Let Us Use White Noise

The following sections are included:

• Preface
• Contents

The starting point of White Noise Analysis¹¹ and^{,14–16,20,21,34,39} is a real separable Hilbert space $\mathscr{H}$ with inner product (·,·) and the corresponding norm |·|, and a nuclear triple

Since nuclear triples are the basis of the whole White Noise Analysis, we start by briefly recalling the main background of the theory of nuclear spaces, due to A. Grothendieck.⁷ For simplicity, instead of general nuclear spaces, cf. e.g.,^40,42,45,50 we just consider nuclear Fréchet spaces, which are the only ones needed in this book. For more details and the proofs see e.g.^2,3,9,14.

The following sections are included:

• Introduction and Preliminaries
• Applications of white noise analysis to canonical quantum field theory with ϕ-bound
• Application of Gaussian white noise analysis to relativistic quantum field models with indefinite metric
• Application of non-Gaussian white noise to constructive quantum field theory
• References

In this chapter we summarize applications of White Noise Analysis within the framework of Feynman Path Integrals. With more than 30 years of intensive study this field can be seen as one of the most important applications. Of course this chapter cannot give an exhaustive summary of all approaches to different path integrals in White Noise Analysis. Here we give a slice through different attemtps to path integrals in the White Noise framework.

In this chapter we give an overview of self intersection local times of Brownian motion paths and related processes in the framework of white noise analysis. More precisely, we show that self intersection local times are well defined Hida distributions as well as k-fold intersections. It turns out that the kernel functions of the chaos expansion are remarkably simple and exhibit clearly the dimension dependence. We use these kernel functions to study the regularity and convergence properties of self intersection local times.

The following sections are included:

• Introduction
• The heuristic Chern-Simons path integral in the torus gauge
• Rigorous realization of the RHS of Eq. (46)
• Open Questions
• Appendix A. Turaev’s shadow invariant
• Appendix B. Construction of and
• References

We present a new approach to the optimal portfolio problem for an insider with logarithmic utility. Our method is based on white noise theory, stochastic forward integrals, Hida-Malliavin calculus and the Donsker delta function.

The following sections are included:

• Introduction
• Revisiting the original ideas
• Three kinds of a noise
• Functionals of a noise
• Some further developments to be proposed

The following section is included:

• Subject Index