Infinite Dimensional Analysis, Quantum Probability and Related Topics

We introduce, for each a ∈ ℝ₊, the Brownian motion associated to the distribution derivative of order a of white noise. We prove that the generator of this Markov process is the exotic Laplacian of order 2a, given by the Cesàro mean of order 2a of the second derivatives along the elements of an orthonormal basis of a suitable Hilbert space (the Cesàro space of order 2a). In particular, for a = 1/2 one finds the usual Lévy Laplacian, but also in this case the connection with the 1/2-derivative of white noise is new. The main technical tool, used to achieve these goals, is a generalization of a result due to Accardi and Smolyanov⁵ extending the well-known Cesàro theorem to higher order arithmetic means. These and other estimates allow to prove existence of the heat semi-group associated to any exotic Laplacian of order ≥ 1/2 and to give its explicit expression in terms of infinite dimensional Fourier transform.

The control landscape for various canonical quantum control problems is considered. For the class of pure-state transfer problems, analysis of the fidelity as a functional over the unitary group reveals no suboptimal attractive critical points (traps). For the actual optimization problem over controls in L²(0, T), however, there are critical points for which the fidelity can assume any value in (0, 1), critical points for which the second order analysis is inconclusive, and traps. For the class of unitary operator optimization problems analysis of the fidelity over the unitary group shows that while there are no traps over U(N), traps already emerge when the domain is restricted to the special unitary group. The traps on the group can be eliminated by modifying the performance index, corresponding to optimization over the projective unitary group. However, again, the set of critical points for the actual optimization problem for controls in L²(0, T) is larger and includes traps, some of which remain traps even when the target time is allowed to vary.

An understanding of the second quantization operator of a constant times the identity operator and the Poissonian Wick product, without using the orthogonal Charlier polynomials, is presented first. We use both understanding, with and without the Charlier polynomials, to prove some inequalities about the norms of Poissonian Wick products. These inequalities are the best ones in the case of L¹, L², and L^∞ norms. We close the paper with some probabilistic interpretations of the Poissonian Wick product.

In this paper, the smoothness and exact modulus of continuity of a class of fractional Brownian fields are studied. These Gaussian random fields satisfy a kind of operator-scaling property and, depending on the choice of their parameters, may share similar fractal properties as those of fractional Brownian sheets or may be smooth in some (or all) directions. It is proved that these Gaussian random fields satisfy the property of sectorial local nondeterminism which is useful for further studying their sample path properties. In addition, the link between these Gaussian random fields and the functional fluctuation limits of branching particle systems is studied.

Following the ideas of F. Russo and P. Vallois, we use the notion of forward integral to introduce a new stochastic integral respect to the cylindrical Wiener process. This integral is an extension of the classical integral. As an application, we prove existence of solution of a parabolic stochastic differential partial equation with anticipating stochastic initial date.

Continuing the study of representations of amenable groups, we discuss one of the simplest cases in which a (not necessarily bounded) Banach space quasirepresentation of an amenable group is close to an ordinary representation of the group in the same space.

We show that if H = ℂ^d then there exists a system of d + 1 measurements which allows to recover an unknown state ρ on H from the probabilities of results of these measurements. Moreover, we show that a randomly chosen system of d + 1 measurements has the above property with probability 1. We also present some novel properties of known mutually unbiased bases.