The Ornstein-Uhlenbeck process \( {X={(X_t)}_{t\in[0,\infty)}} \) on \( {\mathbb{R}^n} \) is the solution of the stochastic differential equation
\[ dX_t=\sqrt{2}dB_t-X_tdt \]
where \( {{(B_t)}_{t\in[0,\infty)}} \) is a standard Brownian motion. Since the diffusion coefficient is constant and the drift is affine, it follows that \( {X} \) is a Gaussian process. The computation of the mean and of the variance of \( {X_t} \) conditional on \( {\{X_0=x\}} \) yields
\[ \mathrm{Law}(X_t\mid X_0=x)=\mathcal{N}(xe^{-t},\sqrt{1-e^{-2t}}I_n). \]
This shows that for any \( {x} \) and conditional on \( {\{X_0=x\}} \), \( {X} \) converges in distribution:
\[ X_t\underset{t\rightarrow\infty}{\overset{d}{\longrightarrow}}\gamma_n \]
where \( {\gamma_n=\mathcal{N}(0,I_n)} \) has density \( {(2\pi)^{-\frac{n}{2}}e^{-\frac{1}{2}|x|^2}} \). This shows also that \( {\gamma_n} \) is invariant:
\[ X_0\sim\gamma_n\quad\Rightarrow\quad\forall t\geq0,\quad X_t\sim\gamma_n. \]
Actually a stronger property holds true: the law \( {\gamma_n} \) is reversible in the sense that
\[ X_0\sim\gamma_n\quad\Rightarrow\quad\forall t\geq0,\quad (X_0,X_t)\overset{d}{=}(X_t,X_0). \]
The explicit law of the process allows computations, for instance for any \( {s,t\geq0} \),
\[ \mathrm{Cov}(X_s,X_t)=e^{-|t-s|}(1-e^{-2\min(s,t)}). \]
For any bounded and measurable \( {f:\mathbb{R}^n\rightarrow\mathbb{R}} \), any \( {x\in\mathbb{R}} \) and \( {t\in[0,+\infty)} \), we set
\[ P_t(f)(x)=\mathbb{E}(f(X_t)\mid X_0=x). \]
We have \( {P_t(\mathbf{1}_A)(x)=\mathbb{P}(X_t\in A\mid X_0=x)} \). The family \( {{(P_t)}_{t\in[0,\infty)}} \) is a semigroup of linear operators acting on continuous and bounded functions, in the sense that
\[ P_0=id, \quad \forall s,t\geq0, \quad P_t\circ P_s = P_{t+s}. \]
These operators are Markov operators, in the sense that for any \( {t\in[0,\infty)} \),
\[ \forall c\in\mathbb{R}, P_t(c)=c, \quad\text{and}\quad \forall f, f\geq0 \Rightarrow P_t(f)\geq0. \]
The explicit law of the process provides the Mehler formula for the semigroup
\[ P_t(f)(x) =\int\!f(xe^{-t}+\sqrt{1-e^{-2t}}y)\gamma_n(dy)\\ =\mathbb{E}(f(xe^{-t}+\sqrt{1-e^{-2t}}Z)). \]
This gives the following commutation with the gradient when \( {f} \) is smooth:
\[ (\nabla P_t f)(x)=e^{-t}P_t(\nabla f)(x), \]
where in the right hand side, \( {P_t} \) acts on each coordinates of the vector \( {\nabla f} \).
