Scholars@Duke
Jonathan Christopher Mattingly

Jonathan Christopher Mattingly

Kimberly J. Jenkins Distinguished University Professor of New Technologies
Mathematics
jonathan.mattingly@duke.edu
105 Math/Physics Building, Box 90320, Durham, NC 27708
120 Science Drive, I, Durham, NC 27708

Selected Publications

PHASE SPACE CONTRACTION OF DEGENERATELY DAMPED RANDOM SPLITTINGS

Journal Article Probability and Mathematical Physics · January 1, 2025 When studying out-of-equilibrium systems, one often excites the dynamics in some degrees of freedom while removing the excitation in others through damping. In order for the system to converge to a statistical steady state, the dynamics must transfer the e ... Full text Cite

Convergence of stratified MCMC sampling of non-reversible dynamics

Journal Article Stochastics and Partial Differential Equations: Analysis and Computations · December 1, 2024 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 assumption ... Full text Open Access Cite

Random splitting of point vortex flows

Journal Article Electronic Communications in Probability · January 1, 2024 We consider a stochastic version of the point vortex system, in which the fluid velocity advects single vortices intermittently for small random times. Such system converges to the deterministic point vortex dynamics as the rate at which single components ... Full text Open Access Cite

Metropolized Forest Recombination for Monte Carlo Sampling of Graph Partitions

Journal Article SIAM Journal on Applied Mathematics · August 31, 2023 Full text Open Access Cite

Optimal enhanced dissipation and mixing for a time-periodic, Lipschitz velocity field on $\mathbb{T}^2$

Journal Article · April 11, 2023 We consider the advection-diffusion equation on $\mathbb{T}^2$ with a Lipschitz and time-periodic velocity field that alternates between two piecewise linear shear flows. We prove enhanced dissipation on the timescale $|\log \nu|$, where $\nu$ is the diffu ... Link to item Cite

Gibbsian dynamics and the generalized Langevin equation

Journal Article Electronic Journal of Probability · January 1, 2023 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 ... Full text Open Access Cite

Mathematically Quantifying Non-responsiveness of the 2021 Georgia Congressional Districting Plan

Conference ACM International Conference Proceeding Series · October 6, 2022 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 c ... Full text Open Access Cite

The Gaussian structure of the singular stochastic Burgers equation

Journal Article Forum of Mathematics, Sigma · September 2, 2022 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 equa ... Full text Open Access Cite

NOISE-INDUCED STRONG STABILIZATION

Journal Article Pure and Applied Functional Analysis · January 1, 2022 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 ... Open Access Cite

Limiting behaviors of high dimensional stochastic spin ensembles

Journal Article Communications in Mathematical Sciences · 2021 Full text Open Access Cite

Numerical Methods For Stochastic Differential Equations Based On Gaussian Mixture

Journal Article Communications in Mathematical Sciences · January 1, 2021 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, t ... Full text Open Access Cite

Singularities of invariant densities for random switching between two linear ODEs in 2D

Journal Article SIAM Journal on Applied Dynamical Systems · January 1, 2021 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. W ... Full text Open Access Cite

Metropolized Multiscale Forest Recombination for Redistricting

Journal Article Multiscale Modeling & Simulation · January 2021 Full text Open Access Cite

Nonlocal stochastic-partial-differential-equation limits of spatially correlated noise-driven spin systems derived to sample a canonical distribution

Journal Article Physical Review E · November 9, 2020 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 ( ... Full text Open Access Cite

Noise-induced strong stabilization

Journal Article · September 22, 2020 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 ... Open Access Link to item Cite

Non-reversible Markov chain Monte Carlo for sampling of districting maps

Journal Article · August 18, 2020 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. ... Open Access Link to item Cite

Multi-Scale Merge-Split Markov Chain Monte Carlo for Redistricting

Journal Article · August 18, 2020 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 spe ... Open Access Link to item Cite

Quantifying Gerrymandering in North Carolina

Journal Article Statistics and Public Policy · January 1, 2020 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 pro ... Full text Open Access Cite

Geometric ergodicity of Langevin dynamics with Coulomb interactions

Journal Article Nonlinearity · January 1, 2020 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 ex ... Full text Open Access Cite

Separating Effect From Significance in Markov Chain Tests

Journal Article Statistics and Public Policy · January 1, 2020 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 re ... Full text Open Access Cite

Optimal Legislative County Clustering in North Carolina

Journal Article Statistics and Public Policy · January 1, 2020 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 Car ... Full text Open Access Cite

Optimal Legislative County Clustering in North Carolina

Internet Publication · November 22, 2019 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 Car ... Open Access Link to item Cite

