## Non-symmetric diffusions on a Riemannian manifold.(English)Zbl 1200.58024

Kotani, Motoko (ed.) et al., Probabilistic approach to geometry. Proceedings of the 1st international conference, Kyoto, Japan, 28th July – 8th August, 2008. Tokyo: Mathematical Society of Japan (MSJ) (ISBN 978-4-931469-58-7/hbk). Advanced Studies in Pure Mathematics 57, 437-461 (2010).
Let $$(M,g)$$ be a smooth and complete $$d$$-dimensional Riemannian manifold (not necessarily compact). Consider a diffusion process on $$M$$ with generator given by ${\mathfrak A}=\frac{1}{2} \Delta + b\,$ where $$\Delta$$ is the Laplace-Beltrami operator, $$b$$ is a smooth vector field. The author gives sufficient conditions such that the closure of $$({\mathfrak A}, C_0^\infty(M))$$ generates a Markovian $$C_0$$ semigroup in $$L^2(m)$$, where $$m$$ denotes the volume on $$M$$. The proof is based on showing that the operator $${\mathfrak A}$$ is maximal dissipative (an operator $$A$$ is called dissipative if $$(Au,u)\leq 0$$ for any $$u\in\text{Dom}(A)$$ and if it has no proper dissipative extension, it is called maximal dissipative). Moreover, the author provides a characterization of the generator domain which states as follows: a necessary and sufficient condition for $$u\in{\text{Dom}}({\mathfrak A})$$ is that $$u\in{\text{Dom}}({\Delta})$$ and $$b u\in L^2(m)$$. The $$L^p$$ setting where $$1<p< \infty$$ is also considered by the author and some examples are given.
For the entire collection see [Zbl 1190.60003].

### MSC:

 58J65 Diffusion processes and stochastic analysis on manifolds 60J60 Diffusion processes 35P15 Estimates of eigenvalues in context of PDEs