The infinitesimal generator is the unbounded operator in \( {L^2(\gamma_n)} \) given by
\[ Af =\partial_{t=0^+}P_tf =\lim_{t\rightarrow0^+}\frac{P_tf-f}{t} =\Delta f-\langle x,\nabla f\rangle. \]
The Chapman-Kolmogorov evolution equation writes
\[ \partial_tP_t=AP_t=P_tA. \]
If we fix \( {f} \) and write \( {u_t(x)=P_t(f)(x)} \) for any \( {x} \) and \( {t} \) then
\[ u_0=f\quad\text{and}\quad \partial_t u_t=Au_t=\Delta u_t-\langle x,\nabla u_t\rangle. \]
The operator \( {A} \) (and \( {P_t} \) for any \( {t\geq0} \)) is symmetric in \( {L^2(\gamma_n)} \), in other words an integration by parts holds, meaning that for any \( {f} \) and \( {g} \),
\[ -\int\!fAg\,d\gamma_n =\int\!\nabla f\cdot\nabla g\,d\gamma_n. \]
If \( {X_0} \) has density \( {f_0} \) with respect to \( {\gamma_n} \) then \( {X_t} \) has also a density with respect to \( {\gamma_n} \) given by \( {f_t=P_tf_0} \). If \( {g} \) is the Lebesgue density of \( {\gamma_n} \), then \( {g_t=f_tg} \) is the Lebesgue density of \( {X_t} \). The evolution of \( {g_t} \) with respect to \( {t} \) is described by the Fokker-Planck equation, dual of the Chapman-Kolmogorov equation,
\[ \partial_tg_t=\Delta g_t+\mathrm{div}(xg_t). \]
If \( {\mu} \) and \( {\nu} \) are probability measures on \( {\mathbb{R}^n} \) with \( {\nu\ll\mu} \) then the Kullback-Leibler divergence or relative entropy of \( {\nu} \) with respect to \( {\mu} \) is defined by
\[ H(\nu\mid\mu)=\int\!f\log f\,d\mu=\int\!\log f\,d\nu \quad\text{where}\quad f=\frac{d\nu}{d\mu}. \]
We take the convention \( {H(\nu\mid\mu)=+\infty} \) if \( {f\log f\not\in L^1(\mu)} \) or if \( {\nu\not\ll\mu} \). Note that Jensen’s inequality shows that \( {H(\nu\mid\mu)\geq0} \) with equality iff \( {\mu=\nu} \).
In the case where \( {\mu} \) is a Boltzmann-Gibbs measure with Lebesgue density \( {g(x)=e^{-V(x)}} \), the quantity \( {H(\nu\mid\mu)} \) becomes a Helmholtz free energy, in the sense that
\[ H(\nu\mid\mu)=\int\!V\,d\nu-S(\nu) \]
where the first term in the right hand side is the mean energy of \( {\nu} \) while the second term in the right hand side is the Boltzmann-Shannon entropy
\[ S(\nu)=\displaystyle\int\!fg\log(fg)\,dx. \]
Suppose that the law \( {\mu_0} \) of \( {X_0} \) has density \( {f_0} \) with respect to \( {\gamma_n} \). Then the law \( {\mu_t} \) of \( {X_t} \) has density \( {f_t=P_tf_0} \) with respect to \( {\gamma_n=\mu_\infty} \). The free energy decays along the time. Namely, using the evolution equation and the integration by parts,
\[ \begin{array}{rcl} \frac{d}{dt}H(\mu_t\mid\gamma_n) &=&\displaystyle\int\!\partial_t(f_t\log f_t)\,d\gamma_n\\ &=&\displaystyle\int\!(1+\log f_t)Af_t\,d\gamma_n\\ &=&\displaystyle-\int\!\frac{|\nabla f_t|^2}{f_t}\,d\gamma_n\\ &=&-J(\mu_t\mid\gamma_n) \leq0. \end{array} \]
This is know as the de Bruijn identity:
\[ \frac{d}{dt}H(\mu_t\mid\gamma_n)=-J(\mu_t\mid\gamma_n)\leq0. \]
The quantity
\[ J(\nu\mid\mu) =\int\!\frac{|\nabla f|^2}{f}\,d\mu=\int\!|\nabla\log f|^2\,d\nu \quad\text{where}\quad f=\frac{d\nu}{d\mu} \]
