El movimiento Browniano y su geometría

Abstract

[en] Brownian motion is probably the most studied stochastic process. This kind of movement can be found in many situations, as in the movement of a particle suspended in water. In this work, we try to understand the brownian motion from a theoretical point of view. First of all, in the first chapter, we introduce the concept of stochastic process and some of its properties, as well as some of the most important theorems, concerning its existence and continuity. The last section of the chapter introduces the concept of Hausdorff measure and dimension. In the second chapter, we approach the Brownian motion. Firstly, we define the process and then we proof its existence. Moreover, we also show some of the most important properties of the Brownian motion, for the one dimensional case and the general case, n > 1. To conclude, we study the Hausdorff dimension of the Brownian motion. The chapter 3 deepens in the concept of the planar Brownian motion. For this purpose, we show some of the classical properties that P. Lévy introduces in one of his famous books ”Processus Stochastiques et Mouvement Brownien”. Then, we study a concept that P. Lévy defined but he did not develop: the convex hull of the planar Brownian motion. Finally, we analyze a paper published in 2017 by James McRedmon and Chang Xu, which delves into the upper and lower bound of the diameter in the convex hull. Ultimately, in the last chapter, we briefly explain how we simulate the Brownian motion in R and we describe the method followed to obtain the figures of this work.