Well-Posedness Threshold and Moment Asymptotics of the Parabolic Anderson Model on Riemannian Manifolds

The subject of this thesis is the parabolic Anderson model, the linear multiplicative stochastic heat equation. We consider the equation on Riemannian manifolds to study how geometry and topology can affect interesting properties of the solution. We focus on two topics: well-posedness and large time moment asymptotics. We first introduce the equation, the basic theory of Gaussian noises and solutions of SPDEs, and relevant existing literature before getting into the work done for this thesis. For both technical and heuristic reasons, we only consider settings with non-positive sectional curvature in this thesis. Specifically, we consider compact manifolds with non-positive sectional curvature and Cartan-Hadamard manifolds, due to our knowledge of how the heat kernel behaves on them. Although both problems are studied in both settings, each setting places emphasis on a different problem. Results in the compact setting, presented in Chapter 3, is based on the paper \cite{CO25}. Here, we allow the initial condition to be a finite measure, and focus on obtaining the sharp regularity requirement (what is known as Dalang's condition in SPDE literature) on the noise in order for the equation to be well-posed. The highlight of this part is that although the result is not different from known ones in Euclidean settings, we can see through the proof why it is not so, based on a delicate treatment of the density of Brownian bridges and how they relate to the geodesic geometry of non-positively curved manifolds. We obtain moment upper bounds as a part of our machinery, and under some technical assumptions, obtain matching lower bounds based on an argument motivated by the ergodicity of Brownian motion. Results in the Cartan-Hadamard manifold setting, presented in Chapter 4, is based on \cite{BCO24}. Here, we make the initial condition a bounded function so that an easier version of the well-posedness argument is accessible (though we doubt that this will be a fundamental difficulty). The focus of this section instead is on the large time moment asymptotics, a topic of interest due to its connection with the phenomenon of intermittency. We prove that if the sectional curvature has a negative upper bound, then there is always a phase transition in the inverse temperature/noise intensity parameter. More specifically, at high temperature the moments would be uniformly bounded or decaying depending on the initial condition, but they grow exponentially at low temperatures. This is contrast to the Euclidean analogue for the class of noises we are considering, which always exhibit exponential growth of the moments in time. Remarkably, we were able to produce a largely matching (though still not sharp) lower bound. We conclude with a corollary involving intermittency characterizations.

Author dc:creator

• Hongyi Chen (2097328)

• Mathematics

Statement dc:rights

• In Copyright

DOI dc:identifier

https://doi.org/10.25417/uic.32991881.v1

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oai:figshare.com:article/32991881