%% Papers Published
@article{fds361709,
Author = {Earle, G and Mattingly, JC},
Title = {Convergence of stratified MCMC sampling of non-reversible
dynamics},
Journal = {Stochastics and Partial Differential Equations: Analysis and
Computations},
Year = {2024},
Month = {January},
url = {http://dx.doi.org/10.1007/s40072-024-00325-0},
Abstract = {We present a form of stratified MCMC algorithm built with
non-reversible stochastic dynamics in mind. It can also be
viewed as a generalization of the exact milestoning method
or form of NEUS. We prove the convergence of the method
under certain assumptions, with expressions for the
convergence rate in terms of the process’s behavior within
each stratum and large-scale behavior between strata. We
show that the algorithm has a unique fixed point which
corresponds to the invariant measure of the process without
stratification. We will show how the convergence speeds of
two versions of the algorithm, one with an extra eigenvalue
problem step and one without, related to the mixing rate of
a discrete process on the strata, and the mixing probability
of the process being sampled within each stratum. The
eigenvalue problem version also relates to local and global
perturbation results of discrete Markov chains, such as
those given by Van Koten, Weare et. al.},
Doi = {10.1007/s40072-024-00325-0},
Key = {fds361709}
}
@article{fds371623,
Author = {Autry, E and Carter, D and Herschlag, GJ and Hunter, Z and Mattingly,
JC},
Title = {METROPOLIZED FOREST RECOMBINATION FOR MONTE CARLO SAMPLING
OF GRAPH PARTITIONS},
Journal = {SIAM Journal on Applied Mathematics},
Volume = {83},
Number = {4},
Pages = {1366-1391},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2023},
Month = {August},
url = {http://dx.doi.org/10.1137/21M1418010},
Abstract = {We develop a new Markov chain on graph partitions that makes
relatively global moves yet is computationally feasible to
be used as the proposal in the Metropolis-Hastings method.
Our resulting algorithm is able to sample from a specified
measure on partitions or spanning forests. Being able to
sample from a specified measure is a requirement of what we
consider as the gold standard in quantifying the extent to
which a particular map is a gerrymander. Our proposal chain
modifies the recently developed method called recombination
(ReCom), which draws spanning trees on joined partitions and
then randomly cuts them to repartition. We improve the
computational efficiency by augmenting the statespace from
partitions to spanning forests. The extra information
accelerates the computation of the forward and backward
proposal probabilities which are required for the
Metropolis-Hastings algorithm. We demonstrate this method by
sampling redistricting plans on several measures of interest
and find promising convergence results on several key
observables of interest. We also explore some limitations in
the measures that are efficient to sample from and
investigate the feasibility of using parallel tempering to
extend this space of measures.},
Doi = {10.1137/21M1418010},
Key = {fds371623}
}
@article{fds361536,
Author = {Herzog, DP and Mattingly, JC and Nguyen, HD},
Title = {Gibbsian dynamics and the generalized Langevin
equation},
Journal = {Electronic Journal of Probability},
Volume = {28},
Year = {2023},
Month = {January},
url = {http://dx.doi.org/10.1214/23-EJP904},
Abstract = {We study the statistically invariant structures of the
nonlinear generalized Langevin equation (GLE) with a
power-law memory kernel. For a broad class of memory
kernels, including those in the subdiffusive regime, we
construct solutions of the GLE using a Gibbsian framework,
which does not rely on existing Markovian approximations.
Moreover, we provide conditions on the decay of the memory
to ensure uniqueness of statistically steady states,
generalizing previous known results for the GLE under
particular kernels as a sum of exponentials.},
Doi = {10.1214/23-EJP904},
Key = {fds361536}
}
@article{fds367803,
Author = {Zhao, Z and Hettle, C and Gupta, S and Mattingly, JC and Randall, D and Herschlag, GJ},
Title = {Mathematically Quantifying Non-responsiveness of the 2021
Georgia Congressional Districting Plan},
Journal = {ACM International Conference Proceeding Series},
Year = {2022},
Month = {October},
ISBN = {9781450394772},
url = {http://dx.doi.org/10.1145/3551624.3555300},
Abstract = {To audit political district maps for partisan
gerrymandering, one may determine a baseline for the
expected distribution of partisan outcomes by sampling an
ensemble of maps. One approach to sampling is to use
redistricting policy as a guide to precisely codify
preferences between maps. Such preferences give rise to a
probability distribution on the space of redistricting
plans, and Metropolis-Hastings methods allow one to sample
ensembles of maps from the specified distribution. Although
these approaches have nice theoretical properties and have
successfully detected gerrymandering in legal settings,
sampling from commonly-used policy-driven distributions is
often computationally difficult. As of yet, there is no
algorithm that can be used off-the-shelf for checking maps
under generic redistricting criteria. In this work, we
mitigate the computational challenges in a
Metropolized-sampling technique through a parallel tempering
method combined with ReCom[11] and, for the first time,
validate that such techniques are effective on these
problems at the scale of statewide precinct graphs for more
policy informed measures. We develop these improvements
through the first case study of district plans in Georgia.
Our analysis projects that any election in Georgia will
reliably elect 9 Republicans and 5 Democrats under the
enacted plan. This result is largely fixed even as public
opinion shifts toward either party and the partisan outcome
of the enacted plan does not respond to the will of the
people. Only 0.12% of the ∼160K plans in our ensemble were
similarly non-responsive.},
Doi = {10.1145/3551624.3555300},
Key = {fds367803}
}
@article{fds361537,
Author = {Mattingly, JC and Romito, M and Su, L},
Title = {The Gaussian structure of the singular stochastic Burgers
equation},
Journal = {Forum of Mathematics, Sigma},
Volume = {10},
Publisher = {Cambridge University Press (CUP)},
Year = {2022},
Month = {September},
url = {http://dx.doi.org/10.1017/fms.2022.64},
Abstract = {We consider the stochastically forced Burgers equation with
an emphasis on spatially rough driving noise. We show that
the law of the process at a fixed time t, conditioned on no
explosions, is absolutely continuous with respect to the
stochastic heat equation obtained by removing the
nonlinearity from the equation. This establishes a form of
ellipticity in this infinite-dimensional setting. The
results follow from a recasting of the Girsanov Theorem to
handle less spatially regular solutions while only proving
absolute continuity at a fixed time and not on path-space.
The results are proven by decomposing the solution into the
sum of auxiliary processes, which are then shown to be
absolutely continuous in law to a stochastic heat equation.
The number of levels in this decomposition diverges to
infinite as we move to the stochastically forced Burgers
equation associated to the KPZ equation, which we conjecture
is just beyond the validity of our results (and certainly
the current proof). The analysis provides insights into the
structure of the solution as we approach the regularity of
KPZ. A number of techniques from singular SPDEs are
employed, as we are beyond the regime of classical solutions
for much of the paper.},
Doi = {10.1017/fms.2022.64},
Key = {fds361537}
}
@article{fds358291,
Author = {Li, L and Lu, J and Mattingly, JC and Wang, L},
Title = {Numerical Methods For Stochastic Differential Equations
Based On Gaussian Mixture},
Journal = {Communications in Mathematical Sciences},
Volume = {19},
Number = {6},
Pages = {1549-1577},
Publisher = {International Press of Boston},
Year = {2021},
Month = {January},
url = {http://dx.doi.org/10.4310/CMS.2021.v19.n6.a5},
Abstract = {We develop in this work a numerical method for stochastic
differential equations (SDEs) with weak second-order
accuracy based on Gaussian mixture. Unlike conventional
higher order schemes for SDEs based on Itô-Taylor expansion
and iterated Itô integrals, the scheme we propose
approximates the probability measure μ(Xn+1|Xn =xn) using a
mixture of Gaussians. The solution at the next time step
Xn+1 is drawn from the Gaussian mixture with complexity
linear in dimension d. This provides a new strategy to
construct efflcient high weak order numerical schemes for
SDEs},
Doi = {10.4310/CMS.2021.v19.n6.a5},
Key = {fds358291}
}
@article{fds360556,
Author = {Autry, EA and Carter, D and Herschlag, GJ and Hunter, Z and Mattingly,
JC},
Title = {METROPOLIZED MULTISCALE FOREST RECOMBINATION for
REDISTRICTING},
Journal = {Multiscale Modeling and Simulation},
Volume = {19},
Number = {4},
Pages = {1885-1914},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2021},
Month = {January},
url = {http://dx.doi.org/10.1137/21M1406854},
Abstract = {We develop a Metropolized Multiscale Forest Recombination
Markov Chain on redistricting plans. The chain is designed
to be usable as the proposal in a Markov Chain Monte Carlo
(MCMC) algorithm. Sampling the space of plans amounts to
dividing a graph into a partition with a specified number of
elements each of which corresponds to a different district
according to a specified probability measure. The districts
satisfy a collection of hard constraints, and the
probability measure may be weighted with regard to a number
of other criteria. The multiscale algorithm is similar to
our previously developed Metropolized Forest Recombination
proposal; however, this algorithm provides improved scaling
properties and may also be used to preserve nested
communities of interest such as counties and precincts. Both
works use a proposal which extends the ReCom algorithm [D.
DeFord, M. Duchin, and J. Solomon, Harvard Data Sci. Rev.,
(2021)] which leveraged spanning trees to merge and split
districts. In this work, we extend the state space so that
each district is defined by a hierarchy of trees. In this
sense, the proposal step in both algorithms can be seen as a
“Forest ReCom.” The collection of plans sampled by the
MCMC algorithm can serve as a baseline against which a
particular plan of interest is compared. If a given plan has
different racial or partisan qualities than what is typical
of the collection of plans, the given plan may have been
gerrymandered and is labeled as an outlier. Metropolizing
relative to a policy driven probability measure removes the
possibility of algorithmically inserted biases.},
Doi = {10.1137/21M1406854},
Key = {fds360556}
}
@article{fds359780,
Author = {Bakhtin, Y and Hurth, T and Lawley, SD and Mattingly,
JC},
Title = {Singularities of invariant densities for random switching
between two linear ODEs in 2D},
Journal = {SIAM Journal on Applied Dynamical Systems},
Volume = {20},
Number = {4},
Pages = {1917-1958},
Year = {2021},
Month = {January},
url = {http://dx.doi.org/10.1137/20M1364345},
Abstract = {We consider a planar dynamical system generated by two
stable linear vector fields with distinct fixed points and
random switching between them. We characterize singularities
of the invariant density in terms of the switching rates and
contraction rates. We prove boundedness away from those
singularities. We also discuss some motivating biological
examples.},
Doi = {10.1137/20M1364345},
Key = {fds359780}
}
@article{fds356175,
Author = {Gao, Y and Kirkpatrick, K and Marzuola, J and Mattingly, J and Newhall,
KA},
Title = {LIMITING BEHAVIORS OF HIGH DIMENSIONAL STOCHASTIC SPIN
ENSEMBLES*},
Journal = {Communications in Mathematical Sciences},
Volume = {19},
Number = {2},
Pages = {453-494},
Year = {2021},
Month = {January},
url = {http://dx.doi.org/10.4310/CMS.2021.v19.n2.a7},
Abstract = {Lattice spin models in statistical physics are used to
understand magnetism. Their Hamiltonians are a discrete form
of a version of a Dirichlet energy, signifying a
relationship to the harmonic map heat flow equation. The
Gibbs distribution, defined with this Hamiltonian, is used
in the Metropolis-Hastings (M-H) algorithm to generate
dynamics tending towards an equilibrium state. In the
limiting situation when the inverse temperature is large, we
establish the relationship between the discrete M-H dynamics
and the continuous harmonic map heat flow associated with
the Hamiltonian. We show the convergence of the M-H dynamics
to the harmonic map heat flow equation in two steps: First,
with fixed lattice size and proper choice of proposal size
in one M-H step, the M-H dynamics acts as gradient descent
and will be shown to converge to a system of Langevin sto
chastic differential equations (SDE). Second, with proper
scaling of the inverse temperature in the Gibbs distribution
and taking the lattice size to infinity, it will be shown
that this SDE system converges to the deterministic harmonic
map heat flow equation. Our results are not unexpected, but
show remarkable connections between the M-H steps and the
SDE Stratonovich formulation, as well as reveal tra
jectory-wise out of equilibrium dynamics to be related to a
canonical PDE system with geometric constraints.},
Doi = {10.4310/CMS.2021.v19.n2.a7},
Key = {fds356175}
}
@article{fds353323,
Author = {Gao, Y and Marzuola, JL and Mattingly, JC and Newhall,
KA},
Title = {Nonlocal stochastic-partial-differential-equation limits of
spatially correlated noise-driven spin systems derived to
sample a canonical distribution},
Journal = {Physical Review E},
Volume = {102},
Number = {5},
Pages = {052112},
Year = {2020},
Month = {November},
url = {http://dx.doi.org/10.1103/PhysRevE.102.052112},
Abstract = {For a noisy spin system, we derive a nonlocal stochastic
version of the overdamped Landau-Lipshitz equation designed
to respect the underlying Hamiltonian structure and sample
the canonical or Gibbs distribution while being driven by
spatially correlated (colored) noise that regularizes the
dynamics, making this Stochastic partial differential
equation mathematically well-posed. We begin from a
microscopic discrete-time model motivated by the
Metropolis-Hastings algorithm for a finite number of spins
with periodic boundary conditions whose values are
distributed on the unit sphere. We thus propose a future
state of the system by adding to each spin colored noise
projected onto the sphere, and then accept this proposed
state with probability given by the ratio of the canonical
distribution at the proposed and current states. For
uncorrelated (white) noise this process is guaranteed to
sample the canonical distribution. We demonstrate that for
colored noise, the method used to project the noise onto the
sphere and conserve the magnitude of the spins impacts the
equilibrium distribution of the system, as coloring
projected noise is not equivalent to projecting colored
noise. In a specific scenario we show this break in symmetry
vanishes with vanishing proposal size; the resulting
continuous-time system of Stochastic differential equations
samples the canonical distribution and preserves the
magnitude of the spins while being driven by colored noise.
