# 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 - A, 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, (2012). Google Scholar [2] V. I. Bogachev, Measure Theory,, V. 1, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1016/S0021-7824(00)01187-9. Google Scholar [14] A. Friedman, Partial Differential Equations of Parabolic Type,, Prentice-Hall, (1964). Google Scholar [15] C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, CRC Press, (1992). Google Scholar [16] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (1977). Google Scholar [17] N. V. Krylov, Controlled Diffusion Processes,, Springer-Verlag, (1980). Google Scholar [18] E. Pardoux and A. Yu. Veretennikov, On the Poisson equation and diffusion approximation. II,, Ann. Probab., 31 (2003), 1166. 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, (1980). Google Scholar [20] S. V. Shaposhnikov, On interior estimates for the Sobolev norms of solutions of elliptic equations,, Matem. Zametki, 83 (2008), 316. 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. Google Scholar [22] N. S. Trudinger, Maximum principles for linear, non-uniformly elliptic operators with measurable coefficients,, Math. Z., 156 (1977), 291. 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. doi: 10.1007/s10958-011-0582-5. Google Scholar [24] W. Ziemer, Weakly Differentiable Functions,, Springer-Verlag, (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, (2012). Google Scholar [2] V. I. Bogachev, Measure Theory,, V. 1, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1016/S0021-7824(00)01187-9. Google Scholar [14] A. Friedman, Partial Differential Equations of Parabolic Type,, Prentice-Hall, (1964). Google Scholar [15] C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, CRC Press, (1992). Google Scholar [16] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (1977). Google Scholar [17] N. V. Krylov, Controlled Diffusion Processes,, Springer-Verlag, (1980). Google Scholar [18] E. Pardoux and A. Yu. Veretennikov, On the Poisson equation and diffusion approximation. II,, Ann. Probab., 31 (2003), 1166. 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, (1980). Google Scholar [20] S. V. Shaposhnikov, On interior estimates for the Sobolev norms of solutions of elliptic equations,, Matem. Zametki, 83 (2008), 316. 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. Google Scholar [22] N. S. Trudinger, Maximum principles for linear, non-uniformly elliptic operators with measurable coefficients,, Math. Z., 156 (1977), 291. 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. doi: 10.1007/s10958-011-0582-5. Google Scholar [24] W. Ziemer, Weakly Differentiable Functions,, Springer-Verlag, (1989). doi: 10.1007/978-1-4612-1015-3. Google Scholar
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