# American Institute of Mathematical Sciences

July  2016, 36(7): 3519-3543. doi: 10.3934/dcds.2016.36.3519

## 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

Received  February 2015 Revised  December 2015 Published  March 2016

We obtain sufficient conditions for the differentiability of solutions to stationary Fokker--Planck--Kolmogorov equations with respect to a parameter. In particular, this gives conditions for the differentiability of stationary distributions of diffusion processes with respect to a parameter.
Citation: Vladimir I. Bogachev, Stanislav V. Shaposhnikov, Alexander Yu. Veretennikov. Differentiability of solutions of stationary Fokker--Planck--Kolmogorov equations with respect to a parameter. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3519-3543. doi: 10.3934/dcds.2016.36.3519
##### References:
 [1] A. Arapostathis, V. S. Borkar and M. K. Ghosh, Ergodic Control of Diffusion Processes, Cambridge University Press, Cambridge, 2012.  Google Scholar [2] V. I. Bogachev, Measure Theory, V. 1, 2, Springer, Berlin - New York, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, New Jersey, 1964.  Google Scholar [15] C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton - London, 1992.  Google Scholar [16] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin - New York, 1977.  Google Scholar [17] N. V. Krylov, Controlled Diffusion Processes, Springer-Verlag, New York, 1980.  Google Scholar [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.  Google Scholar [19] M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, 2nd ed., Academic Press, San Diego - Toronto, 1980.  Google Scholar [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.  Google Scholar [21] N. S. Trudinger, Linear elliptic operators with measurable coefficients, Ann. Scuola Normale Super. Pisa (3), 27 (1973), 265-308.  Google Scholar [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.  Google Scholar [23] A. Yu. Veretennikov, On Sobolev solutions of Poisson equations in $\mathbbR^d$ with a parameter, J. Math. Sci. (New York), 179 (2011), 48-79. doi: 10.1007/s10958-011-0582-5.  Google Scholar [24] W. Ziemer, Weakly Differentiable Functions, Springer-Verlag, New York - Berlin, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

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##### References:
 [1] A. Arapostathis, V. S. Borkar and M. K. Ghosh, Ergodic Control of Diffusion Processes, Cambridge University Press, Cambridge, 2012.  Google Scholar [2] V. I. Bogachev, Measure Theory, V. 1, 2, Springer, Berlin - New York, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, New Jersey, 1964.  Google Scholar [15] C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton - London, 1992.  Google Scholar [16] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin - New York, 1977.  Google Scholar [17] N. V. Krylov, Controlled Diffusion Processes, Springer-Verlag, New York, 1980.  Google Scholar [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.  Google Scholar [19] M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, 2nd ed., Academic Press, San Diego - Toronto, 1980.  Google Scholar [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.  Google Scholar [21] N. S. Trudinger, Linear elliptic operators with measurable coefficients, Ann. Scuola Normale Super. Pisa (3), 27 (1973), 265-308.  Google Scholar [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.  Google Scholar [23] A. Yu. Veretennikov, On Sobolev solutions of Poisson equations in $\mathbbR^d$ with a parameter, J. Math. Sci. (New York), 179 (2011), 48-79. doi: 10.1007/s10958-011-0582-5.  Google Scholar [24] W. Ziemer, Weakly Differentiable Functions, Springer-Verlag, New York - Berlin, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar
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