Martin Grothaus

Construction of relativistic quantum fields in the framework of white noise analysis

We construct a class of Euclidean invariant distributions ΦH indexed by a function H holomorphic at zero. These generalized functions can be considered as generalized densities w.r.t. the white noise measure, and their moments fulfill all Osterwalder–Schrader axioms, except for reflection positivity. The case where F(s)=−(H(is)+12s2), s∈R, is a Levy characteristic is considered in Rev. Math. Phys. 8, 763 (1996). Under this assumption the moments of the Euclidean invariant distributions ΦH can be represented as moments of a generalized white noise measure PH. Here we enlarge this class by convolution with kernels G coming from Euclidean invariant operators G. The moments of the resulting Euclidean invariant distributions ΦHG also fulfill all Osterwalder–Schrader axioms except for reflection positivity. For no nontrivial case we succeeded in proving reflection positivity. Nevertheless, an analytic extension to Wightman functions can be performed. These functions fulfill all Wightman axioms except for the po...

The Feynman integral for time-dependent anharmonic oscillators

We review some basic notions and results of white noise analysis that are used in the construction of the Feynman integrand as a generalized white noise functional. We show that the Feynman integrand for the time-dependent harmonic oscillator in an external potential is a Hida distribution.

Regular generalized functions in Gaussian analysis

The concepts of regular generalized functions in Gaussian analysis are presented. Spaces of regular generalized functions are characterized and their probabilistic structure is worked out. Finally, these concepts are applied to a nonlinear stochastic (Verhulst type) equation. Its solution is shown to be a regular generalized process with martingale property.

ERGODICITY AND RATE OF CONVERGENCE FOR A NONSECTORIAL FIBER LAY-DOWN PROCESS

A stochastic model for the lay-down of fibers on a conveyor belt in the production process of nonwovens is investigated. In particular, convergence of the stochastic process to the stationary solution is proven and estimates on the speed of convergence are given. Numerical results and examples are presented and compared with the analytical estimates on the speed of convergence.

Hypocoercivity for Kolmogorov backward evolution equations and applications

Abstract In this article we extend the modern, powerful and simple abstract Hilbert space strategy for proving hypocoercivity that has been developed originally by Dolbeault, Mouhot and Schmeiser in [16] . As well-known, hypocoercivity methods imply an exponential decay to equilibrium with explicit computable rate of convergence. Our extension is now made for studying the long-time behavior of some strongly continuous semigroup generated by a (degenerate) Kolmogorov backward operator L. Additionally, we introduce several domain issues into the framework. Necessary conditions for proving hypocoercivity need then only to be verified on some fixed operator core of L. Furthermore, the setting is also suitable for covering existence and construction problems as required in many applications. The methods are applicable to various, different, Kolmogorov backward evolution problems. As a main part, we apply the extended framework to the (degenerate) spherical velocity Langevin equation. This equation e.g. also appears in applied mathematics as the so-called fiber lay-down process. For the construction of the strongly continuous contraction semigroup we make use of modern hypoellipticity tools and perturbation theory.

LAGRANGIAN FEYNMAN FORMULAS FOR SECOND-ORDER PARABOLIC EQUATIONS IN BOUNDED AND UNBOUNDED DOMAINS

In this note a class of second-order parabolic equations with variable coefficients, depending on coordinate, is considered in bounded and unbounded domains. Solutions of the Cauchy–Dirichlet and the Cauchy problems are represented in the form of a limit of finite-dimensional integrals of elementary functions (such representations are called Feynman formulas). Finite-dimensional integrals in the Feynman formulas give approximations for functional integrals in the corresponding Feynman–Kac formulas, representing solutions of these problems. Hence, these Feynman formulas give an effective tool to calculate functional integrals with respect to probability measures generated by diffusion processes with a variable diffusion coefficient and absorption on the boundary.

Self-avoiding Fractional Brownian Motion—The Edwards Model

In this work we extend Varadhan’s construction of the Edwards polymer model to the case of fractional Brownian motions in ℝd, for any dimension d≥2, with arbitrary Hurst parameters H≤1/d.

GEOMETRIC LANGEVIN EQUATIONS ON SUBMANIFOLDS AND APPLICATIONS TO THE STOCHASTIC MELT-SPINNING PROCESS OF NONWOVENS AND BIOLOGY

In this paper we develop geometric versions of the classical Langevin equation on regular submanifolds in Euclidean space in an easy, natural way and combine them with a bunch of applications. The equations are formulated as Stratonovich stochastic differential equations on manifolds. The first version of the geometric Langevin equation has already been detected before by Lelievre, Rousset and Stoltz with a different derivation. We propose an additional extension of the models, the geometric Langevin equations with velocity of constant Euclidean norm. The latters are seemingly new and provide a galaxy of new, beautiful and powerful mathematical models. Up to the authors best knowledge there are not many mathematical papers available dealing with geometric Langevin processes. We connect the first version of the geometric Langevin equation via proving that its generator coincides with the generalized Langevin operator proposed by Soloveitchik, Jorgensen or Kolokoltsov. All our studies are strongly motivated by industrial applications in modeling the fiber lay-down dynamics in the production process of nonwovens. We light up the geometry occurring in these models and show up the connection with the spherical velocity version of the geometric Langevin process. Moreover, as a main point, we construct new smooth industrial relevant three-dimensional fiber lay-down models involving the spherical Langevin process. Finally, relations to a class of swarming models are presented and further applications of the geometric Langevin equations are given.

Feynman formulae and phase space Feynman path integrals for tau-quantization of some Lévy-Khintchine type Hamilton functions

Evolution semigroups generated by pseudo-differential operators are considered. These operators are obtained by different (parameterized by a number τ) procedures of quantization from a certain class of functions (or symbols) defined on the phase space. This class contains Hamilton functions of particles with variable mass in magnetic and potential fields and more general symbols given by the Levy-Khintchine formula. The considered semigroups are represented as limits of n-fold iterated integrals when n tends to infinity. Such representations are called Feynman formulae. Some of these representations are constructed with the help of another pseudo-differential operator, obtained by the same procedure of quantization; such representations are called Hamiltonian Feynman formulae. Some representations are based on integral operators with elementary kernels; these are called Lagrangian Feynman formulae. Langrangian Feynman formulae provide approximations of evolution semigroups, suitable for direct computatio...

Quadratic actions, semi-classical approximation, and delta sequences in Gaussian analysis

A mathematically rigorous realization of Feynman integrals is given. The construction works for quadratic actions (there is only a restriction to limited time intervals). These techniques enable the calculation of the semi-classical approximation for a given Feynman propagator. Finally, delta sequences in Gaussian analysis are presented and their connection to semi-classical approximation is discussed.