Geodesic
Maximal inequalities and space-time regularity of stochastic convolutions
Mathematica Bohemica, Tome 123 (1998) no. 1, pp. 7-32

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MR Zbl
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.
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.
DOI : 10.21136/MB.1998.126299
Classification : 60H15
Keywords: stochastic convolutions; maximal inequalities; regularity of stochastic partial differential equations 
Peszat, Szymon; Seidler, Jan. Maximal inequalities and space-time regularity of stochastic convolutions. Mathematica Bohemica, Tome 123 (1998) no. 1, pp. 7-32. doi: 10.21136/MB.1998.126299
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[1] P.-L. Chow J.-L. Jiang: Stochastic partial differential equations in Hölder spaces. Probab. Theory Related Fields 99 (1994), 1-27. | DOI | MR

[2] G. Da Prato S. Kwapień J. Zabczyk: Regularity of solutions of linear stochastic equations in Hilbert spaces. Stochastics 23 (1987), 1-23. | MR

[3] G. Da Prato J. Zabczyk: A note on semilinear stochastic equations. Differential Integral Equations 1 (1988), 143-155. | MR

[4] G. Da Prato J. Zabczyk: A note on stochastic convolution. Stochastic Anal. Appl. 10 (1992), 143-153. | DOI | MR

[5] G. Da Prato J. Zabczyk: Non-explosion, boundedness, and ergodicity for stochastic semilinear equations. J. Differential Equations 98 (1992), 181-195. | DOI | MR

[6] G. Da Prato J. Zabczyk: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge, 1992. | MR

[7] D. A. Dawson: Stochastic evolution equations. Math. Biosci. 15 (1972), 287-316. | DOI | MR | Zbl

[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

[9] T. Funaki: Random motion of strings and related stochastic evolution equations. Nagoya Math. J. 89 (1983), 129-193. | DOI | MR | Zbl

[10] T. Funaki: Regularity properties for stochastic partial diffeгential equations of parabolic type. Osaka J. Math. 28 (1991), 495-516. | MR

[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

[12] I. A. Ibragimov: Sample paths properties of stochastic processes and embedding theorems. Teor. Veroyatnost. i Primenen. 18 (1973), 468-480. (In Russian.) | MR

[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. | DOI | MR | Zbl

[14] P. Kotelenez: Existence, uniqueness and smoothness for a class of function valued stochastic partial differential equations. Stochastics Stochastics Rep. 41 (1992), 177-199. | DOI | MR | Zbl

[15] A. Kufner O. John S. Fučík: Function Spaces. Academia, Praha, 1977. | MR

[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. | DOI | MR

[17] M. Metivier J. Pellaumail: Stochastic Integration. Academic Press, New York, 1980. | MR

[18] S. Peszat: Existence and uniqueness of the solution for stochastic equations on Banach spaces. Stochastics Stochastics Rep. 55 (1995), 167-193. | DOI | MR | Zbl

[19] M. Reed B. Simon: Methods of Modern Mathematical Physics I. Academic Press, New York, 1972. | MR

[20] B. Schmuland: Non-symmetric Ornstein-Uhlenbeck processes in Banach spaces. Canad. J. Math. 45 (1993), 1324-1338. | DOI | MR

[21] J. Seidler: Da Prato-Zabczyk's maximal inequality revisited I. Math. Bohem. 118 (1993), 67-106. | MR | Zbl

[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

[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

[24] H. Tanabe: Equations of Evolution. Pitman, London, 1979. | MR | Zbl

[25] H. Triebel: Interpolation Theory, Function Spaces, Differential Operators. Deutscheг Verlag der Wissenschaften, Berlin, 1978. | MR | Zbl

[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

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