the chaotic representation property. We first construct testing and generalized functionals of $M$ with an appropriate orthonormal basis for $M$'s square integrable functionals. Then we introduce a transform, called the Fock transform, for these functionals and characterize them via the transform. Several characterization theorems are established. Finally we give some applications of these characterization theorems. Our results show that generalized functionals of discrete-time normal martingales can be characterized only by growth condition, which contrasts sharply with the case of some continuous-time processes (e.g., Brownian motion), where both growth condition and analyticity condition are needed to characterize generalized functionals of those continuous-time processes.

math.PR↗