Fractional L\'{e}vy-driven Ornstein--Uhlenbeck processes and stochastic differential equations

Using Riemann-Stieltjes methods for integrators of bounded $p$-variation we define a pathwise integral driven by a fractional L\'{e}vy process (FLP). To explicitly solve general fractional stochastic differential equations (SDEs) we introduce an Ornstein-Uhlenbeck model by a stochastic integral representation, where the driving stochastic process is an FLP. To achieve the convergence of improper integrals, the long-time behavior of FLPs is derived. This is sufficient to define the fractional L\'{e}vy-Ornstein-Uhlenbeck process (FLOUP) pathwise as an improper Riemann-Stieltjes integral. We show further that the FLOUP is the unique stationary solution of the corresponding Langevin equation. Furthermore, we calculate the autocovariance function and prove that its increments exhibit long-range dependence. Exploiting the Langevin equation, we consider SDEs driven by FLPs of bounded $p$-variation for $p<2$ and construct solutions using the corresponding FLOUP. Finally, we consider examples of such SDEs, including various state space transforms of the FLOUP and also fractional L\'{e}vy-driven Cox-Ingersoll-Ross (CIR) models.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ281 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Angular analysis of bremsstrahlung in alpha decay

A new quantum electrodynamical method of calculations of bremsstrahlung spectra in the $\alpha$-decay of heavy nuclei taking into account the angle between the directions of $\alpha$-particle motion (or its tunneling) and photon emission is presented. The angular bremsstrahlung spectra for $^{210}{Po}$ have been obtained for the first time. According to calculations, the bremsstrahlung in the $\alpha$-decay of this nucleus depends extremely weakly on the angle. Taking into account nuclear forces, such dependence is not changed visibly. An analytical formula of the angular dependence of the bremsstrahlung spectra is proposed and gives its harmonic behavior. The extremal values of the angle, at which the bremsstrahlung has maximal and minimal values, has been found.Comment: 15 pages, 1 file of figure in EPS format, LaTeX v.2e with EPJ style. In the new variant of the paper: 1) more attention is given to a convergence problem of computer calculations of the bremsstrahlung spectra; 2) a new section with inclusion of Woods-Saxon component in construction of the total realistic $\alpha$-nucleus potential into our model (with our first brermsstrahlung spectra for $^{210}{Po}$ at such potential) is included into the paper; 3) possible ways of further improvement of the quantum-mechanical models are pointed ou

Accelerating Metropolis-Hastings algorithms: Delayed acceptance with prefetching

MCMC algorithms such as Metropolis-Hastings algorithms are slowed down by the computation of complex target distributions as exemplified by huge datasets. We offer in this paper an approach to reduce the computational costs of such algorithms by a simple and universal divide-and-conquer strategy. The idea behind the generic acceleration is to divide the acceptance step into several parts, aiming at a major reduction in computing time that outranks the corresponding reduction in acceptance probability. The division decomposes the "prior x likelihood" term into a product such that some of its components are much cheaper to compute than others. Each of the components can be sequentially compared with a uniform variate, the first rejection signalling that the proposed value is considered no further, This approach can in turn be accelerated as part of a prefetching algorithm taking advantage of the parallel abilities of the computer at hand. We illustrate those accelerating features on a series of toy and realistic examples.Comment: 20 pages, 12 figures, 2 tables, submitte

Lindstedt series and Hamilton--Jacobi equation for hyperbolic tori in three time scales problems

Interacting systems consisting of two rotators and a pendulum are considered, in a case in which the uncoupled systems have three very different characteristic time scales. The abundance of unstable quasi periodic motions in phase space is studied via Lindstedt series. The result is a strong improvement, compared to our previous results, on the domain of validity of bounds that imply existence of invariant tori, large homoclinic angles, long heteroclinic chains and drift--diffusion in phase space.Comment: TeX 42 pages 2 figure

Pendulum: separatrix splitting

An exact expression for the determinant of the splitting matrix is derived: it allows us to analyze the asympotic behaviour needed to amend the large angles theorem proposed in Ann. Inst. H. Poincar\'e, B-60, 1, 1994. The asymptotic validity of Melnokov's formulae is proved for the class of models considered, which include polynomial perturbations.Comment: 30 pages, one figur