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

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

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

Wrong But Reasonable : The Fourth Amendment Particularity Requirement After United States v. Leon

This Note analyzes the application of the good-faith exception to search warrant particularity violations under the Fourth Amendment. The question compelled by United States v. Leon and Massachusetts v. Sheppard is when, if ever, a particularity-defective warrant will sustain an officer\u27s reasonable reliance.\u27\u27 The Note briefly discusses how particularity traditionally has been assessed under the fourth amendment. The author examines the Supreme Court\u27s holding in Massachusetts v. Sheppard, and contrasts several circuit court cases that have applied Sheppard\u27s objectively reasonable standard of good faith to warrants involving particularity defects. Finally, the Note concludes that the approach taken by the Second Circuit Court of Appeals in United States v. Buck is a preferable approach because it encourages courts to establish clearer standards for the particularity of warrants under the Fourth Amendment

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 $d$-torus, on SU(2) and on SO(3), where we find markedly different behaviour than is the case for the usual heat kernel.Comment: Published in at http://dx.doi.org/10.1214/10-AOP604 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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

Wrong But Reasonable : The Fourth Amendment Particularity Requirement After United States v. Leon

This Note analyzes the application of the good-faith exception to search warrant particularity violations under the Fourth Amendment. The question compelled by United States v. Leon and Massachusetts v. Sheppard is when, if ever, a particularity-defective warrant will sustain an officer\u27s reasonable reliance.\u27\u27 The Note briefly discusses how particularity traditionally has been assessed under the fourth amendment. The author examines the Supreme Court\u27s holding in Massachusetts v. Sheppard, and contrasts several circuit court cases that have applied Sheppard\u27s objectively reasonable standard of good faith to warrants involving particularity defects. Finally, the Note concludes that the approach taken by the Second Circuit Court of Appeals in United States v. Buck is a preferable approach because it encourages courts to establish clearer standards for the particularity of warrants under the Fourth Amendment