Mathematical Physics and Stochastic Analysis

The following sections are included:

• Preface

• Contents

• Ludwig Streit's list of publications

A short review of the scientific work of Ludwig Streit is given. Some recent developments in infinite dimensional analysis, stochastic analysis and mathematical physics stimulated by Streit's work are indicated.

The original interest in the subjects on Mathematical Physics has been entirely inspired by Professor Ludwig Streit since the time when I first met him at RIMS (Research Institute for Mathematical Sciences) of Kyoto University, 1975. Since then, we have been more than a best friend in Mathematics, Physics and everything. Now it seems to be a good opportunity to have a quick review what we have done together and to remind what he has been proposing for the future directions on White Noise Analysis and Physics.

It is shown in our previous work that the large time-weak coupling (stochastic) limit of translation invariant Hamiltonians in quantum field theory leads to the master field which satisfies a new type of commutation relations, the so called entangled (or interacting) commutation relations. These relations extend the interacting Fock relations established earlier in non-relativistic QED and the free (or Boltzmann) commutation relations which have been found in the large N limit of QCD. In this paper a generalization of these results to the case of multiparticle external states is obtained.

We define de Rham complex over a product manifold (infinite product of compact manifolds), and Dirichlet operators on differential forms, associated with differentiable mesures (in particular, Gibbs measures), which generalize the notions of Bochner and de Rham Laplacians. We give probabilistic representations for corresponding semigroups, and study properties of the corresponding stochastic dynamics.

The field of p–adic numbers is a complete metric space under a non–Archimedian metrics. For this reason it is a suitable mathematical framework for the description of hierarchical phenomena appearing in many disciplines. Here we summarize some investigation of the real time random walks on p-adic based on the properties of transition functions obtained by solving the Kolmogorov equations.

We prove a characterization theorem for the test functions in a CKS-space. Some crucial ideas concerning the growth condition are given.

Solutions of the soliton type and their stability conditions obviously appear through polynomial interactions in toy models enlightening relativistic scalar field theories. We show that differential realizations of operators characterizing non-linear Lie algebras connected with deformations of the angular momentum algebra are useful in the study of such polynomial interactions in quantum field theory.

A new approach based on tree approximation is derived in order to estimate the critical parameters of continuous percolation of overlapping disks in R² when the centres of the disks are Poisson distributed. In the case of directed percolation, a lower bound is found for the critical density. The cluster structure near the percolation threshold is analyzed.

The Rigged Hilbert Space (RHS) theory of resonance scattering and decay is reviewed and contrasted with the standard Hilbert space (HS) theory of quantum mechanics. The main difference is in the choice of boundary conditions. Whereas the conventional theory allows for the in-states ϕ⁺ and the out-states (observables) ψ⁻ of the S-matrix elements (ψ⁻, ϕ⁺) = (ψ^out, Sϕⁱⁿ) any elements of the HS , , the RHS theory chooses the boundary conditions : , , where Φ₋ (Φ₊) are Hardy class spaces associated to the lower (upper) half-plane of the second sheet of the analytically continued S-matrix. This can be phenomenologically justified by causality. The two RHS's for states ϕ⁺ and observables ψ⁻ provide new vectors which are not in , e.g. the Dirac-Lippmann-Schwinger kets (solutions of the Lippmann-Schwinger equation with ±iɛ respectively) and the Gamow vectors . The Gamow vectors |E_R – iΓ/2⁻〉 have all the properties that one heuristically needs for quasistable states. In addition, they give rise to asymmetric time evolution expressing irreversibility on the microphysical level.

This contribution reviews the parallel dynamics of Q-Ising neural networks for various architectures: extremely diluted asymmetric, layered feedforward, extremely diluted symmetric, and fully connected. Using a probabilistic signal-to-noise ratio analysis, taking into account all feedback correlations, which are strongly dependent upon these architectures the evolution of the distribution of the local field is found. This leads to a recursive scheme determining the complete time evolution of the order parameters of the network. Arbitrary Q and mainly zero temperature are considered. For the asymmetrically diluted and the layered feedforward network a closed-form solution is obtained while for the symmetrically diluted and fully connected architecture the feedback correlations prevent such a closed-form solution. For these symmetric networks equilibrium fixed-point equations can be derived under certain conditions on the noise in the system. They are the same as those obtained in a thermodynamic replica-symmetric mean-field theory approach.

