Martin Hutzenthaler, Robert Link
In this article we establish strong convergence rates on the whole probability space for explicit full-discrete approximations of stochastic Burgers equations with multiplicative trace-class noise. The key step in our proof is to establish uniform exponential moment estimates for the numerical approximations.
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We construct solutions to Burgers type equations perturbed by a multiplicative space-time white noise in one space dimension. Due to the roughness of the driving noise, solutions are not regular enough to be amenable to classical methods. We use the theory of controlled rough paths to give a meaning to the spatial integrals involved in the definition of a weak solution. Subject to the choice of the correct reference rough path, we prove unique solvability for the equation and we show that our solutions are stable under smooth approximations of the driving noise.
In this paper, we develop low regularity theory for 3D Burgers equation perturbed by a linear multiplicative stochastic force. This method is new and essentially different from the deterministic partial differential equations(PDEs). Our results and method can be widely applied to other stochastic hydrodynamic equations and the deterministic PDEs. As a further study, we establish a random version of maximum principle for random 3D Burgers equations, which will be an important tool for the study of 3D stochastic Burgers equations. As we know establishing moment estimates for highly nonlinear sto…
Central limit theorem and moderate deviation principle for the stochastic generalized Burgers-Huxley equation with multiplicative noise
本文研究了带乘性Gaussian噪声的随机广义Burgers-Huxley方程的CLT与MDP，利用弱收敛方法建立了中偏差原理并推导了相应速率函数。
Stochastic Burgers equation driven by multiplicative Rosenblatt noise: local existence, uniqueness and regularity
本文研究由乘性Rosenblatt噪声驱动的随机Burgers方程，证明了弱解的局部存在性与唯一性，并建立了解的Hölder正则性。
In this work, we focus on the global solvability and uniform large deviations for the solutions of stochastic generalized Burgers-Huxley (SGBH) equation perturbed by a small multiplicative white in time and colored in space noise. The SGBH equation has the nonlinearity of polynomial order and noise considered in this work is infinite dimensional with a coefficient having linear growth. First, we prove the existence of a \textsl{unique local mild solution} in the sense of Walsh to SGBH equation with the help of a truncation argument and contraction mapping principle. Then the global solvability…
We sharpen in this work the tools of paracontrolled calculus in order to provide a complete analysis of the parabolic Anderson model equation and Burgers system with multiplicative noise, in a $3$-dimensional Riemannian setting, in either bounded or unbounded domains. With that aim in mind, we introduce a pair of intertwined space-time paraproducts on parabolic Hölder spaces, with good continuity, that happens to be pivotal and provides one of the building blocks of higher order paracontrolled calculus.
In this project we investigate the stochastic Burgers' equation with multiplicative space-time white noise on an unbounded spatial domain. We give a random field solution to this equation by defining a process via a kind of Feynman-Kac representation which solves a stochastic partial differential equation such that its Hopf-Cole transformation solves Burgers' equation. Finally, we obtain Hölder regularity and moment estimates for the solution to Burgers' equation.