Taking the continuum limit of infinitely many spins we
arrive at the aforementioned version of the overdamped
Landau-Lipshitz equation. Numerical simulations are included
to verify convergence properties and demonstrate the
dynamics.},
Doi = {10.1103/PhysRevE.102.052112},
Key = {fds353323}
}
@article{fds361447,
Author = {Leimbach, M and Mattingly, JC and Scheutzow, M},
Title = {Noise-induced strong stabilization},
Year = {2020},
Month = {September},
Abstract = {We consider a 2-dimensional stochastic differential equation
in polar coordinates depending on several parameters. We
show that if these parameters belong to a specific regime
then the deterministic system explodes in finite time, but
the random dynamical system corresponding to the stochastic
equation is not only strongly complete but even admits a
random attractor.},
Key = {fds361447}
}
@article{fds362598,
Author = {Herschlag, G and Mattingly, JC and Sachs, M and Wyse,
E},
Title = {Non-reversible Markov chain Monte Carlo for sampling of
districting maps},
Year = {2020},
Month = {August},
Abstract = {Evaluating the degree of partisan districting
(Gerrymandering) in a statistical framework typically
requires an ensemble of districting plans which are drawn
from a prescribed probability distribution that adheres to a
realistic and non-partisan criteria. In this article we
introduce novel non-reversible Markov chain Monte-Carlo
(MCMC) methods for the sampling of such districting plans
which have improved mixing properties in comparison to
previously used (reversible) MCMC algorithms. In doing so we
extend the current framework for construction of
non-reversible Markov chains on discrete sampling spaces by
considering a generalization of skew detailed balance. We
provide a detailed description of the proposed algorithms
and evaluate their performance in numerical
experiments.},
Key = {fds362598}
}
@article{fds362599,
Author = {Autry, EA and Carter, D and Herschlag, G and Hunter, Z and Mattingly,
JC},
Title = {Multi-Scale Merge-Split Markov Chain Monte Carlo for
Redistricting},
Year = {2020},
Month = {August},
Abstract = {We develop a Multi-Scale Merge-Split Markov chain on
redistricting plans. The chain is designed to be usable as
the proposal in a Markov Chain Monte Carlo (MCMC) algorithm.
Sampling the space of plans amounts to dividing a graph into
a partition with a specified number of elements which each
correspond to a different district. The districts satisfy a
collection of hard constraints and the measure may be
weighted with regard to a number of other criteria. The
multi-scale algorithm is similar to our previously developed
Merge-Split proposal, however, this algorithm provides
improved scaling properties and may also be used to preserve
nested communities of interest such as counties and
precincts. Both works use a proposal which extends the ReCom
algorithm which leveraged spanning trees merge and split
districts. In this work we extend the state space so that
each district is defined by a hierarchy of trees. In this
sense, the proposal step in both algorithms can be seen as a
"Forest ReCom." We also expand the state space to include
edges that link specified districts, which further improves
the computational efficiency of our algorithm. The
collection of plans sampled by the MCMC algorithm can serve
as a baseline against which a particular plan of interest is
compared. If a given plan has different racial or partisan
qualities than what is typical of the collection of plans,
the given plan may have been gerrymandered and is labeled as
an outlier.},
Key = {fds362599}
}
@article{fds348481,
Author = {Lu, Y and Mattingly, JC},
Title = {Geometric ergodicity of Langevin dynamics with Coulomb
interactions},
Journal = {Nonlinearity},
Volume = {33},
Number = {2},
Pages = {675-699},
Publisher = {IOP Publishing},
Year = {2020},
Month = {January},
url = {http://dx.doi.org/10.1088/1361-6544/ab514a},
Abstract = {This paper is concerned with the long time behavior of
Langevin dynamics of Coulomb gases in with, that is a second
order system of Brownian particles driven by an external
force and a pairwise repulsive Coulomb force. We prove that
the system converges exponentially to the unique
Boltzmann-Gibbs invariant measure under a weighted total
variation distance. The proof relies on a novel construction
of Lyapunov function for the Coulomb system.},
Doi = {10.1088/1361-6544/ab514a},
Key = {fds348481}
}
@article{fds349660,
Author = {Carter, D and Hunter, Z and Teague, D and Herschlag, G and Mattingly,
J},
Title = {Optimal Legislative County Clustering in North
Carolina},
Journal = {Statistics and Public Policy},
Volume = {7},
Number = {1},
Pages = {19-29},
Year = {2020},
Month = {January},
url = {http://dx.doi.org/10.1080/2330443X.2020.1748552},
Abstract = {North Carolina’s constitution requires that state
legislative districts should not split counties. However,
counties must be split to comply with the “one person, one
vote” mandate of the U.S. Supreme Court. Given that
counties must be split, the North Carolina legislature and
the courts have provided guidelines that seek to reduce
counties split across districts while also complying with
the “one person, one vote” criterion. Under these
guidelines, the counties are separated into clusters; each
cluster contains a specified number of districts and that
are drawn independent from other clusters. The primary goal
of this work is to develop, present, and publicly release an
algorithm to optimally cluster counties according to the
guidelines set by the court in 2015. We use this tool to
investigate the optimality and uniqueness of the enacted
clusters under the 2017 redistricting process. We verify
that the enacted clusters are optimal, but find other
optimal choices. We emphasize that the tool we provide lists
all possible optimal county clusterings. We also explore the
stability of clustering under changing statewide populations
and project what the county clusters may look like in the
next redistricting cycle beginning in 2020/2021.
Supplementary materials for this article are available
online.},
Doi = {10.1080/2330443X.2020.1748552},
Key = {fds349660}
}
@article{fds352186,
Author = {Herschlag, G and Kang, HS and Luo, J and Graves, CV and Bangia, S and Ravier, R and Mattingly, JC},
Title = {Quantifying Gerrymandering in North Carolina},
Journal = {Statistics and Public Policy},
Volume = {7},
Number = {1},
Pages = {30-38},
Publisher = {Informa UK Limited},
Year = {2020},
Month = {January},
url = {http://dx.doi.org/10.1080/2330443X.2020.1796400},
Abstract = {By comparing a specific redistricting plan to an ensemble of
plans, we evaluate whether the plan translates individual
votes to election outcomes in an unbiased fashion.
Explicitly, we evaluate if a given redistricting plan
exhibits extreme statistical properties compared to an
ensemble of nonpartisan plans satisfying all legal criteria.
Thus, we capture how unbiased redistricting plans interpret
individual votes via a state’s geo-political landscape. We
generate the ensemble of plans through a Markov chain Monte
Carlo algorithm coupled with simulated annealing based on a
reference distribution that does not include partisan
criteria. Using the ensemble and historical voting data, we
create a null hypothesis for various election results, free
from partisanship, accounting for the state’s
geo-politics. We showcase our methods on two recent
congressional districting plans of NC, along with a plan
drawn by a bipartisan panel of retired judges. We find the
enacted plans are extreme outliers whereas the bipartisan
judges’ plan does not give rise to extreme partisan
outcomes. Equally important, we illuminate anomalous
structures in the plans of interest by developing graphical
representations which help identify and understand instances
of cracking and packing associated with gerrymandering.
These methods were successfully used in recent court cases.
Supplementary materials for this article are available
online.},
Doi = {10.1080/2330443X.2020.1796400},
Key = {fds352186}
}
@article{fds352640,
Author = {Chikina, M and Frieze, A and Mattingly, JC and Pegden,
W},
Title = {Separating Effect From Significance in Markov Chain
Tests},
Journal = {Statistics and Public Policy},
Volume = {7},
Number = {1},
Pages = {101-114},
Year = {2020},
Month = {January},
url = {http://dx.doi.org/10.1080/2330443X.2020.1806763},
Abstract = {We give qualitative and quantitative improvements to
theorems which enable significance testing in Markov chains,
with a particular eye toward the goal of enabling strong,
interpretable, and statistically rigorous claims of
political gerrymandering. Our results can be used to
demonstrate at a desired significance level that a given
Markov chain state (e.g., a districting) is extremely
unusual (rather than just atypical) with respect to the
fragility of its characteristics in the chain. We also
provide theorems specialized to leverage quantitative
improvements when there is a product structure in the
underlying probability space, as can occur due to
geographical constraints on districtings.},
Doi = {10.1080/2330443X.2020.1806763},
Key = {fds352640}
}
@article{fds352949,
Author = {AGAZZI, A and MATTINGLY, JC},
Title = {SEEMINGLY STABLE CHEMICAL KINETICS CAN BE STABLE, MARGINALLY
STABLE, OR UNSTABLE},
Journal = {Communications in Mathematical Sciences},
Volume = {18},
Number = {6},
Pages = {1605-1642},
Publisher = {International Press of Boston},
Year = {2020},
Month = {January},
url = {http://dx.doi.org/10.4310/CMS.2020.v18.n6.a5},
Abstract = {. We present three examples of chemical reaction networks
whose ordinary differential equation scaling limits are
almost identical and in all cases stable. Nevertheless, the
Markov jump processes associated to these reaction networks
display the full range of behaviors: one is stable (positive
recurrent), one is unstable (transient) and one is
marginally stable (null recurrent). We study these
differences and characterize the invariant measures by
Lyapunov function techniques. In particular, we design a
natural set of such functions which scale homogeneously to
infinity, taking advantage of the same scaling behavior of
the reaction rates.},
Doi = {10.4310/CMS.2020.v18.n6.a5},
Key = {fds352949}
}
@article{fds346157,
Author = {Herzog, DP and Mattingly, JC},
Title = {Ergodicity and Lyapunov Functions for Langevin Dynamics with
Singular Potentials},
Journal = {COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS},
Volume = {72},
Number = {10},
Pages = {2231-2255},
Publisher = {WILEY},
Year = {2019},
Month = {October},
url = {http://dx.doi.org/10.1002/cpa.21862},
Doi = {10.1002/cpa.21862},
Key = {fds346157}
}
@article{fds347400,
Author = {Chin, A and Herschlag, G and Mattingly, J},
Title = {The Signature of Gerrymandering in Rucho v. Common
Cause},
Journal = {South Carolina Law Review},
Volume = {70},
Year = {2019},
Key = {fds347400}
}
@article{fds361448,
Author = {Wang, C and Mattingly, J and Lu, YM},
Title = {Scaling Limit: Exact and Tractable Analysis of Online
Learning Algorithms with Applications to Regularized
Regression and PCA},
Year = {2017},
Month = {December},
Abstract = {We present a framework for analyzing the exact dynamics of a
class of online learning algorithms in the high-dimensional
scaling limit. Our results are applied to two concrete
examples: online regularized linear regression and principal
component analysis. As the ambient dimension tends to
infinity, and with proper time scaling, we show that the
time-varying joint empirical measures of the target feature
vector and its estimates provided by the algorithms will
converge weakly to a deterministic measured-valued process
that can be characterized as the unique solution of a
nonlinear PDE. Numerical solutions of this PDE can be
efficiently obtained. These solutions lead to precise
predictions of the performance of the algorithms, as many
practical performance metrics are linear functionals of the
joint empirical measures. In addition to characterizing the
dynamic performance of online learning algorithms, our
asymptotic analysis also provides useful insights. In
particular, in the high-dimensional limit, and due to
exchangeability, the original coupled dynamics associated
with the algorithms will be asymptotically "decoupled", with
each coordinate independently solving a 1-D effective
minimization problem via stochastic gradient descent.
Exploiting this insight for nonconvex optimization problems
may prove an interesting line of future research.},
Key = {fds361448}
}
@article{fds361449,
Author = {Johndrow, JE and Mattingly, JC},
Title = {Error bounds for Approximations of Markov chains used in
Bayesian Sampling},
Year = {2017},
Month = {November},
Abstract = {We give a number of results on approximations of Markov
kernels in total variation and Wasserstein norms weighted by
a Lyapunov function. The results are applied to examples
from Bayesian statistics where approximations to transition
kernels are made to reduce computational
costs.},
Key = {fds361449}
}
@article{fds328807,
Author = {Herschlag, G and Ravier, R and Mattingly, JC},
Title = {Evaluating Partisan Gerrymandering in Wisconsin},
Year = {2017},
Month = {September},
Abstract = {We examine the extent of gerrymandering for the 2010 General
Assembly district map of Wisconsin. We find that there is
substantial variability in the election outcome depending on
what maps are used. We also found robust evidence that the
district maps are highly gerrymandered and that this
gerrymandering likely altered the partisan make up of the
Wisconsin General Assembly in some elections. Compared to
the distribution of possible redistricting plans for the
General Assembly, Wisconsin's chosen plan is an outlier in
that it yields results that are highly skewed to the
Republicans when the statewide proportion of Democratic
votes comprises more than 50-52% of the overall vote (with
the precise threshold depending on the election considered).
