Advanced Studies in Pure Mathematics

Non-symmetric diffusions on a Riemannian manifold

Ichiro Shigekawa

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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 $L^p$.

Article information

Probabilistic Approach to Geometry, M. Kotani, M. Hino and T. Kumagai, eds. (Tokyo: Mathematical Society of Japan, 2010), 437-461

Received: 15 January 2009
Revised: 17 July 2009
First available in Project Euclid: 24 November 2018

Permanent link to this document euclid.aspm/1543086331

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60] 60J60: Diffusion processes [See also 58J65] 35P15: Estimation of eigenvalues, upper and lower bounds

Non-symmetric diffusion Riemannian manifold maximal dissipative operator generator domain


Shigekawa, Ichiro. Non-symmetric diffusions on a Riemannian manifold. Probabilistic Approach to Geometry, 437--461, Mathematical Society of Japan, Tokyo, Japan, 2010. doi:10.2969/aspm/05710437.

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