On a Stochastic Wave Equation Driven by a Non-Gaussian Levy Process
Lijun Bo, Kehua Shi, Yongjin Wang
本文研究了非高斯Levy噪声驱动下的阻尼随机波动方程。证明了该问题的弱解存在且唯一。在较弱的条件下证明了与转移半群相关联的唯一不变测度的存在性。
This paper investigates a damped stochastic wave equation driven by a non-Gaussian Levy noise. The weak solution is proved to exist and be unique. Moreover we show the existence of a unique invariant measure associated with the transition semigroup under mild conditions.
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Maximal regularity for stochastic convolutions driven by Levy noise
我们证明了Da Prato和Lunardi的结果对于Lévy过程驱动的随机卷积是有效的。
We investigate a class of stochastic integro differential equations driven by Levy noise.
Generalized backward doubly stochastic differential equations driven by Lévy processes with non-Lipschitz coefficients
在非Lipschitz假设下，证明了由Lévy过程驱动的广义倒向双重随机微分方程解的存在唯一性.
An L_p-theory of stochastic parabolic equations with the random fractional Laplacian driven by Lévy processes
本文给出了具有随机分式Laplacian算子的随机抛物型方程的$L_p$ -理论。驱动噪声是一般的Lévy过程。
Burkholder-Davis-Gundy type Inequalities of the Itô stochastic integral with respect to Levy noise on Banach spaces
给出了Banach值Levy噪声下伊藤积分的Burkholder-Davis-Gundy型不等式.
In the paper, we consider a type of stochastic differential equations driven by G-Lévy processes. We prove that a kind of their additive functionals has path independence and extend some known results.
Harnack Inequalities for Stochastic Equations Driven by Lévy Noise
利用耦合变元和正则化逼近，建立了一类由Lévy噪声驱动的含从属布朗运动的随机方程的无量纲Harnack不等式. Harnack不等式对于Lévy噪声驱动的线性方程也是新的，并且我们的log-Harnack不等式所隐含的梯度估计在很大程度上推广了最近关于Lévy噪声驱动的Lévy过程或线性方程的梯度估计和耦合性质的一些结果.主要结果也推广到希尔伯特空间中的半线性随机方程。
In this work, we derive sufficient and necessary conditions for the existence of a weak and mild solution of an abstract stochastic Cauchy problem driven by an arbitrary cylindrical Levy process. Our approach requires to establish a stochastic Fubini result for stochastic integrals with respect to cylindrical Levy processes. This approach enables us to conclude that the solution process has almost surely scalarly square integrable paths. Further properties of the solution such as the Markov property and stochastic continuity are derived.
Large deviations for stochastic models of two-dimensional second grade fluids driven by Lévy noise
本文建立了Lévy噪声驱动下二维二阶流体随机模型的大偏差原理。Budhiraja，Dupuis和Maroulas在[5]中提出的弱收敛方法起着关键作用。
Density stability for some Lévy-driven Stochastic Differential Equations
考虑由Lévy过程驱动的随机微分方程，其Lévy测度满足调和稳定控制.我们研究如何扰动的系数反映的解决方案的密度。我们根据系数的接近程度来量化密度之间的距离。这扩展到[KKM 15]的工作的稳定情况，其中噪声是高斯的。
Stochastic logarithmic Schrödinger equations driven by Lévy noise
本文研究带饱和非线性乘性Lévy噪声的随机对数Schrödinger方程，通过正则化方程在Orlicz空间中构造强收敛解，证明了方程的全局适定性。
Strong convergence rate of Euler-Maruyama approximations in temporal-spatial Hölder-norms for Lévy-driven stochastic differential equations
研究莱维驱动随机微分方程中欧拉-马尔日ama近似与精确解在时间-空间霍尔德范数下的误差，分析其收敛性及贡献。
Strong solutions for a stochastic model of 2-D second grade fluids driven by Lévy noise
考虑了Lévy噪声驱动下$\mathbb{R}^2$有界区域上二阶不可压缩非牛顿流体的随机模型.利用变分方法，证明了强概率解的整体存在唯一性。
Stochastic Population Dynamics Driven by Levy Noise
研究了Levy噪声驱动下的随机种群动力学.本文的贡献在于：（a）利用Khasminskiii-Mao定理证明了该模型的随机微分方程存在唯一的整体正解;（b）利用带跳的指数鞅不等式，讨论了该模型的渐近路径估计.
