×

zbMATH — the first resource for mathematics

The Fokker-Planck-Kolmogorov equations with a potential and a non-uniformly elliptic diffusion matrix. (English. Russian original) Zbl 1310.35229
Trans. Mosc. Math. Soc. 2013, 15-29 (2013); translation from Tr. Mosk. Mat. O.-va 74, No. 1, 17-34 (2013).
Summary: We study solutions of the Fokker-Planck-Kolmogorov equation with unbounded coefficients and a non-uniformly elliptic diffusion matrix. Upper bounds for solutions are obtained. In addition, new estimates with a Lyapunov function are obtained.

MSC:
35Q84 Fokker-Planck equations
35K10 Second-order parabolic equations
60J60 Diffusion processes
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] V. I. Bogachev, N. V. Krylov, and M. Rëkner, Elliptic and parabolic equations for measures, Uspekhi Mat. Nauk 64 (2009), no. 6(390), 5 – 116 (Russian, with Russian summary); English transl., Russian Math. Surveys 64 (2009), no. 6, 973 – 1078. · Zbl 1194.35481 · doi:10.1070/RM2009v064n06ABEH004652 · doi.org
[2] V. I. Bogachëv, M. Rëkner, and S. V. Shaposhnikov, Global regularity and estimates for solutions of parabolic equations, Teor. Veroyatn. Primen. 50 (2005), no. 4, 652 – 674 (Russian, with Russian summary); English transl., Theory Probab. Appl. 50 (2006), no. 4, 561 – 581. · Zbl 1203.60095 · doi:10.1137/S0040585X97981986 · doi.org
[3] V. I. Bogachëv, M. Rëkner, and S. V. Shaposhnikov, Estimates for the densities of stationary distributions and transition probabilities of diffusion processes, Teor. Veroyatn. Primen. 52 (2007), no. 2, 240 – 270 (Russian, with Russian summary); English transl., Theory Probab. Appl. 52 (2008), no. 2, 209 – 236. · Zbl 1154.35322 · doi:10.1137/S0040585X97982967 · doi.org
[4] V. I. Bogachav, M. Rëkner, and S. V. Shaposhnikov, Positive densities of transition probabilities of diffusion processes, Teor. Veroyatn. Primen. 53 (2008), no. 2, 213-239; English transl., Theory Probab. Appl. 53 (2009) no. 2, 194-215.
[5] F. O. Porper and S. D. Èĭdel\(^{\prime}\)man, Two-sided estimates of the fundamental solutions of second-order parabolic equations and some applications of them, Uspekhi Mat. Nauk 39 (1984), no. 3(237), 107 – 156 (Russian).
[6] S. V. Shaposhnikov, Regularity and qualitative properties of solutions of parabolic equations for measures, Teor. Veroyatn. Primen. 56 (2011), no. 2, 318-350; English transl., Theory Probab. Appl. 56 (2012), no. 2, 252-279.
[7] S. V. Shaposhnikov, Estimates for solutions of parabolic equations for measures, Dokl. Akad. Nauk 434 (2010), no. 4, 454 – 458 (Russian); English transl., Dokl. Math. 82 (2010), no. 2, 769 – 772. · Zbl 1221.35080 · doi:10.1134/S1064562410050236 · doi.org
[8] D. G. Aronson and James Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Rational Mech. Anal. 25 (1967), 81 – 122. · Zbl 0154.12001 · doi:10.1007/BF00281291 · doi.org
[9] Aissa Aibeche, Karima Laidoune, and Abdelaziz Rhandi, Time dependent Lyapunov functions for some Kolmogorov semigroups perturbed by unbounded potentials, Arch. Math. (Basel) 94 (2010), no. 6, 565 – 577. · Zbl 1200.35027 · doi:10.1007/s00013-010-0124-2 · doi.org
[10] V. I. Bogachev, G. Da Prato, and M. Röckner, On parabolic equations for measures, Comm. Partial Differential Equations 33 (2008), no. 1-3, 397 – 418. · Zbl 1323.35058 · doi:10.1080/03605300701382415 · doi.org
[11] V. I. Bogachev, N. V. Krylov, and M. Röckner, On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions, Comm. Partial Differential Equations 26 (2001), no. 11-12, 2037 – 2080. · Zbl 0997.35012 · doi:10.1081/PDE-100107815 · doi.org
[12] V. I. Bogachev, M. Röckner, and S. V. Shaposhnikov, On uniqueness problems related to the Fokker-Planck-Kolmogorov equation for measures, J. Math. Sci. (N.Y.) 179 (2011), no. 1, 7 – 47. Problems in mathematical analysis. No. 61. · Zbl 1291.35425 · doi:10.1007/s10958-011-0581-6 · doi.org
[13] Karima Laidoune, Giorgio Metafune, Diego Pallara, and Abdelaziz Rhandi, Global properties of transition kernels associated with second-order elliptic operators, Parabolic problems, Progr. Nonlinear Differential Equations Appl., vol. 80, Birkhäuser/Springer Basel AG, Basel, 2011, pp. 415 – 432. · Zbl 1254.35049 · doi:10.1007/978-3-0348-0075-4_21 · doi.org
[14] G. Metafune, D. Pallara, and A. Rhandi, Global properties of transition probabilities of singular diffusions, Teor. Veroyatn. Primen. 54 (2009), no. 1, 116 – 148 (English, with Russian summary); English transl., Theory Probab. Appl. 54 (2010), no. 1, 68 – 96. · Zbl 1206.60072 · doi:10.1137/S0040585X97984012 · doi.org
[15] Jürgen Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101 – 134. · Zbl 0149.06902 · doi:10.1002/cpa.3160170106 · doi.org
[16] Chiara Spina, Kernel estimates for a class of Kolmogorov semigroups, Arch. Math. (Basel) 91 (2008), no. 3, 265 – 279. · Zbl 1161.47028 · doi:10.1007/s00013-008-2676-y · doi.org
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.