Thomas Deck

A Review of White Noise Analysis from a Probabilistic Standpoint

The main notions and tools from white noise analysis are set up on the basis of the calculus of Gaussian random variables and the S -transform. A new proof of the formula for the S -transform of Itô integrals is given. Moreover, measurability and the martingale property with respect to the Brownian filtration are characterized in terms of the S-transform. This allows the extension of these notions to random variables and processes, respectively, in the space of Hida distributions.

String branchings on complex tori and algebraic representations of generalized Krichever-Novikov algebras

The propagation differential for bosonic strings on a complex torus with three symmetric punctures is investigated. We study deformation aspects between two-and three-point differentials as well as the behaviour of the corresponding Krichever-Novikov algebras. The structure constants are calculated and from this we derive a central extension of the Krichever-Novikov algebras by means of b−c systems. The defining cocycle for this central extension deforms to the well-known Virasoro cocycle for certain kinds of degenerations of the torus.

Parabolic Differential Equations with Unbounded Coefficients – A Generalization of the Parametrix Method

We consider uniformly parabolic differential equations with unbounded first- and zero-order coefficients. A fundamental solution is constructed based on the classical parametrix method of E. Levi. From this the existence and uniqueness of the corresponding Cauchy problem is derived. Our approach does not require differentiable coefficients, as is usually assumed in the unbounded case. It only requires Hölder continuous coefficients. In this respect, our new proof also extends known results. We briefly discuss applications which make essential use of this extension.

A geometric Cauchy problem for timelike minimal surfaces

We investigate two-dimensional timelike surfaces σ in a Lorentz manifold (X,g). It is shown that orientable surfaces with two spacelike boundary components γ (homeomorphic toS1) are necessarily of topological type [0,1] x S1. We treat the initial value problem of a string (known from physics) as a purely geometric problem: Find a minimal surface σ which is specified by an initial curve γ and by a distribution of timelike tangent planes along γ. We prove the local existence and uniqueness of ∑ and also obtain global existence for special types (X,g). Global existence does not generally hold; we give a counter-example, which can be interpreted as a string collapsing into a black hole.

Classical string dynamics with non-trivial topology

The possibility of branching processes for classical strings is investigated on the basis of the Nambu-Goto action. We parametrize the world sheet by a Riemann surface M and introduce a C∞-smooth, degenerate metric η on M. Well-known results about the conformal group are generalized to the case of (M, η). We provide a rigorous, infinite dimensional Hamiltonian setting for processes that change the topology of a string. Finally, the classical background for the theory of quantum strings as developed by Krichever and Novikov in 1987 is discussed within this classical framework.

Nonlinear Evolution Equations with Gradient Coupled Noise

We show existence and uniqueness of solutions for a class of nonlinear evolution equations with gradient coupled noise. Our results are obtained by using a simple transformation relating the equation under consideration to an underlying deterministic partial differential equation. Both the Itô and the Stratonovich conventions are treated. Several examples show that the properties of solutions for Itô equations can differ significantly from those of Stratonovich equations.

Explicit strong solutions of SPDE's with applications to non-linear filtering

AbstractA differential calculus for random fields is developed and combined with the S-transform to obtain an explicit strong solution of the Cauchy problem

Hankel Operators over the Complex Wiener Space

{\text{d}}u(t,x) = (Lu + cu)(t,x){\text{ d}}t + \sum\limits_{i = 1}^m {h_i u(t,x){\text{ d}}Y_t^i },