Analytic continuation of holonomy germs of Riccati foliations along Brownian paths

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.

A Novel Non-Stationary Channel Model Utilizing Brownian Random Paths

This paper proposes a non-stationary channel model in which real-time dynamics of the mobile station (MS) are taken into account. We utilize Brownian motion (BM) processes to model targeted and non-targeted dynamics of the MS. The proposed trajectory model consists of both drift and random components to capture both targeted and non-targeted motions of the MS. The Brownian trajectory model is then employed to provide a non-stationary channel model, in which the scattering effects of the propagation area are modelled by a non-centred one-ring geometric scattering model. The starting point of the motion is a fixed point in the propagation environment, whereas its terminating point is a random point along a predetermined drift. The drift component can be controlled by a so-called drift parameter. Tracking the MS on the proposed Brownian path allows us to derive the local angles-of-arrival (AOAs) and local angles-of-motion (AOMs), which are expressed by stochastic processes rather than random variables. We compute the first-order densities of the AOA and AOM processes in closed form. The local power spectral density (PSD) of the Doppler frequencies and the autocorrelation function (ACF) of the complex channel gain are also provided. Given a walking speed scenario, the analytical results are demonstrated and explained in depth. It turns out that the proposed Brownian path model results in a non-stationary non-isotropic channel model. The proposed geometry-based channel model is very useful for the performance analysis of mobile communication systems under non-stationary conditions.