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Keywords:
stochastic convolutions; maximal inequalities; path-continuity; stochastic partial differential equations; $H^\infty $-calculus; $\gamma $-radonifying operators; exponential tail estimates
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.
References:
[1] Albiac, F., Kalton, N. J.: Topics in Banach Space Theory. Graduate Texts in Mathematics, Vol. 233. Springer Berlin (2006). MR 2192298
[2] Amann, H.: Dual semigroups and second order linear elliptic boundary value problems. Isr. J. Math. 45 (1983), 225-254. DOI 10.1007/BF02774019 | MR 0719122 | Zbl 0535.35017
[3] Brze{'z}niak, Z.: Stochastic partial differential equations in $M$-type 2 Banach spaces. Potential Anal. 4 (1995), 1-45. DOI 10.1007/BF01048965 | MR 1313905
[4] Brze{'z}niak, Z.: On stochastic convolution in Banach spaces and applications. Stochastics Stochastics Rep. 61 (1997), 245-295. DOI 10.1080/17442509708834122 | MR 1488138
[5] Brze{'z}niak, Z., Hausenblas, E.: Maximal regularity for stochastic convolutions driven by Lévy processes. Probab. Theory Relat. Fields 145 (2009), 615-637. DOI 10.1007/s00440-008-0181-7 | MR 2529441
[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
[7] Calderón, A.-P.: Intermediate spaces and interpolation, the complex method. Stud. Math. 24 (1964), 113-190. MR 0167830
[8] Cox, S. G., Veraar, M. C.: Vector-valued decoupling and the Burkholder-Davis-Gundy inequality. (2010), (to appear) Ill. J. Math. MR 3006692
[9] Prato, G. Da, Zabczyk, J.: Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications, Vol. 44 Cambridge University Press Cambridge (1992). MR 1207136 | Zbl 0761.60052
[10] Davis, B.: On the $L^{p}$ norms of stochastic integrals and other martingales. Duke Math. J. 43 (1976), 697-704. DOI 10.1215/S0012-7094-76-04354-4 | MR 0418219
[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
[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
[13] Fröhlich, A. M., Weis, L.: $H^\infty$ calculus and dilations. Bull. Soc. Math. Fr. 134 (2006), 487-508. MR 2364942
[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. DOI 10.1007/s00020-005-1416-y | MR 2250161 | Zbl 1138.93361
[15] Haase, M.: The Functional Calculus for Sectorial Operators. Operator Theory: Advances and Applications, Vol. 169. Birkhäuser Basel (2006). MR 2244037
[16] Hausenblas, E., Seidler, J.: A note on maximal inequality for stochastic convolutions. Czechoslovak Math. J. 51(126) (2001), 785-790. DOI 10.1023/A:1013717013421 | MR 1864042 | Zbl 1001.60065
[17] Hausenblas, E., Seidler, J.: Stochastic convolutions driven by martingales: maximal inequalities and exponential integrability. Stochastic Anal. Appl. 26 (2008), 98-119. DOI 10.1080/07362990701673047 | MR 2378512 | Zbl 1153.60035
[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. DOI 10.1007/s11854-008-0051-3 | MR 2448989
[19] Kalton, N. J., Weis, L. W.: The $H^\infty$-calculus and square function estimates. Preprint (2004).
[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. DOI 10.1007/978-3-540-44653-8_2 | MR 2108959 | Zbl 1097.47041
[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. DOI 10.1006/jfan.1999.3393 | MR 1694522
[22] Lenglart, E.: Rélation de domination entre deux processus. Ann. Inst. Henri Poincaré, Nouv. Sér, Sect. B 13 (1977), 171-179 French. MR 0471069 | Zbl 0373.60054
[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. DOI 10.1006/jfan.2001.3909 | MR 1923628 | Zbl 1020.47029
[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
[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
[26] Neerven, J. M. A. M. van, Veraar, M. C., Weis, L.: Stochastic integration in {UMD} Banach spaces. Ann. Probab. 35 (2007), 1438-1478. DOI 10.1214/009117906000001006 | MR 2330977
[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. DOI 10.1016/j.jfa.2008.03.015 | MR 2433958
[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
[29] Pinelis, I.: Optimum bounds for the distributions of martingales in Banach spaces. Ann. Probab. 22 (1994), 1679-1706. DOI 10.1214/aop/1176988477 | MR 1331198 | Zbl 0836.60015
[30] Pisier, G.: Martingales with values in uniformly convex spaces. Isr. J. Math. 20 (1975), 326-350. DOI 10.1007/BF02760337 | MR 0394135 | Zbl 0344.46030
[31] Pisier, G.: Some results on Banach spaces without local unconditional structure. Compos. Math. 37 (1978), 3-19. MR 0501916 | Zbl 0381.46010
[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
[33] Seidler, J.: Exponential estimates for stochastic convolutions in 2-smooth Banach spaces. Electron. J. Probab. 15 (2010), 1556-1573. DOI 10.1214/EJP.v15-808 | MR 2735374
[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
[35] Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators, 2nd ed. Barth Leipzig (1995). MR 1328645
[36] Veraar, M. C.: Continuous local martingales and stochastic integration in {UMD} Banach spaces. Stochastics 79 (2007), 601-618. MR 2368370 | Zbl 1129.60050
[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. DOI 10.1007/3-7643-7601-5_16 | MR 2240065 | Zbl 1111.47020
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