The Ornstein-Uhlenbeck operator $L$ is given by
$$
Lu = \Delta u- \frac{1}{2}x\cdot \nabla u.
$$
Is there a known closed form expression of the Green's function of $L$ on $\mathbb R^d$ (for $d\geq 2$ or at least for $d=2$) ?
Any references?
Thanks a lot!
Best Answer
Denote by $H_k(x)$ the $k$-th Hermite polynomial in one variable,
$$H_k(x) =\delta^k 1,$$
where $\delta f(x)=xf(x)-f'(x)$, $\newcommand{\bR}{\mathbb{R}}$ $\forall f\in C^\infty(\bR)$. $\newcommand{\bx}{\boldsymbol{x}}$ For $\bx=(x_1,\dotsc,x_d)\in\bR^d$ and $\newcommand{\bZ}{\mathbb{Z}}$$\alpha\in\bZ^d_{\geq 0}$ we set
$$ H_\alpha(\bx):=H_{\alpha_1}(x_1)\cdots H_{\alpha_d}(x_d). $$
Denote by $\Gamma$ the standard Gaussian measure on $\bR^d$,
$$ \Gamma(d\bx)=\frac{1}{(2\pi)^{d/2}} e^{-\frac{1}{2}\Vert\bx\Vert^2} d\bx. $$
We have
$$\int_{\bR^d} H_\alpha(\bx)^2\Gamma(\bx)=\alpha!:=\prod_j \alpha_j!, $$
$$ L H_\alpha = -|\alpha|\, H_{\alpha},\;\;\alpha=\sum_j\alpha_j. $$
Moreover, the linear span of the set of polynomials $H_\alpha(\bx)$, $\alpha\in\bZ^d_{\geq 0}$, is dense in the Hilbert space $L^2\bigl(\,\bR^d, \Gamma(d\bx)\,\bigr)$. Thus any $f\in L^2(\bR^d,\Gamma)$ has an orthogonal decomposition
$$ f(\bx)=\sum_\alpha\frac{f_\alpha}{\alpha!} H_\alpha(\bx),\;\;f_\alpha=\int_{\bR^d} f(\bx) H_\alpha(\bx)\Gamma(d\bx). $$
The range of $L$ is the codimension $1$ subspace of $L^2(\bR^d,\Gamma)$ consisting of functions $f$ such that $f_0=0$. Consider the bounded operator
$$G:L^2(\bR^d,\Gamma)\to L^2(\bR^d,\Gamma), $$
given by
$$ G[f](\bx)= -\sum_{\alpha\neq 0}\frac{1}{|\alpha|} \frac{f_\alpha}{\alpha!} H_\alpha(\bx). $$
Then $G[f]$ belongs to the domain of $L$ for any $f\in L^2(\bR^d,\Gamma)$ and
$$ LG [f]=f-f_0. $$
The operator $G$ is an integral operator and its integral kernel has the form $\newcommand{\by}{\boldsymbol{y}}$
$$K_G(\bx,\by)=-\sum_{\alpha\neq 0}\frac{1}{|\alpha|\cdot \alpha!} H_\alpha(\bx) H_\alpha(\by). $$
This means that
$$G[f](\bx)=\int_{\bR^d} K_G(\bx,\by) f(\by) \Gamma(d\bx). $$
You can take $K_G$ as your Green's function. You can simplify the description of $K_G$ a bit by using Mehler's formula
$$\sum_{k\geq 0}H_k(x)H_k(y)\frac{r^k}{k!}=\frac{1}{\sqrt{1-r^2}} \exp\left(-\frac{(rx)^2-2rxy+(ry)^2}{2(1-r^2)}\right). $$
For more details you can check Malliavin's book Integration and Probability.