Sub-elliptic diffusions on compact groups via Dirichlet form perturbation

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.