Continuity versus nonexistence for a class of linear stochastic Cauchy problems driven by a Brownian motion |
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Authors: | Johanna Dettweiler Jan van Neerven |
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Institution: | 1. Mathematisches Institut I, Technische Universit?t Karlsruhe, D-76128, Karlsruhe, Germany 2. Delft Institute of Applied Mathematics, Technical University of Delft, P.O. Box 5031, 2600, GA Delft, The Netherlands
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Abstract: | Let A = d/dθ denote the generator of the rotation group in the space C(Γ), where Γ denotes the unit circle. We show that the stochastic Cauchy problem
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((1)) |
, where b is a standard Brownian motion and f ∈ C(Γ) is fixed, has a weak solution if and only if the stochastic convolution process t ↦ (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 ∈ C(Γ) outside a set of the first category. |
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Keywords: | stochastic linear Cauchy problems nonexistence of weak solutions continuous modifications C 0-groups of linear operators |
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