In \(L^2(\gamma)\) for the standard Gaussian law on \({\mathbb R}\), use the probabilists' Hermite polynomials
The Ornstein–Uhlenbeck generator is \(\mathcal Lf=f''-xf'\). Direct polynomial differentiation gives \(\mathcal LH_n=-nH_n\) for these three polynomials. Their squared norms are \(n!\) and different Hermite degrees are orthogonal.
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The upper stacked bar gives the initial degree-one and degree-two energies. The smaller lower bar gives them at the displayed later time. The legend distinguishes the two degrees and the note gives their total remaining energy.
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- \(t=0\)
- \(t=\log2\)
- \(1\)
- \(2\)
- \(1/4+1/8=3/8\)
Dark: degree 1 energy Light: degree 2 energy
Evidence note from the source edition
For \(f=H_1+H_2\), the two bars are its separate squared-energy contributions. At \(t=\log2\), their sum is \(3/8\), below the spectral-gap bound \(3/4\).
Reconstruction preserves the modes; time changes their weights
The semigroup sends \(f=H_1+H_2\) to
Its mean is zero. The Gaussian gap bounds this by \(e^{-2t}\|f\|_2^2=3e^{-2t}\). It need not be an equality when multiple degrees are present.
For the nonnegative operator \(-\mathcal L\), a square-root evolution replaces rate \(n\) by \(\sqrt n\). On \(g=H_1+H_4\), the coefficients at \(t=\log2\) are \(1/2\) and \(1/4\), whereas ordinary OU evolution gives \(1/2\) and \(1/16\). The eigenfunctions agree; the rates differ.
These calculations concern polynomial elements of Gaussian \(L^2\). The general expansion and generator-domain statements belong to the traditional proof; a finite list of Hermite checks does not establish completeness of the basis.
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Orthogonal packets make the energy accounting exact. A fractional generator alters the spectral clock while keeping the same Gaussian modes.