Browsing by Author "Mattingly, JC"
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A weak trapezoidal method for a class of stochastic differential equations
Anderson, DF; Mattingly, JC (Communications in Mathematical Sciences, 20110301)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 ... 
An adaptive EulerMaruyama scheme for SDEs: Convergence and stability
Lamba, H; Mattingly, JC; Stuart, AM (IMA Journal of Numerical Analysis, 20070101)The understanding of adaptive algorithms for stochastic differential equations (SDEs) is an open area, where many issues related to both convergence and stability (longtime behaviour) of algorithms are unresolved. This ... 
An elementary proof of the existence and uniqueness theorem for the NavierStokes equations
Mattingly, JC; Sinai, Ya G (Communications in Contemporary Mathematics, 19991101) 
Anomalous dissipation in a stochastically forced infinitedimensional system of coupled oscillators
Mattingly, JC; Suidan, TM; VandenEijnden, E (Journal of Statistical Physics, 20070901)We study a system of stochastically forced infinitedimensional coupled harmonic oscillators. Although this system formally conserves energy and is not explicitly dissipative, we show that it has a nontrivial invariant ... 
Approximations of Markov Chains and HighDimensional Bayesian Inference
Mattingly, JC; Johndrow, J; Mukherjee, S; Dunson, D (2015) 
Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations
Hairer, M; Mattingly, JC; Scheutzow, M (Probability Theory and Related Fields, 20110301)There are many Markov chains on infinite dimensional spaces whose onestep transition kernels are mutually singular when starting from different initial conditions. We give results which prove unique ergodicity under minimal ... 
Contractivity and ergodicity of the random map x →
Mattingly, JC (Theory of Probability and its Applications, 20030626)The long time behavior of the random map xn → xn+1 = xnθn 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 ... 
Convergence of numerical timeaveraging and stationary measures via Poisson equations
Mattingly, JC; Stuart, AM; Tretyakov, MV (SIAM Journal on Numerical Analysis, 20100707)Numerical approximation of the long time behavior of a stochastic di.erential equation (SDE) is considered. Error estimates for timeaveraging estimators are obtained and then used to show that the stationary behavior of ... 
Coupling and Decoupling to bound an approximating Markov Chain
Johndrow, JE; Mattingly, JC (20170727)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 ... 
Diffusion limits of the random walk metropolis algorithm in high dimensions
Mattingly, JC; Pillai, NS; Stuart, AM (Annals of Applied Probability, 20120601)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 ... 
Ergodicity and Lyapunov functions for Langevin dynamics with singular potentials
Herzog, DP; Mattingly, JC (20171130)We study Langevin dynamics of $N$ particles on $R^d$ interacting through a singular repulsive potential, e.g.~the wellknown LennardJones type, and show that the system converges to the unique invariant Gibbs measure ... 
Geometric ergodicity of a beadspring pair with stochastic Stokes forcing
Mattingly, JC; McKinley, SA; Pillai, NS (Stochastic Processes and their Applications, 20121201)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 ... 
Geometric ergodicity of Langevin dynamics with Coulomb interactions
Lu, Y; Mattingly, JCThis paper is concerned with the long time behavior of Langevin dynamics of {\em Coulomb gases} in $\mathbf{R}^d$ with $d\geq 2$, that is a second order system of Brownian particles driven by an external force and ... 
Malliavin calculus for the stochastic 2D NavierStokes equation
Mattingly, JC; Pardoux, E (Communications on Pure and Applied Mathematics, 20061201)We consider the incompressible, twodimensional NavierStokes equation with periodic boundary conditions under the effect of an additive, whiteintime, stochastic forcing. Under mild restrictions on the geometry of the ... 
Noiseinduced stabilization of planar flows I
Herzog, DP; Mattingly, JC (Electronic Journal of Probability, 20151022)© 2015 University of Washington. All rights reserved.We show that the complexvalued ODE (n ≥ 1, an+1 6≠ 0): ź = an+1zn+1 + anzn +1zn + a0; which necessarily has trajectories along which the dynamics blows up in finite time, ... 
Nonlocal stochasticpartialdifferentialequation limits of spatially correlated noisedriven spin systems derived to sample a canonical distribution
Gao, Y; Marzuola, JL; Mattingly, JC; Newhall, KA (Physical Review E, 20201109)© 2020 American Physical Society. For a noisy spin system, we derive a nonlocal stochastic version of the overdamped LandauLipshitz equation designed to respect the underlying Hamiltonian structure and sample the canonical ... 
Numerical methods for stochastic differential equations based on Gaussian mixture
Li, L; Lu, J; Mattingly, JC; Wang, LWe develop in this work a numerical method for stochastic differential equations (SDEs) with weak second order accuracy based on Gaussian mixture. Unlike the conventional higher order schemes for SDEs based on It\^oTayl... 
On Unique Ergodicity in Nonlinear Stochastic Partial Differential Equations
GlattHoltz, N; Richards, G; Mattingly, JC (Journal of Statistical Physics, 20160831)© 2016 Springer Science+Business Media New YorkWe 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 ... 
Practical tests for significance in Markov Chains
Chikina, M; Frieze, A; Mattingly, JC; Pegden, WWe give improvements to theorems which enable significance testing in Markov Chains. 
Propagating lyapunov functions to prove noiseinduced stabilization
Athreyaz, A; Kolba, T; Mattingly, JC (Electronic Journal of Probability, 20121119)We investigate an example of noiseinduced 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 ...