Shulan Hu, Ran Wang
For one-dimensional stochastic Burgers equation driven by Brownian motion and Poisson process, we study the $ψ$-uniformly exponential ergodicity with $ψ(x)=1+\|x\|$, the moderate deviation principle and the large deviation principle for the occupation measures.
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A Markov jump process approximation of the stochastic Burgers equation
本文考虑具有周期边界条件的随机Burgers方程$ \dnachd{t} ψ(t,r) = Δψ(t,r) + \nabla ψ^2(t,r)+\sqrt{γψ(t,r)} η(t,r) $，其中$t \ge 0,$ $r \in [0,1],$和$η$是时空白噪声.构造了一个马尔可夫跳过程来逼近这个方程的解。
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。
Convergence to the Stochastic Burgers Equation from a degenerate microscopic dynamics
在本文中，我们证明了收敛到随机Burgers方程的一维相互作用粒子系统，其动力学允许退化的跳跃率。为此，我们提供了[Gonçalves，Jara 2014]中引入的二阶Boltzmann-Gibbs原理的新证明。主要的技术困难是，我们的模型表现出的配置，不发展下的动态-阻塞配置-和局部非遍历。我们的证明不强加任何知识的光谱间隙的微观模型。相反，它依赖于这样一个事实，即在平衡测度下，在一个有限盒子中找到一个阻塞配置的概率在盒子的大小上是指数小的。然后，一个动力学机制允许交换粒子，即使直接交换的跳跃率为零。
On microscopic derivation of a fractional stochastic Burgers equation
我们从一类微观非对称相互作用粒子系统出发，在${\mathbb Z}$上得到了长程跳跃率为$|\cdot|^{-(1+α)}$阶的连续分数阶随机微分方程，对于$0<α<2$.更具体地说，我们显示的零程过程的流体力学质量密度场的平衡波动，取决于结构的不对称性，以及是否与过程特征速度的字段被翻译，是在各种意义上由类型的分数随机热或Burgers方程。主要结果：假设跳跃率是这样的，它的对称性是长程的，但它的（弱）不对称性是最近邻的。然后，当$α<3/2$，时间尺度$1/α:1$中的波动场，以过程特征速度平移，不考虑不对称的强度，收敛到分数阶随机热方程，也是对称过程的极限。然而，当$α\geq 3/2$和弱不对称性的强度在尺度$1-3/2α$中调谐时，相关的极限点满足分数随机Burgers方程的鞅公式。
Stochastic Burgers equation from long range exclusion interactions
考虑一维长跳跃排斥过程，其转移概率为$p_n(\cdot)=s(\cdot)+γ_na(\cdot)$，使得其对称部分$s(\cdot)$是有限方差不可约的，反对称部分由$s(\cdot).$绝对有界.证明了在扩散时间标度和不对称强度$\sqrt n γ_n \to_{n\to\infty} b\neq 0$下，平衡态密度涨落由随机Burgers方程的唯一能量解给出.
On the Hausdorff dimension of regular points of inviscid Burgers equation with stable initial data
考虑一个无粘Burgers方程，其初始数据是一个Levy a-稳定过程Z，且a > 1。我们证明了当Z有正跳时，当a接近1时，与方程相关的拉格朗日正则点集的豪斯多夫维数严格小于1/a.这对Janicki和Woyczynski的一个猜想给出了否定的回答。在此过程中，我们反驳了Z. Shi关于积分稳定过程的下尾.
