A note on maximal inequality for stochastic convolutions

Erika Hausenblas; Jan Seidler

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Let $A = d / d θ$ 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 $d U (t) = A U (t) + f d b_{t}, U (0) = 0, (1)$ where $b$ is a standard Brownian motion and $f \in C (Γ)$ 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...

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