Weak Convergence Methods Research Articles

Well-posedness for the stochastic 2D primitive equations with Lévy noise

The two-dimensional primitive equations with Levy noise are studied in this paper. We proved the existence and uniqueness of the solutions in a fixed probability space which based on a priori estimates, weak convergence method and monotonicity arguments.

Discrete-Time Semi-Markov Random Evolutions and their Applications

In this paper we introduce discrete-time semi-Markov random evolutions (DTSMREs) and study asymptotic properties, namely, averaging, diffusion approximation, and diffusion approximation with equilibrium by the martingale weak convergence method. The controlled DTSMREs are introduced and Hamilton–Jacobi–Bellman equations are derived for them. The applications here concern the additive functionals (AFs), geometric Markov renewal chains (GMRCs), and dynamical systems (DSs) in discrete time. The rates of convergence in the limit theorems for DTSMREs and AFs, GMRCs, and DSs are also presented.

Discrete-Time Semi-Markov Random Evolutions and their Applications

In this paper we introduce discrete-time semi-Markov random evolutions (DTSMREs) and study asymptotic properties, namely, averaging, diffusion approximation, and diffusion approximation with equilibrium by the martingale weak convergence method. The controlled DTSMREs are introduced and Hamilton–Jacobi–Bellman equations are derived for them. The applications here concern the additive functionals (AFs), geometric Markov renewal chains (GMRCs), and dynamical systems (DSs) in discrete time. The rates of convergence in the limit theorems for DTSMREs and AFs, GMRCs, and DSs are also presented.

Weak Convergence Methods for Approximation of the Evaluation of Path-Dependent Functionals

In many applications, one needs to evaluate a path-dependent objective functional $V$ associated with a continuous-time stochastic process $X$. Due to the nonlinearity and possible lack of Markovian property, more often than not, $V$ cannot be evaluated analytically, and only Monte Carlo simulation or numerical approximation is possible. In addition, such calculations often require the handling of stopping times, the usual dynamic programming approach may fall apart, and the continuity of the functional becomes the main issue. Denoting by $h$ the stepsize of the approximation sequence, this work develops a numerical scheme so that an approximating sequence of path-dependent functionals $V^h$ converges to $V$. By a natural division of labors, the main task is divided into two parts. Given a sequence $X^h$ that converges weakly to $X$, the first part provides sufficient conditions for the convergence of the sequence of path-dependent functionals $V^h$ to $V$. The second part constructs a sequence of approximations $X^h$ using Markov chain approximation methods and demonstrates the weak convergence of $X^h$ to $X$, when $X$ is the solution of a stochastic differential equation. As a demonstration, combining the results of the two parts above, approximation of option pricing for the discrete-monitoring-barrier option underlying stochastic volatility model is provided. Different from the existing literature, the weak convergence analysis is carried out by using the Skorohod topology together with the continuous mapping theorem. The advantage of this approach is that the functional under study may be a function of stopping times, projection of the underlying diffusion on a sequence of random times, and/or maximum/minimum of the underlying diffusion.

Calculation of effective slip on rough chemically heterogeneous surfaces using a homogenization approach

We present an expression for the effective slip length of a nanoscale rough chemically heterogeneous surface. A heterogeneous surface may be regarded as having an effective slip length generated by extrapolating the uniform velocity profile found in the far field. We consider two-dimensional steady-state Stokes flow over a surface that has periodic roughness and an intrinsic slip length varying over the same period. Using weak convergence methods for partial differential equations, we derive an expression for the effective slip length in terms of the intrinsic slip length and contact area of the surface. The result predicts that roughness causes a significant reduction in effective slip and that slip effects are dominated by the minimum intrinsic slip length of the system.

Large deviations for multiscale diffusion via weak convergence methods

We study the large deviations principle for locally periodic SDEs with small noise and fast oscillating coefficients. There are three regimes depending on how fast the intensity of the noise goes to zero relative to homogenization parameter. We use weak convergence methods which provide convenient representations for the action functional for all regimes. Along the way, we study weak limits of controlled SDEs with fast oscillating coefficients. We derive, in some cases, a control that nearly achieves the large deviations lower bound at prelimit level. This control is useful for designing efficient importance sampling schemes for multiscale small noise diffusion.