Wisconsin's plan acts to preserve the Republican majority by
providing extra Republican seats even when the Democratic
vote increases into the range when the balance of power
would shift for the vast majority of redistricting
plans.},
Key = {fds328807}
}
@article{fds328808,
Author = {Bakhtin, Y and Hurth, T and Lawley, SD and Mattingly,
JC},
Title = {Smooth invariant densities for random switching on the
torus},
Volume = {31},
Number = {4},
Pages = {1331-1350},
Publisher = {IOP Publishing},
Year = {2017},
Month = {August},
url = {http://dx.doi.org/10.1088/1361-6544/aaa04f},
Abstract = {We consider a random dynamical system obtained by switching
between the flows generated by two smooth vector fields on
the 2d-torus, with the random switchings happening according
to a Poisson process. Assuming that the driving vector
fields are transversal to each other at all points of the
torus and that each of them allows for a smooth invariant
density and no periodic orbits, we prove that the switched
system also has a smooth invariant density, for every
switching rate. Our approach is based on an integration by
parts formula inspired by techniques from Malliavin
calculus.},
Doi = {10.1088/1361-6544/aaa04f},
Key = {fds328808}
}
@article{fds328809,
Author = {Johndrow, JE and Mattingly, JC},
Title = {Coupling and Decoupling to bound an approximating Markov
Chain},
Year = {2017},
Month = {July},
Abstract = {This simple note lays out a few observations which are well
known in many ways but may not have been said in quite this
way before. The basic idea is that when comparing two
different Markov chains it is useful to couple them is such
a way that they agree as often as possible. We construct
such a coupling and analyze it by a simple dominating chain
which registers if the two processes agree or disagree. We
find that this imagery is useful when thinking about such
problems. We are particularly interested in comparing the
invariant measures and long time averages of the processes.
However, since the paths agree for long runs, it also
provides estimates on various stopping times such as hitting
or exit times. We also show that certain bounds are tight.
Finally, we provide a simple application to a Markov Chain
Monte Carlo algorithm and show numerically that the results
of the paper show a good level of approximation at
considerable speed up by using an approximating chain rather
than the original sampling chain.},
Key = {fds328809}
}
@article{fds328810,
Author = {Glatt-Holtz, NE and Herzog, DP and Mattingly, JC},
Title = {Scaling and Saturation in Infinite-Dimensional Control
Problems with Applications to Stochastic Partial
Differential Equations},
Journal = {Annals of PDE},
Year = {2017},
Month = {June},
Abstract = {We establish the dual notions of scaling and saturation from
geometric control theory in an infinite-dimensional setting.
This generalization is applied to the low-mode control
problem in a number of concrete nonlinear partial
differential equations. We also develop applications
concerning associated classes of stochastic partial
differential equations (SPDEs). In particular, we study the
support properties of probability laws corresponding to
these SPDEs as well as provide applications concerning the
ergodic and mixing properties of invariant measures for
these stochastic systems.},
Key = {fds328810}
}
@article{fds361283,
Author = {Bangia, S and Graves, CV and Herschlag, G and Kang, HS and Luo, J and Mattingly, JC and Ravier, R},
Title = {Redistricting: Drawing the Line},
Year = {2017},
Month = {April},
Abstract = {We develop methods to evaluate whether a political
districting accurately represents the will of the people. To
explore and showcase our ideas, we concentrate on the
congressional districts for the U.S. House of
representatives and use the state of North Carolina and its
redistrictings since the 2010 census. Using a Monte Carlo
algorithm, we randomly generate over 24,000 redistrictings
that are non-partisan and adhere to criteria from proposed
legislation. Applying historical voting data to these random
redistrictings, we find that the number of democratic and
republican representatives elected varies drastically
depending on how districts are drawn. Some results are more
common, and we gain a clear range of expected election
outcomes. Using the statistics of our generated
redistrictings, we critique the particular congressional
districtings used in the 2012 and 2016 NC elections as well
as a districting proposed by a bipartisan redistricting
commission. We find that the 2012 and 2016 districtings are
highly atypical and not representative of the will of the
people. On the other hand, our results indicate that a plan
produced by a bipartisan panel of retired judges is highly
typical and representative. Since our analyses are based on
an ensemble of reasonable redistrictings of North Carolina,
they provide a baseline for a given election which
incorporates the geometry of the state's population
distribution.},
Key = {fds361283}
}
@article{fds303552,
Author = {J.C. Mattingly and Cooke, B and Herzog, DP and Mattingly, JC and Mckinle, SA and Schmidler,
SC},
Title = {Geometric ergodicity of two-dimensional hamiltonian systems
with a Lennard-Jones-like repulsive potential},
Journal = {Communications in Mathematical Sciences},
Volume = {15},
Number = {7},
Pages = {1987-2025},
Publisher = {International Press of Boston},
Year = {2017},
Month = {January},
url = {http://arxiv.org/abs/1104.3842v2},
Abstract = {We establish ergodicity of the Langevin dynamics for a
simple two-particle system involving a Lennard-Jones type
potential. Moreover, we show that the dynamics is
geometrically ergodic; that is, the system converges to
stationarity exponentially fast. Methods from stochastic
averaging are used to establish the existence of the
appropriate Lyapunov function.},
Doi = {10.4310/CMS.2017.v15.n7.a10},
Key = {fds303552}
}
@article{fds318321,
Author = {Hairer, M and Mattingly, J},
Title = {The strong Feller property for singular stochastic
PDEs},
Volume = {54},
Number = {3},
Pages = {1314-1340},
Publisher = {Institute of Mathematical Statistics},
Year = {2016},
url = {http://dx.doi.org/10.1214/17-aihp840},
Abstract = {We show that the Markov semigroups generated by a large
class of singular stochastic PDEs satisfy the strong Feller
property. These include for example the KPZ equation and the
dynamical $\Phi^4_3$ model. As a corollary, we prove that
the Brownian bridge measure is the unique invariant measure
for the KPZ equation with periodic boundary
conditions.},
Doi = {10.1214/17-aihp840},
Key = {fds318321}
}
@article{fds318322,
Author = {Tempkin, JOB and Koten, BV and Mattingly, JC and Dinner, AR and Weare,
J},
Title = {Trajectory stratification of stochastic dynamics},
Journal = {SIAM Review},
Volume = {60},
Number = {4},
Pages = {909-938},
Publisher = {Society for Industrial and Applied Mathematics},
Year = {2016},
url = {http://dx.doi.org/10.1137/16m1104329},
Abstract = {We present a general mathematical framework for trajectory
stratification for simulating rare events. Trajectory
stratification involves decomposing trajectories of the
underlying process into fragments limited to restricted
regions of state space (strata), computing averages over the
distributions of the trajectory fragments within the strata
with minimal communication between them, and combining those
averages with appropriate weights to yield averages with
respect to the original underlying process. Our framework
reveals the full generality and flexibility of trajectory
stratification, and it illuminates a common mathematical
structure shared by existing algorithms for sampling rare
events. We demonstrate the power of the framework by
defining strata in terms of both points in time and
path-dependent variables for efficiently estimating averages
that were not previously tractable.},
Doi = {10.1137/16m1104329},
Key = {fds318322}
}
@article{fds300244,
Author = {Johndrow, JE and Mattingly, JC and Mukherjee, S and Dunson,
D},
Title = {Optimal approximating Markov chains for Bayesian
inference},
Year = {2015},
Month = {August},
url = {http://arxiv.org/abs/1508.03387v2},
Abstract = {The Markov Chain Monte Carlo method is the dominant paradigm
for posterior computation in Bayesian analysis. It is common
to control computation time by making approximations to the
Markov transition kernel. Comparatively little attention has
been paid to computational optimality in these approximating
Markov Chains, or when such approximations are justified
relative to obtaining shorter paths from the exact kernel.
We give simple, sharp bounds for uniform approximations of
uniformly mixing Markov chains. We then suggest a notion of
optimality that incorporates computation time and
approximation error, and use our bounds to make
generalizations about properties of good approximations in
the uniformly mixing setting. The relevance of these
properties is demonstrated in applications to a
minibatching-based approximate MCMC algorithm for large $n$
logistic regression and low-rank approximations for Gaussian
processes.},
Key = {fds300244}
}
@article{fds303555,
Author = {Munch, E and Turner, K and Bendich, P and Mukherjee, S and Mattingly, J and Harer, J},
Title = {Probabilistic Fréchet means for time varying persistence
diagrams},
Journal = {Electronic Journal of Statistics},
Volume = {9},
Number = {1},
Pages = {1173-1204},
Publisher = {Institute of Mathematical Statistics},
Year = {2015},
Month = {January},
url = {http://arxiv.org/abs/1307.6530v3},
Abstract = {In order to use persistence diagrams as a true statistical
tool, it would be very useful to have a good notion of mean
and variance for a set of diagrams. In [23], Mileyko and his
collaborators made the first study of the properties of the
Fréchet mean in (D<inf>p</inf>, W<inf>p</inf>), the space
of persistence diagrams equipped with the p-th Wasserstein
metric. In particular, they showed that the Fréchet mean of
a finite set of diagrams always exists, but is not
necessarily unique. The means of a continuously-varying set
of diagrams do not themselves (necessarily) vary
continuously, which presents obvious problems when trying to
extend the Fréchet mean definition to the realm of
time-varying persistence diagrams, better known as
vineyards. We fix this problem by altering the original
definition of Fréchet mean so that it now becomes a
probability measure on the set of persistence diagrams; in a
nutshell, the mean of a set of diagrams will be a weighted
sum of atomic measures, where each atom is itself a
persistence diagram determined using a perturbation of the
input diagrams. This definition gives for each N a map
(D<inf>p</inf>)<sup>N</sup>→ℙ(D<inf>p</inf>). We show
that this map is Hölder continuous on finite diagrams and
thus can be used to build a useful statistic on
vineyards.},
Doi = {10.1214/15-EJS1030},
Key = {fds303555}
}
@article{fds243883,
Author = {Huckemann, S and Mattingly, JC and Miller, E and Nolen,
J},
Title = {Sticky central limit theorems at isolated hyperbolic planar
singularities},
Journal = {Electronic Journal of Probability},
Volume = {20},
Pages = {1-34},
Publisher = {Institute of Mathematical Statistics},
Year = {2015},
url = {http://hdl.handle.net/10161/9516 Duke open
access},
Abstract = {We derive the limiting distribution of the barycenter bn of
an i.i.d. sample of n random points on a planar cone with
angular spread larger than 2π. There are three mutually
exclusive possibilities: (i) (fully sticky case) after a
finite random time the barycenter is almost surely at the
origin; (ii) (partly sticky case) the limiting distribution
of √nb<inf>n</inf> comprises a point mass at the origin,
an open sector of a Gaussian, and the projection of a
Gaussian to the sector’s bounding rays; or (iii)
(nonsticky case) the barycenter stays away from the origin
and the renormalized fluctuations have a fully supported
limit distribution—usually Gaussian but not always. We
conclude with an alternative, topological definition of
stickiness that generalizes readily to measures on general
metric spaces.},
Doi = {10.1214/EJP.v20-3887},
Key = {fds243883}
}
@article{fds300245,
Author = {Glatt-Holtz, N and Richards, G and Mattingly, JC},
Title = {On Unique Ergodicity in Nonlinear Stochastic Partial
Differential Equations},
Volume = {166},
Number = {3-4},
Pages = {618-649},
Publisher = {Springer Nature},
Year = {2015},
url = {http://arxiv.org/abs/1512.04126v1},
Abstract = {We illustrate how the notion of asymptotic coupling provides
a flexible and intuitive framework for proving the
uniqueness of invariant measures for a variety of stochastic
partial differential equations whose deterministic
counterpart possesses a finite number of determining modes.
Examples exhibiting parabolic and hyperbolic structure are
studied in detail. In the later situation we also present a
simple framework for establishing the existence of invariant
measures when the usual approach relying on the
Krylov–Bogolyubov procedure and compactness
fails.},
Doi = {10.1007/s10955-016-1605-x},
Key = {fds300245}
}
@article{fds303549,
Author = {Luo, S and Mattingly, JC},
Title = {Scaling limits of a model for selection at two
scales},
Year = {2015},
url = {http://arxiv.org/abs/1507.00397v1},
Abstract = {The dynamics of a population undergoing selection is a
central topic in evolutionary biology. This question is
particularly intriguing in the case where selective forces
act in opposing directions at two population scales. For
example, a fast-replicating virus strain outcompetes
slower-replicating strains at the within-host scale.
However, if the fast-replicating strain causes host
morbidity and is less frequently transmitted, it can be
outcompeted by slower-replicating strains at the
between-host scale. Here we consider a stochastic
ball-and-urn process which models this type of phenomenon.
