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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.

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    $\begingroup$ "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? $\endgroup$ Jun 17, 2023 at 20:28
  • $\begingroup$ @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. $\endgroup$ Jun 17, 2023 at 21:23

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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 $(M,g)$.

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

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