Convergence of the all-time supremum of a L\'evy process in the heavy-traffic regime

In this paper we derive a technique of obtaining limit theorems for suprema of L\'evy processes from their random walk counterparts. For each $a>0$, let $\{Y^{(a)}_n:n\ge 1\}$ be a sequence of independent and identically distributed random variables and $\{X^{(a)}_t:t\ge 0\}$ be a L\'evy processes such that $X_1^{(a)}\stackrel{d}{=} Y_1^{(a)}$, $\mathbb E X_1^{(a)}<0$ and $\mathbb E X_1^{(a)}\uparrow0$ as $a\downarrow0$. Let $S^{(a)}_n=\sum_{k=1}^n Y^{(a)}_k$. Then, under some mild assumptions, $\Delta(a)\max_{n\ge 0} S_n^{(a)}\stackrel{d}{\to} R\iff\Delta(a)\sup_{t\ge 0} X^{(a)}_t\stackrel{d}{\to} R$, for some random variable $R$ and some function $\Delta(\cdot)$. We utilize this result to present a number of limit theorems for suprema of L\'evy processes in the heavy-traffic regime

Amplification and generation of ultra-intense twisted laser pulses via stimulated Raman scattering

Twisted Laguerre–Gaussian lasers, with orbital angular momentum and characterized by doughnut-shaped intensity profiles, provide a transformative set of tools and research directions in a growing range of fields and applications, from super-resolution microcopy and ultra-fast optical communications to quantum computing and astrophysics. The impact of twisted light is widening as recent numerical calculations provided solutions to long-standing challenges in plasma-based acceleration by allowing for high-gradient positron acceleration. The production of ultra-high-intensity twisted laser pulses could then also have a broad influence on relativistic laser–matter interactions. Here we show theoretically and with ab initio three-dimensional particle-in-cell simulations that stimulated Raman backscattering can generate and amplify twisted lasers to petawatt intensities in plasmas. This work may open new research directions in nonlinear optics and high–energy-density science, compact plasma-based accelerators and light sources

Resonant interaction between gravitational waves, electromagnetic waves and plasma flows

In magnetized plasmas gravitational and electromagnetic waves may interact coherently and exchange energy between themselves and with plasma flows. We derive the wave interaction equations for these processes in the case of waves propagating perpendicular or parallel to the plasma background magnetic field. In the latter case, the electromagnetic waves are taken to be circularly polarized waves of arbitrary amplitude. We allow for a background drift flow of the plasma components which increases the number of possible evolution scenarios. The interaction equations are solved analytically and the characteristic time scales for conversion between gravitational and electromagnetic waves are found. In particular, it is shown that in the presence of a drift flow there are explosive instabilities resulting in the generation of gravitational and electromagnetic waves. Conversely, we show that energetic waves can interact to accelerate particles and thereby \emph{produce} a drift flow. The relevance of these results for astrophysical and cosmological plasmas is discussed.Comment: 12 pages, 1 figure, typos corrected and numerical example adde

An exploration of the effectiveness of artificial mini-magnetospheres as a potential solar storm shelter for long term human space missions

If mankind is to explore the solar system beyond the confines of our Earth and Moon the problem of radiation protection must be addressed. Galactic cosmic rays and highly variable energetic solar particles are an ever-present hazard in interplanetary space. Electric and/or magnetic fields have been suggested as deflection shields in the past, but these treated space as an empty vacuum. In fact it is not empty. Space contains a plasma known as the solar wind; a constant flow of protons and electrons coming from the Sun. In this paper we explore the effectiveness of a “mini-magnetosphere” acting as a radiation protection shield. We explicitly include the plasma physics necessary to account for the solar wind and its induced effects. We show that, by capturing/containing this plasma, we enhance the effectiveness of the shield. Further evidence to support our conclusions can be obtained from studying naturally occurring “mini-magnetospheres” on the Moon. These magnetic anomalies (related to “lunar swirls”) exhibit many of the effects seen in laboratory experiments and computer simulations. If shown to be feasible, this technology could become the gateway to manned exploration of interplanetary space

Limit theorems for weakly subcritical branching processes in random environment

For a branching process in random environment it is assumed that the offspring distribution of the individuals varies in a random fashion, independently from one generation to the other. Interestingly there is the possibility that the process may at the same time be subcritical and, conditioned on nonextinction, 'supercritical'. This so-called weakly subcritical case is considered in this paper. We study the asymptotic survival probability and the size of the population conditioned on non-extinction. Also a functional limit theorem is proven, which makes the conditional supercriticality manifest. A main tool is a new type of functional limit theorems for conditional random walks.Comment: 35 page

