A character of the gradient estimate for diffusion semigroups
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 $|\nabla P_tf|\le \text {\rm {e}} ^{ct}P_t|\nabla f|$ holds for all $t>0$ …