33,483 research outputs found
Analysis of White Noise Limits for Stochastic Systems with Two Fast Relaxation Times
In this paper we present a rigorous asymptotic analysis for stochastic
systems with two fast relaxation times. The mathematical model analyzed in this
paper consists of a Langevin equation for the particle motion with
time-dependent force constructed through an infinite dimensional Gaussian noise
process. We study the limit as the particle relaxation time as well as the
correlation time of the noise tend to zero and we obtain the limiting equations
under appropriate assumptions on the Gaussian noise. We show that the limiting
equation depends on the relative magnitude of the two fast time scales of the
system. In particular, we prove that in the case where the two relaxation times
converge to zero at the same rate there is a drift correction, in addition to
the limiting It\^{o} integral, which is not of Stratonovich type. If, on the
other hand, the colored noise is smooth on the scale of particle relaxation
then the drift correction is the standard Stratonovich correction. If the noise
is rough on this scale then there is no drift correction. Strong (i.e.
pathwise) techniques are used for the proof of the convergence theorems.Comment: 35 pages, 0 figures, To appear in SIAM J. MM
Standing the Test of Time: The Breadth of Majority Coalitions and the Fate of U.S. Supreme Court Precedents
Should a strategic Justice assemble a broader coalition for the majority opinion than is necessary, even if that means accommodating changes that move the opinion away from the author’s ideal holding? If the author’s objective is to durably move the law to his or her ideal holding, the conventional answer is no, because there is a cost and no corresponding benefit. We consider whether attracting a broad majority coalition can placate future courts. Controlling for the size of the coalition, we find that cases with ideologically narrow coalitions are more likely to be treated negatively by later courts. Specifically, adding either ideological breadth or a new member to the majority coalition results in an opinion that is less likely to be overruled, criticized, or questioned by a later court. Our findings contradict the conventional wisdom regarding the coalition-building strategy of a rational and strategic opinion author, establishing that the author has an incentive to go beyond the four most ideologically proximate Justices in building a majority coalition. And because of later interpreters’ negative reactions to narrow coalitions, the law ends up being less ideological than the Justices themselves
Hempstead Union Free School District and United Public Service Employees Union
In the matter of the fact-finding between the Hempstead Union Free School District, employer, and the United Public Service Employees Union, union. PERB case no. M2009-300. Before: Stuart L. Bass, fact finder
Gamma Limit for Transition Paths of Maximal Probability
Chemical reactions can be modelled via diffusion processes conditioned to
make a transition between specified molecular configurations representing the
state of the system before and after the chemical reaction. In particular the
model of Brownian dynamics - gradient flow subject to additive noise - is
frequently used. If the chemical reaction is specified to take place on a given
time interval, then the most likely path taken by the system is a minimizer of
the Onsager-Machlup functional. The Gamma limit of this functional is
determined in the case where the temperature is small and the transition time
scales as the inverse temperatur
Analysis of the 3DVAR Filter for the Partially Observed Lorenz '63 Model
The problem of effectively combining data with a mathematical model
constitutes a major challenge in applied mathematics. It is particular
challenging for high-dimensional dynamical systems where data is received
sequentially in time and the objective is to estimate the system state in an
on-line fashion; this situation arises, for example, in weather forecasting.
The sequential particle filter is then impractical and ad hoc filters, which
employ some form of Gaussian approximation, are widely used. Prototypical of
these ad hoc filters is the 3DVAR method. The goal of this paper is to analyze
the 3DVAR method, using the Lorenz '63 model to exemplify the key ideas. The
situation where the data is partial and noisy is studied, and both discrete
time and continuous time data streams are considered. The theory demonstrates
how the widely used technique of variance inflation acts to stabilize the
filter, and hence leads to asymptotic accuracy
Analysis of SPDEs Arising in Path Sampling Part I: The Gaussian Case
In many applications it is important to be able to sample paths of SDEs
conditional on observations of various kinds. This paper studies SPDEs which
solve such sampling problems. The SPDE may be viewed as an infinite dimensional
analogue of the Langevin SDE used in finite dimensional sampling. Here the
theory is developed for conditioned Gaussian processes for which the resulting
SPDE is linear. Applications include the Kalman-Bucy filter/smoother. A
companion paper studies the nonlinear case, building on the linear analysis
provided here
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