| Title:
|
A note on maximal estimates for stochastic convolutions (English) |
| Author:
|
Veraar, Mark |
| Author:
|
Weis, Lutz |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
61 |
| Issue:
|
3 |
| Year:
|
2011 |
| Pages:
|
743-758 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In stochastic partial differential equations it is important to have pathwise regularity properties of stochastic convolutions. In this note we present a new sufficient condition for the pathwise continuity of stochastic convolutions in Banach spaces. (English) |
| Keyword:
|
stochastic convolutions |
| Keyword:
|
maximal inequalities |
| Keyword:
|
path-continuity |
| Keyword:
|
stochastic partial differential equations |
| Keyword:
|
$H^\infty $-calculus |
| Keyword:
|
$\gamma $-radonifying operators |
| Keyword:
|
exponential tail estimates |
| MSC:
|
35B65 |
| MSC:
|
35R60 |
| MSC:
|
46B09 |
| MSC:
|
47D06 |
| MSC:
|
60H15 |
| idZBL:
|
Zbl 1249.60111 |
| idMR:
|
MR2853088 |
| DOI:
|
10.1007/s10587-011-0023-0 |
| . |
| Date available:
|
2011-09-22T14:44:23Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141635 |
| . |
| Reference:
|
[1] Albiac, F., Kalton, N. J.: Topics in Banach Space Theory. Graduate Texts in Mathematics, Vol. 233.Springer Berlin (2006). MR 2192298 |
| Reference:
|
[2] Amann, H.: Dual semigroups and second order linear elliptic boundary value problems.Isr. J. Math. 45 (1983), 225-254. Zbl 0535.35017, MR 0719122, 10.1007/BF02774019 |
| Reference:
|
[3] Brze{'z}niak, Z.: Stochastic partial differential equations in $M$-type 2 Banach spaces.Potential Anal. 4 (1995), 1-45. MR 1313905, 10.1007/BF01048965 |
| Reference:
|
[4] Brze{'z}niak, Z.: On stochastic convolution in Banach spaces and applications.Stochastics Stochastics Rep. 61 (1997), 245-295. MR 1488138, 10.1080/17442509708834122 |
| Reference:
|
[5] Brze{'z}niak, Z., Hausenblas, E.: Maximal regularity for stochastic convolutions driven by Lévy processes.Probab. Theory Relat. Fields 145 (2009), 615-637. MR 2529441, 10.1007/s00440-008-0181-7 |
| Reference:
|
[6] Brze'zniak, Z., Peszat, S.: Maximal inequalities and exponential estimates for stochastic convolutions in Banach spaces.Stochastic Processes, Physics and Geometry: New Interplays I (Leipzig, 1999). CMS Conf. Proc., Vol. 28 American Mathematical Society Providence (2000), 55-64. MR 1803378 |
| Reference:
|
[7] Calderón, A.-P.: Intermediate spaces and interpolation, the complex method.Stud. Math. 24 (1964), 113-190. MR 0167830, 10.4064/sm-24-2-113-190 |
| Reference:
|
[8] Cox, S. G., Veraar, M. C.: Vector-valued decoupling and the Burkholder-Davis-Gundy inequality.(2010), (to appear) Ill. J. Math. MR 3006692 |
| Reference:
|
[9] Prato, G. Da, Zabczyk, J.: Stochastic equations in infinite dimensions.Encyclopedia of Mathematics and its Applications, Vol. 44 Cambridge University Press Cambridge (1992). Zbl 0761.60052, MR 1207136 |
| Reference:
|
[10] Davis, B.: On the $L^{p}$ norms of stochastic integrals and other martingales.Duke Math. J. 43 (1976), 697-704. MR 0418219, 10.1215/S0012-7094-76-04354-4 |
| Reference:
|
[11] Denk, R., Dore, G., Hieber, M., Prüss, J., Venni, A.: New thoughts on old results of R. T. Seeley.Math. Ann. 328 (2004), 545-583. MR 2047641 |
| Reference:
|
[12] Deville, R., Godefroy, G., Zizler, V.: Smoothness and Renormings in Banach Spaces. Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 64.Longman Scientific & Technical Harlow (1993). MR 1211634 |
| Reference:
|
[13] Fröhlich, A. M., Weis, L.: $H^\infty$ calculus and dilations.Bull. Soc. Math. Fr. 134 (2006), 487-508. MR 2364942, 10.24033/bsmf.2520 |
| Reference:
|
[14] Haak, B. H., Kunstmann, P. C.: Admissibility of unbounded operators and well-posedness of linear systems in Banach spaces.Integral Equations Oper. Theory 55 (2006), 497-533. Zbl 1138.93361, MR 2250161, 10.1007/s00020-005-1416-y |
| Reference:
|
[15] Haase, M.: The Functional Calculus for Sectorial Operators. Operator Theory: Advances and Applications, Vol. 169.Birkhäuser Basel (2006). MR 2244037 |
| Reference:
|
[16] Hausenblas, E., Seidler, J.: A note on maximal inequality for stochastic convolutions.Czechoslovak Math. J. 51(126) (2001), 785-790. Zbl 1001.60065, MR 1864042, 10.1023/A:1013717013421 |
| Reference:
|
[17] Hausenblas, E., Seidler, J.: Stochastic convolutions driven by martingales: maximal inequalities and exponential integrability.Stochastic Anal. Appl. 26 (2008), 98-119. Zbl 1153.60035, MR 2378512, 10.1080/07362990701673047 |
