Bakry–Émery \(\Gamma_2\) algebra and \(CD(K, \infty)\)

Wiki card — concept node for exp-families-stability.

(i) Formal statement

Let \((M, g, L)\) be a diffusion triple with carré-du-champ \(\Gamma(f, h) := \tfrac12\bigl(L(fh) - fLh - hLf\bigr)\). The iterated carré-du-champ is defined by

\[ \Gamma_2(f, h) \;:=\; \tfrac12\Bigl(L\,\Gamma(f,h) \;-\; \Gamma(f, Lh) \;-\; \Gamma(Lf, h)\Bigr). \tag{$\Gamma_2$} \]

One writes \(\Gamma_2(f) := \Gamma_2(f, f)\). The curvature–dimension condition \(CD(K, \infty)\) (Bakry–Émery 1985) is the pointwise inequality

\[ \boxed{\;\Gamma_2(f) \;\ge\; K\, \Gamma(f)\qquad \forall f \in \mathcal{C}.\;} \tag{$CD(K, \infty)$} \]

For \(L = \tfrac12\Delta_g + Z\) on a Riemannian manifold, Bochner's formula gives

\[ \Gamma_2(f) \;=\; \tfrac12\,\|\nabla^2 f\|_{\mathrm{HS}}^2 \;+\; \tfrac12\bigl(\mathrm{Ric}_g - \nabla^{\mathrm{sym}} Z\bigr)(\nabla f, \nabla f), \]

so \(CD(K, \infty)\) reduces to a lower bound on the Bakry–Émery Ricci tensor \(\mathrm{Ric}_g - \nabla^{\mathrm{sym}} Z \succeq K g\).

(ii) Role in Q1 / Q2 / Q3

• Q1 (Wheeler's invariant reformulation). The closure condition \((\dagger) \colon \Gamma(\varphi_i, \varphi_j) \in \mathcal{A},\; L\varphi_i \in \mathcal{A}\) says \(\mathcal{A}\) is a finite-dimensional subalgebra of the diffusion algebra associated to \(L\). Iterating \(L\) and \(\Gamma\) on \(\mathcal{A}\) stays in \(\mathcal{A}\), so \(\Gamma_2\) is computable intrinsically inside \(\mathcal{A}\) — a finite-dimensional linear-algebra shadow of the infinite-dimensional diffusion.
• Q3 (Hamilton–Jacobi equivalence, sufficiency). \(CD(K, \infty)\) is the structural condition that turns formal Cole–Hopf calculus into quantitative statements: monotonicity of Fisher information along the heat flow (Shannon / Barron), log-Sobolev inequalities, hypercontractivity — the tools that control the ODE (5.6)/(3.5) along the flow.
• Q2 (quadratic case). For \(L = \tfrac12\Delta\) on \(\mathbb{R}^d\) (flat), \(\mathrm{Ric} = 0\) and \(CD(0, \infty)\) holds; for Ornstein–Uhlenbeck \(L = \tfrac12\Delta - x\cdot\nabla\), \(CD(1, \infty)\) holds. Both environments host the quadratic-linear stable class.

(iii) References

• Bakry, Émery (1985) — Diffusions hypercontractives. Séminaire de Probabilités XIX. doi:10.1007/BFb0075847.
• Bakry, Gentil, Ledoux (2014) — Analysis and Geometry of Markov Diffusion Operators, ch. 1 §1.16, ch. 3, §C.6. doi:10.1007/978-3-319-00227-9.
• Amari, Nagaoka (2000) — Methods of Information Geometry, §3 (dual connections; Bakry–Émery meets information geometry).

(iv) Worked miniature — \(\Gamma_2\) for OU on \(\mathbb{R}\)

Take \(L = \tfrac12 \partial_{xx} - x \partial_x\) (Ornstein–Uhlenbeck, \(\gamma_1\) invariant). Then \(\Gamma(f, h) = \tfrac12 f' h'\), so \(\Gamma(f) = \tfrac12 (f')^2\).

Compute \(\Gamma_2\). First, \(L \Gamma(f) = L(\tfrac12 (f')^2) = \tfrac12 \cdot 2 f' \cdot L(f') + \Gamma(f', f') = f' \, L f' + \tfrac12 (f'')^2.\)

Here \(L f' = \tfrac12 f''' - x f'' - f'\) (differentiate \(Lf\); the \(-f'\) comes from \(\partial_x(-xf') = -f' - xf''\)).

And \(\Gamma(f, Lf) = \tfrac12 f' (Lf)' = \tfrac12 f' \,(Lf)'\) where \((Lf)' = \tfrac12 f''' - f' - xf''\). So \(\Gamma(f, Lf) = \tfrac12 f' \bigl(\tfrac12 f''' - f' - xf''\bigr)\).

Subtracting and halving yields, after simplification,

\[ \Gamma_2(f) \;=\; \tfrac12\,(f'')^2 \;+\; \tfrac12\,(f')^2 \;=\; \tfrac12 (f'')^2 \;+\; \Gamma(f). \]

Hence \(\Gamma_2(f) \ge \Gamma(f)\) pointwise: \(CD(1, \infty)\) holds for OU — with equality iff \(f'' \equiv 0\), i.e. \(f\) is affine. The term \(\tfrac12 (f'')^2\) is the Hessian energy (Bochner term) and the \(+\Gamma(f)\) is the Ricci contribution from the drift \(-x\partial_x\), which has symmetric gradient \(-I\), hence effective Bakry–Émery Ricci \(= +I \succeq 1 \cdot \mathrm{Id}\). ✓

Takeaway. \(CD(1, \infty)\) is the analytic footprint of Hermite spectral structure (BGL §2.7.2); the quadratic-linear class stable under \((L, \Gamma)\) is the degree-\(\le 2\) sector of the Hermite decomposition.