Publications of Jonathan C. Mattingly :chronological combined listing:
%% Papers Published
@article{fds163256,
Author = {Hairer, Martin and Mattingly, Jonathan C.},
Title = {Slow energy dissipation in anharmonic oscillator
chains},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {62},
Number = {8},
Pages = {999--1032},
Year = {2009},
ISSN = {0010-3640},
MRCLASS = {82C20 (35K55 37Lxx)},
MRNUMBER = {MR2531551},
Abstract = {We study the dynamic behaviour 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 $V_1(q_i) =|q_i|^{2k}/2k$ and harmonic coupling
potentials $V_2(q_i- q_{i-1}) = (q_i- q_{i-1})^2/2$ between
itself and its nearest neighbours. We consider the case $k >
1$ when the pinning potential is stronger then 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 three 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
zero 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 zero 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 which
confirm the theory.},
Key = {fds163256}
}
@article{fds152882,
Author = {J.C. Mattingly and Martin Hairer},
Title = {Spectral gaps in Wasserstein distances and the 2D stochastic
Navier-Stokes equations},
Journal = {Annals of Probability},
Number = {6},
Pages = {993--1032},
Year = {2008},
MRNUMBER = {2478676},
Abstract = {We develop a general method that allows to show 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 L^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 to total
variation convergence, regularly fail to hold. In the first
part of this paper, we consider semigroups that have uniform
behaviour which one can view as an extension of Doeblin's
condition. We then proceed to study situations where the
behaviour 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-Stokers equations,
even in situations where the forcing is extremely
degenerate. Using the convergence result, we show shat the
stochastic Navier-Stokes equations' invariant measures
depend continuously on the viscosity and the structure of
the forcing.},
Key = {fds152882}
}
@article{fds155602,
Author = {J.C. Mattingly and Iyer, Gautam and Mattingly, Jonathan},
Title = {A stochastic-{L}agrangian particle system for the
{N}avier-{S}tokes equations},
Journal = {Nonlinearity},
Volume = {21},
Number = {11},
Pages = {2537--2553},
Year = {2008},
ISSN = {0951-7715},
MRCLASS = {76D05 (35Q30 35R60 60H10 60H30)},
MRNUMBER = {MR2448230 (2009h:76060)},
Key = {fds155602}
}
@article{fds157004,
Author = {Mattingly, Jonathan C. and Suidan, Toufic
M.},
Title = {Transition measures for the stochastic {B}urgers
equation},
Journal = {, Integrable systems and random matrices},
Volume = {458},
Series = {Contemp. Math.},
Pages = {409--418},
Booktitle = {Integrable systems and random matrices},
Publisher = {Amer. Math. Soc.},
Address = {Providence, RI},
Year = {2008},
MRCLASS = {60Hxx (35Q53 35R60 60Jxx 76M35)},
MRNUMBER = {MR2411921},
Key = {fds157004}
}
@article{fds139724,
Author = {David Anderson and Jonathan C. Mattingly},
Title = {Propagation of fluctuations in biochemical systems, II:
nonlinear chains},
Journal = {IET Systems Biology},
Volume = {1},
Number = {6},
Pages = {313-325},
Year = {2007},
Month = {November},
Key = {fds139724}
}
@article{fds139695,
Author = {Mattingly, Jonathan C. and Suidan, Toufic and Vanden-Eijnden, Eric},
Title = {Simple systems with anomalous dissipation and energy
cascade},
Journal = {Communications in Mathematical Physics},
Volume = {276},
Number = {1},
Pages = {189--220},
Year = {2007},
ISSN = {0010-3616},
MRCLASS = {37L99 (37C99 37N10 76D05 76F02)},
MRNUMBER = {MR2342292 (2008m:37135)},
Abstract = {We analyze a class of dynamical systems of the type
\dot{a}_n(t) = c_{n−1} a_{n−1}(t) − c_n a_{n+1}(t) +
f_n(t), n ∈ N, a0 = 0, where f_n(t)is a forcing term with
f_n(t) ̸= 0onlyforn ≤ n⋆ < ∞ and the coupling coef-
ficients c_n satisfy a condition ensuring the formal
conservation of energy 1 ␣ |a_n(t)|2. 2n 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 fn(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 cn. 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 cn 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.},
Key = {fds139695}
}
@article{fds139688,
Author = {Lamba, H. and Mattingly, J. C. and Stuart, A.
M.},
Title = {An adaptive {E}uler-{M}aruyama scheme for {SDE}s:
convergence and stability},
Journal = {IMA Journal of Numerical Analysis},
Volume = {27},
Number = {3},
Pages = {479--506},
Year = {2007},
ISSN = {0272-4979},
MRCLASS = {60H35 (60H10 65C30)},
MRNUMBER = {MR2337577},
Key = {fds139688}
}
@article{fds139690,
Author = {Anderson, David F. and Mattingly, Jonathan C. and Nijhout,
H. Frederik and Reed, Michael C.},
Title = {Propagation of fluctuations in biochemical systems. {I}.