Large deviations for locally monotone stochastic partial differential equations driven by Levy noise
本文建立了一类Lévy噪声驱动的局部单调系数随机偏微分方程（SPDE）的大偏差原理.弱收敛方法在其中起着重要的作用。
Anticipating Linear Stochastic Differential Equations Driven by a Lévy Process
本文研究了由Lévy过程驱动的线性随机微分方程解的存在性，其中初始条件和系数是随机的，且不一定适应于潜在的过滤.为此，我们将Buckdahn在Wiener空间上发展的Girsanov变换方法推广到正则Lévy空间。
Stochastic integration with respect to cylindrical Levy processes in Hilbert spaces: an L^2 approach
在这项工作中，随机积分的圆柱Levy过程的弱二阶矩。众所周知，确定性Hilbert-Schmidt算子将柱面随机变量radonize，即它将柱面随机变量映射到经典希尔伯特空间值随机变量。我们的方法是基于这一结果的推广到随机希尔伯特-施密特算子的圆柱列维过程的圆柱增量的radonification。这一推广使我们能够引入一个希尔伯特空间值随机变量作为一个可预测的随机过程的随机积分相对于一个圆柱Levy过程。我们完成这项工作，通过推导伊藤等距和考虑短期随机偏微分方程驱动的圆柱Levy过程。
Spatial asymptotic behaviors of fractional stochastic heat equations driven by additive Lévy white noise
该研究针对加性Lévy白噪声驱动的分数阶随机热方程，建立了其空间渐近行为的显式积分检验准则，证明轻尾噪声下方程在全维度具有加性物理间歇性质，证明依赖分数阶Laplacian热核估计与泊松泛函精确尾部行为。
Building upon the well-posedness results in \cite{snse1}, in this note we prove the existence of invariant measures for the stochastic Navier-Stokes equations with stable Lévy noise. The crux of our proof relies on the assumption of finite dimensional Lévy noise.
Approximations for solutions of Lévy-type stochastic differential equations
研究了跳跃-扩散方程具有给定收敛阶的强逼近的构造问题。构造了Lévy型随机微分方程的一般逼近格式。特别地，本文推广了Platen Kloeden和Gardo \n的结果.给出了有限和无限Lévy测度下的Euler和Milstein格式。
Blow-up of stochastic semilinear parabolic equations driven by Lévy noise
针对加性与线性乘性Lévy噪声驱动的随机半线性抛物方程，本文改进随机情形下的凹性方法，证明了有界区域上系统解的爆破现象。
In this work, we consider the stochastic Cauchy problem driven by the canonical $α$-stable cylindrical Lévy process. This noise naturally generalises the cylindrical Brownian motion or space-time Gaussian white noise. We derive a sufficient and necessary condition for the existence of the weak and mild solution of the stochastic Cauchy problem and establish the temporal irregularity of the solution.
Multivalued stochastic partial differential-integral equations via backward doubly stochastic differential equations driven by a Lévy process
本文研究了一类由Teugels鞅驱动的带凸函数的次微分算子的倒向双随机微分方程（简称BDSDEs）.利用Yosida逼近证明了该解的存在唯一性。作为应用，我们给出了一类多值随机偏微分-积分方程（简称MSPIDE）的随机黏性解的存在性。
The work is about multiscale stochastic dynamical systems driven by Lévy processes. First, we prove that these systems can approximate low-dimensional systems on random invariant manifolds. Second, we establish that nonlinear filterings of multiscale stochastic dynamical systems also approximate that of reduced low-dimensional systems. Finally, we investigate the reduction for $\e=0$ and obtain that these reduced systems does not approximate these multiscale stochastic dynamical systems.
Dirichlet heat kernel estimates for isotropic Levy processes with Gaussian components in Lipschitz open sets
针对含高斯分量的不连续各向同性Levy过程，研究其在Lipschitz开集中的Dirichlet热核估计，给出双侧界及Varopoulos型估计成立的充要条件。
We study the existence and uniqueness of solutions to stochastic differential equations with Volterra processes driven by Lévy noise. For this purpose, we study in detail smoothness properties of these processes. Special attention is given to two kinds of Volterra-Gaussian processes that generalize the compact interval representation of fractional Brownian motion and to stochastic equations with such processes.
