zbMATH — the first resource for mathematics

A maximal inequality for stochastic convolution integrals on Hilbert spaces and space-time regularity of linear stochastic partial differential equations. (English) Zbl 0622.60065
The author considers continuity properties of stochastic convolution integrals \[ Y(t)=\int^{t}_{0}U(t-s)dM(s), \] where U(\(\cdot)\) is an analytic semigroup on a complex separable Hilbert space H and M a cylindrical martingale. Let A be the generator of U(\(\cdot)\) and choose \(\beta >\) spectral radius of A. Assume that for some \(\gamma\in [0,)\) \((\beta -A)^{-\gamma}M\) is an H-valued martingale.
Under an integrability condition Y - considered as a stochastic process with values in the space \(dom((\beta -A)^{\alpha})\), endowed with the norm \(\| (\beta -A)^{\alpha}\cdot \|\)- has a.s. continuous paths if \(\alpha\) is sufficiently small (possibly 0). For suitable \(\alpha\) and \(\gamma\) Y is even Hölder continuous.
This generalizes the result of D. A. Dawson [Math. Biosci. 15, 287- 316 (1972; Zbl 0251.60040)], who proved continuity for the case where A is self-adjoint, \(A^{-1}\) Hilbert-Schmidt, M the cylindrical Wiener process, and \(\alpha =0\).
Reviewer: K.U.Schaumlöffel

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
Full Text: DOI
[1] DOI: 10.1137/0323051 · Zbl 0587.93068 · doi:10.1137/0323051
[2] Gihman, I. I. and Skorohod, A. V. 1972. ”Stochastic differential equations”. Berlin: Springer-Verlag. · Zbl 0242.60003
[3] Gilbarg, D. and Trudinger, N. S. 1977. ”Elliptic partial differential equations of second order”. Berlin: Springer-Verlag. · Zbl 0361.35003
[4] Harrison, J. M. 1985. ”Brownian motion and stochastic flow systems”. New York: John Wiley &Sons. · Zbl 0659.60112
[5] DOI: 10.1007/BF01442115 · Zbl 0438.93078 · doi:10.1007/BF01442115
[6] DOI: 10.2307/1426435 · Zbl 0511.93076 · doi:10.2307/1426435
[7] DOI: 10.1137/0322054 · Zbl 0551.93078 · doi:10.1137/0322054
[8] DOI: 10.1137/0323028 · Zbl 0573.93078 · doi:10.1137/0323028
[9] Shreve S. E., Stochastics 18 pp 245– (1986)
[10] J. M. Lasry, Un probleme de controle stochastique avec critere asymptotique, Cahiers de Math, de la Decision, no. 7715, Universite de Paris IX-Dauphine,1977
[11] DOI: 10.1109/TAC.1984.1103433 · Zbl 0554.93076 · doi:10.1109/TAC.1984.1103433
[12] DOI: 10.1093/imamci/1.4.309 · Zbl 0638.93079 · doi:10.1093/imamci/1.4.309
[13] DOI: 10.1090/S0002-9947-1983-0701523-2 · doi:10.1090/S0002-9947-1983-0701523-2
[14] DOI: 10.1080/07362998508809053 · Zbl 0561.93068 · doi:10.1080/07362998508809053
[15] Menaldi, J. L. and Robin, M. 1984. ”Some singular control problem with long term average criterion, Lecture Notes in Control and Information Sciences”. Vol. 59, 424–432. Berlin: Springer-Verlag. · Zbl 0548.93079
[16] J. L. Menaldi and E. Rofman, A continuous multi-echelon inventory problem, Proceedings of the Fourth IFAC-IFIP Symposium on Information Control Problems in Manufacturing Technology, Gaithersburg, Maryland, USA, Oct., 1982, 41-49
[17] DOI: 10.1007/BF01362380 · Zbl 0064.35703 · doi:10.1007/BF01362380
[18] DOI: 10.1137/0322005 · Zbl 0535.93071 · doi:10.1137/0322005
[19] DOI: 10.1016/0167-6911(86)90076-9 · Zbl 0608.60047 · doi:10.1016/0167-6911(86)90076-9
[20] M. Sun and J. L. Menaldi, Monotone control of a damped oscillator under random perturbations, preprint · Zbl 0850.93883
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.