The Feynman-Kac formula and applications to PDEs

The thesis presents a probabilistic approach to the theory of semigroups of operators, with particular attention to the Markov and Feller semigroups. The first goal of this work is the proof of the fundamental Feynman-Kac formula, which gives the solution of certain parabolic Cauchy problems, in terms of the expected value of the initial condition computed at the associated stochastic diffusion processes. The second target is the characterization of the principal eigenvalue of the generator of a semigroup with Markov transition probability function and of second order elliptic operators with real coefficients not necessarily self-adjoint. The thesis is divided into three chapters. In the first chapter we study the Brownian motion and some of its main properties, the stochastic processes, the stochastic integral and the Itô formula in order to finally arrive, in the last section, at the proof of the Feynman-Kac formula. The second chapter is devoted to the probabilistic approach to the semigroups theory and it is here that we introduce Markov and Feller semigroups. Special emphasis is given to the Feller semigroup associated with the Brownian motion. The third and last chapter is divided into two sections. In the first one we present the abstract characterization of the principal eigenvalue of the infinitesimal generator of a semigroup of operators acting on continuous functions over a compact metric space. In the second section this approach is used to study the principal eigenvalue of elliptic partial differential operators with real coefficients. At the end, in the appendix, we gather some of the technical results used in the thesis in more details. Appendix A is devoted to the Sion minimax theorem, while in appendix B we prove the Chernoff product formula for not necessarily self-adjoint operators.

Measurement of ttbar production differential cross section at \sqrt{s}=8 TeV at ATLAS

Top quark studies play an important role in the physics program of the Large Hadron Collider (LHC). The energy and luminosity reached allow the acquisition of a large amount of data especially in kinematic regions never studied before. In this thesis is presented the measurement of the ttbar production differential cross section on data collected by ATLAS in 2012 in proton proton collisions at \sqrt{s} = 8 TeV, corresponding to an integrated luminosity of 20.3 fb^{−1}. The measurement is performed for ttbar events in the semileptonic channel where the hadronically decaying top quark has a transverse momentum above 300 GeV. The hadronic top quark decay is reconstructed as a single large radius jet and identified using jet substructure properties. The final differential cross section result has been compared with several theoretical distributions obtaining a discrepancy of about the 25% between data and predictions, depending on the MC generator. Furthermore the kinematic distributions of the ttbar production process are very sensitive to the choice of the parton distribution function (PDF) set used in the simulations and could provide constraints on gluons PDF. In particular in this thesis is performed a systematic study on the PDF of the protons, varying several PDF sets and checking which one better describes the experimental distributions. The boosted techniques applied in this measurement will be fundamental in the next data taking at \sqrt{s}=13 TeV when will be produced a large amount of heavy particles with high momentum.

Measurement of the inclusive and differential t/t cross sections at 13 TeV in the all-jets boosted regime with CMS

The t/t production cross section is measured with the CMS detector in the all-jets channel in $pp$ collisions at the centre-of-mass energy of 13 TeV. The analysis is based on the study of t/t events in the boosted topology, namely events in which decay products of the quark top have a high Lorentz boost and are thus reconstructed in the detector as a single, wide jet. The data sample used in this analysis corresponds to an integrated luminosity of 2.53 fb-1. The inclusive cross section is found to be sigma(t/t) = 727 +- 46 (stat.) +115-112 (sys.) +- 20~(lumi.) pb, a value which is consistent with the theoretical predictions. The differential, detector-level cross section is measured as a function of the transverse momentum of the leading jet and compared to the QCD theoretical predictions. Finally, the differential, parton-level cross section is reported, measured as a function of the transverse momentum of the leading parton, extrapolated to the full phase space and compared to the QCD predictions.

Mineral paragenesis and fluid inclusion constraints on the Torre di Rio skarn, Island of Elba

The historical iron ore deposits of eastern Elba held great importance for the region and were its primary source of iron. The Torre di Rio skarn, despite its easily accessible outcrop and vicinity to the larger Rio Marina deposit, was never properly characterized. The results of petrographic and microthermometric study presented in this work provide new constraints on the Torre di Rio skarn. Mineral assemblage of ilvaite, calcite, quartz, iron oxides and sulphides combined with textural evidence indicate that Torre di Rio skarn does not fit into classical skarn model. The complex paragenetic sequence and overlapping of skarn and ore mineralogy is result of fast formation at relatively low temperatures evidenced by the silicon enrichment and pervasive nature of limonite alteration. Hematite-magnetite textural relationship points to boundary conditions of the ore fluid in terms of oxygen fugacity. Eutectic temperatures range from -16 to -33 °C indicating complex fluids. Calculated salinities range from 1.4 to 17.4 wt% NaCleq suggesting multiple fluids of different compositions. Total homogenization temperatures vary from 330 °C to 150 °C with both homogeneously and heterogeneously trapped FIAs. Ore deposition is concentrated where skarn formation was controlled primarily by phase separation during boiling. Calculated fluid pressure at boiling suggest shallow formation depth of a few hundred meters and constrains maximum temperature of ore deposition to c. 260 °C. This work suggest that relatively low salinities of fluid inclusions could indicate dominant marine origin of the hydrothermal fluids that were activated by the Porto Azzurro pluton emplacement and that scavenged Fe from sedimentary host rocks. During boiling at shallow depths and decreasing iron solubility, these fluids started precipitating Fe-minerals at Torre di Rio mineralization. Mixing with batches of more saline fluids at around 236 °C increased salinity abruptly and marked the end of ore deposition.