{L}inear {SSC} networks},
Journal = {Bulletin of Mathematical Biology},
Volume = {69},
Number = {6},
Pages = {1791--1813},
Year = {2007},
ISSN = {0092-8240},
MRCLASS = {92E20 (34F05 60H10 60H30)},
MRNUMBER = {MR2329180},
Key = {fds139690}
}
@article{fds139685,
Author = {Mattingly, Jonathan C. and Suidan, Toufic M. and Vanden-Eijnden, Eric},
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},
Year = {2007},
ISSN = {0022-4715},
MRCLASS = {37Lxx (60H10 82C05)},
MRNUMBER = {MR2348788},
Abstract = {We analyze a class of linear shell models subject to
stochastic forcing in finitely many degrees of freedom. The
unforced systems considered formally conserve energy.
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 is nonzero. This claim
is demonstrated via the complete characterization of the
solutions of the system above for specific choices of the
coupling coefficients. The mechanism of anomalous
dissipations is shown to arise via a cascade of the energy
towards the modes ($a_n$) with higher $n$; this is
responsible for solutions with interesting energy spectra,
namely $\EE |a_n|^2$ scales as $n^{-\alpha}$ as
$n\to\infty$. Here the exponents $\alpha$ depend on the
coupling coefficients $c_n$ and $\EE$ 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.},
Key = {fds139685}
}
@article{fds139686,
Author = {Bakhtin, Yuri and Mattingly, Jonathan C.},
Title = {Malliavin calculus for infinite-dimensional systems with
additive noise},
Journal = {Journal of Functional Analysis},
Volume = {249},
Number = {2},
Pages = {307--353},
Year = {2007},
ISSN = {0022-1236},
MRCLASS = {60H07 (76D05 76M35)},
MRNUMBER = {MR2345335},
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 system we develop a counterpart of Hormander's
classical theory in this setting. 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 Stochastic Reaction Diffusion Equation and
Stochastic 2D Navier--Stokes System.},
Key = {fds139686}
}
@article{fds49711,
Author = {Jonathan C. Mattingly and Etienne Pardoux},
Title = {Malliavin calculus for the Stochastic 2D Navier Stokes
Equation},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {59},
Number = {12},
Pages = {1742 - 1790},
Year = {2006},
Month = {December},
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 density with respect to
Lebesgue measure. We also show that under natural
assumptions the density of such a projection is everywhere
strictly positive. In particular, our conditions are
viscosity independent. We are mainly interested in forcing
which excites a very small number of modes. All of the
results rely on the nondegeneracy of the infinite
dimensional Malliavin matrix.},
Key = {fds49711}
}
@article{fds49712,
Author = {Martin Hairer and J.C. Mattingly},
Title = {Ergodicity of the 2D Navier-Stokes Equations with Degenerate
Stochastic Forcing},
Journal = {Annals of Mathematics},
Volume = {164},
Number = {3},
Year = {2006},
Month = {November},
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 L^2 of the torus. Unlike in 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 Hormander-type
condition. This requires some interesting non-adapted
stochastic analysis.},
Key = {fds49712}
}
@article{fds49730,
Author = {H. Frederik Nijhout and Michael C. Reed and David F. Anderson and Jonathan C. Mattingly and S. Jill james and Cornelia M.
Ulrich},
Title = {Long-Range Allosteric Interactions between the Folate and
Methionine Cycles Stabilize DNA Methylation Reaction Rate,
Epigenetics},
Journal = {Epigenetics},
Volume = {1},
Number = {2},
Pages = {81-87},
Year = {2006},
Key = {fds49730}
}
@article{fds43490,
Author = {Bakhtin, Yuri and Mattingly, Jonathan C.},
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},
Year = {2005},
MRNUMBER = {MR2175090},
Abstract = {We explore ItÆo stochastic di erential equations where the
drift term has possibly infinite dependence on the past.
