Transición de sistemas de partículas con dinámicas probabilistas a ecuaciones diferenciales parciales.

Abstract

In this research project, we will begin by conducting a brief study of an intuitive procedure, where the Central Limit Theorem (CLT) states that the Brownian motion eventually emerges like the hydrodynamic limit of any random walk with jumps of finite variance. A tool that will be used in this procedure is the Fourier transform and it will allow us to appreciate a known result, which is that the density functions of the Brownian motion solve the traditional reaction-diffusion equation. In this sense, we are interested in knowing the kind of stochastic process that will eventually emerge from a random walk with jumps that follow a Pareto distribution or power law. Here we will find a problem, the jumps of a random walk with Pareto jumps have infinite variance, which in this case prevents us from using the TLC. However, by means of a suitable Taylor series expansion, it is possible to use the methodology made for the Brownian motion to show that this type of random walks eventually behaves like a stable process and its density functions resolve a particular type of model known as anomalous reaction-diffusion equation or simply anomalous diffusion equation. The anomalous diffusion model involves a temporal derivative and a fractional spatial derivative, so it will be necessary to study some basic properties of fractional calculus to justify that the density functions of a stable process solve the equations of the anomalous diffusion model. Although it is true, the methodology that is followed for a random walk with finite variance jumps or Pareto jumps provide evidence that connects the theory of differential equations with the theory of stochastic processes, it is necessary to formalize this through the appropriate theory. In this sense, the adequate results of the theory of stable distributions and semigroups will be studied, which will allow us to connect in an elegant and formal way the fractional calculus and the stochastic processes involved in this work.