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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

Vladimir I. Bogachev 1, , Stanislav V. Shaposhnikov 1, and  Alexander Yu. Veretennikov 2,

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.
Keywords: Stationary Fokker--Planck--Kolmogorov equation, differentiability with respect to a parameter..
Mathematics Subject Classification: Primary: 35J15; Secondary: 60J6.
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

show all references

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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