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Stationary diffusion processes with discontinuous drift coefficients

Author: A. I. Noarov
Translated by: S. V. Kislyakov
Original publication: Algebra i Analiz, tom 24 (2012), nomer 5.
Journal: St. Petersburg Math. J. 24 (2013), 795-809
MSC (2010): Primary 60J60; Secondary 35J99
Published electronically: July 24, 2013
MathSciNet review: 3087824
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Abstract: The paper is devoted to the stationary Fokker-Planck equation $ \Delta u - \mathrm {div} (u \mathbf {f})=0$ with a locally bounded measurable vector field $ \mathbf {f}$ defined on the entire $ \mathbb{R}^n$. The existence of a positive (not necessarily integrable) solution is proved. Various conditions on the vector field $ \mathbf {f}$ are deduced that suffice for the existence of a solution that is a probability density. Under these conditions, the corresponding diffusion has an invariant probability measure with density $ u$. In particular, the assumptions of the theorems proved here are fulfilled for certain vector fields $ \mathbf {f}$ with trajectories tending to infinity. In this situation, the approach in question turns out to be more efficient than the earlier methods based on the construction of a Lyapunov function.

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Additional Information

A. I. Noarov
Affiliation: Institute of Numerical Mathematics, ul. Gubkina 8, Moscow 119333, Russia

Keywords: Elliptic equation for measures, invariant measure, averaging method
Received by editor(s): September 16, 2011
Published electronically: July 24, 2013
Article copyright: © Copyright 2013 American Mathematical Society