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.

Recycling of end of life concrete fines (0 - 4 mm) into silica and cement

The purpose of this work is to find a methodology in order to make possible the recycling of fines (0 - 4 mm) in the Construction and Demolition Waste (CDW) process. At the moment this fraction is a not desired by-product: it has high contaminant content, it has to be separated from the coarse fraction, because of its high water absorption which can affect the properties of the concrete. In fact, in some countries the use of fines recycled aggregates is highly restricted or even banned. This work is placed inside the European project C2CA (from Concrete to Cement and Clean Aggregates) and it has been held in the Faculty of Civil Engineering and Geosciences of the Technical University of Delft, in particular, in the laboratory of Resources And Recycling. This research proposes some procedures in order to close the loop of the entire recycling process. After the classification done by ADR (Advanced Dry Recovery) the two fractions "airknife" and "rotor" (that together constitute the fraction 0 - 4 mm) are inserted in a new machine that works at high temperatures. The temperatures analysed in this research are 600 °C and 750 °C, cause at that temperature it is supposed that the cement bounds become very weak. The final goal is "to clean" the coarse fraction (0,250 - 4 mm) from the cement still attached to the sand and try to concentrate the cement paste in the fraction 0 - 0,250 mm. This new set-up is able to dry the material in very few seconds, divide it into two fractions (the coarse one and the fine one) thanks to the air and increase the amount of fines (0 - 0,250 mm) promoting the attrition between the particles through a vibration device. The coarse fraction is then processed in a ball mill in order to improve the result and reach the final goal. Thanks to the high temperature it is possible to markedly reduce the milling time. The sand 0 - 2 mm, after being heated and milled is used to replace 100% of norm sand in mortar production. The results are very promising: the mortar made with recycled sand reaches an early strength, in fact the increment with respect to the mortar made with norm sand is 20% after three days and 7% after seven days. With this research it has been demonstrated that once the temperature is increased it is possible to obtain a clean coarse fraction (0,250 - 4 mm), free from cement paste that is concentrated in the fine fraction 0 - 0,250 mm. The milling time and the drying time can be largely reduced. The recycled sand shows better performance in terms of mechanical properties with respect to the natural one.