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Geometric convolution and non-Gaussian kernels for hypoelliptic diffusions

Banff International Research Station for Mathematical Innovation and Discovery

When considering a reasonable Brownian motion in a subriemannian manifold, we have a good idea of what to expect its kernel to look like for small times. It should be more or less a time singularity $t^{-Q/2}$ for some $Q$, multiplied by a Gaussian function where the subriemannian distance $d$ replaces what would be the norm in Euclidean space. Along a suitable decomposition of the tangent space, the distance $d$ behaves more or less like the $k$th power of a given smooth distance, where $k$ depends on the chosen factor in the decomposition. In many strictly hypoelliptic settings, we do not expect the kernel to be well approximated by a Gaussian. However, the grading phenomenon still occurs. In this talk, I will suggest a way to encode this geometric information in appropriate function spaces, so that we can consider non-Gaussian models, and apply some Duhamel formula to show the exact kernel vanishes at a rate prescribed by the grading.

Mathematics; Probability Theory And Stochastic Processes; Global Analysis; Analysis On Manifolds

video/mp4

Author affiliation: University of Notre Dame

2021-09-08

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/79661

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/