zbMATH — the first resource for mathematics

Positiveness of invariant measures of diffusion processes. (English. Russian original) Zbl 1173.35048
Dokl. Math. 76, No. 1, 533-538 (2007); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 415, No. 2, 174-179 (2007).
This short note studies measures \(\mu\) on \(\mathbb{R}^n\), which are solutions of \({\mathcal L}^* \mu\), where \({\mathcal L}\) is a linear second order differential operator. This means that for all smooth functions \(u\) one has \(\int Lu \;d\mu=0\).
The main results of the paper give sufficient conditions that ensure the positivity of the density of \(\mu\).
35J25 Boundary value problems for second-order elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35J15 Second-order elliptic equations
Full Text: DOI
[1] V. I. Bogachev and M. Röckner, J. Funct. Anal. 133(1), 168–223 (1995). · Zbl 0840.60069 · doi:10.1006/jfan.1995.1123
[2] V. I. Bogachev, N. V. Krylov, and M. Röckner, Commun. Partial Differ. Equations 26, 2037–2080 (2001). · Zbl 0997.35012 · doi:10.1081/PDE-100107815
[3] V. I. Bogachev, M. Röckner, and S. V. Shaposhnikov, Teor. Veroyatn. Ee Primen. 50(4), 3–26 (2005). · doi:10.4213/tvp156
[4] V. I. Bogachev, N. V. Krylov, and M. Röckner, J. Math. Pures Appl. 85, 743–757 (2006). · Zbl 1206.35242 · doi:10.1016/j.matpur.2005.11.006
[5] G. Metafune, D. Pallara, and A. Rhandi, J. Funct. Anal. 223, 396–424 (2005). · Zbl 1131.35318 · doi:10.1016/j.jfa.2005.02.001
[6] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer-Verlag, Berlin, 1983; Nauka, Moscow, 1989). · Zbl 0562.35001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.