On diffusion semigroups preserving the log-concavity (Q5949998)

Let \(\sum_{i,j=1}^{n} {a_{ij}\partial_i\partial_j} +\sum_{i=1}^{n}\beta_i\partial_i\), \(a_{ij},\beta_i\in C^{2+\delta}_{\text{loc}}({\mathbb R}^n)\), \(\delta>0\), be the generator of the one-parameter semigroup \(T(t)\) which preserves the log-concavity. It is proved that the matrix \((a_{ij})\) is constant and the vector \((\beta_i)\) is an affine mapping.