Browsing by Author "Mattingly, Jonathan C"
Now showing items 1-20 of 27
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A practical criterion for positivity of transition densities
(Nonlinearity, 2015-07-10)© 2015 IOP Publishing Ltd & London Mathematical Society.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 ... -
A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs
(Electronic Journal of Probability, 2011-05-09)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. ... -
Ergodic properties of highly degenerate 2D stochastic Navier-Stokes equations
(Comptes Rendus Mathématique. Académie des Sciences. Paris, 2004)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 ... -
Ergodicity for the navier-stokes equation with degenerate random forcing: Finite-dimensional approximation
(Communications on Pure and Applied Mathematics, 2001-11-01)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. ... -
Error bounds for Approximations of Markov chains
(2017-11-30)The first part of this article gives error bounds for approximations of Markov kernels under Foster-Lyapunov conditions. The basic idea is that when both the approximating kernel and the original kernel satisfy a Foster-Lyapunov ... -
Geometric Ergodicity of Two–dimensional Hamiltonian systems with a Lennard–Jones–like Repulsive Potential
(arXiv preprint arXiv:1104.3842, 2011) -
Invariant measure selection by noise. An example
(Discrete and Continuous Dynamical Systems- Series A, 2014-01-01)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. ... -
Multi-Scale Merge-Split Markov Chain Monte Carlo for Redistricting
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 ... -
Noise-induced stabilization of planar flows ii
(Electronic Journal of Probability, 2015-10-25)© 2015 University of Washington. All rights reserved.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 ... -
Noise-induced strong stabilization
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 ... -
Non-local SPDE limits of spatially-correlated-noise driven spin systems derived to sample a canonical distribution
We study the macroscopic behavior of a stochastic spin ensemble driven by a discrete Markov jump process motivated by the Metropolis-Hastings algorithm where the proposal is made with spatially correlated (colored) noise, ... -
Non-reversible Markov chain Monte Carlo for sampling of districting maps
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 ... -
Propagation of fluctuations in biochemical systems, I: Linear SSC networks
(Bulletin of Mathematical Biology, 2007-08-01)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 ... -
Rare Transition Events in Nonequilibrium Systems with State-Dependent Noise: Application to Stochastic Current Switching in Semiconductor Superlattices
(arXiv preprint arXiv:1008.4037, 2010) -
Redistricting and the Will of the People
(arXiv preprint arXiv:1410.8796, 2014) -
Redistricting: Drawing the Line
(2017-04-12)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 ... -
Regularity of invariant densities for 1D systems with random switching
(Nonlinearity, 2015-09-30)© 2015 IOP Publishing Ltd & London Mathematical Society.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 ... -
Sensitivity to switching rates in stochastically switched ODEs
(Communications in Mathematical Sciences, 2014-01-01)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 ... -
Spectral gaps in wasserstein distances and the 2d stochastic navier-stokes equations
(Annals of Probability, 2008-11-01)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 ...
