Cole–Hopf transform

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

(i) Formal statement

Let \(u \in C^{1,2}(\mathbb{R}_+ \times M)\) with \(u > 0\) solve the linear parabolic equation \(\partial_t u = L u\) for a diffusion generator \(L\) (in the BGL sense with carré-du-champ \(\Gamma\)). Define \(f := -\log u\). Then

\[ \boxed{\;\partial_t f \;=\; L f \;-\; \Gamma(f, f)\;} \tag{CH} \]

Euclidean specialisation. For \(M = \mathbb{R}^d\) and \(L = \tfrac12\Delta\), \(\Gamma(f,f) = \tfrac12\|\nabla f\|^2\), so (CH) becomes the viscous Hamilton–Jacobi equation

\[ \partial_t f \;=\; \tfrac12 \Delta f \;-\; \tfrac12 \|\nabla f\|^2. \]

Equivalently, the Cole–Hopf operator \(\mathcal{L}_{\mathrm{CH}} f := \Gamma(f,f) - Lf\) satisfies \(\partial_t f = -\mathcal{L}_{\mathrm{CH}} f\).

Historical note. Hopf 1950 and Cole 1951 independently used \(u = e^{-f/\nu}\) to linearise Burgers' equation \(\partial_t f + f \partial_x f = \nu \partial_{xx}f\) into the heat equation \(\partial_t u = \nu \partial_{xx} u\). The identity transposes verbatim to any diffusion \((M, g, L, \Gamma)\) by the chain rule.

(ii) Role in Q1 / Q2 / Q3

• Q3 (⇒). (CH) is the analytic identity that derives Hamilton–Jacobi (6) of docs/problem.md. Starting from stability \(p_{\theta_t} = p_\theta * \gamma_{\sqrt{2t}}\) and writing \(f_t := -\log p_{\theta_t} = \theta_t^\top \varphi + \log Z_{\theta_t}\), (CH) gives exactly equation (6) after identifying terms in \(\varphi\).
• Q1 (algebraic closure). The RHS of (CH) involves \(\Gamma(\varphi_i, \varphi_j)\) and \(L\varphi_i\). Requiring (CH) to hold identically in \(x\) with \(f_t\) in the affine span \(\mathcal{A} = \mathrm{Span}\{\varphi_1,\ldots,\varphi_r,\mathbf{1}\}\) forces the closure condition \((\dagger)\): \(\Gamma(\varphi_i, \varphi_j) \in \mathcal{A}\) and \(L\varphi_i \in \mathcal{A}\) (see invariant-reformulation.md §3).
• Q2 (quadratic marchepied). For \(\varphi(x) = \mathrm{vec}(xx^\top) \oplus x\), (CH) produces the Riccati flow \(\dot M = -M^2\), \(\dot b = -Mb\) — the canonical quadratic specialisation.

(iii) References

• Hopf (1950) — The Partial Differential Equation \(u_t + u u_x = \mu u_{xx}\). doi:10.1002/cpa.3160030302.
• Cole (1951) — On a Quasi-Linear Parabolic Equation Occurring in Aerodynamics. doi:10.1090/qam/42889.
• Bakry, Gentil, Ledoux (2014) — Analysis and Geometry of Markov Diffusion Operators, ch. 1 §1.4–§1.11 (chain rule, \(\Gamma\) algebra). doi:10.1007/978-3-319-00227-9.

(iv) Worked miniature — heat kernel ↦ Hamilton–Jacobi

Take \(M = \mathbb{R}\), \(L = \tfrac12\partial_{xx}\), and \(u(t,x) = (2\pi t)^{-1/2} e^{-x^2/(2t)}\) (heat kernel centred at \(0\)). Then \(f(t,x) = -\log u = \tfrac12 \log(2\pi t) + x^2/(2t)\).

Check (CH) directly:

• \(\partial_t f = \tfrac{1}{2t} - \tfrac{x^2}{2 t^2}\).
• \(L f = \tfrac12 \partial_{xx} f = \tfrac{1}{2t}\).
• \(\Gamma(f,f) = \tfrac12 (\partial_x f)^2 = \tfrac12 \cdot \tfrac{x^2}{t^2} = \tfrac{x^2}{2t^2}\).

Then \(Lf - \Gamma(f,f) = \tfrac{1}{2t} - \tfrac{x^2}{2t^2} = \partial_t f\). ✓

The nonlinear HJ equation on \(f\) is exactly equivalent to the linear heat equation on \(u = e^{-f}\) — no approximation. This is the lift that underlies the whole program: the quadratic family \(\varphi(x) = x^2\) is stable under \(*\gamma_\sigma\) because \(x^2\) (up to a linear and a constant) is closed under \(f \mapsto \Gamma(f,f) - Lf\) evaluated on its span.