Wong-Zakai approximations of SDEs and SPDEs with jump noise
具有跳跃噪声的 SDE 和 SPDE 的 Wong-Zakai 近似
基本信息
- 批准号:315297061
- 负责人:
- 金额:--
- 依托单位:
- 依托单位国家:德国
- 项目类别:Research Grants
- 财政年份:2016
- 资助国家:德国
- 起止时间:2015-12-31 至 2018-12-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
Stochastic ordinary or partial differential equations driven by a Brownian motion or Lévy processes are indispensable for the modelling of various real world phenomena. However their solutions are already by construction just convenient mathematical idealizations of real processes. About 50 years ago, Wong and Zakai suggested to treat stochastic differential equations as limits of the ordinary random equations driven by path-wise regular (e.g. smooth) approximations of the noise process. In this approach, Brownian motions can be seen as idealization of short range chaotic motions (diffusion), whereas jumps appear as idealizations of very fast continuous long range anomalous transitions. It is known in case of stochastic differential equations, that the limiting process solves the Stratonovich equation in the Brownian case, and the canonical (Marcus) equation in the general case with jump noise.Motivated by examples form physics, hydrology and engineering, we are going to underpin the Wong--Zakai type approximations for stochastic ordinary and partial differential equations driven by Lévy noise. The main emphasis will be made on the convergence of the regular approximations to a discontinuous limit in the non-standard Skorokhod topology and to the identification of proper correction terms in the limiting stochastic equation. One focus of the project will be set on the advection-diffusion equations in the whole space with Lévy noise acting on the transport term, which can be related to the turbulent diffusivity. For advection-diffusion equations on bounded domains, Lévy noise on the boundary will mimic an instantaneous release of a contaminant into a ground water. Finally, we explore the numerical methods of solving SPDEs with the help of deterministic solvers applied to the Wong-Zakai approximations.The results obtained in the project, besides their mathematical value, should contribute to a deeper understanding of the Lévy driven dynamics and numerics in physics and applied sciences.
由布朗运动或L过程驱动的随机常微分方程或偏微分方程对于各种现实世界现象的建模是不可或缺的。然而,它们的解已经是通过构造得到的,只是对真实过程的方便的数学理想化。大约50年前,Wong和Zakai建议将随机微分方程视为由噪声过程的路径规则(例如光滑)近似驱动的普通随机方程的极限。在这种方法中,布朗运动可以被看作是短程混沌运动(扩散)的理想化,而跳跃则表现为非常快的连续长程反常跃迁的理想化。在随机微分方程的情况下,极限过程在布朗情形下求解Stratonovich方程,在一般情况下求解带有跳跃噪声的正则(Marcus)方程。在物理、水文学和工程学的例子的激励下,我们将支持由L噪声驱动的随机常微分方程和偏微分方程组的Wong-Zakai型近似。主要着重于非标准Skorokhod拓扑中对不连续极限的正则逼近的收敛,以及极限随机方程中适当修正项的识别。该项目的一个重点将放在整个空间的对流-扩散方程上,其中L噪声作用于传输项,这可以与湍流扩散率有关。对于有界域上的对流扩散方程,边界上的L噪声将模拟污染物向地下水中的瞬时释放。最后,我们探索了应用于Wong-Zakai近似的确定性求解器求解SPDEs的数值方法,所获得的结果除了其数学价值外,还有助于在物理和应用科学中更深入地理解L驱动的动力学和数值计算。
项目成果
期刊论文数量(1)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Advection-diffusion equation on a half-line with boundary Lévy noise
具有边界 Lévy 噪声的半线上平流扩散方程
- DOI:10.3934/dcdsb.2018200
- 发表时间:2019
- 期刊:
- 影响因子:0
- 作者:L.-S. Hartmann;I. Pavlyukevich
- 通讯作者:I. Pavlyukevich
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Professor Dr. Ilya Pavlyukevich其他文献
Professor Dr. Ilya Pavlyukevich的其他文献
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{{ truncateString('Professor Dr. Ilya Pavlyukevich', 18)}}的其他基金
Asymptotic analysis of multiscale Lévy-driven stochastic Cucker-Smale and non-linear friction models
多尺度 Lévy 驱动的随机 Cucker-Smale 和非线性摩擦模型的渐近分析
- 批准号:
418509727 - 财政年份:2018
- 资助金额:
-- - 项目类别:
Research Grants
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- 批准号:11971186
- 批准年份:2019
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- 项目类别:面上项目
相似海外基金
Research Initiation: Numerical Integration Methods For the Robust Zakai Equation of Nonlinear Filtering Theory
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8307433 - 财政年份:1983
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