Diffusions on Manifolds and Curvature

6D.4 Fokker–Planck, Time Reversal and Diffusion Models proved that the Fokker–Planck flow of $dX = -\nabla V\,dt + \sqrt2\,dB$ converges to its Gibbs measure at a rate set by the convexity of $V$, and stated the Bakry–Émery criterion behind it. This chapter moves Brownian motion off flat space onto a Riemannian manifold, where the Laplacian is the Laplace–Beltrami operator (6B.10 Harmonic Coordinates and the Regularity of Metrics, 9A.6 The Laplacian and the Bochner Formula) and convexity of the potential is joined, and partly replaced, by the Ricci curvature of the space. The guiding principle is the one found in 6C.11 Optimal Transport and Ricci Curvature from the side of optimal transport: Ricci curvature measures how much the volume of a small ball changes as its points are transported along geodesics, so it governs how fast diffusing particles forget where they started, and how fast two diffusions started at different points can be brought together.

The chapter has three parts. The first constructs Brownian motion on a manifold, which already has geometric content (in harmonic coordinates, the drift of its stochastic differential equation vanishes). The second develops the Bakry–Émery calculus: the carré du champ $\Gamma$, its iterate $\Gamma_2$, the Bochner formula $\Gamma_2(f) = |\nabla^2f|^2 + \operatorname{Ric}(\nabla f, \nabla f)$, and the curvature-dimension inequality $CD(K, N)$, from which gradient estimates, Poincaré and log-Sobolev inequalities and the Li–Yau inequality all follow by one interpolation argument along the heat semigroup. The third runs Brownian motion along a Ricci flow, where the conjugate heat kernel and Perelman's reduced distance appear as the probabilistic objects they are, and ends with Haslhofer and Naber's characterisation of the Ricci flow by gradient estimates on path space.

By the end of this chapter you will be able to:

• construct Brownian motion on a manifold by a stochastic differential equation in coordinates, and explain why its drift is $-\Gamma^k$ and vanishes in harmonic coordinates;
• describe the Eells–Elworthy–Malliavin construction and geodesic random walks, and state Yau's stochastic completeness theorem;
• define $\Gamma$ and $\Gamma_2$ for a diffusion operator, compute $\Gamma_2$ for the weighted Laplacian on $\mathbb{R}^n$, and state the Bochner formula and the condition $CD(K, N)$;
• prove the gradient estimate $\Gamma(P_tf) \leq e^{-2Kt}P_t\Gamma(f)$ from $CD(K, \infty)$, and deduce the Poincaré inequality with constant $K$;
• explain couplings of Brownian motions, the contraction $d(X_t, Y_t) \leq e^{-Kt}d(x, y)$ for synchronous couplings under $\operatorname{Ric} \geq K$, and reflection coupling;
• describe Brownian motion along a backward Ricci flow, Varadhan's formula and its Ricci flow analogue, Perelman's inequality $f \leq \ell$, and Haslhofer–Naber's characterisation.

5.1Brownian motion on a manifold

On $\mathbb{R}^n$, Brownian motion with generator $\Delta$ is $\sqrt2B_t$, and its transition density is the heat kernel of 6A.3 The Heat Equation on ℝⁿ. On a Riemannian manifold $(M, g)$, Brownian motion is the diffusion with generator $\Delta_g$, the Laplace–Beltrami operator: a continuous process $X_t$ such that $f(X_t) - \int_0^t\Delta_gf(X_s)\,ds$ is a martingale for every smooth $f$ with compact support (6D.2 Martingales and the Itô Integral, 6D.3 Stochastic Differential Equations and Feynman–Kac). Its transition density with respect to the volume measure is the heat kernel $p_t(x, y)$ of the manifold (9B.7 The Heat Equation on a Manifold).

In coordinates. The generator in coordinates is $\Delta_g = g^{ij}\partial_i\partial_j - \Gamma^k\partial_k$ (6B.10 Harmonic Coordinates and the Regularity of Metrics), with diffusion matrix $a = 2g^{-1}$ and first-order coefficient $-\Gamma^k$. So, in a chart, Brownian motion solves the Itô equation

with $B$ a Brownian motion in $\mathbb{R}^n$. The drift $-\Gamma^k = \Delta_gx^k$ is not a force: it is the Itô correction that makes the process isotropic with respect to $g$ rather than with respect to the coordinates. In harmonic coordinates ($\Delta_gx^k = 0$) the drift vanishes and the coordinate functions of Brownian motion are local martingales: harmonic coordinates are the coordinates in which Brownian motion has no drift, which is one more reason they are natural (6B.10 Harmonic Coordinates and the Regularity of Metrics).

