The dual Yamada–Watanabe theorem for mild solutions to stochastic partial differential equations
Author:
S. Tappe
Journal:
Theor. Probability and Math. Statist. 105 (2021), 51-68
MSC (2020):
Primary 60H15; Secondary 60H10, 60H05
DOI:
https://doi.org/10.1090/tpms/1155
Published electronically:
December 7, 2021
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Additional Information
Abstract: We provide the dual result of the Yamada–Watanabe theorem for mild solutions to semilinear stochastic partial differential equations with path-dependent coefficients. An essential tool is the so-called “method of the moving frame”, which allows us to reduce the proof to infinite dimensional stochastic differential equations.
References
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References
- M. Barczy, Z. Li, and G. Pap, Yamada–Watanabe results for stochastic differential equations with jumps, Int. J. Stoch. Anal. 2015 (2015), Article ID 460472, 23 pages. MR 3298537
- A. S. Cherny, On the uniqueness in law and the pathwise uniqueness for stochastic differential equations, Theory Probab. Appl. 46 (2002), no. 3, 406–419. MR 1978664
- P. L. Chow and J. L. Jiang, Almost sure convergence of stochastic integrals in Hilbert spaces, Stoch. Anal. Appl. 10 (1992), no. 5, 533–543. MR 1185047
- D. Criens, A dual Yamada–Watanabe theorem for Lévy driven stochastic differential equations, Electron. Commun. Probab. 26 (2021), no. 18, 1–10. MR 4240146
- D. Criens and M. Ritter, On a theorem by A. S. Cherny for semilinear stochastic partial differential equations, J Theor Probab (2021), https://doi.org/10.1007/s10959-021-01107-3
- G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Second Edition, Cambridge University Press, Cambridge, 2014. MR 3236753
- E. B. Davies, Quantum theory of open systems, Academic Press, London, 1976. MR 0489429
- D. Filipović, S. Tappe, and J. Teichmann, Jump-diffusions in Hilbert spaces: Existence, stability and numerics, Stochastics 82 (2010), no. 5, 475–520. MR 2739608
- L. Gawarecki and V. Mandrekar, Stochastic differential equations in infinite dimensions with applications to SPDEs, Springer, Berlin, 2011. MR 2560625
- T. G. Kurtz, The Yamada–Watanabe–Engelbert theorem for general stochastic equations and inequalities, Electron. J. Probab. 12 (2007), no. 33, 951–965. MR 2336594
- T. G. Kurtz, Weak and strong solutions of general stochastic models, Electron. Commun. Probab. 19 (2014), no. 58, 1–16. MR 3254737
- W. Liu and M. Röckner, Stochastic partial differential equations: An introduction, Springer, Heidelberg, 2015. MR 3410409
- M. Ondreját, Uniqueness for stochastic evolution equations in Banach spaces, Dissertationes Mathematicae 426 (2004), 1–63. MR 2067962
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer, New York, 1983. MR 710486
- C. Prévôt and M. Röckner, A concise course on stochastic partial differential equations, Springer, Berlin, 2007. MR 2329435
- H. Qiao, A theorem dual to Yamada–Watanabe theorem for stochastic evolution equations, Stoch. Dyn. 10 (2010), no. 3, 367–374. MR 2671381
- M. Rehmeier, On Cherny’s results in infinite dimensions: a theorem dual to Yamada–Watanabe, Stoch. PDE: Anal. Comp. 9 (2021), no. 1, 33–70. MR 4218787
- M. Riedle, Cylindrical Wiener processes, Sém. Probab. XLIII, Lect. Notes in Math. vol. 2006, Springer, Berlin, 2011, pp. 191–214. MR 2790373
- M. Röckner, B. Schmuland, and X. Zhang, Yamada–Watanabe theorem for stochastic evolution equations in infinite dimensions, Condens. Matter Phys. 11 (2008), no. 2, 247–259.
- B. Sz.-Nagy, C. Foias, H. Bercovici, and L. Kérchy, Harmonic analysis of operators on Hilbert space, Revised and Enlarged Edition, Springer, New York, 2010. MR 2760647
- S. Tappe, Some refinements of existence results for SPDEs driven by Wiener processes and Poisson random measures, Int. J. Stoch. Anal. 2012 (2012), Article ID 236327, 24 pages. MR 3008827
- S. Tappe, The Yamada–Watanabe theorem for mild solutions to stochastic partial differential equations, Electron. Commun. Probab. 18 (2013), no. 24, 1–13. MR 3044472
- S. Tappe, Mild solutions to semilinear stochastic partial differential equations with locally monotone coefficients, Theor. Probability and Math. Statist. 104 (2021), 113–122.
- T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ. 11 (1971), no. 1, 155–167. MR 278420
- H. Zhao, Yamada–Watanabe theorem for stochastic evolution equation driven by Poisson random measure, ISRN Probab. Statist. 2014 (2014), Article ID 982190, 7 pages.
- H. Zhao, C. Hu, and S. Xu, Equivalence of uniqueness in law and joint uniqueness in law for SDEs driven by Poisson processes, Appl. Math. 7 (2016), 784–792.
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Additional Information
S. Tappe
Affiliation:
Department of Mathematical Stochastics, Albert Ludwig University of Freiburg, Ernst-Zermelo-Straße 1, D-79104 Freiburg, Germany
Email:
stefan.tappe@math.uni-freiburg.de
Keywords:
Stochastic partial differential equation,
martingale solution,
mild solution,
dual Yamada–Watanabe theorem,
uniqueness in law,
joint uniqueness in law,
pathwise uniqueness
Received by editor(s):
May 18, 2021
Published electronically:
December 7, 2021
Additional Notes:
The author gratefully acknowledges financial support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – project number 444121509
Article copyright:
© Copyright 2021
Taras Shevchenko National University of Kyiv