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Keywords:
stochastic linear Cauchy problems; nonexistence of weak solutions; continuous modifications; $C_0$-groups of linear operators
stochastic linear Cauchy problems; nonexistence of weak solutions; continuous modifications; $C_0$-groups of linear operators
Summary:
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.
References:
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