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Archetypal analysis is an unsupervised machine learning method that summarizes data using a convex polytope. In its original formulation, for fixed k, the method finds a convex polytope with k vertices, called archetype points, such that the polytope is contained in the convex hull of the data and the mean squared Euclidean distance between the data and the polytope is minimal. In the present work, we consider an alternative formulation of archetypal analysis based on the Wasserstein metric, which we call Wasserstein archetypal analysis (WAA). In one dimension, there exists a unique solution of WAA and, in two dimensions, we prove the existence of a solution, as long as the data distribution is absolutely continuous with respect to the Lebesgue measure. We discuss obstacles to extending our result to higher dimensions and general data distributions. We then introduce an appropriate regularization of the problem, via a Rényi entropy, which allows us to obtain the existence of solutions of the regularized problem for general data distributions, in arbitrary dimensions. We prove a consistency result for the regularized problem, ensuring that if the data are iid samples from a probability measure, then as the number of samples is increased, a subsequence of the archetype points converges to the archetype points for the limiting data distribution, almost surely. Finally, we develop and implement a gradient-based computational approach for the two-dimensional problem, based on the semi-discrete formulation of the Wasserstein metric. Detailed numerical experiments are provided to support our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, the authors study a class of general mean-field BDSDEs whose coefficients satisfy some stochastic conditions. Specifically, the authors prove the existence and uniqueness theorem of solution under stochastic Lipschitz condition and obtain the related comparison theorem. Besides, the authors further relax the conditions and deduce the existence theorem of solutions under stochastic linear growth and continuous conditions, and the authors also prove the associated comparison theorem. Finally, an asset pricing problem is discussed, which demonstrates the application of the general mean-field BDSDEs in finance. [ABSTRACT FROM AUTHOR]

Physical activity has long been shown to be associated with biological and physiological performance and risk of diseases. It is of great interest to assess whether the effect of an exposure or intervention on an outcome is mediated through physical activity measured by modern wearable devices such as actigraphy. However, existing methods for mediation analysis focus almost exclusively on mediation variable that is in the Euclidean space, which cannot be applied directly to physical activity measured by wearable devices. Such data is best summarized in the form of a random distribution. In this paper, we develop a structural equation model to the setting where a random distribution is treated as the mediator. We provide sufficient conditions for identifying the average causal effects of a distribution mediator and present methods for estimating the direct and mediating effects of a random distribution mediator on the outcome. We apply our method to analyze the iCOMPARE data set that compares flexible duty-hour policies and standard duty-hour policies on interns' sleep related outcomes and to investigate the mediation effect of physical activity on the causal path between flexible duty-hour policies and sleep related outcomes. [ABSTRACT FROM AUTHOR]

We study the computation of the zero set of the Bargmann transform of a signal contaminated with complex white noise, or, equivalently, the computation of the zeros of its short-time Fourier transform with Gaussian window. We introduce the adaptive minimal grid neighbors algorithm (AMN), a variant of a method that has recently appeared in the signal processing literature, and prove that with high probability it computes the desired zero set. More precisely, given samples of the Bargmann transform of a signal on a finite grid with spacing δ , AMN is shown to compute the desired zero set up to a factor of δ in the Wasserstein error metric, with failure probability O (δ 4 log 2 (1 / δ)) . We also provide numerical tests and comparison with other algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we consider a data-driven distributionally robust two-stage stochastic linear optimization problem over 1-Wasserstein ball centered at a discrete empirical distribution. Differently from the traditional two-stage stochastic programming which involves the expected recourse function as the preference criterion and hence is risk-neutral, we take the conditional value-at-risk (CVaR) as the risk measure in order to model its effects on decision making problems. We mainly explore tractable reformulations for the proposed robust two-stage stochastic programming with mean-CVaR criterion by analyzing the first case where uncertainties are only in the objective function and then the second case where uncertainties are only in the constraints. We demonstrate that the first model can be exactly reformulated as a deterministic convex programming. Furthermore, it is shown that under several different support sets, the resulting convex optimization problems can be converted into computationally tractable conic programmings. Besides, the second model is generally NP-hard since checking constraint feasibility can be reduced to a norm maximization problem over a polytope. However, even with the case of uncertainty in constraints, tractable conic reformulations can be established when the extreme points of the polytope are known. Finally, we present numerical results to discuss how to control the risk for the best decisions and illustrate the computational effectiveness and superiority of the proposed models. [ABSTRACT FROM AUTHOR]

