Comparison theorem for solutions of parabolic stochastic equations with an absorber

Mel’nik, S.

doi:10.1090/tpms/957

Comparison theorem for solutions of parabolic stochastic equations with an absorber

Author: S. A. Mel’nik
Translated by: S. Kvasko
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 Free Access

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

E. Pardoux, Equations aux derivees partielles stochastiques non lineaires monotones, These doct. math., Univ. Paris. Sud., 1975.

N. V. Krylov and B. L. Rozovskiĭ, Stochastic evolution equations, Current problems in mathematics, Vol. 14 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1979, pp. 71–147, 256 (Russian). MR 570795

Claudia Prévôt and Michael Röckner, A concise course on stochastic partial differential equations, Lecture Notes in Mathematics, vol. 1905, Springer, Berlin, 2007. MR 2329435

V. M. Radchenko, Properties of integrals with respect to a general stochastic measure in a stochastic heat equation, Teor. Ĭmovīr. Mat. Stat. 82 (2010), 104–114 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 82 (2011), 103–114. MR 2790486, DOI https://doi.org/10.1090/S0094-9000-2011-00830-7

V. M. Radchenko, A cable equation with a general stochastic measure, Teor. Ĭmovīr. Mat. Stat. 84 (2011), 123–130 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 84 (2012), 131–138. MR 2857423, DOI https://doi.org/10.1090/S0094-9000-2012-00856-9

V. Radchenko, Riemann integral of a random function and the parabolic equation with a general stochastic measure, Theory Probab. Math. Statist. 87 (2013), 185–198. Translation of Teor. Ǐmovīr. Mat. Stat. No. 87 (2012), 163–175. MR 3241455, DOI https://doi.org/10.1090/S0094-9000-2014-00912-6

Alexander A. Samarskii, Victor A. Galaktionov, Sergei P. Kurdyumov, and Alexander P. Mikhailov, Blow-up in quasilinear parabolic equations, De Gruyter Expositions in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 1995. Translated from the 1987 Russian original by Michael Grinfeld and revised by the authors. MR 1330922

Robert Dalang, Davar Khoshnevisan, Carl Mueller, David Nualart, and Yimin Xiao, A minicourse on stochastic partial differential equations, Lecture Notes in Mathematics, vol. 1962, Springer-Verlag, Berlin, 2009. Held at the University of Utah, Salt Lake City, UT, May 8–19, 2006; Edited by Khoshnevisan and Firas Rassoul-Agha. MR 1500166

Laurent Denis, Anis Matoussi, and Lucretiu Stoica, Maximum principle and comparison theorem for quasi-linear stochastic PDE’s, Electron. J. Probab. 14 (2009), no. 19, 500–530. MR 2480551, DOI https://doi.org/10.1214/EJP.v14-629

Sigurd Assing, Comparison of systems of stochastic partial differential equations, Stochastic Process. Appl. 82 (1999), no. 2, 259–282. MR 1700009, DOI https://doi.org/10.1016/S0304-4149%2899%2900031-9

S. A. Mel′nik, A comparison theorem for quasilinear parabolic SPDEs with weak sources, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki 1 (2011), 17–22 (Russian, with English and Ukrainian summaries). MR 3113877

Mathematical Encyclopedia, vol. 1, “Soviet Encyclopedia”, Moscow, 1977. (Russian)

Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981. MR 637061