Fractional moment bounds and disorder relevance for pinning models

We study the critical point of directed pinning/wetting models with quenched disorder. The distribution K(.) of the location of the first contact of the (free) polymer with the defect line is assumed to be of the form K(n)=n^{-\alpha-1}L(n), with L(.) slowly varying. The model undergoes a (de)-localization phase transition: the free energy (per unit length) is zero in the delocalized phase and positive in the localized phase. For \alpha<1/2 it is known that disorder is irrelevant: quenched and annealed critical points coincide for small disorder, as well as quenched and annealed critical exponents. The same has been proven also for \alpha=1/2, but under the assumption that L(.) diverges sufficiently fast at infinity, an hypothesis that is not satisfied in the (1+1)-dimensional wetting model considered by Forgacs et al. (1986) and Derrida et al. (1992), where L(.) is asymptotically constant. Here we prove that, if 1/21, then quenched and annealed critical points differ whenever disorder is present, and we give the scaling form of their difference for small disorder. In agreement with the so-called Harris criterion, disorder is therefore relevant in this case. In the marginal case \alpha=1/2, under the assumption that L(.) vanishes sufficiently fast at infinity, we prove that the difference between quenched and annealed critical points, which is known to be smaller than any power of the disorder strength, is positive: disorder is marginally relevant. Again, the case considered by Forgacs et al. (1986) and Derrida et al. (1992) is out of our analysis and remains open.Comment: 20 pages, 1 figure; v2: few typos corrected, references revised. To appear on Commun. Math. Phy

Barotropic tides in the South Atlantic Bight

The characteristics of the principal barotropic diurnal and semidiurnal tides are examined for the South Atlantic Bight (SAB) of the eastern United States coast. We combine recent observations from pressure gauges and ADCPs on fixed platforms and additional short-term deployments off the Georgia and South Carolina coasts together with National Ocean Service coastal tidal elevation harmonics. These data have shed light on the regional tidal propagation, particularly off the Georgia/South Carolina coast, which is perforated by a dense estuary/tidal inlet complex (ETIC). We have computed tidal solutions for the western North Atlantic Ocean on two model domains. One includes a first-order representation of the ETIC in the SAB, and the other does not include the ETIC. We find that the ETIC is highly dissipative and affects the regional energy balance of the semidiurnal tides. Nearshore, inner, and midshelf model skill at semidiurnal frequencies is sensitive to the inclusion of the ETIC. The numerical solution that includes the ETIC shows significantly improved skill compared to the solution that does not include the ETIC. For the M2 constituent, the largest tidal frequency in the SAB, overall amplitude and phase error is reduced from 0.25 m to 0.03 m and 13.8° to 2.8° for coastal observation stations. Similar improvement is shown for midshelf stations. Diurnal tides are relatively unaffected by the ETIC

Operator renewal theory and mixing rates for dynamical systems with infinite measure

We develop a theory of operator renewal sequences in the context of infinite ergodic theory. For large classes of dynamical systems preserving an infinite measure, we determine the asymptotic behaviour of iterates $L^n$ of the transfer operator. This was previously an intractable problem. Examples of systems covered by our results include (i) parabolic rational maps of the complex plane and (ii) (not necessarily Markovian) nonuniformly expanding interval maps with indifferent fixed points. In addition, we give a particularly simple proof of pointwise dual ergodicity (asymptotic behaviour of $\sum_{j=1}^nL^j$) for the class of systems under consideration. In certain situations, including Pomeau-Manneville intermittency maps, we obtain higher order expansions for $L^n$ and rates of mixing. Also, we obtain error estimates in the associated Dynkin-Lamperti arcsine laws.Comment: Preprint, August 2010. Revised August 2011. After publication, a minor error was pointed out by Kautzsch et al, arXiv:1404.5857. The updated version includes minor corrections in Sections 10 and 11, and corresponding modifications of certain statements in Section 1. All main results are unaffected. In particular, Sections 2-9 are unchanged from the published versio

Apparatus for investigating non-linear microwave interactions in magnetised plasma

Plasma, as a non-linear medium supporting a rich and diverse range of electromagnetic and electrostatic oscillations, can enable a range of multi-wave interactions when excited by multiple injected propagating electromagnetic waves. Electromagnetic wave injection plays a dominant role in the introduction of energy in laser plasma interactions and in the heating of magnetically confined fusion reactors. In magnetically confined plasma, the EM waves tend to fall in the RF to microwave range, whilst in laser plasma interactions the signals are typically near the optical part of the spectrum