Tychonoff uniqueness for the heat equation

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

(i) Formal statement

Theorem (Tychonoff 1935). Let $u : [0, T] \times \mathbb{R}^d \to \mathbb{R}$ be a classical solution of the heat equation $\partial_t u = \tfrac12 \Delta u$ with $u(0, \cdot) = 0$. If there exist $C, a > 0$ such that

\[ |u(t, x)| \;\le\; C\, e^{a\|x\|^2} \qquad \forall (t, x) \in [0, T] \times \mathbb{R}^d, \tag{Tych} \]

then $u \equiv 0$ on $[0, T/(2a)] \times \mathbb{R}^d$.

The sub-Gaussian growth hypothesis (Tych) is sharp: Tychonoff exhibited a non-zero solution violating (Tych) (growth $e^{a\|x\|^{2+\varepsilon}}$ for any $\varepsilon > 0$), showing the heat Cauchy problem is not uniquely solvable in full generality.

Corollary (Widder 1944). If $u \ge 0$ (positivity restriction), uniqueness holds without any growth hypothesis on $[0, T] \times \mathbb{R}^d$.

(ii) Role in Q1 / Q2 / Q3

Q3 (⇐, converse implication). To go from the Hamilton–Jacobi equation (6) on $f_t$ back to stability $p_{\theta_t} = p_\theta * \gamma_{\sqrt{2t}}$, one constructs $u_t := e^{-f_t} = Z_{\theta_t}\, p_{\theta_t}$ and verifies $u_t$ solves $\partial_t u = \tfrac12 \Delta u$ with the same initial condition as $p_\theta * \gamma_{\sqrt{2t}}$. Uniqueness then forces equality. Tychonoff is the hidden hypothesis Feynman flags as "saut caché n°2".
Q1 (integrability layer). The sub-Gaussian bound (Tych) on $u_t = Z_{\theta_t} p_{\theta_t}$ translates to a growth condition on $f_t = \theta_t^\top\varphi
- \log Z$. For $\varphi$ with $\deg = 2$, $f_t$ is quadratic in $x$, so $u_t = e^{-f_t}$ decays faster than any exponential — (Tych) holds trivially. For higher-degree $\varphi$, compatibility with (Tych) is a genuine obstruction.
Widder's positivity loophole. Densities $p_{\theta_t} > 0$ by construction, so Widder 1944 applies directly — no sub-Gaussian hypothesis needed on every $\theta$. This is the cleaner route for the converse in Q3.

(iii) References

Tychonoff (1935) — Théorèmes d'unicité pour l'équation de la chaleur. Mat. Sb. 42(2):199–216. MathNet:mi.mathnet.ru/msb5744.
Widder (1944) — Positive Temperatures on an Infinite Rod. doi:10.1090/S0002-9947-1944-0009795-2.
Bakry, Gentil, Ledoux (2014) — §3.1 Theorem 3.1.1 (modern formulation in the diffusion-semigroup framework).

(iv) Worked miniature — Tychonoff's counterexample (sketch)

Tychonoff's non-uniqueness example: define, for $a \in \mathbb{R}$, \[ g(t) \;=\; \begin{cases} e^{-1/t^2} & t > 0 \\ 0 & t \le 0 \end{cases}, \qquad u(t, x) \;=\; \sum_{n = 0}^\infty \frac{g^{(n)}(t)}{(2n)!}\, x^{2n}. \]

Claim. The series converges for all $(t, x)$, $\partial_t u = \tfrac12 \partial_{xx} u$ holds classically, and $u(0, x) = 0$ for all $x$, yet $u \not\equiv 0$.

Sketch. $g$ is $C^\infty$ with all derivatives vanishing at $0$; Cauchy estimates show $|g^{(n)}(t)| \le n!\,(2 e/t^2)^n$, so the series converges. Term- by-term differentiation is justified by majorant geometric series; the identity $\partial_t u = \tfrac12 \partial_{xx} u$ is checked shift-by-shift. At $t = 0$, every $g^{(n)}(0) = 0$, so $u(0, x) = 0$.

Yet $u(t, x)$ grows like $e^{c x^2/t}$ as $x \to \infty$ for fixed $t > 0$, violating (Tych) for any $a$ once $x^2/t \gg 2a$. So the growth condition is necessary.

Take-away for Q3. The converse "HJ $\Rightarrow$ stability" needs either a growth certificate on $u_t = Z_{\theta_t} p_{\theta_t}$, or the Widder positivity argument. Since $p_{\theta_t} > 0$, Widder is the clean path — and the proof of Q3 converse thereby avoids any hand-waving about integrability.