We present the $L_p$-solvability for stochastic time fractional Burgers' equations driven by multiplicative space-time white noise: $$ \partial_t^αu = a^{ij}u_{x^ix^j} + b^{i}u_{x^i} + cu + \bar b^i u u_{x^i} + \partial_t^β\int_0^t σ(u)dW_t,\,t>0;\,\,u(0,\cdot) = u_0, $$ where $α\in(0,1)$, $β< 3α/4+1/2$, and $d< 4 - 2(2β-1)_+/α$. The operators $\partial_t^α$ and $\partial_t^β$ are the Caputo fractional derivatives of order $α$ and $β$, respectively. The process $W_t$ is an $L_2(\mathbb{R}^d)$-valued cylindrical Wiener process, and the coefficients $a^{ij}, b^i, c$ and $σ(u)$ are random. In add…
In this paper, we study the strong and weak convergence rates for multi-scale one-dimensional stochastic Burgers equation. Based on the techniques of Galerkin approximation, Kolmogorov equation and Poisson equation, we obtain the slow component strongly and weakly converges to the solution of the corresponding averaged equation with optimal orders 1/2 and 1 respectively. The highly nonlinear term in system brings us huge difficulties, we develop new technique to overcome these difficulties. To the best of our knowledge, this work seems to be the first result in which the optimal convergence or…
In this work, we consider the stochastic Burgers-Huxley equation perturbed by multiplicative Gaussian noise, and discuss about the global solvability results and asymptotic behavior of solutions. We show the existence of a global strong solution of the stochastic Burgers-Huxley equation, by making use of a local monotonicity property of the linear and nonlinear operators and a stochastic generalization of localized version of the Minty-Browder technique. We then discuss about the inviscid limit of the stochastic Burgers-Huxley equation to Burgers as well as Huxley equations. By considering the…
We introduce the uniqueness, existence, $L_p$-regularity, and maximal Hölder regularity of the solution to semilinear stochastic partial differential equation driven by a multiplicative space-time white noise: $$ u_t = au_{xx} + bu_{x} + cu + \bar b|u|^λu_{x} + σ(u)\dot W,\quad (t,x)\in(0,\infty)\times\mathbb{R}; \quad u(0,\cdot) = u_0, $$ where $λ> 0$. The function $σ(u)$ is either bounded Lipschitz or super-linear in $u$. The noise $\dot W$ is a space-time white noise. The coefficients $a,b,c$ depend on $(ω,t,x)$, and $\bar b$ depends on $(ω,t)$. The coefficients $a,b,c,\bar{b}$ are uniforml…
In this paper we prove the convergence to the stochastic Burgers equation from one-dimensional interacting particle systems, whose dynamics allow the degeneracy of the jump rates. To this aim, we provide a new proof of the second order Boltzmann-Gibbs principle introduced in [Gonçalves, Jara 2014]. The main technical difficulty is that our models exhibit configurations that do not evolve under the dynamics - the blocked configurations - and are locally non-ergodic. Our proof does not impose any knowledge on the spectral gap for the microscopic models. Instead, it relies on the fact that, under…
Invariant measures for stochastic Burgers equation in unbounded domains with space-time white noise
本文研究全实轴上带乘性时空白噪声的随机阻尼Burgers方程，证明了加权空间中温和解的存在唯一性与依概率有界性，并借助Krylov-Bogolioubov定理得到了不变测度的存在性。
Regional Fractional Stochastic Burgers from random interactions
本文研究带长跳跃的边界驱动排斥过程，根据非对称强度证明其收敛于Ornstein-Uhlenbeck或区域分数阶随机Burgers方程的平稳解，后者部分收敛性由已有能量解唯一性结果支撑。
This article deals with stochastic partial differential equations with quadratic nonlinearities perturbed by small additive and multiplicative noise. We present the approximate solution of the original equation via the amplitude equation and give the rigorous error analysis. For illustration, we apply our main theorems to stochastic Burger's equation.
A Freidlin-Wentzell type large deviation principle is established for stochastic partial differential equations with slow and fast time-scales, where the slow component is a one-dimensional stochastic Burgers equation with small noise and the fast component is a stochastic reaction-diffusion equation. Our approach is via the weak convergence criterion developed in [3].
A moderate deviations principle for the law of a stochastic Burgers equation is proved via the weak convergence approach. In addition, some useful estimates toward a central limit theorem are established.
