Infinite Dimensional Pathwise Volterra Processes Driven by Gaussian Noise -- Probabilistic Properties and Applications

We investigate the probabilistic and analytic properties of Volterra processes constructed as pathwise integrals of deterministic kernels with respect to the H\"older continuous trajectories of Hilbert-valued Gaussian processes. To this end, we extend the Volterra sewing lemma from \cite{HarangTindel} to the two dimensional case, in order to construct two dimensional operator-valued Volterra integrals of Young type. We prove that the covariance operator associated to infinite dimensional Volterra processes can be represented by such a two dimensional integral, which extends the current notion of representation for such covariance operators. We then discuss a series of applications of these results, including the construction of a rough path associated to a Volterra process driven by Gaussian noise with possibly irregular covariance structures, as well as a description of the irregular covariance structure arising from Gaussian processes time-shifted along irregular trajectories. Furthermore, we consider an infinite dimensional fractional Ornstein-Uhlenbeck process driven by Gaussian noise, which can be seen as an extension of the volatility model proposed by Rosenbaum et al. in \cite{ElEuchRosenbaum}.Comment: 38 page

Representation by Integrating Reproducing Kernels

Based on direct integrals, a framework allowing to integrate a parametrised family of reproducing kernels with respect to some measure on the parameter space is developed. By pointwise integration, one obtains again a reproducing kernel whose corresponding Hilbert space is given as the image of the direct integral of the individual Hilbert spaces under the summation operator. This generalises the well-known results for finite sums of reproducing kernels; however, many more special cases are subsumed under this approach: so-called Mercer kernels obtained through series expansions; kernels generated by integral transforms; mixtures of positive definite functions; and in particular scale-mixtures of radial basis functions. This opens new vistas into known results, e.g. generalising the Kramer sampling theorem; it also offers interesting connections between measurements and integral transforms, e.g. allowing to apply the representer theorem in certain inverse problems, or bounding the pointwise error in the image domain when observing the pre-image under an integral transform

Control of electron spin and orbital resonance in quantum dots through spin-orbit interactions

Influence of resonant oscillating electromagnetic field on a single electron in coupled lateral quantum dots in the presence of phonon-induced relaxation and decoherence is investigated. Using symmetry arguments it is shown that spin and orbital resonance can be efficiently controlled by spin-orbit interactions. The control is possible due to the strong sensitivity of Rabi frequency to the dot configuration (orientation of the dot and a static magnetic field) as a result of the anisotropy of the spin-orbit interactions. The so called easy passage configuration is shown to be particularly suitable for magnetic manipulation of spin qubits, ensuring long spin relaxation time and protecting the spin qubit from electric field disturbances accompanying on-chip manipulations.Comment: 11 pages, 5 figures; v2: introduction and conclusions broadened, moderate structure and content change

Spectrum of Low-Lying Excitations in a Supersymmetric Extended Hubbard Model

We continue the study of the $u(2|2)$-supersymmetric extension of the Hubbard model in one dimension. We determine the excitation spectrum at zero temperature even in the sectors where the ground states are $u(2|2)$-descendants of Bethe states. The excitations include spinons, holons, electrons, localons (local electrons pairs, moving coherently through the lattice) and their bound states. The spectra are found to be very different for repulsive and attractive on-site interaction. We also study the thermodynamics of the model.Comment: 37 pages, uuencoded compressed postscript fil

Fuelling quasars with hot gas

We consider a model for quasar formation in which massive black holes are formed and fuelled largely by the accretion of hot gas during the process of galaxy formation. In standard hierarchical collapse models, objects about the size of normal galaxies and larger form a dense hot atmosphere when they collapse. We show that if such an atmosphere forms a nearly "maximal" cooling flow, then a central black hole can accrete at close to its Eddington limit. This leads to exponential growth of a seed black hole, resulting in a quasar in some cases. In this model, the first quasars form soon after the first collapses to produce hot gas. The hot gas is depleted as time progresses, mostly by cooling, so that the accretion rate eventually falls below the threshold for advection-dominated accretion, at which stage radiative efficiency plummets and any quasar turns off. A simple implementation of this model, incorporated into a semi-analytical model for galaxy formation, over-produces quasars when compared with observed luminosity functions, but is consistent with models of the X-ray Background which indicate that most accretion is obscured. It produces few quasars at high redshift due to the lack of time needed to grow massive black holes. Quasar fuelling by hot gas provides a minimum level, sufficient to power most quasars at redshifts between one and two, to which other sources of fuel can be added. The results are sensitive to feedback effects, such as might be due to radio jets and other outflows.Comment: 12 pages, 6 figures, MN Latex style, accepted for publication in MNRA

The role of cooling flows in galaxy formation

The present structure of galaxies is governed by the radiative dissipation of the gravitational and supernova energy injected during formation. A crucial aspect of this process is whether the gas cools as fast as it falls into the gravitational potential well. If it does then rapid normal star formation is assumed to ensue. If not, and the gas can still cool by the present time, then the situation resembles that of a cooling flow, such as commonly found in clusters of galaxies. The cooled matter is assumed to accumulate as very cold clouds and/or low mass stars, i.e. as baryonic dark matter. In this paper we investigate the likelihood of a cooling flow phase during the hierarchical formation of galaxies. We concentrate on the behaviour of the gas, using a highly simplified treatment of the evolution of the dark matter potential within which the gas evolves. We assume that normal star formation is limited by how much gas the associated supernovae can unbind and allow the gas profile to flatten as a consequence of supernova energy injection. We find that cooling flows are an important phase in the formation of most galaxies with total (dark plus luminous) masses approxgt 10^12 Msun , creating about 20 per cent of the total dark halo in a galaxy such as our own and a smaller but comparable fraction of an elliptical galaxy of similar mass. The onset of a cooling flow determines the upper mass limit for the formation of a visible spheroid from gas, setting a characteristic mass scale for normal galaxies. We argue that disk formation requires a cooling flow phase and that dissipation in the cooling flow phase is the most important factor in the survival of normal galaxies during subsequent hierarchical mergers.Comment: uuencoded compressed postscript. The preprint is also available at http://www.ast.cam.ac.uk/preprint/PrePrint.htm

Existence of mesons after deconfinement

We investigate the possibility for a quark-antiquark pair to form a bound state at temperatures higher than the critical one ($T>T_c$), thus after deconfinement. Our main goal is to find analytical criteria constraining the existence of such mesons. Our formalism relies on a Schr\"{o}dinger equation for which we study the physical consequences of both using the free energy and the internal energy as potential term, assuming a widely accepted temperature-dependent Yukawa form for the free energy and a recently proposed nonperturbative form for the screening mass. We show that using the free energy only allows for the 1S bottomonium to be bound above $T_c$, with a dissociation temperature around $1.5\times T_c$. The situation is very different with the internal energy, where we show that no bound states at all can exist in the deconfined phase. But, in this last case, quasi-bound states could be present at higher temperatures because of a positive barrier appearing in the potential.Comment: 14 pages, 3 figures; only the case T>T_c is discussed in v