# American Institue of Mathematical Sciences

2017, 37(8): 4507-4542. doi: 10.3934/dcds.2017193

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

 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 2016 Revised  March 2017 Published  April 2017

Fund Project: The author was supported by JSPS KAKENHI Grant Number 20140047

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.
Citation: Tôn Việt Tạ. Non-autonomous stochastic evolution equations in Banach spaces of martingale type 2: Strict solutions and maximal regularity. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4507-4542. doi: 10.3934/dcds.2017193
##### References:
 [1] P. Acquistapace, 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. [3] Z. Brzeźniak, On stochastic convolution in Banach spaces and applications, Stochastics Stochastics Rep., 61 (1997), 245-295. [4] G. Da Prato, F. Flandoli, Pathwise uniqueness for a class of SDE in Hilbert spaces and applications, J. Funct. Anal., 259 (2010), 243-267. [5] G. Da Prato, F. Flandoli, E. Priola, M. Röckner, Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift, Ann. Probab., 41 (2013), 3306-3344. [6] G. Da Prato, F. Flandoli, E. Priola, M. Röckner, Strong uniqueness for stochastic evolution equations with unbounded measurable drift term, J. Theoret. Probab., 28 (2015), 1571-1600. [7] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge, 1992. [8] J. M. A. M. van Neerven, M. C. Veraar, L. Weis, Stochastic integration in UMD Banach spaces, Ann. Probab., 35 (2007), 1438-1478. [9] J. M. A. M. van Neerven, M. C. Veraar, L. Weis, Stochastic evolution equations in UMD Banach spaces, J. Funct. Anal., 255 (2008), 940-993. [10] J. M. A. M. van Neerven, M. C. Veraar, L. Weis, Maximal Lp -regularity for stochastic evolution equations, SIAM J. Math. Anal., 44 (2012), 1372-1414. [11] J. M. A. M. van Neerven, M. C. Veraar, L. Weis, Stochastic maximal Lp -regularity, Ann. Probab., 40 (2012), 788-812. [12] J. M. A. M. van Neerven, M. C. Veraar, L. Weis, Maximal γ -regularity, J. Evol. Equ., 15 (2015), 361-402. [13] G. Pisier, Probabilistic methods in the geometry of Banach spaces, Probability and Analysis, 1206 (1986), 167-241. [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. [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. [25] A. Yagi, Fractional powers of operators and evolution equations of parabolic type, Proc. Japan Acad. Ser. A Math. Sci., 64 (1988), 227-230. [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.

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##### References:
 [1] P. Acquistapace, 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. [3] Z. Brzeźniak, On stochastic convolution in Banach spaces and applications, Stochastics Stochastics Rep., 61 (1997), 245-295. [4] G. Da Prato, F. Flandoli, Pathwise uniqueness for a class of SDE in Hilbert spaces and applications, J. Funct. Anal., 259 (2010), 243-267. [5] G. Da Prato, F. Flandoli, E. Priola, M. Röckner, Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift, Ann. Probab., 41 (2013), 3306-3344. [6] G. Da Prato, F. Flandoli, E. Priola, M. Röckner, Strong uniqueness for stochastic evolution equations with unbounded measurable drift term, J. Theoret. Probab., 28 (2015), 1571-1600. [7] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge, 1992. [8] J. M. A. M. van Neerven, M. C. Veraar, L. Weis, Stochastic integration in UMD Banach spaces, Ann. Probab., 35 (2007), 1438-1478. [9] J. M. A. M. van Neerven, M. C. Veraar, L. Weis, Stochastic evolution equations in UMD Banach spaces, J. Funct. Anal., 255 (2008), 940-993. [10] J. M. A. M. van Neerven, M. C. Veraar, L. Weis, Maximal Lp -regularity for stochastic evolution equations, SIAM J. Math. Anal., 44 (2012), 1372-1414. [11] J. M. A. M. van Neerven, M. C. Veraar, L. Weis, Stochastic maximal Lp -regularity, Ann. Probab., 40 (2012), 788-812. [12] J. M. A. M. van Neerven, M. C. Veraar, L. Weis, Maximal γ -regularity, J. Evol. Equ., 15 (2015), 361-402. [13] G. Pisier, Probabilistic methods in the geometry of Banach spaces, Probability and Analysis, 1206 (1986), 167-241. [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. [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. [25] A. Yagi, Fractional powers of operators and evolution equations of parabolic type, Proc. Japan Acad. Ser. A Math. Sci., 64 (1988), 227-230. [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.
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