Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-24T12:11:44.287Z Has data issue: false hasContentIssue false
  • English
  • Français

Regularity theory of Kolmogorov operator revisited

Part of: Special classes of linear operators Stochastic analysis Groups and semigroups of linear operators, their generalizations and applications

Published online by Cambridge University Press:  24 August 2020

Damir Kinzebulatov
Damir Kinzebulatov*
Affiliation:
Département de mathématiques et de statistique, Université Laval, 1045 av. de la Médecine, Québec, QC G1V 0A6, Canada
*
e-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider Kolmorogov operator $-\Delta +b \cdot \nabla $ with drift b in the class of form-bounded vector fields (containing vector fields having critical-order singularities). We characterize quantitative dependence of the Sobolev and Hölder regularity of solutions to the corresponding elliptic equation on the value of the form-bound of b.

Keywords

Elliptic operatorsform-bounded vector fieldsregularity of solutionsFeller semigroups

MSC classification

Primary: 47B44: Accretive operators, dissipative operators, etc.
Secondary: 47D07: Markov semigroups and applications to diffusion processes 60H10: Stochastic ordinary differential equations
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 4 , December 2021 , pp. 725 - 736
Check for updates
Copyright
© Canadian Mathematical Society 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The research of the author is supported by grants from NSERC and FRQNT.

References

Bass, R. and Chen, Z.-Q., Brownian motion with singular drift . Ann. Probab. 31(2003), 791–817. https://doi.org/10.1214/aop/1048516536 CrossRefGoogle Scholar
Blumenthal, R. M. and Getoor, R. K., Markov processes and potential theory . Pure and Applied Mathematics, 29, Academic Press, New York, NY, 1968.Google Scholar
Chang, S. Y. A., Wilson, J. M., and Wolff, T. H., Some weighted norm inequalities concerning the Schrödinger operator . Comment. Math. Helvetici 60(1985), 217–246. https://doi.org/10.1007/BF02567411 CrossRefGoogle Scholar
Kato, T., Perturbation theory for linear operators . Classics in Mathematics, 132, Springer-Verlag, Berlin, Heidelberg/Germany, 1995.CrossRefGoogle Scholar
Kato, T., Remarks on pseudo-resolvents and infinitesimal generators . Proc. Japan. Acad. 35(1959), 467–468.Google Scholar
Kinzebulatov, D., A new approach to the ${L}^p$ -theory of $-\varDelta +b\cdot \nabla$ , and its applications to Feller processes with general drifts . Ann. Sc. Norm. Sup. Pisa 17(2017), no. 5, 685–711.Google Scholar
Kinzebulatov, D. and Semënov, Y. A., On the theory of the Kolmogorov operator in the spaces ${L}^p$ and C∞ . Ann. Sc. Norm. Sup. Pisa 5(2017), to appear.Google Scholar
Kinzebulatov, D. and Semënov, Yu. A., Brownian motion with general drift . Stoc. Proc. Appl. 130(2020), 2737–2750. https://doi.org/10.1016/j.spa.2019.08.003 CrossRefGoogle Scholar
Kovalenko, V. F., Perelmuter, M. A., and Semënov, Yu. A., Schrödinger operators with ${L}_W^{1/ 2}\left({R}^l\right)$ -potentials . J. Math. Phys. 22(1981), 1033–1044. https://doi.org/10.1063/1.525009 CrossRefGoogle Scholar
Kovalenko, V. F. and Semënov, Yu. A., C0 -semigroups in ${L}^p\left({\mathbb{R}}^d\right)$ and ${C}_{\infty}\left({\mathbb{R}}^d\right)$ spaces generated by differential expression $\varDelta +b\cdot \nabla$ (Russian). Teor. Veroyatnost. i Primenen 35(1990), 449–458 [in Russian]; English translation, Theory Probab. Appl. 35(1990), 443–453. https://doi.org/10.1137/1135064 Google Scholar
Krylov, N. V., On diffusion processes with drift in ${L}_d$ . Preprint, 2020. arXiv:2001.04950Google Scholar
Krylov, N. V., On time inhomogeneous stochastic Itô equations with drift in ${L}_{d+1}$ . Preprint, 2020. arXiv:2005.08831Google Scholar
Krylov, N. V. and Röckner, M., Strong solutions of stochastic equations with singular time dependent drift . Probab. Theory Rel. Fields 131(2005), 154–196. https://doi.org/10.1007/s00440-004-0361-z CrossRefGoogle Scholar
Portenko, N. I., Generalized diffusion processes . American Mathematical Society, Providence, RI, 1990. https://doi.org/10.1090/mmono/083 CrossRefGoogle Scholar
Xia, P., Xie, L., Zhang, X., and Zhao, G., Lp (Lq)-theory of stochastic differential equations . Stoc. Proc. Appl. 130(2020), 5188–5211. https://doi.org/10.1016/j.spa.2020.03-004 CrossRefGoogle Scholar
Zhang, X., Stochastic homeomorphism flows of SDEs with singular drifts and Sobolev diffusion coefficients . Electr. J. Prob. 16(2011), 1096–1116. https://doi.org/10.1214/E/jp.v16-887 Google Scholar