Without coordinates. James Eells and David Elworthy (1971) and Paul Malliavin (1978) constructed Brownian motion globally by rolling without slipping. Lift a path to the bundle of orthonormal frames by parallel transport; solve, on that bundle, the Stratonovich equation driven by a Euclidean Brownian motion in the directions of the horizontal vector fields (moving the frame's base point in the direction of each frame vector, transporting the frame in parallel); and project to $M$. The result is Brownian motion on $M$, together with a stochastic parallel transport along its paths, which is how one transports vectors and tensors along Brownian paths, though the paths are nowhere differentiable. An elementary approximation is the geodesic random walk: from the current point, move a distance $\sqrt{2n\,h}$ along a geodesic in a uniformly random direction, and repeat every time step $h$; as $h \to 0$ these walks converge to Brownian motion (Jørgensen, 1975), which is how Brownian motion on curved spaces is simulated in practice.

Stochastic completeness. On a complete manifold, Brownian motion might reach infinity in finite time (explode), in which case $\int p_t(x, y)\,dV(y) < 1$: heat leaks out through infinity. Shing-Tung Yau (1978) proved that this cannot happen if the Ricci curvature is bounded below; Alexander Grigor'yan (1986) gave the sharp volume criterion $\int^\infty\frac{r\,dr}{\log\operatorname{Vol}B_r} = \infty$. The Ricci flow theory on complete manifolds with bounded curvature (11A.9 Flows on Complete Manifolds) always works in this stochastically complete setting, which is why its maximum principles, heat kernels and conjugate heat kernels behave as on closed manifolds.

5.2The Bakry–Émery calculus

Let $L$ be a diffusion operator: on a manifold, $L = \Delta_g$, or more generally $L = \Delta_g - \nabla V\cdot\nabla$, which is symmetric for the measure $e^{-V}dV$. Dominique Bakry and Michel Émery (1985) built a calculus of curvature from $L$ alone. Define the carré du champ

and its iterate

Write $\Gamma(f) = \Gamma(f, f)$, $\Gamma_2(f) = \Gamma_2(f, f)$. For $L = \Delta_g - \nabla V\cdot\nabla$, the product rule gives $\Gamma(f) = |\nabla f|^2$: the carré du champ is the squared gradient, recovered from the operator. And $\Gamma_2$ is given by the Bochner formula.

So for the weighted Laplacian, the "Ricci curvature" seen by the operator is $\operatorname{Ric} + \nabla^2V$, the Bakry–Émery tensor of 6C.11 Optimal Transport and Ricci Curvature, and Perelman's $\operatorname{Ric} + \nabla^2f$. Since $|\nabla^2f|^2 \geq \frac1n(\Delta f)^2$ (Cauchy–Schwarz on the eigenvalues of the Hessian), an unweighted manifold with $\operatorname{Ric} \geq K$ satisfies the curvature-dimension condition

with $N = n$. For the weighted Laplacian, $CD(K, \infty)$ is $\operatorname{Ric} + \nabla^2V \geq Kg$. The condition makes sense for any diffusion operator, including ones on spaces with no Riemannian structure, and it is equivalent, on smooth spaces, to the Lott–Sturm–Villani condition of 6C.11 Optimal Transport and Ricci Curvature: the analytic and the transport definitions of Ricci curvature bounds agree.

Everything follows by interpolation. Let $P_t = e^{tL}$ be the heat semigroup, $P_tf(x) = \mathbb{E}_xf(X_t)$ (6B.13 Semigroups and Evolution Equations, 6D.3 Stochastic Differential Equations and Feynman–Kac). The method is to interpolate between a quantity at time $0$ and at time $t$ by a function of $s \in [0, t]$, and differentiate.

The gradient bound says that the heat flow contracts gradients at the rate $e^{-Kt}$: positive curvature makes diffusion forget its starting point exponentially fast, and nonnegative curvature at least does not let gradients grow. Integrating it in time gives the functional inequalities.

