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

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 ${\mathbb{R}}^{\mathrm{\infty }}$, and find an explicit formula for this measure.