Consider a Riccati foliation whose monodromy representation is non-elementary and parabolic and consider a non-invariant section of the fibration whose associated developing map is onto. We prove that any holonomy germ from any non-invariant fibre to the section can be analytically continued along a generic Brownian path. To prove this theorem, we prove a dual result about complex projective structures. Let
$\unicode[STIX]{x1D6F4}$
be a hyperbolic Riemann surface of finite type endowed with a branched complex projective structure: such a structure gives rise to a non-constant holomorphic map
${\mathcal{D}}:\tilde{\unicode[STIX]{x1D6F4}}\rightarrow \mathbb{C}\mathbb{P}^{1}$
, from the universal cover of
$\unicode[STIX]{x1D6F4}$
to the Riemann sphere
$\mathbb{C}\mathbb{P}^{1}$
, which is
$\unicode[STIX]{x1D70C}$
-equivariant for a morphism
$\unicode[STIX]{x1D70C}:\unicode[STIX]{x1D70B}_{1}(\unicode[STIX]{x1D6F4})\rightarrow \mathit{PSL}(2,\mathbb{C})$
. The dual result is the following. If the monodromy representation
$\unicode[STIX]{x1D70C}$
is parabolic and non-elementary and if
${\mathcal{D}}$
is onto, then, for almost every Brownian path
$\unicode[STIX]{x1D714}$
in
$\tilde{\unicode[STIX]{x1D6F4}}$
,
${\mathcal{D}}(\unicode[STIX]{x1D714}(t))$
does not have limit when
$t$
goes to
$\infty$
. If, moreover, the projective structure is of parabolic type, we also prove that, although
${\mathcal{D}}(\unicode[STIX]{x1D714}(t))$
does not converge, it converges in the Cesàro sense.