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Differentiability of solutions of stationary Fokker--Planck--Kolmogorov equations with respect to a parameter
1. | National Research University Higher School of Economics, Vavilova 7, Moscow, 117312, Russian Federation, Russian Federation |
2. | University of Leeds, Leeds, LS2 9JT, United Kingdom |
References:
[1] |
A. Arapostathis, V. S. Borkar and M. K. Ghosh, Ergodic Control of Diffusion Processes, Cambridge University Press, Cambridge, 2012. |
[2] |
V. I. Bogachev, Measure Theory, V. 1, 2, Springer, Berlin - New York, 2007.
doi: 10.1007/978-3-540-34514-5. |
[3] |
V. I. Bogachev, A. I. Kirillov and S. V. Shaposhnikov, On probability and integrable solutions to the stationary Kolmogorov equation, Dokl. Russian Acad. Sci., 438 (2011), 154-159 (in Russian); English transl.: Dokl. Math., 83 (2011), 309-313.
doi: 10.1134/S1064562411030112. |
[4] |
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 Diff. Eq., 26 (2001), 2037-2080.
doi: 10.1081/PDE-100107815. |
[5] |
V. I. Bogachev, N. V. Krylov and M. Röckner, Elliptic equations for measures: Regularity and global bounds of densities, J. Math. Pures Appl., 85 (2006), 743-757.
doi: 10.1016/j.matpur.2005.11.006. |
[6] |
V. I. Bogachev, N. V. Krylov and M. Röckner, Elliptic and parabolic equations for measures, Uspehi Mat. Nauk, 64 (2009), 5-116 (in Russian); English transl.: Russian Math. Surveys, 64 (2009), 973-1078.
doi: 10.1070/RM2009v064n06ABEH004652. |
[7] |
V. I. Bogachev and M. Röckner, A generalization of Khasminskii's theorem on the existence of invariant measures for locally integrable drifts, Teor. Verojatn. i Primen., 45 (2000), 417-436; correction: Ibid. 46 (2001), p600 (in Russian); English transl.: Theory Probab. Appl., 45 (2001), 363-378.
doi: 10.1137/S0040585X97978348. |
[8] |
V. I. Bogachev, M. Röckner and S. V. Shaposhnikov, Estimates of densities of stationary distributions and transition probabilities of diffusion processes, Teor. Verojatn. i Primen., 52 (2007), 240-270 (in Russian); English transl.: Theory Probab. Appl., 52 (2008), 209-236.
doi: 10.1137/S0040585X97982967. |
[9] |
V. I. Bogachev, M. Röckner and S. V. Shaposhnikov, On uniqueness problems related to elliptic equations for measures, J. Math. Sci. (New York), 176 (2011), 759-773.
doi: 10.1007/s10958-011-0434-3. |
[10] |
V. I. Bogachev, M. Röckner and S. V. Shaposhnikov, On positive and probability solutions of the stationary Fokker-Planck-Kolmogorov equation, Dokl. Akad. Nauk, 444 (2012), 245-249 (in Russian); English transl.: Dokl. Math., 85 (2012), 350-354.
doi: 10.1134/S1064562412030143. |
[11] |
V. I. Bogachev, M. Röckner and S. V. Shaposhnikov, On existence of Lyapunov functions for a stationary Kolmogorov equation with a probability solution, Dokl. Akad. Nauk, 457 (2014), 136-140 (in Russian); English transl.: Dokl. Math., 90 (2014), 424-428. |
[12] |
V. I. Bogachev, M. Röckner and W. Stannat, Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions, Matem. Sb., 193 (2002), 3-36 (in Russian); English transl.: Sbornik Math., 193 (2002), 945-976.
doi: 10.1070/SM2002v193n07ABEH000665. |
[13] |
V. I. Bogachev, M. Röckner and F.-Y. Wang, Elliptic equations for invariant measures on finite and infinite dimensional manifolds, J. Math. Pures Appl., 80 (2001), 177-221.
doi: 10.1016/S0021-7824(00)01187-9. |
[14] |
A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, New Jersey, 1964. |
[15] |
C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton - London, 1992. |
[16] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin - New York, 1977. |
[17] |
N. V. Krylov, Controlled Diffusion Processes, Springer-Verlag, New York, 1980. |
[18] |
E. Pardoux and A. Yu. Veretennikov, On the Poisson equation and diffusion approximation. II, Ann. Probab., 31 (2003), 1166-1192.
doi: 10.1214/aop/1055425774. |
[19] |
M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, 2nd ed., Academic Press, San Diego - Toronto, 1980. |
[20] |
S. V. Shaposhnikov, On interior estimates for the Sobolev norms of solutions of elliptic equations, Matem. Zametki, 83 (2008), 316-320 (in Russian); English transl.: Math. Notes, 83 (2008), 285-289.
doi: 10.1134/S0001434608010318. |
[21] |
N. S. Trudinger, Linear elliptic operators with measurable coefficients, Ann. Scuola Normale Super. Pisa (3), 27 (1973), 265-308. |
[22] |
N. S. Trudinger, Maximum principles for linear, non-uniformly elliptic operators with measurable coefficients, Math. Z., 156 (1977), 291-301.
doi: 10.1007/BF01214416. |
[23] |
A. Yu. Veretennikov, On Sobolev solutions of Poisson equations in $\mathbbmathbb{R}^{d}$ with a parameter, J. Math. Sci. (New York), 179 (2011), 48-79.
