Strong renewal theorems and local large deviations for multivariate random walks and renewals

We study a random walk $\mathbf{S}_n$ on $\mathbb{Z}^d$ ($d\geq 1$), in the domain of attraction of an operator-stable distribution with index $\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_d) \in (0,2]^d$: in particular, we allow the scalings to be different along the different coordinates. We prove a strong renewal theorem, $i.e.$ a sharp asymptotic of the Green function $G(\mathbf{0},\mathbf{x})$ as $\|\mathbf{x}\|\to +\infty$, along the "favorite direction or scaling": (i) if $\sum_{i=1}^d \alpha_i^{-1} < 2$ (reminiscent of Garsia-Lamperti's condition when $d=1$ [Comm. Math. Helv. $\mathbf{37}$, 1962]); (ii) if a certain $local$ condition holds (reminiscent of Doney's condition [Probab. Theory Relat. Fields $\mathbf{107}$, 1997] when $d=1$). We also provide uniform bounds on the Green function $G(\mathbf{0},\mathbf{x})$, sharpening estimates when $\mathbf{x}$ is away from this favorite direction or scaling. These results improve significantly the existing literature, which was mostly concerned with the case $\alpha_i\equiv \alpha$, in the favorite scaling, and has even left aside the case $\alpha\in[1,2)$ with non-zero mean. Most of our estimates rely on new general (multivariate) local large deviations results, that were missing in the literature and that are of interest on their own.Comment: 46 pages, comments are welcom

THE SETTING OF MILTON\u27S EIGHTEENTH SONNET

...But this sonnet immortalizes an aspect, perhaps only an insignificant aspect of that age; yet its greatness lies less in its subject, which is hardly a universal one, which is, in fact, little more than the angered outburst of a passing moment, than in the splendour and rolling grandeur of its words. Who can show a finer example of emotion recollected in tranquility

Pinning model in random correlated environment: appearance of an infinite disorder regime

We study the influence of a correlated disorder on the localization phase transition in the pinning model. When correlations are strong enough, a strong disorder regime arises: large and frequent attractive regions appear in the environment. We present here a pinning model in random binary ({-1,1}-valued) environment. Defining strong disorder via the requirement that the probability of the occurrence of a large attractive region is sub-exponential in its size, we prove that it coincides with the fact that the critical point is equal to its minimal possible value. We also stress that in the strong disorder regime, the phase transition is smoother than in the homogeneous case, whatever the critical exponent of the homogeneous model is: disorder is therefore always relevant. We illustrate these results with the example of an environment based on the sign of a Gaussian correlated sequence, in which we show that the phase transition is of infinite order in presence of strong disorder. Our results contrast with results known in the literature, in particular in the case of an IID disorder, where the question of the influence of disorder on the critical properties is answered via the so-called Harris criterion, and where a conventional relevance/irrelevance picture holds.Comment: 27 pages, some corrections made in v

Singular measure traveling waves in an epidemiological model with continuous phenotypes

We consider the reaction-diffusion equation \begin{equation*} u_t=u_{xx}+\mu\left(\int_\Omega M(y,z)u(t,x,z)dz-u\right) + u\left(a(y)-\int_\Omega K(y,z) u(t,x,z)dz\right) , \end{equation*} where $u=u(t,x,y)$ stands for the density of a theoretical population with a spatial ($x\in\mathbb R$) and phenotypic ($y\in\Omega\subset \mathbb R^n$) structure, $M(y,z)$ is a mutation kernel acting on the phenotypic space, $a(y)$ is a fitness function and $K(y,z)$ is a competition kernel. Using a vanishing viscosity method, we construct measure-valued traveling waves for this equation, and present particular cases where singular traveling waves do exist. We determine that the speed of the constructed traveling waves is the expected spreading speed $c^*:=2\sqrt{-\lambda_1}$, where $\lambda_1$ is the principal eigenvalue of the linearized equation. As far as we know, this is the first construction of a measure-valued traveling wave for a reaction-diffusion equation

Mechanistic target of rapamycin (mTOR) activation during ruminant mammary development and function : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Animal Science at Massey University, Palmerston North, New Zealand

This thesis examines the abundance of total and activated mechanistic target of rapamycin (mTOR) pathway components in the developing and functional ruminant mammary gland. mTOR pathway activation is stimulated by a wide range of intra- and extracellular signals, such as amino acids (AA) and hormones, making the mTOR pathway a potential candidate for the development of intervention strategies designed to increase ruminant lactation potential. Tissues from two trials shown to improve lactation potential; dam-fetal nutrition and exogenous growth hormone (GH) administration during lactation, were used to measure changes in total and activated mTOR pathway protein abundance. Results show mammary glands of d 140 fetal lambs carried by maintenance fed dams and dairy cows administered exogenous GH, had increased abundance of total and activated mTOR and mitogen activated protein kinase (MAPK) pathway proteins. Increased abundance was associated with changes in biochemical indices. In the GH study MAPK pathway activation was stimulated by IGF-1 signaling whilst mTOR pathway activation was proposed to be mediated by AA signalling. Data from the GH study shows, L-arginine a known activator of the mTOR pathway, was the only AA reduced in both plasma and the lactating gland. Upstream factors were not identified for the phenotype observed in the dam-fetal nutrition study, but similar mechanisms were proposed. To elucidate the potential regulation of mTOR pathway activation by L-arginine and examine the effect on milk production, in vitro bovine cell culture models were evaluated. Results show that none of the models evaluated produced a lactating phenotype – a pre-requisite to accurately study the lactating gland in vitro. Finally, this thesis shows L-arginine administration from d 100 to d 140 of pregnancy, in twin bearing ewes had no effect on mTOR protein abundance or activation. However, administration from d 100 to parturition improved maternal gland health. In summary, this thesis associates improved lactation potential with increased total and activated mTOR pathway protein abundance, and the administration of L-arginine during late gestation with improved gland health. These findings provide fundamental knowledge that may lead to the development of novel technologies to increase ruminant gland performance and health

Testing Lorentz Symmetry with Gravity

In this talk, results from the gravitational sector of the Standard-Model Extension (SME) are discussed. The weak-field phenomenology of the resulting modified gravitational field equations is explored. The application of the results to a variety of modern gravity experiments, including lunar laser ranging, Gravity Probe B, binary pulsars, and Earth-laboratory tests, shows promising sensitivity to gravitational coefficients for Lorentz violation in the SME.Comment: 7 pages, presented at the Fourth Meeting on CPT and Lorentz Symmetry, Bloomington, Indiana, August 200

Optimal Testing for Planted Satisfiability Problems

We study the problem of detecting planted solutions in a random satisfiability formula. Adopting the formalism of hypothesis testing in statistical analysis, we describe the minimax optimal rates of detection. Our analysis relies on the study of the number of satisfying assignments, for which we prove new results. We also address algorithmic issues, and give a computationally efficient test with optimal statistical performance. This result is compared to an average-case hypothesis on the hardness of refuting satisfiability of random formulas