Quantitative approximation of the Burgers and Keller-Segel equations by moderately interacting particles
本文研究具有奇异核的适度相互作用粒子系统对粘性 Burgers 方程和 Keller-Segel 方程的定量逼近，在非局部可积核条件下分别获得了 $N^{-1/6}$ 和 $N^{-\frac{1}{2(d+1)}}$ 阶的几乎必然收敛速率。
Invariant measures for stochastic Burgers equation on unbounded domains
针对整个实线上带乘性噪声的随机阻尼 Burgers 方程，证明 mild 解唯一存在且时间一致有界，并借助尾部估计、分布胎紧性与 Krylov-Bogolioubov 定理得到不变测度。
Large deviation principles for multiscale stochastic Burgers equations with reflection
研究具有反射的多尺度随机伯格斯方程，利用弱收敛方法，通过惩罚方法等关键技术工具，严格建立慢分量的大偏差原理。
We consider the stationary O'Connell-Yor model of semi-discrete directed polymers in a Brownian environment in the intermediate disorder regime and show convergence of the increments of the log-partition function to the energy solutions of the stochastic Burgers equation. The proof does not rely on the Cole-Hopf transform and avoids the use of spectral gap estimates for the discrete model. The key technical argument is a second-order Boltzmann-Gibbs principle.
It has been known for a while that a nonlinear equation driven by singular noise must be interpreted in the re-normalized, or Wick, form. For the stochastic Burgers equation, Wick nonlinearity forces the solution to be a generalized process no matter how regular the random perturbation is, whence the curse. On the other hand, certain multiplicative random perturbations of the deterministic Burgers equation can only be interpreted in the Wick form, whence the cure. The analysis is based on the study of the coefficients of the chaos expansion of the solution at different stochastic scales.
In this paper, we consider the averaging principle for one dimensional stochastic Burgers equation with slow and fast time-scales. Under some suitable conditions, we show that the slow component strongly converges to the solution of the corresponding averaged equation. Meanwhile, when there is no noise in the slow component equation, we also prove that the slow component weakly converges to the solution of the corresponding averaged equation with the order of convergence $1-r$, for any $0<r<1$.
Absolute continuity of the solution to stochastic generalized Burgers-Huxley equation
本文研究乘性时空白噪声驱动的随机广义Burgers-Huxley方程，证明其整体可解性，并借助比较定理与Malliavin弱可微性得到解的分布关于Lebesgue测度的绝对连续性及密度存在性。
The main result of this article establishes strong convergence rates on the whole probability space for explicit space-time discrete numerical approximations for a class of stochastic evolution equations with possibly non-globally monotone coefficients such as stochastic Burgers equations with additive trace-class noise. The key idea in the proof of our main result is (i) to bring the classical Alekseev-Gröbner formula from deterministic analysis into play and (ii) to employ uniform exponential moment estimates for the numerical approximations.
In the paper, we consider a stochastic hybrid Korteweg - de Vries - Burgers type equation with multiplicative noise in the form of cylindrical Wiener process. We prove the existence of a martingale solution to the equation studied. The proof of the existence of the solution is based on two approximations of the considered problem and the compactness method. First, we introduce an auxiliary problem corresponding to the equation studied. Then, we prove the existence of a martingale solution to this problem. Finally, we show that the solution of the auxiliary problem converges, in some sense, to …
Consider the viscous Burgers equation on a bounded interval with inhomogeneous Dirichlet boundary conditions. Following the variational framework introduced by Bertini-De Sole-Gabrielli-Jona-Lasinio-Landim C, we analyze a Lyapunov functional for such equation which gives the large deviations asymptotics of a stochastic interacting particles model associated to the Burgers equation. We discuss the asymptotic behavior of this energy functional, whose minimizer is given by the unique stationary solution, as the length of the interval diverges. We focus on boundary data corresponding to a standing…
Optimal rate of convergence for stochastic Burgers-type equations
本文研究了加性噪声下随机Burgers型方程的数值解收敛性，证明了其收敛率可接近1/2，改进了之前关于收敛率接近1/6的结果。
Ergodicity for the randomly forced Korteweg-de Vries-Burgers equation
本文研究非可加白噪声驱动的 KdVB 方程遍历性，利用新解析性质与耦合及渐近耦合方法，证明局部化噪声下的指数遍历性和乘性噪声下的遍历性。
Complex systems may be subject to various uncertainties. A great effort has been concentrated on predicting the dynamics under uncertainty in initial conditions. In the present work, we consider the well-known Burgers equation with random boundary forcing or with random body forcing. Our goal is to attempt to understand the stochastic Burgers dynamics by predicting or estimating the solution processes in various diagnostic metrics, such as mean length scale, correlation function and mean energy. First, for the linearized model, we observe that the important statistical quantities like mean ene…
In this paper, we study a class of slow-fast stochastic partial differential equations with multiplicative Wiener noise. Under some appropriate conditions, we prove the slow component converges to the solution of the corresponding averaged equation with optimal orders 1/2 and 1 in the strong and weak sense respectively. The main technique is based on the Poisson equation.
