Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers' law and beyond

We prove a Kramers-type law for metastable transition times for a class of one-dimensional parabolic stochastic partial differential equations (SPDEs) with bistable potential. The expected transition time between local minima of the potential energy depends exponentially on the energy barrier to overcome, with an explicit prefactor related to functional determinants. Our results cover situations where the functional determinants vanish owing to a bifurcation, thereby rigorously proving the results of formal computations announced in [Berglund and Gentz, J. Phys. A 42:052001 (2009)]. The proofs rely on a spectral Galerkin approximation of the SPDE by a finite-dimensional system, and on a potential-theoretic approach to the computation of transition times in finite dimension.Comment: 64 pages, 4 figure

Pathwise description of dynamic pitchfork bifurcations with additive noise

The slow drift (with speed \eps) of a parameter through a pitchfork bifurcation point, known as the dynamic pitchfork bifurcation, is characterized by a significant delay of the transition from the unstable to the stable state. We describe the effect of an additive noise, of intensity $\sigma$, by giving precise estimates on the behaviour of the individual paths. We show that until time \sqrt\eps after the bifurcation, the paths are concentrated in a region of size \sigma/\eps^{1/4} around the bifurcating equilibrium. With high probability, they leave a neighbourhood of this equilibrium during a time interval [\sqrt\eps, c\sqrt{\eps\abs{\log\sigma}}], after which they are likely to stay close to the corresponding deterministic solution. We derive exponentially small upper bounds for the probability of the sets of exceptional paths, with explicit values for the exponents.Comment: 47 pages, 3 figure

CSTI Earth-to-orbit propulsion research and technology program overview

NASA supports a vigorous Earth-to-orbit (ETO) research and technology program as part of its Civil Space Technology Initiative. The purpose of this program is to provide an up-to-date technology base to support future space transportation needs for a new generation of lower cost, operationally efficient, long-lived and highly reliable ETO propulsion systems by enhancing the knowledge, understanding and design methodology applicable to advanced oxygen/hydrogen and oxygen/hydrocarbon ETO propulsion systems. Program areas of interest include analytical models, advanced component technology, instrumentation, and validation/verification testing. Organizationally, the program is divided between technology acquisition and technology verification as follows: (1) technology acquisition; and (2) technology verification

On the noise-induced passage through an unstable periodic orbit I: Two-level model

We consider the problem of stochastic exit from a planar domain, whose boundary is an unstable periodic orbit, and which contains a stable periodic orbit. This problem arises when investigating the distribution of noise-induced phase slips between synchronized oscillators, or when studying stochastic resonance far from the adiabatic limit. We introduce a simple, piecewise linear model equation, for which the distribution of first-passage times can be precisely computed. In particular, we obtain a quantitative description of the phenomenon of cycling: The distribution of first-passage times rotates around the unstable orbit, periodically in the logarithm of the noise intensity, and thus does not converge in the zero-noise limit. We compute explicitly the cycling profile, which is universal in the sense that in depends only on the product of the period of the unstable orbit with its Lyapunov exponent.Comment: 32 pages, 7 figure

Hunting French Ducks in a Noisy Environment

We consider the effect of Gaussian white noise on fast-slow dynamical systems with one fast and two slow variables, containing a folded-node singularity. In the absence of noise, these systems are known to display mixed-mode oscillations, consisting of alternating large- and small-amplitude oscillations. We quantify the effect of noise and obtain critical noise intensities above which the small-amplitude oscillations become hidden by fluctuations. Furthermore we prove that the noise can cause sample paths to jump away from so-called canard solutions with high probability before deterministic orbits do. This early-jump mechanism can drastically influence the local and global dynamics of the system by changing the mixed-mode patterns.Comment: 60 pages, 9 figure

From random Poincar\'e maps to stochastic mixed-mode-oscillation patterns

We quantify the effect of Gaussian white noise on fast--slow dynamical systems with one fast and two slow variables, which display mixed-mode oscillations owing to the presence of a folded-node singularity. The stochastic system can be described by a continuous-space, discrete-time Markov chain, recording the returns of sample paths to a Poincar\'e section. We provide estimates on the kernel of this Markov chain, depending on the system parameters and the noise intensity. These results yield predictions on the observed random mixed-mode oscillation patterns. Our analysis shows that there is an intricate interplay between the number of small-amplitude oscillations and the global return mechanism. In combination with a local saturation phenomenon near the folded node, this interplay can modify the number of small-amplitude oscillations after a large-amplitude oscillation. Finally, sufficient conditions are derived which determine when the noise increases the number of small-amplitude oscillations and when it decreases this number.Comment: 56 pages, 14 figures; revised versio

Promising State Policies for Personalized Learning

This report is a valuable resource for state policymakers—whether they are seeking to create conditions in state policy to support personalized learning, moving forward with initiatives to develop personalized learning pilot programs, hosting task forces to explore policy issues and needs, or taking a comprehensive policy approach for supporting advanced personalized learning models.Personalized learning is where instruction is tailored to each student's strengths, needs, and interests—including enabling student voice and choice in what, how, when, and where they learn—to provide flexibility and supports to ensure mastery of the highest standards possible