Fundamental Solutions and Decay of Fully Non-local Problems

In this paper, we study a fully non-local reaction-diffusion equation which is non-local both in time and space. We apply subordination principles to construct the fundamental solutions of this problem, which we use to find a representation of the mild solutions. Moreover, using techniques of Harmonic Analysis and Fourier Multipliers, we obtain the temporal decay rates for the mild solutions

Brownian yet non-Gaussian diffusion: from superstatistics to subordination of diffusing diffusivities

A growing number of biological, soft, and active matter systems are observed to exhibit normal diffusive dynamics with a linear growth of the mean squared displacement, yet with a non-Gaussian distribution of increments. Based on the Chubinsky-Slater idea of a diffusing diffusivity we here establish and analyze a minimal model framework of diffusion processes with fluctuating diffusivity. In particular, we demonstrate the equivalence of the diffusing diffusivity process with a superstatistical approach with a distribution of diffusivities, at times shorter than the diffusivity correlation time. At longer times a crossover to a Gaussian distribution with an effective diffusivity emerges. Specifically, we establish a subordination picture of Brownian but non-Gaussian diffusion processes, that can be used for a wide class of diffusivity fluctuation statistics. Our results are shown to be in excellent agreement with simulations and numerical evaluations.Comment: 19 pages, 6 figures, RevTeX. Physical Review X, at pres

Langevin formulation of a subdiffusive continuous time random walk in physical time

Systems living in complex non equilibrated environments often exhibit subdiffusion characterized by a sublinear power-law scaling of the mean square displacement. One of the most common models to describe such subdiffusive dynamics is the continuous time random walk (CTRW). Stochastic trajectories of a CTRW can be described mathematically in terms of a subordination of a normal diffusive process by an inverse Levy-stable process. Here, we propose a simpler Langevin formulation of CTRWs without subordination. By introducing a new type of non-Gaussian noise, we are able to express the CTRW dynamics in terms of a single Langevin equation in physical time with additive noise. We derive the full multi-point statistics of this noise and compare it with the noise driving scaled Brownian motion (SBM), an alternative stochastic model describing subdiffusive behaviour. Interestingly, these two noises are identical up to the level of the 2nd order correlation functions, but different in the higher order statistics. We extend our formalism to general waiting time distributions and force fields, and compare our results with those of SBM.Comment: 11 pages, 4 figures - The new version contains corrected figures and new paragraphs in the main tex

Functional Inequalities and Subordination: Stability of Nash and Poincar\'e inequalities

We show that certain functional inequalities, e.g.\ Nash-type and Poincar\'e-type inequalities, for infinitesimal generators of $C_0$ semigroups are preserved under subordination in the sense of Bochner. Our result improves \cite[Theorem 1.3]{BM} by A.\ Bendikov and P.\ Maheux for fractional powers, and it also holds for non-symmetric settings. As an application, we will derive hypercontractivity, supercontractivity and ultracontractivity of subordinate semigroups.Comment: 15 page

Restructuring Venezuela\u27s Debt Using Pari Passu

Given the depth of Venezuela\u27s economic crisis, many fear that the government and the state-owned oil company Petroleos de Venezuela, S.A. ( PDVSA ) are on the brink of insolvency. In this paper, we introduce a restructuring plan that would allow Venezuela to restructure its external debt in an orderly manner. We propose that Venezuela restructure both PDVSA debt and its own external debt via Exchange Offers. To maximize the number of participating bondholders and receive sufficient debt relief, we suggest that Venezuela primarily utilize the pari passu clauses included in the vast majority of PDVSA and Venezuelan bonds, which are modified versions of a typical pari passu clause and can be read to allow the subordination of the bonds in accordance with Venezuelan law. To minimize the number of holdout creditors, Venezuela can introduce a law that subordinates non-exchanged debt to exchanged debt, making timely or full payment of holdout debt unlikely. This tactic would minimize the need to rely solely on alternative restructuring techniques, such as exit consents and Collective Action Clauses (CACs). We argue that while these techniques might alone prove insufficient to successfully restructure Venezuela\u27s debt, they could supplement the restructuring options we propose here. Because the parties contracted for debt subordination in the bond contracts, we predict that using a debt subordination technique would be more viable in Venezuela\u27s case than it has been in past sovereign debt restructurings. Ironically, the pari passu clause that doomed Argentina might be what saves Venezuela

Fractional statistical dynamics and fractional kinetics

We apply the subordination principle to construct kinetic fractional statistical dynamics in the continuum in terms of solutions to Vlasov-type hierarchies. As a by-product we obtain the evolution of the density of particles in the fractional kinetics in terms of a non-linear Vlasov-type kinetic equation. As an application we study the intermittency of the fractional mesoscopic dynamics.Comment: Published in Methods of Functional Analysis and Topology (MFAT), available at http://mfat.imath.kiev.ua/article/?id=890. arXiv admin note: text overlap with arXiv:1604.0380