Levy processes and stochastic integrals in Banach spaces

We review in¯nite divisibility and Levy processes in Banach spaces and discuss the relationship with notions of type and cotype. The Levy-It^o decomposition is described. Strong, weak and Pettis-style notions of stochastic integral are introduced and applied to construct generalised Ornstein-Uhlenbeck processes

Extending stochastic resonance for neuron models to general Levy noise

A recent paper by Patel and Kosko (2008) demonstrated stochastic resonance (SR) for general feedback continuous and spiking neuron models using additive Levy noise constrained to have finite second moments. In this brief, we drop this constraint and show that their result extends to general Levy noise models. We achieve this by showing that �¿large jump�¿ discontinuities in the noise can be controlled so as to allow the stochastic model to tend to a deterministic one as the noise dissipates to zero. SR then follows by a �¿forbidden intervals�¿ theorem as in Patel and Kosko's paper

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

Some L2 properties of semigroups of measures on Lie groups

We investigate the induced action of convolution semigroups of probability measures on Lie groups on the L 2-space of Haar measure. Necessary and sufficient conditions are given for the infinitesimal generator to be self-adjoint and the associated symmetric Dirichlet form is constructed. We show that the generated Markov semigroup is trace-class if and only if the measures have a square-integrable density. Two examples are studied in some depth where the spectrum can be explicitly computed, these being the n-torus and Riemannian symmetric pairs of compact type

Universal Malliavin calculus in Fock and Levy-Ito spaces

We review and extend Lindsay's work on abstract gradient and divergence operators in Fock space over a general complex Hilbert space. Precise expressions for the domains are given, the L2-equivalence of norms is proved and an abstract version of the It^o-Skorohod isometry is established. We then outline a new proof of It^o's chaos expansion of complex Levy-It^o space in terms of multiple Wiener-Levy integrals based on Brownian motion and a compensated Poisson random measure. The duality transform now identies Levy-It^o space as a Fock space. We can then easily obtain key properties of the gradient and divergence of a general Levy process. In particular we establish maximal domains of these operators and obtain the It^o-Skorohod isometry on its maximal domain

A Lévy-Ciesielski expansion for quantum Brownian motion and the construction of quantum Brownian bridges

We introduce "probabilistic" and "stochastic Hilbertian structures". These seem to be a suitable context for developing a theory of "quantum Gaussian processes". The Schauder system is utilised to give a Lévy-Ciesielski representation of quantum (bosonic) Brownian motion as operators in Fock space over a space of square summable sequences. Similar results hold for non-Fock, fermion, free and monotone Brownian motions. Quantum Brownian bridges are defined and a number of representations of these are given

Covariant Mehler semigroups in Hilbert space

We find necessary and sufficient conditions for a generalised Mehler semigroup to be covariant under the action of a locally compact group. These are then applied to implement "noise reduction" for Hilbert-space valued Ornstein - Uhlenbeck processes driven by Levy processes

Levy-type stochastic integrals with regularly varying tails

Levy-type stochastic integrals M = (M(t), t ≥ 0) are obtained by integrating suitable predictable mappings against Brownian motion B and an independent Poisson random measure N. We establish conditions under which teh right tails of M are of regular variation. In particular, we require that the intensity measure associated to N is the product of a regularly varying Lvy measure with Lebesgue measure. Both univariate and multivariate versions of the problem are considered

Probability measures on compact groups which have square-integrable densities

We apply Peter–Weyl theory to obtain necessary and sufficient conditions for a probability measure on a compact group to have a square-integrable density. Applications are given to measures on the d-dimensional torus

Probabilistic trace and Poisson summation formulae on locally compact abelian groups

We investigate convolution semigroups of probability measures with continuous densities on locally compact abelian groups, which have a discrete subgroup such that the factor group is compact. Two interesting examples of the quotient structure are the d-dimensional torus, and the adèlic circle. Our main result is to show that the Poisson summation formula for the density can be interpreted as a probabilistic trace formula, linking values of the density on the factor group to the trace of the associated semigroup on L2-space. The Gaussian is a very important example. For rotationally invariant α-stable densities, the trace formula is valid, but we cannot verify the Poisson summation formula. To prepare to study semistable laws on the adèles, we first investigate these on the p-adics, where we show they have continuous densities which may be represented as series expansions. We use these laws to construct a convolution semigroup on the adèles whose densities fail to satisfy the probabilistic trace formula