New Trends in Stochastic Analysis and Related Topics - A Volume in Honour of Professor K D Elworthy
New Trends in Stochastic Analysis and Related Topics - A Volume in Honour of Professor K D Elworthy
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随机分析和相关主题的新趋势 - 纪念 K D Elworthy 教授的卷
DOI：
10.1142/9789814360920_0001
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
Brzezniak Z
中科院分区：
文献类型：
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作者：
Brzezniak Z
在过去的二十年中，随机偏微分方程（简称SPDE）的探索已经成为数学和物理学中迅速扩展的领域。除了数学，物理和生命科学中的一些基本问题的应用外，对此类研究的兴趣还源于对理解和控制出现在自然科学和社会科学许多领域的复杂系统行为的渴望。小的随机波动，如热，存在于所有复杂系统中，即使它们的基本理论是确定性的。例如，一维非线性薛定谔方程（NLSE）出现在光波导传播和光通信中，参见Falkovich等人27。一般来说，它们模拟理想的物理情况（波的传播或热方程），忽略外部影响，如湍流或随机影响。由于这些影响的数量巨大且不可预测，因此难以对其进行详细建模。因此，引入随机扰动来模拟外力和随机波动引起的统计误差，计算机模拟证实了这种类型的模型的准确性。在过去的60年里，人们已经清楚地认识到，有时通过向特定的微分方程添加随机扰动可以获得更好和更现实的描述性结果。
Over the past two decades the exploration of Stochastic Partial Differential Equations (briefly SPDEs) has become a rapidly expanding area in Mathematics and Physics. In addition to applications to some fundamental problems in Mathematical, Physical and Life Sciences, interest in such studies is motivated by a desire to understand and control the behaviour of complex systems that appear in many areas of natural and social sciences. Small random fluctuations such as thermal are present in all complex systems even if their fundamental theory is deterministic. For example, the 1D Nonlinear Schrödinger Equation (NLSE) arises in optical waveguide propagation and in optical communication, see eg Falkovich et al. 27It is generally accepted that differential equations serve as a mathematical and rigorous support of models in natural sciences. In general, they model ideal physical situations (propagation of waves or heat equations) neglecting external influences such as turbulence or random impacts. These effects are too complicated to be modeled in detail due to its immense quantity and unpredictable character. Therefore, stochastic perturbations are introduced to model the statistical errors caused by the external forces and random fluctuations, and computer simulations corroborate the accuracy of models of this type. During the last sixty years, it has become clear that sometimes better and more realistic descriptive results can be achieved by adding a stochastic perturbation to a particular differential equation.