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183
Galerkin Methods for Linear and Nonlinear Elliptic Stochastic Partial Differential Equations
, 2003
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THE KARDARPARISIZHANG EQUATION AND UNIVERSALITY CLASS
, 2011
"... Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or regularity) and expanding the breadth of its universality class. Over the past twenty five years a new univ ..."
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Cited by 97 (15 self)
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Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or regularity) and expanding the breadth of its universality class. Over the past twenty five years a new universality class has emerged to describe a host of important physical and probabilistic models (including one dimensional interface growth processes, interacting particle systems and polymers in random environments) which display characteristic, though unusual, scalings and new statistics. This class is called the KardarParisiZhang (KPZ) universality class and underlying it is, again, a continuum object – a nonlinear stochastic partial differential equation – known as the KPZ equation. The purpose of this survey is to explain the context for, as well as the content of a number of mathematical breakthroughs which have culminated in the derivation of the exact formula for the distribution function of the KPZ equation started with narrow wedge initial data. In particular we emphasize three topics: (1) The approximation of the KPZ equation through the weakly asymmetric simple exclusion process; (2) The derivation of the exact onepoint distribution of the solution to the KPZ equation with narrow wedge initial data; (3) Connections with directed polymers in random media. As the purpose of this article is to survey and review, we make precise statements but provide only heuristic arguments with indications of the technical complexities necessary to make such arguments mathematically rigorous.
Solving the KPZ equation
 Ann. of Math
, 2013
"... We introduce a new concept of solution to the KPZ equation which is shown to extend the classical ColeHopf solution. This notion provides a factorisation of the ColeHopf solution map into a “universal ” measurable map from the probability space into an explicitly described auxiliary metric space, ..."
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Cited by 57 (9 self)
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We introduce a new concept of solution to the KPZ equation which is shown to extend the classical ColeHopf solution. This notion provides a factorisation of the ColeHopf solution map into a “universal ” measurable map from the probability space into an explicitly described auxiliary metric space, composed with a new solution map that has very good continuity properties. The advantage of such a formulation is that it essentially provides a pathwise notion of a solution, together with a very detailed approximation theory. In particular, our construction completely bypasses the ColeHopf transform, thus laying the groundwork for proving that the KPZ equation describes the fluctuations of systems in the KPZ universality class. As a corollary of our construction, we obtain very detailed new regularity results about the solution, as well as its derivative with respect to the initial condition. Other byproducts of the proof include an explicit approximation to the stationary solution of the KPZ equation, a wellposedness result for the Fokker
Stochastic NavierStokes equations for turbulent flows
 SIAM J. Math. Anal
"... This paper concerns the fluid dynamics modelled by the stochastic flow ..."
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Cited by 53 (4 self)
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This paper concerns the fluid dynamics modelled by the stochastic flow
Sample path properties of anisotropic Gaussian random fields
, 2008
"... Anisotropic Gaussian random fields arise in probability theory and in various applications. Typical examples are fractional Brownian sheets, operatorscaling Gaussian fields with stationary increments, and the solution to the stochastic heat equation. This paper is concerned with sample path propert ..."
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Cited by 46 (16 self)
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Anisotropic Gaussian random fields arise in probability theory and in various applications. Typical examples are fractional Brownian sheets, operatorscaling Gaussian fields with stationary increments, and the solution to the stochastic heat equation. This paper is concerned with sample path properties of anisotropic Gaussian random fields in general. Let X = {X(t), t ∈ RN} be a Gaussian random field with values in Rd and with parameters H1,..., HN. Our goal is to characterize the anisotropic nature of X in terms of its parameters explicitly. Under some general conditions, we establish results on the modulus of continuity, small ball probabilities, fractal dimensions, hitting probabilities and local times of anisotropic Gaussian random fields. An important tool for our study is the various forms of strong local nondeterminism.
A general fractional white noise theory and applications to finance
 Mathematical Finance
, 2003
"... We present a new framework for fractional Brownian motion in which processes with all indices can be considered under the same probability measure. Our results extend recent contributions by Hu, Øksendal, Duncan, PasikDuncan, and others. As an application we develop option pricing in a fractional B ..."
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Cited by 43 (1 self)
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We present a new framework for fractional Brownian motion in which processes with all indices can be considered under the same probability measure. Our results extend recent contributions by Hu, Øksendal, Duncan, PasikDuncan, and others. As an application we develop option pricing in a fractional BlackScholesmarket with a noise process driven by a sum of fractional Brownian motions with various Hurst indices.
Statistical Validation of Engineering and Scientific Models: Background
, 1999
"... A tutorial is presented discussing the basic issues associated with propagation of uncertainty analysis and statistical validation of engineering and scientific models. The propagation of uncertainty tutorial illustrates the use of the sensitivity method and the Monte Carlo method to evaluate the un ..."
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Cited by 35 (4 self)
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A tutorial is presented discussing the basic issues associated with propagation of uncertainty analysis and statistical validation of engineering and scientific models. The propagation of uncertainty tutorial illustrates the use of the sensitivity method and the Monte Carlo method to evaluate the uncertainty in predictions for linear and nonlinear models. Four example applications are presented; a linear model, a model for the behavior of a damped springmass system, a transient thermal conduction model, and a nonlinear transient convectivediffusive model based on Burger’s equation. Correlated and uncorrelated model input parameters are considered. The model validation tutorial builds on the material presented in the propagation of uncertainty tutorial and uses the damp springmass system as the example application. The validation tutorial illustrates several concepts associated with the application of statistical inference to test model predictions against experimental observations. Several validation methods are presented including error band based, multivariate, sum of squares of residuals, and optimization methods. After completion of the tutorial, a survey of statistical model validation literature is presented and recommendations for future work are made.
ON SOLVING ELLIPTIC STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS
"... A model elliptic boundary value problem of second order, with stochastic coefficients described by the Karhunen–Loève expansion is addressed. This problem is transformed into an equivalent deterministic one. The perturbation method and the method of successive approximations is analyzed. Rigorous er ..."
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Cited by 30 (0 self)
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A model elliptic boundary value problem of second order, with stochastic coefficients described by the Karhunen–Loève expansion is addressed. This problem is transformed into an equivalent deterministic one. The perturbation method and the method of successive approximations is analyzed. Rigorous error estimates in the framework of Sobolev spaces are given.
An introduction to white noise theory and Malliavin calculus for fractional Brownian motion
, 2004
"... Fractional Brownian motion (FBM) with Hurst parameter index between 0 and 1 is a stochastic process originally introduced by Kolmogorov in a study of turbulence. Many other applications have subsequently been suggested. In order to obtain good mathematical models based on FBM, it is necessary to hav ..."
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Cited by 29 (0 self)
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Fractional Brownian motion (FBM) with Hurst parameter index between 0 and 1 is a stochastic process originally introduced by Kolmogorov in a study of turbulence. Many other applications have subsequently been suggested. In order to obtain good mathematical models based on FBM, it is necessary to have a stochastic calculus for such processes. The purpose of this paper is to give an introduction to this newly developed theory of stochastic integration for FBM based on whitenoise theory and (Malliavintype) differentiation.