A variational approach to the stationary solutions of Burgers equation
考虑有界区间上粘性Burgers方程的非齐次Dirichlet边界条件.在Bertini-De Sole-Gabrielli-Jona-Lasinio-Landim C提出的变分框架下，我们分析了该方程的一个Lyapunov泛函，该泛函给出了与Burgers方程相关联的随机相互作用粒子模型的大偏差渐近性.我们讨论了这个能量泛函的渐近行为，其极小值由唯一的稳态解给出，随着区间的长度发散。我们专注于边界数据对应的驻波解决方案的Burgers方程在全线。在这种情况下，限制功能实际上有一个单参数家庭的极小值，我们分析所谓的发展伽玛收敛，这相当于计算尖锐的渐近成本对应于一个给定的移位的固定解决方案。
Stochastic Burgers Equation from Non-Product Stationary Measures via a Generalised Second-Order Boltzmann-Gibbs Principle
本文证明广义二阶Boltzmann-Gibbs原理，适用于非乘积平稳测度的保守粒子系统，并应用于Katz-Lebowitz-Spohn模型，证明其密度涨落在扩散标度下收敛到随机Burgers方程的平稳能量解。
In this paper, we develop a new method to obtain the accessibility of stochastic partial differential equations driven by additive pure jump noise. An important novelty of this paper is to allow the driving noises to be degenerate. As an application, for the first time, we obtain the accessibility of a class of stochastic equations driven by pure jump degenerate noise, which cover 2D stochastic Navier-Stokes equations, stochastic Burgers type equations, singular stochastic p-Laplace equations, stochastic fast diffusion equations, etc. As a further application, we establish the ergodicity of si…
We are concerned about the averaging principle for the stochastic Burgers equation with slow-fast time scale. This slow-fast system is driven by Lévy processes. Under some appropriate conditions, we show that the slow component of this system strongly converges to a limit, which is characterized by the solution of stochastic Burgers equation whose coefficients are averaged with respect to the stationary measure of the fast-varying jump-diffusion. To illustrate our theoretical result, we provide some numerical simulations.
Approximations of Stochastic Partial Differential Equations
本文证明了布朗运动驱动的随机偏微分方程的解可以用纯跳跃噪声/随机反冲驱动的随机偏微分方程逼近。随机Burgers方程的应用进行了讨论。
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…
A new mechanism leading to a random version of Burgers' equation is introduced: it is shown that the Totally Asymmetric Exclusion Process in discrete time (TASEP) can be understood as an intrinsically stochastic, non-entropic weak solution of Burgers' equation on $\mathbb{R}$. In this interpretation, the appearance of randomness in the Burgers' dynamics is caused by random additions of jumps to the solution, corresponding to the random effects in TASEP.
Asymptotics of the truncated variation of model-free price paths and semimartingales with jumps
我们证明了典型的（在无模型金融设置）跳跃的价格路径可以一致近似的准确度$c>0$的路径，其全变差是为了$1/c.$一个更精确的结果得到半鞅跳跃。
Stochastic ranking process with space-time dependent intensities
我们考虑粒子跳跃率与空间-不定常相关的随机秩过程。该过程是排名时间演变的简化模型，例如在线书店的销售排名。我们证明了跳跃率和标度位置的联合经验分布几乎必然收敛到一个确定性分布，并且在无限粒子极限下，标记粒子过程几乎必然收敛。极限分布的特点是一个系统的无粘Burgers型积分-偏微分方程的蒸发项，和一个标记粒子的极限过程是一个运动沿特征曲线的微分方程除了在其泊松时间的跳跃到原点。
Local asymptotic properties for the growth rate of a jump-type CIR process
