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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

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  • $\begingroup$ What's a reference or link to the Molchanov paper? $\endgroup$ – Noam D. Elkies Jul 26 '14 at 23:49
  • $\begingroup$ Diffusion Processes and Riemannian Geometry, Russian Math. Surveys 30:1 (1975), 1-63. $\endgroup$ – Cristopher Moore Jul 29 '14 at 5:24
  • $\begingroup$ Good news: I can show that for diffusion on $S_n$ after time t, the angle $\theta$ away from the starting point has second moment $\mathbb{E}[\theta^2] \le 2nt$, i.e., it's bounded below what it would be in flat space. It turns out this is easy: we just use the Laplacian to compute the time derivative of $\mathbb{E}[\theta^2]$, and integrate by parts. In particular, this shows that $\mathbb{E}[\theta] \le \sqrt{2nt}$. Note that this applies for all $t$, not just for short times. $\endgroup$ – Cristopher Moore Jul 29 '14 at 5:26
  • $\begingroup$ Of course, the second moment is special. A similar integration doesn't work for other (higher) moments. So I'm still curious whether $\theta$ on $S_n$ is stochastically dominated by $\theta$ on $\mathbb{R}^n$. $\endgroup$ – Cristopher Moore Jul 29 '14 at 13:27
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Yes, it is true that $\theta$ on $S_n$ is dominated by $\theta$ on $\mathbb{R}^n$.

Let $(B_t)$ be a Brownian motion on the sphere. The radial process $\theta_t=d(x,B_t)$ is a Jacobi process, that is a Markov process with generator

$ L=\frac{n-1}{2} \text{cotan} (r) \frac{d}{dr} +\frac{1}{2} \frac{d^2}{dr^2 } $

Since $ \text{cotan} (r) \le \frac{1}{r}$, we deduce from the comparison theorem for stochastic differential equations that

$ \theta_t \le \beta_t $

where $\beta_t$ is a Bessel process, that is a Markov process with generator

$ L=\frac{n-1}{2r} \frac{d}{dr} +\frac{1}{2} \frac{d^2}{dr^2 } $

The Bessel process is the radial part of the Euclidean Brownian motion in $\mathbb{R}^n$, so you get the desired stochastic domination.

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