Sub-elliptic diffusions on compact groups via Dirichlet form perturbation

TL;DR

This paper extends sub-elliptic theory to infinite-dimensional compact groups, showing how comparison inequalities between intrinsic distances imply domination of Dirichlet forms and transfer of heat kernel properties, with examples on infinite products of SU(2).

Contribution

It introduces a method to relate sub-Laplacians and bi-invariant Laplacians on infinite-dimensional groups via Dirichlet form domination, extending classical sub-elliptic results.

Findings

Comparison inequality implies Dirichlet form domination.

Heat kernel properties transfer under certain conditions.

Explicit examples on infinite products of SU(2).

Abstract

This work provides an extension of parts of the classical finite dimensional sub-elliptic theory in the context of infinite dimensional compact connected metrizable groups. Given a well understood and well behaved bi-invariant Laplacian, $\Delta$, and a sub-Laplacian, $L$, to which intrinsic distances, $d_\Delta$, $d_L$, are naturally attached, we show that a comparison inequality of the form $d_L\le C(d_\Delta)^c$ (for some $0<c\le 1$) implies that the Dirichlet form of a fractional power of $\Delta$ is dominated by the Dirichlet form associated with $L$. We use this result to show that, under additional assumptions, certain good properties of the heat kernel for $\Delta$ are then passed to the heat kernel associated with $L$. Explicit examples on the infinite product of copies of $SU(2)$ are discussed to illustrate these results.