Sub-elliptic diffusions on compact groups via Dirichlet form perturbation
Qi Hou, Laurent Saloff-Coste
TL;DR
The paper extends finite-dimensional sub-elliptic theory to infinite-dimensional compact groups by relating intrinsic distances and Dirichlet forms of two left-invariant (sub-)Laplacians. It proves that a distance comparison $d_L\le C(d_Δ)^c$ together with CK$λ$ bounds for $Δ$ implies fractional Dirichlet-form domination of $Δ^{ε}$ by $L$, enabling transfer of heat-kernel properties from $Δ$ to $L$. Under precise conditions, it yields CK$γ$ bounds and Gaussian-type upper bounds for the heat kernel of $L$, along with corollaries such as a parabolic Harnack principle and CK$0^+$ stability. The paper then provides explicit infinite-dimensional examples on $G=\prod SU(2)$ with detailed coefficient conditions that ensure the distance and Dirichlet-form comparisons, illustrating the applicability to and limitations of the theory in a concrete noncentral, infinite-product setting.
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, $Δ$, and a sub-Laplacian, $L$, to which intrinsic distances, $d_Δ$, $d_L$, are naturally attached, we show that a comparison inequality of the form $d_L\le C(d_Δ)^c$ (for some $0<c\le 1$) implies that the Dirichlet form of a fractional power of $Δ$ 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 $Δ$ 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.