Infinite Dimensional Analysis, Quantum Probability and Related Topics

On p. 1096 of Ref. 31, we remark that the Witten current defined in Ref. 13 is an analogue of Hida distribution, Wiener chaos being replaced by Chen iterated integrals. We clarify this remark in the case of a Feynman path integral on a manifold.

We present a mathematical model for a Black–Scholes market driven by fractional Brownian motion B_H(t) with Hurst parameter . The interpretation of the integrals with respect to B_H(t) is in the sense of Itô (Skorohod–Wick), not pathwise (which is known to lead to arbitrage).

We find explicitly the optimal consumption rate and the optimal portfolio in such a market for an agent with utility functions of power type. When H → 1/2+ the results converge to the corresponding (known) results for standard Brownian motion.

Let A = (A_x)_{x ∈ X} be a family of commuting normal operators in a separable Hilbert space H₀. Obtaining the spectral expansion of A involves constructing of the corresponding joint resolution of identity E. The support supp E is not, in general, a set of full measure. This causes numerous difficulties, in particular, when proving the projection spectral theorem, i.e. the main theorem about the expansion in generalized joint eigenvectors. In this work, we show that supp E has a full outer measure under the conditions of the projection spectral theorem. Using this result, we simplify the proof of the theorem and refine its assertions.

Martingale characterizations and the related martingale problem are studied for processes with independent (not necessarily stationary) increments in an arbitrary locally compact group. In the special case of a compact Lie group, a Lévy-type characterization is given in terms of a faithful finite dimensional representation of the group and its tensor square. For the proofs noncommutative Fourier theory is applied for the convolution hemigroups associated with the increment processes.

A purely atomic superprocess with dependent spatial motion is characterized as the pathwise unique solution of a stochastic partial differential equation, which is driven by a time-space white noise defining the spatial motion and a sequence of independent Brownian motions defining the branching mechanism.

We give an effective definition of higher dimensional parameter Poisson noise. The idea comes from the observation of the conditional probablistic behavior of a Poisson process under the condition that it jumps over the unit time interval exactly as many times as n. Also, taking the optimality into account we can give effective determination of a Poisson noise with multi-dimensional parameter, which is in agreement with the formal generalization of a Poisson noise in terms of the characteristic functional.

In this paper we shall give combinatorial remarks on the r-free convolution. In particular, we shall introduce the set partition statistic on non-crossing partitions, which gives the r-free deformed moment-cumulant formula. We shall also give the probability measure of the r-free Poisson law and its moments, exactly.

This is a survey article on Brownian motions on compact connected groups and the associated Gaussian convolution semigroups. The emphasize is on infinite dimensional groups such as the infinite dimensional torus and infinite products of special orthogonal groups. We discuss the existence of Brownian motions having nice properties such as marginales having a continuous density with respect to Haar measure. We relate the existence of these Brownian motions to the algebraic structure of the group. The results we describe reflect the conflicting effects of, on the one hand, the infinite dimensionality and, on the other hand, the compact nature of the underlying group.