I'm interested in diffusion, a.k.a. the heat kernel driven by the Laplace-Beltrami operator, on the $n$-dimensional sphere. There are lots of bounds showing that, for small times, it behaves in a way close to the heat kernel in $\mathbb{R}^n$: that is, the probability $p_t(\theta)$ that we have moved an angle $\theta$ away from the starting point, at time $t$, is bounded by a Gaussian of variance $t$. But I need some control on how the errors or leading constants in these bounds depend on $n$.

The most precise estimates I've found (where the sphere has radius $1$) is due to Molchanov, $$ p_t(\theta) \sim \frac{\mathrm{e}^{-\theta^2/2t}}{(2\pi t)^{n/2}} \left( \frac{\theta}{\sin \theta} \right)^{(n-1)/2} $$ This is the first term of an asymptotic series, and if I understand his paper correctly, for fixed $n$ the next term would give a multiplicative error of $1+O(t)$. But does the constant hidden in $O(t)$ grow rapidly with $n$?

Another family of bounds gives $$ p_t(\theta) \le C \,t^{-n/2} \,\mathrm{e}^{-\theta^2/(4+\delta)t} $$ Any constant value of $\delta$ would be fine with me. But we need $C$ to decay roughly as $(4\pi)^{-n/2}$ to match the normalization of the flat-space Gaussian.

All we really need for our application is the following. If we think of the heat kernel as a stochastic process, let $\theta$ be the angular distance from the initial point. We know that for small enough $t$ we have $$ \mathbb{E}(\theta) \le C' \sqrt{nt} $$ for some constant $C'$, as it would be in flat space; but we need to know this holds for all $t$ up to $1/n$ or so. In other words, we need to know that the error term doesn't do something horrible like $$ \mathbb{E}(\theta) = C' \sqrt{nt} + O(2^n t) \, . $$ This would follow, for instance, from a bound on the error term in Molchanov's estimate above, although this might be overkill.

Thanks!

Cris