The geometry of differential constraints for a class of evolution PDEs

The problem of computing differential constraints for a family of evolution PDEs is discussed from a constructive point of view. A new method, based on the existence of generalized characteristics for evolution vector fields, is proposed in order to obtain explicit differential constraints for PDEs belonging to this family. Several examples, with applications in non-linear stochastic filtering theory, stochastic perturbation of soliton equations and non-isospectral integrable systems, are discussed in detail to verify the effectiveness of the method

LIE SYMMETRY ANALYSIS AND GEOMETRICAL METHODS FOR FINITE AND INFINITE DIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS

The main aim of the thesis is a systematic application (via suitable generalizations) of Lie symmetry analysis, or more generally, of the various geometric techniques for differential equations, to the study of finite and infinite dimensional stochastic differential equations (SDEs). The work can be divided in three main parts. In the first part a new geometric approach to finite dimensional SDEs driven by a multidimensional Brownian motion is proposed, which is based on a new notion of random transformations of a stochastic process called stochastic transformations. After having studied the probabilistic and geometric properties of stochastic transformations, we provide a useful generalization of the well-known results of reduction and reconstruction of symmetric ODEs to the stochastic setting. We give many applications of previous results to some interesting SDEs among which the two dimensional Brownian motion, the Kolmogorov-Pearson equation, a generalized Langevin equation and the SABR model. Finally, using the previous theorems, we propose a symmetry-adapted numerical scheme whose effectiveness is verified through both theoretical estimates and numerical simulations. The second part contains an extension of the results obtained in the first part to finite dimensional SDEs driven by a general semimartingale taking values in a Lie group. In order to provide such an extension we use the notion of geometrical SDEs introduced by Serge Choen, and we introduce some new notions of stochastic invariance for semimartingales called gauge and time symmetries of a semimartingale. Using these mathematical tools we generalize the notion of stochastic transformations in this setting and we propose the natural definition of symmetry based on this group of transformations. The formulated theory allows us to analyze in detail an important class of SDEs with possible relevant applications to iterated random maps theory. In the third part we take advantage of the geometry of the infinite jets bundle to develop a convenient algorithm for the explicit determination of finite dimensional solutions to stochastic partial differential equations (SPDEs). In this setting we are able to propose a generalization of Frobenius theorem in the infinite jet bundles setting, which, exploiting the classical notion of characteristics of a PDE, allows us to find some sufficient conditions for the existence of finite dimensional solutions to an SPDE and then to explicitly reduce the SPDE to a finite dimensional SDE. Our techniques permits to individuate new finite dimensional solutions to interesting SPDEs among which the proportional volatility equation in Heath-Jarrow-Morton framework, a stochastic perturbation of Hunter-Saxton equation and a filtering problem related to affine type processes

Symmetries of stochastic differential equations using Girsanov transformations

Aiming at enlarging the class of symmetries of an SDE, we introduce a family of stochastic transformations able to change also the underlying probability measure exploiting Girsanov Theorem and we provide new determining equations for the infinitesimal symmetries of the SDE. The well-defined subset of the previous class of measure transformations given by Doob transformations allows us to recover all the Lie point symmetries of the Kolmogorov equation associated with the SDE. This gives the first stochastic interpretation of all the deterministic symmetries of the Kolmogorov equation. The general theory is applied to some relevant stochastic models

Weak symmetries of stochastic differential equations driven by semimartingales with jumps

Stochastic symmetries and related invariance properties of \ufb01nite dimensional SDEs driven by general c`adl`ag semimartingales taking values in Lie groups are de\ufb01ned and investigated. The considered set of SDEs, \ufb01rst introduced by S. Cohen, includes a\ufb03ne and Marcus type SDEs as well as smooth SDEs driven by L\ub4evy processes and iterated random maps. A natural extension to this general setting of reduction and reconstruction theory for symmetric SDEs is provided. Our theorems imply as special cases non trivial invariance results concerning a class of a\ufb03ne iterated random maps as well as (weak) symmetries for numerical schemes (of Euler and Milstein type) for Brownian motion driven SDEs

A note on symmetries of diffusions within a martingale problem approach

A geometric reformulation of the martingale problem associated with a set of diffusion processes is proposed. This formulation, based on second-order geometry and It\uf4 integration on manifolds, allows us to give a natural and effective definition of Lie symmetries for diffusion processes

