Ludwig Streit

A characterization of Hida distributions

The space (S)∗ of Hida distributions is characterized in terms of analytic properties of the Fourier transformation of its elements.

ENERGY FORMS, HAMILTONIANS, AND DISTORTED BROWNIAN PATHS

We study the Hamiltonians for nonrelativistic quantum mechanics—and for the heat equation—in terms of energy forms ∫∇f∇fd’gm, where dμ is a positive, not necessarily finite measure on Rn. We cover the cases of very singular interactions (e.g., N particles in R3 interacting by two‐body ’’δ potentials’’). We also exhibit, on the other hand, regularity conditions for μ in order that H be realized as a perturbation of the Laplacian by a measurable or generalized functon. The Hamiltonians defined by energy forms alwasy generate Markov semigroups, and the associated processes are symmetric homogeneous strong Markov diffusion Hunt processes with continuous paths realizations. Ergodicity, transiency, and recurrency are also discussed. The associated stochastic differential equaiton is discussed in the situaion were μ is finite but the drift coefficient is only restricted to be l (Rr,dμ). These results provide a large class of examples where solutions of the heat equaion can be expressed by averages with respect t...

Analysis on Poisson and Gamma Spaces

We study the spaces of Poisson, compound Poisson and Gamma noises as special cases of a general approach to non-Gaussian white noise calculus, see Ref. 18. We use a known unitary isomorphism between Poisson and compound Poisson spaces in order to transport analytic structures from Poisson space to compound Poisson space. Finally we study a Fock type structure of chaos decomposition on Gamma space.

Wick calculus in Gaussian analysis

We define an extension of the distribution spaces conventionally used in Gaussian analysis. This space, characterized by analytic properties of S-transforms, allows for a calculus based on the Wick product. We indicate some of its features.

Spaces of White Noise distributions: constructions, descriptions, applications. I☆

The paper introduces a scale of spaces of test functions and associated distributions in White Noise Analysis. Special cases of this scale include Hida and Meyer–Yan triples. We give a characterization of the spaces considered in terms of analytic and growth properties of the corresponding S-transforms.

The Feynman integrand as a Hida distribution

It is shown that Feynman integrals are weighted averages over Brownian paths, with suitable Hida distributions as (complex) weights. For simple cases, the construction of these distributions is be performed explicitly, for a much larger class of interactions, their existence is shown.

On quantum theory in terms of white noise

It has often been pointed out that a much more manageable structure is obtained from quantum theory if the time parameter t is chosen imaginary instead of real. Under a replacement of t by i·t the Schrodinger equation turns into a generalized heat equation, time ordered correlation functions transform into the moments of a probability measure, etc. More recently this observation has become extremely important for the construction of quantum dynamical models, where criteria were developed by E. Nelson, by K. Osterwalder and R. Schrader and others [8] which would permit the reverse transition to real time after one has constructed an imaginary time (“Euclidean”) model.

Capacity and quantum mechanical tunneling

We connect the notion of capacity of sets in the theory of symmetric Markov process and Dirichlet forms with the notion of tunneling through the boundary of sets in quantum mechanics. In particular we show that for diffusion processes the notion appropriate to a boundary without tunneling is more refined than simply capacity zero. We also discuss several examples in ℝd.

Feynman integrals for a class of exponentially growing potentials

We construct the Feynman integrands for a class of exponentially growing time-dependent potentials as white noise functionals. We show that they solve the Schrodinger equation. The Morse potential is considered as a special case.

Dirichlet forms and white noise analysis

We use the white noise calculus as a framework for the introduction of Dirichlet forms in infinite dimensions. In particular energy forms associated with positive generalized white noise functionals are considered and we prove criteria for their closability. If the forms are closable, we show that their closures are Markovian (in the sense of Fukushima).