Solving the KPZ equation
TL;DR: In this article, the authors introduce a new solution to the KPZ equation which is shown to extend the classical Cole-Hopf solution, providing a pathwise notion of a solution, together with a very detailed approximation theory.
Abstract: We introduce a new concept of solution to the KPZ equation which is shown to extend the classical Cole-Hopf solution. This notion provides a factorisation of the Cole-Hopf 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 Cole-Hopf transform, thus laying the groundwork for proving that the KPZ equation describes the uctuations 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 well-posedness result for the Fokker-Planck equation associated to a particle diusing in a rough space-time dependent potential, and a new periodic homogenisation result for the heat equation with a space-time periodic potential. One ingredient in our construction is an example of a non-Gaussian rough path such that the area process of its natural approximations needs to be renormalised by a diverging term for the approximations to converge.
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TL;DR: In this article, the authors studied a random Groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process and showed that shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.
Abstract: We study a certain random groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.
1,031 citations
TL;DR: In this paper, the authors introduce a regularity structure for describing functions and distributions via a kind of "jet" or local Taylor expansion around each point, which allows to describe the local behaviour not only of functions but also of large classes of distributions.
Abstract: We introduce a new notion of “regularity structure” that provides an algebraic framework allowing to describe functions and/or distributions via a kind of “jet” or local Taylor expansion around each point. The main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose-built for the problem at hand. In particular, this allows to describe the local behaviour not only of functions but also of large classes of distributions. We then build a calculus allowing to perform the various operations (multiplication, composition with smooth functions, integration against singular kernels) necessary to formulate fixed point equations for a very large class of semilinear PDEs driven by some very singular (typically random) input. This allows, for the first time, to give a mathematically rigorous meaning to many interesting stochastic PDEs arising in physics. The theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems, possibly modified by the addition of diverging counterterms. These counterterms arise naturally through the action of a “renormalisation group” which is defined canonically in terms of the regularity structure associated to the given class of PDEs. Our theory also allows to easily recover many existing results on singular stochastic PDEs (KPZ equation, stochastic quantisation equations, Burgers-type equations) and to understand them as particular instances of a unified framework. One surprising insight is that in all of these instances local solutions are actually “smooth” in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions/distributions that play the role of “polynomials” in the theory. As an example of a novel application, we solve the long-standing problem of building a natural Markov process that is symmetric with respect to the (finite volume) measure describing the $$\Phi ^4_3$$
Euclidean quantum field theory. It is natural to conjecture that the Markov process built in this way describes the Glauber dynamic of $$3$$
-dimensional ferromagnets near their critical temperature.
768 citations
TL;DR: In this article, a regularity structure is introduced to describe functions and distributions via a kind of "jet" or local Taylor expansion around each point, which allows to describe the local behaviour not only of functions but also of large classes of distributions.
Abstract: We introduce a new notion of "regularity structure" that provides an algebraic framework allowing to describe functions and / or distributions via a kind of "jet" or local Taylor expansion around each point. The main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose-built for the problem at hand. In particular, this allows to describe the local behaviour not only of functions but also of large classes of distributions.
We then build a calculus allowing to perform the various operations (multiplication, composition with smooth functions, integration against singular kernels) necessary to formulate fixed point equations for a very large class of semilinear PDEs driven by some very singular (typically random) input. This allows, for the first time, to give a mathematically rigorous meaning to many interesting stochastic PDEs arising in physics. The theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems, possibly modified by the addition of diverging counterterms. These counterterms arise naturally through the action of a "renormalisation group" which is defined canonically in terms of the regularity structure associated to the given class of PDEs.
As an example of a novel application, we solve the long-standing problem of building a natural Markov process that is symmetric with respect to the (finite volume) measure describing the $\Phi^4_3$ Euclidean quantum field theory. It is natural to conjecture that the Markov process built in this way describes the Glauber dynamic of 3-dimensional ferromagnets near their critical temperature.
580 citations
Cites background from "Solving the KPZ equation"
...For example, in the case of the KPZ equation, it was already remarked in [59] that regularisation via a non-symmetric mollifier can cause the appearance in the limiting solution of an additional transport term, thus breaking the invariance under left/right reflection....
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...Amore robust concept of solution for theKPZ equationwhere g4 = g1 = 1 and g2 = g3 = 0, as well as for a number of other equations belonging to the class (KPZ) was given recently in the series of articles [57–59,72], using ideas from the theory of rough paths that eventually lead to the development of the theory presented here....
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...8 in [59]....
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...Over the past few years, it has transpired that the theory of controlled rough paths [54,55,82] could be used in certain situations to provide a meaning to the ill-posed nonlinearities arising in a class of Burgers-type equations [57,63,70,72], as well as in the KPZ equation [59]....
[...]
...As a matter of fact, it was shown in [20] that the WNA solution to the KPZ equation exhibits a physically incorrect large-time behaviour, while the Cole–Hopf solution (which can also be obtained via a suitable regularity structure, see [59]) is the physically relevant solution [8]....
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TL;DR: In this article, the authors introduce an approach to study singular partial differential equations (PDEs) which is based on techniques from paradifferential calculus and on ideas from the theory of controlled rough paths.
Abstract: We introduce an approach to study certain singular partial differential equations (PDEs) which is based on techniques from paradifferential calculus and on ideas from the theory of controlled rough paths. We illustrate its applicability on some model problems such as differential equations driven by fractional Brownian motion, a fractional Burgers-type stochastic PDE (SPDE) driven by space-time white noise, and a nonlinear version of the parabolic Anderson model with a white noise potential.