The Dirac equation in (2+1)-dimensional spacetime for a particle interacting with a combined Aharonov Bohm field and Coulomb potential is evaluated by the path integral method. To do this, a modified Biedenharn transformation is used to reduce the path integral to a form similar to the non-relativistic Coulomb problem.

The methods of differential geometry applied to probability and statistics theories open a new domain of investigation the statistical manifold or Information Geometry. This framwork provides a geometrical description of statistical quantities and leads to a new approach of complex statistical problems. A pecular feature is that statistical manifolds are naturaly associated to a family of affine-metric geometries. The differential geometry contents of a particular statistical model, the finite probability manifold, is used to exhibit the relations of some dynamical proprties of the relative entropy and self-parallel curves. Applications are done to the learning process of stochastic neural networks.

This note is a tribute to both Ludwig Streit and Sergio Albeverio. Given the schedules of the conferences in their honor and the publications of their festschrifts, as well as our current interest in their works, it is more appropriate for us to make our contributions to these celebrations a joint one.

With modified Bogoliubov replacement we study the fluctuation of trapped condensate, N₀ atoms in average, taking into account the interaction between particles inside and outside the condensate. It is indicated that the operator should in effect be of for large N₀ if the interaction is repulsive, based on some anticipation of numerical results still to be worked out.

The aim of the paper is to stress the effectiveness of point interactions for the construction of models of quantum systems whose dynamics is generated by time dependent or nonlinear hamiltonians. Examples of time dependent point interaction hamiltonians which could have some importance in applications are given. In particular nonlinear interactions of range zero are analyzed in dimensions one and three. Results on local and global existence of the solution and conditions for the blow-up are given and an explicit blow-up solution is constructed in the critical case.

An elementary yet remarkable similarity between the Cole-Hopf transformation relating the Burgers and heat equation and Miura's transformation connecting the KdV and mKdV equations is studied in detail. In the special (1 + 1)-dimensional case, our considerations apply to the entire hierarchy of Burgers evolution equations.

Using the functional integral techniques homogeneous limits of the perturbations of thermal states (describing nonrelativistic Bose Matter at the thermal equilibrium) by bounded cocycles are constructed rigorously. Additionally some elementary properties of these limiting states are discussed and in particular the preservation of the nonpurity in the critical case is proved.

To describe a continuous unitary representation of the group of diffeomorphisms of physical space, we typically specify a quasi-invariant measure together with a unitary 1-cocycle on some configuration space. Lebesgue measure on the space of N-point configurations describes N-particle quantum mechanics, where noncohomologous cocycles (obtained from unitary representations of the symmetric group or the braid group) give the statistics of bosons, fermions, paraparticles, anyone, or plektons, and even suggest possibilities for quantum nonlinearity. Poissonian or more general measures on spaces of infinite but locally finite configurations describe the infinite-volume limit of a gas of particles. Then some cocycles of interest are associated with infinite permutations or braids. A class of quasi-invariant measures, obtained from self-similar random processes, requires spaces of configurations having accumulation points. A program of research on this topic is discussed, including ongoing joint work with Moschella.

We discuss a representation of quantum dynamics in terms of Markov processes. It is shown that a holomorphic continuation of the Schrödinger wave function ψ_t(x) → ψ_t(z) determines a complex Markov process q_t(z). Feynman integration at a finite temperature is briefly discussed.

We consider a single production device which delivers a single type of items. The operating policy is of the type make-to-stock with a single hedging point. We calculate the optimal position of the hedging level in two different contexts: a) a discrete flow model describing a reliable machine operating with a random cycle time and serving non-markovian random demand and b) a fluid model, (i.e. continuous flow), describing a failure non-markovian prone machine serving a constant demand. The influence of the variability of the underlying stochastic processes on the position of the hedging level is explicitly observable as closed form expressions can be derived for these simple configurations.

