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Random attractors for stochastic parabolic equations with additive noise in weighted spaces
Existence results for linear evolution equations of parabolic type
Center for Promotion of International Education and Research, Faculty of Agriculture, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan |
We study a stochastic parabolic evolution equation of the form $ dX+AXdt = F(t)dt+G(t)dW(t)$ in Banach spaces. Existence of mild and strict solutions and their space-time regularity are shown in both the deterministic and stochastic cases. Abstract results are applied to a nonlinear stochastic heat equation.
References:
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J. M. Ball,
Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc., 63 (1977), 370-373.
|
[2] |
Z. Brzeźniak and E. Hausenblas,
Maximal regularity for stochastic convolutions driven by Levy processes, Probab. Theory Related Fields, 145 (2009), 615-637.
|
[3] |
G. Da Prato and P. Grisvard,
Maximal regularity for evolution equations by interpolation and extrapolation, J. Funct. Anal., 58 (1984), 107-124.
|
[4] |
G. Da Prato, S. Kwapien and J. Zabczyk,
Regularity of solutions of linear stochastic equations in Hilbert spaces, Stochastic, 23 (1987), 1-23.
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[5] |
G. Da Prato and J. Zabczyk,
Stochastic Equations in Infinite Dimensions, Second edition, Cambridge University Press, Cambridge, 2014. |
[6] |
G. Da Prato and A. Lunardi,
Maximal regularity for stochastic convolutions in $ L_p$ spaces, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 9 (1998), 25-29.
|
[7] |
A. Favini and A. Yagi,
Degenerate Differential Equations in Banach Spaces, Marcel-Dekker, 1999. |
[8] |
M. Hairer, An introduction to stochastic PDEs,
arXiv e-prints (2009), arXiv: 0907.4178. |
[9] |
N. V. Krylov,
An analytic approach to SPDEs, in stochastic partial differential equations: Six perspectives, Math. Surveys Monogr. Amer. Math. Soc., 64 (1999), 185-242.
|
[10] |
N. V. Krylov and S. V. Lototsky,
A Sobolev space theory of SPDEs with constant coefficients in a half space, SIAM J. Math. Anal., 31 (1999), 19-33.
|
[11] |
A. Lunardi,
Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, 1995. |
[12] |
R. Mikulevicius,
On the Cauchy problem for parabolic SPDEs in Hölder classes, Ann. Probab., 28 (2000), 74-103.
|
[13] |
C. Mueller and D. Nualart,
Regularity of the density for the stochastic heat equation, Electron. J. Probab., 13 (2008), 2248-2258.
|
[14] |
E. Pardoux and T. Zhang,
Absolute continuity of the law of the solution of a parabolic SPDE, J. Functional Anal., 13 (2008), 2248-2258.
|
[15] |
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983. |
[16] |
B. L. Rozovskii,
Stochastic Evolution Systems. Linear Theory and Applications to Nonlinear Filtering, Kluwer Academic Publishers Group, Dordrecht, 1990. |
[17] |
E. Sinestrari,
On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. Math. Anal. Appl., 107 (1985), 16-66.
|
[18] |
T. Shiga,
Two contrasting properties of solutions for one-dimensional stochastic partial differential equations, Can. J. Math., 46 (1994), 415-437.
|
[19] |
H. Tanabe,
Remarks on the equations of evolution in a Banach space, Osaka J. Math., 12 (1960), 145-166.
|
[20] |
H. Tanabe,
Note on singular pertubation for abstract differential equations, Osaka J. Math., 1 (1964), 239-252.
|
[21] |
H. Tanabe,
Equation of Evolution, Iwanami (in Japanese), 1975. English translation, Pitman, 1979. |
[22] |
H. Tanabe,
Functional Analytical Methods for Partial Differential Equations, Marcel-Dekker, 1997. |
[23] |
T. V. Tạ,
Regularity of solutions of abstract linear evolution equations, Lith. Math. J., 56 (2016), 268-290.
|
[24] |
T. V. Tạ,
Non-autonomous stochastic evolution equations in Banach spaces of martingale type 2: strict solutions and maximal regularity, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 4507-4542.
|
[25] |
T. V. Tạ, Stochastic parabolic evolution equations in M-type 2 Banach spaces,
arXiv e-prints, (2015), arXiv: 1508.07340. |
[26] |
T. V. Tạ, Y. Yamamoto and A. Yagi, Strict solutions to stochastic linear evolution equations in M-type 2 Banach spaces, Funkc. Ekvacioj. (in press) (arXiv: 1508.07431). |
[27] |
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.
|
[28] |
J. M. A. M. van Neerven, M. C. Veraar and L. Weis,
Stochastic maximal $ L_p$-regularity, Ann. Probab., 40 (2012), 788-812.
|
[29] |
J. M. A. M. van Neerven, M. C. Veraar and L. Weis,
Maximal $γ$-regularity, J. Evol. Equ., 15 (2015), 361-402.
|
[30] |
M. C. Veraar,
Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations, J. Evol. Equ., 10 (2010), 85-127.
|
[31] |
J. B. Walsh,
An Introduction to Stochastic Partial Differential Equations,
École d'été de probabilités de Saint-Flour, XIV-1984,265-439, Lecture Notes in Mathematics 1180, Springer, Berlin, 1986. |
[32] |
A. Yagi,
Fractional powers of operators and evolution equations of parabolic type, Proc. Japan Acad. Ser. A Math. Sci., 64 (1988), 227-230.
|
[33] |
A. Yagi,
Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups, Funkcial. Ekvac., 32 (1989), 107-124.
