Infinite Dimensional Analysis, Quantum Probability and Related Topics

We show $L^r({ℝ}^d,\mu )$-uniqueness for any $r\in (1,2]$ and the essential self-adjointness of a Dirichlet operator $Lf=\mathrm{\Delta }f+〈\frac{1}{\rho }\nabla \rho ,\nabla f〉$, $f\in C_0^{\infty }({ℝ}^d)$ with $d\ge 3$ and $\mu =\rho dx$. In particular, $\nabla \rho$ is allowed to be in $L_{\mathrm{\text{loc}}}^d({ℝ}^d,{ℝ}^d)$ or in $L_{\mathrm{\text{loc}}}^{2+𝜀}({ℝ}^d,{ℝ}^d)$ for some $𝜀>0$, while $\rho$ is required to be locally bounded below and above by strictly positive constants. The main tools in this paper are elliptic regularity results for divergence and non-divergence type operators and basic properties of Dirichlet forms and their resolvents.

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.

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.

In the present paper, we introduce stopping rules and related notions for quantum Markov chains on trees (QMCT). We prove criteria for recurrence, accessibility and irreducibility for QMCT. This work extends to trees the notion of stopping times for quantum Markov chains (QMC) introduced by Accardi and Koroliuk, which plays a key role in the study of many properties of QMC. Moreover, we illustrate the obtained results for a concrete model of XY-Ising type.

Let $K$ be an algebraic number field. With $K$ we associate the ring of finite adeles ${𝔸}_K.$ In this paper we give a path integral formula for the propagator of a quantum mechanical system over the abelian group ${𝔸}_K^n.$ Specifically, we consider matrix-valued Hamiltonian operators $H_{{𝔸}_K^n}={\mathrm{\Delta }}_{{𝔸}_K^n}\otimes \mathrm{\text{Id}}+V,$ where ${\mathrm{\Delta }}_{{𝔸}_K^n}$ is the Vladimirov operator and $V$ is a non-negative definite potential. The free part of the Hamiltonian gives rise to a measure on the Skorokhod space of paths which allows us to prove the Feynman–Kac formula for the Schrödinger semigroup generated by $-H_{{𝔸}_K^n}.$ This formula is given in terms of the ordered time exponentials.

We set up a general theory leading to a quantum Wasserstein distance of order 1 between channels in an operator algebraic framework. This gives a metric on the set of channels from one composite system to another, which is deeply connected to reductions of the channels. The additivity and stability properties of this metric are studied.