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!

  • 2
    $\begingroup$ 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. $\endgroup$ Jan 11, 2020 at 8:20


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