The Yamada-Watanabe theorem for mild solutions to stochastic partial differential equations

Stefan Tappe (Leibniz Universität Hannover)

Abstract


We prove the Yamada-Watanabe theorem for semilinear stochastic partial differential equations with path-dependent coefficients. The so-called "method of the moving frame" allows us to reduce the proof to the Yamada-Watanabe theorem for stochastic differential equations in infinite dimensions.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-13

Publication Date: April 3, 2013

DOI: 10.1214/ECP.v18-2392

References

  • Da Prato, G.; Flandoli, F. Pathwise uniqueness for a class of SDE in Hilbert spaces and applications. J. Funct. Anal. 259 (2010), no. 1, 243--267. MR2610386
  • Da Prato, Giuseppe; Zabczyk, Jerzy. Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1992. xviii+454 pp. ISBN: 0-521-38529-6 MR1207136
  • Davies, E. B. Quantum theory of open systems. Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. x+171 pp. MR0489429
  • Engel, Klaus-Jochen; Nagel, Rainer. One-parameter semigroups for linear evolution equations. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000. xxii+586 pp. ISBN: 0-387-98463-1 MR1721989
  • Filipović, Damir; Tappe, Stefan; Teichmann, Josef. Jump-diffusions in Hilbert spaces: existence, stability and numerics. Stochastics 82 (2010), no. 5, 475--520. MR2739608
  • Gawarecki, Leszek; Mandrekar, Vidyadhar. Stochastic differential equations in infinite dimensions with applications to stochastic partial differential equations. Probability and its Applications (New York). Springer, Heidelberg, 2011. xvi+291 pp. ISBN: 978-3-642-16193-3 MR2560625
  • Hausenblas, Erika; Seidler, Jan. A note on maximal inequality for stochastic convolutions. Czechoslovak Math. J. 51(126) (2001), no. 4, 785--790. MR1864042
  • Hausenblas, Erika; Seidler, Jan. Stochastic convolutions driven by martingales: maximal inequalities and exponential integrability. Stoch. Anal. Appl. 26 (2008), no. 1, 98--119. MR2378512
  • Ikeda, Nobuyuki; Watanabe, Shinzo. Stochastic differential equations and diffusion processes. North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981. xiv+464 pp. ISBN: 0-444-86172-6 MR0637061
  • Jacod, Jean; Shiryaev, Albert N. Limit theorems for stochastic processes. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288. Springer-Verlag, Berlin, 2003. xx+661 pp. ISBN: 3-540-43932-3 MR1943877
  • Ondreját, Martin. Uniqueness for stochastic evolution equations in Banach spaces. Dissertationes Math. (Rozprawy Mat.) 426 (2004), 63 pp. MR2067962
  • Prévôt, Claudia; Röckner, Michael. A concise course on stochastic partial differential equations. Lecture Notes in Mathematics, 1905. Springer, Berlin, 2007. vi+144 pp. ISBN: 978-3-540-70780-6; 3-540-70780-8 MR2329435
  • Röckner, M., Schmuland, B. and Zhang, X.: Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions. textitCond. Matt. Phys. 11, (2008), 247--259.
  • Shiryaev, A. N. Probability. Translated from the first (1980) Russian edition by R. P. Boas. Second edition. Graduate Texts in Mathematics, 95. Springer-Verlag, New York, 1996. xvi+623 pp. ISBN: 0-387-94549-0 MR1368405
  • Sz.-Nagy, Béla; Foias, Ciprian; Bercovici, Hari; Kérchy, László. Harmonic analysis of operators on Hilbert space. Second edition. Revised and enlarged edition. Universitext. Springer, New York, 2010. xiv+474 pp. ISBN: 978-1-4419-6093-1 MR2760647
  • Tappe, S.: Some refinements of existence results for SPDEs driven by Wiener processes and Poisson random measures. textitInt. J. Stoch. Anal. 2012, Art. ID 236327, 24 pp.
  • Yamada, Toshio; Watanabe, Shinzo. On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11 1971 155--167. MR0278420
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.