Large deviations for stochastic PDE with Levy noise
本文证明了抽象随机发展方程在小Levy噪声扰动下解的一个大偏差原理结果。我们使用Varadhan和Bryc的一般大偏差定理，Hilbert空间中积分-偏微分方程的黏性解，以及确定性最优控制方法。拉普拉斯极限被确定为一个相关的控制问题的Hamilton-Jacobi-Bellman方程的黏性解。我们还建立了Levy噪声驱动的随机发展方程解的指数矩估计。将一般结果应用于受从属Wiener过程扰动的随机双曲型方程。
Long-time behaviors of some stochastic differential equations driven by Lévy noise
研究Lévy噪声驱动的随机微分方程，利用Itô公式、矩估计和正则变化函数，给出解几乎必然渐近等价于非随机函数的条件。
In this article, well-posedness of stochastic anisotropic $p$-Laplace equation driven by Lévy noise is shown. Such an equation in deterministic setting was considered by Lions [7]. The results obtained in this article can be applied to solve a large class of semilinear and quasilinear stochastic partial differential equations.
On the location of the maximum of a process: Lévy, Gaussian and multidimensional cases
在这篇简短的文章中，我们展示了arXiv：1207.4469中提出的技术如何扩展到各种非连续和多变量过程。作为例子，我们证明了谱正Lévy过程，Ornstein-Uhlenbeck过程，分数阶布朗运动和布朗单等高斯过程的极大值位置的唯一性.
Excursions away from a regular point for one-dimensional symmetric Levy processes without Gaussian part
研究了一类不含高斯部分的对称Lévy过程偏离正则点的特征测度.证明了killed过程的调和变换具有Feller性质。应用所得结果证明了漂移测度的极值性，并证明了漂移过程和$ h $ -路过程的几个样本路行为。
We establish the existence and uniqueness for a one-dimensional stochastic differential equation driven by a Brownian motion and a pure jump {\levy} process. It is shown that under fairly general conditions on the coefficients, pathwise uniqueness holds based on the methods of weak uniqueness and local time technique.
Solutions of martingale problems for Lévy-type operators and stochastic differential equations driven by Lévy processes with discontinuous coefficients
我们证明了一维空间中Lévy型随机过程的存在性，其特征三联体在阈值处是不连续的，或者是稳定的，其稳定性指标函数的不连续集的闭包是可数的。为此，我们制定的Skorokhod空间鞅问题与非局部算子的不连续系数的问题。这些运营商近似沿着一系列的光滑的非本地运营商产生的Feller过程一致控制的符号。它们一致收敛于极限算子的奇点所在的勒贝格零集的越来越小的邻域之外。
In this article, the existence of a unique solution in the variational approach of the stochastic evolution equation $$\dX(t) = F(X(t)) \dt + G(X(t)) \dL(t)$$ driven by a cylindrical Lévy process $L$ is established. The coefficients $F$ and $G$ are assumed to satisfy the usual monotonicity and coercivity conditions. The noise is modelled by a cylindrical Lévy processes which is assumed to belong to a certain subclass of cylindrical Lévy processes and may not have finite moments.
This work is devoted to deriving the Onsager-Machlup action functional for stochastic partial differential equations with (non-Gaussian) Levy process as well as Gaussian Brownian motion. This is achieved by applying the Girsanov transformation for probability measures and then by a path representation. This enables the investigation of the most probable transition path for infinite dimensional stochastic dynamical systems modeled by stochastic partial differential equations, by minimizing the Onsager-Machlup action functional.
