Fractal Mehler kernels and nonlinear geometric flows

TL;DR

The paper introduces a two-parameter generalized Mehler kernel $G_{\ alpha,\beta}$ and shows it encodes the heat kernel of a fractal Baouendi-Grushin flow on Heisenberg-type groups when $\beta=\alpha+k$, linking the linear heat kernel to a fractal dimension $m_\alpha=2\alpha$ via a conformal-type energy identity. Through Fourier analysis, Bochner/Gegenbauer identities, and hypergeometric transformations, it provides an explicit kernel representation and proves the energy identity $\mathscr E_{\alpha,\alpha+k}(z,\sigma) \propto N(z,\sigma)^{-(2\alpha+2k-2)}$, illustrating a deep connection between sub-Riemannian heat kernels and fractal geometry. The work also outlines a nonlinear direction by proposing a sub-Riemannian analogue of the normalized $p$-Laplacian flow, including a candidate kernel $G_p((z,\sigma),t)$, and it presents two open problems to advance this nonlinear, fractal-geometry framework in sub-Riemannian spaces.

Abstract

In this paper we introduce a two-parameter family of Mehler kernels and connect them to a class of Baouendi-Grushin flows in fractal dimension. We also highlight a link with a geometric fully nonlinear equation and formulate two questions.

Theorems & Definitions (10)

• Theorem 1.1
• Proposition 1.2
• Theorem 1.3
• Proposition 2.1
• Lemma 2.2
• proof
• proof : Proof of Proposition \ref{['P:limit']}
• Proposition 3.1
• Proposition 4.1
• proof