Stochastic Burgers Research Articles

Diffusion Approximation for Self-Similarity of Stochastic Advection in Burgers’ Equation

Self-similarity of Burgers’ equation with stochastic advection is studied. In self-similar variables a stationary solution is constructed which establishes the existence of a stochastically self-similar solution for the stochastic Burgers’ equation. The analysis assumes that the stochastic coefficient of advection is transformed to a white noise in the self-similar variables. Furthermore, by a diffusion approximation, the long time convergence to the self-similar solution is proved in the sense of distribution.

Large deviations for a stochastic Burgers' equation

Large deviations for a stochastic Burgers' equation

On a high-dimensional nonlinear stochastic partial differential equation

In this paper we investigate a nonlinear stochastic partial differential equation perturbed by a space-correlated Gaussian noise in arbitrary dimension , with a non-Lipschitz coefficient noisy term. The equation studied coincides in one dimension with the stochastic Burgers equation. Existence of a weak solution is established through an approximation procedure.

Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise

In this paper we prove the local existence and uniqueness of solutions for a class of stochastic fractional partial differential equations driven by multiplicative noise. We also establish that for this class of equations adding linear multiplicative noise provides a regularizing effect: the solutions will not blow up with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large. As applications our main results are applied to various types of SPDE such as stochastic reaction–diffusion equations, stochastic fractional Burgers equation, stochastic fractional Navier–Stokes equation, stochastic quasi-geostrophic equations and stochastic surface growth PDE.

On a time-splitting method for a scalar conservation law with a multiplicative stochastic perturbation and numerical experiments

In this paper, we present a numerical scheme for a first-order hyperbolic equation of nonlinear type perturbed by a multiplicative noise. The problem is set in a bounded domain D of \({\mathbb{R}^{d}}\) and with homogeneous Dirichlet boundary condition. Using a time-splitting method, we are able to show the existence of an approximate solution. The result of convergence of such a sequence is based on the work of Bauzet–Vallet–Wittbold (J Funct Anal, 2013), where the authors used the concept of measure-valued solution and Kruzhkov’s entropy formulation to show the existence and uniqueness of the stochastic weak entropy solution. Then, we propose numerical experiments by applying this scheme to the stochastic Burgers’ equation in the one-dimensional case.

Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise

Motivated by applications to various semilinear and quasi-linear stochastic partial differential equations (SPDEs) appeared in real world models, we establish the existence and uniqueness of strong solutions to coercive and locally monotone SPDEs driven by Lévy processes. We illustrate the main results of our paper by showing how it can be applied to a large class of SPDEs such as stochastic reaction–diffusion equations, stochastic Burgers type equations, stochastic 2D hydrodynamical systems and stochastic equations of non-Newtonian fluids, which generalize many existing results in the literature.

Multidimensional Stochastic Burgers Equation

We consider multidimensional stochastic Burgers equation on the torus $\mathbb{T}^d$ and the whole space ${\mathbb R}^d$. In both cases we show that for positive viscosity $\nu>0$ there exists a unique strong global solution in $L^p$ for $p>d$. In the case of the torus we also establish a uniform in $\nu$ a priori estimate and consider a limit $\nu\searrow 0$ for potential solutions. In the case of ${\mathbb R}^d$ uniform with respect to $\nu$ a priori estimate established if a Beale--Kato--Majda-type condition is satisfied.

Kolmogorov operator and Fokker–Planck equation associated to a stochastic Burgers equation driven by Lévy noise

In this paper, we consider a stochastic Burgers equation driven by Levy noise and study the transition semigroup of the solution to the initial value problem for the equation in the space of continuous functions weighted by a proper potential. We show that the infinitesimal generator is the closure of the Kolmogorov operator associated to the equation in a suitable topology. We also prove existence and uniqueness results for the associated Fokker–Planck equation.

Nonlinear Fluctuations of Weakly Asymmetric Interacting Particle Systems

We introduce what we call the second-order Boltzmann-Gibbs principle, which allows to replace local functionals of a conservative, one-dimensional stochastic process by a possibly nonlinear function of the conserved quantity. This replacement opens the way to obtain nonlinear stochastic evolutions as the limit of the fluctuations of the conserved quantity around stationary states. As an application of this second-order Boltzmann-Gibbs principle, we introduce the notion of energy solutions of the KPZ and stochastic Burgers equations. Under minimal assumptions, we prove that the density fluctuations of one-dimensional, stationary, weakly asymmetric, conservative particle systems are sequentially compact and that any limit point is given by energy solutions of the stochastic Burgers equation. We also show that the fluctuations of the height function associated to these models are given by energy solutions of the KPZ equation in this sense. Unfortunately, we lack a uniqueness result for these energy solutions. We conjecture these solutions to be unique, and we show some regularity results for energy solutions of the KPZ/Burgers equation, supporting this conjecture.