Stochastic partial differential equations driven by Levy space-time white noise
在本文中，我们开发了一个白噪声框架的d参数（纯跳）Levy白噪声驱动的随机偏微分方程的研究。作为一个例子，我们使用这个理论来解决随机泊松方程关于任何维d的Levy白噪声。解是一个显式给出的随机分布过程。我们还证明了，如果d \leq 3，那么这个解可以表示为L2（μ）中的经典随机场，其中μ是Levy过程的概率律.我们的理论的出发点是广义Charlier多项式的混沌展开。在此基础上定义了Kondratiev空间和Levy Hermite变换。
Large deviations for stochastic heat equations with memory driven by Levy-type noise
对于一类由Lévy型噪声驱动的带记忆项的热方程，我们证明了其唯一解的存在性.本文的主体部分主要研究了在Lévy型噪声驱动下具有记忆的热传导方程解的Freidlin-Wentzell大偏差原理。为此，我们利用最近推出的弱收敛方法。
In this paper we prove that the stochastic Navier-Stokes equations with stable Lévy noise generates a random dynamical systems. Then we prove the existence of random attractor for the Navier-Stokes equations on 2D spheres under stable Lévy noise (finite dimensional). We also deduce the existence of Feller Markov invariant measure.
In this paper, BDG-type inequality for G-stochastic calculus with respect to G-Levy process is obtained and solutions of stochastic differential equations driven by G-Levy process under non-Lipschitz condition are constructed. Moreover, we establish the mean square exponential stability and quasi sure exponential stability of the solutions be means of G-Lyapunov function method. An example is presented to illustrate the efficiency of the obtained results.
On stochastic integration for volatility modulated Lévy-driven Volterra processes
本文建立了波动调制Lévy-driven Volterra（VMLV）过程的随机积分理论。它扩展了最近的文献中的结果，允许随机波动和纯跳跃过程的积分。新的积分算子是基于Malliavin演算，并描述了一个预期的积分。导出了积分的基本性质，并给出了重要的应用。
Asymptotics of transition densities for Lévy processes with stochastic resetting
研究具有随机重置的Lévy过程转移密度渐近性，通过推导显式表示，该方法适用于相对论稳定过程等，给出了新的渐近估计。
We analyze the spatial asymptotic properties of the solution to the stochastic heat equation driven by an additive Lévy space-time white noise. For fixed time $t > 0$ and space $x \in \mathbb{R}^d$ we determine the exact tail behavior of the solution both for light-tailed and for heavy-tailed Lévy jump measures. Based on these asymptotics we determine for any fixed time $t> 0$ the almost-sure growth rate of the solution as $|x| \to \infty$.
We give a new definition of a Lévy driven CARMA random field, defining it as a generalized solution of a stochastic partial differential equation (SPDE). Furthermore, we give sufficient conditions for the existence of a mild solution of our SPDE. Our model finds a connection between all known definitions of CARMA random fields, and especially for dimension 1 we obtain the classical CARMA process.
Weak convergence of Galerkin approximations of stochastic partial differential equations driven by additive Lévy noise
本文研究了Lévy噪声驱动的随机偏微分方程的弱逼近问题。手头的SPDE是抛物线与加性噪声过程。给出了相应的Galerkin逼近的弱收敛速度。收敛结果是通过使用Malliavin导数而不是通过Kolmogorov向后方程的普通方法导出的。
Stochastic integration with respect to cylindrical Lévy processes
一个圆柱Levy过程不享受半鞅分解的圆柱版本，这导致需要开发一个全新的方法来随机集成。本文在希尔伯特空间中引入了关于圆柱Levy过程的随机被积函数的随机积分。容许被积函数空间由值在希尔伯特-施密特算子空间中的适应随机过程组成。被积函数和积分器都不需要满足任何矩或有界性条件。积分过程的特点是适应，希尔伯特空间值半鞅与cadlag轨迹。
Generalized backward doubly stochastic differential equations driven by Lévy processes with continuous coefficients
研究了一类新的由Teugels鞅驱动的广义倒向双重随机微分方程（简称GBDSDEs）.我们建立了一个比较定理，使我们能够得到的连续和线性增长条件下的解的存在性结果。
We prove a dual Yamada-Watanabe theorem for one-dimensional stochastic differential equations driven by quasi-left continuous semimartingales with independent increments. In particular, our result covers stochastic differential equations driven by (time-inhomogeneous) Levy processes. More precisely, we prove that weak uniqueness, i.e. uniqueness in law, implies weak joint uniqueness, i.e. joint uniqueness in law for the solution process and its driver.
