The stochastic heat equation: Feynman-Kac formula and intermittence
TLDR
In this paper, the heat equation with a random potential that is a white noise in space and time was studied in one space dimension and the statistical properties of the solution were investigated.Abstract:
We study, in one space dimension, the heat equation with a random potential that is a white noise in space and time. This equation is a linearized model for the evolution of a scalar field in a space-time-dependent random medium. It has also been related to the distribution of two-dimensional directed polymers in a random environment, to the KPZ model of growing interfaces, and to the Burgers equation with conservative noise. We show how the solution can be expressed via a generalized Feynman-Kac formula. We then investigate the statistical properties: the two-point correlation function is explicitly computed and the intermittence of the solution is proven. This analysis is carried out showing how the statistical moments can be expressed through local times of independent Brownian motions.read more
Citations
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The kardar-parisi-zhang equation and universality class
TL;DR: In this article, the authors present a survey of the development of the Kardar-Parisi-Zhang (KPZ) universality class and its application to a wide class of physical and probabilistic models.
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Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions
TL;DR: In this paper, the authors considered the solution of the stochastic heat equation @TZ D 1 @ 2 X ZZ P W with delta function initial condition Z and obtained explicit formulas for the one-dimensional marginal distributions, the crossover distributions, which interpolate between a standard Gaussian dis- tribution (small time) and the GUE Tracy-Widom distribution (large time).
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Stochastic Burgers and KPZ equations from particle systems
TL;DR: In this paper, it was shown that the weakly asymmetric exclusion process converges to the Kardar-Parisi-Zhang equation with a random noise on the density current.
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Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
TL;DR: In this article, the authors consider the crossover behavior between the symmetric and asymmetric exclusion processes and obtain explicit formulas for the one-dimensional marginal distributions, which interpolate between a standard Gaussian distribution and the GUE Tracy-Widom distribution.
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Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem
David Aldous,Persi Diaconis +1 more
TL;DR: In this paper, a simple one-person card game, called patience sorting, is described and its analysis leads to a broad circle of ideas linking Young tableaux with the longest increasing subsequence of a random permutation via the Schensted correspondence.
References
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Book
Continuous martingales and Brownian motion
Daniel Revuz,Marc Yor +1 more
TL;DR: In this article, the authors present a comprehensive survey of the literature on limit theorems in distribution in function spaces, including Girsanov's Theorem, Bessel Processes, and Ray-Knight Theorem.
Journal ArticleDOI
Dynamic Scaling of Growing Interfaces
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.
Book
Stochastic differential equations
TL;DR: In this article, the authors introduce the notion of a stochastic differential equation and prove general theorems concerning the existence and uniqueness of solutions of these equations, which is a generalization of the notions of integral integral integral functions.
Journal ArticleDOI
Scaling of directed polymers in random media.
Mehran Kardar,Yi-Cheng Zhang +1 more
TL;DR: Directed polymers subject to quenched external impurities (as in a polyelectrolyte in a gel matrix) are examined analytically, and numerically to suggest a superuniversal exponent of $\ensuremath{
u}=\frac{2}{3}$.