Multi-energy virtual power plant (MEVPP) has attracted more and more attention due to its advantages in renewable energy consumption and carbon emission reduction. However, the characteristics of multi-energy coupling and the access of renewable energy may lead to some challenges in the operation of MEVPP. In this paper, a data-driven distributionally robust chance constraints optimization model (DD-DRCCO) is proposed for the dispatching of MEVPP. Firstly, the uncertainties of wind power and photovoltaic output forecasting errors are modeled as an ambiguity set based on the Wasserstein metric. Secondly, combined with the chance constraint, the expected probability of the inequality constraint with uncertain variables is limited to the lowest allowable confidence level to improve the reliability of the model. Thirdly, the forecast errors of wind power and photovoltaic are considered in the constraint conditions, so that the system can effectively resist the interference of uncertain output. Besides, based on the strong duality theory, the DD-DRCCO model is equivalent to a MILP problem which is easy to solve. Finally, simulations implemented on a typical MEVPP are delivered to show that our proposed model: 1) The model is data-driven, and the conservativeness is kept at a low level, and the solution time is about 7s~8s; 2) The MEVPP system can achieve a balance between economy and low-carbon, making the total operation cost reduced by 0.89% compared with no increase of electric boiler; 3) The CO2 emission during the operation of the MEVPP system was significantly reduced by about 87.33 kg. [ABSTRACT FROM AUTHOR]

We develop a class of hierarchically low-rank, scalable optimal transport dissimilarity measures for structured data, bringing the current state-of-the-art optimal transport solvers to a higher level of performance. Given two n-dimensional discrete probability measures supported on two structured grids in R d , we present a fast method for computing an entropically regularized optimal transport distance, referred to as the debiased Sinkhorn distance. The method combines Sinkhorn's matrix scaling iteration with a low-rank hierarchical representation of the scaling matrices to achieve a near-linear complexity O (n ln 4 n) . This provides a fast, scalable, and easy-to-implement algorithm for computing a class of optimal transport dissimilarity measures, enabling their applicability to large-scale optimization problems, where the computation of the classical Wasserstein metric is not feasible. We carry out a rigorous error-complexity analysis for the proposed algorithm and present several numerical examples to verify the accuracy and efficiency of the algorithm and to demonstrate its applicability in tackling real-world problems. [ABSTRACT FROM AUTHOR]

We propose a distributionally robust optimization formulation with a Wasserstein-based uncertainty set for selecting grouped variables under perturbations on the data for both linear regression and classification problems. The resulting model offers robustness explanations for grouped least absolute shrinkage and selection operator algorithms and highlights the connection between robustness and regularization. We prove probabilistic bounds on the out-of-sample loss and the estimation bias, and establish the grouping effect of our estimator, showing that coefficients in the same group converge to the same value as the sample correlation between covariates approaches 1. Based on this result, we propose to use the spectral clustering algorithm with the Gaussian similarity function to perform grouping on the predictors, which makes our approach applicable without knowing the grouping structure a priori. We compare our approach to an array of alternatives and provide extensive numerical results on both synthetic data and a real large dataset of surgery-related medical records, showing that our formulation produces an interpretable and parsimonious model that encourages sparsity at a group level and is able to achieve better prediction and estimation performance in the presence of outliers. [ABSTRACT FROM AUTHOR]

In this paper we wish to tackle stochastic programs affected by ambiguity about the probability law that governs their uncertain parameters. Using optimal transport theory, we construct an ambiguity set that exploits the knowledge about the distribution of the uncertain parameters, which is provided by: (1) sample data and (2) a-priori information on the order among the probabilities that the true data-generating distribution assigns to some regions of its support set. This type of order is enforced by means of order cone constraints and can encode a wide range of information on the shape of the probability distribution of the uncertain parameters such as information related to monotonicity or multi-modality. We seek decisions that are distributionally robust. In a number of practical cases, the resulting distributionally robust optimization (DRO) problem can be reformulated as a finite convex problem where the a-priori information translates into linear constraints. In addition, our method inherits the finite-sample performance guarantees of the Wasserstein-metric-based DRO approach proposed by Mohajerin Esfahani and Kuhn (Math Program 171(1–2):115–166. https://doi.org/10.1007/s10107-017-1172-1, 2018), while generalizing this and other popular DRO approaches. Finally, we have designed numerical experiments to analyze the performance of our approach with the newsvendor problem and the problem of a strategic firm competing à la Cournot in a market. [ABSTRACT FROM AUTHOR]

