Infinite Dimensional Analysis, Quantum Probability and Related Topics

We study detailed balance and non-equilibrium steady states of a Markov generator of weak coupling limit type, modeling absorption and simultaneous emission of $m$- and $n$-photons, with $m,n\ge 1$. In the case $n=4,m=2$, under natural constraints on the absorption and emission rates, there exist infinitely many non-equilibrium steady states which are convex linear combination of even and odd states.

Functional Itô calculus was introduced in order to expand a functional $F(t,X_{\cdot +t},X_t)$ depending on time $t$, past and present values of the process $X$. Another possibility to expand $F(t,X_{\cdot +t},X_t)$ consists in considering the path $X_{\cdot +t}=\{X_{x+t},x\in [-T,0]\}$ as an element of the Banach space of continuous functions on $C([-T,0])$ and to use Banach space stochastic calculus. The aim of this paper is threefold. (1) To reformulate functional Itô calculus, separating time and past, making use of the regularization procedures which match more naturally the notion of horizontal derivative which is one of the tools of that calculus. (2) To exploit this reformulation in order to discuss the (not obvious) relation between the functional and the Banach space approaches. (3) To study existence and uniqueness of smooth solutions to path-dependent partial differential equations which naturally arise in the study of functional Itô calculus. More precisely, we study a path-dependent equation of Kolmogorov type which is related to the window process of the solution to an Itô stochastic differential equation with path-dependent coefficients. We also study a semilinear version of that equation.

We investigate one-dimensional three-state quantum walks. We find a formula for the moments of the weak limit distribution via a vacuum expectation of powers of a self-adjoint operator. We use this formula to fully characterize the localization of three-state quantum walks in one dimension. The localization is also characterized by investing the eigenvectors of the evolution operator for the quantum walk. As a byproduct we clarify the concepts of localization differently used in the literature. We also study the continuous part of the limit distribution. For typical examples we show that the continuous part is the same kind as that of two-state quantum walks. We provide with explicit expressions for the density of the weak limits of some three-state quantum walks.

We present the expansion of the multifractional Brownian motion (mBm) local time in higher dimensions, in terms of Wick powers of white noises (or multiple Wiener integrals). If a suitable number of kernels is subtracted, they exist in the sense of generalized white noise functionals. Moreover, we show the convergence of the regularized truncated local times for mBm in the sense of Hida distributions.

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.

Semi-classical asymptotic expansions up to any finite orders of the partition function of an abstract Bose field model is considered.

In this paper, we give Hurwitz–Lerch zeta distributions with $0<\sigma \ne 1$ by using the Gamma function. Moreover, we define Hurwitz–Lerch type of Euler–Zagier double zeta distributions not only in the region of absolute convergence but also outside the region of absolute convergence.

This paper presents a closed-form expression for the integral kernels associated with the derivatives of the Ornstein–Uhlenbeck semigroup ${\mathrm{\text{e}}}^{tL}$. Our approach is to expand the Mehler kernel into Hermite polynomials and apply the powers $L^N$ of the Ornstein–Uhlenbeck operator to it, where we exploit the fact that the Hermite polynomials are eigenfunctions for $L$. As an application we give an alternative proof of the kernel estimates by Ref. 10, making all relevant quantities explicit.