We prove the weak convergence of this process under two
natural scalings. The first scaling leads to a deterministic
nonlinear integro-partial differential equation on the
interval $[0,1]$ with dependence on a single parameter,
$\lambda$. We show that the fixed points of this
differential equation are Beta distributions and that their
stability depends on $\lambda$ and the behavior of the
initial data around $1$. The second scaling leads to a
measure-valued Fleming-Viot process, an infinite dimensional
stochastic process that is frequently associated with a
population genetics.},
Key = {fds303549}
}
@article{fds337964,
Author = {Herzog, DP and Mattingly, JC},
Title = {Noise-Induced Stabilization of Planar Flows
II},
Year = {2014},
Month = {April},
Key = {fds337964}
}
@article{fds243876,
Author = {J.C. Mattingly and Lawley, SD and Mattingly, JC and Reed, MC},
Title = {Sensitivity to switching rates in stochastically switched
ODEs},
Journal = {Communications in Mathematical Sciences},
Volume = {12},
Number = {7},
Pages = {1343-1352},
Publisher = {International Press of Boston},
Year = {2014},
ISSN = {1539-6746},
url = {http://hdl.handle.net/10161/9515 Duke open
access},
Abstract = {We consider a stochastic process driven by a linear ordinary
differential equation whose right-hand side switches at
exponential times between a collection of different
matrices. We construct planar examples that switch between
two matrices where the individual matrices and the average
of the two matrices are all Hurwitz (all eigenvalues have
strictly negative real part), but nonetheless the process
goes to infinity at large time for certain values of the
switching rate. We further construct examples in higher
dimensions where again the two individual matrices and their
averages are all Hurwitz, but the process has arbitrarily
many transitions between going to zero and going to infinity
at large time as the switching rate varies. In order to
construct these examples, we first prove in general that if
each of the individual matrices is Hurwitz, then the process
goes to zero at large time for sufficiently slow switching
rate and if the average matrix is Hurwitz, then the process
goes to zero at large time for sufficiently fast switching
rate. We also give simple conditions that ensure the process
goes to zero at large time for all switching rates. © 2014
International Press.},
Doi = {10.4310/CMS.2014.v12.n7.a9},
Key = {fds243876}
}
@article{fds243878,
Author = {Mattingly, JC and Pardoux, E},
Title = {Invariant measure selection by noise. An
example},
Journal = {Discrete and Continuous Dynamical Systems. Series
A},
Volume = {34},
Number = {10},
Pages = {4223-4257},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2014},
ISSN = {1078-0947},
url = {http://hdl.handle.net/10161/9511 Duke open
access},
Abstract = {We consider a deterministic system with two conserved
quantities and infinity many invariant measures. However the
systems possess a unique invariant measure when enough
stochastic forcing and balancing dissipation are added. We
then show that as the forcing and dissipation are removed a
unique limit of the deterministic system is selected. The
exact structure of the limiting measure depends on the
specifics of the stochastic forcing.},
Doi = {10.3934/dcds.2014.34.4223},
Key = {fds243878}
}
@article{fds243880,
Author = {J.C. Mattingly and Bakhtin, Y and Hurth, T and Mattingly, JC},
Title = {Regularity of invariant densities for 1D-systems with random
switching},
Journal = {arXiv preprint arXiv:1406.5425},
Volume = {28},
Number = {11},
Pages = {3755-3787},
Publisher = {IOP Publishing},
Year = {2014},
ISSN = {0951-7715},
url = {http://hdl.handle.net/10161/9514 Duke open
access},
Abstract = {This is a detailed analysis of invariant measures for
one-dimensional dynamical systems with random switching. In
particular, we prove the smoothness of the invariant
densities away from critical points and describe the
asymptotics of the invariant densities at critical
points.},
Doi = {10.1088/0951-7715/28/11/3755},
Key = {fds243880}
}
@article{fds243881,
Author = {Lawley, SD and Mattingly, JC and Reed, MC},
Title = {Stochastic switching in infinite dimensions with
applications to random parabolic PDEs},
Journal = {arXiv preprint arXiv:1407.2264},
Volume = {47},
Number = {4},
Pages = {3035-3063},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2014},
ISSN = {0036-1410},
url = {http://hdl.handle.net/10161/9517 Duke open
access},
Abstract = {We consider parabolic PDEs with randomly switching boundary
conditions. In order to analyze these random PDEs, we
consider more general stochastic hybrid systems and prove
convergence to, and properties of, a stationary
distribution. Applying these general results to the heat
equation with randomly switching boundary conditions, we
find explicit formulae for various statistics of the
solution and obtain almost sure results about its regularity
and structure. These results are of particular interest for
biological applications as well as for their significant
departure from behavior seen in PDEs forced by disparate
Gaussian noise. Our general results also have applications
to other types of stochastic hybrid systems, such as ODEs
with randomly switching right-hand sides.},
Doi = {10.1137/140976716},
Key = {fds243881}
}
@article{fds243882,
Author = {Herzog, DP and Mattingly, JC},
Title = {A practical criterion for positivity of transition
densities},
Journal = {arXiv preprint arXiv:1407.3858},
Volume = {28},
Number = {8},
Pages = {2823-2845},
Publisher = {IOP Publishing},
Year = {2014},
ISSN = {0951-7715},
url = {http://hdl.handle.net/10161/9510 Duke open
access},
Abstract = {We establish a simple criterion for locating points where
the transition density of a degenerate diffusion is strictly
positive. Throughout, we assume that the diffusion satisfies
a stochastic differential equation (SDE) on Rd with additive
noise and polynomial drift. In this setting, we will see
that it is often the case that local information of the
flow, e.g. the Lie algebra generated by the vector fields
defining the SDE at a point x ∈ Rd, determines where the
transition density is strictly positive. This is surprising
in that positivity is a more global property of the
diffusion. This work primarily builds on and combines the
ideas of Arous and Lé andre (1991 Décroissance
exponentielle du noyau de la chaleur sur la diagonale. II
Probab. Theory Relat. Fields 90 377-402) and Jurdjevic and
Kupka (1985 Polynomial control systems Math. Ann. 272
361-8).},
Doi = {10.1088/0951-7715/28/8/2823},
Key = {fds243882}
}
@article{fds243884,
Author = {Herzog, DP and Mattingly, JC},
Title = {Noise-Induced Stabilization of Planar Flows
I},
Journal = {arXiv preprint arXiv:1404.0957},
Volume = {20},
Publisher = {Institute of Mathematical Statistics},
Year = {2014},
url = {http://hdl.handle.net/10161/9512 Duke open
access},
Abstract = {We continue the work started in Part I [6], showing how the
addition of noise can stabilize an otherwise unstable
system. The analysis makes use of nearly optimal Lyapunov
functions. In this continuation, we remove the main limiting
assumption of Part I by an inductive procedure as well as
establish a lower bound which shows that our construction is
radially sharp. We also prove a version of Peskir’s [7]
generalized Tanaka formula adapted to patching together
Lyapunov functions. This greatly simplifies the analysis
used in previous works.},
Doi = {10.1214/EJP.v20-4048},
Key = {fds243884}
}
@article{fds303554,
Author = {J.C. Mattingly and Mattingly, JC and Vaughn, C},
Title = {Redistricting and the Will of the People},
Journal = {arXiv preprint arXiv:1410.8796},
Year = {2014},
url = {http://arxiv.org/abs/1410.8796v1},
Abstract = {We introduce a non-partisan probability distribution on
congressional redistricting of North Carolina which
emphasizes the equal partition of the population and the
compactness of districts. When random districts are drawn
and the results of the 2012 election were re-tabulated under
the drawn districtings, we find that an average of 7.6
democratic representatives are elected. 95% of the randomly
sampled redistrictings produced between 6 and 9 Democrats.
Both of these facts are in stark contrast with the 4
Democrats elected in the 2012 elections with the same vote
counts. This brings into serious question the idea that such
elections represent the "will of the people." It underlines
the ability of redistricting to undermine the democratic
process, while on the face allowing democracy to
proceed.},
Key = {fds303554}
}
@article{fds243877,
Author = {Hotz, T and Huckemann, S and Le, H and Marron, JS and Mattingly, JC and Miller, E and Nolen, J and Owen, M and Patrangenaru, V and Skwerer,
S},
Title = {Sticky central limit theorems on open books},
Journal = {The Annals of Applied Probability},
Volume = {23},
Number = {6},
Pages = {2238-2258},
Publisher = {Institute of Mathematical Statistics},
Year = {2013},
ISSN = {1050-5164},
url = {http://dx.doi.org/10.1214/12-AAP899},
Abstract = {Given a probability distribution on an open book (a metric
space obtained by gluing a disjoint union of copies of a
half-space along their boundary hyperplanes), we define a
precise concept of when the Fr\'{e}chet mean (barycenter) is
sticky. This nonclassical phenomenon is quantified by a law
of large numbers (LLN) stating that the empirical mean
eventually almost surely lies on the (codimension 1 and
hence measure 0) spine that is the glued hyperplane, and a
central limit theorem (CLT) stating that the limiting
distribution is Gaussian and supported on the spine. We also
state versions of the LLN and CLT for the cases where the
mean is nonsticky (i.e., not lying on the spine) and partly
sticky (i.e., is, on the spine but not sticky).},
Doi = {10.1214/12-AAP899},
Key = {fds243877}
}
@article{fds243874,
Author = {J.C. Mattingly and Mattingly, JC and McKinley, SA and Pillai, NS},
Title = {Geometric ergodicity of a bead-spring pair with stochastic
Stokes forcing},
Journal = {Stochastic Processes and their Applications},
Volume = {122},
Number = {12},
Pages = {3953-3979},
Publisher = {Elsevier BV},
Year = {2012},
Month = {December},
ISSN = {0304-4149},
MRCLASS = {Preliminary Data},
MRNUMBER = {2971721},
url = {http://hdl.handle.net/10161/9524 Duke open
access},
Abstract = {We consider a simple model for the fluctuating hydrodynamics
of a flexible polymer in a dilute solution, demonstrating
geometric ergodicity for a pair of particles that interact
with each other through a nonlinear spring potential while
being advected by a stochastic Stokes fluid velocity field.
This is a generalization of previous models which have used
linear spring forces as well as white-in-time fluid velocity
fields. We follow previous work combining control theoretic
arguments, Lyapunov functions, and hypo-elliptic diffusion
theory to prove exponential convergence via a Harris chain
argument. In addition we allow the possibility of excluding
certain "bad" sets in phase space in which the assumptions
are violated but from which the system leaves with a
controllable probability. This allows for the treatment of
singular drifts, such as those derived from the
Lennard-Jones potential, which is a novel feature of this
work. © 2012 Elsevier B.V. All rights reserved.},
Doi = {10.1016/j.spa.2012.07.003},
Key = {fds243874}
}
@article{fds243855,
Author = {Luo, S and Reed, M and Mattingly, JC and Koelle, K},
Title = {The impact of host immune status on the within-host and
population dynamics of antigenic immune escape.},
Journal = {J R Soc Interface},
Volume = {9},
Number = {75},
Pages = {2603-2613},
Publisher = {The Royal Society},
Year = {2012},
Month = {October},
url = {http://www.ncbi.nlm.nih.gov/pubmed/22572027},
Abstract = {Antigenically evolving pathogens such as influenza viruses
are difficult to control owing to their ability to evade
host immunity by producing immune escape variants.
Experimental studies have repeatedly demonstrated that viral
immune escape variants emerge more often from immunized
hosts than from naive hosts. This empirical relationship
between host immune status and within-host immune escape is
not fully understood theoretically, nor has its impact on
antigenic evolution at the population level been evaluated.
Here, we show that this relationship can be understood as a
trade-off between the probability that a new antigenic
variant is produced and the level of viraemia it reaches
within a host. Scaling up this intra-host level trade-off to
a simple population level model, we obtain a distribution
for variant persistence times that is consistent with
influenza A/H3N2 antigenic variant data. At the within-host
level, our results show that target cell limitation, or a
functional equivalent, provides a parsimonious explanation
for how host immune status drives the generation of immune
escape mutants. At the population level, our analysis also
offers an alternative explanation for the observed tempo of
antigenic evolution, namely that the production rate of
immune escape variants is driven by the accumulation of herd
immunity. Overall, our results suggest that disease control
strategies should be further assessed by considering the
impact that increased immunity--through vaccination--has on
the production of new antigenic variants.},
Doi = {10.1098/rsif.2012.0180},
Key = {fds243855}
}
@article{fds243875,
Author = {J.C. Mattingly and Athreyaz, A and Kolba, T and Mattingly, JC},
Title = {Propagating lyapunov functions to prove noise-induced
stabilization},
Journal = {Electronic Journal of Probability},
Volume = {17},
Pages = {1-38},
Publisher = {Institute of Mathematical Statistics},
Year = {2012},
ISSN = {1083-6489},
url = {http://hdl.handle.net/10161/9518 Duke open
access},
Abstract = {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.},
Doi = {10.1214/EJP.v17-2410},
Key = {fds243875}
}
@article{fds243854,
Author = {Porporato, A and Kramer, PR and Cassiani, M and Daly, E and Mattingly,
J},
Title = {Local kinetic interpretation of entropy production through
reversed diffusion.},
Journal = {Phys Rev E Stat Nonlin Soft Matter Phys},
Volume = {84},
Number = {4 Pt 1},
Pages = {041142},
Year = {2011},
Month = {Fall},
url = {http://www.ncbi.nlm.nih.gov/pubmed/22181122},
Abstract = {The time reversal of stochastic diffusion processes is
revisited with emphasis on the physical meaning of the
time-reversed drift and the noise prescription in the case
of multiplicative noise. The local kinematics and mechanics
of free diffusion are linked to the hydrodynamic
description. These properties also provide an interpretation
of the Pope-Ching formula for the steady-state probability
density function along with a geometric interpretation of
the fluctuation-dissipation relation. Finally, the
statistics of the local entropy production rate of diffusion
are discussed in the light of local diffusion properties,
and a stochastic differential equation for entropy
production is obtained using the Girsanov theorem for
reversed diffusion. The results are illustrated for the
Ornstein-Uhlenbeck process.},
Doi = {10.1103/PhysRevE.84.041142},
Key = {fds243854}
}
@article{fds243853,
Author = {Koelle, K and Ratmann, O and Rasmussen, DA and Pasour, V and Mattingly,
J},
Title = {A dimensionless number for understanding the evolutionary
dynamics of antigenically variable RNA viruses},
Journal = {Proceedings of the Royal Society B: Biological
Sciences},
Volume = {278},
Number = {1725},
Pages = {3723-3730},
Year = {2011},
ISSN = {0962-8452},
url = {http://hdl.handle.net/10161/9525 Duke open
access},
Abstract = {Antigenically variable RNA viruses are significant
contributors to the burden of infectious disease worldwide.
One reason for their ubiquity is their ability to escape
herd immunity through rapid antigenic evolution and thereby
to reinfect previously infected hosts. However, the ways in
which these viruses evolve antigenically are highly diverse.
Some have only limited diversity in the long-run, with every
emergence of a new antigenic variant coupled with a
replacement of the older variant. Other viruses rapidly
accumulate antigenic diversity over time. Others still
exhibit dynamics that can be considered evolutionary
intermediates between these two extremes. Here, we present a
theoretical framework that aims to understand these
differences in evolutionary patterns by considering a
virus's epidemiological dynamics in a given host population.
Our framework, based on a dimensionless number,
probabilistically anticipates patterns of viral antigenic
diversification and thereby quantifies a virus's
evolutionary potential. It is therefore similar in spirit to
the basic reproduction number, the well-known dimensionless
number which quantifies a pathogen's reproductive potential.