The generative adversarial network (GAN) is a well-known model for learning high-dimensional distributions, but the mechanism for its generalization ability is not understood. In particular, GAN is vulnerable to the memorization phenomenon, the eventual convergence to the empirical distribution. We consider a simplified GAN model with the generator replaced by a density and analyze how the discriminator contributes to generalization. We show that with early stopping, the generalization error measured by Wasserstein metric escapes from the curse of dimensionality, despite that in the long term, memorization is inevitable. In addition, we present a hardness of learning result for WGAN. [ABSTRACT FROM AUTHOR]

Loop closure detection (LCD) is a key step in visual simultaneous localization and mapping systems to correct the map and relocalize the vehicle. However, it may fail when illumination variations and shifting dynamics present in the scenes. One way to improve the precision is to effectively combine multiple image features that supply complementary information. In this paper, a general method to quantitatively measure the efficacy of individual image features as well as feature combinations for LCD is proposed by calculating the statistical distance considering the distributions of feature vectors. Based on different statistical distances including Kullback-Leibler divergence, Bhattacharyya divergence and Wasserstein metric, various numerical indices capable of evaluating feature combinations are obtained and compared. An unsupervised algorithm is further proposed to optimize feature combinations by maximizing any of the indices. Experiments show that the proposed indices can measure the efficacies of image features and the resulting feature combinations maximizing the indices can improve the precision of LCD. [ABSTRACT FROM AUTHOR]

The generative adversarial network (GAN) is a well-known model for learning high-dimensional distributions, but the mechanism for its generalization ability is not understood. In particular, GAN is vulnerable to the memorization phenomenon, the eventual convergence to the empirical distribution. We consider a simplified GAN model with the generator replaced by a density and analyze how the discriminator contributes to generalization. We show that with early stopping, the generalization error measured by Wasserstein metric escapes from the curse of dimensionality, despite that in the long term, memorization is inevitable. In addition, we present a hardness of learning result for WGAN. [ABSTRACT FROM AUTHOR]

We study distributionally robust chance-constrained programming (DRCCP) optimization problems with data-driven Wasserstein ambiguity sets. The proposed algorithmic and reformulation framework applies to all types of distributionally robust chance-constrained optimization problems subjected to individual as well as joint chance constraints, with random right-hand side and technology vector, and under two types of uncertainties, called uncertain probabilities and continuum of realizations. For the uncertain probabilities (UP) case, we provide new mixed-integer linear programming reformulations for DRCCP problems. For the continuum of realizations case with random right-hand side, we propose an exact mixed-integer second-order cone programming (MISOCP) reformulation and a linear programming (LP) outer approximation. For the continuum of realizations (CR) case with random technology vector, we propose two MISOCP and LP outer approximations. We show that all proposed relaxations become exact reformulations when the decision variables are binary or bounded general integers. For DRCCP with individual chance constraint and random right-hand side under both the UP and CR cases, we also propose linear programming reformulations which need the ex-ante derivation of the worst-case value-at-risk via the solution of a finite series of linear programs determined via a bisection-type procedure. We evaluate the scalability and tightness of the proposed MISOCP and (MI)LP formulations on a distributionally robust chance-constrained knapsack problem. [ABSTRACT FROM AUTHOR]