doi: 10.1007/s10958-011-0582-5. |
[24] |
W. Ziemer, Weakly Differentiable Functions, Springer-Verlag, New York - Berlin, 1989.
doi: 10.1007/978-1-4612-1015-3. |
show all references
References:
[1] |
A. Arapostathis, V. S. Borkar and M. K. Ghosh, Ergodic Control of Diffusion Processes, Cambridge University Press, Cambridge, 2012. |
[2] |
V. I. Bogachev, Measure Theory, V. 1, 2, Springer, Berlin - New York, 2007.
doi: 10.1007/978-3-540-34514-5. |
[3] |
V. I. Bogachev, A. I. Kirillov and S. V. Shaposhnikov, On probability and integrable solutions to the stationary Kolmogorov equation, Dokl. Russian Acad. Sci., 438 (2011), 154-159 (in Russian); English transl.: Dokl. Math., 83 (2011), 309-313.
doi: 10.1134/S1064562411030112. |
[4] |
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 Diff. Eq., 26 (2001), 2037-2080.
doi: 10.1081/PDE-100107815. |
[5] |
V. I. Bogachev, N. V. Krylov and M. Röckner, Elliptic equations for measures: Regularity and global bounds of densities, J. Math. Pures Appl., 85 (2006), 743-757.
doi: 10.1016/j.matpur.2005.11.006. |
[6] |
V. I. Bogachev, N. V. Krylov and M. Röckner, Elliptic and parabolic equations for measures, Uspehi Mat. Nauk, 64 (2009), 5-116 (in Russian); English transl.: Russian Math. Surveys, 64 (2009), 973-1078.
doi: 10.1070/RM2009v064n06ABEH004652. |
[7] |
V. I. Bogachev and M. Röckner, A generalization of Khasminskii's theorem on the existence of invariant measures for locally integrable drifts, Teor. Verojatn. i Primen., 45 (2000), 417-436; correction: Ibid. 46 (2001), p600 (in Russian); English transl.: Theory Probab. Appl., 45 (2001), 363-378.
doi: 10.1137/S0040585X97978348. |
[8] |
V. I. Bogachev, M. Röckner and S. V. Shaposhnikov, Estimates of densities of stationary distributions and transition probabilities of diffusion processes, Teor. Verojatn. i Primen., 52 (2007), 240-270 (in Russian); English transl.: Theory Probab. Appl., 52 (2008), 209-236.
doi: 10.1137/S0040585X97982967. |
[9] |
V. I. Bogachev, M. Röckner and S. V. Shaposhnikov, On uniqueness problems related to elliptic equations for measures, J. Math. Sci. (New York), 176 (2011), 759-773.
doi: 10.1007/s10958-011-0434-3. |
[10] |
V. I. Bogachev, M. Röckner and S. V. Shaposhnikov, On positive and probability solutions of the stationary Fokker-Planck-Kolmogorov equation, Dokl. Akad. Nauk, 444 (2012), 245-249 (in Russian); English transl.: Dokl. Math., 85 (2012), 350-354.
doi: 10.1134/S1064562412030143. |
[11] |
V. I. Bogachev, M. Röckner and S. V. Shaposhnikov, On existence of Lyapunov functions for a stationary Kolmogorov equation with a probability solution, Dokl. Akad. Nauk, 457 (2014), 136-140 (in Russian); English transl.: Dokl. Math., 90 (2014), 424-428. |
[12] |
V. I. Bogachev, M. Röckner and W. Stannat, Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions, Matem. Sb., 193 (2002), 3-36 (in Russian); English transl.: Sbornik Math., 193 (2002), 945-976.
doi: 10.1070/SM2002v193n07ABEH000665. |
[13] |
V. I. Bogachev, M. Röckner and F.-Y. Wang, Elliptic equations for invariant measures on finite and infinite dimensional manifolds, J. Math. Pures Appl., 80 (2001), 177-221.
doi: 10.1016/S0021-7824(00)01187-9. |
[14] |
A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, New Jersey, 1964. |
[15] |
C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton - London, 1992. |
[16] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin - New York, 1977. |
[17] |
N. V. Krylov, Controlled Diffusion Processes, Springer-Verlag, New York, 1980. |
[18] |
E. Pardoux and A. Yu. Veretennikov, On the Poisson equation and diffusion approximation. II, Ann. Probab., 31 (2003), 1166-1192.
doi: 10.1214/aop/1055425774. |
[19] |
M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, 2nd ed., Academic Press, San Diego - Toronto, 1980. |
[20] |
S. V. Shaposhnikov, On interior estimates for the Sobolev norms of solutions of elliptic equations, Matem. Zametki, 83 (2008), 316-320 (in Russian); English transl.: Math. Notes, 83 (2008), 285-289.
doi: 10.1134/S0001434608010318. |
[21] |
N. S. Trudinger, Linear elliptic operators with measurable coefficients, Ann. Scuola Normale Super. Pisa (3), 27 (1973), 265-308. |
[22] |
N. S. Trudinger, Maximum principles for linear, non-uniformly elliptic operators with measurable coefficients, Math. Z., 156 (1977), 291-301.
doi: 10.1007/BF01214416. |
[23] |
A. Yu. Veretennikov, On Sobolev solutions of Poisson equations in $\mathbbmathbb{R}^{d}$ with a parameter, J. Math. Sci. (New York), 179 (2011), 48-79.
doi: 10.1007/s10958-011-0582-5. |
[24] |
W. Ziemer, Weakly Differentiable Functions, Springer-Verlag, New York - Berlin, 1989.
doi: 10.1007/978-1-4612-1015-3. |
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