We further outline how our theoretical framework can be
applied to empirical viral systems, using influenza A/H3N2
as a case study. We end with predictions of our framework
and work that remains to be done to further integrate viral
evolutionary dynamics with disease ecology. © 2011 The
Royal Society.},
Doi = {10.1098/rspb.2011.0435},
Key = {fds243853}
}
@article{fds243870,
Author = {J.C. Mattingly and Hairer, M and Mattingly, JC and Scheutzow, M},
Title = {Asymptotic coupling and a general form of Harris' theorem
with applications to stochastic delay equations},
Journal = {Probability Theory and Related Fields},
Volume = {149},
Number = {1},
Pages = {223-259},
Publisher = {Springer Nature},
Year = {2011},
ISSN = {0178-8051},
MRNUMBER = {2531551},
url = {http://hdl.handle.net/10161/10831 Duke open
access},
Abstract = {There are many Markov chains on infinite dimensional spaces
whose one-step transition kernels are mutually singular when
starting from different initial conditions. We give results
which prove unique ergodicity under minimal assumptions on
one hand and the existence of a spectral gap under
conditions reminiscent of Harris' theorem. The first uses
the existence of couplings which draw the solutions together
as time goes to infinity. Such "asymptotic couplings" were
central to (Mattingly and Sinai in Comm Math Phys
219(3):523-565, 2001; Mattingly in Comm Math Phys
230(3):461-462, 2002; Hairer in Prob Theory Relat Field
124:345-380, 2002; Bakhtin and Mattingly in Commun Contemp
Math 7:553-582, 2005) on which this work builds. As in
Bakhtin and Mattingly (2005) the emphasis here is on
stochastic differential delay equations. Harris' celebrated
theorem states that if a Markov chain admits a Lyapunov
function whose level sets are "small" (in the sense that
transition probabilities are uniformly bounded from below),
then it admits a unique invariant measure and transition
probabilities converge towards it at exponential speed. This
convergence takes place in a total variation norm, weighted
by the Lyapunov function. A second aim of this article is to
replace the notion of a "small set" by the much weaker
notion of a "d-small set," which takes the topology of the
underlying space into account via a distance-like function
d. With this notion at hand, we prove an analogue to Harris'
theorem, where the convergence takes place in a
Wasserstein-like distance weighted again by the Lyapunov
function. This abstract result is then applied to the
framework of stochastic delay equations. In this framework,
the usual theory of Harris chains does not apply, since
there are natural examples for which there exist no small
sets (except for sets consisting of only one point). This
gives a solution to the long-standing open problem of
finding natural conditions under which a stochastic delay
equation admits at most one invariant measure and transition
probabilities converge to it. © 2009 Springer-Verlag.},
Doi = {10.1007/s00440-009-0250-6},
Key = {fds243870}
}
@article{fds243872,
Author = {J.C. Mattingly and Anderson, DF and Mattingly, JC},
Title = {A weak trapezoidal method for a class of stochastic
differential equations},
Journal = {Communications in Mathematical Sciences},
Volume = {9},
Number = {1},
Pages = {301-318},
Publisher = {International Press of Boston},
Year = {2011},
ISSN = {1539-6746},
url = {http://hdl.handle.net/10161/9520 Duke open
access},
Abstract = {We present a numerical method for the approximation of
solutions for the class of stochastic differential equations
driven by Brownian motions which induce stochastic variation
in fixed directions. This class of equations arises
naturally in the study of population processes and chemical
reaction kinetics. We show that the method constructs paths
that are second order accurate in the weak sense. The method
is simpler than many second order methods in that it neither
requires the construction of iterated It̂o integrals nor
the evaluation of any derivatives. The method consists of
two steps. In the first an explicit Euler step is used to
take a fractional step. The resulting fractional point is
then combined with the initial point to obtain a higher
order, trapezoidal like, approximation. The higher order of
accuracy stems from the fact that both the drift and the
quadratic variation of the underlying SDE are approximated
to second order. © 2011 International Press.},
Doi = {10.4310/CMS.2011.v9.n1.a15},
Key = {fds243872}
}
@article{fds243873,
Author = {J.C. Mattingly and Hairer, M and Mattingly, JC},
Title = {A theory of hypoellipticity and unique ergodicity for
semilinear stochastic PDEs},
Journal = {Electronic Journal of Probability},
Volume = {16},
Number = {23},
Pages = {658-738},
Publisher = {Institute of Mathematical Statistics},
Year = {2011},
ISSN = {1083-6489},
url = {http://hdl.handle.net/10161/9521 Duke open
access},
Abstract = {We present a theory of hypoellipticity and unique ergodicity
for semilinear parabolic stochastic PDEs with "polynomial"
nonlinearities and additive noise, considered as abstract
evolution equations in some Hilbert space. It is shown that
if Hörmander's bracket condition holds at every point of
this Hilbert space, then a lower bound on the Malliavin
covariance operatorμt can be obtained. Informally, this
bound can be read as "Fix any finite-dimensional projection
on a subspace of sufficiently regular functions. Then the
eigenfunctions of μt with small eigenvalues have only a
very small component in the image of Π." We also show how
to use a priori bounds on the solutions to the equation to
obtain good control on the dependency of the bounds on the
Malliavin matrix on the initial condition. These bounds are
sufficient in many cases to obtain the asymptotic strong
Feller property introduced in [HM06]. One of the main novel
technical tools is an almost sure bound from below on the
size of "Wiener polynomials," where the coefficients are
possibly non-adapted stochastic processes satisfying a Lips
chitz condition. By exploiting the polynomial structure of
the equations, this result can be used to replace Norris'
lemma, which is unavailable in the present context. We
conclude by showing that the two-dimensional stochastic
Navier-Stokes equations and a large class of
reaction-diffusion equations fit the framework of our
theory.},
Doi = {10.1214/EJP.v16-875},
Key = {fds243873}
}
@article{fds303551,
Author = {Hairer, M and Mattingly, JC},
Title = {Yet another look at Harris’ ergodic theorem for Markov
chains},
Volume = {63},
Pages = {109-117},
Booktitle = {Progress in Probability},
Publisher = {Birkhäuser/Springer Basel AG, Basel},
Year = {2011},
url = {http://arxiv.org/abs/0810.2777v1},
Abstract = {The aim of this note is to present an elementary proof of a
variation of Harris’ ergodic theorem of Markov
chains.},
Doi = {10.1007/978-3-0348-0021-1_7},
Key = {fds303551}
}
@article{fds303548,
Author = {J.C. Mattingly and Heymann, M and Teitsworth, SW and Mattingly, JC},
Title = {Rare Transition Events in Nonequilibrium Systems with
State-Dependent Noise: Application to Stochastic Current
Switching in Semiconductor Superlattices},
Year = {2010},
Month = {August},
url = {http://arxiv.org/abs/1008.4037v2},
Abstract = {Using recent mathematical advances, a geometric approach to
rare noise-driven transition events in nonequilibrium
systems is given, and an algorithm for computing the maximum
likelihood transition curve is generalized to the case of
state-dependent noise. It is applied to a model of
electronic transport in semiconductor superlattices to
investigate transitions between metastable electric field
distributions. When the applied voltage $V$ is varied near a
saddle-node bifurcation at $V_th$, the mean life time $<T>$
of the initial metastable state is shown to scale like
$log<T> \propto |V_th - V|^{3/2}$ as $V\to
V_th$.},
Key = {fds303548}
}
@article{fds303553,
Author = {J.C. Mattingly and Mattingly, JC and Pillai, NS and Stuart, AM},
Title = {Diffusion limits of the random walk Metropolis algorithm in
high dimensions},
Journal = {Annals of Applied Probability},
Volume = {22},
Number = {3},
Pages = {881-930},
Publisher = {Institute of Mathematical Statistics},
Year = {2010},
Month = {March},
url = {http://arxiv.org/abs/1003.4306v4},
Abstract = {Diffusion limits of MCMC methods in high dimensions provide
a useful theoretical tool for studying computational
complexity. In particular, they lead directly to precise
estimates of the number of steps required to explore the
target measure, in stationarity, as a function of the
dimension of the state space. However, to date such results
have mainly been proved for target measures with a product
structure, severely limiting their applicability. The
purpose of this paper is to study diffusion limits for a
class of naturally occurring high-dimensional measures found
from the approximation of measures on a Hilbert space which
are absolutely continuous with respect to a Gaussian
reference measure. The diffusion limit of a random walk
Metropolis algorithm to an infinite-dimensional Hilbert
space valued SDE (or SPDE) is proved, facilitating
understanding of the computational complexity of the
algorithm.},
Doi = {10.1214/10-AAP754},
Key = {fds303553}
}
@article{fds243871,
Author = {J.C. Mattingly and Mattingly, JC and Stuart, AM and Tretyakov, MV},
Title = {Convergence of numerical time-averaging and stationary
measures via Poisson equations},
Journal = {SIAM Journal on Numerical Analysis},
Volume = {48},
Number = {2},
Pages = {552-577},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2010},
ISSN = {0036-1429},
MRCLASS = {65C30 (37Hxx 60H10 60H35)},
MRNUMBER = {2669996},
url = {http://hdl.handle.net/10161/4314 Duke open
access},
Abstract = {Numerical approximation of the long time behavior of a
stochastic di.erential equation (SDE) is considered. Error
estimates for time-averaging estimators are obtained and
then used to show that the stationary behavior of the
numerical method converges to that of the SDE. The error
analysis is based on using an associated Poisson equation
for the underlying SDE. The main advantages of this approach
are its simplicity and universality. It works equally well
for a range of explicit and implicit schemes, including
those with simple simulation of random variables, and for
hypoelliptic SDEs. To simplify the exposition, we consider
only the case where the state space of the SDE is a torus,
and we study only smooth test functions. However, we
anticipate that the approach can be applied more widely. An
analogy between our approach and Stein's method is
indicated. Some practical implications of the results are
discussed. Copyright © by SIAM. Unauthorized reproduction
of this article is prohibited.},
Doi = {10.1137/090770527},
Key = {fds243871}
}
@article{fds243869,
Author = {Hairer, M and Mattingly, JC},
Title = {Slow energy dissipation in anharmonic oscillator
chains},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {62},
Number = {8},
Pages = {999-1032},
Publisher = {WILEY},
Year = {2009},
ISSN = {0010-3640},
MRCLASS = {82C20 (35K55 37Lxx)},
MRNUMBER = {MR2531551},
url = {http://dx.doi.org/10.1002/cpa.20280},
Abstract = {We study the dynamic behavior at high energies of a chain of
anharmonic oscillators coupled at its ends to heat baths at
possibly different temperatures. In our setup, each
oscillator is subject to a homogeneous anharmonic pinning
potential V 1(qi) = |qi| 2k/2k and harmonic coupling
potentials V 2(qi-qi-1) = (qi-q i-1) 2/2 between itself and
its nearest neighbors. We consider the case k > 1 when
the pinning potential is stronger than the coupling
potential. At high energy, when a large fraction of the
energy is located in the bulk of the chain, breathers appear
and block the transport of energy through the system, thus
slowing its convergence to equilibrium. In such a regime, we
obtain equations for an effective dynamics by averaging out
the fast oscillation of the breather. Using this
representation and related ideas, we can prove a number of
results. When the chain is of length 3 and k > 3/2, we
show that there exists a unique invariant measure. If k >
2 we further show that the system does not relax
exponentially fast to this equilibrium by demonstrating that
0 is in the essential spectrum of the generator of the
dynamics. When the chain has five or more oscillators and k
> 3/2, we show that the generator again has 0 in its
essential spectrum. In addition to these rigorous results, a
theory is given for the rate of decrease of the energy when
it is concentrated in one of the oscillators without
dissipation. Numerical simulations are included that confirm
the theory. © 2009 Wiley Periodicals, Inc.},
Doi = {10.1002/cpa.20280},
Key = {fds243869}
}
@article{fds243868,
Author = {J.C. Mattingly and Hairer, M and Mattingly, JC},
Title = {Spectral gaps in wasserstein distances and the 2d stochastic
navier-stokes equations},
Journal = {Annals of Probability},
Volume = {36},
Number = {6},
Pages = {2050-2091},
Publisher = {Institute of Mathematical Statistics},
Year = {2008},
Month = {November},
MRNUMBER = {2478676},
url = {http://dx.doi.org/10.1214/08-AOP392},
Abstract = {We develop a general method to prove the existence of
spectral gaps for Markov semigroups on Banach spaces. Unlike
most previous work, the type of norm we consider for this
analysis is neither a weighted supremum norm nor an Ł
p-type norm, but involves the derivative of the observable
as well and hence can be seen as a type of 1-Wasserstein
distance. This turns out to be a suitable approach for
infinite-dimensional spaces where the usual Harris or
Doeblin conditions, which are geared toward total variation
convergence, often fail to hold. In the first part of this
paper, we consider semigroups that have uniform behavior
which one can view as the analog of Doeblin's condition. We
then proceed to study situations where the behavior is not
so uniform, but the system has a suitable Lyapunov
structure, leading to a type of Harris condition. We finally
show that the latter condition is satisfied by the
two-dimensional stochastic Navier-Stokes equations, even in
situations where the forcing is extremely degenerate. Using
the convergence result, we show that the stochastic
Navier-Stokes equations' invariant measures depend
continuously on the viscosity and the structure of the
forcing. © Institute of Mathematical Statistics,
2008.},
Doi = {10.1214/08-AOP392},
Key = {fds243868}
}
@article{fds194398,
Author = {J.C. Mattingly and Martin Hairer},
Title = {Yet another look at Harris' ergodic theorem for Markov
chains},
Year = {2008},
Month = {August},
url = {http://arxiv.org/abs/0810.2777},
Abstract = {The aim of this note is to present an elementary proof of a
variation of Harris' ergodic theorem of Markov chains. This
theorem, dating back to the fifties essentially states that
a Markov chain is uniquely ergodic if it admits a "small"
set which is visited infinitely often. This gives an
extension of the ideas of Doeblin to the unbounded state
space setting. Often this is established by finding a
Lyapunov function with "small" level sets. This topic has
been studied by many authors (cf. Harris, Hasminskii,
Nummelin, Meyn and Tweedie). If the Lyapunov function is
strong enough, one has a spectral gap in a weighted supremum
norm (cf. Meyn and Tweedie). Traditional proofs of this
result rely on the decomposition of the Markov chain into
excursions away from the small set and a careful analysis of
the exponential tail of the length of these excursions.
