Uniqueness for diffusions degenerating at the boundary of a smooth bounded set

For continuous \gamma, g:[0,1]\to(0,\infty), consider the degenerate stochastic differential equation dX_t=[1-|X_t|^2]^{1/2}\gamma(|X_t|) dB_t-g(|X_t|)X_t dt in the closed unit ball of R^n. We introduce a new idea to show pathwise uniqueness holds when \gamma and g are Lipschitz and \frac{g(1)}{\gamma^2(1)}>\sqrt2-1. When specialized to a case studied by Swart [Stochastic Process. Appl. 98 (2002) 131-149] with \gamma=\sqrt2 and g\equiv c, this gives an improvement of his result. Our method applies to more general contexts as well. Let D be a bounded open set with C^3 boundary and suppose h:\barD\to R Lipschitz on \barD, as well as C^2 on a neighborhood of \partial D with Lipschitz second partials there. Also assume h>0 on D, h=0 on \partial D and |\nabla h|>0 on \partial D. An example of such a function is h(x)=d(x,\partial D). We give conditions which ensure pathwise uniqueness holds for dX_t=h(X_t)^{1/2}\sigma(X_t) dB_t+b(X_t) dt in \barD.Comment: Published at http://dx.doi.org/10.1214/009117904000000810 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Data Snapshot: “Trump Towns” Swung Democratic in New Hampshire Midterms

New Hampshire municipalities with fewer college-educated residents, which generally offered strong support for Donald Trump two years ago, swung toward the opposing party in the 2018 midterms

GRDC milestone updates for BACI 2.0

A report on the INTEGRATED DISEASE MANAGEMENT TOOLS TO MANAGE SUMMER CROP DISEASES IN THE NORTHERN REGION, (USQ-1907-001RTX

Life at the edge: Complexity and criticality in biological function

Why life is complex and - most importantly - what is the origin of the over abundance of complexity in nature? This is a fundamental scientific question which, paraphrasing the late Per Bak, “is screaming to be answered but seldom is even being asked”. In this article, we review recent attempts across several scales to understand the origins of complex biological problems from the perspective of critical phenomena. To illustrate the approach, three cases are discussed, namely the large scale brain dynamics, the characterization of spontaneous fluctuations of proteins, and the physiological complexity of the cell mitochondria network.Fil: Chialvo, Dante Renato. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de San Martín. Escuela de Ciencia y Tecnología; Argentin

Iterated Brownian motion in an open set

Suppose a solid has a crack filled with a gas. If the crack reaches the surrounding medium, how long does it take the gas to diffuse out of the crack? Iterated Brownian motion serves as a model for diffusion in a crack. If \tau is the first exit time of iterated Brownian motion from the solid, then P(\tau>t) can be viewed as a measurement of the amount of contaminant left in the crack at time t. We determine the large time asymptotics of P(\tau>t) for both bounded and unbounded sets. We also discuss a strange connection between iterated Brownian motion and the parabolic operator {1/8}\Delta^2-\frac{\partial}{\partial t}

Changes in New Hampshire’s republican party: evolving footprint in presidential politics, 1960-2008

This brief describes a series of dramatic changes in New Hampshire\u27s political landscape over the past four decades. Examining presidential elections from 1960 to 2008, author Dante Scala uncovers a series of significant shifts in New Hampshire\u27s political geography at the county level. He reports that historically Republican counties Grafton and Merrimack have both tilted Democratic consistently in recent decades and that New Hampshire has become less Republican overall. All of these changes have impacted not just general elections in New Hampshire, but the Republican presidential primary as well

Poverty, inequality, and welfare in a rapid-growth economy: The Chilean experience

Poverty reduction, Hunger, Inequality, Poverty, Equal opportunity, Education,