本文针对由布朗运动和从属过程驱动的跳型CIR过程，在连续与高频离散观测下分别证明其增长率在次临界、临界和超临界情形具有LAN、LAQ和LAMN三种局部渐近性质，核心工具为Malliavin calculus与跳跃结构分析。
Averaging principles for nonautonomous multiscale stochastic Burgers equations with reflection
本文在最小假设下建立带反射的非自治多尺度随机Burgers方程的一般 averaging 原理，通过周期或渐近条件得到两个系数不依赖小参数的独立平均方程，并给出实例验证。
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…
Asymptotic equivalence of jumps Lévy processes and their discrete counterpart
建立了时间区间$[0,T]$上的纯跳Lévy过程$\{X_t\}$与参数与Lévy测度$ν$相关的$2m^2$ Poisson独立随机变量的观测值之间的全局渐近等价性.当$m$趋于无穷大时，等价结果是渐近的.时间$T$保持固定，并连续观察样品路径。这个结果证明了这样一个观点，即从统计学的角度来看，知道有多少跳跃落入区间网格中，与观察$\{X_t\}$得到的信息量是渐进相同的。
Tail asymptotics of the stationary distribution of a two dimensional reflecting random walk with unbounded upward jumps
考虑二维非负整数象限上的反射随机游动。假设该随机游动在到象限边界的方向上是无跳跃的，但是可以在相反方向上具有无界跳跃，其被称为向上跳跃。我们感兴趣的是它的平稳分布的尾渐近行为，只要它存在。假设向上跳跃的大小分布具有轻尾，我们完全找到了边缘平稳分布在所有方向上的粗尾渐近性。这推广了Miyazawa（2009）[Mathematics of运筹学34，547-575]中无跳反射随机游动的相应结果。我们证明了这些结果的两个节点的网络与外源批次到达。
Weak and strong versions of the Kolmogorov 4/5-law for stochastic Burgers equation
本文针对空间周期随机1d Burgers方程建立了Kolmogorov 4/5定律的弱强两个版本，证明了Landau对湍流普适性反对意见的类比，并指出第三矩是唯一具有普适渐近展开的矩。
Uniform Asymptotics for Compound Poisson Processes with Regularly Varying Jumps and Vanishing Drift
研究了一类谱单边Lévy过程泛函分布尾的重尾大偏差估计.我们的贡献是表明，这些估计仍然有效，在近临界状态。这补充了最近的类似结果，已获得的所有时间的上确界，这样的过程。具体来说，我们考虑的所有时间的上确界，直到退出$[0,\infty)$的过程的上确界的局部渐近性，最大跳跃，直到该时间，直到退出$[0,\infty)$的时间。除其他外，证明依赖于尺度函数的性质。
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。
Concave majorant of stochastic processes and Burgers turbulence
初始势为$ψ$的无粘Burgers方程的渐近解与$ψ$的图的凸壳密切相关。在本文中，我们研究这个凸壳，更准确地说，它的极值点，如果$ψ$是一个随机过程。到达这些极值点的时间称为极值时间，形成了Lévy过程、它们的积分过程和伊藤过程的可忽略集合。我们更仔细地研究了有界变差的Lévy过程的情况。它的极值点几乎肯定是可数的，只有在极值附近才有累加。这些结果是从对$ψ+f$极值时间的一般研究中得到的，其中$ψ$是Lévy过程，$f$是光滑的确定性漂移.这些结果使我们能够证明，对于具有紧支初始势$ψ$的无粘Burgers湍流，唯一能够成为拉格朗日正则的点是$ψ$达到最大值的时刻$T$，并且这确实是一个正则点当且仅当0对于两条半线都是正则的。因此，如果湍流发生在非紧凑间隔上，则存在a.s.没有拉格朗日正则点。
The purpose of this paper is to investigate properties of self-exciting jump processes. We derive the Laplace transform of SDE driven self-exciting processes with independent, identically distributed jump sizes. By using this Laplace transform, we find a recursive formula for the moments of the self-exciting process. The formula for the moments allow us to derive expressions for the expectation and variance of the self-exciting process. We show that self-exciting processes can exhibit both finite and infinite activity behaviour. Furthermore, we show that the scaling limit of the intensity proc…
Donsker-Varadhan asymptotics for degenerate jump Markov processes
考虑紧致度量空间上的一类连续时间马尔可夫链，该链允许开集上的不变测度严格正，且具有吸收状态。证明了经验测度与流量的联合大偏差原理。由于速率函数不具有一致遍历性，其零水平集不是单点集。作为推论，我们得到的Donsker-Varadhan率函数的经验措施和变分表达式的经验流量的率函数。
Asymptotics for the ruin time of a piecewise exponential Markov process with jumps
