Previous |  Up |  Next


Title: Maximal inequalities and space-time regularity of stochastic convolutions (English)
Author: Peszat, Szymon
Author: Seidler, Jan
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959 (print)
ISSN: 2464-7136 (online)
Volume: 123
Issue: 1
Year: 1998
Pages: 7-32
Summary lang: English
Category: math
Summary: Space-time regularity of stochastic convolution integrals J = {\int^\cdot_0 S(\cdot-r)Z(r)W(r)} driven by a cylindrical Wiener process $W$ in an $L^2$-space on a bounded domain is investigated. The semigroup $S$ is supposed to be given by the Green function of a $2m$-th order parabolic boundary value problem, and $Z$ is a multiplication operator. Under fairly general assumptions, $J$ is proved to be Holder continuous in time and space. The method yields maximal inequalities for stochastic convolutions in the space of continuous functions as well. (English)
Keyword: stochastic convolutions
Keyword: maximal inequalities
Keyword: regularity of stochastic partial differential equations
MSC: 60H15
idZBL: Zbl 0903.60047
idMR: MR1618707
DOI: 10.21136/MB.1998.126299
Date available: 2009-09-24T21:28:51Z
Last updated: 2020-07-29
Stable URL:
Reference: [1] P.-L. Chow J.-L. Jiang: Stochastic partial differential equations in Hölder spaces.Probab. Theory Related Fields 99 (1994), 1-27. MR 1273740, 10.1007/BF01199588
Reference: [2] G. Da Prato S. Kwapień J. Zabczyk: Regularity of solutions of linear stochastic equations in Hilbert spaces.Stochastics 23 (1987), 1-23. MR 0920798
Reference: [3] G. Da Prato J. Zabczyk: A note on semilinear stochastic equations.Differential Integral Equations 1 (1988), 143-155. MR 0922558
Reference: [4] G. Da Prato J. Zabczyk: A note on stochastic convolution.Stochastic Anal. Appl. 10 (1992), 143-153. MR 1154532, 10.1080/07362999208809260
Reference: [5] G. Da Prato J. Zabczyk: Non-explosion, boundedness, and ergodicity for stochastic semilinear equations.J. Differential Equations 98 (1992), 181-195. MR 1168978, 10.1016/0022-0396(92)90111-Y
Reference: [6] G. Da Prato J. Zabczyk: Stochastic Equations in Infinite Dimensions.Cambridge University Press, Cambridge, 1992. MR 1207136
Reference: [7] D. A. Dawson: Stochastic evolution equations.Math. Biosci. 15 (1972), 287-316. Zbl 0251.60040, MR 0321178, 10.1016/0025-5564(72)90039-9
Reference: [8] S. D. Eideľman S. D. Ivasishen: Investigation of the Green matrix of a homogeneous parabolic boundary value problem.Trudy Moskov. Mat. Obshch. 23 (1970), 179-234. (In Russian.) MR 0367455
Reference: [9] T. Funaki: Random motion of strings and related stochastic evolution equations.Nagoya Math. J. 89 (1983), 129-193. Zbl 0531.60095, MR 0692348, 10.1017/S0027763000020298
Reference: [10] T. Funaki: Regularity properties for stochastic partial diffeгential equations of parabolic type.Osaka J. Math. 28 (1991), 495-516. MR 1144470
Reference: [11] B. Gołdys: On weak solutions of stochastic evolution equations with unbounded coefficients.Miniconference on probability and analysis (Sydney, 1991). Proc. Centre Math. Appl. Austral. Nat. Univ. 29, Austral Nat. Univ., Canberra, 1992, pp. 116-128. MR 1188889
Reference: [12] I. A. Ibragimov: Sample paths properties of stochastic processes and embedding theorems.Teor. Veroyatnost. i Primenen. 18 (1973), 468-480. (In Russian.) MR 0326827
Reference: [13] P. Kotelenez: A maximal inequality for stochastic convolution integrals on Hilbert spaces and space-time regularity of linear stochastic partial differential equations.Stochastics 21 (1987), 345-358. Zbl 0622.60065, MR 0905052, 10.1080/17442508708833463
Reference: [14] P. Kotelenez: Existence, uniqueness and smoothness for a class of function valued stochastic partial differential equations.Stochastics Stochastics Rep. 41 (1992), 177-199. Zbl 0766.60078, MR 1275582, 10.1080/17442509208833801
Reference: [15] A. Kufner O. John S. Fučík: Function Spaces.Academia, Praha, 1977. MR 0482102
Reference: [16] R. Manthey: Existence and uniqueness of solutions of a reaction-diffusion equation with polynomial nonlinearity and white noise disturbance.Math. Nachr. 125 (1986), 121-133. MR 0847354, 10.1002/mana.19861250108
Reference: [17] M. Metivier J. Pellaumail: Stochastic Integration.Academic Press, New York, 1980. MR 0578177
Reference: [18] S. Peszat: Existence and uniqueness of the solution for stochastic equations on Banach spaces.Stochastics Stochastics Rep. 55 (1995), 167-193. Zbl 0886.60064, MR 1378855, 10.1080/17442509508834024
Reference: [19] M. Reed B. Simon: Methods of Modern Mathematical Physics I.Academic Press, New York, 1972. MR 0751959
Reference: [20] B. Schmuland: Non-symmetric Ornstein-Uhlenbeck processes in Banach spaces.Canad. J. Math. 45 (1993), 1324-1338. MR 1247550, 10.4153/CJM-1993-075-6
Reference: [21] J. Seidler: Da Prato-Zabczyk's maximal inequality revisited I.Math. Bohem. 118 (1993), 67-106. Zbl 0785.35115, MR 1213834
Reference: [22] V. A. Solonnikov: On boundary value problems foг lineaг paгabolic systems of differential equations of geneгal foгm.Trudy Mat. Inst. Steklov 83 (1965), 3-162. (In Russian.) MR 0211083
Reference: [23] V. A. Solonnikov: On the Green matrices for parabolic boundary value problems.Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 14 (1969), 256-287. (In Russian.) MR 0296527
Reference: [24] H. Tanabe: Equations of Evolution.Pitman, London, 1979. Zbl 0417.35003, MR 0533824
Reference: [25] H. Triebel: Interpolation Theory, Function Spaces, Differential Operators.Deutscheг Verlag der Wissenschaften, Berlin, 1978. Zbl 0387.46033, MR 0500580
Reference: [26] J. B. Walsh: An intгoduction to stochastic partial diffeгential equations.École d'été de pгobabilités de Saint-Flour XIV-1984. Lectuгe Notes in Math. 1180, Spгingeг-Verlag, Berlin, 1986, pp. 265-439. MR 0876085


Files Size Format View
MathBohem_123-1998-1_2.pdf 1.708Mb application/pdf View/Open
Back to standard record
Partner of
EuDML logo