  Let $M$ be a complete connected Riemannian manifold with boundary $\partial
M$, and let $P_t$ be the Neumann semigroup generated by $\frac{ 1}{ 2} L$ where
$L=\Delta+Z$ for a $C^1$-vector field $Z$ on $M$. We establish Bismut type
formulae for $LP_t f$ and ${\rm Hess}_{P_tf}$ and present estimates of these
quantities under suitable curvature conditions. In case when $P_t$ is symmetric
in $L^2(\mu)$ for some probability measure $\mu$, a new type of log-Sobolev
inequality is established which links the relative entropy $H$, the Stein
discrepancy $S$, and relative Fisher information $I$, generalizing the authors'
recent work in the case without boundary.
