Paper VI: Gaussian transfer and Hermite packets

E7 Gaussian transfer and Hermite packets

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Let \(\gamma\) be the standard Gaussian probability measure on \({\mathbb R}\). For \(t\geq0\), define the Ornstein Uhlenbeck transfer

\[ ({\mathcal O}_t f)(x) = \int_{\mathbb R} f\left( {\mathrm e}^{-t}x+\sqrt{1-{\mathrm e}^{-2t}}\,y \right) \gamma({\,\mathrm d} y). \]

Let \(H_n\) be the probabilists' Hermite polynomial defined by

\[ \exp\left(ux-\frac{u^2}{2}\right) = \sum_{n=0}^\infty H_n(x)\frac{u^n}{n!}, \]

and set \(h_n=H_n/\sqrt{n!}\).

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Theorem E7.1: Ornstein Uhlenbeck semigroup

The operators \(({\mathcal O}_t)_{t\geq0}\) form a positive Markov contraction semigroup on \(L^p(\gamma)\) for every \(1\leq p\leq\infty\). On \(L^2(\gamma)\), the semigroup is self adjoint and

\[ {\mathcal O}_t h_n={\mathrm e}^{-nt}h_n. \]

The family \((h_n)_{n\geq0}\) is an orthonormal basis of \(L^2(\gamma)\).

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Proof of Theorem E7.1

Positivity and preservation of constants follow from the integral formula. If \(X\) and \(Y\) are independent standard Gaussian variables, then

\[ {\mathrm e}^{-s} \left( {\mathrm e}^{-t}x+\sqrt{1-{\mathrm e}^{-2t}}X \right) + \sqrt{1-{\mathrm e}^{-2s}}Y \]

has the same distribution as

\[ {\mathrm e}^{-(s+t)}x + \sqrt{1-{\mathrm e}^{-2(s+t)}}Z \]

with \(Z\) standard Gaussian. This proves the semigroup law. Jensen's inequality and invariance of \(\gamma\) give the \(L^p\) contraction. The joint Gaussian pair

\[ \left( X, {\mathrm e}^{-t}X+\sqrt{1-{\mathrm e}^{-2t}}Y \right) \]

is symmetric under interchange, which proves self adjointness.

Apply \({\mathcal O}_t\) to the Hermite generating function. Gaussian integration gives

\[ \begin{aligned} {\mathcal O}_t \exp\left(ux-\frac{u^2}{2}\right) &= \exp\left( {\mathrm e}^{-t}ux- \frac{{\mathrm e}^{-2t}u^2}{2} \right)\\ &= \sum_{n=0}^\infty {\mathrm e}^{-nt}H_n(x)\frac{u^n}{n!}. \end{aligned} \]

Comparison of coefficients proves the eigenvalue formula. The same generating function shows

\[ \int H_mH_n{\,\mathrm d}\gamma=n!\delta_{mn}. \]

The Hermite polynomials span all polynomials, and polynomials are dense in \(L^2(\gamma)\). Hence the normalized family is an orthonormal basis. See [Janson1997] for the general Gaussian Hilbert space setting.

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Corollary E7.2: Hermite packet reconstruction

Let \(P_n\) be the orthogonal projection onto \({\mathbb C} h_n\). Then

\[ I=\sum_{n=0}^\infty P_n \]

in the strong operator topology and

\[ {\mathcal O}_t = \sum_{n=0}^\infty{\mathrm e}^{-nt}P_n. \]

For \(f\in L^2(\gamma)\),

\[ {\left\lVert{\mathcal O}_t f\right\rVert}_2^2 = \sum_{n=0}^\infty {\mathrm e}^{-2nt}{\left\lVert P_nf\right\rVert}_2^2. \]

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Proof of Corollary E7.2

Apply Parseval's identity and the eigenvalue formula in Theorem E7.1.

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Proposition E7.3: Generator and Dirichlet form

On the polynomial subspace, the infinitesimal generator of \(({\mathcal O}_t)\) is the differential operator

\[ {\mathcal L} f=f''-xf'. \]

It satisfies

\[ {\mathcal L} h_n=-nh_n. \]

For every polynomial \(f\),

\[ {\left\langle-{\mathcal L} f,f\right\rangle}_{L^2(\gamma)} = \int_{\mathbb R}{\left\lvert f'(x)\right\rvert}^2\gamma({\,\mathrm d} x). \]

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Proof of Proposition E7.3

Write

\[ \rho_t={\mathrm e}^{-t}=1-t+O(t^2), \qquad \sigma_t^2=1-{\mathrm e}^{-2t}=2t+O(t^2). \]

For a polynomial \(f\), Taylor expansion at \(x\) and Gaussian integration give

\[ \begin{aligned} {\mathcal O}_tf(x) ={}&f(x) +f'(x)(\rho_t-1)x +\frac12f''(x)\sigma_t^2 +o(t)\\ ={}&f(x)+t\bigl(f''(x)-xf'(x)\bigr)+o(t). \end{aligned} \]

Every omitted term is a polynomial in the Gaussian variable multiplied by a quantity that is \(o(t)\), so the expansion may be integrated term by term. Therefore

\[ \left.\frac{{\,\mathrm d}}{{\,\mathrm d} t}{\mathcal O}_tf\right|_{t=0} =f''-xf'. \]

The Hermite eigenvalue identity also follows by differentiating \( {\mathcal O}_th_n={\mathrm e}^{-nt}h_n \) at zero. Gaussian integration by parts gives

\[ \int_{\mathbb R} f'(x)\overline{g(x)}\gamma({\,\mathrm d} x) = \int_{\mathbb R} f(x)\overline{xg(x)-g'(x)}\gamma({\,\mathrm d} x). \]

Set \(g=f'\) and rearrange.

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Corollary E7.4: Gaussian spectral gap

If \(f\in L^2(\gamma)\) has mean zero, then

\[ {\left\lVert{\mathcal O}_tf\right\rVert}_2 \leq {\mathrm e}^{-t}{\left\lVert f\right\rVert}_2. \]

For every polynomial \(f\),

\[ {\operatorname{Var}}_\gamma(f) \leq \int_{\mathbb R}{\left\lvert f'(x)\right\rvert}^2\gamma({\,\mathrm d} x). \]

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Proof of Corollary E7.4

A mean zero function has no \(h_0\) component, so every surviving Hermite multiplier is at most \({\mathrm e}^{-t}\). For a polynomial

\[ f-{\mathbb E}_\gamma f=\sum_{n\geq1}c_nh_n. \]

Then

\[ {\operatorname{Var}}_\gamma(f)=\sum_{n\geq1}{\left\lvert c_n\right\rvert}^2 \leq \sum_{n\geq1}n{\left\lvert c_n\right\rvert}^2 = {\left\langle-{\mathcal L} f,f\right\rangle}. \]

Apply Proposition E7.3.

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Corollary E7.5: Fractional Hermite transfer

Let \(-{\mathcal L}\) be the nonnegative Ornstein Uhlenbeck number operator with

\[ -{\mathcal L} h_n=nh_n. \]

For every \(\alpha>0\),

\[ {\mathrm e}^{-t(-{\mathcal L})^\alpha}f = \sum_{n=0}^\infty {\mathrm e}^{-tn^\alpha}P_nf. \]

This is a contraction semigroup on \(L^2(\gamma)\).

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Proof of Corollary E7.5

The spectral theorem applies to the orthonormal Hermite basis. Every multiplier has modulus at most one and obeys the scalar semigroup law.

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Hermite packet multipliers for the Ornstein Uhlenbeck semigroup and two fractional powers of its number operator.

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