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May  2021, 20(5): 1867-1891. doi: 10.3934/cpaa.2021050

Strict solutions to stochastic semilinear evolution equations in M-type 2 Banach spaces

Tôn Việt Tạ

Center for Promotion of International Education and Research, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan

Received  August 2020 Revised  December 2020 Published  May 2021 Early access  March 2021

Fund Project: This work was supported by JSPS KAKENHI Grant Number 19K14555

This paper is devoted to studying stochastic semilinear evolution equations in Banach spaces of M-type 2. First, we prove existence, uniqueness and regularity of strict solutions. Then, we give an application to stochastic partial differential equations.

Keywords: Stochastic evolution equations, M-type 2 Banach spaces, strict and mild solutions, analytic semigroups.
Mathematics Subject Classification: Primary: 60H15, 35R60; Secondary: 47D06.
Citation: Tôn Việt Tạ. Strict solutions to stochastic semilinear evolution equations in M-type 2 Banach spaces. Communications on Pure & Applied Analysis, 2021, 20 (5) : 1867-1891. doi: 10.3934/cpaa.2021050
References:
[1]

Z. Brzeźniak, Stochastic partial differential equations in M-type 2 Banach spaces, Potential Anal., 4 (1995), 1-45.  doi: 10.1007/BF01048965.  Google Scholar

[2]

Z. Brzeźniak, On stochastic convolution in Banach spaces and applications, Stochastics Stochastics Rep., 61 (1997), 245-295.  doi: 10.1080/17442509708834122.  Google Scholar

[3]

R. F. Curtain and P. L. Falb, Stochastic differential equations in Hilbert space, J. Differential Equations, 10 (1971), 412-430.  doi: 10.1016/0022-0396(71)90004-0.  Google Scholar

[4] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar
[5]

E. Dettweiler, On the martingale problem for Banach space valued stochastic differential equations, J. Theoret. Probab., 2 (1989), 159-191.  doi: 10.1007/BF01053408.  Google Scholar

[6]

M. Hairer, An introduction to stochastic PDEs, preprint, (2009) arXiv: 0907.4178. Google Scholar

[7]

B. H. Haak and J. M. A. M. van Neerven, Uniformly $\gamma$-radonifying families of operators and the stochastic Weiss conjecture, Oper. Matrices, 6 (2012), 767-792.  doi: 10.7153/oam-06-50.  Google Scholar

[8]

O. Martin, Uniqueness for stochastic evolution equations in Banach spaces, Dissertationes Math. (Rozprawy Mat.), 426 (2004), 63 pp. Google Scholar

[9]

G. Pisier, Probabilistic methods in the geometry of Banach spaces, in Probability and Analysis, Lecture Notes in Math., vol. 1206, Springer, Berlin, 1986,167–241. doi: 10.1007/BFb0076302.  Google Scholar

[10]

T. V. Tạ, Existence results for linear evolution equations of parabolic type, Commun. Pure Appl. Anal., 17 (2018), 751-785.  doi: 10.3934/cpaa.2018039.  Google Scholar

[11]

T. V. Tạ, Non-autonomous stochastic evolution equations in Banach spaces of martingale type 2: strict solutions and maximal regularity, Discrete Contin. Dyn. Syst., 37 (2017), 4507-4542.  doi: 10.3934/dcds.2017193.  Google Scholar

[12]

T. V. Tạ, Note on abstract stochastic semilinear evolution equations, J. Korean Math. Soc., 54 (2017), 909-943.  doi: 10.4134/JKMS.j160311.  Google Scholar

[13]

T. V. Tạ, Y. Yamamoto and A. Yagi, Strict solutions to stochastic linear evolution equations in M-type 2 Banach spaces, Funkcial. Ekvac., 61 (2018), 191-217.  doi: 10.1619/fesi.61.191.  Google Scholar

[14]

J. M. A. M. van Neerven, M. C. Veraar and L. Weis, Stochastic evolution equations in UMD Banach spaces, J. Funct. Anal., 255 (2008), 940-993.  doi: 10.1016/j.jfa.2008.03.015.  Google Scholar

[15]

J. M. A. M. van Neerven, M. C. Veraar and L. Weis, Stochastic maximal $L^p$-regularity, Ann. Probab., 40 (2012), 788-812.  doi: 10.1214/10-AOP626.  Google Scholar

[16]

A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.  Google Scholar

show all references

References:
[1]

Z. Brzeźniak, Stochastic partial differential equations in M-type 2 Banach spaces, Potential Anal., 4 (1995), 1-45.  doi: 10.1007/BF01048965.  Google Scholar

[2]

Z. Brzeźniak, On stochastic convolution in Banach spaces and applications, Stochastics Stochastics Rep., 61 (1997), 245-295.  doi: 10.1080/17442509708834122.  Google Scholar

[3]

R. F. Curtain and P. L. Falb, Stochastic differential equations in Hilbert space, J. Differential Equations, 10 (1971), 412-430.  doi: 10.1016/0022-0396(71)90004-0.  Google Scholar

[4] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar
[5]

E. Dettweiler, On the martingale problem for Banach space valued stochastic differential equations, J. Theoret. Probab., 2 (1989), 159-191.  doi: 10.1007/BF01053408.  Google Scholar

[6]

M. Hairer, An introduction to stochastic PDEs, preprint, (2009) arXiv: 0907.4178. Google Scholar

[7]

B. H. Haak and J. M. A. M. van Neerven, Uniformly $\gamma$-radonifying families of operators and the stochastic Weiss conjecture, Oper. Matrices, 6 (2012), 767-792.  doi: 10.7153/oam-06-50.  Google Scholar

[8]

O. Martin, Uniqueness for stochastic evolution equations in Banach spaces, Dissertationes Math. (Rozprawy Mat.), 426 (2004), 63 pp. Google Scholar

[9]

G. Pisier, Probabilistic methods in the geometry of Banach spaces, in Probability and Analysis, Lecture Notes in Math., vol. 1206, Springer, Berlin, 1986,167–241. doi: 10.1007/BFb0076302.  Google Scholar

[10]

T. V. Tạ, Existence results for linear evolution equations of parabolic type, Commun. Pure Appl. Anal., 17 (2018), 751-785.  doi: 10.3934/cpaa.2018039.  Google Scholar

[11]

T. V. Tạ, Non-autonomous stochastic evolution equations in Banach spaces of martingale type 2: strict solutions and maximal regularity, Discrete Contin. Dyn. Syst., 37 (2017), 4507-4542.  doi: 10.3934/dcds.2017193.  Google Scholar

[12]

T. V. Tạ, Note on abstract stochastic semilinear evolution equations, J. Korean Math. Soc., 54 (2017), 909-943.  doi: 10.4134/JKMS.j160311.  Google Scholar

[13]

T. V. Tạ, Y. Yamamoto and A. Yagi, Strict solutions to stochastic linear evolution equations in M-type 2 Banach spaces, Funkcial. Ekvac., 61 (2018), 191-217.  doi: 10.1619/fesi.61.191.  Google Scholar

[14]

J. M. A. M. van Neerven, M. C. Veraar and L. Weis, Stochastic evolution equations in UMD Banach spaces, J. Funct. Anal., 255 (2008), 940-993.  doi: 10.1016/j.jfa.2008.03.015.  Google Scholar

[15]

J. M. A. M. van Neerven, M. C. Veraar and L. Weis, Stochastic maximal $L^p$-regularity, Ann. Probab., 40 (2012), 788-812.  doi: 10.1214/10-AOP626.  Google Scholar

[16]

A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.  Google Scholar

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