A character of the gradient estimate for diffusion semigroups

Let $P_t$ be the semigroup of the diffusion process generated by $L:= \sum_{i,j}a_{ij}\partial_i\partial_j +\sum_ib_i\partial_i$ on $\mathbb{R} ^d$. It is proved that there exists $c\in \mathbb{R}$ and an $\mathbb{R} ^d$-valued function $b=(b_i)$ such that $\vert\nabla P_tf\vert\le \text{\rm {e}} ^{ct}P_t\vert\nabla f\vert$ holds for all $t>0$ and all $f\in C_b^1(\mathbb{R} ^d)$ if and only if $a=(a_{ij})$ satisfies the formula $\partial_k a_{ij}+\partial_ja_{ki} +\partial_i a_{kj}=0$ for all $i,j,k.$