Yilmaz, Atilla2024-09-122024-09-122024-08http://hdl.handle.net/20.500.12613/10708Hamilton-Jacobi equations are a class of partial differential equations that arise in many areas of science and engineering. Originating from classical mechanics, they are widely used in various fields such as optimal control theory, quantitative finance, and game theory. Stochastic homogenization is a phenomenon used to study the behavior of solutions to partial differential equations in stationary ergodic media, aiming to understand how these solutions average out or 'homogenize' over large scales. This process results in effective deterministic descriptions, called effective Hamiltonians, which capture the essential behavior of the system. We consider nonconvex Hamilton-Jacobi equations in one space dimension. We provide a fully constructive proof of homogenization, which yields a formula for the effective Hamiltonian. Our proof employs sublinear correctors, functions extensively discussed in the literature. The proof involves strong induction: we first show homogenization for our base cases, then use gluing processes to generalize the solution for the strong induction. Finally, we extend the result to a wide class of functions. We study the properties of the resulting effective Hamiltonian and investigate the occurrence of flat pieces. Additionally, we develop a Python-based computational tool that performs the same homogenization steps in a computing environment, returning the effective Hamiltonian along with its graph and properties.134 pagesengIN COPYRIGHT- This Rights Statement can be used for an Item that is in copyright. Using this statement implies that the organization making this Item available has determined that the Item is in copyright and either is the rights-holder, has obtained permission from the rights-holder(s) to make their Work(s) available, or makes the Item available under an exception or limitation to copyright (including Fair Use) that entitles it to make the Item available.http://rightsstatements.org/vocab/InC/1.0/MathematicsHamilton-Jacobi EquationsPartial differential equationsProbabilityStochastic homogenizationText158702024-08-30Demirelli_temple_0225E_15870.pdf