Extinction for two parabolic stochastic PDE's on the lattice

It is well known that, starting with finite mass, the super-Brownian motion dies out in finite time. The goal of this article is to show that with some additional work, one can prove finite time die-out for two types of systems of stochastic differential equations on the lattice Z^d. Our first system involves the heat equation on the lattice Z^d, with a nonlinear noise term u(t,x)^gamma dB_x(t), with 1/2 <= gamma < 1. The B_x are independent Brownian motions. When gamma = 1/2, the measure which puts mass u(t,x) at x is a super-random walk and it is well-known that the process becomes extinct in finite time a.s. Finite-time extinction is known to be a.s. false if gamma = 1. For 1/2 < gamma < 1, we show finite-time die-out by breaking up the solution into pieces, and showing that each piece dies in finite time. Our second example involves the mutually catalytic branching system of stochastic differential equations on Z^d, which was first studied by Dawson and Perkins. Roughly speaking, this process consists of 2 superprocesses with the continuous time simple random walk as the underlying spatial motion. Furthermore, each process stimulates branching and dying in the other process. By using a somewhat different argument, we show that, depending on the initial conditions, finite time extinction of one type may occur with probability 0, or with probability arbitrarily close to 1

Entrepreneurship and Growth

In the year 2000 at a meeting in Lisbon, leaders of the European Union (EU) articulated a set of goals for the Union, which have come to be called the Lisbon Strategy or Lisbon Agenda. The agenda had three main goals: to promote growth through innovation, to create a learning economy, and to bring about social and environmental renewal. Exactly what the last goal implies is not clear, at least to me, but the intent and substance behind the first two certainly is. Research spending was to rise across the EU, university enrollments would rise with them, and a more friendly environment for innovation would be created as markets continued to be liberalized and integrated. The EU leaders meeting in Lisbon set the year 2010 as their goal for fulfilling this agenda. The year 2010 has come and gone. Today, growth rates in Europe are even lower than they were in 2000. Research and university budgets have been cut – sometimes drastically – across the EU. These developments are, of course, largely a response to the recent financial crisis and its impact on state finances. But the crisis would not have been nearly as severe as it has been, if EU countries had been well on their way to fulfilling the goals of the Lisbon Agenda when the crisis hit. The EU’s failure to come anywhere near meeting the goals set out in the year 2000 stems, I shall argue, to underlying structural factors and ideological perspectives, which constitute major obstacles to the kind of knowledge-based, innovative society that the EU leaders dreamed of in Lisbon more than a decade ago. This paper attempts to identify what these obstacles are.Entrepreneurship; Economic Growth; Human Capital; European Union

NASA 50 amp hour nickel cadmium battery waste heat determination

A process for determining the waste heat generated in a 50-ampere-hour, nickel cadmium battery as a function of the discharge rate is described and results are discussed. The technique involved is essentially calibration of the battery as a heat transfer rate calorimeter. The tests are run at three different levels of battery activity, one at 40-watts of waste heat generated, one at 60, and one at 100. Battery inefficiency ranges from 14 to 18 percent at discharge rates of 284 to 588 watts, respectively and top-of-cell temperatures of 20 C

Multiple points of the Brownian sheet in critical dimensions

It is well known that an $N$-parameter $d$-dimensional Brownian sheet has no $k$-multiple points when $(k-1)d>2kN$, and does have such points when $(k-1)d<2kN$. We complete the study of the existence of $k$-multiple points by showing that in the critical cases where $(k-1)d=2kN$, there are a.s. no $k$-multiple points.Comment: Published at http://dx.doi.org/10.1214/14-AOP912 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Some non-linear s.p.d.e.'s that are second order in time

We extend Walsh's theory of martingale measures in order to deal with hyperbolic stochastic partial differential equations that are second order in time, such as the wave equation and the beam equation, and driven by spatially homogeneous Gaussian noise. For such equations, the fundamental solution can be a distribution in the sense of Schwartz, which appears as an integrand in the reformulation of the s.p.d.e. as a stochastic integral equation. Our approach provides an alternative to the Hilbert space integrals of Hilbert-Schmidt operators. We give several examples, including the beam equation and the wave equation, with nonlinear multiplicative noise terms

Hitting properties of parabolic s.p.d.e.'s with reflection

We study the hitting properties of the solutions $u$ of a class of parabolic stochastic partial differential equations with singular drifts that prevent $u$ from becoming negative. The drifts can be a reflecting term or a nonlinearity $cu^{-3}$, with $c>0$. We prove that almost surely, for all time $t>0$, the solution $u_t$ hits the level 0 only at a finite number of space points, which depends explicitly on $c$. In particular, this number of hits never exceeds 4 and if $c>15/8$, then level 0 is not hit.Comment: Published at http://dx.doi.org/10.1214/009117905000000792 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Studying pion effects on the chiral phase transition

We investigate the chiral phase transition at finite temperatures and zero chemical potential with Dyson-Schwinger equations. Our truncation for the quark-gluon interaction includes mesonic degrees of freedom, which allows us to study the impact of the pions on the nature of the phase transition. Within the present scheme we find a five percent change of the critical temperature due to the pion backreaction whereas the mean field character of the transition is not changed.Comment: 2 pages, 2 figures, talk given by J.A.M. at the 30th International School of Nuclear Physics, Erice, Sicily from 16 - 24 September 200