Assuming the existence of a Lyapunov function, we prove the
existence of a stationary solution assuming only minimal
continuity of the coe cients. Uniqueness of the stationary
solution is proved if the dependence on the past decays su -
ciently fast. The results of this paper are then applied to
stochastically forced dissipative partial di erential
equations such as the stochastic Navier-Stokes equation and
stochastic Ginsburg-Landau equation.},
Key = {fds43490}
}
@article{b:MattinglySuidan05,
Author = {Mattingly, Jonathan C. and Suidan, Toufic
M.},
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},
Year = {2005},
MRNUMBER = {MR2122959},
Abstract = {We prove the small scale structure of the stochastically
forced Navier Stokes equations converge to those of the
naturally associated Ornstien-Uhlenbeck process as the scale
is taken to zero. In addition, we prove that the Navier
Stokes equations with su cient hyperviscosity and its
natural Ornstien-Uhlenbeck process induce equivalent
measures on path space. This gives a simple proof of unique
ergodicity for the hyperviscous Navier Stokes
system.},
Key = {b:MattinglySuidan05}
}
@article{b:HairerMattingly04b,
Author = {Hairer, Martin and Mattingly, Jonathan C.},
Title = {Ergodic properties of highly degenerate {2D} stochastic
{N}avier-{S}tokes equations},
Journal = {Comptes Rendus Math\'ematique. Acad\'emie des Sciences.
Paris},
Volume = {339},
Number = {12},
Pages = {879--882},
Year = {2004},
MRNUMBER = {MR2111726},
Key = {b:HairerMattingly04b}
}
@article{fds29478,
Author = {Martin Hairer and Jonathan C. Mattingly and Étienne
Pardoux},
Title = {Malliavin calculus for highly degenerate 2D stochastic
Navier–Stokes equations},
Journal = {Comptes Rendus Mathematique},
Volume = {339},
Number = {11},
Pages = {793-796},
Year = {2004},
MRNUMBER = {MR2110383},
Key = {fds29478}
}
@article{fds13566,
Author = {J.C. Mattingly},
Title = {On Recent Progress for the Stochastic Navier Stokes
Equations},
Journal = {Journées "Équations aux Dérivées Partielles"
(Forges-les-Eaux, 2003)},
Volume = {XV},
Pages = {viii+298},
Publisher = {Universit\'e de 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 = {fds13566}
}
@article{fds10349,
Author = {Mattingly, Jonathan C.},
Title = {The dissipative scale of the stochastics Navier-Stokes
equation: regularization and analyticity},
Journal = {J. Statist. Phys.},
Volume = {108},
Number = {5-6},
Pages = {1157--1179},
Year = {2002},
MRNUMBER = {2004e:76035},
url = {http://www.math.duke.edu/~jonm/PaperArchive/01/SnsGevery/snsGevrey.pdf},
Abstract = {We prove that the two dimensional Naiver-Stokes equations
posses 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
prove the central results.},
Key = {fds10349}
}
@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{fds10329,
Author = {Mattingly, J. C. and Stuart, A. M.},
Title = {Geometric ergodicity of some hypo-elliptic diffusions for
particle motions},
Journal = {Markov Process. Related Fields},
Volume = {8},
Number = {2},
Pages = {199--214},
Year = {2002},
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 = {fds10329}
}
@article{fds10352,
Author = {Mattingly, J. C. and Stuart, A. M. and Higham, D.
J.},
Title = {Ergodicity for {SDE}s and approximations: locally Lipschitz
vector fields and degenerate noise},
Journal = {Stochastic Process. Appl.},
Volume = {101},
Number = {2},
Pages = {185--232},
Year = {2002},
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. Application of these
Markov chain results leads to straightforward proofs of
ergodicity for a variety of SDEs, in particular for problems
with degenerate noise and for problems with locally
Lipschitz vector fields. The key points which need to be
verified are the existence of a Lyapunov function inducing
returns to a compact set, a uniformly reachable point from
within that set, and some smoothness of the probability
densities; the last two points imply a minorization
condition. Together the minorization condition and Lyapunov
structure give geometric ergodicity. Applications include
the Langevin equation, the Lorenz equation with degenerate
noise and gradient systems. The ergodic theorems proved are
strong, yielding exponential convergence of expectations for
classes of measurable functions restricted only by the
condition that they grow no faster than the Lyapunov
function. The same Markov chain theory is then used to study
time-discrete approximations of these SDEs. 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; 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.},
Key = {fds10352}
}
@article{fds23989,
Author = {J.C. Mattingly},
Title = {Exponential convergence for the stochastically forced
Navier-Stokes equations and other partially dissipative
dynamics},
Journal = {Comm. Math. Phys.},
Volume = {230},
Number = {3},
Pages = {421--462},
Year = {2002},
MRNUMBER = {2004a:76039},
url = {http://www.math.duke.edu/~jonm/PaperArchive/01/NsMixing/nsMixing.pdf},
Key = {fds23989}
}
@article{fds10311,
Author = {E, Weinan and Mattingly, Jonathan C.},
Title = {Ergodicity for the Navier-Stokes equation with degenerate
random forcing: finite-dimensional approximation},
Journal = {Comm. Pure Appl. Math.},
Volume = {54},
Number = {11},
Pages = {1386--1402},
Year = {2001},
MRNUMBER = {2002g:76075},
url = {http://www.math.duke.edu/~jonm/PaperArchive/00/GalerkinNS/galerkinNS.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.},
Key = {fds10311}
}
@article{fds10443,
Author = {Holmes, P. J. and Mattingly, J. C. and Wittenberg, R.