|
[34] |
A. Yagi,
Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups, Ⅱ, Funkcial. Ekvac., 33 (1990), 139-150.
|
[35] |
A. Yagi,
Abstract Parabolic Evolution Equations and their Applications, Springer, Berlin, 2010. |
show all references
References:
[1] |
J. M. Ball,
Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc., 63 (1977), 370-373.
|
[2] |
Z. Brzeźniak and E. Hausenblas,
Maximal regularity for stochastic convolutions driven by Levy processes, Probab. Theory Related Fields, 145 (2009), 615-637.
|
[3] |
G. Da Prato and P. Grisvard,
Maximal regularity for evolution equations by interpolation and extrapolation, J. Funct. Anal., 58 (1984), 107-124.
|
[4] |
G. Da Prato, S. Kwapien and J. Zabczyk,
Regularity of solutions of linear stochastic equations in Hilbert spaces, Stochastic, 23 (1987), 1-23.
|
[5] |
G. Da Prato and J. Zabczyk,
Stochastic Equations in Infinite Dimensions, Second edition, Cambridge University Press, Cambridge, 2014. |
[6] |
G. Da Prato and A. Lunardi,
Maximal regularity for stochastic convolutions in $ L_p$ spaces, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 9 (1998), 25-29.
|
[7] |
A. Favini and A. Yagi,
Degenerate Differential Equations in Banach Spaces, Marcel-Dekker, 1999. |
[8] |
M. Hairer, An introduction to stochastic PDEs,
arXiv e-prints (2009), arXiv: 0907.4178. |
[9] |
N. V. Krylov,
An analytic approach to SPDEs, in stochastic partial differential equations: Six perspectives, Math. Surveys Monogr. Amer. Math. Soc., 64 (1999), 185-242.
|
[10] |
N. V. Krylov and S. V. Lototsky,
A Sobolev space theory of SPDEs with constant coefficients in a half space, SIAM J. Math. Anal., 31 (1999), 19-33.
|
[11] |
A. Lunardi,
Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, 1995. |
[12] |
R. Mikulevicius,
On the Cauchy problem for parabolic SPDEs in Hölder classes, Ann. Probab., 28 (2000), 74-103.
|
[13] |
C. Mueller and D. Nualart,
Regularity of the density for the stochastic heat equation, Electron. J. Probab., 13 (2008), 2248-2258.
|
[14] |
E. Pardoux and T. Zhang,
Absolute continuity of the law of the solution of a parabolic SPDE, J. Functional Anal., 13 (2008), 2248-2258.
|
[15] |
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983. |
[16] |
B. L. Rozovskii,
Stochastic Evolution Systems. Linear Theory and Applications to Nonlinear Filtering, Kluwer Academic Publishers Group, Dordrecht, 1990. |
[17] |
E. Sinestrari,
On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. Math. Anal. Appl., 107 (1985), 16-66.
|
[18] |
T. Shiga,
Two contrasting properties of solutions for one-dimensional stochastic partial differential equations, Can. J. Math., 46 (1994), 415-437.
|
[19] |
H. Tanabe,
Remarks on the equations of evolution in a Banach space, Osaka J. Math., 12 (1960), 145-166.
|
[20] |
H. Tanabe,
Note on singular pertubation for abstract differential equations, Osaka J. Math., 1 (1964), 239-252.
|
[21] |
H. Tanabe,
Equation of Evolution, Iwanami (in Japanese), 1975. English translation, Pitman, 1979. |
[22] |
H. Tanabe,
Functional Analytical Methods for Partial Differential Equations, Marcel-Dekker, 1997. |
[23] |
T. V. Tạ,
Regularity of solutions of abstract linear evolution equations, Lith. Math. J., 56 (2016), 268-290.
|
[24] |
T. V. Tạ,
Non-autonomous stochastic evolution equations in Banach spaces of martingale type 2: strict solutions and maximal regularity, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 4507-4542.
|
[25] |
T. V. Tạ, Stochastic parabolic evolution equations in M-type 2 Banach spaces,
arXiv e-prints, (2015), arXiv: 1508.07340. |
[26] |
T. V. Tạ, Y. Yamamoto and A. Yagi, Strict solutions to stochastic linear evolution equations in M-type 2 Banach spaces, Funkc. Ekvacioj. (in press) (arXiv: 1508.07431). |
[27] |
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.
|
[28] |
J. M. A. M. van Neerven, M. C. Veraar and L. Weis,
Stochastic maximal $ L_p$-regularity, Ann. Probab., 40 (2012), 788-812.
|
[29] |
J. M. A. M. van Neerven, M. C. Veraar and L. Weis,
Maximal $γ$-regularity, J. Evol. Equ., 15 (2015), 361-402.
|
[30] |
M. C. Veraar,
Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations, J. Evol. Equ., 10 (2010), 85-127.
|
[31] |
J. B. Walsh,
An Introduction to Stochastic Partial Differential Equations,
École d'été de probabilités de Saint-Flour, XIV-1984,265-439, Lecture Notes in Mathematics 1180, Springer, Berlin, 1986. |
[32] |
A. Yagi,
Fractional powers of operators and evolution equations of parabolic type, Proc. Japan Acad. Ser. A Math. Sci., 64 (1988), 227-230.
|
[33] |
A. Yagi,
Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups, Funkcial. Ekvac., 32 (1989), 107-124.
|
[34] |
A. Yagi,
Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups, Ⅱ, Funkcial. Ekvac., 33 (1990), 139-150.
|
[35] |
A. Yagi,
Abstract Parabolic Evolution Equations and their Applications, Springer, Berlin, 2010. |
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