本文考虑一类由复合Poisson过程驱动的Ornstein-Uhlenbeck过程.跳跃以指数等待时间到达，并且允许是双侧的。假设跳跃形成一个独立于其他一切的指数分布的混合分布（不一定是凸的）的iid序列。允许向下跳跃的事实使得通过连续性和跳跃都可能通过给定的较低水平。考虑了这种通过的时间和可能的下冲（在跳跃情况下）。通过求过程的无穷小生成元的部分特征函数，得到了通过时间和下冲的联合拉普拉斯变换的表达式，并由拉普拉斯变换得到了过水平的破产概率.当漂移为负时，这个概率小于1，并且当过程的初态趋于无穷大时，它的渐近行为被明确地确定。交叉能级降到负无穷大的情况更加复杂：交叉能级在联合拉普拉斯变换的表达式中比过程的初态起着更基本的作用。给出了正漂移情形下的破产概率的极限和负漂移情形下的下冲分布的极限。
On Asymptotics of Solutions of Stochastic Differential Equations with Jumps
本文研究带跳的一维随机微分方程，证明了在系数满足特定幂渐近性与增长条件时，其解在无穷远处几乎必然与确定性方程解具有相同渐近性。
Motivated by the multilevel Monte Carlo method introduced by Giles [5], we study the asymptotic behavior of the normalized error process $u_{n,m}(X^n-X^{nm})$ where $X^n$ and $X^{nm}$ are respectively Euler approximations with time steps $1/n$ and $1/nm$ of a given stochastic differential equation $X$ driven by a pure jump Lévy process. In this paper, we prove that this normalized multilevel error converges to different non-trivial limiting processes with various sharp rates $u_{n,m}$ depending on the behavior of the Lévy measure around zero. Our results are consistent with those of Jacod [9] …
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…
Asymptotic behavior of random walks on a half-line with a jump at the origin
研究了非负整数上的离散随机游动，使得当到达0时，以给定的概率跳到任意位置。我们得到了一个渐近公式的期望位置在大的时候，依赖于跳跃概率和起始位置。我们证明这一结果显示的相关性的过渡算子的谱分析的随机过程，其特征值和共振。
Large deviations principle for invariant measures of stochastic Burgers equations
针对带Dirichlet边界条件的(0,1)上随机Burgers方程，研究小噪声渐近性，考虑比时空白噪声更奇异的噪声，证明解的大偏差原理（关于初值一致），并在特定Q_ε收敛条件下证明不变测度的大偏差原理。
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。
Thermodynamic limit for directed polymers and stationary solutions of the Burgers equation
本文的第一个目的是证明时间离散和空间连续的聚合物模型在随机势场中的随机游动的多个渐近结果。这些成果包括：确定性的自由能密度的存在，在无限体积极限的每一个固定的渐近斜率;浓度不等式的自由能意味着一个约束其波动指数;直线估计意味着一个约束的横向波动指数。这个程序的高潮是几乎必然存在性和唯一性的聚合物措施的单边无限路径与给定的端点和斜率，并解释这些无限体积吉布斯措施的热力学极限。此外，我们证明了具有相同斜率和不同端点的聚合测度的边缘是彼此渐近的。本文的第二个目标是在不作任何紧性假设的情况下，发展了具有正粘性和随机反冲力的Burgers方程的遍历论。也就是说，我们证明了一个力-一个解决方案的原则，使用无限体积聚合物措施，以构建一个家庭的稳定的整体解决方案，该系统，并证明这些解决方案是一个一点拉回吸引子的初始条件下具有相同的空间平均。这提供了一个自然扩张的相同的程序实现的无粘Burgers方程的帮助下的行动极小，可以被视为零温度限制的聚合物措施。
Parameter estimation from local measurements for a class of stochastic Burgers equations
针对由时空白噪声驱动的含随机Burgers方程的半线性SPDE的扩散系数估计问题，基于空间局部观测，借助解的新正则性结果，证明了已有估计量的强相合性和渐近正态性。
In this paper, we study the asymptotic of exit problem for controlled Markov diffusion processes with random jumps and vanishing diffusion terms, where the random jumps are introduced in order to modify the evolution of the controlled diffusions by switching from one mode of dynamics to another. That is, depending on the state-position and state-transition information, the dynamics of the controlled diffusions randomly switches between the different drift and diffusion terms. Here, we specifically investigate the asymptotic exit problem concerning such controlled Markov diffusion processes in …