There have been other variations which have made use of
Poisson equations or worked at getting explicit constants.
The present proof is very direct, and relies instead on
introducing a family of equivalent weighted norms indexed by
a parameter $\beta$ and to make an appropriate choice of
this parameter that allows to combine in a very elementary
way the two ingredients (existence of a Lyapunov function
and irreducibility) that are crucial in obtaining a spectral
gap. The original motivation of this proof was the authors'
work on spectral gaps in Wasserstein metrics. The proof
presented in this note is a version of our reasoning in the
total variation setting which we used to guide the
calculations in arXiv:math/0602479. While we initially
produced it for that purpose, we hope that it will be of
interest in its own right.},
Key = {fds194398}
}
@article{fds243839,
Author = {Mattingly, JC and Suidan, TM},
Title = {Transition measures for the stochastic Burgers
equation},
Volume = {458},
Series = {Contemp. Math.},
Pages = {409-418},
Booktitle = {Integrable systems and random matrices},
Publisher = {Amer. Math. Soc., Providence, RI},
Address = {Providence, RI},
Year = {2008},
ISBN = {9780821842409},
ISSN = {0271-4132},
MRCLASS = {60Hxx (35Q53 35R60 60Jxx 76M35)},
MRNUMBER = {MR2411921},
url = {http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000256557400025&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=47d3190e77e5a3a53558812f597b0b92},
Doi = {10.1090/conm/458/08950},
Key = {fds243839}
}
@article{fds243867,
Author = {J.C. Mattingly and Iyer, G and Mattingly, J},
Title = {A stochastic-Lagrangian particle system for the
Navier-Stokes equations},
Journal = {Nonlinearity},
Volume = {21},
Number = {11},
Pages = {2537-2553},
Publisher = {IOP Publishing},
Year = {2008},
ISSN = {0951-7715},
MRCLASS = {76D05 (35Q30 35R60 60H10 60H30)},
MRNUMBER = {MR2448230 (2009h:76060)},
url = {http://dx.doi.org/10.1088/0951-7715/21/11/004},
Abstract = {This paper is based on a formulation of the Navier-Stokes
equations developed by Constantin and the first author
(Commun. Pure Appl. Math. at press, arXiv:math.PR/0511067),
where the velocity field of a viscous incompressible fluid
is written as the expected value of a stochastic process. In
this paper, we take N copies of the above process (each
based on independent Wiener processes), and replace the
expected value with 1/N times the sum over these N copies.
(We note that our formulation requires one to keep track of
N stochastic flows of diffeomorphisms, and not just the
motion of N particles.) We prove that in two dimensions,
this system of interacting diffeomorphisms has (time) global
solutions with initial data in the space C1,α which
consists of differentiable functions whose first derivative
is α Hölder continuous (see section 3 for the precise
definition). Further, we show that as N → ∞ the system
converges to the solution of Navier-Stokes equations on any
finite interval [0, T]. However for fixed N, we prove that
this system retains roughly O(1/N) times its original energy
as t → ∞. Hence the limit N → ∞ and T → ∞ do not
commute. For general flows, we only provide a lower bound to
this effect. In the special case of shear flows, we compute
the behaviour as t → ∞ explicitly. © 2008 IOP
Publishing Ltd and London Mathematical Society.},
Doi = {10.1088/0951-7715/21/11/004},
Key = {fds243867}
}
@article{fds243836,
Author = {Anderson, DF and Mattingly, JC},
Title = {Propagation of fluctuations in biochemical systems, II:
Nonlinear chains.},
Journal = {IET systems biology},
Volume = {1},
Number = {6},
Pages = {313-325},
Publisher = {Institution of Engineering and Technology
(IET)},
Year = {2007},
Month = {November},
ISSN = {1751-8849},
url = {http://hdl.handle.net/10161/11278 Duke open
access},
Abstract = {We consider biochemical reaction chains and investigate how
random external fluctuations, as characterised by variance
and coefficient of variation, propagate down the chains. We
perform such a study under the assumption that the number of
molecules is high enough so that the behaviour of the
concentrations of the system is well approximated by
differential equations. We conclude that the variances and
coefficients of variation of the fluxes will decrease as one
moves down the chain and, through an example, show that
there is no corresponding result for the variances of the
concentrations of the chemical species. We also prove that
the fluctuations of the fluxes as characterised by their
time averages decrease down reaction chains. The results
presented give insight into how biochemical reaction systems
are buffered against external perturbations solely by their
underlying graphical structure and point out the benefits of
studying the out-of-equilibrium dynamics of
systems.},
Doi = {10.1049/iet-syb:20060063},
Key = {fds243836}
}
@article{fds243863,
Author = {Mattingly, JC and Suidan, T and Vanden-Eijnden,
E},
Title = {Simple systems with anomalous dissipation and energy
cascade},
Journal = {Communications in Mathematical Physics},
Volume = {276},
Number = {1},
Pages = {189-220},
Publisher = {Springer Nature},
Year = {2007},
Month = {November},
ISSN = {0010-3616},
MRCLASS = {37L99 (37C99 37N10 76D05 76F02)},
MRNUMBER = {MR2342292 (2008m:37135)},
url = {http://dx.doi.org/10.1007/s00220-007-0333-0},
Abstract = {We analyze a class of dynamical systems of the type ȧn(t) =
cn-1 an-1(t) - cn an+1(t) + f n(t), n ∈ ℕ, a 0=0, where
f n (t) is a forcing term with fn(t) ≠ = 0 only for ≤n
n* < ∞ and the coupling coefficients c n satisfy a
condition ensuring the formal conservation of energy 1/2 Σn
|a n(t)|2. Despite being formally conservative, we show that
these dynamical systems support dissipative solutions
(suitably defined) and, as a result, may admit unique
(statistical) steady states when the forcing term f n (t) is
nonzero. This claim is demonstrated via the complete
characterization of the solutions of the system above for
specific choices of the coupling coefficients c n . The
mechanism of anomalous dissipations is shown to arise via a
cascade of the energy towards the modes with higher n; this
is responsible for solutions with interesting energy
spectra, namely E |an|2 scales as n-α as n→∞. Here the
exponents α depend on the coupling coefficients c n and E
denotes expectation with respect to the equilibrium measure.
This is reminiscent of the conjectured properties of the
solutions of the Navier-Stokes equations in the inviscid
limit and their accepted relationship with fully developed
turbulence. Hence, these simple models illustrate some of
the heuristic ideas that have been advanced to characterize
turbulence, similar in that respect to the random passive
scalar or random Burgers equation, but even simpler and
fully solvable. © 2007 Springer-Verlag.},
Doi = {10.1007/s00220-007-0333-0},
Key = {fds243863}
}
@article{fds243864,
Author = {Bakhtin, Y and Mattingly, JC},
Title = {Malliavin calculus for infinite-dimensional systems with
additive noise},
Journal = {Journal of Functional Analysis},
Volume = {249},
Number = {2},
Pages = {307-353},
Publisher = {Elsevier BV},
Year = {2007},
Month = {August},
ISSN = {0022-1236},
MRCLASS = {60H07 (76D05 76M35)},
MRNUMBER = {MR2345335},
url = {http://dx.doi.org/10.1016/j.jfa.2007.02.011},
Abstract = {We consider an infinite-dimensional dynamical system with
polynomial nonlinearity and additive noise given by a finite
number of Wiener processes. By studying how randomness is
spread by the dynamics, we develop in this setting a partial
counterpart of Hörmander's classical theory of Hypoelliptic
operators. We study the distributions of finite-dimensional
projections of the solutions and give conditions that
provide existence and smoothness of densities of these
distributions with respect to the Lebesgue measure. We also
apply our results to concrete SPDEs such as a Stochastic
Reaction Diffusion Equation and the Stochastic 2D
Navier-Stokes System. © 2007 Elsevier Inc. All rights
reserved.},
Doi = {10.1016/j.jfa.2007.02.011},
Key = {fds243864}
}
@article{fds243861,
Author = {Anderson, DF and Mattingly, JC and Nijhout, HF and Reed,
MC},
Title = {Propagation of fluctuations in biochemical systems, I:
linear SSC networks.},
Journal = {Bulletin of mathematical biology},
Volume = {69},
Number = {6},
Pages = {1791-1813},
Publisher = {Springer Nature},
Year = {2007},
Month = {August},
ISSN = {0092-8240},
MRCLASS = {92E20 (34F05 60H10 60H30)},
MRNUMBER = {MR2329180},
url = {http://www.ncbi.nlm.nih.gov/pubmed/17457656},
Abstract = {We investigate the propagation of random fluctuations
through biochemical networks in which the number of
molecules of each species is large enough so that the
concentrations are well modeled by differential equations.
We study the effect of network topology on the emergent
properties of the reaction system by characterizing the
behavior of variance as fluctuations propagate down chains
and studying the effect of side chains and feedback loops.
We also investigate the asymptotic behavior of the system as
one reaction becomes fast relative to the
others.},
Doi = {10.1007/s11538-007-9192-2},
Key = {fds243861}
}
@article{fds243862,
Author = {Lamba, H and Mattingly, JC and Stuart, AM},
Title = {An adaptive Euler-Maruyama scheme for SDEs: Convergence and
stability},
Journal = {IMA Journal of Numerical Analysis},
Volume = {27},
Number = {3},
Pages = {479-506},
Publisher = {Oxford University Press (OUP)},
Year = {2007},
Month = {January},
ISSN = {0272-4979},
MRCLASS = {60H35 (60H10 65C30)},
MRNUMBER = {MR2337577},
url = {http://dx.doi.org/10.1093/imanum/drl032},
Abstract = {The understanding of adaptive algorithms for stochastic
differential equations (SDEs) is an open area, where many
issues related to both convergence and stability (long-time
behaviour) of algorithms are unresolved. This paper
considers a very simple adaptive algorithm, based on
controlling only the drift component of a time step. Both
convergence and stability are studied. The primary issue in
the convergence analysis is that the adaptive method does
not necessarily drive the time steps to zero with the
user-input tolerance. This possibility must be quantified
and shown to have low probability. The primary issue in the
stability analysis is ergodicity. It is assumed that the
noise is nondegenerate, so that the diffusion process is
elliptic, and the drift is assumed to satisfy a coercivity
condition. The SDE is then geometrically ergodic (averages
converge to statistical equilibrium exponentially quickly).
If the drift is not linearly bounded, then explicit fixed
time step approximations, such as the Euler-Maruyama scheme,
may fail to be ergodic. In this work, it is shown that the
simple adaptive time-stepping strategy cures this problem.
In addition to proving ergodicity, an exponential moment
bound is also proved, generalizing a result known to hold
for the SDE itself. © The author 2006. Published by Oxford
University Press on behalf of the Institute of Mathematics
and its Applications. All rights reserved.},
Doi = {10.1093/imanum/drl032},
Key = {fds243862}
}
@article{fds243866,
Author = {Mattingly, JC and Suidan, TM and Vanden-Eijnden,
E},
Title = {Anomalous dissipation in a stochastically forced
infinite-dimensional system of coupled oscillators},
Journal = {Journal of Statistical Physics},
Volume = {128},
Number = {5},
Pages = {1145-1152},
Publisher = {Springer Nature},
Year = {2007},
ISSN = {0022-4715},
MRCLASS = {37Lxx (60H10 82C05)},
MRNUMBER = {MR2348788},
url = {http://dx.doi.org/10.1007/s10955-007-9351-8},
Abstract = {We study a system of stochastically forced
infinite-dimensional coupled harmonic oscillators. Although
this system formally conserves energy and is not explicitly
dissipative, we show that it has a nontrivial invariant
probability measure. This phenomenon, which has no finite
dimensional equivalent, is due to the appearance of some
anomalous dissipation mechanism which transports energy to
infinity. This prevents the energy from building up locally
and allows the system to converge to the invariant measure.
The invariant measure is constructed explicitly and some of
its properties are analyzed. © 2007 Springer
Science+Business Media, LLC.},
Doi = {10.1007/s10955-007-9351-8},
Key = {fds243866}
}
@article{fds304491,
Author = {Nijhout, HF and Reed, MC and Anderson, DF and Mattingly, JC and James,
SJ and Ulrich, CM},
Title = {Erratum to H. Frederik Nijhout, et al. Epigenetics Volume 1,
Issue 2; pp. 81-87.},
Journal = {Epigenetics},
Volume = {1},
Number = {3},
Pages = {115-115},
Publisher = {Informa UK Limited},
Year = {2006},
Month = {July},
ISSN = {1559-2294},
url = {http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000207063900001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=47d3190e77e5a3a53558812f597b0b92},
Doi = {10.4161/epi.1.3.3281},
Key = {fds304491}
}
@article{fds243857,
Author = {Mattingly, JC and Pardoux, É},
Title = {Malliavin calculus for the stochastic 2D Navier-Stokes
equation},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {59},
Number = {12},
Pages = {1742-1790},
Publisher = {WILEY},
Year = {2006},
ISSN = {0010-3640},
url = {http://dx.doi.org/10.1002/cpa.20136},
Abstract = {We consider the incompressible, two-dimensional
Navier-Stokes equation with periodic boundary conditions
under the effect of an additive, white-in-time, stochastic
forcing. Under mild restrictions on the geometry of the
scales forced, we show that any finite-dimensional
projection of the solution possesses a smooth, strictly
positive density with respect to Lebesgue measure. In
particular, our conditions are viscosity independent. We are
mainly interested in forcing that excites a very small
number of modes. All of the results rely on proving the
nondegeneracy of the infinite-dimensional Malliavin matrix.
