Geodesic
Estimates of distances between solutions of Fokker--Planck--Kolmogorov equations with partially degenerate diffusion matrices
Teoriâ slučajnyh processov, Tome 23 (2018) no. 2, pp. 41-54.

Voir la notice de l'article provenant de la source Math-Net.Ru

Using a metric which interpolates between the Kantorovich metric and the total variation norm we estimate the distance between solutions to Fokker–Planck–Kolmogorov equations with degenerate diffusion matrices. Some relations between the degeneracy of the diffusion matrix and the regularity of the drift coefficient are analysed. Applications to nonlinear Fokker–Planck–Kolmogorov equations are given.
Keywords: Degenerate diffusion matrix.
Mots-clés : Fokker–Planck–Kolmogorov equation 
@article{THSP_2018_23_2_a4,
     author = {Oxana A. Manita and Maxim S. Romanov and Stanislav V. Shaposhnikov},
     title = {Estimates of distances between solutions of {Fokker--Planck--Kolmogorov} equations with partially degenerate diffusion matrices},
     journal = {Teori\^a slu\v{c}ajnyh processov},
     pages = {41--54},
     publisher = {mathdoc},
     volume = {23},
     number = {2},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/THSP_2018_23_2_a4/}
}
TY  - JOUR
AU  - Oxana A. Manita
AU  - Maxim S. Romanov
AU  - Stanislav V. Shaposhnikov
TI  - Estimates of distances between solutions of Fokker--Planck--Kolmogorov equations with partially degenerate diffusion matrices
JO  - Teoriâ slučajnyh processov
PY  - 2018
SP  - 41
EP  - 54
VL  - 23
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/THSP_2018_23_2_a4/
LA  - en
ID  - THSP_2018_23_2_a4
ER  - 
%0 Journal Article
%A Oxana A. Manita
%A Maxim S. Romanov
%A Stanislav V. Shaposhnikov
%T Estimates of distances between solutions of Fokker--Planck--Kolmogorov equations with partially degenerate diffusion matrices
%J Teoriâ slučajnyh processov
%D 2018
%P 41-54
%V 23
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/THSP_2018_23_2_a4/
%G en
%F THSP_2018_23_2_a4
Oxana A. Manita; Maxim S. Romanov; Stanislav V. Shaposhnikov. Estimates of distances between solutions of Fokker--Planck--Kolmogorov equations with partially degenerate diffusion matrices. Teoriâ slučajnyh processov, Tome 23 (2018) no. 2, pp. 41-54. http://geodesic.mathdoc.fr/item/THSP_2018_23_2_a4/

[1] V. I. Bogachev, Measure theory, Springer-Verlag, Berlin, 2007 | MR | Zbl

[2] V. I. Bogachev, N. V. Krylov, M. Röckner, S. V. Shaposhnikov, Fokker–Planck–Kolmogorov equations, Amer. Math. Soc., Providence, Rhode Island, 2015 | MR | Zbl

[3] V. I. Bogachev, M. Röckner, S. V. Shaposhnikov, “Uniqueness problems for degenerate Fokker–Planck–Kolmogorov equations”, J. Math. Sci. (New York), 207:2 (2015), 147–165 | DOI | MR | Zbl

[4] V. I. Bogachev, G. Da Prato, M. Röckner, S. V. Shaposhnikov, “An analytic approach to infinite dimensional continuity and Fokker–Planck–Kolmogorov equations”, Annali della Scuola Normale Super. di Pisa, 14:3 (2015), 983–1023 | MR | Zbl

[5] V. I. Bogachev, G. Da Prato, M. Röckner, S. V. Shaposhnikov, “On the uniqueness of solutions to continuity equations”, J. Differ. Equ., 259:8 (2015), 3854–3873 | DOI | MR | Zbl

[6] V. I. Bogachev, M. Röckner, S. V. Shaposhnikov, “Distances between transition probabilities of diffusions and applications to nonlinear Fokker–Planck–Kolmogorov equations”, J. Funct. Anal., 271 (2016), 1262–1300 | DOI | MR | Zbl

[7] T. D. Frank, Nonlinear Fokker–Planck equations. Fundamentals and applications, Springer-Verlag, Berlin, 2005 | MR | Zbl

[8] V. Konakov, A. Kozhina, S. Menozzi, “Stability of densities for perturbed diffusions and Markov chains”, ESAIM: Probab. Stat., 21 (2017), 88–112 | DOI | MR | Zbl

[9] H. Li, D. Luo, “Quantitative stability estimates for solutions of Fokker–Planck equations”, J. Math. Pures Appl., 122 (2019), 125–163 | DOI | MR | Zbl

[10] O. A. Manita, “Estimates for transportation costs along solutions to Fokker–Planck–Kolmogorov equations with dissipative drifts”, Rendiconti Lincei – Matem. Appl., 28:3 (2018), 601–618 | DOI | MR

[11] O. A. Manita, M. S. Romanov, S. V. Shaposhnikov, “On uniqueness of solutions to nonlinear Fokker–Planck–Kolmogorov equations”, Nonlinear Analysis: Theory, Methods and Applications, 128 (2015), 199–226 | DOI | MR | Zbl

[12] O. A. Manita, M. S. Romanov, S. V. Shaposhnikov, “Fokker–Planck–Kolmogorov equations with a partially degenerate diffusion matrix”, Doklady Math., 96:1 (2017), 384–388 | DOI | MR | Zbl

[13] O. A. Manita, S. V. Shaposhnikov, “Nonlinear parabolic equations for measures”, St. Petersburg Math. J., 25:1 (2014), 43–62 | DOI | MR | Zbl

[14] O. A. Manita, S. V. Shaposhnikov, “On the Cauchy problem for Fokker–Planck–Kolmogorov equations with potential terms on arbitrary domains”, J. Dynamics Differ. Equ., 28:2 (2016), 493–518 | DOI | MR | Zbl

[15] H. Risken, The Fokker–Planck equation, Springer–Verlag, Berlin, 1996 | MR | Zbl

[16] D. W. Stroock, S. R. S. Varadhan, Multidimensional diffusion processes, Springer-Verlag, Berlin – New York, 1979 | MR | Zbl