Context: Boltzmann entropy, also called thermodynamic entropy, has long been suggested and recently reemphasized as a basis for achieving a profound understanding of landscape dynamics with thermodynamic insights. The difficulty in practically applying this entropy lies in its computation with landscapes, and many solutions have attempted to address this. The latest solution for landscape mosaics is the Wasserstein metric-based method. Objectives: The first objective is to provide a clarification of and a correction to the Wasserstein metric-based method. The second is to evaluate the method in terms of thermodynamic consistency using different implementations. Methods: Two implementation methods, namely the von Neumann and the Moore neighborhood, were used, which led to two different Wasserstein metric-based entropies. Thermodynamic consistency, the fundamental property of entropy, was used as the evaluation principle. Three criteria (validity, reliability, and ability) were designed in terms of thermodynamic consistency, and corresponding indicators were proposed. Boltzmann entropies computed using all existing methods were used as benchmarks. Results: The three indicators of the five Boltzmann entropies (including two based on the Wasserstein metric and three using existing methods) against 100,000 landscapes were computed and investigated. The reasons for both the good and poor performance of the Wasserstein metric-based entropies were identified. Conclusions: The Wasserstein metric-based method can be safely used with the von Neumann neighborhood. Compared with the entropies produced by existing methods, the Wasserstein metric-based entropy has worse reliability but better ability (i.e., working range). The most reliable entropy was computed using the total edge-based method. [ABSTRACT FROM AUTHOR]

Natural gradients have been widely used in the optimization of loss functionals over probability space, with important examples such as Fisher–Rao gradient descent for Kullback–Leibler divergence, Wasserstein gradient descent for transport-related functionals, and Mahalanobis gradient descent for quadratic loss functionals. This note considers the situation in which the loss is a convex linear combination of these examples. We propose a new natural gradient algorithm by utilizing compactly supported wavelets to diagonalize approximately the Hessian of the combined loss. Numerical results are included to demonstrate the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Ensemble Kalman inversion (EKI) has been a very popular algorithm used in Bayesian inverse problems (Iglesias et al. in Inverse Probl 29: 045001, 2013). It samples particles from a prior distribution and introduces a motion to move the particles around in pseudo-time. As the pseudo-time goes to infinity, the method finds the minimizer of the objective function, and when the pseudo-time stops at 1, the ensemble distribution of the particles resembles, in some sense, the posterior distribution in the linear setting. The ideas trace back further to ensemble Kalman filter and the associated analysis (Evensen in J Geophys Res: Oceans 99: 10143–10162, 1994; Reich in BIT Numer Math 51: 235–249, 2011), but to today, when viewed as a sampling method, why EKI works, and in what sense with what rate the method converges is still largely unknown. In this paper, we analyze the continuous version of EKI, a coupled SDE system, and prove the mean-field limit of this SDE system. In particular, we will show that 1. as the number of particles goes to infinity, the empirical measure of particles following SDE converges to the solution to a Fokker–Planck equation in Wasserstein 2-distance with an optimal rate, for both linear and weakly nonlinear case; 2. the solution to the Fokker–Planck equation reconstructs the target distribution in finite time in the linear case, as suggested in Iglesias et al. (Inverse Probl 29: 045001, 2013). [ABSTRACT FROM AUTHOR]

We rigorously justify the mean-field limit of an N-particle system subject to Brownian motions and interacting through the Newtonian potential in R 3 . Our result leads to a derivation of the Vlasov–Poisson–Fokker–Planck (VPFP) equations from the regularized microscopic N-particle system. More precisely, we show that the maximal distance between the exact microscopic trajectories and the mean-field trajectories is bounded by N - 1 3 + ε ( 1 63 ≤ ε < 1 36 ) with a blob size of N - δ ( 1 3 ≤ δ < 19 54 - 2 ε 3 ) up to a probability of 1 - N - α for any α > 0 . Moreover, we prove the convergence rate between the empirical measure associated to the regularized particle system and the solution of the VPFP equations. The technical novelty of this paper is that our estimates rely on the randomness coming from the initial data and from the Brownian motions. [ABSTRACT FROM AUTHOR]

We consider the problem of finding an optimal transport plan between an absolutely continuous measure and a finitely supported measure of the same total mass when the transport cost is the unsquared Euclidean distance. We may think of this problem as closest distance allocation of some resource continuously distributed over Euclidean space to a finite number of processing sites with capacity constraints. This article gives a detailed discussion of the problem, including a comparison with the much better studied case of squared Euclidean cost. We present an algorithm for computing the optimal transport plan, which is similar to the approach for the squared Euclidean cost by Aurenhammer et al. (Algorithmica 20(1):61–76, 1998) and Mérigot (Comput Graph Forum 30(5):1583–1592, 2011). We show the necessary results to make the approach work for the Euclidean cost, evaluate its performance on a set of test cases, and give a number of applications. The later include goodness-of-fit partitions, a novel visual tool for assessing whether a finite sample is consistent with a posited probability density. [ABSTRACT FROM AUTHOR]

