Previous |  Up |  Next


stochastic linear Cauchy problems; nonexistence of weak solutions; continuous modifications; $C_0$-groups of linear operators
Let $A={\mathrm d}/{\mathrm d}\theta $ denote the generator of the rotation group in the space $C(\Gamma )$, where $\Gamma $ denotes the unit circle. We show that the stochastic Cauchy problem \[ {\mathrm d}U(t) = AU(t)+ f\mathrm{d}b_t, \quad U(0)=0, \qquad \mathrm{(1)}\] where $b$ is a standard Brownian motion and $f\in C(\Gamma )$ is fixed, has a weak solution if and only if the stochastic convolution process $t\mapsto (f * b)_t$ has a continuous modification, and that in this situation the weak solution has a continuous modification. In combination with a recent result of Brzeźniak, Peszat and Zabczyk it follows that (1) fails to have a weak solution for all $f\in C(\Gamma )$ outside a set of the first category.
[1] Z.  Brzeźniak: Some remarks on stochastic integration in 2-smooth Banach spaces. Probabilistic Methods in Fluids, I. M.  Davies, A.  Truman  et. al. (eds.), World Scientific, New Jersey, 2003, pp. 48–69. MR 2083364
[2] Z.  Brzeźniak, J. M. A. M.  van Neerven: Stochastic convolution in separable Banach spaces and the stochastic linear Cauchy problem. Studia Math. 143 (2000), 43–74. MR 1814480
[3] Z.  Brzeźniak, Sz.  Peszat, and J.  Zabczyk: Continuity of stochastic convolutions. Czechoslovak Math.  J. 51 (2001), 679–684. DOI 10.1023/A:1013752526625 | MR 1864035
[4] G.  Da Prato, J.  Zabczyk: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1992. MR 1207136
[5] G.  Da Prato and J.  Zabczyk: Ergodicity for Infinit-Dimensional Systems. London Math. Soc. Lect. Note Series, Vol.  229. Cambridge University Press, Cambridge, 1996. MR 1417491
[6] E.  Hausenblas, J.  Seidler: A note on maximal inequality for stochastic convolutions. Czechoslovak Math.  J. 51 (2001), 785–790. DOI 10.1023/A:1013717013421 | MR 1864042
[7] J.-P.  Kahane: Some Random Series of Functions. Second edition. Cambridge Studies in Advanced Mathematics, Vol.  5. Cambridge University Press, Cambridge, 1985. MR 0833073
[8] O.  Kallenberg: Foundations of Modern Probability. Second edition. Probability and its Applications. Springer-Verlag, New York, 2002. MR 1876169
[9] S.  Kwapień, W. A.  Woyczyński: Random Series and Stochastic Integrals: Single and Multiple. Probability and its Applications. Birkhäuser-Verlag, Boston, 1992. MR 1167198
[10] M.  Ledoux, M.  Talagrand: Probability in Banach Spaces Ergebnisse der Math. und ihrer Grenzgebiete, Vol.  23. Springer-Verlag, Berlin, 1991. MR 1102015
[11] J. M. A. M.  van Neerven, L.  Weis: Stochastic integration of functions with values in a Banach space. Studia Math. 166 (2005), 131–170. DOI 10.4064/sm166-2-2 | MR 2109586
Partner of
EuDML logo