Infinite Dimensional Analysis, Quantum Probability and Related Topics

We consider a 2D Navier–Stokes equation with Dirichlet boundary conditions perturbed by a stochastic term of the form , where Q is a non-negative operator and Ẇ is a spacetime white noise. We solve the corresponding Kolmogorov equation in the space L²(H,ν) where ν is an invariant measure and prove the "carré du champs" identity. The key tool are some sharp estimates on the derivatives of the solution. Our main assumption is that the viscosity is large compared with the norm of Q. As a byproduct, we give a new simple proof of the uniqueness of the invariant measure and obtain an existence result for a Hamilton–Jacobi equation related to a control problem.

We determine the structure of the *-endomorphisms of the RSWN algebra, induced by linear maps in the 1-particle Hilbert algebra, introduce the RSWN analogue of the free evolutions and find the explicit form of the KMS states associated with some of them.

Some a priori bounds for measures on configuration spaces are considered. We establish relations between them and consequences for corresponding measures (such as support properties etc.). Applications to Gibbs measures are discussed.

We show that, for each value of α∈(-1, 1), the only Riemannian metrics on the space of positive definite matrices for which the ∇^(α) and ∇^(-α) connections are mutually dual are matrix multiples of the Wigner–Yanase–Dyson metric. If we further impose that the metric be monotone, then this set is reduced to scalar multiples of the Wigner–Yanase–Dyson metric.

A Fischer pair (FP) for a vector space E is a pair (u, v) of linear maps on E, not necessarily everywhere defined, such that E=ker u⊕Im v (Fischer decomposition). Thus, in particular, every densely defined closed operator u on a Hilbert space E forms a Fischer pair together with its adjoint u*, whenever Im u, or equivalently, Im u* is closed since then Im u*=ker u^⊥. The question of when a given pair of maps (u, v) is a FP is related to the well-posedness of the (abstract) Cauchy–Goursat problem for u, v in E.

We establish some Fischer pairs, for spaces that are built up by homogeneous Hilbert–Schmidt polynomials on a Hilbert space, consisting of differential and multiplication operators. In particular we study Fischer decompositions of the space of entire functions of Hilbert–Schmidt type. As a basis we generalize Fischers theorem for homogeneous polynomials in n variables to Hilbert–Schmidt polynomials.

We prove the essential self-adjointness of Ornstein–Uhlenbeck operators perturbed by a dissipative drift and a potential with polynomial growth, by establishing certain gradient estimates and using Berezanskii's parabolic criterion.

We provide an explicit formula which gives natural extensions of piecewise monotonic Markov maps defined on an interval of the real line. These maps are exact endomorphisms and define chaotic discrete dynamical systems.

We give a new criterion for asymptotic completeness in Kümmerer–Maassen-scattering theory which is more directly in terms of the coupling. To this end we define the semigroup of dual extended transition operators. Asymptotic completeness holds if and only if this semigroup is ergodic. This may also be interpreted in terms of entanglement.

Analytic and probabilistic properties of symmetric or non-symmetric Dirichlet forms are well studied. But the processes with parabolic generators are out of the framework of symmetric Dirichlet forms. To cover these cases, we have introduced the time-dependent Dirichlet forms and studied their properties so far. In this expository article, we intend to explain in detail the analytic and probabilistic properties for time-dependent Dirichlet forms parallel to the symmetric Dirichlet forms. New results on a characterization of the minimal α-excessive function dominating a quasi-continuous function as well as the correspondence between additive functionals and smooth mesures are given. In particular, we emphasized the existence of the nontrivial semipolar sets under our settings.