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, stoch…
Strong convergence of a fully discrete scheme for stochastic Burgers equation with fractional-type noise
本文研究了由加性圆柱分数布朗运动驱动的随机Burgers方程的数值近似，采用谱伽辽金方法空间离散，结合非线性 tame 加速指数欧拉方法构建全离散方案，通过证明分数布朗运动的随机卷积指数可积性，得出半离散和全离散近似的矩有界性，并利用停止时间技术证明方案的强收敛性。
Weak error estimates of Galerkin approximations for the stochastic Burgers equation driven by additive trace-class noise
本文研究了由可加迹类噪声驱动的随机粘性Burgers方程的谱Galerkin逼近，证明了其弱收敛速率为2，是已知强收敛速率的两倍，并利用相关Kolmogorov方程的新正则性结果完成证明。
In this paper, we establish the existence and uniqueness of invariant measures for a class of semilinear stochastic partial differential equations driven by multiplicative noise on a bounded domain. The main results can be applied to SPDEs of various types such as the stochastic Burgers equation and the reaction-diffusion equations perturbed by space-time white noise.
Convergence of processes time-changed by Gaussian multiplicative chaos
本文在$L^2$区域中证明了由高斯乘性混沌进行时间变换的过程的收敛性，并给出Liouville简单随机游走和Liouville $α$-稳定过程的收敛实例。
We investigate the problem of the rate of convergence to equilibrium for ergodic stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H\in (1/3,1)$ and multiplicative noise component $σ$. When $σ$ is constant and for every $H\in (0,1)$, it was proved in [19] that, under some mean-reverting assumptions, such a process converges to its equilibrium at a rate of order $t^{-α}$ where $α\in (0,1)$ (depending on $H$). In [11], this result has been extended to the multiplicative case when $H\textgreater{}1/2$. In this paper, we obtain these types of results in t…
In the present paper we study the convergence of the solution of the two dimensional (2-D) stochastic Leray-$α$ model to the solution of the 2-D stochastic Navier-Stokes equations. We are mainly interested in the rate of convergence, as $α$ tends to 0, of the error function which is the difference between the two solutions in an appropriate topology. We show that when properly localized the error function converges in mean square as $α\to 0$ and the convergence is of order $O(α)$. We also prove that the error converges in probability to zero with order at most $O(α)$.