is the Fisher information. How it behaves along the O.-U. dynamics? Well, using commutation, two times Jensen’s inequality, and the invariance of \( {\gamma_n} \), we get
\[ \begin{array}{rcl} J(\mu_t\mid\gamma_n) &=&\int\!\frac{|\nabla f_t|^2}{f_t}\,d\gamma_n\\ &=&e^{-2t}\int\!\frac{|(\nabla f)_t|^2}{f_t}\,d\gamma_n\\ &\leq& e^{-2t}\int\!\frac{(|\nabla f|)_t^2}{f_t}\,d\gamma_n\\ &\leq& e^{-2t}\int\!\left(\frac{|\nabla f_0|^2}{f_0}\right)_t\,d\gamma_n\\ &=& e^{-2t}J(\mu_0\mid\gamma_n), \end{array} \]
in other words the Fisher information decays exponentially:
\[ \forall \mu_0\ll\gamma_n, \forall t\geq0,\quad J(\mu_t\mid\gamma_n)\leq e^{-2t}J(\mu_0\mid\gamma_n). \]
In particular we get \begin{align*} H(\mu_0\mid\gamma_n) =-\int_0^\infty\!\frac{d}{dt}H(\mu_t\mid\gamma_n)\,dt =\int_0^\infty\!J(\mu_t\mid\gamma_n)\,dt \leq \frac{1}{2}J(\mu_0\mid\gamma_n). \end{align*} This inequality is known as a logarithmic Sobolev inequality:
\[ \forall \nu\ll\gamma_n,\quad H(\nu\mid\gamma_n)\leq\frac{1}{2}J(\nu\mid\gamma_n). \]
This inequality is optimal in the sense that equality is achieved when \( {d\nu(x)/d\gamma_n(x)=e^{ax}} \) for some \( {a\in\mathbb{R}} \). Using this inequality for \( {\nu=\mu_t} \) yields
\[ \frac{d}{dt}H(\mu_t\mid\gamma_n) =-J(\mu_t\mid\gamma_n) \leq -\frac{1}{2}H(\mu_t\mid\gamma_n). \]
which gives, by Gronwall’s lemma, an exponential decay of the free energy, namely
\[ \forall\mu_0\ll\gamma_n, \forall t\geq0, H(\mu_t\mid\gamma_n)\leq e^{-2t}H(\mu_0\mid\gamma_n). \]
Since both sides are equal for \( {t=0} \), taking the derivative at time \( {t=0} \) allows to recover from this exponential decay the logarithmic Sobolev inequality!
Hypercontractivity. For any \( {t\in[0,\infty)} \) and any \( {p\in[1,\infty]} \), Mehler’s formula shows immediately that \( {P_t} \) can be extended into a linear operator on \( {L^p(\gamma_n)} \). In fact \( {P_t} \) is always a contraction:
\[ \forall p\geq1, \forall t\in[0,\infty), \forall f\in L^p(\gamma_n),\quad \Vert P_tf\Vert_p\leq\Vert f\Vert_p. \]
Namely, using Jensen’s inequality and the invariance of \( {\gamma_n} \),
\[ \begin{array}{rcl} \Vert P_tf\Vert_p^p &=&\int\!|\mathbb{E}(f(X_t)\mid X_0=x)|^p\,d\gamma_n(x)\\ &\leq& \int\!\mathbb{E}(|f(X_t)|^p\mid X_0=x)\,d\gamma_n(x)\\ &=&\int\!P_t(|f|^p)\,d\gamma_n(x)\\ &=&\int\!|f|^p\,d\gamma_n(x)\\ &=&\Vert f\Vert_p^p. \end{array} \]
Since equality is achieved for constant functions, it follows that \( {\Vert P_t\Vert_{p\rightarrow p}=1} \). The semigroup \( {{(P_t)}_{t\in[0,\infty)}} \) is in fact hypercontractive:
\[ \forall p\geq1, \forall t\geq0, \forall f\in L^p(\gamma_n), \quad \Vert P_t f \Vert_{p(t)} \leq \Vert f \Vert_p, \]
where \( {p(t) = 1 + (p-1)e^{2t}} \), in other words \( {\Vert P_t\Vert_{p\rightarrow p(t)}=1} \), and moreover this value \( {p(t)} \) is critical in the sense that if \( {q > p(t)} \) then \( {\Vert P_t\Vert_{p\rightarrow q}=+\infty} \).
