How long does it take a Brownian motion particle to be uniformly distributed on a compact manifold?

How long does it take for a Brownian motion on a (compact) smooth manifold to reach the uniform distribution (it's steady-state)?

More precisely, let $$X_t$$ be a Brownian motion on a manifold $$(M, g)$$ in the sense that $$dX_t = \frac12 \nabla_M \cdot g^{-1}(X_t) dt + \sqrt{g^{-1}(X_t)} dB_t,$$ where $$\nabla_M$$ is the manifold-divergence. It is well known that that the steady-state measure $$p_\infty(x)dx$$ of $$X_\infty$$ is proportional to $$\sqrt{\det g(x)}dx$$, i.e. the volume measure on $$M$$, hence $$X_\infty$$ is uniformly distributed on $$M$$.

For practical purposes, it would be nice to know (even heuristically!), how large must $$T$$ be for a simulated sample path $$\{X_{t_i}\}_{i=1}^n$$ to obtain approximate uniformity for all $$t_j > T$$. By approximate uniformity, we mean the total variation distance between the observed measure and the true uniform measure is small; if $$U(dx)= \frac{1}{\nu(M)} \nu(dx)$$ is the exact uniform measure, where $$\nu(dx) = \sqrt{\det g(x)} dx$$ is the volume measure, and $$P(T)$$ is empirical measure (or PDF) associated to $$X_{t_i}$$ for $$t_i>T$$, $$T$$ large, then the total variation distance $$\delta(P(T), U)$$ should become exceedingly small as $$T$$ grows indefinitely large. Can we pick $$T=T(\epsilon)$$ so that $$\delta(P(T), U)<\epsilon$$? I am definitely not opposed to using other probability metrics to measure the "approximate uniformity", however!

I am familiar with similar problems in discrete settings, like how many steps of a random walk it takes for the visits to be uniform on finite groups. There is a nice cut-off phenomena, and a slew of results in this setting (see Diaconis' book). It is not obvious if these are adaptable to SDEs, in this setting but this is where I will look into first, while I pose the question out in the open.

• "Approximately uniformly distributed" in which sense? Do you have a preferred metric regarding this or are you just looking for an answer regarding some probability metric? Jun 17, 2023 at 20:28
• @SmallDeviation I am familiar with total variation distance metric of probabilities, so that might be preferable as a first choice. But I am not opposed to other options, certainly. Jun 17, 2023 at 21:23

Yes, there is a "slew" of results in this area. You could start with:

Saloff-Coste, L., Precise estimates on the rate at which certain diffusions tend to equilibrium, Math. Z. 217, No. 4, 641-677 (1994). ZBL0815.60074.

Saloff-Coste, L., Convergence to equilibrium and logarithmic Sobolev constant on manifolds with Ricci curvature bounded below, Colloq. Math. 67, No. 1, 109-121 (1994). ZBL0816.53027.

Very roughly speaking, what you get is that $$\delta(P(T), U) \leq C e^{-T \lambda}$$ where $$\lambda$$ is the spectral gap, i.e. the first nonzero eigenvalue of the Laplace-Beltrami operator, and so the distance decreases exponentially fast.

There are explicit lower bounds for $$\lambda$$ available in certain settings, e.g.

• if $$(M,g)$$ has nonnegative Ricci curvature (see Theorem 5 of the second paper cited above)

• if it is a homogeneous space - see Li, Peter, Eigenvalue estimates on homogeneous manifolds, Comment. Math. Helv. 55, 347-363 (1980). ZBL0451.53036. for the remarkable estimate $$\lambda \ge \frac{\pi^2}{4 d^2}$$ where $$d$$ is the diameter of $$(M,g)$$.

• +1 Wow! This is fascinating. Thank you for the many references. Jun 18, 2023 at 1:19