We use the martingale method of Cox and Huang to solve explicitly the optimal portfolio problem in a Black & Scholes type of market driven by fractional Brownian motion B_H(t) with Hurst parameter . The results are compared to the corresponding well-known results in the standard Black & Scholes market.

A redesigned starting point for covariant , models is suggested that takes the form of an alternative lattice action and which may have the virtue of leading to a nontrivial quantum field theory in the continuum limit. The lack of conventional scattering for such theories is understood through an interchange of limits.

With the Schwinger model as example I discuss properties of lattice Dirac operators, with some emphasis on Monte Carlo results for topological charge, chiral fermions and eigenvalue spectra.

The description of both classical and quantum states with fluctuations, in view of the concept of the tomographic -probability distribtution, is presented. Time-dependent integrals of motion (invariants) for classical and quantum systems with noise are discussed. Relation of the invariants to the propagator of the classical Boltzmann equation and quantum evolution equation for density operator written in the tomographic-probability representation is elucidated. Examples of a free particle and quantum harmonic oscillator are given in detail.

The following sections are included:

• Introduction

• Transformation chaotique

• Espaces de fonctions test et de distributions

• Noyaux et symboles d'opérateurs

• Généralisation des théorèmes 1.2.3.4 pour des espaces de types ε_θ(N′)

• Application aux equations aux dérivées partielles stochastiques

• Bibliographie

It is shown that Donsker's delta function can be defined via the properties of its integral against a smooth function. Moreover, the composites of tempered distributions with (non–degenerate) elements in the first chaos are constructed with the help of the characterization theorem.

We give an analytic proof for L¹-uniqueness of a class of diffusion operators on arbitrary measurable state spaces. In particular, this generalizes a recent result, proved by probabilistic means by L. Wu, to the non-symmetric case. The method we use is based on the classical DuHamel formula which is applicable here due to results on (non-symmetric) Dirichlet forms on merely measurable state spaces, and a recent result by W. Stannat.

A stochastic Glauber dynamics is introduced in the two models, in the high temperature region. This dynamics drives the systems towards the equilibrium state. An estimate is given for the spectral gap of the spectrum of the generator.

In this talk we will show a transparent relation between the intrinsic pre-Dirichlet form and the extrinsic one corresponding to the Gibbs measure μ on the configuration space Γ_X. This extends the result obtained in ¹ (see also ²) for Poisson measure π_σ. As a consequence we prove the closability of on L²(Γ_X, μ) under very general assumptions on the interaction potential of the Gibbs measures μ, see also. ³

The Schrödinger-Weyl representation of the canonical commutation relation is circulary invariant in a sense. Here, as a continuation of ⁵, we show the converse.

Newton-type methods provide a significant computacional economy for simulations done with totally implicit numerical schemes. Several numerical results are presented for the nonlinear Schrödinger equation, showing the efficiency and robustness of the methods.

We consider a spin coherent states description of a general quantum spin system. It is shown that it is possible to use the spin-1/2 representation to study the general spin-J case. We identify the 1/ 2 spinor components as the homogeneous coordinates of the projective space associated to the complex variable that labels the coherent states and establish a relation between the two-component spinor and the bosonic Schwinger representation of a spin operator. We rewrite the equations of motion, obtained from the path integral for the evolution operator or partition function, in terms of the 1/2 spinor and define the effective Hamiltonian of its evolution.

Phenomenological evidence and analytic approximations to the QCD ground state suggest a complex gluon condensate structure. Exclusion of elementary fermion excitations by the generation of infinite mass corrections is a consequence. In addition the existence of vacuum condensates in unbroken non-abelian gauge theories, endows SU(3) and higher order groups with a non-trivial structure in the manifold of possible vacuum solutions, which is not present in SU(2). This may be related to the existence of particle generations.

Let T be an ergodic, measurable and measure preserving automorphism of a probability space , f = m + g − g ○ T where m ∊ L², g ∊ L^p, g − g ○ T ∊ L^r, 0 < p < 2 ≤ r; (m ○ Tⁱ) is a martingale difference sequence. We shall study for which values of p, r the (Donsker) invariance principle and the law of iterated logarithm hold for the process (f ○ Tⁱ).