Let us give a proof. One can assume that \( {f\geq0} \) since \( {|P_t f|\leq P_t|f|} \) by Jensen’s inequality. Note that \( {p(0)=0} \) and \( {p(t)>p} \) if \( {t>0} \). Set \( {\alpha(t)=\log\Vert P_t f\Vert_{p(t)}} \). To lighten the notation, let us set \( {f_t=P_tf} \). We have, for any \( {t\geq0} \),
\[ \begin{array}{rcl} \alpha'(t) &=&\left(\frac{1}{p(t)}\log\int\!f_t^{p(t)}\,d\gamma_n\right)’\\ &=&-\frac{p'(t)}{p(t)^2}\log\int\!f_t^{p(t)}\,d\gamma_n +\frac{1}{p(t)}\frac{\left(\displaystyle\int\!f_t^{p(t)}\,d\gamma_n\right)’}{\displaystyle\int\!f_t^{p(t)}\,d\gamma_n}\\ &=&-\frac{p'(t)}{p(t)^2}\log\int\!(f_t)^{p(t)}\,d\gamma_n +\frac{1}{p(t)}\frac{\displaystyle\int\!\left(p'(t)\log f_t+p(t)\frac{Af_t}{f_t}\right)f_t^{p(t)}\,d\gamma_n}{\displaystyle\int\!f_t^{p(t)}\,d\gamma_n}\\ &=&-\frac{p'(t)}{p(t)^2}\log\int\!f_t^{p(t)}\,d\gamma_n +\frac{p'(t)}{p(t)^2}\frac{\displaystyle\int\!f_t^{p(t)}\log f_t^{p(t)}\,d\gamma_n}{\displaystyle\int\!f_t^{p(t)}\,d\gamma_n} +\frac{\displaystyle\int\!(Af_t)f_t^{p(t)-1}\,d\gamma_n}{\displaystyle\int\!f_t^{p(t)}\,d\gamma_n}\\ &=&\frac{p'(t)}{p(t)^2}\left(H(h_t^{p(t)}\gamma_n\mid\gamma_n)+\frac{p(t)^2}{p'(t)}\int\!(Ah_t)h_t^{p(t)-1}\,d\gamma_n\right) \end{array} \]
where \( {h_t=f_t/\Vert f_t\Vert_{p(t)}} \). Now the logarithmic Sobolev inequality and the integration by parts give, for any \( {h\geq0} \) such that \( {h^p} \) is a probability density with respect to \( {\gamma_n} \),
\[ \begin{array}{rcl} \mathrm{H}(h^p\gamma_n\mid\gamma_n) &\leq&\frac{1}{2}\int\!\frac{|\nabla h^p|^2}{h^p}\,d\gamma_n\\ &=&\frac{p^2}{2}\int\!|\nabla h|^2h^{p-2}\,d\gamma_n\\ &=&\frac{p^2}{2(p-1)}\int\!\left<\nabla h,\nabla h^{p-1}\right>\,d\gamma_n\\ &=&-\frac{p^2}{2(p-1)}\int\!(Ah)h^{p-1}\,d\gamma_n. \end{array} \]
Using this inequality for \( {h=h_t} \) and \( {p=p(t)} \), and using \( {2(p(t)-1)=p'(t)} \), we obtain that \( {\alpha'(t)\leq0} \) for any \( {t\geq0} \), and as a consequence
\[ \log\Vert P_tf\Vert_{p(t)} = \alpha(t) \leq\alpha(0)=\log\Vert f\Vert_p. \]
Finally, if now \( {q>p(t)} \) then taking \( {f_\lambda(x)=e^{\langle\lambda,x\rangle}} \) for some \( {\lambda\in\mathbb{R}^n} \) gives
\[ \Vert f_\lambda\Vert_p=e^{p|\lambda|^2/2} \quad\text{and}\quad P_t f_\lambda=e^{|\lambda|^2(1-e^{-2t})/2}f_{\lambda e^{-t}} \]
and therefore
\[ \frac{\Vert P_t f_\lambda\Vert_{q}}{\Vert f_\lambda\Vert_p} =e^{|\lambda|^2(e^{-2t}(q-1)+1-p)/2}, \]
a quantity which tends to \( {+\infty} \) as \( {|\lambda|\rightarrow\infty} \) since \( {q>p(t)=1+(p-1)e^{2t}} \).
The proof shows that conversely, from the hypercontractive statement, one can extract the logarithmic Sobolev inequality by taking the derivative at \( {t=0} \).
