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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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.

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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