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.

show all references

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.

[1]

Fatihcan M. Atay, Lavinia Roncoroni. Lumpability of linear evolution equations in Banach spaces. Evolution Equations & Control Theory, 2017, 6 (1) : 15-34. doi: 10.3934/eect.2017002

[2]

Goro Akagi, Mitsuharu Ôtani. Evolution equations and subdifferentials in Banach spaces. Conference Publications, 2003, 2003 (Special) : 11-20. doi: 10.3934/proc.2003.2003.11

[3]

Zdzisław Brzeźniak, Paul André Razafimandimby. Irreducibility and strong Feller property for stochastic evolution equations in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1051-1077. doi: 10.3934/dcdsb.2016.21.1051

[4]

Wolfgang Arendt, Patrick J. Rabier. Linear evolution operators on spaces of periodic functions. Communications on Pure & Applied Analysis, 2009, 8 (1) : 5-36. doi: 10.3934/cpaa.2009.8.5

[5]

Mustapha Mokhtar-Kharroubi. On permanent regimes for non-autonomous linear evolution equations in Banach spaces with applications to transport theory. Kinetic & Related Models, 2010, 3 (3) : 473-499. doi: 10.3934/krm.2010.3.473

[6]

Laura Levaggi. Existence of sliding motions for nonlinear evolution equations in Banach spaces. Conference Publications, 2013, 2013 (special) : 477-487. doi: 10.3934/proc.2013.2013.477

[7]

Pengyu Chen, Yongxiang Li, Xuping Zhang. On the initial value problem of fractional stochastic evolution equations in Hilbert spaces. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1817-1840. doi: 10.3934/cpaa.2015.14.1817

[8]

Alfredo Lorenzi, Gianluca Mola. Identification of a real constant in linear evolution equations in Hilbert spaces. Inverse Problems & Imaging, 2011, 5 (3) : 695-714. doi: 10.3934/ipi.2011.5.695

[9]

Jin Liang, James H. Liu, Ti-Jun Xiao. Condensing operators and periodic solutions of infinite delay impulsive evolution equations. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 475-485. doi: 10.3934/dcdss.2017023

[10]

Alexandre Nolasco de Carvalho, Stefanie Sonner. Pullback exponential attractors for evolution processes in Banach spaces: Theoretical results. Communications on Pure & Applied Analysis, 2013, 12 (6) : 3047-3071. doi: 10.3934/cpaa.2013.12.3047

[11]

Alexandre Nolasco de Carvalho, Stefanie Sonner. Pullback exponential attractors for evolution processes in Banach spaces: Properties and applications. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1141-1165. doi: 10.3934/cpaa.2014.13.1141

[12]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2553-2581. doi: 10.3934/dcdsb.2015.20.2553

[13]

Alfredo Lorenzi, Ioan I. Vrabie. An identification problem for a linear evolution equation in a Banach space and applications. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 671-691. doi: 10.3934/dcdss.2011.4.671

[14]

Giuseppe Da Prato, Alessandra Lunardi. Maximal dissipativity of a class of elliptic degenerate operators in weighted $L^2$ spaces . Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 751-760. doi: 10.3934/dcdsb.2006.6.751

[15]

Masahiro Kubo. Quasi-subdifferential operators and evolution equations. Conference Publications, 2013, 2013 (special) : 447-456. doi: 10.3934/proc.2013.2013.447

[16]

Alain Haraux, Mitsuharu Ôtani. Analyticity and regularity for a class of second order evolution equations. Evolution Equations & Control Theory, 2013, 2 (1) : 101-117. doi: 10.3934/eect.2013.2.101

[17]

Tomás Caraballo, Leonid Shaikhet. Stability of delay evolution equations with stochastic perturbations. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2095-2113. doi: 10.3934/cpaa.2014.13.2095

[18]

Yalçin Sarol, Frederi Viens. Time regularity of the evolution solution to fractional stochastic heat equation . Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 895-910. doi: 10.3934/dcdsb.2006.6.895

[19]

Kim Dang Phung, Gengsheng Wang, Xu Zhang. On the existence of time optimal controls for linear evolution equations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 925-941. doi: 10.3934/dcdsb.2007.8.925

[20]

Pascal Auscher, Sylvie Monniaux, Pierre Portal. The maximal regularity operator on tent spaces. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2213-2219. doi: 10.3934/cpaa.2012.11.2213

2016 Impact Factor: 1.099

Metrics

  • PDF downloads (2)
  • HTML views (1)
  • Cited by (0)

Other articles
by authors

[Back to Top]