We consider the symmetric scan Gibbs sampler, and give some explicit estimates of convergence rates on the Wasserstein distance for this Markov chain Monte Carlo under the Dobrushin uniqueness condition. [ABSTRACT FROM AUTHOR]

Context: Entropy is an important concept traditionally associated with thermodynamics and is widely used to describe the degree of disorder in a substance, system, or process. Configurational entropy has received more attention because it better reflects the thermodynamic properties of physical and biological processes. However, as the number of configuration combinations increases, configurational entropy becomes too complex to calculate, and its value is too large to be accurately represented in practical applications. Objectives: To calculate the spatial configurational entropy of a landscape mosaic based on a statistical metric. Methods: We proposed a relative entropy using histograms to compare two ecosystems with the Wasserstein metric, and used six digital elevation models and five simulated data to calculate the entropy of the complex ecosystems. Results: The calculation and simulation showed that the purposed metric captured disorder in the spatial landscape, and the result was consistent with the general configurational entropy. By calculating several spatial scale landscapes, we found that relative entropy can be a trade-off between the rationality of results and the cost of calculation. Conclusions: Our results show that the Wasserstein metric is suitable to capture the discrepancy using complex landscape mosaic data sets, which provides a numerically efficient approximation for the similarity in the histograms, reducing excessive expansion of the calculated result. [ABSTRACT FROM AUTHOR]

The concept of distance rationalizability of voting rules has been explored in recent years by several authors. Roughly speaking, we first choose a consensus set of elections (defined via preferences of voters over candidates) for which the result is specified a priori (intuitively, these are elections on which all voters can easily agree on the result). We also choose a measure of distance between elections. The result of an election outside the consensus set is defined to be the result of the closest consensual election under the distance measure. Most previous work has dealt with a definition in terms of preference profiles. However, most voting rules in common use are anonymous and homogeneous. In this case there is a much more succinct representation (using the voting simplex) of the inputs to the rule. This representation has been widely used in the voting literature, but rarely in the context of distance rationalizability. We show exactly how to connect distance rationalizability on profiles for anonymous and homogeneous rules to geometry in the simplex. We develop the connection for the important special case of votewise distances, recently introduced and studied by Elkind, Faliszewski and Slinko in several papers. This yields a direct interpretation in terms of well-developed mathematical concepts not seen before in the voting literature, namely Kantorovich (also called Wasserstein) distances and the geometry of Minkowski spaces. As an application of this approach, we prove some positive and some negative results about the decisiveness of distance rationalizable anonymous and homogeneous rules. The positive results connect with the recent theory of hyperplane rules, while the negative ones deal with distances that are not metrics, controversial notions of consensus, and the fact that the ℓ1-norm is not strictly convex. We expect that the above novel geometric interpretation will aid the analysis of rules defined by votewise distances, and the discovery of new rules with desirable properties. [ABSTRACT FROM AUTHOR]

We refine the classical Lindeberg-Feller central limit theorem by obtaining asymptotic bounds on the Kolmogorov distance, the Wasserstein distance, and the parameterized Prokhorov distances in terms of a Lindeberg index. We thus obtain more general approximate central limit theorems, which roughly state that the row-wise sums of a triangular array are approximately asymptotically normal if the array approximately satisfies Lindeberg’s condition. This allows us to continue to provide information in nonstandard settings in which the classical central limit theorem fails to hold. Stein’s method plays a key role in the development of this theory. [ABSTRACT FROM AUTHOR]

This paper studies the expected optimal value of a mixed 0-1 programming problem with uncertain objective coefficients following a joint distribution. We assume that the true distribution is not known exactly, but a set of independent samples can be observed. Using the Wasserstein metric, we construct an ambiguity set centered at the empirical distribution from the observed samples and containing the true distribution with a high statistical guarantee. The problem of interest is to investigate the bound on the expected optimal value over the Wasserstein ambiguity set. Under standard assumptions, we reformulate the problem into a copositive program, which naturally leads to a tractable semidefinite-based approximation. We compare our approach with a moment-based approach from the literature on three applications. Numerical results illustrate the effectiveness of our approach. [ABSTRACT FROM AUTHOR]