The same interpolation applied to $P_s\big(P_{t-s}f\log P_{t-s}f\big)$, with the stronger gradient bound $\sqrt{\Gamma(P_tf)} \leq e^{-Kt}P_t\sqrt{\Gamma(f)}$ that $CD(K, \infty)$ also implies, gives the logarithmic Sobolev inequality with constant $K$, the Bakry–Émery criterion of 6D.4 Fokker–Planck, Time Reversal and Diffusion Models; on the Gaussian space, where $K = 1$, this is Gross's inequality (6A.12 Entropy, Information and Diffusion), and on the unit sphere $S^n$, where $\operatorname{Ric} = (n - 1)g$, it gives log-Sobolev with constant $n - 1$ (improved by the dimension term to $n$). Using the full $CD(0, n)$ condition, with its dimensional term $\frac1n(Lf)^2$, the interpolation of $P_s\big(P_{t-s}f\,\Gamma(\log P_{t-s}f)\big)$ gives the Li–Yau inequality $\Delta\log u \geq -\frac{n}{2t}$ for positive solutions of the heat equation on manifolds with $\operatorname{Ric} \geq 0$ (6A.12 Entropy, Information and Diffusion, 9B.7 The Heat Equation on a Manifold), with the Euclidean heat kernel as the equality case. Bakry and Ledoux (2006) gave this semigroup proof; Li and Yau's 1986 proof used the maximum principle.

5.3Couplings

A probabilist's proof that two diffusions forget their starting points is to run them together. A coupling of Brownian motions started at $x$ and $y$ is a pair $(X_t, Y_t)$ on one probability space, each marginally a Brownian motion from its starting point. Then for any bounded $f$,

so a coupling in which the two processes meet quickly gives a strong bound on the gradient of $P_tf$.

Synchronous coupling. On $\mathbb{R}^n$, drive both processes by the same Brownian motion: $X_t = x + \sqrt2B_t$, $Y_t = y + \sqrt2B_t$, and the distance stays $|x - y|$. On a manifold, drive both by the same Euclidean noise, transported to each by parallel transport along the geodesic between them. Positive Ricci curvature now pulls the processes together: the second variation of distance along a family of geodesics (9A.7 Jacobi Fields and Curvature versus Topology) produces a drift $-\operatorname{Ric}$-proportional in the distance, and Max-Karl von Renesse and Karl-Theodor Sturm (2005) proved that $\operatorname{Ric} \geq K$ is equivalent to the existence, for all $x$, $y$, of couplings with

Averaging, this is the Wasserstein contraction $W_2(\mu P_t, \nu P_t) \leq e^{-Kt}W_2(\mu, \nu)$ of 6C.11 Optimal Transport and Ricci Curvature, now path by path. It is the probabilistic form of the gradient bound: $|\nabla P_tf| \leq e^{-Kt}P_t|\nabla f|$ follows by differentiating $\mathbb{E}f(X_t) - \mathbb{E}f(Y_t)$ as $y \to x$.

Reflection coupling. To make the processes meet, drive them by reflected noise: the component of the noise along the geodesic from $X$ to $Y$ is reversed for $Y$, so the distance performs a one-dimensional Brownian motion (plus curvature drift) and hits zero. Wilfrid Kendall (1986) and Michael Cranston (1991) used reflection coupling on manifolds to prove gradient estimates for harmonic functions and Harnack inequalities with constants depending only on a lower Ricci bound, reproving Cheng–Yau type estimates by probability. Coupling is also an algorithmic tool: the convergence of Markov chain Monte Carlo is certified by couplings, and James Propp and David Wilson's coupling from the past (1996) produces exact samples from the stationary distribution of some chains by running coupled copies from the distant past until they coalesce.

