Positivity of Heat Kernels

Abstract: Partial di˙erential equations are a well-studied field of mathematics, and in this thesis we attempt to use some of the newer methods, including path integrals (also known as Feynman path integrals) and the so-called geometric approach, to find conditions for the heat kernel of a di˙erential operator on a certain form to be zero. We also derive a maximum principle, more general than the classical one, that allows for degenerate di˙erential operators, where the degeneracy is controlled by a Muckenhoupt weight. While we find the path integral method too deeply flawed to use, the geometric approach yields some results which show that the zeros of the coeÿcient of the second order term in the di˙erential operator control information transfer within the domain of the solution.