Time correlations for the parabolic Anderson model

We derive exact asymptotics of time correlation functions for the parabolic Anderson model with homogeneous initial condition and time-independent tails that decay more slowly than those of a double exponential distribution and have a finite cumulant generating function. We use these results to give precise asymptotics for statistical moments of positive order. Furthermore, we show what the potential peaks that contribute to the intermittency picture look like and how they are distributed in space. We also investigate for how long intermittency peaks remain relevant in terms of ageing properties of the model.Comment: 28 page

On Envelope Theorems in Economics: Inspired by a Revival of a Forgotten Lecture

This paper studies how envelope theorems have been used in Economics, their history and also who first introduced them. The existing literature is full of them and the reason is that all families of optimal value functions can produce them. The paper is driven by curiosity, but hopefully it will give the reader some new insights.Envelope theorems; names and history; value functions

Conservation laws for under determined systems of differential equations

This work extends the Ibragimov's conservation theorem for partial differential equations [{\it J. Math. Anal. Appl. 333 (2007 311-328}] to under determined systems of differential equations. The concepts of adjoint equation and formal Lagrangian for a system of differential equations whose the number of equations is equal to or lower than the number of dependent variables are defined. It is proved that the system given by an equation and its adjoint is associated with a variational problem (with or without classical Lagrangian) and inherits all Lie-point and generalized symmetries from the original equation. Accordingly, a Noether theorem for conservation laws can be formulated

Eigenvector localization in the heavy-tailed random conductance model

We generalize our former localization result about the principal Dirichlet eigenvector of the i.i.d. heavy-tailed random conductance Laplacian to the first $k$ eigenvectors. We overcome the complication that the higher eigenvectors have fluctuating signs by invoking the Bauer-Fike theorem to show that the $k$th eigenvector is close to the principal eigenvector of an auxiliary spectral problem.Comment: 14 pages. Generalizes the results of article arXiv:1608.02415 to higher order eigenvectors. For better readability, we have copied the main definition

Modelling wall shear stress in small arteries using LBM and FVM

This paper was presented at the 2nd Micro and Nano Flows Conference (MNF2009), which was held at Brunel University, West London, UK. The conference was organised by Brunel University and supported by the Institution of Mechanical Engineers, IPEM, the Italian Union of Thermofluid dynamics, the Process Intensification Network, HEXAG - the Heat Exchange Action Group and the Institute of Mathematics and its Applications.In this study a finite-volume discretisation of a Lattice Boltzmann equation over unstructured grids is presented. The new scheme is based on the idea of placing the unknown fields at the nodes of the mesh and evolve them based on the fluxes crossing the surfaces of the corresponding control volumes. The method, named unstructured Lattice Boltzmann equation (ULBE) is compared with the classical finite volume method (FVM) and is applied here to the problem of blood flow over the endothelium in small arteries. The study shows a significant variation and a high sensitivity of wall shear stress to the endothelium corrugation degree

The quenched limiting distributions of a one-dimensional random walk in random scenery

For a one-dimensional random walk in random scenery (RWRS) on Z, we determine its quenched weak limits by applying Strassen's functional law of the iterated logarithm. As a consequence, conditioned on the random scenery, the one-dimensional RWRS does not converge in law, in contrast with the multi-dimensional case

Brownian motion in attenuated or renormalized inverse-square Poisson potential

We consider the parabolic Anderson problem with random potentials having inverse-square singularities around the points of a standard Poisson point process in $\mathbb{R}^d$, $d \geq 3$. The potentials we consider are obtained via superposition of translations over the points of the Poisson point process of a kernel $\mathfrak{K}$ behaving as $\mathfrak{K}(x) \approx \theta |x|^{-2}$ near the origin, where $\theta \in (0,(d-2)^2/16]$. In order to make sense of the corresponding path integrals, we require the potential to be either attenuated (meaning that $\mathfrak{K}$ is integrable at infinity) or, when $d=3$, renormalized, as introduced by Chen and Kulik in [8]. Our main results include existence and large-time asymptotics of non-negative solutions via Feynman-Kac representation. In particular, we settle for the renormalized potential in $d=3$ the problem with critical parameter $\theta = 1/16$, left open by Chen and Rosinski in [arXiv:1103.5717].Comment: 36 page