Ergodicity and Lyapunov Functions for Langevin Dynamics with Singular Potentials

Journal Article COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS · October 1, 2019 Full text Open Access Link to item Cite

The Signature of Gerrymandering in Rucho v. Common Cause

Journal Article South Carolina Law Review · 2019 Open Access Cite

Scaling Limit: Exact and Tractable Analysis of Online Learning Algorithms with Applications to Regularized Regression and PCA

Journal Article · December 7, 2017 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. ... Open Access Link to item Cite

Error bounds for Approximations of Markov chains used in Bayesian Sampling

Journal Article · November 14, 2017 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 re ... Open Access Link to item Cite

Evaluating Partisan Gerrymandering in Wisconsin

Journal Article · September 2, 2017 Featured Publication 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 ... Open Access Link to item Cite

Smooth invariant densities for random switching on the torus

Journal Article · August 1, 2017 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 transversa ... Open Access Link to item Cite

Coupling and Decoupling to bound an approximating Markov Chain

Journal Article · July 31, 2017 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 ... Open Access Link to item Cite

Scaling and Saturation in Infinite-Dimensional Control Problems with Applications to Stochastic Partial Differential Equations

Journal Article Annals of PDE · June 30, 2017 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 al ... Open Access Link to item Cite

Redistricting: Drawing the Line

Journal Article · April 9, 2017 Featured Publication 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 C ... Open Access Link to item Cite

Geometric ergodicity of two-dimensional hamiltonian systems with a Lennard-Jones-like repulsive potential

Journal Article Communications in Mathematical Sciences · January 1, 2017 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. Meth ... Full text Open Access Cite

The strong Feller property for singular stochastic PDEs

Journal Article · 2016 Featured Publication 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 m ... Open Access Link to item Cite

Trajectory stratification of stochastic dynamics

Journal Article SIAM Review · 2016 Featured Publication 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) ... Open Access Link to item Cite

Optimal approximating Markov chains for Bayesian inference

Journal Article · August 13, 2015 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 ... Open Access Link to item Cite

Sticky central limit theorems at isolated hyperbolic planar singularities

Journal Article Electronic Journal of Probability · 2015 Featured Publication Full text Open Access Cite

Scaling limits of a model for selection at two scales

Journal Article · 2015 Featured Publication 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 v ... Open Access Link to item Cite

Probabilistic Fréchet means for time varying persistence diagrams

Journal Article Electronic Journal of Statistics · January 1, 2015 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 ... Full text Open Access Cite

Invariant measure selection by noise. An example

Journal Article Discrete and Continuous Dynamical Systems. Series A · 2014 Featured Publication Full text Open Access Cite

Sensitivity to switching rates in stochastically switched ODEs

Journal Article Communications in Mathematical Sciences · 2014 Featured Publication Full text Open Access Cite

Noise-Induced Stabilization of Planar Flows I

Journal Article arXiv preprint arXiv:1404.0957 · 2014 Featured Publication Open Access Cite

Regularity of invariant densities for 1D-systems with random switching

Journal Article arXiv preprint arXiv:1406.5425 · 2014 Featured Publication Open Access Cite

A practical criterion for positivity of transition densities

Journal Article arXiv preprint arXiv:1407.3858 · 2014 Featured Publication Open Access Cite

Redistricting and the Will of the People

Journal Article arXiv preprint arXiv:1410.8796 · 2014 Featured Publication Open Access Cite

Sticky central limit theorems on open books

Journal Article The Annals of Applied Probability · 2013 Featured Publication Full text Open Access Cite

Geometric ergodicity of a bead-spring pair with stochastic Stokes forcing

Journal Article Stochastic Processes and their Applications · December 1, 2012 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 ... Full text Open Access Cite

The impact of host immune status on the within-host and population dynamics of antigenic immune escape.

Journal Article J R Soc Interface · October 7, 2012 Featured Publication 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 emerg ... Full text Open Access Link to item Cite

Propagating lyapunov functions to prove noise-induced stabilization

Journal Article Electronic Journal of Probability · 2012 Featured Publication 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 addit ... Full text Open Access Cite

Local kinetic interpretation of entropy production through reversed diffusion.