© 2006 Wiley Periodicals, Inc.},
Doi = {10.1002/cpa.20136},
Key = {fds243857}
}
@article{fds243865,
Author = {Hairer, M and Mattingly, JC},
Title = {Ergodicity of the 2D Navier-Stokes equations with degenerate
stochastic forcing},
Journal = {Annals of Mathematics},
Volume = {164},
Number = {3},
Pages = {993-1032},
Publisher = {Annals of Mathematics, Princeton U},
Year = {2006},
ISSN = {0003-486X},
url = {http://dx.doi.org/10.4007/annals.2006.164.993},
Abstract = {The stochastic 2D Navier-Stokes equations on the torus
driven by degenerate noise are studied. We characterize the
smallest closed invariant subspace for this model and show
that the dynamics restricted to that subspace is ergodic. In
particular, our results yield a purely geometric
characterization of a class of noises for which the equation
is ergodic in L02(double struck T sighn2). Unlike previous
works, this class is independent of the viscosity and the
strength of the noise. The two main tools of our analysis
are the asymptotic strong Feller property, introduced in
this work, and an approximate integration by parts formula.
The first, when combined with a weak type of irreducibility,
is shown to ensure that the dynamics is ergodic. The second
is used to show that the first holds under a Hörmander-type
condition. This requires some interesting nonadapted
stochastic analysis.},
Doi = {10.4007/annals.2006.164.993},
Key = {fds243865}
}
@article{fds243860,
Author = {Bakhtin, Y and Mattingly, JC},
Title = {Stationary solutions of stochastic differential equations
with memory and stochastic partial differential
equations},
Journal = {Communications in Contemporary Mathematics},
Volume = {7},
Number = {5},
Pages = {553-582},
Publisher = {World Scientific Pub Co Pte Lt},
Year = {2005},
Month = {October},
ISSN = {0219-1997},
MRNUMBER = {MR2175090},
url = {http://dx.doi.org/10.1142/S0219199705001878},
Abstract = {We explore Itô stochastic differential equations where the
drift term possibly depends on the infinite past. Assuming
the existence of a Lyapunov function, we prove the existence
of a stationary solution assuming only minimal continuity of
the coefficients. Uniqueness of the stationary solution is
proven if the dependence on the past decays sufficiently
fast. The results of this paper are then applied to
stochastically forced dissipative partial differential
equations such as the stochastic Navier-Stokes equation and
stochastic Ginsburg-Landau equation. © World Scientific
Publishing Company.},
Doi = {10.1142/S0219199705001878},
Key = {fds243860}
}
@article{b:MattinglySuidan05,
Author = {Mattingly, JC and Suidan, TM},
Title = {The small scales of the stochastic Navier-Stokes equations
under rough forcing},
Journal = {Journal of Statistical Physics},
Volume = {118},
Number = {1-2},
Pages = {343-364},
Publisher = {Springer Nature},
Year = {2005},
Month = {January},
ISSN = {0022-4715},
MRNUMBER = {MR2122959},
url = {http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000227233700013&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=47d3190e77e5a3a53558812f597b0b92},
Abstract = {We prove that the small scale structures of the
stochastically forced Navier-Stokes equations approach those
of the naturally associated Ornstein-Uhlenbeck process as
the scales get smaller. Precisely, we prove that the
rescaled kth spatial Fourier mode converges weakly on path
space to an associated Ornstein-Uhlenbeck process as |k| →
∞. In addition, we prove that the Navier-Stokes equations
and the naturally associated Ornstein-Uhlenbeck process
induce equivalent transition densities if the viscosity is
replaced with hyperviscosity. This gives a simple proof of
unique ergodicity for the hyperviscous Navier-Stokes system.
We show how different strengthened hyperviscosity produce
varying levels of equivalence.},
Doi = {10.1007/s10955-004-8787-3},
Key = {b:MattinglySuidan05}
}
@article{b:HairerMattingly04b,
Author = {Hairer, M and Mattingly, JC},
Title = {Ergodic properties of highly degenerate 2D stochastic
Navier-Stokes equations},
Journal = {Comptes Rendus Mathématique. Académie des Sciences.
Paris},
Volume = {339},
Number = {12},
Pages = {879-882},
Publisher = {Elsevier BV},
Year = {2004},
ISSN = {1631-073X},
MRNUMBER = {MR2111726},
url = {http://dx.doi.org/10.1016/j.crma.2004.09.035},
Abstract = {This Note presents the results from "Ergodicity of the
degenerate stochastic 2D Navier-Stokes equation"; by M.
Hairer and J.C. Mattingly. We study the Navier-Stokes
equation on the two-dimensional torus when forced by a
finite dimensional Gaussian white noise and give conditions
under which the system is ergodic. In particular, our
results hold for specific choices of four-dimensional
Gaussian white noise. © 2004 Académie des sciences.
Published by Elsevier SAS. All rights reserved.},
Doi = {10.1016/j.crma.2004.09.035},
Key = {b:HairerMattingly04b}
}
@article{fds243842,
Author = {Hairer, M and Mattingly, JC and Pardoux, É},
Title = {Malliavin calculus for highly degenerate 2D stochastic
Navier-Stokes equations},
Journal = {Comptes Rendus Mathématique. Académie des Sciences.
Paris},
Volume = {339},
Number = {11},
Pages = {793-796},
Publisher = {Elsevier BV},
Year = {2004},
ISSN = {1631-073X},
MRNUMBER = {MR2110383},
url = {http://dx.doi.org/10.1016/j.crma.2004.09.002},
Abstract = {This Note mainly presents the results from "Malliavin
calculus and the randomly forced Navier-Stokes equation" by
J.C. Mattingly and E. Pardoux. It also contains a result
from "Ergodicity of the degenerate stochastic 2D
Navier-Stokes equation" by M. Hairer and J.C. Mattingly. We
study the Navier-Stokes equation on the two-dimensional
torus when forced by a finite dimensional Gaussian white
noise. We give conditions under which the law of the
solution at any time t > 0, projected on a finite
dimensional subspace, has a smooth density with respect to
Lebesgue measure. In particular, our results hold for
specific choices of four dimensional Gaussian white noise.
Under additional assumptions, we show that the preceding
density is everywhere strictly positive. This Note's results
are a critical component in the ergodic results discussed in
a future article. © 2004 Académie des sciences. Published
by Elsevier SAS. All rights reserved.},
Doi = {10.1016/j.crma.2004.09.002},
Key = {fds243842}
}
@article{fds303550,
Author = {Hairer, M and Mattingly, JC and Pardoux, E},
Title = {Malliavin calculus and ergodic properties of highly
degenerate 2D stochastic Navier–Stokes
equation},
Journal = {arXiv preprint math/0409057},
Year = {2004},
url = {http://arxiv.org/abs/math/0409057v1},
Abstract = {The objective of this note is to present the results from
the two recent papers. We study the Navier--Stokes equation
on the two--dimensional torus when forced by a finite
dimensional white Gaussian noise. We give conditions under
which both the law of the solution at any time t>0,
projected on a finite dimensional subspace, has a smooth
density with respect to Lebesgue measure and the solution
itself is ergodic. In particular, our results hold for
specific choices of four dimensional white Gaussian noise.
Under additional assumptions, we show that the preceding
density is everywhere strictly positive.},
Key = {fds303550}
}
@article{fds243859,
Author = {Mattingly, JC},
Title = {On recent progress for the stochastic Navier Stokes
equations},
Volume = {XV},
Pages = {Exp. No. XI-52},
Publisher = {Univ. Nantes, Nantes},
Year = {2003},
Month = {Summer},
MRNUMBER = {MR2050586(2004j:00022)},
url = {http://www.math.duke.edu/~jonm/PaperArchive/03/ForgesLesEaux/forgesLesEaux2003.pdf},
Abstract = {We give an overview of the ideas central to some recent
developments in the ergodic theory of the stochastically
forced Navier Stokes equations and other dissipative
stochastic partial differential equations. Since our desire
is to make the core ideas clear, we will mostly work with a
specific example: the stochastically forced Navier Stokes
equations. To further clarify ideas, we will also examine in
detail a toy problem. A few general theorems are given.
Spatial regularity, ergodicity, exponential mixing, coupling
for a SPDE, and hypoellipticity are all discussed.},
Key = {fds243859}
}
@article{fds10350,
Author = {Mattingly, J. C.},
Title = {Contractivity and ergodicity of the random map
{$x\mapsto\vert x-\theta\vert $}},
Journal = {Teor. Veroyatnost. i Primenen.},
Volume = {47},
Number = {2},
Pages = {388--397},
Year = {2002},
MRNUMBER = {2004f:60148},
url = {http://www.math.duke.edu/~jonm/PaperArchive/01/AbsMap/absMap.pdf},
Abstract = {The long time behavior of the random map $x_n \mapsto
x_{n+1} |x_n-\theta_n|$ is studied under various assumptions
on the distribution of the $\theta_n$. One of the
interesting features of this random dynamical system is that
for a single fixed deterministic $\theta$ the map is not a
contraction, while the composition is almost surely a
contraction if $\theta$ is picked randomly with only mild
assumptions on the distribution of the $\theta$'s. The
system is useful as an explicit model where more abstract
ideas can be explored concretely. We explore various
measures of convergence rates, hyperbolically from
randomness, and the structure of the random
attractor.},
Key = {fds10350}
}
@article{fds243838,
Author = {Mattingly, JC and Stuart, AM},
Title = {Geometric ergodicity of some hypo-elliptic diffusions for
particle motions},
Journal = {Markov Processes and Related Fields},
Volume = {8},
Number = {2},
Pages = {199-214},
Year = {2002},
ISSN = {1024-2953},
MRNUMBER = {2003g:60101},
url = {https://www.math.duke.edu/~jonm/PaperArchive/01/StuartParticles/cergy.pdf},
Abstract = {Two degenerate SDEs arising in statistical physics are
studied. The first is a Langevin equation with
state-dependent noise and damping. The second is the
equation of motion for a particle obeying Stokes' law in a
Gaussian random field; this field is chosen to mimic certain
features of turbulence. Both equations are hypo-elliptic and
smoothness of probability densities may be established. By
developing appropriate Lyapunov functions and by studying
the necessary control problems, geometric ergodicity is
proved.},
Key = {fds243838}
}
@article{fds243844,
Author = {Mattingly, JC},
Title = {Exponential convergence for the stochastically forced
Navier-Stokes equations and other partially dissipative
dynamics},
Journal = {Communications in Mathematical Physics},
Volume = {230},
Number = {3},
Pages = {421-462},
Year = {2002},
ISSN = {0010-3616},
MRNUMBER = {2004a:76039},
url = {http://www.math.duke.edu/~jonm/PaperArchive/01/NsMixing/nsMixing.pdf},
Abstract = {We prove that the two dimensional Navier-Stokes equations
possess an exponentially attracting invariant measure. This
result is in fact the consequence of a more general
"Harris-like" ergodic theorem applicable to many dissipative
stochastic PDEs and stochastic processes with memory. A
simple iterated map example is also presented to help build
intuition and showcase the central ideas in a less
encumbered setting. To analyze the iterated map, a general
"Doeblin-like" theorem is proven. One of the main features
of this paper is the novel coupling construction used to
examine the ergodic theory of the non-Markovian
processes.},
Doi = {10.1007/s00220-002-0688-1},
Key = {fds243844}
}
@article{fds243846,
Author = {Mattingly, JC},
Title = {Contractivity and ergodicity of the random map
$x\mapsto\vert x-θ\vert $},
Journal = {Rossi\u\i skaya Akademiya Nauk. Teoriya Veroyatnoste\u\i i
ee Primeneniya},
Volume = {47},
Number = {2},
Pages = {388-397},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2002},
ISSN = {0040-585X},
url = {http://hdl.handle.net/10161/10833 Duke open
access},
Abstract = {The long time behavior of the random map xn → xn+1 =
|xn-θn| is studied under various assumptions on the
distribution of the θn. One of the interesting features of
this random dynamical system is that for a single fixed
deterministic θ the map is not a contraction, while the
composition is almost surely a contraction if θ is chosen
randomly with only mild assumptions on the distribution of
the θ's. The system is useful as an explicit model where
more abstract ideas can be explored concretely. We explore
various measures of convergence rates, hyperbolically from
randomness, and the structure of the random
attractor.},
Doi = {10.1137/S0040585X97979767},
Key = {fds243846}
}
@article{fds243848,
Author = {Mattingly, JC and Stuart, AM and Higham, DJ},
Title = {Ergodicity for SDEs and approximations: locally Lipschitz
vector fields and degenerate noise},
Journal = {Stochastic Processes and their Applications},
Volume = {101},
Number = {2},
Pages = {185-232},
Publisher = {Elsevier BV},
Year = {2002},
ISSN = {0304-4149},
MRNUMBER = {2003i:60103},
url = {https://www.math.duke.edu/~jonm/PaperArchive/00/HarrisNumerics/harrisNumerics.pdf},
Abstract = {The ergodic properties of SDEs, and various time
discretizations for SDEs, are studied. The ergodicity of
SDEs is established by using techniques from the theory of
Markov chains on general state spaces, such as that
expounded by Meyn-Tweedie. Application of these Markov chain
results leads to straightforward proofs of geometric
ergodicity for a variety of SDEs, including problems with
degenerate noise and for problems with locally Lipschitz
vector fields. Applications where this theory can be
usefully applied include damped-driven Hamiltonian problems
(the Langevin equation), the Lorenz equation with degenerate
noise and gradient systems. The same Markov chain theory is
then used to study time-discrete approximations of these
SDEs. The two primary ingredients for ergodicity are a
minorization condition and a Lyapunov condition. It is shown
that the minorization condition is robust under
approximation. For globally Lipschitz vector fields this is
also true of the Lyapunov condition. However in the locally
Lipschitz case the Lyapunov condition fails for explicit
methods such as Euler-Maruyama; for pathwise approximations
it is, in general, only inherited by specially constructed
implicit discretizations. Examples of such discretization
based on backward Euler methods are given, and approximation
of the Langevin equation studied in some detail. © 2002
Elsevier Science B.V. All rights reserved.},
Doi = {10.1016/S0304-4149(02)00150-3},
Key = {fds243848}
}
@article{fds243850,
Author = {Mattingly, JC},
Title = {The dissipative scale of the stochastics Navier-Stokes
equation: regularization and analyticity},
Journal = {Journal of Statistical Physics},
Volume = {108},
Number = {5-6},
Pages = {1157-1179},
Year = {2002},
ISSN = {0022-4715},
MRNUMBER = {2004e:76035},
url = {http://www.math.duke.edu/~jonm/PaperArchive/01/SnsGevery/snsGevrey.pdf},
Abstract = {We prove spatial analyticity for solutions of the
stochastically forced Navier-Stokes equation, provided that
the forcing is sufficiently smooth spatially. We also give
estimates, which extend to the stationary regime, providing
strong control of both of the expected rate of dissipation
and fluctuations about this mean. Surprisingly, we could not
obtain non-random estimates of the exponential decay rate of
the spatial Fourier spectra.},
Doi = {10.1023/A:1019799700126},
Key = {fds243850}
}
@article{fds318323,
Author = {Mattingly, JC},
Title = {Contractivity and ergodicity of the random map
$x\mapsto|x-\theta|$},
Journal = {Teoriya Veroyatnostei i ee Primeneniya},
Volume = {47},
Number = {2},
Pages = {388-397},
Publisher = {Steklov Mathematical Institute},
Year = {2002},
url = {http://dx.doi.org/10.4213/tvp3671},
Doi = {10.4213/tvp3671},
Key = {fds318323}
}
@article{fds243845,
Author = {E, W and Mattingly, JC},
Title = {Ergodicity for the Navier-Stokes equation with degenerate
random forcing: finite-dimensional approximation},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {54},
Number = {11},
Pages = {1386-1402},
Publisher = {WILEY},
Year = {2001},
ISSN = {0010-3640},
MRNUMBER = {2002g:76075},
url = {http://www.math.duke.edu/~jonm/PaperArchive/00/GalerkinNS/galerkinNS.pdf},
Abstract = {We study Galerkin truncations of the two-dimensional
Navier-Stokes equation under degenerate, large-scale,
stochastic forcing. We identify the minimal set of modes
that has to be forced in order for the system to be ergodic.