Exponential Ergodicity of Stochastic Burgers Equations Driven by α-Stable Processes

In this work, we prove the strong Feller property and the exponential ergodicity of stochastic Burgers equations driven by α/2-subordinated cylindrical Brownian motions with α∈(1,2). To prove the results, we truncate the nonlinearity and use the derivative formula for SDEs driven by α-stable noises established in (Zhang in Stoch. Process. Appl. 123(4):1213–1228, 2013).

Strong solution of the stochastic Burgers equation

This work introduces a pathwise notion of solution for the stochastic Burgers equation, in particular, our approach encompasses the Cole–Hopf solution. The developments are based on regularization arguments from the theory of distributions.

Regularization by noise and stochastic Burgers equations

We study a generalized 1d periodic SPDE of Burgers type: $$\begin{aligned} \partial _t u =- A^\theta u + \partial _x u^2 + A^{\theta /2} \xi \end{aligned}$$where \(\theta > 1/2\), \(-A\) is the 1d Laplacian, \(\xi \) is a space–time white noise and the initial condition \(u_0\) is taken to be (space) white noise. We introduce a notion of weak solution for this equation in the stationary setting. For these solutions we point out how the noise provide a regularizing effect allowing to prove existence and suitable estimates when \(\theta >1/2\). When \(\theta >5/4\) we obtain pathwise uniqueness. We discuss the use of the same method to study different approximations of the same equation and for a model of stationary 2d stochastic Navier–Stokes evolution.

Well-posedness of stochastic partial differential equations with Lyapunov condition

In this paper we show the existence and uniqueness of strong solutions for a large class of SPDE where the coefficients satisfy the local monotonicity and Lyapunov condition (one-sided linear growth condition). Moreover, some new invariance result and stronger regularity estimate are also established for the solutions. As examples, the main result is applied to stochastic tamed 3D Navier–Stokes equations, stochastic generalized curve shortening flow, singular stochastic p-Laplace equations, stochastic fast diffusion equations, stochastic Burgers type equations and stochastic reaction–diffusion equations.

Stochastic Burgers PDEs with random coefficients and a generalization of the Cole–Hopf transformation

This paper studies forward and backward versions of the random Burgers equation (RBE) with stochastic coefficients. First, the celebrated Cole–Hopf transformation reduces the forward RBE to a forward random heat equation (RHE) that can be treated pathwise. Next we provide a connection between the backward Burgers equation and a system of FBSDEs. Exploiting this connection, we derive a generalization of the Cole–Hopf transformation which links the backward RBE with the backward RHE and investigate the range of its applicability. Stochastic Feynman–Kac representations for the solutions are provided. Explicit solutions are constructed and applications in stochastic control and mathematical finance are discussed.

A stochastic Burgers-Zeldovich model for the formation of planetary ring systems and the satellites of Jupiter and Saturn

We consider a proto-ring nebula of a gas giant such as Neptune as a cloud of gas/dust particles whose behaviour is governed by the stochastic mechanics associated to the Kepler problem. This leads to a stochastic Burgers-Zeldovich type model for the formation of planetesimals involving a stochastic Burgers equation with vorticity which could help to explain the turbulent behaviour observed in ring systems. The Burgers fluid density and the distribution of the mass M(T) of a spherical planetesimal of radius δ are computed for times T. For circular orbits, sufficient conditions on certain time averages of δ2 are given ensuring that VarM(T) ∼ 0 as T ∼ ∞. Some applications are given to the satellites of Jupiter and Saturn, in particular giving a possible explanation of the equal mass families of satellites.

Stochastic Burgers equation with polynomial nonlinearity driven by Lévy process

Stochastic Burgers equation with polynomial nonlinearity driven by Lévy process

Scaling Limits of Additive Functionals of Interacting Particle Systems

AbstractUsing the renormalization method introduced by the authors, we prove what we call the local Boltzmann‐Gibbs principle for conservative, stationary interacting particle systems in dimension d = 1. As applications of this result, we obtain various scaling limits of additive functionals of particle systems, like the occupation time of a given site or extensive additive fields of the dynamics. As a by‐product of these results, we also construct a novel process, related to the stationary solution of the stochastic Burgers equation. © 2013 Wiley Periodicals, Inc.

Instanton filtering for the stochastic Burgers equation

We address the question of whether one can identify instantons in direct numerical simulations of the stochastically driven Burgers equation. For this purpose, we first solve the instanton equations using the Chernykh–Stepanov method (2001 Phys. Rev. E 64 026306). These results are then compared to direct numerical simulations by introducing a filtering technique to extract prescribed rare events from massive data sets of realizations. Using this approach we can extract the entire time history of the instanton evolution, which allows us to identify the different phases predicted by the direct method of Chernykh and Stepanov with remarkable agreement.

Full discretization of the stochastic Burgers equation with correlated noise

Full discretization of the stochastic Burgers equation with correlated noise

Ergodicity of Stochastic Burgers’ System with Dissipative Term

A 2-dimensional stochastic Burgers equation with dissipative term perturbed by Wiener noise is considered. The aim is to prove the well-posedness, existence, and uniqueness of invariant measure as well as strong law of large numbers and convergence to equilibrium.