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Register a new userSpace-time max-stable models with spectral separability
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Natural disasters may have considerable impact on society as well as on (re)insurance industry. Max-stable processes are ideally suited for the modeling of the spatial extent of such extreme events, but it is often assumed that there is no temporal dependence. Only a few papers have introduced spatio-temporal max-stable models, extending the Smith, Schlather and Brown-Resnick spatial processes. These models suffer from two major drawbacks: time plays a similar role as space and the temporal dynamics is not explicit. In order to overcome these defects, we introduce spatio-temporal max-stable models where we partly decouple the influence of time and space in their spectral representations. We introduce both continuous and discrete-tim
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Max-stable processes are central models for spatial extremes. In this paper, we focus on some space-time max-stable models introduced in Embrechts et al. (2016). The processes considered induce discrete-time Markov chains taking values in the space of continuous functions from the unit sphere of $mathbb{R}^3$ to $(0, infty)$. We show that these Markov chains are geometrically ergodic. An interesting feature lies in the fact that the state space is not locally compact, making the classical methodology inapplicable. Instead, we use the fact that the state space is Polish and apply results presented in Hairer (2010).
The risk of extreme environmental events is of great importance for both the authorities and the insurance industry. This paper concerns risk measures in a spatial setting, in order to introduce the spatial features of damages stemming from environmental events into the measure of the risk. We develop a new concept of spatial risk measure, based on the spatially aggregated loss over the region of interest, and propose an adapted set of axioms for these spatial risk measures. These axioms quantify the sensitivity of the risk measure with respect to the space and are especially linked to spatial diversification. The proposed model for the cost underlying our definition of spatial risk measure involves applying a damage function to the environmental variable considered. We build and theoretically study concrete examples of spatial risk measures based on the indicator function of max-stable processes exceeding a given threshold. Some interpretations in terms of insurance are provided.
Partially motivated by the recent papers of Conus, Joseph and Khoshnevisan [Ann. Probab. 41 (2013) 2225-2260] and Conus et al. [Probab. Theory Related Fields 156 (2013) 483-533], this work is concerned with the precise spatial asymptotic behavior for the parabolic Anderson equation [cases{displaystyle {frac{partial u}{partial t}}(t,x)={frac{1}{2}}Delta u(t,x)+V(t,x)u(t,x),cr u(0,x)=u_0(x),}] where the homogeneous generalized Gaussian noise $V(t,x)$ is, among other forms, white or fractional white in time and space. Associated with the Cole-Hopf solution to the KPZ equation, in particular, the precise asymptotic form [lim_{Rtoinfty}(log R)^{-2/3}logmax_{|x|le R}u(t,x)={frac{3}{4}}root 3of {frac{2t}{3}}qquad a.s.] is obtained for the parabolic Anderson model $partial_tu={frac{1}{2}}partial_{xx}^2u+dot{W}u$ with the $(1+1)$-white noise $dot{W}(t,x)$. In addition, some links between time and space asymptotics for the parabolic Anderson equation are also pursued.
We derive concentration inequalities for functions of the empirical measure of large random matrices with infinitely divisible entries and, in particular, stable ones. We also give concentration results for some other functionals of these random matrices, such as the largest eigenvalue or the largest singular value.
Electromagnetic pulses are typically treated as space-time (or space-frequency) separable solutions of Maxwells equations, where spatial and temporal (spectral) dependence can be treated separately. In contrast to this traditional viewpoint, recent advances in structured light and topological optics have highlighted the non-trivial wave-matter interactions of pulses with complex topology and space-time non-separable structure, as well as their potential for energy and information transfer. A characteristic example of such a pulse is the Flying Doughnut (FD), a space-time non-separable toroidal few-cycle pulse with links to toroidal and non-radiating (anapole) excitations in matter. Here, we propose a quantum-mechanics-inspired methodology for the characterization of space-time non-separability in structured pulses. In analogy to the non-separability of entangled quantum systems, we introduce the concept of space-spectrum entangled states to describe the space-time non-separability of classical electromagnetic pulses and develop a method to reconstruct the corresponding density matrix by state tomography. We apply our method to the FD pulse and obtain the corresponding fidelity, concurrence, and entanglement of formation. We demonstrate that such properties dug out from quantum mechanics quantitatively characterize the evolution of the general spatiotemporal structured pulse upon propagation.
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