Jonathan C. Mattingly

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

The ergodic properties of SDEs, and various time discretizations for SDEs, are studied. The ergodicity of SDEs is established by using techniques from the theory of Markov chains on general state spaces, such as that expounded by Meyn-Tweedie. Application of these Markov chain results leads to straightforward proofs of geometric ergodicity for a variety of SDEs, including problems with degenerate noise and for problems with locally Lipschitz vector fields. Applications where this theory can be usefully applied include damped-driven Hamiltonian problems (the Langevin equation), the Lorenz equation with degenerate noise and gradient systems. The same Markov chain theory is then used to study time-discrete approximations of these SDEs. The two primary ingredients for ergodicity are a minorization condition and a Lyapunov condition. It is shown that the minorization condition is robust under approximation. For globally Lipschitz vector fields this is also true of the Lyapunov condition. However in the locally Lipschitz case the Lyapunov condition fails for explicit methods such as Euler-Maruyama; for pathwise approximations it is, in general, only inherited by specially constructed implicit discretizations. Examples of such discretization based on backward Euler methods are given, and approximation of the Langevin equation studied in some detail.

Low-dimensional models of coherent structures in turbulence

Abstract 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 simulation certainly yields a “solution”, but it provides little understanding of the process per se .) However, three developments are beginning to bear fruit: (1) The discovery, by experimental fluid mechanicians, of coherent structures in certain fully developed turbulent flows; (2) the suggestion, by Ruelle, Takens and others, that strange attractors and other ideas from dynamical systems theory might play a role in the analysis of the governing equations, and (3) the introduction of the statistical technique of Karhunen-Loeve or proper orthogonal decomposition, by Lumley in the case of turbulence. Drawing on work on modeling the dynamics of coherent structures in turbulent flows done over the past ten years, and concentrating on the near-wall region of the fully developed boundary layer, we describe how these three threads can be drawn together to weave low-dimensional models which yield new qualitative understanding. We focus on low wave number phenomena of turbulence generation, appealing to simple, conventional modeling of inertial range transport and energy dissipation.

Yet another look at Harris' Ergodic Theorem for Markov Chains

The aim of this note is to present an elementary proof of a variation of Harris’ ergodic theorem of Markov chains.

AN ELEMENTARY PROOF OF THE EXISTENCE AND UNIQUENESS THEOREM FOR THE NAVIER–STOKES EQUATIONS

Here ν is the viscosity, p is the pressure, and f1, f2 are the components of an external forcing which may be time-dependent. As our setting is periodic, the functions u1, u2, ∇p, f1, and f2 are all periodic in x. For simplicity, we take the period to be one. The first existence and uniqueness theorems for weak solutions of (1) were proven by Leray ([Ler34]) in whole plane R. Later these results were extended by E. Hopf (see [Hop51]). In 1962, Ladyzenskaya proved existence and uniqueness results for strong solutions for general two-dimensional domains [Lad69]. V. Yudovich, C. Foias, R. Teman, P. Constantin, and others developed strong methods which provided deep insights into the dynamics described by (1) (see [Yud89, Tem79, Tem95, CF88]). The purpose of this paper is to present elementary proofs of three theorems. These theorems imply the existence and uniqueness of smooth solutions of (1) and shed some additional light on the dissipative character of the dynamics. We will also discuss what our techniques can give in the three-dimensional setting. In two-dimensions, it is useful to consider the vorticity ω(x1, x2, t) = ∂u1(x1,x2,t) ∂x2 − ∂u2(x1,x2,t) ∂x1 . The equation governing ω has the form ( see [CM93, DG95] )

Long-Range Allosteric Interactions between the Folate and Methionine Cycles Stabilize DNA Methylation Reaction Rate

Several metabolites in the folate and methionine cycles influence the activities of distant enzymes involved in one-carbon metabolism. Many hypotheses have been advanced about the functional impact of these long-range interactions. Using both steady-state and fluctuation analyses of a mathematical model of methionine metabolism, we investigate the biochemical basis for several of these hypotheses. We show that the long-range interactions provide remarkable stabilization of the DNA methylation rate in the face of large fluctuations in methionine input. In particular, they enable the system to maintain methylation in the face of low and extremely low protein input. These interactions may therefore have evolved primarily to stabilize DNA methylation under conditions of methionine starvation. In silico experimentation allows us to evaluate the independent effects of various combinations of the long-range interactions, and thereby propose a plausible evolutionary scenario.

Sticky central limit theorems on open books

Given a probability distribution on an open book (a metric space obtained by gluing a disjoint union of copies of a half-space along their boundary hyperplanes), we define a precise concept of when the Frechet mean (barycenter) is sticky. This nonclassical phenomenon is quantified by a law of large numbers (LLN) stating that the empirical mean eventually almost surely lies on the (codimension 1 and hence measure 0) spine that is the glued hyperplane, and a central limit theorem (CLT) stating that the limiting distribution is Gaussian and supported on the spine. We also state versions of the LLN and CLT for the cases where the mean is nonsticky (i.e., not lying on the spine) and partly sticky (i.e., is, on the spine but not sticky).

Convergence of Numerical Time-Averaging and Stationary Measures via Poisson Equations

Numerical approximation of the long time behavior of a stochastic differential 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 that of the SDE. The error analysis is based on using an associated Poisson equation for the underlying SDE. The main advantages of this approach are its simplicity and universality. It works equally well for a range of explicit and implicit schemes, including those with simple simulation of random variables, and for hypoelliptic SDEs. To simplify the exposition, we consider only the case where the state space of the SDE is a torus, and we study only smooth test functions. However, we anticipate that the approach can be applied more widely. An analogy between our approach and Steins method is indicated. Some practical implications of the results are discussed.

Diffusion limits of the random walk Metropolis algorithm in high dimensions.

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 stationarity, as a function of the dimension of the state space. However, to date such results have mainly been proved for target measures with a product structure, severely limiting their applicability. The purpose of this paper is to study diffusion limits for a class of naturally occurring high-dimensional measures found from the approximation of measures on a Hilbert space which are absolutely continuous with respect to a Gaussian reference measure. The diffusion limit of a random walk Metropolis algorithm to an infinite-dimensional Hilbert space valued SDE (or SPDE) is proved, facilitating understanding of the computational complexity of the algorithm.

STATIONARY SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH MEMORY AND STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

We explore Ito 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. Uniqueness of the stationary solution is proven if the dependence on the past decays sufficiently fast. The results of this paper are then applied to stochastically forced dissipative partial differential equations such as the stochastic Navier–Stokes equation and stochastic Ginsburg–Landau equation.

The Dissipative Scale of the Stochastics Navier-Stokes Equation: Regularization and Analyticity

We prove spatial analyticity for solutions of the stochastically forced Navier–Stokes equation, provided that the forcing is sufficiently smooth spatially. We also give estimates, which extend to the stationary regime, providing strong control of both of the expected rate of dissipation and fluctuations about this mean. Surprisingly, we could not obtain non-random estimates of the exponential decay rate of the spatial Fourier spectra.