Now showing items 1-20 of 29

• #### A weak trapezoidal method for a class of stochastic differential equations ﻿

(Communications in Mathematical Sciences, 2011-03-01)
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 Euler-Maruyama scheme for SDEs: Convergence and stability ﻿

(IMA Journal of Numerical Analysis, 2007-01-01)
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 ...
• #### An elementary proof of the existence and uniqueness theorem for the Navier-Stokes equations ﻿

(Communications in Contemporary Mathematics, 1999-11-01)
• #### Anomalous dissipation in a stochastically forced infinite-dimensional system of coupled oscillators ﻿

(Journal of Statistical Physics, 2007-09-01)
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 ...

(2015)
• #### Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations ﻿

(Probability Theory and Related Fields, 2011-03-01)
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 ...
• #### Contractivity and ergodicity of the random map x → ﻿

(Theory of Probability and its Applications, 2003-06-26)
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 ...
• #### Convergence of numerical time-averaging and stationary measures via Poisson equations ﻿

(SIAM Journal on Numerical Analysis, 2010-07-07)
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 ...
• #### Coupling and Decoupling to bound an approximating Markov Chain ﻿

(2017-07-27)
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 ﻿

(Annals of Applied Probability, 2012-06-01)
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 ﻿

(2017-11-30)
We study Langevin dynamics of $N$ particles on $R^d$ interacting through a singular repulsive potential, e.g.~the well-known Lennard-Jones type, and show that the system converges to the unique invariant Gibbs measure ...
• #### Geometric ergodicity of a bead-spring pair with stochastic Stokes forcing ﻿

(Stochastic Processes and their Applications, 2012-12-01)
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 ﻿

This 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 Navier-Stokes equation ﻿

(Communications on Pure and Applied Mathematics, 2006-12-01)
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 ...
• #### Noise-induced stabilization of planar flows I ﻿

(Electronic Journal of Probability, 2015-10-22)
© 2015 University of Washington. All rights reserved.We show that the complex-valued 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 stochastic-partial-differential-equation limits of spatially correlated noise-driven spin systems derived to sample a canonical distribution ﻿

(Physical Review E, 2020-11-09)
© 2020 American Physical Society. 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 ...
• #### Numerical methods for stochastic differential equations based on Gaussian mixture ﻿

We 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\^o-Tayl...
• #### On Unique Ergodicity in Nonlinear Stochastic Partial Differential Equations ﻿

(Journal of Statistical Physics, 2016-08-31)
© 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 ﻿

We give improvements to theorems which enable significance testing in Markov Chains.
• #### Propagating lyapunov functions to prove noise-induced stabilization ﻿

(Electronic Journal of Probability, 2012-11-19)
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 ...
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