Sub-elliptic diffusions on compact groups via Dirichlet form perturbation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910850807234560 |
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| author | Hou, Qi Saloff-Coste, Laurent |
| author_facet | Hou, Qi Saloff-Coste, Laurent |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_21152 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sub-elliptic diffusions on compact groups via Dirichlet form perturbation Hou, Qi Saloff-Coste, Laurent Probability Analysis of PDEs Functional Analysis 22E66, 31C25, 35K08, 35H20, 35K65, 60B15, 60G51 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. |
| title | Sub-elliptic diffusions on compact groups via Dirichlet form perturbation |
| topic | Probability Analysis of PDEs Functional Analysis 22E66, 31C25, 35K08, 35H20, 35K65, 60B15, 60G51 |
| url | https://arxiv.org/abs/2502.21152 |