The classical Birkhoff ergodic theorem states that for an ergodic Markov process the limiting behaviour of the time average of a function (having finite p-th moment, $$p\ge 1$$ , with respect to the invariant measure) along the trajectories of the process, starting from the invariant measure, is a.s. and in the p-th mean constant and equals to the space average of the function with respect to the invariant measure. The crucial assumption here is that the process starts from the invariant measure, which is not always the case. In this paper, under the assumptions that the underlying process is a Markov process on Polish space, that it admits an invariant probability measure and that its marginal distributions converge to the invariant measure in the $$L^{1}$$ -Wasserstein metric, we show that the assertion of the Birkhoff ergodic theorem holds in the p-th mean, $$p\ge 1$$ , for any bounded Lipschitz function and any initial distribution of the process. [ABSTRACT FROM AUTHOR]

In this article, we consider mean field games between a dominating player and a group of representative agents, each of which acts similarly and also interacts with each other through a mean field term being substantially influenced by the dominating player. We first provide the general theory and discuss the necessary condition for the optimal controls and equilibrium condition by adopting adjoint equation approach. We then present a special case in the context of linear-quadratic framework, in which a necessary and sufficient condition can be asserted by stochastic maximum principle; we finally establish the sufficient condition that guarantees the unique existence of the equilibrium control. The proof of the convergence result of finite player game to mean field counterpart is provided in Appendix. [ABSTRACT FROM AUTHOR]

The aim of the paper is to estimate the density functions or distribution functions measured by Wasserstein metric, a typical kind of statistical distances, which is usually required in the statistical learning. Based on the classical Bernstein approximation, a scheme is presented. To get the error estimates of the scheme, the problem turns to estimating the L norm of the Bernstein approximation for monotone C functions, which was rarely discussed in the classical approximation theory. Finally, we get a probability estimate by the statistical distance. [ABSTRACT FROM AUTHOR]

We introduce and study a rough (approximate) curvature-dimension condition for metric measure spaces, applicable especially in the framework of discrete spaces and graphs. This condition extends the one introduced by Karl-Theodor Sturm, in his 2006 article On the geometry of metric measure spaces II, to a larger class of (possibly non-geodesic) metric measure spaces. The rough curvature-dimension condition is stable under an appropriate notion of convergence, and stable under discretizations as well. For spaces that satisfy a rough curvature-dimension condition we prove a generalized Brunn-Minkowski inequality and a Bonnet-Myers type theorem. [ABSTRACT FROM AUTHOR]

We obtain nonuniform estimates depending on the interval length for Poisson and compound Poisson approximations to 2-runs. We calculate exact constants for the estimates in the Wasserstein metric. [ABSTRACT FROM AUTHOR]

We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all finite subspaces sampled from these spaces converge. This topology is metrized following Gromov’s idea of embedding two metric spaces isometrically into a common metric space combined with the Prohorov metric between probability measures on a fixed metric space. We show that for this topology convergence in distribution follows—provided the sequence is tight—from convergence of all randomly sampled finite subspaces. We give a characterization of tightness based on quantities which are reasonably easy to calculate. Subspaces of particular interest are the space of real trees and of ultra-metric spaces equipped with a probability measure. As an example we characterize convergence in distribution for the (ultra-)metric measure spaces given by the random genealogies of the Λ-coalescents. We show that the Λ-coalescent defines an infinite (random) metric measure space if and only if the so-called “dust-free”-property holds. [ABSTRACT FROM AUTHOR]

Under the bounded geometry assumption on Riemannian manifold M, a variational approximation for Fokker–Planck equation on M is constructed by the scheme of Jordan et al. in SIAM J Math Anal 29(1):1–17, 1998. Moreover, the uniqueness and global L p -estimate of the solution for 1 < p < dim( M)/(dim( M) − 1) are obtained for a broad class of potential. [ABSTRACT FROM AUTHOR]

We study the long time behavior of an Ornstein–Uhlenbeck process under the influence of a periodic drift. We prove that, under the standard diffusive rescaling, the law of the particle position converges weakly to the law of a Brownian motion whose covariance can be expressed in terms of the solution of a Poisson equation. We also derive upper bounds on the convergence rate in several metrics. [ABSTRACT FROM AUTHOR]