David Applebaum

Fermion Ito's Formula and Stochastic Evolutions*

An Ito product formula is proved for stochastic integrals against Fermion Brownian motion, and used to construct unitary processes satisfying stochastic differential equations. As in the corresponding Boson theory [10, 11] these give rise to stochastic dilations of completely positive semigroups.

Probability and Information: An Integrated Approach

This new and updated textbook is an excellent way to introduce probability and information theory to students new to mathematics, computer science, engineering, statistics, economics, or business studies. Only requiring knowledge of basic calculus, it begins by building a clear and systematic foundation to probability and information. Classic topics covered include discrete and continuous random variables, entropy and mutual information, maximum entropy methods, the central limit theorem and the coding and transmission of information. Newly covered for this edition is modern material on Markov chains and their entropy. Examples and exercises are included to illustrate how to use the theory in a wide range of applications, with detailed solutions to most exercises available online for instructors.

Martingale-valued measures, Ornstein-Uhlenbeck processes with jumps and operator self-decomposability in Hilbert space

We investigate a class of Hilbert space valued martingale-valued measures whose covariance structure is determined by a trace class positive operator valued measure. The paradigm example is the martingale part of a Levy process. We develop both weak and strong stochastic integration with respect to such martingale-valued measures. As an application, we investigate the stochastic convolution of a C0-semigroup with a Levy process and the associated Ornstein-Uhlenbeck process. We give an in¯nite dimensional generalisation of the concept of operator self-decomposability and conditions for random variables of this type to be embedded into a stationary Ornstein-Uhlenbeck process.

Ideas and Methods in Mathematical Analysis, Stochastics and Applications

Preface Picture of Raphael Hoegh-Krohn Bibliography of Raphael Hoegh-Krohn 1. On the scientific work of Raphael Hoegh-Krohn Part I. Stochastic Analysis 2. Some Euclidean integer-valued random fields with Markov properties 3. A Beurling-Deny type structure theorem for Dirichlet forms on general state spaces 4. Log-concavity of radial Schrodinger wave functions and convergence of planetesimal diffusions 5. Dirichlet forms, diffusion processes and spectral dimensions for nested fractals 6. Large deviations for weak solutions of stochastic differential equations 7. On a class of probabilistic integrodifferential equations 8. Operator integrals and martingale integrals with general parameter 9. Wick multiplication and Ito-Skorohod stochastic differential equations 10. Multiscale analysis in random dynamics 11. A limiting distribution connected with fractional parts of linear forms 12. Rapport sur les representations chaotiques 13. Markov properties of solutions of stochastic partial differential equations in a finite volume 14. On stochastic evolution equations with non-homogeneous boundary conditions 15. Grey noise 16. Existence of invariant measures for diffusion process with infinite dimensional state space Part II. Infinite Dimensional Groups 17. On nonlinear equations associated with Lie algebras of diffeomorphism groups of two-dimensional manifolds 18. Fields of left-invariant standard Brownian motion processes on a smooth bundle of compact-simple Lie groups 19. Unitary highest weight representations of gauge groups Part III. Operator Algebras 20. Some non-commutative orbifolds 21. Ergodic actions of non-abelian compact groups 22. Positive projections onto Jordan algebras and their enveloping von Neumann algebras Part IV. Nonlinear Analysis and Applications 23. How many singularities can there be in an energy minimizing map from the ball to the sphere? 24. Front tracking for petroleum reservoirs 25. Quasi-periodic, finite-gap solutions of the modified Korteweg-de Vries equations 26. A new representation of soliton solutions of the Kadomtsev-Petviashvili equation 27. On scalar conservation laws in one dimension.

STOCHASTIC STABILIZATION OF DYNAMICAL SYSTEMS USING LÉVY NOISE

We investigate the perturbation of the nonlinear differential equation by random noise terms consisting of Brownian motion and an independent Poisson random measure. We find conditions under which the perturbed system is almost surely exponentially stable and estimate the corresponding Lyapunov exponents.

Cylindrical Lévy processes in Banach spaces

Cylindrical probability measures are flnitely additive measures on Banach spaces that have sigma-additive projections to Euclidean spaces of all dimensions. They are naturally associated to notions of weak (cylindrical) random variable and hence weak (cylindrical) stochastic processes. In this paper we focus on cylindrical Levy processes. These have (weak) Levy-It^o decompositions and an associated Levy- Khintchine formula. If the process is weakly square integrable, its covariance oper- ator can be used to construct a reproducing kernel Hilbert space in which the pro- cess has a decomposition as an inflnite series built from a sequence of uncorrelated bona flde one-dimensional Levy processes. This series is used to deflne cylindrical stochastic integrals from which cylindrical Ornstein-Uhlenbeck processes may be constructed as unique solutions of the associated Cauchy problem. We demonstrate that such processes are cylindrical Markov processes and study their (cylindrical) invariant measures.

Infinitely divisible central probability measures on compact Lie groups—regularity, semigroups and transition kernels

We introduce a class of central symmetric infinitely divisible probability measures on compact Lie groups by lifting the characteristic exponent from the real line via the Casimir operator. The class includes Gauss, Laplace and stable-type measures. We find conditions for such a measure to have a smooth density and give examples. The Hunt semigroup and generator of convolution semigroups of measures are represented as pseudo-differential operators. For sufficiently regular convolution semigroups, the transition kernel has a tractable Fourier expansion and the density at the neutral element may be expressed as the trace of the Hunt semigroup. We compute the short time asymptotics of the density at the neutral element for the Cauchy distribution on the

The strong Markov property for fermion Brownian motion

d

Probability on compact lie groups

-torus, on SU(2) and on SO(3), where we find markedly different behaviour than is the case for the usual heat kernel.

Lévy Processes in Stochastic Differential Geometry

Abstract A concept of independence of W 2 ∗ -algebras is developed which is the fermion analogue of independence of von Neumann algebras for bosons. For a fermion Brownian motion process {A(t), A † (t); t ∈ R + } ( D. Applebaum, “Fermion Stochastic Calculus”, Ph. D. thesis, University of Nottingham, 1984 ) and a Markov time T = ∝0∞ λ dE(λ), it is shown that the process {A(T + t) − A(T), A † (T + t) − A † (T); t ∈ R + } , whose components are defined as spectral integrals with operator-valued integrands as in ( R. L. Hudson, J. Funct. Anal. 34 (1979) , 266), constitutes a new fermion Brownian motion process such that the “pre-T” and “post-T” W 2 ∗ -algebras are independent.