5.4Brownian motion along a Ricci flow

Now let the metric move. If $g(\tau)$, $0 \leq \tau \leq T$, is a family of metrics, a $g(\tau)$-Brownian motion is a diffusion whose generator at time $\tau$ is $\Delta_{g(\tau)}$. Its transition densities, with respect to the moving volume $dV_{g(\tau)}$, satisfy the forward equation $\partial_\tau u = \Delta_{g(\tau)}u - \frac12\operatorname{tr}_g(\partial_\tau g)\,u$, the correction compensating the change of volume. For a backward Ricci flow, $\partial_\tau g = 2\operatorname{Ric}$, the correction is $-Ru$ and the forward equation is Perelman's conjugate heat equation $\partial_\tau u = \Delta u - Ru$: the conjugate heat kernel is the transition density of Brownian motion along the backward Ricci flow (6D.3 Stochastic Differential Equations and Feynman–Kac, 6D.4 Fokker–Planck, Time Reversal and Diffusion Models). Marc Arnaudon, Koléhè Abdoulaye Coulibaly and Anton Thalmaier (2008) constructed this Brownian motion with its stochastic parallel transport, and observed a striking fact: along a backward Ricci flow, the damped parallel transport that appears in the probabilistic Bochner formula coincides with the ordinary stochastic parallel transport and is an isometry, because the time derivative of the metric exactly cancels the Ricci term. That cancellation is the probabilistic face of the computation behind Perelman's monotonicity formulas.

Varadhan's formula and the reduced distance. On a fixed manifold, Srinivasa Varadhan (1967) proved that the heat kernel at small times is governed by the distance:

the analogue of the Gaussian exponent $\frac{|x - y|^2}{4t}$, by large deviations for Brownian paths (the probability of a Brownian bridge following a path is controlled by the path's energy). Along a backward Ricci flow, the energy of a path is replaced by Perelman's $\mathcal L$-length $\int_0^\tau\sqrt s\big(R + |\dot\gamma|^2\big)ds$, and the role of $\frac{d^2}{4t}$ is played by the reduced distance $\ell(q, \tau)$ (12A.5 Reduced Distance and Reduced Volume). Perelman proved that the conjugate heat kernel $u = (4\pi\tau)^{-n/2}e^{-f}$ based at a point satisfies

(his Corollary 9.5), so that $u \geq (4\pi\tau)^{-n/2}e^{-\ell}$: the conjugate heat kernel is bounded below by the "Gaussian" built from the reduced distance, the inequality that links his two monotone quantities. In flat space with the trivial flow, $f = \ell = \frac{|q - p|^2}{4\tau}$ and equality holds.

The Ricci flow among all flows. Perelman remarked (his §9.6) that the Ricci flow is characterised among all evolutions of a metric by the infinitesimal behaviour of the fundamental solutions of the conjugate heat equation: for a general evolution $\partial_tg = A$, a natural quantity built from the fundamental solution is of order $1$ near the base point, and it is small exactly when $A = -2\operatorname{Ric}$. Robert Haslhofer and Aaron Naber (2018) found a characterisation on path space. For a family of metrics, consider martingales on the space of $g(\tau)$-Brownian paths and their gradients (in the sense of Malliavin calculus, using stochastic parallel transport). They proved that the family evolves by the Ricci flow exactly when a sharp infinite-dimensional gradient estimate holds for all functions on the path space of space-time, and gave further characterisations by the regularity of martingales on path space and by log-Sobolev and spectral-gap inequalities there; the gradient estimate generalises the Bakry–Émery estimate $|\nabla P_tf| \leq P_t|\nabla f|$ (the case $K = 0$ of Proposition 5.2) from functions of the endpoint to functions of the whole path. Earlier in the same programme, Naber had characterised bounded Ricci curvature of a fixed manifold by such estimates. In this sense the Ricci flow is the flow along which Brownian motion has "zero Ricci curvature in space-time", the probabilistic form of the slogan that the Ricci flow is the heat equation for the metric.

5.5History

Itô constructed diffusions on manifolds in the 1950s; Eells and Elworthy (1971) and Malliavin (1978) gave the frame-bundle construction, and Jørgensen proved the convergence of geodesic random walks in 1975. Debye's rotational diffusion dates from 1929 and Perrin's from 1934. Varadhan's formula appeared in 1967. Yau proved stochastic completeness under a Ricci lower bound in 1978, and Grigor'yan his volume criterion in 1986. Bakry and Émery introduced $\Gamma_2$ in 1985; Kendall's coupling paper appeared in 1986 and Cranston's in 1991. Li and Yau's inequality dates from 1986, and the semigroup proof from the work of Bakry and Ledoux (2006). Propp and Wilson's coupling from the past appeared in 1996. Von Renesse and Sturm's coupling characterisation of Ricci bounds appeared in 2005. Perelman's §9 appeared in 2002, Arnaudon, Coulibaly and Thalmaier's Brownian motion along a Ricci flow in 2008, and Haslhofer and Naber's path-space characterisation of the Ricci flow in 2018.

