August  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 and B. Terreni, A unified approach to abstract linear nonautonomous parabolic equations, Rend. Sem. Mat. Univ. Padova, 78 (1987), 47-107. Google Scholar

[2]

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

[3]

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

[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. Google Scholar

[5]

G. Da PratoF. FlandoliE. 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. Google Scholar

[6]

G. Da PratoF. FlandoliE. 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. Google Scholar

[7]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge, 1992. doi: 10.1017/CBO9780511666223. Google Scholar

[8]

J. M. A. M. van NeervenM. C. Veraar and L. Weis, Stochastic integration in UMD Banach spaces, Ann. Probab., 35 (2007), 1438-1478. doi: 10.1214/009117906000001006. Google Scholar

[9]

J. M. A. M. van NeervenM. 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

[10]

J. M. A. M. van NeervenM. C. Veraar and L. Weis, Maximal Lp -regularity for stochastic evolution equations, SIAM J. Math. Anal., 44 (2012), 1372-1414. doi: 10.1137/110832525. Google Scholar

[11]

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

[12]

J. M. A. M. van NeervenM. C. Veraar and L. Weis, Maximal γ -regularity, J. Evol. Equ., 15 (2015), 361-402. doi: 10.1007/s00028-014-0264-0. Google Scholar

[13]

G. Pisier, Probabilistic methods in the geometry of Banach spaces, Probability and Analysis, 1206 (1986), 167-241. doi: 10.1007/BFb0076302. Google Scholar

[14]

J. Seidler, Da Prato-Zabczyk's maximal inequality revisited, Ⅰ, Math. Bohem., 118 (1993), 67-106. Google Scholar

[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. Google Scholar

[16]

H. Tanabe, Remarks on the equations of evolution in a Banach space, Osaka J. Math., 12 (1960), 145-166. Google Scholar

[17]

H. Tanabe, Note on singular pertubation for abstract differential equations, Osaka J. Math., 1 (1964), 239-252. Google Scholar

[18]

H. Tanabe, Equation of Evolution, Iwanami (in Japanese), 1975; Pitman (English translation), 1979. Google Scholar

[19]

H. Tanabe, Functional Analytical Methods for Partial Differential Equations, Marcel-Dekker, 1997. Google Scholar

[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. Google Scholar

[21]

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

[22]

T. V. Tạ, Stochastic parabolic evolution equations in M-type 2 Banach spaces, ArXiv e-prints [arXiv: 1508.07340v2].Google Scholar

[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).Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

A. Yagi, Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups, Funkcial. Ekvac., 32 (1989), 107-124. Google Scholar

[27]

A. Yagi, Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups, Ⅱ, Funkcial. Ekvac., 33 (1990), 139-150. Google Scholar

[28]

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]

P. Acquistapace and B. Terreni, A unified approach to abstract linear nonautonomous parabolic equations, Rend. Sem. Mat. Univ. Padova, 78 (1987), 47-107. Google Scholar

[2]

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

[3]

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

[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. Google Scholar

[5]

G. Da PratoF. FlandoliE. 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. Google Scholar

[6]

G. Da PratoF. FlandoliE. 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. Google Scholar

[7]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge, 1992. doi: 10.1017/CBO9780511666223. Google Scholar

[8]

J. M. A. M. van NeervenM. C. Veraar and L. Weis, Stochastic integration in UMD Banach spaces, Ann. Probab., 35 (2007), 1438-1478. doi: 10.1214/009117906000001006. Google Scholar

[9]

J. M. A. M. van NeervenM. 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

[10]

J. M. A. M. van NeervenM. C. Veraar and L. Weis, Maximal Lp -regularity for stochastic evolution equations, SIAM J. Math. Anal., 44 (2012), 1372-1414. doi: 10.1137/110832525. Google Scholar

[11]

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

[12]

J. M. A. M. van NeervenM. C. Veraar and L. Weis, Maximal γ -regularity, J. Evol. Equ., 15 (2015), 361-402. doi: 10.1007/s00028-014-0264-0. Google Scholar

[13]

G. Pisier, Probabilistic methods in the geometry of Banach spaces, Probability and Analysis, 1206 (1986), 167-241. doi: 10.1007/BFb0076302. Google Scholar

[14]

J. Seidler, Da Prato-Zabczyk's maximal inequality revisited, Ⅰ, Math. Bohem., 118 (1993), 67-106. Google Scholar

[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. Google Scholar

[16]

H. Tanabe, Remarks on the equations of evolution in a Banach space, Osaka J. Math., 12 (1960), 145-166. Google Scholar

[17]

H. Tanabe, Note on singular pertubation for abstract differential equations, Osaka J. Math., 1 (1964), 239-252. Google Scholar

[18]

H. Tanabe, Equation of Evolution, Iwanami (in Japanese), 1975; Pitman (English translation), 1979. Google Scholar

[19]

H. Tanabe, Functional Analytical Methods for Partial Differential Equations, Marcel-Dekker, 1997. Google Scholar

[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. Google Scholar

[21]

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

[22]

T. V. Tạ, Stochastic parabolic evolution equations in M-type 2 Banach spaces, ArXiv e-prints [arXiv: 1508.07340v2].Google Scholar

[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).Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

A. Yagi, Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups, Funkcial. Ekvac., 32 (1989), 107-124. Google Scholar

[27]

A. Yagi, Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups, Ⅱ, Funkcial. Ekvac., 33 (1990), 139-150. Google Scholar

[28]

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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