Poisson process Fock space representation, chaos expansion and covariance inequalities

We consider a Poisson process $\eta$ on an arbitrary measurable space with an arbitrary sigma-finite intensity measure. We establish an explicit Fock space representation of square integrable functions of $\eta$. As a consequence we identify explicitly, in terms of iterated difference operators, the integrands in the Wiener-Ito chaos expansion. We apply these results to extend well-known variance inequalities for homogeneous Poisson processes on the line to the general Poisson case. The Poincare inequality is a special case. Further applications are covariance identities for Poisson processes on (strictly) ordered spaces and Harris-FKG-inequalities for monotone functions of $\eta$.Comment: 25 page

On the capacity functional of the infinite cluster of a Boolean model

Consider a Boolean model in $\R^d$ with balls of random, bounded radii with distribution $F_0$, centered at the points of a Poisson process of intensity $t>0$. The capacity functional of the infinite cluster $Z_\infty$ is given by \theta_L(t) = \BP\{ Z_\infty\cap L\ne\emptyset\}, defined for each compact $L\subset\R^d$. We prove for any fixed $L$ and $F_0$ that $\theta_L(t)$ is infinitely differentiable in $t$, except at the critical value $t_c$; we give a Margulis-Russo type formula for the derivatives. More generally, allowing the distribution $F_0$ to vary and viewing $\theta_L$ as a function of the measure $F:=tF_0$, we show that it is infinitely differentiable in all directions with respect to the measure $F$ in the supercritical region of the cone of positive measures on a bounded interval. We also prove that $\theta_L(\cdot)$ grows at least linearly at the critical value. This implies that the critical exponent known as $\beta$ is at most 1 (if it exists) for this model. Along the way, we extend a result of H.~Tanemura (1993), on regularity of the supercritical Boolean model in $d \geq 3$ with fixed-radius balls, to the case with bounded random radii.Comment: 23 pages, 24 references, 1 figure in Annals of Applied Probability, 201

Moments and central limit theorems for some multivariate Poisson functionals

This paper deals with Poisson processes on an arbitrary measurable space. Using a direct approach, we derive formulae for moments and cumulants of a vector of multiple Wiener-It\^o integrals with respect to the compensated Poisson process. Second, a multivariate central limit theorem is shown for a vector whose components admit a finite chaos expansion of the type of a Poisson U-statistic. The approach is based on recent results of Peccati et al.\ combining Malliavin calculus and Stein's method, and also yields Berry-Esseen type bounds. As applications, moment formulae and central limit theorems for general geometric functionals of intersection processes associated with a stationary Poisson process of $k$-dimensional flats in $\R^d$ are discussed

Martingale representation for Poisson processes with applications to minimal variance hedging

AbstractWe consider a Poisson process η on a measurable space equipped with a strict partial ordering, assumed to be total almost everywhere with respect to the intensity measure λ of η. We give a Clark–Ocone type formula providing an explicit representation of square integrable martingales (defined with respect to the natural filtration associated with η), which was previously known only in the special case, when λ is the product of Lebesgue measure on R+ and a σ-finite measure on another space X. Our proof is new and based on only a few basic properties of Poisson processes and stochastic integrals. We also consider the more general case of an independent random measure in the sense of Itô of pure jump type and show that the Clark–Ocone type representation leads to an explicit version of the Kunita–Watanabe decomposition of square integrable martingales. We also find the explicit minimal variance hedge in a quite general financial market driven by an independent random measure

Bloch electron in a magnetic field and the Ising model

The spectral determinant det(H-\epsilon I) of the Azbel-Hofstadter Hamiltonian H is related to Onsager's partition function of the 2D Ising model for any value of magnetic flux \Phi=2\pi P/Q through an elementary cell, where P and Q are coprime integers. The band edges of H correspond to the critical temperature of the Ising model; the spectral determinant at these (and other points defined in a certain similar way) is independent of P. A connection of the mean of Lyapunov exponents to the asymptotic (large Q) bandwidth is indicated.Comment: 4 pages, 1 figure, REVTE

Wittgensteinian spatial practices between architecture and philosophy

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Architecture and Planning, February 1999.Includes bibliographical references (p. 217-218).This thesis explores the deep spatio-linguistic relationship between the Austrian born philosopher Ludwig Wittgenstein's practices of philosophy and of architecture. Wittgenstein's philosophy of language is notable for its sharply distinguished early and late work: with the early work most strongly associated with his Tractatus Logico-Philosophicus (1922) and the later frequently designated by his posthumously published Philosophical Investigations (1953). Following the completion of the early work Wittgenstein abandoned philosophy for a period of ten years, spending the years from 1926 to 1929 engaged in the design and construction of a house in Vienna for his sister Margarethe Stonborough. The thesis considers the ways in which the intervening practice of architecture infiltrated, altered, influenced and manifested itself in the later philosophy by focusing on the spatial. temporal. conceptual and cognitive gaps in the philosophy. The importance and the prevalence of the practice of architecture for Wittgenstein's later philosophy are exhibited in a variety of ways that together broaden, reconceive and resituate the functioning of language and philosophy. The thesis considers these developments in the philosophy as they are revealed in the visual and spatial language, thinking and construction of the philosophical texts. This analysis reveals a shift from the removed, idealized and flattened picture theory of the Tractatus to the production of the spatially complex and ambiguous images of entanglement in the Investigations. The Stonborough house, itself, is analyzed through its production of cognitive and spatial practices and problematics. Wittgenstein's practice of architecture is shown to utilize. develop, challenge and reveal related spatial concepts found in the philosophy. These include the ideas of limits, boundaries, inner/outer dichotomies, the relationship between showing and saying, the idea of correspondence and the practices of representation, assembly. resemblance, construction, building and rearrangement.by Nana D. Last.Ph.D