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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
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