Consider $\mathbb{R}^d$ with Gibbs measure $d\mu=Z^{-1}\exp(-V(x))dx$, where the potential $V(x)$ is strongly convex ($\nabla^2 V(x) \ge \lambda Id $). We can assume the regularity of $V$ is as good as we need, and the Hessian is also bounded above.

Now, for “Laplace equation” in $(\mathbb{R}^d, d\mu)$: $$\nabla^* \nabla u=f,$$ Here $\nabla^*$ is the $L^2(d\mu)$-adjoint of $\nabla$. Assume $f\in L^2(d\mu)$ and solvability condition $\mathbb{E}_{\mu} f=0$, do we always have well-posedness of the equation in the space of mean-zero functions? If yes, do we have $u \in H^2(d\mu)$, and $\| D^2 u\|_{L^2(d\mu)} \le C \|f\|_{L^2(d\mu)}$? Finally, if yes, what is the constant $C$?

Thanks!