Non-autonomous stochastic evolution equations in Banach spaces of martingale type 2: Strict solutions and maximal regularity

Tôn Việt Tạ

doi:10.3934/dcds.2017193

Article Contents

2017, Volume 37, Issue 8: 4507-4542. Doi: 10.3934/dcds.2017193

This issue Previous Article Existence, nonexistence and uniqueness of positive stationary solutions of a singular Gierer-Meinhardt system Next Article Measurable sensitivity via Furstenberg families

Non-autonomous stochastic evolution equations in Banach spaces of martingale type 2: Strict solutions and maximal regularity

Tôn Việt Tạ

Center for Promotion of International Education and Research, Faculty of Agriculture, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan

Received: July 1, 2016

Revised: March 1, 2017

Published online: April 1, 2017

The author was supported by JSPS KAKENHI Grant Number 20140047.

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

This paper is devoted to studying a non-autonomous stochastic linear evolution equation in Banach spaces of martingale type 2. We construct unique strict solutions to the equation and show their maximal regularity. The abstract results are then applied to stochastic diffusion equations.

Keywords:

Mathematics Subject Classification: Primary: 60H15, 35R60; Secondary: 47D06.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	P. Acquistapace and B. Terreni, A unified approach to abstract linear nonautonomous parabolic equations, Rend. Sem. Mat. Univ. Padova, 78 (1987), 47-107.
[2]	Z. Brzeźniak, Stochastic partial differential equations in M-type 2 Banach spaces, Potential Anal., 4 (1995), 1-45. doi: 10.1007/BF01048965.
[3]	Z. Brzeźniak, On stochastic convolution in Banach spaces and applications, Stochastics Stochastics Rep., 61 (1997), 245-295. doi: 10.1080/17442509708834122.
[4]	G. Da Prato and F. Flandoli, Pathwise uniqueness for a class of SDE in Hilbert spaces and applications, J. Funct. Anal., 259 (2010), 243-267. doi: 10.1016/j.jfa.2009.11.019.
[5]	G. Da Prato, F. Flandoli, E. Priola and M. Röckner, Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift, Ann. Probab., 41 (2013), 3306-3344. doi: 10.1214/12-AOP763.
[6]	G. Da Prato, F. Flandoli, E. Priola and M. Röckner, Strong uniqueness for stochastic evolution equations with unbounded measurable drift term, J. Theoret. Probab., 28 (2015), 1571-1600. doi: 10.1007/s10959-014-0545-0.
[7]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge, 1992. doi: 10.1017/CBO9780511666223.
[8]	J. M. A. M. van Neerven, M. C. Veraar and L. Weis, Stochastic integration in UMD Banach spaces, Ann. Probab., 35 (2007), 1438-1478. doi: 10.1214/009117906000001006.
[9]	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.
[10]	J. M. A. M. van Neerven, M. C. Veraar and L. Weis, Maximal L^p -regularity for stochastic evolution equations, SIAM J. Math. Anal., 44 (2012), 1372-1414. doi: 10.1137/110832525.
[11]	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.
[12]	J. M. A. M. van Neerven, M. C. Veraar and L. Weis, Maximal γ -regularity, J. Evol. Equ., 15 (2015), 361-402. doi: 10.1007/s00028-014-0264-0.
[13]	G. Pisier, Probabilistic methods in the geometry of Banach spaces, Probability and Analysis, 1206 (1986), 167-241. doi: 10.1007/BFb0076302.
[14]	J. Seidler, Da Prato-Zabczyk's maximal inequality revisited, Ⅰ, Math. Bohem., 118 (1993), 67-106.
[15]	P. E. Sobolevskii, Parabolic equation in Banach space with an unbounded variable operator, a fractional power of which has a constant domain of definition, (Russian), Dokl. Akad. Nauk SSSR, 138 (1961), 59-62.
[16]	H. Tanabe, Remarks on the equations of evolution in a Banach space, Osaka J. Math., 12 (1960), 145-166.
[17]	H. Tanabe, Note on singular pertubation for abstract differential equations, Osaka J. Math., 1 (1964), 239-252.
[18]	H. Tanabe, Equation of Evolution, Iwanami (in Japanese), 1975; Pitman (English translation), 1979.
[19]	H. Tanabe, Functional Analytical Methods for Partial Differential Equations, Marcel-Dekker, 1997.
[20]	T. V. Tạ, Regularity of solutions of abstract linear evolution equations, Lith. Math. J., 56 (2016), 268-290. doi: 10.1007/s10986-016-9318-z.
[21]	T. V. Tạ, Note on abstract stochastic semilinear evolution equations, J. Korean Math. Soc., 54 (2017), 909-943.
[22]	T. V. Tạ, Stochastic parabolic evolution equations in M-type 2 Banach spaces, ArXiv e-prints [arXiv: 1508.07340v2].
[23]	T. V. Tạ, Y. Yamamoto and A. Yagi, Strict solutions to stochastic parabolic evolution equations in M-type 2 Banach spaces, Funkcial. Ekvac. 26 pages (to appear).
[24]	M. C. Veraar, Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations, J. Evol. Equ., 10 (2010), 85-127. doi: 10.1007/s00028-009-0041-7.
[25]	A. Yagi, Fractional powers of operators and evolution equations of parabolic type, Proc. Japan Acad. Ser. A Math. Sci., 64 (1988), 227-230. doi: 10.3792/pjaa.64.227.
[26]	A. Yagi, Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups, Funkcial. Ekvac., 32 (1989), 107-124.
[27]	A. Yagi, Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups, Ⅱ, Funkcial. Ekvac., 33 (1990), 139-150.
[28]	A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.