Mathematical Sciences: Randomly Perturbed Infinite Dimensional Systems

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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.