Moment estimates for the stochastic heat equation on Cartan-Hadamard manifolds

Fabrice Baudoin; Hongyi Chen; Cheng Ouyang

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arXiv · 2411.09614

Moment estimates for the stochastic heat equation on Cartan-Hadamard manifolds

Fabrice Baudoin·Hongyi Chen·Cheng Ouyang

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Abstract

We study the effect of curvature on the Parabolic Anderson model by posing it over a Cartan-Hadamard manifold. We first construct a family of noises white in time and colored in space parameterized by a regularity parameter $\alpha$, which we use to explore regularity requirements for well-posedness. Then, we show that conditions on the heat kernel imply an exponential in time upper bound for the moments of the solution, and a lower bound for sectional curvature imply a corresponding lower bound. These results hold if the noise is strong enough, where the needed strength of the noise is affected by sectional curvature.

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Fabrice Baudoin, Hongyi Chen, Cheng Ouyang. 2024-11-14. Moment estimates for the stochastic heat equation on Cartan-Hadamard manifolds. https://arxiv.org/abs/2411.09614

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