Hybrid deterministic-stochastic gradient Langevin dynamics for Bayesian learning

We propose a new algorithm to obtain Bayesian posterior distribution by a hybrid deterministic-stochastic gradient Langevin dynamics. To speed up convergence and reduce computational costs, it is common to use stochastic gradient method to approximate the full gradient by sampling a subset of the large dataset. Stochastic gradient methods make progress fast initially, however, they often become slow in the late stage as the iterations approach the desired solution. The conventional gradient methods converge better eventually however at the expense of evaluating the full gradient at each iteration. Our hybrid method has the advantages of both approaches for constructing the Bayesian posterior distribution. We prove that our algorithm converges based on the weak convergence methods, and illustrate numerically its effectiveness and improved accuracy.

Large deviation properties of weakly interacting processes via weak convergence methods

We study large deviation properties of systems of weakly interacting particles modeled by It\^{o} stochastic differential equations (SDEs). It is known under certain conditions that the corresponding sequence of empirical measures converges, as the number of particles tends to infinity, to the weak solution of an associated McKean-Vlasov equation. We derive a large deviation principle via the weak convergence approach. The proof, which avoids discretization arguments, is based on a representation theorem, weak convergence and ideas from stochastic optimal control. The method works under rather mild assumptions and also for models described by SDEs not of diffusion type. To illustrate this, we treat the case of SDEs with delay.

Large deviation properties of weakly interacting processes via weak convergence methods

Large deviation properties of weakly interacting processes via weak convergence methods

Discrete-Time Semi-Markov Random Evolutions – Average and Diffusion Approximation of Difference Equations and Additive Functionals

This article introduces discrete-time semi-Markov random evolution and studies asymptotic properties, namely, averaging and diffusion approximation by weak convergence methods. The cases considered here concern difference equations and additive functionals with application in estimation of stationary probability of semi-Markov chains.

Stochastic Recursive Algorithms for Networked Systems with Delay and Random Switching: Multiscale Formulations and Asymptotic Properties

Motivated by consensus control of networked systems with communication latency and randomly switching topologies, this paper studies stochastic approximation (SA) algorithms for systems with time delays and randomly switching dynamics. To accommodate realistic time delay systems, our formulation of the discrete-time systems does not impose bounds on delays when the sampling intervals become small. The switching dynamics are modeled as a finite-state Markov chain. The transition probability matrix of the Markov chain contains a small parameter, termed the transition frequency parameter, while the SA algorithm defines its updating speed by another small parameter called the adaptation stepsize. The interplay of the two parameters introduces multiscale system dynamics, whose interaction with time delay further complicates system analysis. Using weak convergence methods, convergence of the algorithm is obtained. The limit behavior of the scaled estimation errors is also analyzed. It is shown that, depending o...

Endogeny for the Logistic Recursive Distributional Equation

In this article we prove the endogeny and bivariate uniqueness property for a particular “max-type” recursive distributional equation (RDE). The RDE we consider is the so called logistic RDE, which appears in the proof of the ζ(2)-limit of the random assignment problem using the local weak convergence method proved by D. Aldous [Probab. Theory Related Fields 93 (1992)(4), 507–534]. This article provides a non-trivial application of the general theory developed by D. Aldous and A. Bandyopadhyay [Ann. Appl. Probab. 15 (2005)(2), 1047–1110]. The proofs involve analytic arguments, which illustrate the need to develop more analytic tools for studying such max-type RDEs.

Probabilistic Representation of Weak Solutions of Partial Differential Equations with Polynomial Growth Coefficients

In this paper we develop a new weak convergence and compact embedding method to study the existence and uniqueness of the $L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{1}})\otimes L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{d}})$ valued solution of backward stochastic differential equations with p-growth coefficients. Then we establish the probabilistic representation of the weak solution of PDEs with p-growth coefficients via corresponding BSDEs.

On the number of binary-minded individuals required to compute [formula omitted

We recently obtained partial results on the computational power of population protocols when the population is assumed to be large. We studied in particular a particular protocol that we proved to converge towards 1 2 , using weak-convergence methods for stochastic processes. In this paper, we prove that it is possible to compute 1 2 with precision ϵ > 0 in a time polynomial in 1 ϵ using a number of agents polynomial in 1 ϵ , with individuals that can have only two states. This is established through a general result on approximation of stochastic differential equations by a stochastic Euler-like discretization algorithm, of general interest.

