### An extension theorem to rough paths

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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...

We show how to construct a canonical choice of stochastic area for paths of reversible Markov processes satisfying a weak Hölder condition, and hence demonstrate that the sample paths of such processes are rough paths in the sense of Lyons. We further prove that certain polygonal approximations to these paths and their areas converge in $p$-variation norm. As a corollary of this result and standard properties of rough paths, we are able to provide a significant generalization of the classical result...

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