Spectral families of quantum stochastic integrals

We develop a theory of spectral integration for quantum stochastic integrals of certain families of processes driven by creation, conservation and annihilation processes in Fock space. These give a non-commutative generalisation of classical stochastic integrals driven by Poisson random measures. A stochastic calculus for these processes is developed and used to obtain unitary operator valued solutions of stochastic differential equations. As an application we construct stochastic flows on operator algebras driven by Lévy processes with finite Lévy measure.     

Brownian Motion on So(3) Factorises in Fock Space

Techniques of quantum stochastic calculus are employed to factorisecanonical Brownian motion on SO(3) into two non-commutativeunitary operator valued processes acting in a suitable Fockspace.     

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.Work completed in part while the first author was supported by an SERC research studentship, and in part while the second author was visiting the Physics Department of the University of Texas at Austin supported by NSF grant PHY 81-07381     

$$L^2$$ L 2 Properties of Lévy Generators on Compact Riemannian Manifolds

Journal of Theoretical Probability - We consider isotropic Lévy processes on a compact Riemannian manifold, obtained from an $${\mathbb {R}}^d$$ -valued Lévy process through rolling...     

The positive maximum principle on symmetric spaces

Positivity - We investigate the Courrège theorem in the context of linear operators that satisfy the positive maximum principle on a space of continuous functions over a symmetric space....     

Transition Densities and Traces for Invariant Feller Processes on Compact Symmetric Spaces

We find necessary and sufficient conditions for a finite K–bi–invariant measure on a compact Gelfand pair (G,K) to have a square–integrable density. For convolution semigroups, this is equivalent to having a continuous density in positive time. When (G,K) is a compact Riemannian symmetric pair, we study the induced transition density for G–invariant Feller processes on the symmetric space X = G/K. These are obtained as projections of K–bi–invariant Lévy processes on G, whose laws form a convolution semigroup. We obtain a Fourier series expansion for the density, in terms of spherical functions, where the spectrum is described by Gangolli’s Lévy–Khintchine formula. The density of returns to any given point on X is given by the trace of the transition semigroup, and for subordinated Brownian motion, we can calculate the short time asymptotics of this quantity using recent work of Bañuelos and Baudoin. In the case of the sphere, there is an interesting connection with the Funk–Hecke theorem.     

Stochastic evolution of Yang-Mills connections on the noncommutative two-torus

We study quantum stochastic parallel transport processes where the noise terms arise from quantum Brownian motion in Fock space and the connection is chosen to minimize the Yang-Mills functional on a Heisenberg module over the smooth algebra of the noncommutative two-torus. Each such process yields a dilation of a quantum dynamical semigroup whose action on components of the connection induces a family of transformations of the moduli space. From a physical point of view, this describes a highly singular interaction between quantized Yang-Mills fields and the free boson field.     

On the Infinitesimal Generators of Ornstein–Uhlenbeck Processes with Jumps in Hilbert Space

We study Hilbert space valued Ornstein–Uhlenbeck processes (Y(t), t ≥ 0) which arise as weak solutions of stochastic differential equations of the type dY = JY + CdX(t) where J generates a C ₀ semigroup in the Hilbert space H, C is a bounded operator and (X(t), t ≥ 0) is an H-valued Lévy process. The associated Markov semigroup is of generalised Mehler type. We discuss an analogue of the Feller property for this semigroup and explicitly compute the action of its generator on a suitable space of twice-differentiable functions. We also compare the properties of the semigroup and its generator with respect to the mixed topology and the topology of uniform convergence on compacta.      

Asynchronous code-division random access using convex optimization

Many applications in cellular systems and sensor networks involve a random subset of a large number of users asynchronously reporting activity to a base station. This paper examines the problem of multiuser detection (MUD) in random access channels for such applications. Traditional orthogonal signaling ignores the random nature of user activity in this problem and limits the total number of users to be on the order of the number of signal space dimensions. Contention-based schemes, on the other hand, suffer from delays caused by colliding transmissions and the hidden node problem. In contrast, this paper presents a novel pairing of an asynchronous non-orthogonal code-division random access scheme with a convex optimization-based MUD algorithm that overcomes the issues associated with orthogonal signaling and contention-based methods. Two key distinguishing features of the proposed MUD algorithm are that it does not require knowledge of the delay or channel state information of every user and it has polynomial-time computational complexity. The main analytical contribution of this paper is the relationship between the performance of the proposed MUD algorithm in the presence of arbitrary or random delays and two simple metrics of the set of user codewords. The study of these metrics is then focused on two specific sets of codewords, random binary codewords and specially constructed algebraic codewords, for asynchronous random access. The ensuing analysis confirms that the proposed scheme together with either of these two codeword sets significantly outperforms the orthogonal signaling-based random access in terms of the total number of users in the system.