23 research outputs found
Propagating Lyapunov Functions to Prove Noise--induced Stabilization
We investigate an example of noise-induced stabilization in the plane that
was also considered in (Gawedzki, Herzog, Wehr 2010) and (Birrell, Herzog, Wehr
2011). We show that despite the deterministic system not being globally stable,
the addition of additive noise in the vertical direction leads to a unique
invariant probability measure to which the system converges at a uniform,
exponential rate. These facts are established primarily through the
construction of a Lyapunov function which we generate as the solution to a
sequence of Poisson equations. Unlike a number of other works, however, our
Lyapunov function is constructed in a systematic way, and we present a
meta-algorithm we hope will be applicable to other problems. We conclude by
proving positivity properties of the transition density by using Malliavin
calculus via some unusually explicit calculations.Comment: 41 pages, 3 figures Added picture to this version and simplified the
control theory discussion significantl
A nonparametric two-sample hypothesis testing problem for random dot product graphs
We consider the problem of testing whether two finite-dimensional random dot
product graphs have generating latent positions that are independently drawn
from the same distribution, or distributions that are related via scaling or
projection. We propose a test statistic that is a kernel-based function of the
adjacency spectral embedding for each graph. We obtain a limiting distribution
for our test statistic under the null and we show that our test procedure is
consistent across a broad range of alternatives.Comment: 24 pages, 1 figure
Discovering underlying dynamics in time series of networks
Understanding dramatic changes in the evolution of networks is central to
statistical network inference, as underscored by recent challenges of
predicting and distinguishing pandemic-induced transformations in
organizational and communication networks. We consider a joint network model in
which each node has an associated time-varying low-dimensional latent vector of
feature data, and connection probabilities are functions of these vectors.
Under mild assumptions, the time-varying evolution of the constellation of
latent vectors exhibits low-dimensional manifold structure under a suitable
notion of distance. This distance can be approximated by a measure of
separation between the observed networks themselves, and there exist consistent
Euclidean representations for underlying network structure, as characterized by
this distance, at any given time. These Euclidean representations permit the
visualization of network evolution and transform network inference questions
such as change-point and anomaly detection into a classical setting. We
illustrate our methodology with real and synthetic data, and identify change
points corresponding to massive shifts in pandemic policies in a communication
network of a large organization.Comment: 31 pages, 7 figure
Asymptotic Analysis of Microtubule-Based Transport by Multiple Identical Molecular Motors
We describe a system of stochastic differential equations (SDEs) which model
the interaction between processive molecular motors, such as kinesin and
dynein, and the biomolecular cargo they tow as part of microtubule-based
intracellular transport. We show that the classical experimental environment
fits within a parameter regime which is qualitatively distinct from conditions
one expects to find in living cells. Through an asymptotic analysis of our
system of SDEs, we develop a means for applying in vitro observations of the
nonlinear response by motors to forces induced on the attached cargo to make
analytical predictions for two parameter regimes that have thus far eluded
direct experimental observation: 1) highly viscous in vivo transport and 2)
dynamics when multiple identical motors are attached to the cargo and
microtubule