Infinite Dimensional Analysis, Quantum Probability and Related Topics

We discuss some examples of complementary observables, in the sense of Accardi. We show that the trace property is not sufficient to determine such a pair of observables.

The viscous hydrodynamics is investigated via the studying diffusion processes on groups of H^s Sobolev diffeomorphisms of a flat n-dimensional torus (s > ½n + 1). A certain stochastic perturbation of the curve on the above groups, describing the motion of perfect incompressible fluid (or of the diffuse matter), is constructed such that it satisfies a certain stochastic analogue of the geodesic equation and the expectation of its "backward velocity" in the tangent space at identical diffeomorphism (algebra of the group) is a solution of Navier–Stokes (or Burgers, respectively) equations. In particular, the existence of solutions of those equations for lower s is proved. Some other approaches to stochastic presentation of viscous hydrodynamics, using the groups of diffeomorphisms, are also discussed.

In recent papers the authors presented the key ideas involved in their approach to Feynman's operational calculi for a system of not necessarily commuting bounded linear operators acting on a Banach space. The central objects of the theory are the disentangling algebra, a commutative Banach algebra, and the disentangling map which carries this commutative structure into the noncommutative algebra of operators. Under assumptions concerning the growth of disentangled exponential expressions, the associated functional calculus for the system of operators is a distribution with compact support which we view as the joint spectrum of the operators with respect to the disentangling map. The spectral properties of the disentangling maps are studied in this paper.

We develop a combinatorial version of harmonic analysis on configuration spaces over Riemannian manifolds. Our constructions are based on the use of a lifting operator which can be considered as a kind of (combinatorial) Fourier transform in the configuration space analysis. The latter operator gives us a natural lifting of the geometry from the underlying manifold onto the configuration space. Properties of correlation measures for given states (i.e. probability measures) on configuration spaces are studied including a characterization theorem for correlation measures.

Below a new kind of convolution is introduced for probability measures, whose combinatorics is related to non-crossing partitions without inner blocks other than singletons — the partitions corresponding to the fermionic creation and annihilation operators and Pauli's principle.

In this paper we study the solution of the stochastic heat equation where the potential V and the initial condition f are generalized stochastic processes. We construct explicitly the solution and we prove that it belongs to the generalized function space .

We study a homogeneous infinite dimensional Dirichlet problem in a half-space of a Hilbert space involving a second-order elliptic operator with Hölder continuous coefficients. Thanks to a new explicit formula for the solution in the constant coefficients case, we prove an optimal regularity result of Schauder type. The proof uses nonstandard techniques from semigroups and interpolation theory and involves extensive computations on Gaussian integrals.

We consider the Gaussian process X_λ defined by parameterizing a singular kernel of Volterra-type introduced in Ref. 1. The kernel has a close connection with the noncanonical representation. The result is that the representation is canonical (resp. noncanonical) if λ < 1/2 (resp. λ > 1/2), being independent of the choice of g of a class of functions (Theorem 3).