Mathematical Sciences: Randomly Perturbed Infinite Dimensional Systems

9414153: Kotelenez摘要。本提案中描述的研究是试图利用两个机构在随机分析和相关领域的不同方法和优势：凯斯西储大学和基辅的乌克兰科学院数学研究所。随机分析这门学科是由爱因斯坦和斯摩鲁霍夫斯基，以及后来的奥恩斯坦和乌伦贝克对轻粒子介质中（重的）布朗粒子的研究而产生的。在北美，重点一直放在粒子系统的随机建模上：特别是相互作用的粒子和所谓的“超过程”。在后一个领域，人们试图为扩散和分支粒子的分布建立物理上有意义的随机方程。到目前为止，质量分布的严格随机偏微分方程（SPDE’s）仅在非常特殊的情况下（空间维度为1）才得到，这是由于噪声的非常奇异的特性（粒子水平上的独立性假设的结果）。美国方面的P.I. （P. Kotelenez）最近开发了一种新的SPDE方法，该方法可以在任何空间维度上产生物理上有意义且数学上可解的SPDE。由于这种方法从粒子位置和动量的随机常微分方程开始，然后从微观方程推导出SPDE作为介观方程，因此这些方程在物理科学中的适用性是立即的。对于远距离相互作用和质量守恒的情况，得到了严格的结果。我们将与拥有有限和无限维度专业知识的乌克兰研究人员一起，尝试将这种方法扩展到具有短程相互作用和湮灭的相互作用和扩散粒子系统的大型物理意义方程中。提出的研究的另一个重点是所谓的预期积分，出现在椭圆方程和某些积分方程中。预期整合实际上起源于基辅，近年来在西欧得到了很多关注（在那里它被称为“Skorokhod整合”），但在北美做得很少。本提案中描述的研究是试图利用两个机构在随机分析和相关领域的不同方法和优势：凯斯西储大学和基辅的乌克兰科学院数学研究所。随机分析这门学科是由爱因斯坦和斯摩鲁霍夫斯基，以及后来的奥恩斯坦和乌伦贝克对布朗运动等过程的研究而产生的。在北美，重点一直放在粒子系统的随机建模上：特别是相互作用的粒子和所谓的“超过程”。在后一个领域，人们试图为扩散和分支粒子的分布建立物理上有意义的随机方程。到目前为止，严格的随机偏微分方程（SPDE's）只在非常特殊的情况下才得到。美国方面的P.I. （P. Kotelenez）最近开发了一种新的SPDE方法，该方法可以在任何空间维度上产生物理上有意义且数学上可解的SPDE。这种方法在物理科学中有直接的适用性。我们将与在有限和无限方面都有专门知识的乌克兰调查人员一道，努力进一步扩大这一办法。提出的研究的另一个重点是所谓的预期积分，出现在椭圆方程和某些积分方程中。预期整合实际上起源于基辅，近年来在西欧得到了很多关注（在那里它被称为“Skorokhod整合”），但在北美做得很少。

9414153: Kotelenez Abstract. The research described in this proposal is an attempt to capitalize on the different approaches and strengths in stochastic analysis and related areas at two institutions: Case Western Reserve University and the Institute of Mathematics of the Ukrainian Academy of Sciences in Kiev. The subject of stochastic analysis is motivated by the work of Einstein and Smoluchowski, and later Ornstein and Uhlenbeck, on the (heavy) "Brownian" particle in a medium of light particles. In North America much emphasis has been on stochastic modeling in particle systems: in particular, interacting particles and so-called "superprocesses". In the latter area, attempts have been made to develop physically meaningful stochastic equations for the distribution of diffusing and branching particles. So far, rigorous stochastic partial differential equations (SPDE's) for the mass distribution have been obtained only in very special cases (space dimension one), due to the very singular character of the noise (a result of the independence assumptions on the level of particles). The P.I. on the American side (P. Kotelenez) recently developed a new approach to SPDE's which leads to physically meaningful and mathematically solvable SPDE's in any space dimension. Since this approach starts with stochastic ordinary differential equations for the positions and momenta of the particles and then derives the SPDE's as mezoscopic equations from the microscopic equations, the applicability of these equations in the physical sciences is immediate. Rigorous results have been obtained for the case of long range interaction and mass conservation. Together with the Ukrainian investigators, who have expertise in both finite and infinite dimensions, we wil l attempt to extend this approach to a large class of physically meaningful equations for interacting and diffusing particle systems with short range interaction and annihilation. Another focus of the proposed research is on so-called anticipating integration, arising in elliptic equations and certain integral equations. Anticipating integration actually originated in Kiev and has been given much attention in Western Europe (where it is called the "Skorokhod integral") in recent years but much less has been done in North America. The research described in this proposal is an attempt to capitalize on the different approaches and strengths in stochastic analysis and related areas at two institutions: Case Western Reserve University and the Institute of Mathematics of the Ukrainian Academy of Sciences in Kiev. The subject of stochastic analysis is motivated by the work of Einstein and Smoluchowski, and later Ornstein and Uhlenbeck, on processes like Brownian motion. In North America much emphasis has been on stochastic modeling in particle systems: in particular, interacting particles and so-called "superprocesses". In the latter area, attempts have been made to develop physically meaningful stochastic equations for the distribution of diffusing and branching particles. So far, rigorous stochastic partial differential equations (SPDE's) have been obtained only in very special cases . The P.I. on the American side (P. Kotelenez) recently developed a new approach to SPDE's which leads to physically meaningful and mathematically solvable SPDE's in any space dimension. This approach has immediate applicability in the physical sciences. Together with the Ukrainian investigators, who have expertise in both finite and infinite dimensions, we will attempt to extend the approach still further. Another focus of the proposed research is on so-called anticipating integration, arising in elliptic equations and certain integral equations. Anticipating integration actually originated in Kiev and has been given much attention in Western Europe (where it is called the "Skorokhod integral") in recent years but much less has been done in North America.