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Strict solutions to stochastic semilinear evolution equations in M-type 2 Banach spaces
Center for Promotion of International Education and Research, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan |
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.
References:
[1] |
Z. Brzeźniak,
Stochastic partial differential equations in M-type 2 Banach spaces, Potential Anal., 4 (1995), 1-45.
doi: 10.1007/BF01048965. |
[2] |
Z. Brzeźniak,
On stochastic convolution in Banach spaces and applications, Stochastics Stochastics Rep., 61 (1997), 245-295.
doi: 10.1080/17442509708834122. |
[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. |
[4] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.
doi: 10.1017/CBO9780511666223.![]() ![]() ![]() |
[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. |
[6] |
M. Hairer, An introduction to stochastic PDEs, preprint, (2009) arXiv: 0907.4178. |
[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. |
[8] |
O. Martin, Uniqueness for stochastic evolution equations in Banach spaces, Dissertationes Math. (Rozprawy Mat.), 426 (2004), 63 pp. |
[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. |
[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. |
[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. |
[12] |
T. V. Tạ,
Note on abstract stochastic semilinear evolution equations, J. Korean Math. Soc., 54 (2017), 909-943.
doi: 10.4134/JKMS.j160311. |
[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. |
[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. |
[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. |
[16] |
A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-04631-5. |
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. |
[2] |
Z. Brzeźniak,
On stochastic convolution in Banach spaces and applications, Stochastics Stochastics Rep., 61 (1997), 245-295.
doi: 10.1080/17442509708834122. |
[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. |
[4] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.
doi: 10.1017/CBO9780511666223.![]() ![]() ![]() |
[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. |
[6] |
M. Hairer, An introduction to stochastic PDEs, preprint, (2009) arXiv: 0907.4178. |
[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. |
[8] |
O. Martin, Uniqueness for stochastic evolution equations in Banach spaces, Dissertationes Math. (Rozprawy Mat.), 426 (2004), 63 pp. |
[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. |
[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. |
[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. |
[12] |
T. V. Tạ,
Note on abstract stochastic semilinear evolution equations, J. Korean Math. Soc., 54 (2017), 909-943.
doi: 10.4134/JKMS.j160311. |
[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. |
[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. |
[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. |
[16] |
A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-04631-5. |
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