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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
DOI: https://doi.org/10.1090/S1061-0022-2013-01266-0
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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  • 1. A. I. Noarov, On the solvability of stationary Fokker-Planck equations that are close to the Laplace equation, Differ. Uravn. 42 (2006), no. 4, 521-530; English transl., Differ. Equ. 42 (2006), no. 4, 556-566. MR 2296526 (2007j:35042)
  • 2. -, Generalized solvability of the stationary Fokker-Planck equation, Differ. Uravn. 43 (2007), no. 6, 813-819; English transl., Differ. Equ. 43 (2007), no. 6, 833-839. MR 2383830 (2008h:35039)
  • 3. -, Unique solvability of the stationary Fokker-Planck equation in the class of positive functions, Differ. Uravn. 45 (2009), no. 2, 191-202; English transl., Differ. Equ. 45 (2009), no. 2, 197-208. MR 2596729 (2010m:35099)
  • 4. -, On some diffusion processes with stationary distributions, Teor. Veroyatnost. i Primenen. 54 (2009) , no. 3, 589-598; English transl., Theory Probab. Appl. 54 (2009), no. 3, 525-533. MR 2766352 (2011k:60270)
  • 5. E. C. Zeeman, Stability of dynamical systems, Nonlinearity 1 (1988), 115-155. MR 928950 (89d:58070)
  • 6. R. Z. Khas'minskiĭ, Stability of systems of differential equations under random perturbations of their parameters, Nauka, Moscow, 1969. (Russian) MR 0259283 (41:3925)
  • 7. V. I. Bogachev and M. Roeckner, A generalization of Khas'minskiĭ's theorem on the existence of invariant measures for locally integrable drifts, Teor. Veroyatnost. i Primenen. 45 (2000), no. 3, 417-436; English transl., Theory Probab. Appl. 45 (2002), no. 3, 363-378. MR 1967783 (2004h:60086)
  • 8. K. Yosida, Functional analysis, Springer-Verlag, Berlin, 1995. MR 1336382 (96a:46001)
  • 9. E. Lieb and M. Loss, Analysis, Grad. Stud. Math., vol. 14, Amer. Math. Soc., Providence, RI, 1997. MR 1415616 (98b:00004)
  • 10. V. I. Bogachev, M. Roeckner, and S. V. Shaposhnikov, Estimates for the densities of stationary distributions and transition probabilities of diffusion processes, Teor. Veroyatnost. i Primenen. 52 (2007), no. 2, 240-270; English transl., Theory Probab. Appl. 52 (2008), no. 2, 209-236. MR 2742501 (2011i:60143)
  • 11. V. I. Bogachev, M. Roeckner, and V. Shtannat, Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions, Mat. Sb. 193 (2002), no. 7, 3-36; English transl., Sb. Math. 193 (2002), no. 7-8, 945-976. MR 1936848 (2003i:35050)

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Additional Information

A. I. Noarov
Affiliation: Institute of Numerical Mathematics, ul. Gubkina 8, Moscow 119333, Russia
Email: ligrans@mail.ru

DOI: https://doi.org/10.1090/S1061-0022-2013-01266-0
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

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