W.},
Title = {Low-dimensional models of turbulence or, The dynamics of
coherent structures},
Journal = {From finite to infinite dimensional dynamical systems
(Cambridge, 1995), vol. 19, pp. 177--215, 2001, Kluwer Acad.
Publ., Dordrecht, From finite to infinite dimensional
dynamical systems (Cambridge, 1995)},
Volume = {19},
Pages = {177--215},
Booktitle = {From finite to infinite dimensional dynamical systems
(Cambridge, 1995)},
Publisher = {Kluwer Acad. Publ., Dordrecht},
Year = {2001},
MRNUMBER = {2004h:76106},
Key = {fds10443}
}
@article{fds23992,
Author = {W. E and J.C. Mattingly and Ya Sinai},
Title = {Gibbsian dynamics and ergodicity for the stochastically
forced Navier-Stokes equation},
Journal = {Comm. Math. Phys.},
Volume = {224},
Number = {1},
Pages = {83--106},
Year = {2001},
MRNUMBER = {2002m:76024},
url = {http://www.math.duke.edu/~jonm/PaperArchive/00/Gibbsian/gibbsian.pdf},
Key = {fds23992}
}
@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{fds10314,
Author = {Mattingly, J. C. and Sinai, Ya. G.},
Title = {An elementary proof of the existence and uniqueness theorem
for the Navier-Stokes equations},
Journal = {Commun. Contemp. Math., vol. 1, no. 4, pp. 497--516,
1999},
MRNUMBER = {2000j:35226},
url = {http://www.math.duke.edu/~jonm/PaperArchive/98/ElementaryExistanceNS/elementaryExistanceNS.pdf},
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.},
Key = {fds10314}
}
@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}
}
@article{fds10315,
Author = {Holmes, Philip J. and Lumley, John L. and Berkooz, Gal and Mattingly, Jonathan C. and Wittenberg, Ralf
W.},
Title = {Low-dimensional models of coherent structures in
turbulence},
Journal = {Phys. Rep., vol. 287, no. 4, pp. 337--384,
1997},
MRNUMBER = {98j:76065},
url = {http://www.math.duke.edu/~jonm/PaperArchive/98/PhysRep/physrep.pdf},
Key = {fds10315}
}
%% Papers Accepted
@article{fds163255,
Author = {J.C. Mattingly and Martin Hairer and Michael Scheutzow},
Title = {Asymptotic coupling and a weak form of Harris' theorem with
applications to stochastic delay equations},
Year = {2009},
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 recent work on SPDEs on which this work builds.
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.},
Key = {fds163255}
}
@article{fds163257,
Author = {J.C. Mattingly and Martin Hairer},
Title = {Yet another look at Harris' ergodic theorem for Markov
chains},
Year = {2008},
Month = {August},
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 = {fds163257}
}
%% Papers Submitted
@article{fds164683,
Author = {J.C. Mattingly and Andrew M. Stuart and M.V. Tretyakov},
Title = {Convergence of Numerical Time-Averaging and Stationary
Measures via Poisson Equations},
Year = {2009},
Abstract = {Numerical approximation of the long time behavior of a
stochastic differential 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 advantage of this approach
is 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 general
hypoelliptic SDEs. An analogy between this approach and
Stein's method is indicated. Some practical implications of
the results are discussed.},
Key = {fds164683}
}
@article{fds160937,
Author = {J.C. Mattingly and David Anderson},
Title = {A weak trapezoidal method for a class of stochastic
differential equations},
Year = {2009},
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 Ito 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. This 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.},
Key = {fds160937}
}
@article{fds156904,
Author = {J.C. Mattingly and Scott A. McKinley and Natesh S. Pillai},
Title = {Geometric ergodicity of a bead-spring pair with stochastic
Stokes forcing},
Year = {2009},
Abstract = {We consider a simple model for the fluctuating hydrodynamics
of a flexible polymer in 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. To this, we add the possibility of excluding
certain "bad" sets in phase space in which the assumptions
are violated but from which the systems leaves with a
controllable probability. This allows for the treatment of
singular drifts, such as those derived from the
Lennard-Jones potential, which is an novel feature of this
work.},
Key = {fds156904}
}
@article{fds151344,
Author = {J.C. Mattingly and Martin Hairer},
Title = {A Theory of Hypoellipticity and Unique Ergodicity for
Semilinear Stochastic PDEs},
Year = {2008},
Key = {fds151344}
}
%% 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},
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