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372 results on '"Shaposhnikov S"'

8. Log-Sobolev-type inequalities for solutions to stationary Fokker-Planck-Kolmogorov equations

Author

Bogachev, V. I., Shaposhnikov, A. V., and Shaposhnikov, S. V.

Subjects
Mathematics - Probability, Mathematics - Analysis of PDEs, Mathematics - Functional Analysis, 35J15
Abstract

We prove that every probability measure $\mu$ satisfying the stationary Fokker-Planck-Kolmogorov equation obtained by a $\mu$-integrable perturbation $v$ of the drift term $-x$ of the Ornstein-Uhlenbeck operator is absolutely continuous with respect to the corresponding Gaussian measure $\gamma$ and for the density $f=d\mu/d\gamma$ the integral of $f |\log (f+1)|^\alpha$ against $\gamma$ is estimated via $\|v\|_{L^1(\mu)}$ for all $\alpha<1/4$, which is a weakened $L^1$-analog of the logarithmic Sobolev inequality. This means that stationary measures of diffusions whose drifts are integrable perturbations of $-x$ are absolutely continuous with respect to Gaussian measures.

Published
2018

9. Convergence in variation of solutions of nonlinear Fokker-Planck-Kolmogorov equations to stationary measures

Author

Bogachev, V. I., Röckner, M., and Shaposhnikov, S. V.

Subjects
Mathematics - Probability, 60J75, 47G20, 60G52
Abstract

We study convergence in variation of probability solutions of nonlinear Fokker-Planck-Kolmogorov equations to stationary solutions. We obtain sufficient conditions for the exponential convergence of solutions to the stationary solution in case of coefficients that can have an arbitrary growth at infinity and depend on the solutions through convolutions with unbounded discontinuous kernels. In addition, we study a more difficult case where the nonlinear equation has several stationary solutions and convergence to a stationary solution depends on initial data. Finally, we obtain sufficient conditions for solvability of nonlinear Fokker-Planck-Kolmogorov equations.

Published
2018

11. On the uniqueness of solutions to continuity equations

Author

Bogachev, V. I., Da Prato, G., Röckner, M., and Shaposhnikov, S. V.

Subjects
Mathematics - Analysis of PDEs, Mathematics - Functional Analysis
Abstract

We obtain sufficient conditions for the uniqueness of solutions to the Cauchy problem for the continuity equation in classes of measures that need not be absolutely continuous.

Published
2014

34. Vladimir Igorevich Bogachev

Author

Borodin, P. A., primary, Ibragimov, I. A., additional, Kashin, B. S., additional, Kozlov, V. V., additional, Kolesnikov, A. V., additional, Konyagin, S. V., additional, Kosov, E. D., additional, Smolyanov, O. G., additional, Tolmachev, N. A., additional, Treshchev, D. V., additional, Shaposhnikov, A. V., additional, Shaposhnikov, S. V., additional, Shiryaev, A. N., additional, and Shkalikov, A. A., additional

Published
2021
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