533 citations
TL;DR: In this article, a nonlinear version of fluctuating hydrodynamics is developed, in which the Euler currents are kept to second order in the deviations from equilibrium and dissipation plus noise are added.
Abstract: With focus on anharmonic chains, we develop a nonlinear version of fluctuating hydrodynamics, in which the Euler currents are kept to second order in the deviations from equilibrium and dissipation plus noise are added. The required model-dependent parameters are written in such a way that they can be computed numerically within seconds, once the interaction potential, pressure, and temperature are given. In principle the theory is applicable to any one-dimensional system with local conservation laws. The resulting nonlinear stochastic field theory is handled in the one-loop approximation. Some of the large scale predictions can still be worked out analytically. For more details one has to rely on numerical simulations of the corresponding mode-coupling equations. In this way we arrive at detailed predictions for the equilibrium time correlations of the locally conserved fields of an anharmonic chain.
303 citations
References
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TL;DR: A model is proposed for the evolution of the profile of a growing interface that exhibits nontrivial relaxation patterns, and the exact dynamic scaling form obtained for a one-dimensional interface is in excellent agreement with previous numerical simulations.
Abstract: A model is proposed for the evolution of the profile of a growing interface. The deterministic growth is solved exactly, and exhibits nontrivial relaxation patterns. The stochastic version is studied by dynamic renormalization-group techniques and by mappings to Burgers's equation and to a random directed-polymer problem. The exact dynamic scaling form obtained for a one-dimensional interface is in excellent agreement with previous numerical simulations. Predictions are made for more dimensions.
4,299 citations
Book•
01 Dec 1992TL;DR: In this paper, the existence and uniqueness of nonlinear equations with additive and multiplicative noise was investigated. But the authors focused on the uniqueness of solutions and not on the properties of solutions.
Abstract: Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.
4,042 citations
Book•
09 May 1995
TL;DR: The Malliavin calculus as mentioned in this paper is an infinite-dimensional differential calculus on a Gaussian space, originally developed to provide a probabilistic proof to Hormander's "sum of squares" theorem, but it has found a wide range of applications in stochastic analysis.
Abstract: The Malliavin calculus (or stochastic calculus of variations) is an infinite-dimensional differential calculus on a Gaussian space. Originally, it was developed to provide a probabilistic proof to Hormander's "sum of squares" theorem, but it has found a wide range of applications in stochastic analysis. This monograph presents the main features of the Malliavin calculus and discusses in detail its main applications. The author begins by developing the analysis on the Wiener space, and then uses this to establish the regularity of probability laws and to prove Hormander's theorem. The regularity of the law of stochastic partial differential equations driven by a space-time white noise is also studied. The subsequent chapters develop the connection of the Malliavin with the anticipating stochastic calculus, studying anticipating stochastic differential equations and the Markov property of solutions to stochastic differential equations with boundary conditions.
The second edition of this monograph includes recent applications of the Malliavin calculus in finance and a chapter devoted to the stochastic calculus with respect to the fractional Brownian motion.
3,883 citations
TL;DR: In this article, the authors define a notion of "two-scale" convergence, which is aimed at a better description of sequences of oscillating functions, and prove that bounded sequences in $L^2 (Omega )$ are relatively compact with respect to this new type of convergence.
Abstract: Following an idea of G. Nguetseng, the author defines a notion of “two-scale” convergence, which is aimed at a better description of sequences of oscillating functions. Bounded sequences in $L^2 (\Omega )$ are proven to be relatively compact with respect to this new type of convergence. A corrector-type theorem (i.e., which permits, in some cases, replacing a sequence by its “two-scale” limit, up to a strongly convergent remainder in $L^2 (\Omega )$) is also established. These results are especially useful for the homogenization of partial differential equations with periodically oscillating coefficients. In particular, a new method for proving the convergence of homogenization processes is proposed, which is an alternative to the so-called energy method of Tartar. The power and simplicity of the two-scale convergence method is demonstrated on several examples, including the homogenization of both linear and nonlinear second-order elliptic equations.
2,279 citations
TL;DR: In this paper, the Navier-Stokes equations for one-dimensional non-stationary flow of a compressible viscous fluid are compared to the shock wave theory of a model of turbulence.
Abstract: where u — u(x, t) in some domain and v is a parameter. The occurrence of the first derivative in t and the second in x clearly indicates the equation is parabolic, similar to the heat equation, while the interesting additional feature is the occurrence of the non-linear term u du/dx. The equation thus shows a structure roughly similar to that of the Navier-Stokes equations and has actually appeared in two separate problems in aerodynamics. An equation simply related to (1) appears in the approximate theory of a weak non-stationary shock wave in a real fluid. This is discussed in Ref. 1 (pp. 146-154) where a general solution of (1) is given. The equation is also given in J. Burgers' theory of a model of turbulence (Ref. 2) where he notes the relationship between the model theory and the shock wave. Historically, the equation (1) first appears in a paper by H. Bateman (Ref. 3) in 1915 when he mentioned it as worthy of study and gave a special solution. Eq. (1) is of some mathematical interest in itself and may have applications in the theory of stochastic processes. The aim of this paper is to study the general properties of (1) and relate the various applications. I wish to thank Professor P. A. Lagerstrom and F. K. Chuang for helpful collaboration. 2. Relationship of (1) to Shock Wave Theory. The solutions to Eq. (1) can approximately describe the flow through a shock wave in a viscous fluid. They can be related to the shock wave in several ways. In Ref. 1 an approximation based on the NavierStokes equations for one-dimensional non-stationary flow of a compressible viscous fluid gives
1,615 citations