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Stochastic Homogenization of Nonconvex Hamilton-Jacobi Equations in One Dimension

Demirelli, Abdurrahman
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Thesis/Dissertation
Date
2024-08
Advisor
Yilmaz, Atilla
Committee member
Rider, Brian (Brian C.)
Grabovsky, Yury
Futer, David
Kosygina, Elena
Group
Department
Mathematics
Subject
Mathematics
 Hamilton-Jacobi Equations
 Partial differential equations
 Probability
 Stochastic homogenization
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http://hdl.handle.net/20.500.12613/10708
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Demirelli_temple_0225E_15870.pdf
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http://dx.doi.org/10.34944/dspace/10670
Abstract
Hamilton-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.
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