Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)

 
 

 

Comparison theorem for solutions of parabolic stochastic equations with an absorber


Author: S. A. Mel’nik
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 90 (2014).
Journal: Theor. Probability and Math. Statist. 90 (2015), 161-173
MSC (2010): Primary 60F10; Secondary 62F05
DOI: https://doi.org/10.1090/tpms/957
Published electronically: August 10, 2015
MathSciNet review: 3242028
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A comparison theorem is proved for solutions of the Cauchy problem for a quasi-linear parabolic stochastic equation. The drift and diffusion coefficients of this equation do not necessarily satisfy the Lipschitz condition. The drift coefficient is assumed to be an absorber.


References [Enhancements On Off] (What's this?)

  • 1. E. Pardoux, Equations aux derivees partielles stochastiques non lineaires monotones, These doct. math., Univ. Paris. Sud., 1975.
  • 2. N. V. Krylov and B. L. Rozovskiĭ, Stochastic evolution equations, Current Problems in Mathematics, vol. 14, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1979, pp. 71-147. (Russian) MR 570795 (81m:60116)
  • 3. C. Prevot and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Springer, Berlin, 2007. MR 2329435 (2009a:60069)
  • 4. V. M. Radchenko, Properties of integrals with respect to a general stochastic measure in a stochastic heat equation, Teor. Imovirnost. Matem. Statyst. 82 (2010), 104-115; English transl. in Theor. Probability and Math. Statist. 82 (2011), 103-114. MR 2790486 (2011m:60166)
  • 5. V. M. Radchenko, Cable equation with a general stochastic measure, Teor. Imovirnost. Matem. Statyst. 84 (2011), 126-133; English transl. in Theor. Probability and Math. Statist. 84 (2012), 131-138. MR 2857423 (2012j:60169)
  • 6. V. Radchenko, Riemann integral of a random function and the parabolic equation with a general stochastic measure, Teor. Imovirnost. Matem. Statyst. 87 (2012), 163-185; English transl. in Theor. Probability and Math. Statist. 87 (2013), 185-198. MR 3241455
  • 7. A. A. Samarskiĭ, S. P. Galaktionov, S. P. Kurdyumov, and A. P. Mikhaĭlov, Blow-up in Quasilinear Parabolic Equations, ``Nauka'', Moscow, 1987; English transl., Walter de Gruyter & Co., Berlin, 1995. MR 1330922 (96b:35003)
  • 8. R. C. Dalang, D. Khoshnevisan, C. Müller, D. Nualart, and Y. Xiao, A Minicourse on Stochastic Partial Differential Equations, Utah, Salt Lake City, 2006. MR 1500166 (2009k:60009)
  • 9. L. Denis, A. Matoussi, and L. Stoica, Maximum principle and comparison theorem for quasi-linear SPDE's, Electronic J. Probab. 14 (2009), no. 19, 500-530. MR 2480551 (2010d:60146)
  • 10. S. Assing, Comparison of systems of stochastic partial differential equations, Stoch. Process. Appl. 82 (1999), 259-282. MR 1700009 (2000g:60104)
  • 11. S. A. Mel'nik, Comparison theorem for solutions of quasi-linear partial stochastic differential equations with weak sources, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki 1 (2011), 13-17. (Russian) MR 3113877
  • 12. Mathematical Encyclopedia, vol. 1, ``Soviet Encyclopedia'', Moscow, 1977. (Russian)
  • 13. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Publishing Co., Amsterdam-New York/Kodansha, Ltd., Tokyo, 1981. MR 637061 (84b:60080)

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2010): 60F10, 62F05

Retrieve articles in all journals with MSC (2010): 60F10, 62F05


Additional Information

S. A. Mel’nik
Affiliation: Department of Probability Theory and Mathematical Statistics, Institute for Applied Mathematics and Mechanics, National Academy of Science of Ukraine, R. Luxembyrg Street, 74, Donetsk, 83114, Ukraine
Email: s.a.melnik@yandex.ua

DOI: https://doi.org/10.1090/tpms/957
Keywords: Stochastic partial differential equation, comparison theorem
Received by editor(s): November 23, 2012
Published electronically: August 10, 2015
Additional Notes: This research was partially supported by the State Foundation for Fundamental Researches of Ukraine and RFFI (Russian Federation), grant $Φ$40.1/023
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society