Jürgen Potthoff

On a dual pair of spaces of smooth and generalized random variables

A dual pairG andG* of smooth and generalized random variables, respectively, over the white noise probability space is studied.G is constructed by norms involving exponentials of the Ornstein-Uhlenbeck operator,G* is its dual. Sufficient criteria are proved for when a function onL(ℝ) is theL-transform of an element inG orG*.

Interacting and Annealing Particle Filters: Mathematics and a Recipe for Applications

Abstract Interacting and annealing are two powerful strategies that are applied in different areas of stochastic modelling and data analysis. Interacting particle systems approximate a distribution of interest by a finite number of particles where the particles interact between the time steps. In computer vision, they are commonly known as particle filters. Simulated annealing, on the other hand, is a global optimization method derived from statistical mechanics. A recent heuristic approach to fuse these two techniques for motion capturing has become known as annealed particle filter. In order to analyze these techniques, we rigorously derive in this paper two algorithms with annealing properties based on the mathematical theory of interacting particle systems. Convergence results and sufficient parameter restrictions enable us to point out limitations of the annealed particle filter. Moreover, we evaluate the impact of the parameters on the performance in various experiments, including the tracking of articulated bodies from noisy measurements. Our results provide a general guidance on suitable parameter choices for different applications.

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.

On the Martingale Property for Generalized Stochastic Processes

In the recent years, several groups have studied stochastic equations (e.g. SDEs, SPDEs, stochastic Volterra equations) outside the framework of the Ito calculus. Often, this led to solutions in spaces of generalized random processes or fields. It is therefore of interest to study the probabilistic properties of generalized stochastic processes, and the present paper makes some (rather naive) first steps into this direction. If we think of a generalized process as a mapping from the real line (or an interval) into aspace of generalized random variables (with some additional properties), then there is a wide range of choices for the latter: e.g., the space (S)* of Hida distributions (e.g. [HKPS]), the space (S)-1 of Kondratiev distributions (e.g. [KLS]), the Sobolev type space D* which is used within the Malliavin calculus and so on. Often, the generalized processes coming up in applications have a chaos decomposition with kernels which belong to L2(IRn), and in this paper we shall focus our attention on this situation. It will be convenient to work with a space G* which is larger than D*. This space has already been considered by several authors, cf. e.g. [PT] and the references quoted there. It turns out, that basic notions from the theory of stochastic processes like conditional expectation, martingales, sub- (super-) martingales etc. have a rather natural generalization to mappings from the realline into G* . The paper is organized as follows. In Section 2 we shall give a construction of the Ito integral (with respect to Brownian motion) of generalized stochastic processes, and compare it with the Hitsuda-Skorokhod integral (e.g. [HKPS]). In Section 3 we define generalized martingales and derive a number of properties. In particular, we prove that the generalized Ito integrals of Section 2 are indeed generalized martingales. Moreover, a representation of generalized martingales (in analogy with the Clark-Haussmann formula) will be shown, and we prove that the Wick product of two generalized martingales is again one. Finally, in Section 4 we define the notion of a generalized semimartingale and give a class of examples. In the remainder of this Introduction we provide the necessary background.

Brownian motions on metric graphs

Brownian motions on a metric graph are defined. Their generators are characterized as Laplace operators subject to Wentzell boundary at every vertex. Conversely, given a set of Wentzell boundary conditions at the vertices of a metric graph, a Brownian motion is constructed pathwise on this graph so that its generator satisfies the given boundary conditions.

Almost sure convergence of a semidiscrete Milstein scheme for SPDEs of Zakai type

A semidiscrete Milstein scheme for stochastic partial differential equations of Zakai type on a bounded domain of is derived. It is shown that the order of convergence of this scheme is 1 for convergence in mean square sense. For almost sure convergence, the order of convergence is proved to be for any .

Stochastic Volterra equations with singular kernels

Existence, uniqueness and continuity properties of solutions of stochastic Volterra equations with singular integral kernels (driven by Brownian motion) are proven.

Stochastic analysis and applications in physics

Infinite Dimensional Diffusions, Markcov Fields, Quantum Fields and Stochastic Quantization, S. Albeverio, Y.G. Kondratiev, M. Rockner. Burgers Equation forced by Conservative or Nonconservative Noise, L. Bertini, N. Cancrini, G. Jona-Lasinio. Statistical Properties of Piecewise Expanding Maps of the Interval, P. Collet. Euclidean Quantum Mechanics. An Outline A.B. Cruzeiro, J.C. Zambrini. Analysis on Loop Groups, L. Gross. White Noise Analysis and Applications, T. Hida. Computer Stochastics in Scalar Quantum Field Theory, C.B. Lang. Brownian Motion over a Kahler Manifold and Elliptic Genera of Level N, R. Leandre. On Lyapunov Stability Theorems for Stochastic (Deterministic) Evolution Equations, V. Mandrekar. Stochastic Techniques in Condensed Matter Physics, R. Vilela Mendes. Stochastic Partial Differential Equations and Applications to Hydrodynamics, B. Oksendal. White Noise Approach to Parabolic Stochastic Partial Differential Equations, J. Potthoff. Colombeau Generalized Functions and Stochastic Analysis, F. Russo. Constructive Field Theoretic Methods and Stochastic Partial Differential Equations, R. Seneor. Quantum Yang-Mills Theory on Compact Surfaces, A. Sengupta. Detailed Balance and Free Energy, R.F. Streater. An Introduction to White Noise Analysis, L. Streit. Excursion Measures for One-Dimensional Time-Homogeneous Diffusions with Inaccessible and Accessible Boundaries, A. Truman.

Hedging of Spatial Temperature Risk with Market-Traded Futures

Abstract The main objective of this work is to construct optimal temperature futures from available market-traded contracts to hedge spatial risk. Temperature dynamics are modelled by a stochastic differential equation with spatial dependence. Optimal positions in market-traded futures minimizing the variance are calculated. Examples with numerical simulations based on a fast algorithm for the generation of random fields are presented.

White Noise Approach to Parabolic Stochastic Partial Differential Equations

A certain dass of stochastic partial differential equations of parabolic type is studied within white noise analysis.