Ergodicity of Stochastic two-phase Stefan problem driven by pure jump Lévy noise
本文研究由纯跳Lévy噪声驱动的随机两相Stefan问题，证明其强解存在唯一并建立遍历性，同时刻画不变测度支撑集以给出平稳解的正则性。
Schauder estimates for stochastic transport-diffusion equations with Lévy processes
考虑一类带Lévy噪声和Hölder连续系数的迁移-扩散方程.利用热核估计，我们得到了温和解的Schauder估计。此外，当输运项为零时，当$p=2$为零时，我们证明了在空间变量上的Hölder指数是最优的.
Second derivative of the log-likelihood in the model given by a Levy driven stochastic differential equations
利用Malliavin演算，给出了一类基于离散时间观测的Levy过程驱动的随机系统解的对数似然函数二阶导数的整表示.
In this work, we present sufficient conditions for the existence of a stationary solution of an abstract stochastic Cauchy problem driven by an arbitrary cylindrical Lévy process, and show that these conditions are also necessary if the semigroup is stable, in which case the invariant measure is unique. For typical situations such as the heat equation, we significantly simplify these conditions without assuming any further restrictions on the driving cylindrical Lévy process and demonstrate their application in some examples.
Scaling limit of the step-reinforced and stochastic Levy--Lorentz model on weakly entangled integer lattice
本文研究了弱关联整数晶格上步强化和随机Levy-Lorentz模型的标度极限，探讨了随机游走的记忆性及中心极限定理的获得。
The existence-uniqueness theory for solutions to stochastic dynamic systems is always a significant theme and has received a huge attention. The objective of this article is to study the mentioned theory for stochastic functional differential equations (SFDEs) driven by G-Lévy process. The existence-uniqueness theorem for solutions to SFDEs driven by G-Lévy process has been determined. The error estimation between the exact solution and Picard approximate solutions has been shown. In addition, the exponential estimate has been derived.
In this article we give sufficient and necessary conditions for the existence of a weak and mild solution to stochastic evolution equations with (general) Lévy noise taking values in the dual of a nuclear space. As part of our approach we develop a theory of stochastic integration with respect to a Lévy process taking values in the dual of a nuclear space. We also derive further properties of the solution such as the existence of a solution with square moments, the Markov property and path regularity of the solution. In the final part of the paper we give sufficient conditions for the weak con…
For stochastic reaction-diffusion equations with Lévy noises and non-Lipschitz reaction terms, we prove that $W_1H$ transportation cost inequalities hold for their invariant probability measures and for their process-level laws on the path space with respect to the $L^1$-metric. The proofs are based on the Galerkin approximations.
A Functional Limit Theorem for stochastic integrals driven by a time-changed symmetric α-stable Lévy process
在适当的尺度和分布假设下，证明了由时变对称α-稳定Lévy过程驱动的确定函数的随机积分序列在具有M_1-拓扑的Skorokhod空间中的收敛性.时变由逆β稳定从属子给出。
The work concerns nonlinear filtering problems of stochastic differential equations with correlated Lévy noises. First, we establish the Kushner-Stratonovich and Zakai equations through martingale representation theorems and the Kallianpur-Striebel formula. Second, we show the pathwise uniqueness and uniqueness in joint law of weak solutions for the Zakai equation. Finally, we investigate the uniqueness in joint law of weak solutions to the Kushner-Stratonovich equation.
We study stochastic Korteweg - de Vries equation driven by Lévy noise consisting of the compensated time homogeneous Poisson random measure and a cylindrical Wiener process. We prove the existence of a martingale solution to the equation studied. In proof of the existence theorem we use the Galerkin approximation and several auxiliary results suitable for the problem considered.
In this article we show that a finite dimensional stochastic differential equation driven by a Lévy process can be formulated as a stochastic partial differential equation. We prove the existence and uniqueness of strong solutions of such stochastic PDEs. The solutions that we construct have the `translation invariance' property. The special case of this correspondence for diffusion processes was proved in [Rajeev, Translation invariant diffusion in the space of tempered distributions, Indian J. Pure Appl. Math. 44 (2013), no.~2, 231--258].
Reflected Backward Stochastic Differential Equations Driven by Lévy Process
研究了一类与下半连续凸函数的次微分算子相关联的反射倒向随机微分方程，该次微分算子由与Lévy过程相关联的Teugels鞅驱动.利用罚函数法得到了这类方程解的存在唯一性。作为应用，我们给出了一类偏微分-积分包含解的概率解释。
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