In $\mathbb{R}^d$, consider the following equation
$$\Delta u -x\cdot \nabla u = f $$
where $f$ can be $C^\infty$ and decay exponentially fast.

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!