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

By Jonathan C. Mattingly, Andrew M. Stuart, Desmond John Higham. 2002

680 citations recorded by the source corpus.

Subfields: Finance, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c5a59af819b1211dd88b0ee22846f562ec94a53f

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Citations in the corpus

Bridging the gap between constant step size stochastic gradient descent and Markov chains (2017, methodology): “In the context of discretization of ergodic diffusions, weak error estimates between the stationary distribution of the discretization and the invariant distribution of the associated diffusion have been first shown by [59] and [37] in the case of the Euler-Maruyama scheme.”
The Bayesian Approach to Inverse Problems (2013, result): “The ergodicity of the noisy gradient flow is proved in [MSH02], although the almost sure pathwise result is not stated explicitly, but rather is a consequence of the exponential convergence of expectatations, using the ideas in Chapter 17 of [MT93]; see also [PV01, PV03, PV05] for results on converg”
High-dimensional Bayesian inference via the unadjusted Langevin algorithm (2016, methodology): “The ULA algorithm has already been studied in depth for constant sequence of step sizes in [36], [33] and [27].”
Exponential Convergence for the Stochastically Forced Navier-Stokes Equations and Other Partially Dissipative Dynamics (2002, background): “3 of [MSH01] or [Has80] for the continuous setting.”
CONVERGENCE AND STABILITY OF IMPLICIT METHODS FOR JUMP-DIFFUSION SYSTEMS (2005, background): “for introducing and studying (2) is that for the ODE and SDE cases, it has been found that the class ofiers good linear stability properties [8, 9, 23] and excellent potential for capturing long time dynamics [ 20 , 25].”
Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations (2002, methodology): “The method is also studied in [15], where it is effective for inheriting ergodicity; for related reasons it is effective here in enabling the derivation of moment bounds.”
Propagating Lyapunov functions to prove noise-induced stabilization (2011, methodology): “This is the method employed in [25, 24] in various forms.”
Physics constrained nonlinear regression models for time series (2012, background): “4 of Mattingly-Stuart-Higham [27]) to establish the necessary and sufficient conditions for nondegenerate solutions of the nonlinear regression model in (4), (5).”
Hypocoercivity (2006, methodology): “It is for such equations that theorems of exponentially fast convergence to equilibrium were first established via probabilistic tools [36, 47, 40, 41], taking their roots in the Meyn-Tweedie theory of the asymptotic behavior of Markov chains.”
The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics (2013, methodology): “3) is ergodic for the Boltzmann–Gibbs distribution (see, for instance, Mattingly et al., 2002; Talay, 2002; Cancès et al., 2007 and references therein). We will also be interested in nonequilibrium situations where a given system is subject to nonconservative driving and dissipative perturbations. I”
Couplings and quantitative contraction rates for Langevin dynamics (2017, methodology): “The choice of the Lyapunov function is motivated by Mattingly, Stuart and Higham [32], see also [39, 46, 1].”
An adaptive scheme for the approximation of dissipative systems (2005, neutral): “Under this condition and Assumption 1, Mattingly, Stuart and Higham proved the geometric ergodicity of (1) when � is constant (see [ 9 ] for details).”
The Numerical Approximation of Stochastic Partial Differential Equations (2009, methodology): “9) was handled by Higham et al [56] and Mattingly et al [94] using dissipativity and ergodic arguments.”
Explicit numerical approximations for stochastic differential equations in finite and infinite horizons: truncation methods, convergence in pth moment, and stability (2019, background): “Mattingly et al. (2002) showed that for a class of nonlinear SDEs the mean square of the EM numerical solutions in the infinite interval tends to infinity but the mean square of the exact solutions is bounded.”
Strong convergence of projective integration schemes for singularly perturbed stochastic differential systems (2006, background): “[24] for further elaboration and applications to various situations.”