Polynomials. The set of polynomials \( {\mathbb{R}[X]} \) is dense in \( {L^2(\gamma_1)} \). To see it, let us take \( {f\in L^2(\gamma_1)} \), then the Laplace transform \( {\varphi_\mu} \) of the signed measure \( {\mu(dx)=f(x)\gamma_1(dx)} \) is finite on \( {\mathbb{R}} \) since for any \( {\theta\in\mathbb{R}} \), by the Cauchy-Schwarz inequality,
\[ (\varphi_\mu(\theta))^2=\left(\int\! \exp(\theta x)\,\mu(dx)\right)^2 \leq \int\!f^2\,d\gamma_1\int\!\exp(2\theta x)\,\gamma_1(dx)<+\infty, \]
and in particular, \( {\varphi_\mu} \) is analytic on a neighborhood of \( {0} \). Now since for any \( {k\in\mathbb{N}} \),
\[ \varphi_\mu^{(k)}(0) =\int\!x^kf(x)\,\gamma_1(dx) =\langle P_k,f\rangle_{L^2(\gamma_1)}\quad\text{where}\quad P_{k}(x) =x^{k}, \]
and if \( {f\perp\mathbb{R}[X_1,\ldots,X_n]} \) in \( {L^2(\mathbb{R})} \), then the derivatives of any order of \( {\varphi_\mu} \) vanish at \( {0} \), and since \( {\varphi_\mu} \) is analytic, we get \( {\varphi_\mu\equiv0} \) and then \( {\mu=0} \) and then \( {f=0} \) in \( {L^2(\gamma_n)} \).
Hermite polynomials. Hermite’s polynomials \( {{(H_k)}_{k\in\mathbb{N}}} \) are the orthogonal polynomials obtained using the Gram-Schmidt algorithm in \( {L^2(\gamma_1)} \) from the canonical basis of \( {\mathbb{R}[X]} \). They are normalized in such a way that the coefficient of the term of highest degree in \( {H_k} \) is \( {1} \) for any \( {k\geq0} \). We find
\[ H_0(x)=1,\quad H_1(x)=x,\quad H_2(x)=x^2-1,\quad\ldots \]
It can be checked that Hermite’s polynomials \( {{(H_k)}_{k\geq0}} \) satisfy
- Generating series: for any \( {k\geq0} \) and \( {x\in\mathbb{R}} \),
\[ H_k(x)=\partial^k_1G(0,x) \quad\text{where}\quad G(s,x)=e^{sx-\frac{1}{2}s^2}=\sum_{k=0}^\infty\frac{s^k}{k!}H_k(x); \]
- Three terms recursion formula: for any \( {k\geq0} \) and \( {x\in\mathbb{R}} \),
\[ H_{k+1}(x)= xH_{k}(x) – kH_{k-1}(x); \]
- Recursive differential equation: for any \( {k\geq0} \) and \( {x\in\mathbb{R}} \),
\[ H_k'(x)=kH_{k-1}(x); \]
- Differential equation: for any \( {k\geq0} \) and \( {x\in\mathbb{R}} \),
\[ H_k”(x)-xH_k'(x)+kH_k(x)=0. \]
Using the generating series and Plancherel’s formula, we get
\[ \sum_{k=0}^\infty \frac{s^{2k}}{k!^2}\Vert H_k\Vert_2^2 =\int\!G(s,x)^2\,\gamma_1(dx)=\exp(-s^2)\int\!e^{2sx}\,\gamma_1(dx) =e^{s^2} =\sum_{k=0}^\infty\frac{s^{2k}}{k!}, \]
which gives \( {\Vert H_k\Vert_2^2=k!} \) by identifying the series coefficients. It follows that \( {{(H_k/\sqrt{k!})}_{k\in\mathbb{N}}} \) is a dense orthonormal sequence in the Hilbert space \( {L^2(\gamma_1)} \).
For any \( {f\in L^2(\gamma_1)} \), we have
\[ f=\sum_{k\geq0}a_kH_k\quad\text{where}\quad k!a_k=\int\!fH_k\,d\gamma_1. \]
In particular
\[ \Vert f\Vert_2^2=\int\!f^2\,d\gamma_1=\sum_{k\geq0}k!a_k^2. \]
Note that \( {a_0=\displaystyle\int\!f\,d\gamma_1=\gamma_1(f)} \) is the mean of \( {f} \) under \( {\gamma_1} \).