5.6Exercises

Solution

(a) $u(t, x) = \mathbb{E}_xX^3_t$ solves $\partial_tu = D\Delta u$ with $u(0) = x_3$ (the backward equation, 6D.3 Stochastic Differential Equations and Feynman–Kac), and $e^{-2Dt}x_3$ is a solution, since $x_3$ is an eigenfunction (a spherical harmonic of degree one, eigenvalue $\ell(\ell + 1) = 2$, 9A.6 The Laplacian and the Bochner Formula). By uniqueness on the compact sphere, $u = e^{-2Dt}x_3$. (b) By rotational symmetry the same holds for every linear function $\langle\cdot, v\rangle$, so $\mathbb{E}_x\langle X_t, x\rangle = e^{-2Dt}\langle x, x\rangle = e^{-2Dt}$.

Solution

By Itô's formula with generator $\Delta$, $dr_t = \Delta r\,dt + dM_t = \frac{1}{r_t}dt + \sqrt2\,dW_t$, where $W$ is a one-dimensional Brownian motion ($|\nabla r| = 1$). This is a Bessel process of dimension two (in the normalisation with generator $\partial_r^2 + \frac1r\partial_r$). The outward drift is the Itô correction: the circles of radius $r + \delta$ are longer than those of radius $r - \delta$, so a symmetric random step from radius $r$ lands on average at slightly larger radius; it is the drift $-\Gamma^r = \frac1r$ of the coordinate system, not a force.

Solution

(a) $\Gamma(f) = |\nabla f|^2$ and, by Theorem 5.1 with $\nabla^2V = I$, $\Gamma_2(f) = |\nabla^2f|^2 + |\nabla f|^2 \geq \Gamma(f)$: $CD(1, \infty)$. (b) $L x_1 = -x_1$, so $P_tx_1 = e^{-t}x_1$ and $|\nabla P_tx_1| = e^{-t}$, while $P_t|\nabla x_1| = P_t1 = 1$: equality. Linear functions are the extremal case, as for the Gaussian Poincaré inequality.

Solution

$\Gamma_2 = |\nabla^2\phi|^2 + (n - 1)|\nabla\phi|^2 \geq \frac1n(\Delta\phi)^2 + (n - 1)|\nabla\phi|^2$. Integrate over $S^n$: $\int L\Gamma(\phi) = 0$, and $\int\Gamma(\phi, \Delta\phi) = \int\nabla\phi\cdot\nabla\Delta\phi = -\int(\Delta\phi)^2$, so $\int\Gamma_2(\phi) = \int(\Delta\phi)^2 = \lambda^2\int\phi^2$. And $\int\Gamma(\phi) = \lambda\int\phi^2$. So $\lambda^2 \geq \frac{\lambda^2}{n} + (n - 1)\lambda$, that is $\lambda(1 - \frac1n) \geq n - 1$, $\lambda \geq n$. Equality holds for the linear functions restricted to the sphere (eigenvalue $n$).

Solution

$-4t\log p_t = 2nt\log(4\pi t) + |x - y|^2$, and $t\log t \to 0$. For the trivial flow ($R = 0$, $\partial_\tau g = 0$), the conjugate heat equation is the heat equation in $\tau$, its kernel based at $p$ is $(4\pi\tau)^{-n/2}e^{-|q-p|^2/4\tau}$, so $f = \frac{|q - p|^2}{4\tau}$. On flat space the $\mathcal L$-length of a path is $\int_0^\tau\sqrt s|\dot\gamma|^2ds$, minimised (with $\sigma = 2\sqrt s$) by straight lines, giving $L = \frac{|q - p|^2}{2\sqrt\tau}$ and $\ell = \frac{L}{2\sqrt\tau} = \frac{|q - p|^2}{4\tau}$ (12A.5 Reduced Distance and Reduced Volume). So $f = \ell$, the equality case of Perelman's Corollary 9.5.