An explicit formula of the limit of the heat kernel measures on the spheres embedded in $\mathbb {R}^\infty$

TL;DR

The paper demonstrates that heat-kernel measures on spheres of radius $\sqrt N$ embedded in $\mathbb{R}^\infty$, centered via a time-dependent shift, converge to a Gaussian measure on $\mathbb{R}^\infty$ as $N\to\infty$. By deriving an explicit limiting parabolic PDE and solving it, the authors obtain a closed-form Gaussian kernel $u^{\infty}_t$ on finite coordinates and extend it to $\mathbb{R}^\infty$ using Kolmogorov extension. The approach combines a PDE-based heuristic with a rigorous operator-theoretic analysis, including a Baker–Campbell–Hausdorff argument, to establish kernel-level convergence of heat kernels on scaled spheres. The result provides an explicit, time-dependent Gaussian description of the limit, connecting geometric diffusion on high-dimensional spheres to infinite-dimensional Gaussian processes with precise variance parameters. This explicit kernel has potential implications for understanding diffusion on large manifolds and for probabilistic representations in infinite-dimensional settings.

Abstract

We show that the heat kernel measures based at the north pole of the spheres $S^{N-1}(\sqrt N)$, with properly scaled radius $\sqrt N$ and adjusted center, converge to a Gaussian measure in $\mathbb R^\infty$, and find an explicit formula for this measure.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1: PeSen
• Proposition 2.2: Coifman1971
• Definition 2.3
• Theorem 2.4
• Proposition 2.5
• proof
• Corollary 2.6
• Remark 2.7