The Hypoelliptic Heat Kernel of Infinite-dimensional Lie Groups: Heisenberg-like Quasi-invariance and the Taylor Isomorphism

The Gaussian distribution on R^n translates to infinite-dimensional (separable) Banach spaces by assuming the structure of an abstract Wiener space. The equivalent of the Gaussian distribution on Lie groups is called the heat kernel measure, named for its connection to a version of the Lie group equivalent of the heat equation. In this work, we will investigate combining these ideas to define what it means for G to be a (simply connected graded nilpotent) abstract Wiener Lie group. We will impose 2 major complications. Firstly, we restrict our attention to the hypoelliptic setting, in which the diffusion is only infinitesimally generated by a subset of the possible directions, called ``horizontal'' directions. Secondly, we allow for the possibility that there are infinitely-many ``vertical'' directions. Imposing both of these restrictions simultaneously complicates the analysis, and will require specifying a generalization of the Hörmander (bracket-generating) condition. Presented here are 2 primary results. The first is a quasi-invariance result for Heisenberg-like groups, meaning that we restrict to when G is nilpotent of step 2. There, we show that, under the right conditions, the infinite-dimensional heat kernel measure is invariant under shifts of a certain group, which we call the Cameron-Martin subgroup. The second result is a Taylor isomorphism that allows for G to be of arbitrary step. It provides a classification of the ``L^2 holomorphic'' functions on G. While there are a number of works that illustrate similar results, this work is the first to show such results for infinite-dimensional hypoelliptic diffusions in the presence of infinitely-many vertical directions.