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Continuity versus nonexistence for a class of linear stochastic Cauchy problems driven by a Brownian motion

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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

$$dU(t) = AU(t) + f db_t , U(0) = 0$$
((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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Authors and Affiliations

  1. Mathematisches Institut I, Technische Universität Karlsruhe, D-76128, Karlsruhe, Germany

    Johanna Dettweiler

  2. Delft Institute of Applied Mathematics, Technical University of Delft, P.O. Box 5031, 2600, GA Delft, The Netherlands

    Jan van Neerven

Authors
  1. Johanna Dettweiler
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  2. Jan van Neerven
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Dettweiler, J., van Neerven, J. Continuity versus nonexistence for a class of linear stochastic Cauchy problems driven by a Brownian motion. Czech Math J 56, 579–586 (2006). https://doi.org/10.1007/s10587-006-0038-0

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  • DOI: https://doi.org/10.1007/s10587-006-0038-0

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