# Reference (foundamental sol. and grad estimate, etc.): a particular elliptic PDE

In $$\mathbb{R}^d$$, consider the following equation $$\Delta u -x\cdot \nabla u = f$$ where $$f$$ can be $$C^\infty$$ and decay like $$e^{-\frac{c|x|^2}{2}}$$.

I would like to know fundamental sol. to this equation or some gradient estimate, preferably pointwise. In particular I'm interested in the following quantity \begin{align*} \int_{\mathbb{R}^d}|\nabla u|^2e^{-\frac{|x|^2}{2}}dx. \end{align*}

One useful observation might be $$\nabla \cdot (e^{-\frac{|x|^2}{2}}\nabla u)=f e^{-\frac{|x|^2}{2}} .$$

Thank you!

• This is called Ornstein–Uhlenbeck evolution, and it has been studied a lot. In particular, an explicit solution is known in terms of the usual heat semigroup. Jan 11, 2020 at 8:20