Study of the doubly charmed tetraquark T+cc

Quantum chromodynamics, the theory of the strong force, describes interactions of coloured quarks and gluons and the formation of hadronic matter. Conventional hadronic matter consists of baryons and mesons made of three quarks and quark-antiquark pairs, respectively. Particles with an alternative quark content are known as exotic states. Here a study is reported of an exotic narrow state in the D0D0π+ mass spectrum just below the D*+D0 mass threshold produced in proton-proton collisions collected with the LHCb detector at the Large Hadron Collider. The state is consistent with the ground isoscalar T+cc tetraquark with a quark content of ccu⎯⎯⎯d⎯⎯⎯ and spin-parity quantum numbers JP = 1+. Study of the DD mass spectra disfavours interpretation of the resonance as the isovector state. The decay structure via intermediate off-shell D*+ mesons is consistent with the observed D0π+ mass distribution. To analyse the mass of the resonance and its coupling to the D*D system, a dedicated model is developed under the assumption of an isoscalar axial-vector T+cc state decaying to the D*D channel. Using this model, resonance parameters including the pole position, scattering length, effective range and compositeness are determined to reveal important information about the nature of the T+cc state. In addition, an unexpected dependence of the production rate on track multiplicity is observed

Helium identification with LHCb

The identification of helium nuclei at LHCb is achieved using a method based on measurements of ionisation losses in the silicon sensors and timing measurements in the Outer Tracker drift tubes. The background from photon conversions is reduced using the RICH detectors and an isolation requirement. The method is developed using pp collision data at √(s) = 13 TeV recorded by the LHCb experiment in the years 2016 to 2018, corresponding to an integrated luminosity of 5.5 fb-1. A total of around 105 helium and antihelium candidates are identified with negligible background contamination. The helium identification efficiency is estimated to be approximately 50% with a corresponding background rejection rate of up to O(10^12). These results demonstrate the feasibility of a rich programme of measurements of QCD and astrophysics interest involving light nuclei

Momentum scale calibration of the LHCb spectrometer

For accurate determination of particle masses accurate knowledge of the momentum scale of the detectors is crucial. The procedure used to calibrate the momentum scale of the LHCb spectrometer is described and illustrated using the performance obtained with an integrated luminosity of 1.6 fb-1 collected during 2016 in pp running. The procedure uses large samples of J/ψ → μ + μ - and B+ → J/ψ K + decays and leads to a relative accuracy of 3 × 10-4 on the momentum scale

Curvature-bias corrections using a pseudomass method

Momentum measurements for very high momentum charged particles, such as muons from electroweak vector boson decays, are particularly susceptible to charge-dependent curvature biases that arise from misalignments of tracking detectors. Low momentum charged particles used in alignment procedures have limited sensitivity to coherent displacements of such detectors, and therefore are unable to fully constrain these misalignments to the precision necessary for studies of electroweak physics. Additional approaches are therefore required to understand and correct for these effects. In this paper the curvature biases present at the LHCb detector are studied using the pseudomass method in proton-proton collision data recorded at centre of mass energy √(s)=13 TeV during 2016, 2017 and 2018. The biases are determined using Z→μ + μ - decays in intervals defined by the data-taking period, magnet polarity and muon direction. Correcting for these biases, which are typically at the 10-4 GeV-1 level, improves the Z→μ + μ - mass resolution by roughly 18% and eliminates several pathological trends in the kinematic-dependence of the mean dimuon invariant mass

The LHCb upgrade I

The LHCb upgrade represents a major change of the experiment. The detectors have been almost completely renewed to allow running at an instantaneous luminosity five times larger than that of the previous running periods. Readout of all detectors into an all-software trigger is central to the new design, facilitating the reconstruction of events at the maximum LHC interaction rate, and their selection in real time. The experiment's tracking system has been completely upgraded with a new pixel vertex detector, a silicon tracker upstream of the dipole magnet and three scintillating fibre tracking stations downstream of the magnet. The whole photon detection system of the RICH detectors has been renewed and the readout electronics of the calorimeter and muon systems have been fully overhauled. The first stage of the all-software trigger is implemented on a GPU farm. The output of the trigger provides a combination of totally reconstructed physics objects, such as tracks and vertices, ready for final analysis, and of entire events which need further offline reprocessing. This scheme required a complete revision of the computing model and rewriting of the experiment's software