From the classical to the stochastic heat equation

Abstract

[en] This bachelor’s thesis revolves around the connection between stochastic processes and the heat equation. The main goal is to carry out a thorough study of the transition from the classical to the stochastic one-dimensional heat equation. In order to develop the mathematical framework for linear stochastic partial differential equations, we use tools of basic probability theory, calculus and functional analysis. We start with a concise study of the classical deterministic heat equation, from its physical derivation to the search for explicit solutions under specific conditions. Then, we describe the mathematical foundations of the stochastic version of this partial differential equation, focusing on Gaussian stochastic processes. On that basis, we define the stochastic heat equation on $\mathbb{R}$. Finally, we conclude this project with a comprehensive analysis of its solutions’ continuity properties.