Spectral gap properties for linear random walks and Pareto's asymptotics for affine stochastic recursions

Let $V=\mathbb R^d$ be the Euclidean $d$-dimensional space, $\mu$ (resp $\lambda$) a probability measure on the linear (resp affine) group $G=G L (V)$ (resp H= \Aff (V)) and assume that $\mu$ is the projection of $\lambda$ on $G$. We study asymptotic properties of the iterated convolutions $\mu^n *\delta\_{v}$ (resp $\lambda^n*\delta\_{v})$ if $v\in V$, i.e asymptotics of the random walk on $V$ defined by $\mu$ (resp $\lambda$), if the subsemigroup $T\subset G$ (resp.\ $\Sigma \subset H$) generated by the support of $\mu$ (resp $\lambda$) is "large". We show spectral gap properties for the convolution operator defined by $\mu$ on spaces of homogeneous functions of degree $s\geq 0$ on $V$, which satisfy H{\"o}lder type conditions. As a consequence of our analysis we get precise asymptotics for the potential kernel $\Sigma\_{0}^{\infty} \mu^k * \delta\_{v}$, which imply its asymptotic homogeneity. Under natural conditions the $H$-space $V$ is a $\lambda$-boundary; then we use the above results and radial Fourier Analysis on $V\setminus \{0\}$ to show that the unique $\lambda$-stationary measure $\rho$ on $V$ is "homogeneous at infinity" with respect to dilations $v\rightarrow t v$ (for t\textgreater{}0), with a tail measure depending essentially of $\mu$ and $\Sigma$. Our proofs are based on the simplicity of the dominant Lyapunov exponent for certain products of Markov-dependent random matrices, on the use of renewal theorems for "tame" Markov walks, and on the dynamical properties of a conditional $\lambda$-boundary dual to $V$

On the homogeneity at infinity of the stationary probability for an affine random walk

Proceedings of the International Conference in honor of S. G. Dani's 65th birthday held at the Maharaja Sayajirao University of Baroda, Vadodara, December 26–29, 2012, including two surveys on the work of GaniInternational audienceWe consider an affine random walk on $\mathbb R$. We assume the existence of a stationary probability $\nu$ on $\mathbb R$ and we describe the shape at infinity of $\nu$, if $\nu$ has unbounded support. We discuss the connections of the result with geometrical or probabilistic problems

Exact travelling wave solutions in viscoelastic channel flow

Elasto-inertial turbulence (EIT) is a new, two-dimensional chaotic flow state observed in polymer solutions with possible connections to inertialess elastic turbulence and drag-reduced Newtonian turbulence. In this Letter, we argue that the origins of EIT are fundamentally different from Newtonian turbulence by finding a dynamical connection between EIT and an elasto-inertial linear instability recently found at high Weissenberg numbers (Garg et al. Phys. Rev. Lett. 121, 024502, 2018). This link is established by isolating the first known exact coherent structures in viscoelastic parallel flows - nonlinear elasto-inertial travelling waves (TWs) - borne at the linear instability and tracking them down to substantially lower Weissenberg numbers where EIT exists. These TWs have a distinctive ``arrowhead'' structure in the polymer stretch field and can be clearly recognised, albeit transiently, in EIT, as well as being attractors for EIT dynamics if the Weissenberg number is sufficiently large. Our findings suggest that the dynamical systems picture in which Newtonian turbulence is built around the co-existence of many (unstable) simple invariant solutions populating phase space carries over to EIT, though these solutions rely on elasticity to exist

Multistability of elasto-inertial two-dimensional channel flow

Elasto-inertial turbulence (EIT) is a recently discovered two-dimensional chaotic flow state observed in dilute polymer solutions. It has been hypothesised that the dynamical origins of EIT are linked to a center-mode instability, whose nonlinear evolution leads to a travelling wave with an 'arrowhead' structure in the polymer conformation, a structure also observed instantaneously in simulations of EIT. In this work we conduct a suite of two-dimensional direct numerical simulations spanning a wide range of polymeric flow parameters to examine the possible dynamical connection between the arrowhead and EIT. Our calculations reveal (up to) four co-existent attractors: the laminar state and a steady arrowhead, along with EIT and a 'chaotic arrowhead'. The steady arrowhead is stable for all parameters considered here, while the final pair of (chaotic) flow states are visually very similar and can be distinguished only by the presence of a weak polymer arrowhead structure in the 'chaotic arrowhead' regime. Analysis of energy transfers between the flow and the polymer indicates that both chaotic regimes are maintained by an identical near-wall mechanism and that the weak arrowhead does not play a role. Our results suggest that the arrowhead is a benign flow structure that is disconnected from the self-sustaining mechanics of EIT.Comment: 17 pages, 10 figure

Retrosplenial and postsubicular head direction cells compared during visual landmark discrimination

Background: Visual landmarks are used by head direction (HD) cells to establish and help update the animal’s representation of head direction, for use in orientation and navigation. Two cortical regions that are connected to primary visual areas, postsubiculum (PoS) and retrosplenial cortex (RSC), possess HD cells: we investigated whether they differ in how they process visual landmarks. Methods: We compared PoS and RSC HD cell activity from tetrode-implanted rats exploring an arena in which correct HD orientation required discrimination of two opposing landmarks having high, moderate or low discriminability. Results: RSC HD cells had higher firing rates than PoS HD cells and slightly lower modulation by angular head velocity, and anticipated actual head direction by ~48 ms, indicating that RSC spiking leads PoS spiking. Otherwise, we saw no differences in landmark processing, in that HD cells in both regions showed equal responsiveness to and discrimination of the cues, with cells in both regions having unipolar directional tuning curves and showing better discrimination of the highly discriminable cues. There was a small spatial component to the signal in some cells, consistent with their role in interacting with the place cell navigation system, and there was also slight modulation by running speed. Neither region showed theta modulation of HD cell spiking. Conclusions: That the cells can immediately respond to subtle differences in spatial landmarks is consistent with rapid processing of visual snapshots or scenes; similarities in PoS and RSC responding may be due either to similar computations being performed on the visual inputs, or to rapid sharing of information between these regions. More generally, this two-cue HD cell paradigm may be a useful method for testing rapid spontaneous visual discrimination capabilities in other experimental settings