| Reference:
|
[18] Hytönen, T., Neerven, J. van, Portal, P.: Conical square function estimates in {UMD} Banach spaces and applications to $H^\infty$-functional calculi.J. Anal. Math. 106 (2008), 317-351. MR 2448989, 10.1007/s11854-008-0051-3 |
| Reference:
|
[19] Kalton, N. J., Weis, L. W.: The $H^\infty$-calculus and square function estimates.Preprint (2004). |
| Reference:
|
[20] Kunstmann, P. C., Weis, L. W.: Maximal $L_p$-regularity for parabolic equations, Fourier multiplier theorems and $H_\infty$-functional calculus.Functional Analytic Methods for Evolution Equations. Lecture Notes Math., Vol. 1855 Springer Berlin (2004), 65-311. Zbl 1097.47041, MR 2108959, 10.1007/978-3-540-44653-8_2 |
| Reference:
|
[21] Langer, M., Maz'ya, V.: On $L^p$-contractivity of semigroups generated by linear partial differential operators.J. Funct. Anal. 164 (1999), 73-109. MR 1694522, 10.1006/jfan.1999.3393 |
| Reference:
|
[22] Lenglart, E.: Rélation de domination entre deux processus.Ann. Inst. Henri Poincaré, Nouv. Sér, Sect. B 13 (1977), 171-179 French. Zbl 0373.60054, MR 0471069 |
| Reference:
|
[23] Liskevich, V., Sobol, Z., Vogt, H.: On the $L_p$-theory of $C_0$-semigroups associated with second-order elliptic operators. II.J. Funct. Anal. 193 (2002), 55-76. Zbl 1020.47029, MR 1923628, 10.1006/jfan.2001.3909 |
| Reference:
|
[24] McIntosh, A.: Operators which have an $H_\infty$ functional calculus.Miniconference Operator Theory and Partial Differential Equations (North Ryde, 1986), Proc. Cent. Math. Anal. Austr. Nat. Univ., Vol. 14 Austr. Nat. Univ. Canberra (1986), 210-231. MR 0912940 |
| Reference:
|
[25] Neerven, J. M. A. M. van: $\gamma$-Radonifying operators---a survey.Spectral Theory and Harmonic Analysis (Canberra, 2009). Proc. Cent. Math. Anal. Austr. Nat. Univ., Vol. 44 Austral. Nat. Univ. Canberra (2010), 1-62. MR 2655391 |
| Reference:
|
[26] Neerven, J. M. A. M. van, Veraar, M. C., Weis, L.: Stochastic integration in {UMD} Banach spaces.Ann. Probab. 35 (2007), 1438-1478. MR 2330977, 10.1214/009117906000001006 |
| Reference:
|
[27] Neerven, J. M. A. M. van, Veraar, M. C., Weis, L.: Stochastic evolution equations in {UMD} Banach spaces.J. Funct. Anal. 255 (2008), 940-993. MR 2433958, 10.1016/j.jfa.2008.03.015 |
| Reference:
|
[28] Neerven, J. M. A. M. van, Weis, L.: Weak limits and integrals of Gaussian covariances in Banach spaces.Probab. Math. Stat. 25 (2005), 55-74. MR 2211356 |
| Reference:
|
[29] Pinelis, I.: Optimum bounds for the distributions of martingales in Banach spaces.Ann. Probab. 22 (1994), 1679-1706. Zbl 0836.60015, MR 1331198, 10.1214/aop/1176988477 |
| Reference:
|
[30] Pisier, G.: Martingales with values in uniformly convex spaces.Isr. J. Math. 20 (1975), 326-350. Zbl 0344.46030, MR 0394135, 10.1007/BF02760337 |
| Reference:
|
[31] Pisier, G.: Some results on Banach spaces without local unconditional structure.Compos. Math. 37 (1978), 3-19. Zbl 0381.46010, MR 0501916 |
| Reference:
|
[32] Rosiński, J., Suchanecki, Z.: On the space of vector-valued functions integrable with respect to the white noise.Colloq. Math. 43 (1980), 183-201. MR 0615985, 10.4064/cm-43-1-183-201 |
| Reference:
|
[33] Seidler, J.: Exponential estimates for stochastic convolutions in 2-smooth Banach spaces.Electron. J. Probab. 15 (2010), 1556-1573. MR 2735374, 10.1214/EJP.v15-808 |
| Reference:
|
[34] Suárez, J., Weis, L.: Interpolation of Banach spaces by the $\gamma$-method.Methods in Banach space theory. London Math. Soc. Lecture Note Series Vol. 337. Proc. Conf. on Banach Spaces, Cácres, Spain, Sept. 13-18, 2004 J. M. F. Castillo Cambridge Univ. Press Cambridge (2006), 293-306. MR 2326391 |
| Reference:
|
[35] Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators, 2nd ed.Barth Leipzig (1995). MR 1328645 |
| Reference:
|
[36] Veraar, M. C.: Continuous local martingales and stochastic integration in {UMD} Banach spaces.Stochastics 79 (2007), 601-618. Zbl 1129.60050, MR 2368370, 10.1080/17442500701594516 |
| Reference:
|
[37] Weis, L. W.: The $H^\infty$ holomorphic functional calculus for sectorial operators---a survey.Partial differential equations and functional analysis. Oper. Theory Adv. Appl., Vol. 168 Birkhäuser Basel (2006), 263-294. Zbl 1111.47020, MR 2240065, 10.1007/3-7643-7601-5_16 |
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