Infinite Dimensional Analysis, Quantum Probability and Related Topics

Let X be a Markov process, which is assumed to be associated with a symmetric Dirichlet form . For , the extended Dirichlet space, we have the classical Fukushima's decomposition: , where is a quasi-continuous version of u, the martingale part and the zero energy part. In this paper, we investigate two important transformations for X, the Girsanov transform induced by and the generalized Feynman–Kac transform induced by . For the Girsanov transform, we present necessary and sufficient conditions for which to induce a positive supermartingale and hence to determine another Markov process . Moreover, we characterize the symmetric Dirichlet form associated with the Girsanov transformed process . For the generalized Feynman–Kac transform, we give a necessary and sufficient condition for the generalized Feynman–Kac semigroup to be strongly continuous.

We introduce the notion of Markov states and chains on the Canonical Anticommutation Relations algebra over ℤ, emphasizing some remarkable differences with the infinite tensor product case. We describe the structure of the Markov states on this algebra and show that, contrarily to the infinite tensor product case, not all these states are diagonalizable. A general method to construct nontrivial quantum Markov chains on the CAR algebra is also proposed and illustrated by some pivotal examples. This analysis provides a further step for a satisfactory theory of quantum Markov processes.

We construct a new equilibrium dynamics of infinite particle systems in a Riemannian manifold X. This dynamics is an analog of the Kawasaki dynamics of lattice spin systems. The Kawasaki dynamics now is a process where interacting particles randomly hop over X. We establish conditions on the a priori explicitly given symmetrizing measure and the generator of this dynamics, under which a corresponding conservative Markov processes exists. We also outline two types of scaling limit of the equilibrium Kawasaki dynamics: one leading to an equilibrium Glauber dynamics in continuum (a birth-and-death process), and the other leading to a diffusion dynamics of interacting particles (in particular, the gradient stochastic dynamics).

In this paper we give a necessary and sufficient condition on the interacting Fock (IFF) space by which the vacuum distribution of the position operator is rotation invariant.

We develop linear stochastic Schrödinger equations driven by standard cylindrical Brownian motions (LSSs) that unravel quantum master equations in Lindblad form into quantum trajectories. More precisely, this paper establishes the existence and uniqueness of the smooth strong solution X_t to a LSS with regular initial condition. Moreover, we obtain that the mean value of the square norm of X_t is constant. We also treat the approximation of LSSs by ordinary stochastic differential equations. We apply our results to:

(i) models of quantum measurements of position and momentum; and

(ii) a system formed by fermions.

In this paper we present a construction of an infinite dimensional separable Hilbert space associated with a norm induced from the Lévy trace. The space is slightly different from the Cesàro Hilbert space introduced in Ref. 1. The Lévy Laplacian is discussed with a suitable domain which is constructed by a rigging of Fock spaces based on a rigging of Hilbert spaces with the Lévy trace. Then the Lévy Laplacian can be considered as the Gross Laplacian acting on a certain countable Hilbert space. By constructing one-parameter group of operators of which the infinitesimal generator is the Lévy Laplacian, we study the existence and uniqueness of solution of heat equation associated with the Lévy Laplacian. Moreover we give an infinite dimensional stochastic process generated by the Lévy Laplacian.

Bi-Poisson processes are defined as a two-parameter family of processes with linear regressions and linear conditional variances. We give conditions for the existence and uniqueness of such processes. We point out that one-dimensional distributions of the bi-Poisson process are closed under a generalized free convolution introduced by Bożejko and Speicher.⁷.

We present in this work a specific construction of raising and lowering operators for 2-orthogonal quasi-monomial polynomials associated with continuous and discrete natural exponential families. We use these operators in order to characterize the real class of cubic natural exponential families.