Non-symmetric diffusions on a Riemannian manifold

Abstract

We consider a non-symmetric diffusion on a Riemannian manifold generated by $\mathfrak{A} = \frac{1}{2}\triangle + b$. We give a sufficient condition for which $\mathfrak{A}$ generates a $C_0$-semigroup in $L^2$. To do this, we show that $\mathfrak{A}$ is maximal dissipative. Further we give a characterization of the generator domain.

We also discuss the same issue in $L^p$ ($1 \lt p \lt \infty$) setting and give a sufficient condition for which $\mathfrak{A}$ generates a $C_0$-semigroup in $L^p$.