Journal Article Phys Rev E Stat Nonlin Soft Matter Phys · October 2011 Featured Publication 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 linke ... Full text Open Access Link to item Cite

Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations

Journal Article Probability Theory and Related Fields · 2011 Featured Publication 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 ... Full text Open Access Cite

A weak trapezoidal method for a class of stochastic differential equations

Journal Article Communications in Mathematical Sciences · 2011 Featured Publication 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 p ... Open Access Cite

A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs

Journal Article Electronic Journal of Probability · 2011 Featured Publication 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 bra ... Open Access Cite

A dimensionless number for understanding the evolutionary dynamics of antigenically variable RNA viruses

Journal Article Proceedings of the Royal Society B: Biological Sciences · 2011 Featured Publication 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 infec ... Full text Open Access Cite

Rare Transition Events in Nonequilibrium Systems with State-Dependent Noise: Application to Stochastic Current Switching in Semiconductor Superlattices

Journal Article · August 24, 2010 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. ... Open Access Link to item Cite

Diffusion limits of the random walk Metropolis algorithm in high dimensions

Journal Article Annals of Applied Probability · March 22, 2010 Featured Publication 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 stationari ... Open Access Link to item Cite

Convergence of numerical time-averaging and stationary measures via Poisson equations

Journal Article SIAM Journal on Numerical Analysis · 2010 Featured Publication 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 t ... Full text Open Access Cite

Slow energy dissipation in anharmonic oscillator chains

Journal Article Communications on Pure and Applied Mathematics · 2009 Featured Publication 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| ... Full text Open Access Cite

Spectral gaps in wasserstein distances and the 2d stochastic navier-stokes equations

Journal Article Annals of Probability · November 1, 2008 Featured Publication 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 ... Full text Open Access Cite

Propagation of fluctuations in biochemical systems, II: Nonlinear chains.

Journal Article IET systems biology · November 2007 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 hig ... Full text Cite

Simple systems with anomalous dissipation and energy cascade

Journal Article Communications in Mathematical Physics · November 1, 2007 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 cons ... Full text Open Access Cite

Malliavin calculus for infinite-dimensional systems with additive noise

Journal Article Journal of Functional Analysis · August 15, 2007 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 Hör ... Full text Cite

Propagation of fluctuations in biochemical systems, I: linear SSC networks.

Journal Article Bulletin of mathematical biology · August 2007 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 topolo ... Full text Open Access Cite

Anomalous dissipation in a stochastically forced infinite-dimensional system of coupled oscillators

Journal Article Journal of Statistical Physics · 2007 Featured Publication 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, ... Full text Open Access Cite

An adaptive Euler-Maruyama scheme for SDEs: Convergence and stability

Journal Article IMA Journal of Numerical Analysis · January 1, 2007 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 adapti ... Full text Open Access Cite

Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing

Journal Article Annals of Mathematics · 2006 Featured Publication 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 resul ... Cite

Malliavin calculus for the stochastic 2D Navier-Stokes equation

Journal Article Communications on Pure and Applied Mathematics · 2006 Featured Publication 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 fini ... Full text Open Access Cite

Stationary solutions of stochastic differential equations with memory and stochastic partial differential equations

Journal Article Communications in Contemporary Mathematics · October 1, 2005 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 ... Full text Open Access Cite

The small scales of the stochastic Navier-Stokes equations under rough forcing

Journal Article Journal of Statistical Physics · January 1, 2005 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 co ... Full text Cite

Malliavin calculus for highly degenerate 2D stochastic Navier-Stokes equations

Journal Article Comptes Rendus Mathématique. Académie des Sciences. Paris · 2004 Full text Cite

Ergodic properties of highly degenerate 2D stochastic Navier-Stokes equations

Journal Article Comptes Rendus Mathématique. Académie des Sciences. Paris · 2004 Full text Open Access Cite

Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise

Journal Article Stochastic Processes and their Applications · 2002 Full text Cite

Contractivity and ergodicity of the random map $x\mapsto\vert x-θ\vert $

Journal Article Rossi\u\i skaya Akademiya Nauk. Teoriya Veroyatnoste\u\i i ee Primeneniya · 2002 Full text Open Access Cite

Geometric ergodicity of some hypo-elliptic diffusions for particle motions

Journal Article Markov Processes and Related Fields · 2002 Cite

Contractivity and ergodicity of the random map $x\mapsto|x-\theta|$

Journal Article Теория вероятностей и ее применения · 2002 Full text Open Access Cite

Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation

Journal Article Communications in Mathematical Physics · 2001 Full text Cite

Ergodicity of $2$D Navier-Stokes equations with random forcing and large viscosity

Journal Article Communications in Mathematical Physics · 1999 Full text Cite

Low-dimensional models of coherent structures in turbulence

Journal Article Physics Report · January 1, 1997 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 simulat ... Full text Cite

A Merge-Split Proposal for Reversible Monte Carlo Markov Chain Sampling of Redistricting Plans

Journal Article We describe a Markov chain on redistricting plans that makes relatively global moves. 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 partitio ... Open Access Link to item Cite