Maximum Weight Partial Colorings on Sparse Random Graphs

We examine the asymptotic behavior of optimal values and optimal configurations for the maximum weight partial coloring (MWPC) problem on sparse random graphs. The MWPC problem is a weighted version of the partial coloring problem, and it is a generalization of the maximum weight independent set (MWIS) problem. The MWPC problem is an illustrative example of how to employ the local weak convergence method on combinatorial optimization problems that are defined via a graphical model and have a nonbinary alphabet. Many of the results are inspired by Gamarnik, Nowicki, and Swirscsz [Random Structures Algorithms, 28 (2005), pp. 76–106]. They showed that if vertex weights were sampled independently from a nonatomic distribution and if a particular fixed point equation had a unique solution, then the scaled MWIS value on sparse Erdös–Rényi graphs converges in probability. Under the same conditions, Sanghavi and Shah [http://arxiv.org/pdf/cs/0508097v2 (12 April 2008)] showed how the max-product message passing algorithm can be used to produce approximations with probability . There are major difficulties in applying the above techniques to other natural combinatorial optimization problems. We show how to strengthen them so that they can be applied to a new range of problems which includes the MWPC problem. In the MWPC case, we show how our techniques can be used to get new convergence results on sparse Erdös–Rényi graphs—even in the case where the weights are sampled from a discrete distribution. This depends on new results on how certain long range independence properties occur almost surely for branching processes.

Large deviation principle for semilinear stochastic evolution equations with monotone nonlinearity and multiplicative noise

We demonstrate the large deviation property for the mild solutions of stochastic evolution equations with monotone nonlinearity and multiplicative noise. This is achieved using the recently developed weak convergence method, in studying the large deviation principle. An Itô-type inequality is a main tool in the proofs. We also give two examples to illustrate the applications of the theorems.

Large deviations for the stochastic quasigeostrophic equation with multiplicative noise

This paper proves a large deviation principle for the stochastic quasigeostrophic equation with multiplicative noise. The proof is based on the Laplace principle and on weak convergence methods. We also give a rigorous mathematical result on the existence of a strong solution and uniqueness of mild solutions to the stochastic quasigeostrophic equation in the dimension d=2,3. Finally, large deviations for short time are considered.

Representing measures in potential theory and an ideal boundary

A primary motivating application of the author’s work on measure theory was a nonstandard construction of standard representing measures for positive harmonic functions. That work yielded new standard weak convergence methods for constructing such measures on spaces of extreme harmonic functions in very general settings. The search for a Martin-type ideal boundary for the placement of those measures resulted in a new almost everywhere regular boundary that supported the representing measures for a large proper subclass of all nonnegative harmonic functions. In this note, we outline the construction of the rich measure spaces that are now called Loeb measure spaces in the literature. We then review the application of these measure spaces to the construction of representing measures. We finish with the problem of constructing an appropriate boundary associated with the nonstandard construction of general representing measures that supports all of those measures.

Stochastic approximation: a survey

AbstractStochastic recursive algorithms, also known as stochastic approximation, take many forms and have numerous applications. It is the asymptotic properties that are of interest. The early history, starting with the work of Robbins and Monro, is discussed. An approach to proofs of convergence with probability one is illustrated by a stability‐type argument. For general noise processes and algorithms, the most powerful current approach is what is called the ordinary differential equations (ODE) method. The algorithm is interpolated into a continuous‐time process, which is shown to converge to the solution of an ODE, whose asymptotic properties are those of the algorithm. There are probability one and weak convergence methods, the latter being the easiest to use and the most powerful. After discussing the basic ideas and giving some standard proofs, extensions are outlined. These include multiple time scales, tracking of time changing systems, state‐dependent noise, rate of convergence, and random direction methods for high‐dimensional problems. Copyright © 2009 John Wiley & Sons, Inc.This article is categorized under: Algorithms and Computational Methods > Stochastic Optimization

Stochastic 2D Hydrodynamical Type Systems: Well Posedness and Large Deviations

We deal with a class of abstract nonlinear stochastic models, which covers many 2D hydrodynamical models including 2D Navier-Stokes equations, 2D MHD models and the 2D magnetic Bénard problem and also some shell models of turbulence. We state the existence and uniqueness theorem for the class considered. Our main result is a Wentzell-Freidlin type large deviation principle for small multiplicative noise which we prove by a weak convergence method.