We investigate the problem of the rate of convergence to equilibrium for ergodic stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H\textgreater{}1/2$ and multiplicative noise component $σ$. When $σ$ is constant and for every $H\in(0,1)$, it was proved in \cite{hairer} that, under some mean-reverting assumptions, such a process converges to its equilibrium at a rate of order $t^{-α}$ where $α\in(0,1)$ (depending on $H$). The aim of this paper is to extend such types of results to some multiplicative noise setting. More precisely, we show that we can r…
The present work deals with the global solvability as well as asymptotic analysis of stochastic generalized Burgers-Huxley (SGBH) equation perturbed by space-time white noise in a bounded interval of $\mathbb{R}$. We first prove the existence of unique mild as well as strong solution to SGBH equation and then obtain the existence of an invariant measure. Later, we establish two major properties of the Markovian semigroup associated with the solutions of SGBH equation, that is, irreducibility and strong Feller property. These two properties guarantees the uniqueness of invariant measures and er…
We study the convergence of Langevin-Simulated Annealing type algorithms with multiplicative noise, i.e. for $V : \mathbb{R}^d \to \mathbb{R}$ a potential function to minimize, we consider the stochastic differential equation $dY_t = - σσ^\top \nabla V(Y_t) dt + a(t)σ(Y_t)dW_t + a(t)^2Υ(Y_t)dt$, where $(W_t)$ is a Brownian motion, where $σ: \mathbb{R}^d \to \mathcal{M}_d(\mathbb{R})$ is an adaptive (multiplicative) noise, where $a : \mathbb{R}^+ \to \mathbb{R}^+$ is a function decreasing to $0$ and where $Υ$ is a correction term. Allowing $σ$ to depend on the position brings faster convergence…
In this review we discuss the weak KPZ universality conjecture for a class of 1-d systems whose dynamics conserves one or more quantities. As a prototype example for the former case, we will focus on weakly asymmetric simple exclusion processes, for which the density is preserved and the equilibrium fluctuations are shown to cross from the Edwards-Wilkinson universality class to the KPZ universality class. The crossover depends on the strength of the asymmetry. For the latter case, we will present an exclusion process with three species of particles, known as the ABC model, for which we aim to…
In this paper, we establish a central limit theorem (CLT) and the moderate deviation principles (MDP) for a class of semilinear stochastic partial differential equations driven by multiplicative noise on a bounded domain. The main results can be applied to stochastic partial differential equations of various types such as the stochastic Burgers equation and the reaction-diffusion equations perturbed by space-time white noise.
In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the so-called derivative martingale, introduced in the context of branching Brownian motions and branching random walks, converges almost surely (in all dimensions) to a random measure with full support. We also show that the limiting measure has no atom. In connection with the derivative martingale, we write explicit conjectures about the glassy phase of log-correlated Gaussian potentials and the relation with the asymptotic expansion of the maximum of log-correlated Gaussian random variables.
Self-similarity of Burgers' equation with some 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.
Strong and weak convergence rates for slow-fast system driven by multiplicative Lévy noises
本文研究了受α-稳定过程驱动的慢-快系统强收敛性和弱收敛率，通过耦合方法和空间周期方法推导指数遍历性，并利用热核渐进行为得到梯度估计，最终获得最优强收敛率和弱收敛率。
We study the convergence of Langevin-Simulated Annealing type algorithms with multiplicative noise, i.e. for $V : \mathbb{R}^d \to \mathbb{R}$ a potential function to minimize, we consider the stochastic equation $dY_t = - σσ^\top \nabla V(Y_t) dt + a(t)σ(Y_t)dW_t + a(t)^2Υ(Y_t)dt$, where $(W_t)$ is a Brownian motion, where $σ: \mathbb{R}^d \to \mathcal{M}_d(\mathbb{R})$ is an adaptive (multiplicative) noise, where $a : \mathbb{R}^+ \to \mathbb{R}^+$ is a function decreasing to $0$ and where $Υ$ is a correction term. This setting can be applied to optimization problems arising in Machine Learn…
The Fourier coefficients of the holomorphic multiplicative chaos in the limit of large frequency
本文研究全纯乘法混沌的大频率 Fourier 系数，证明归一化后依分布收敛于由 Gaussian 乘法混沌总质量缩放的复正态变量，并推广到连续系数的过程收敛，推出 circular-β-系综次线性特征系数的收敛律。
This paper concerns the convergence of an iterative scheme for 2D stochastic primitive equations on a bounded domain. The stochastic system is split into two equations: a deterministic 2D primitive equations with random initial value and a linear stochastic parabolic equation, which are both simpler for numerical computations. An estimate of approximation error is given, which implies that the strong speed rate of the convergence in probability is almost $\frac{1}{2}$.
