Mathematical Sciences: Randomly Perturbed Infinite Dimensional Systems

9414153：Kotelenez摘要。 本提案中描述的研究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乌克兰科学院数学研究所。 随机分析的主题是由爱因斯坦和Smoluchowski的工作，后来奥恩斯坦和乌伦贝克，对（重）“布朗”粒子在轻粒子介质的动机。 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，已经尝试开发物理上有意义的随机方程的扩散和分支粒子的分布。 到目前为止，严格的随机偏微分方程（SPDE）的质量分布已获得只有在非常特殊的情况下（空间维度1），由于非常奇异的字符的噪声（粒子的水平上的独立假设的结果）。 私家侦探最近，美国的P.Kotelenez开发了一种新的SPDE方法，该方法导致在任何空间维度上物理上有意义和数学上可解的SPDE。 由于这种方法开始与随机常微分方程的位置和动量的粒子，然后推导出SPDE的介观方程从微观方程，这些方程在物理科学中的适用性是直接的。 严格的结果 在长程相互作用和质量的情况下， 节约 与乌克兰调查人员一起， 在有限和无限维度的专业知识，我们将尝试 将这种方法扩展到一大类有物理意义的方程 用于短距离的相互作用和扩散粒子系统 相互作用和湮灭。 拟议研究的另一个重点是 在所谓的预期积分，产生于椭圆方程和 某些积分方程 预期一体化实际上起源于基辅，并在西欧（在 称为“Skorokhod积分”），但近年来 在北美完成。 本提案中描述的研究试图利用不同的方法和优势， 在两个机构的随机分析和相关领域：凯斯西方 储备大学和乌克兰数学研究所 科学院在基辅。 随机分析的主题是 受到爱因斯坦和Smoluchowski，以及后来的Ornstein和Uhlenbeck关于布朗运动等过程的工作的启发。 在北美，粒子系统中的随机建模一直受到重视： 特别是相互作用的粒子和所谓的“超过程”。 在 在后一个领域，已经尝试开发物理上有意义的随机方程的扩散和分支的分布 粒子 到目前为止，严格随机偏微分方程 (SPDE的）仅在非常特殊的情况下才获得。 私家侦探上 美国方面（P. Kotelenez）最近开发了一种新的方法，导致物理上有意义的和数学上可解的SPDE的任何空间维的SPDE的。 这种方法在物理科学中具有直接的适用性。 我们将与在有限和无限维度方面都有专长的乌克兰调查人员一起，努力进一步扩大这一方法。 拟议研究的另一个重点是所谓的预测 积分，产生于椭圆方程和某些积分方程。 预期一体化实际上起源于基辅，近年来在西欧受到了很大的关注（在那里被称为“Skorokhod积分”），但在北美却做得很少。

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.