### Cones of Lower Semicontinuous Functions and a Characterisation of Finely Hyperharmonic Functions.

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We give a new proof of a Phragmén Lindelöf theorem due to W.H.J. Fuchs and valid for an arbitrary open subset $U$ of the complex plane: if $f$ is analytic on $U$, bounded near the boundary of $U$, and the growth of $j$ is at most polynomial then either $f$ is bounded or $U\supset \left\{\right|z|\>r\}$ for some positive $r$ and $f$ has a simple pole.

This paper aims to provide a systematic approach to the treatment of differential equations of the type
dy_{t} = Σ_{i} f^{i}(y_{t}) dx_{t}
^{i}
where the driving signal x_{t} is a rough path. Such equations are very common and occur particularly frequently in probability where the driving signal might be a vector valued Brownian...

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