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We consider a stochastic partial differential equation (SPDE) which describes the velocity field of a viscous, incompressible non-Newtonian fluid subject to a random force. Here, the extra stress tensor of the fluid is given by a polynomial of degree p — 1 of the rate of strain tensor, while the colored noise is considered as a random force. We focus on the shear thickening case, more precisely, on the case $p \\in [1 + \\frac{d}{2},\\frac{{2d}}{{d - 2}})$ , where d is the dimension of the space. We prove that the Galerkin scheme approximates the velocity field in a strong sense. As a consequence, we establish the energy equality for the velocity field.

We consider branching random walks in d-dimensional integer lattice with time-space i.i.d. offspring distributions. When d ≥ 3 and the fluctuation of the environment is well moderated by the random walk, we prove a central limit theorem for the density of the population, together with upper bounds for the density of the most populated site and the replica overlap. We also discuss the phase transition of this model in connection with directed polymers in random environment.

The 2014 release of a new set of purchasing power parity (PPP) conversion factors (PPPs) for 2011 has prompted a revision of the World Bank’s international poverty line. In revising the line, we have sought to minimize changes to the real purchasing power of the earlier $1.25 line (in 2005 PPPs), so as to preserve the integrity of the goalposts for international targets such as the Sustainable Development Goals (SDGs) and the World Bank’s twin goals – which were set with respect to that line. In particular, the new line was obtained by inflating the same fifteen national poverty lines – originally used by Ravallion et al. (World Bank Econ. Rev. 23 (2): 163–184 2009 ) to construct the $1.25 line – to 2011 prices in local currency units, and then converting them to US dollars using 2011 PPP conversion factors. With a small approximation, this procedure yields a new international poverty line of $1.90 per person per day. In combination with other changes described in the paper, this revision leads to relatively small changes in global poverty incidence for 2011: from 14.5 % using the old method to 14.1 % using the new method. In 2012, the new reference year for the global count, we find 12.7 % of the world’s population, or 897 million people, are living in extreme poverty. There are changes in the regional composition of poverty, but they are also relatively small. This paper documents methodological decisions taken in the process of updating both the poverty line and the consumption and income distributions at the country level, including issues of inter-temporal and spatial price adjustments. It also describes various caveats and limitations of the approach taken.

We consider a stochastic partial differential equation (SPDE) which describes the velocity field of a viscous, incompressible non-Newtonian fluid subject to a random force. Here the extra stress tensor of the fluid is given by a polynomial of degree p − 1 of the rate of strain tensor, while the colored noise is considered as a random force. We investigate the existence and the uniqueness of weak solutions to this SPDE.

We consider directed polymers in a random environment. Under some mild assumptions on the environment, we prove equivalence between the decay rate of the partition function and some natural localization properties of the path; some quantitative estimates of the decay of the partition function in one or two dimensions, or at sufficiently low temperature; and the existence of quenched free energy. In particular, we generalize to general environments the results recently obtained by Carmona and Hu for a Gaussian environment. Our approach is based on martingale decomposition and martingale analysis. It leads to a natural, asymptotic relation between the partition function, and the probability that two polymers in the same environment, but otherwise independent, end up at the same point.

In this paper we consider directed polymers in random environment with discrete space and time. For transverse dimension at least equal to 3, we prove that diffusivity holds for the path in the full weak disorder region, that is, where the partition function differs from its annealed value only by a non-vanishing factor. Deep inside this region, we also show that the quenched averaged energy has fluctuations of order 1. In complete generality (arbitrary dimension and temperature), we prove monotonicity of the phase diagram in the temperature.

We consider a ferromagnetic spin system with unbounded interactions on the d-dimensional integer lattice (d > 1). Under mild assumptions on the one-body interactions (so that arbitrarily deep double wells are allowed), we prove that if the coupling constants are small enough, then the finite volume Gibbs states satisfy the log-Sobolev inequality uniformly in the volume and the boundary condition.

Friendly walkers is a stochastic model obtained from independent one-dimensional simple random walks {Skj}j≥0, k=1,2,…,d by introducing ``non-crossing condition'': and ``reward for collisions'' characterized by parameters . Here, the reward for collisions is described as follows. If, at a given time n, a site in ℤ is occupied by exactly m≥2 walkers, then the site increases the probabilistic weight for the walkers by multiplicative factor exp (βm)≥1. We study the localization transition of this model in terms of the positivity of the free energy and describe the location and the shape of the critical surface in the (d−1)-dimensional space for the parameters .

We consider an operator \\(P_V=(1+V)P\\) on \\(\\ell^2(Z^d)\\), where \\(P\\) is the transition operator of a symmetric irreducible random walk, and \\(V\\) is a ``sparse'' potential. We first characterize the essential spectra of this operator. Secondly, we prove that all the eigenfunctions which correspond to discrete spectra decay exponentially fast. Thirdly, we give a sufficient condition for this operator to have an absolute spectral gap at the right edge of the spectra. Finally, as an application of the absolute spectral gap and the exponential decay of the eigenfunctions, we prove a limit theorem for the random walk under the Gibbs measure associated to the potential \\(V\\).