Stochastic Differential Equations in a Scale of Hilbert Spaces

This thesis consists of two main sections. Section II is motivated by studies of stochastic differential equations in infinite dimensional spaces. Here we consider an SDE with coefficients defined in a scale of Hilbert spaces and prove existence, uniqueness and path-continuity of infinite-time solutions using a variation of Ovsjannikov’s method. Markov property and several norm estimates are also established. Our findings are then applied to a system of equations describing non-equilibrium stochastic dynamics of (real-valued) spins of an infinite particle system on a typical realization of a Poisson or Gibbs point process in Rn. Section III is motivated by studies of stochastic systems describing non-equilibrium dynamics of (real-valued) spins of an infinite particle system in Rn. Here we consider a row-finite system of stochastic differential equations with dissipative drift. The existence and uniqueness of infinite-time solutions is proved via finite volume approximations and a version of Ovsjannikov’s method.