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 , 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.
$$dU(t) = AU(t) + f db_t , U(0) = 0$$
(1)
Keywords
stochastic linear Cauchy problems nonexistence of weak solutions continuous modifications C0-groups of linear operatorsPreview
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© Mathematical Institute, Academy of Sciences of Czech Republic 2006