On the solvability of stationary Fokker-Planck equations close to the Laplace equation (Q2461906)

This paper provides a sufficient condition on vector fields such that the associated stationary Fokker-Planck equation has a solution. More precisely, for any \(n,p \in \mathbb N\) such that \(p\geq [n/2]+1,\) there exists a number \(m>0\) such that if a \(C^\infty\)-vector field \(f\) on \(\mathbb R^n\) satisfies \[ \sum_{| \alpha| =0}^p \Big(\| (1+| \cdot| )D^\alpha \| _{L^2(\mathbb R^n)}+ \sup_x| D^\alpha f| \Big)<m \] then there exists a smooth function \(u\) on \(\mathbb R^n\) such that \[ \Delta u- \text{div} (uf)=0, \qquad \nabla u\in L^2(\mathbb R^n), \] holds.