Pathwise central limit theorem and moderate deviations via rough paths for SPDEs with multiplicative noise
本研究借助粗糙路径理论提出通用框架，推导小乘性线性噪声扰动的随机偏微分方程的路径中心极限定理与中偏差原理，给出最优收敛速度，并应用于多类典型随机偏微分方程。
We study the total mass of high points in a random model for the Riemann-Zeta function. We consider the same model as in [8], [2], and build on the convergence to 'Gaussian' multiplicative chaos proved in [14]. We show that the total mass of points which are a linear order below the maximum divided by their expectation converges almost surely to the Gaussian multiplicative chaos of the approximating Gaussian process times a random function. We use the second moment method together with a branching approximation to establish this convergence.
The convergence of the totally asymmetric simple exclusion process to the solution of the Burgers equation is a classical result. In his seminal 1981 paper, Herman Rost proved the convergence of the density fields and local equilibrium when the limiting solution of the equation is a rarefaction fan. An important tool of his proof is the subadditive ergodic theorem. We prove his results by showing how second class particles transport the rarefaction-fan solution, as characteristics do for the Burgers equation, avoiding subadditivity. In the way we show laws of large numbers for tagged particles…
Quantitative Contraction Rates for McKean-Vlasov Stochastic Differential Equations with Multiplicative Noise
本文采用耦合方法推导乘性噪声 McKean-Vlasov SDE 在 Wasserstein 距离下的显式收缩率，并据此证明时间一致的混沌传播与指数遍历性。
The familiar cascade measures are sequences of random positive measures obtained on $[0,1]$ via $b$-adic independent cascades. To generalize them, this paper allows the random weights invoked in the cascades to take real or complex values. This yields sequences of random functions whose possible strong or weak limits are natural candidates for modeling multifractal phenomena. Their asymptotic behavior is investigated, yielding a sufficient condition for almost sure uniform convergence to nontrivial statistically self-similar limits. Is the limit function a monofractal function in multifractal …
A stochastic transport linear equation (STLE) with multiplicative space-time dependent noise is studied. It is shown that, under suitable assumptions on the noise, a multiplicative renormalization leads to convergence of the solutions of STLE to the solution of a deterministic parabolic equation. Existence and uniqueness for STLE are also discussed. Our method works in dimension $d\geq 2$; the case $d=1$ is also investigated but no conclusive answer is obtained.
Galerkin approximations for the stochastic Burgers equation
本文研究了随机偏微分方程中Galerkin近似的存在性、唯一性和收敛性，特别针对随机Burgers方程，提出了在L^∞拓扑下统一估计的新型方法。
We consider the large time behavior of strong solutions to a kind of stochastic Burgers equation, where the position x is perturbed by a Brownian noise. It is well known that both the rarefaction wave and viscous shock wave are time-asymptotically stable for deterministic Burgers equation since the pioneer work of A. Ilin and O. Oleinik [21] in 1964. However, the stability of these wave patterns under stochastic perturbation is not known until now. In this paper, we give a definite answer to the stability problem of the rarefaction and viscous shock waves for the 1-d stochastic Burgers equatio…
In this paper, we find a regularized approximate solution for an inverse problem for the Burgers' equation. The solution of the inverse problem for the Burgers' equation is ill-posed, i.e., the solution does not depend continuously on the data. The approximate solution is the solution of a regularized equation with randomly perturbed coefficients and randomly perturbed final value and source functions. To find the regularized solution, we use the modified quasi-reversibility method associated with the truncated expansion method with nonparametric regression. We also investigate the convergence…
We define a notion of a viscous shock solution of the stochastic Burgers equation that connects "top" and "bottom" spatially stationary solutions of the same equation. Such shocks generally travel in space, but we show that they admit time-invariant measures when viewed in their own reference frames. Under such a measure, the viscous shock is a deterministic function of the bottom and top solutions and the shock location. However, the measure of the bottom and top solutions must be tilted to account for the change of reference frame. We also show a convergence result to these stationary shock …
Pathwise Holder convergence of the implicit Euler scheme for semi-linear SPDEs with multiplicative noise
本文证明在UMD Banach空间中，乘性噪声半线性抛物型随机微分方程的隐式欧拉格式具有最优路径wise Hölder收敛速率，并应用于乘性时空白噪声驱动的二阶抛物型SPDEs。
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