We investigate a simple velocity jump process in the regime of large deviation asymptotics. New velocities are taken randomly at a constant, large, rate from a Gaussian distribution with vanishing variance. The Kolmogorov forward equation associated with this process is the linear BGK kinetic transport equation. We derive a new type of Hamilton-Jacobi equation which is nonlocal with respect to the velocity variable. We introduce a suitable notion of viscosity solution, and we prove well-posedness in the viscosity sense. We also prove convergence of the logarithmic transformation towards this l…
Asymptotic equivalence for inhomogeneous jump diffusion processes and white noise
我们证明了由离散（高频）或连续观察时间非均匀跳跃扩散过程的路径和高斯白噪声实验产生的实验之间的全局渐近等价性。这里，所考虑的参数是漂移函数，并且我们假设观测时间$T$趋向于$\infty$。近似的Le Cam $Δ$距离的意义下，光滑条件下的未知漂移函数。这些渐近等价性是通过构建显式马尔可夫核来建立的，这些显式马尔可夫核可以用来从另一个实验中重现一个实验。
We study asymptotic properties of maximum likelihood estimators of drift parameters for a jump-type Heston model based on continuous time observations, where the jump process can be any purely non-Gaussian Lévy process of not necessarily bounded variation with a Lévy measure concentrated on $(-1,\infty)$. We prove strong consistency and asymptotic normality for all admissible parameter values except one, where we show only weak consistency and mixed normal (but non-normal) asymptotic behavior. It turns out that the volatility of the price process is a measurable function of the price process. …
Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models
令$σ_t(x)$表示到期时的隐含波动率$t$，其中$K=S_0 e^{xt}$和$x\in\bbR$和$S_0$是标的资产的当前价值。我们发现，$σ_t(x)$有一个统一的（在$x$）的限制，成熟$t$趋于无穷大，由公式$σ_\infty(x)=\sqrt{2}(h^*(x)^{1/2}+(h^*(x)-x)^{1/2})$，$x$在一些紧凑的邻域中的仿射随机波动率模型。函数$h^*$是标度对数光点过程的极限累积量生成函数$h$的凸对偶。我们表达$h$的功能特性的基础模型。极限公式的证明依赖于当时间趋于无穷大时标度对数点过程的大偏差行为。我们应用我们的结果获得了几类具有跳跃的随机波动率模型的极限微笑的应用（例如具有状态独立跳跃的Heston，具有状态依赖跳跃的Bates和Barndorff-Nielsen-Shephard模型）.
Small noise asymptotics for a class of jump-diffusions with heavy tails for large times
本文研究了具有重尾性的跳跃扩散在小噪声和大时间下的渐进行为，通过确定性控制问题的最优值来描述极限分布行为。
Asymptotic equivalence for pure jump Lévy processes with unknown Lévy density and Gaussian white noise
本文的目的是建立一个全局渐近等价的实验产生的离散（高频）或连续观察的L{é}vy过程的路径和高斯白噪声实验观察到的时间T，T趋于$\infty$。这些近似是在Le Cam距离的意义下，在未知L{é}vy密度的某些光滑条件下给出的。所有的渐近等价性都是通过构造显式马尔可夫核来建立的，这些显式马尔可夫核可以用来从另一个实验中重现一个实验。
In this paper, we define an extension of the supersymmetric hyperbolic nonlinear sigma model introduced by Zirnbauer. We show that it arises as a weak joint limit of a time-changed version introduced by Sabot and Tarrès of the vertex-reinforced jump process. It describes the asymptotics of rescaled crossing numbers, rescaled fluctuations of local times, asymptotic local times on a logarithmic scale, endpoints of paths, and last exit trees.
Asymptotic Analysis of Stochastic Variational Inequalities Modeling an Elasto-Plastic Problem with Vanishing Jumps
本文通过引入人工小跳跃近似随机变分不等式，证明当跳跃尺寸趋于零时近似解在任意有限时间区间上收敛，从而调和了弹塑性振荡器模型中相位难以描述与需要相位信息之间的矛盾。
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