Our results rely heavily on the structure of the
nonlinearity. © 2001 John Wiley & Sons,
Inc.},
Doi = {10.1002/cpa.10007},
Key = {fds243845}
}
@article{fds243849,
Author = {E, W and Mattingly, JC and Sinai, Y},
Title = {Gibbsian dynamics and ergodicity for the stochastically
forced Navier-Stokes equation},
Journal = {Communications in Mathematical Physics},
Volume = {224},
Number = {1},
Pages = {83-106},
Year = {2001},
ISSN = {0010-3616},
MRNUMBER = {2002m:76024},
url = {http://www.math.duke.edu/~jonm/PaperArchive/00/Gibbsian/gibbsian.pdf},
Abstract = {We study stationary measures for the two-dimensional
Navier-Stokes equation with periodic boundary condition and
random forcing. We prove uniqueness of the stationary
measure under the condition that all "determining modes" are
forced. The main idea behind the proof is to study the
Gibbsian dynamics of the low modes obtained by representing
the high modes as functionals of the time-history of the low
modes.},
Doi = {10.1007/s002201224083},
Key = {fds243849}
}
@article{fds372199,
Author = {Holmes, PJ and Mattingly, JC and Wittenberg, RW},
Title = {Low-Dimensional Models of Turbulence},
Pages = {177-215},
Publisher = {Springer Netherlands},
Year = {2001},
ISBN = {9780792369769},
url = {http://dx.doi.org/10.1007/978-94-010-0732-0_7},
Doi = {10.1007/978-94-010-0732-0_7},
Key = {fds372199}
}
@article{fds243847,
Author = {Mattingly, JC},
Title = {Ergodicity of $2$D Navier-Stokes equations with random
forcing and large viscosity},
Journal = {Communications in Mathematical Physics},
Volume = {206},
Number = {2},
Pages = {273-288},
Publisher = {Springer Nature},
Year = {1999},
ISSN = {0010-3616},
url = {http://dx.doi.org/10.1007/s002200050706},
Abstract = {The stochastically forced, two-dimensional, incompressable
Navier-Stokes equations are shown to possess an unique
invariant measure if the viscosity is taken large enough.
This result follows from a stronger result showing that at
high viscosity there is a unique stationary solution which
attracts solutions started from arbitrary initial
conditions. That is to say, the system has a trivial random
attractor. Along the way, results controling the expectation
and averaging time of the energy and enstrophy are
given.},
Doi = {10.1007/s002200050706},
Key = {fds243847}
}
@article{fds243856,
Author = {Mattingly, JC and Sinai, YG},
Title = {An elementary proof of the existence and uniqueness theorem
for the Navier-Stokes equations},
Journal = {Communications in Contemporary Mathematics},
Volume = {1},
Number = {4},
Pages = {497-516},
Publisher = {World Scientific Pub Co Pte Lt},
Year = {1999},
MRNUMBER = {2000j:35226},
url = {http://hdl.handle.net/10161/9459 Duke open
access},
Abstract = {The purpose of this paper is to show that some results
concerning solutions of the Navier-Stokes systems can be
proven by purely elementary methods using imagery from
Dynamical Systems.},
Doi = {10.1142/S0219199799000183},
Key = {fds243856}
}
@article{fds243843,
Author = {Holmes, PJ and Lumley, JL and Berkooz, G and Mattingly, JC and Wittenberg, RW},
Title = {Low-dimensional models of coherent structures in
turbulence},
Journal = {Physics Report},
Volume = {287},
Number = {4},
Pages = {337-384},
Publisher = {Elsevier BV},
Year = {1997},
Month = {January},
ISSN = {0370-1573},
MRNUMBER = {98j:76065},
url = {http://www.math.duke.edu/~jonm/PaperArchive/98/PhysRep/physrep.pdf},
Abstract = {For fluid flow one has a well-accepted mathematical model:
the Navier-Stokes equations. Why, then, is the problem of
turbulence so intractable? One major difficulty is that the
equations appear insoluble in any reasonable sense. (A
direct numerical simulation certainly yields a "solution",
but it provides little understanding of the process per se.)
However, three developments are beginning to bear fruit: (1)
The discovery, by experimental fluid mechanicians, of
coherent structures in certain fully developed turbulent
flows; (2) the suggestion, by Ruelle, Takens and others,
that strange attractors and other ideas from dynamical
systems theory might play a role in the analysis of the
governing equations, and (3) the introduction of the
statistical technique of Karhunen-Loève or proper
orthogonal decomposition, by Lumley in the case of
turbulence. Drawing on work on modeling the dynamics of
coherent structures in turbulent flows done over the past
ten years, and concentrating on the near-wall region of the
fully developed boundary layer, we describe how these three
threads can be drawn together to weave low-dimensional
models which yield new qualitative understanding. We focus
on low wave number phenomena of turbulence generation,
appealing to simple, conventional modeling of inertial range
transport and energy dissipation.},
Doi = {10.1016/S0370-1573(97)00017-3},
Key = {fds243843}
}
@article{fds10313,
Author = {Mattingly, Jonathan C.},
Title = {Ergodicity of 2D Navier-Stokes equations with random forcing
and large viscosity},
Journal = {Comm. Math. Phys., vol. 206, no. 2, pp. 273--288,
1999},
MRNUMBER = {2000k:76040},
url = {http://www.math.duke.edu/~jonm/PaperArchive/98/LargeNu/largeNu.pdf},
Key = {fds10313}
}
@article{fds10320,
Author = {Jonathan C. Mattingly},
Title = {The Stochastic Navier-Stokes Equation: Energy Estimates and
Phase Space Contraction},
Journal = {PhD Thesis, Princeton University 1998},
url = {http://www.math.duke.edu/~jonm/PaperArchive/98/Thesis/thesisC1.ps},
Key = {fds10320}
}
%% Papers Submitted
@article{fds225248,
Author = {J.C. Mattingly and Stephan Huckemann and Ezra Miller and James Nolen},
Title = {Sticky central limit theorems at isolated hyperbolic planar
singularities},
Year = {2014},
Month = {October},
url = {http://arxiv.org/abs/1410.6879},
Abstract = {We derive the limiting distribution of the barycenter bn of
an i.i.d. sample of n random points on a planar cone with
angular spread larger than 2π. There are three mutually
exclusive possibilities: (i) (fully sticky case) after a
finite random time the barycenter is almost surely at the
origin; (ii) (partly sticky case) the limiting distribution
of n‾√bn comprises a point mass at the origin, an open
sector of a Gaussian, and the projection of a Gaussian to
the sector's bounding rays; or (iii) (nonsticky case) the
barycenter stays away from the origin and the renormalized
fluctuations have a fully supported limit
distribution---usually Gaussian but not always. We conclude
with an alternative, topological definition of stickiness
that generalizes readily to measures on general metric
spaces.},
Key = {fds225248}
}
@article{fds223434,
Author = {J.C. Mattingly and David P. Herzog},
Title = {Noise-Induced Stabilization of Planar Flows
II},
Year = {2014},
Month = {April},
url = {http://arxiv.org/abs/1404.0955},
Key = {fds223434}
}
@article{fds223435,
Author = {J.C. Mattingly and David P. Herzog},
Title = {Noise-Induced Stabilization of Planar Flows
I},
Year = {2014},
Month = {April},
url = {http://arxiv.org/abs/1404.0957},
Key = {fds223435}
}
@article{fds224076,
Author = {J.C. Mattingly and Sean D. Lawley and Michael C. Reed},
Title = {Stochastic switching in infinite dimensions with
applications to random parabolic PDEs},
Year = {2014},
url = {http://arxiv.org/abs/1407.2264},
Abstract = {We consider parabolic PDEs with randomly switching boundary
conditions. In order to analyze these random PDEs, we
consider more general stochastic hybrid systems and prove
convergence to, and properties of, a stationary
distribution. Applying these general results to the heat
equation with randomly switching boundary conditions, we
find explicit formulae for various statistics of the
solution and obtain almost sure results about its regularity
and structure. These results are of particular interest for
biological applications as well as for their significant
departure from behavior seen in PDEs forced by disparate
Gaussian noise. Our general results also have applications
to other types of stochastic hybrid systems, such as ODEs
with randomly switching right-hand sides.},
Key = {fds224076}
}
@article{fds221260,
Author = {Elizabeth Munch and Paul Bendich and Katharine Turner and Sayan
Mukherjee, Jonathan Mattingly and John Harer},
Title = {Probabilistic Fréchet Means and Statistics on
Vineyards},
Year = {2013},
url = {http://arxiv.org/abs/1307.6530},
Abstract = {In order to use persistence diagrams as a true statistical
tool, it would be very useful to have a good notion of mean
and variance for a set of diagrams. Mileyko and his
collaborators made the first study of the properties of the
Fr\'{e}chet mean in (Dp,Wp), the space of persistence
diagrams equipped with the p-th Wasserstein metric. In
particular, they showed that the Fr\'{e}chet mean of a
finite set of diagrams always exists, but is not necessarily
unique. As an unfortunate consequence, one sees that the
means of a continuously-varying set of diagrams do not
themselves vary continuously, which presents obvious
problems when trying to extend the Fr\'{e}chet mean
definition to the realm of vineyards. We fix this problem by
altering the original definition of Fr\'{e}chet mean so that
it now becomes a probability measure on the set of
persistence diagrams; in a nutshell, the mean of a set of
diagrams will be a weighted sum of atomic measures, where
each atom is itself the (Fr\'{e}chet mean) persistence
diagram of a perturbation of the input diagrams. We show
that this new definition defines a (H\"older) continuous
map, for each k, from (Dp)k→P(Dp), and we present several
examples to show how it may become a useful statistic on
vineyards.},
Key = {fds221260}
}
%% Preprints
@article{fds139692,
Author = {Martin Hairer and Jonathan C. Mattingly and Etienne
Pardoux},
Title = {Malliavin calculus and ergodic properties of highly
degenerate 2D stochastic Navier--Stokes equation},
Journal = {Comptes rendus Mathematique (CRAS), In press},
Year = {2004},
Month = {Summer},
url = {http://arxiv.org/abs/math/0409057},
Abstract = {The objective of this note is to present the results from
the two recent papers. We study the Navier--Stokes equation
on the two--dimensional torus when forced by a finite
dimensional white Gaussian noise. We give conditions under
which both the law of the solution at any time t>0,
projected on a finite dimensional subspace, has a smooth
density with respect to Lebesgue measure and the solution
itself is ergodic. In particular, our results hold for
specific choices of four dimensional white Gaussian noise.
Under additional assumptions, we show that the preceding
density is everywhere strictly positive.},
Key = {fds139692}
}
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Mathematics Department