An addendum to “Mild solutions to semilinear stochastic partial differential equations with locally monotone coefficients”
Author:
Stefan Tappe
Journal:
Theor. Probability and Math. Statist. 107 (2022), 173-184
MSC (2020):
Primary 60H15; Secondary 60H10
DOI:
https://doi.org/10.1090/tpms/1181
Published electronically:
November 8, 2022
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: In this addendum we provide an existence and uniqueness result for mild solutions to semilinear stochastic partial differential equations driven by Wiener processes and Poisson random measures in the framework of the semigroup approach with locally monotone coefficients, where the semigroup is allowed to be pseudo-contractive. This improves an earlier paper of the author, where the equation was only driven by Wiener processes, and where the semigroup was only allowed to be a semigroup of contractions.
References
- Zdzisław Brzeźniak, Wei Liu, and Jiahui Zhu, Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise, Nonlinear Anal. Real World Appl. 17 (2014), 283–310. MR 3158475, DOI 10.1016/j.nonrwa.2013.12.005
- Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimensions, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 152, Cambridge University Press, Cambridge, 2014. MR 3236753, DOI 10.1017/CBO9781107295513
- E. B. Davies, Quantum theory of open systems, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0489429
- Damir Filipović, Stefan Tappe, and Josef Teichmann, Jump-diffusions in Hilbert spaces: existence, stability and numerics, Stochastics 82 (2010), no. 5, 475–520. MR 2739608, DOI 10.1080/17442501003624407
- Leszek Gawarecki and Vidyadhar Mandrekar, Stochastic differential equations in infinite dimensions with applications to stochastic partial differential equations, Probability and its Applications (New York), Springer, Heidelberg, 2011. MR 2560625, DOI 10.1007/978-3-642-16194-0
- Wei Liu and Michael Röckner, Stochastic partial differential equations: an introduction, Universitext, Springer, Cham, 2015. MR 3410409, DOI 10.1007/978-3-319-22354-4
- Claudia Prévôt and Michael Röckner, A concise course on stochastic partial differential equations, Lecture Notes in Mathematics, vol. 1905, Springer, Berlin, 2007. MR 2329435
- E. Salavati and B. Z. Zangeneh, Stochastic evolution equations with multiplicative Poisson noise and monotone nonlinearity, Bull. Iranian Math. Soc. 43 (2017), no. 5, 1287–1299. MR 3730642
- Béla Sz.-Nagy, Ciprian Foias, Hari Bercovici, and László Kérchy, Harmonic analysis of operators on Hilbert space, Revised and enlarged edition, Universitext, Springer, New York, 2010. MR 2760647, DOI 10.1007/978-1-4419-6094-8
- Stefan Tappe, The Yamada-Watanabe theorem for mild solutions to stochastic partial differential equations, Electron. Commun. Probab. 18 (2013), no. 24, 13. MR 3044472, DOI 10.1214/ECP.v18-2392
- S. Tappe, Mild solutions to semilinear stochastic partial differential equations with locally monotone coefficients, Theory Probab. Math. Statist. 104 (2021), 113–122. MR 4421357, DOI 10.1090/tpms/1159
References
- Z. Brzeźniak, W. Liu and J. Zhu, Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise, Nonlinear Anal. Real World Appl. 17 (2014), 283–310. MR 3158475
- G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Second Edition, Encyclopedia of Mathematics and its Applications, 152, 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, Probability and its Applications (New York), Springer, Heidelberg, 2011. MR 2560625
- W. Liu and M. Röckner, Stochastic partial differential equations: An introduction, Universitext, Springer, Cham, 2015. MR 3410409
- C. Prévôt and M. Röckner, A concise course on stochastic partial differential equations, Lecture Notes in Mathematics, 1905, Springer, Berlin, 2007. MR 2329435
- E. Salavati and B. Z. Zangeneh, Stochastic evolution equations with multiplicative Poisson noise and monotone nonlinearity, Bull. Iranian Math. Soc. 43 (2017), no. 5, 1287–1299. MR 3730642
- B. Sz.-Nagy, C. Foias, H. Bercovici, and L. Kérchy, Harmonic analysis of operators on Hilbert space, second edition, revised and enlarged edition, Universitext, Springer, New York, 2010. MR 2760647
- S. Tappe, The Yamada–Watanabe theorem for mild solutions to stochastic partial differential equations, Electron. Commun. Probab. 18 (2013), no. 24, 13 pp. MR 3044472
- S. Tappe, Mild solutions to semilinear stochastic partial differential equations with locally monotone coefficients, Theory Probab. Math. Statist. No. 104 (2021), 113–122. MR 4421357
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2020):
60H15,
60H10
Retrieve articles in all journals
with MSC (2020):
60H15,
60H10
Additional Information
Stefan 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,
variational approach,
semigroup approach,
pseudo-contractive semigroup,
mild solution,
monotonicity condition,
coercivity condition
Received by editor(s):
November 12, 2021
Accepted for publication:
March 10, 2022
Published electronically:
November 8, 2022
Additional Notes:
I gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) — project number 444121509.
Article copyright:
© Copyright 2022
Taras Shevchenko National University of Kyiv