Hermite’s polynomials and Ornstein-Uhlenbeck process. Hermite’s polynomials are eigenvectors of the operators \( {P_t} \) and \( {A} \), namely for any \( {k\in\mathbb{N}} \) and \( {t\geq0} \),
\[ P_tH_k=e^{-kt}H_k \quad\text{and}\quad AH_k=-kH_k. \]
The property for \( {A} \) is immediate from the differential equation satisfied by Hermite’s polynomials. To establish the property for \( {P_t} \), we note that for any \( {Z\sim\gamma_1} \),
\[ P_t(G(s,\cdot))(x) =e^{se^{-t}x-\frac{1}{2}s^2}\mathbb{E}\bigr(e^{s\sqrt{1-e^{-2t}}Z}\bigr), \]
and since the Laplace transform of \( {Z} \) is given by \( {\mathbb{E}(e^{\theta Y})=e^{\frac{1}{2}\theta^2}} \) we get
\[ P_t(G(s,\cdot))(x)=G(se^{-t},x), \]
therefore, by the generating series property of Hermite’s polynomials,
\[ \begin{array}{rcl} P_t(H_k)(x) &=&P_t(\partial_1^kG(0,\cdot))(x)\\ &=&\partial_{s}^k P_t(G(s,\cdot))(x)_{\vert s=0}\\ &=&\partial_{s}^k G(se^{-t},x)_{\vert s=0}\\ &=&e^{-kt}\partial_{1}^k G(se^{-t},x)_{\vert s=0}\\ &=&e^{-kt}H_k(x). \end{array} \]
This shows that Hermite’s polynomials are eigenvectors of \( {P_t} \).
Exponential decay. If \( {f=\sum_{k\geq0}a_kH_k\in L^2(\gamma_1)} \) then for any \( {t\geq0} \),
\[ P_tf=\sum_{k\geq 0}e^{-kt}a_kH_k, \]
and thus
\[ \Vert P_t f-\gamma_1(f)\Vert_2^2 =\sum_{k\geq 1}a_k^2 e^{-2kt}k! \leq e^{-2t}\sum_{k\geq 1}a_k^2 k! =e^{-2t}\Vert f-\gamma_1(f)\Vert_2^2. \]
We have obtained the exponential decay in \( {L^2(\gamma_1)} \): for any \( {t\geq0} \) and \( {f\in L^2(\gamma_1)} \),
\[ \Vert P_t f-\gamma_1(f)\Vert_2 \leq e^{-t}\Vert f-\gamma_1(f)\Vert_2. \]
Using the invariance of \( {\gamma_1} \), we get \( {\displaystyle\int\!P_tf\,d\gamma_1=\int\!f\,d\gamma_1=\gamma_1(f)} \) and therefore
\[ \mathrm{Var}_{\gamma_1}(P_tf)\leq e^{-2t}\mathrm{Var}_{\gamma_1}(f), \]
which is is equivalent to the Poincaré inequality with constant \( {1} \) (optimal for \( {H_1} \)):
\[ \mathrm{Var}_{\gamma_1}(f)\leq-\int\!fAf\,d\gamma_1=\int\!f’^2\,d\gamma_1. \]
This inequality is the linearization at \( {h=1+\varepsilon f} \) of the logarithmic Sobolev inequality
\[ \int\!h^2\log(h^2)\,d\gamma_1 -\int\!h^2\,d\gamma_1\log\int\!h^2\,d\gamma_1 \leq -2\int\!hAh\,d\gamma_1=2\int\!h’^2\,d\gamma_1. \]
The gap between the first eigenvalue \( {0} \) and the second eigenvalue \( {-1} \) of \( {A} \) is of length \( {1} \). This spectral gap produces the exponential convergence. More generally, the semigroup preserves the spectral decomposition. If \( {f\perp\mathrm{Vect}\{H_1,\ldots,H_{k-1}\}} \) in \( {L^2(\gamma_1)} \) then \( {P_t(f)\perp\mathrm{Vect}\{H_1,\ldots,H_{k-1}\}} \) for any \( {t\geq0} \) and for any \( {t\geq0} \),
\[ \Vert P_t f-\gamma_1(f)\Vert_2 \leq e^{-k t}\Vert f-\gamma_1(f)\Vert_2. \]
Dimension \( {n} \). The operator \( {A} \) is a sum of operators acting on one variable:
\[ Af =\Delta f-\langle x,\nabla f\rangle =A_1f+\cdots+A_nf \quad\text{where}\quad A_kf=\partial_k^2f-x_k\partial_kf. \]
The eigenvectors of \( {A} \) are products of univariate Hermite’s polynomials. Namely, for any \( {k\in\mathbb{N}^n} \), if we denote, for any \( {x\in\mathbb{R}^n} \),
\[ H_k(x)=H_{k_1}(x_1)\cdots H_{k_n}(x_n), \]
then
\[ AH_k=(k_1+\cdots+k_n)H_k. \]
Quantum harmonic oscillator. Let \( {g_n} \) be the density of \( {\gamma_n} \). Consider the isometry
\[ \Phi:f\in L^2(dx)\rightarrow \Phi(f)=g_n^{-1/2}f\in L^2(\gamma_n). \]
One can define the operators \( {K} \) on \( {L^2(dx)} \) from the operator \( {A} \) on \( {L^2(\gamma_n)} \), namely
\[ Kf=(\Phi^{-1}\circ A\circ\Phi)(f) =g_n^{1/2}A(fg_n^{-1/2}) =\Delta f+\Bigr(\frac{n}{2}-\frac{1}{4}|x|^2\Bigr)f. \]
This is the quantum harmonic oscillator, a special kind of Schrödinger operator. We have \( {\partial_t Q_t=KQ_t} \) where \( {{(Q_t)}_{t\in[0,\infty)}} \) is the semigroup of operators defined by
\[ Q_t(f) =(\Phi^{-1}\circ P_t\circ \Phi)(f) =g_n^{1/2}P_t(g_n^{-1/2}f) \]
The eigenvectors of \( {K} \) are Hermite’s wave functions: for any \( {k\in\mathbb{N}^n} \),
\[ \psi_k(x)=g_n^{1/2}(x)H_k(x)=(2\pi)^{-\frac{n}{2}}e^{-\frac{1}{4}|x|^2}H_k(x). \]
For instance, for \( {k=(0,1,\ldots,n-1)} \), we get the wave function
\[ \psi(x_1,\ldots,x_n)=g_n^{1/2}(x)e^{-\frac{1}{4}|x|^2}H_0(x_1)\cdots H_{n-1}(x_n). \]
A bosonic wave function is obtained by symmetrization over \( {x_1,\ldots,x_n} \). A fermionic wave function is obtained by anti-symmetrization (implies nullity on the diagonal):
\[ \begin{array}{rcl} \psi_{\mathrm{fermions}}(x_1,\ldots,x_n) &=&g_n^{1/2}(x)\sum_{\sigma\in\Sigma_n}(-1)^{\mathrm{signature}(\sigma)}H_{\sigma(1)-1}(x_1)\cdots H_{\sigma(n)-1}(x_n)\\ &=&g_n^{1/2}(x)\det \begin{pmatrix} H_0(x_1)&\ldots&H_0(x_n)\\ \vdots &\vdots&\vdots\\ H_{n-1}(x_1)&\ldots&H_{n-1}(x_n) \end{pmatrix}\\ &=&g_n^{1/2}(x)\det \begin{pmatrix} x_1^0&\ldots&x_n^0\\ \vdots &\vdots&\vdots\\ x_1^{n-1}&\ldots&x_n^{n-1} \end{pmatrix}\\ &=&g_n^{1/2}\prod_{1\leq i<j\leq n}(x_i-x_j). \end{array} \]
The Slater determinant is here proportional to a Vandermonde determinant. Now
\[ |\psi_{\mathrm{fermions}}(x_1,\ldots,x_n)|^2 =(2\pi)^{-\frac{n}{2}}e^{-\frac{1}{2}(x_1^2+\cdots+x_n^2)}\prod_{1\leq i<j\leq n}(x_i-x_j)^2. \]
We recognize up to normalization the formula of the density of the Gaussian Unitary Ensemble (GUE) namely the density of the eigenvalues of a Gaussian \( {n\times n} \) Hermitian random matrix with Lebesgue density in \( {\mathbb{R}^{n+n^2-n}=\mathbb{R}^{n^2}} \) proportional to
\[ H\mapsto e^{-\frac{1}{2}\mathrm{Tr}(H^2)}. \]
Notes. By pure provocation, we used the Cauchy-Schwarz inequality only once. We have learned the link with the GUE during a talk by Satya Majumdar.
