Stochastic Ordinary Differential Equations: Existence, Uniqueness, and Flows Properties

The principal steps in defining “mesoscopic” equations for the positions of particles in R ^d are as follows: We first construct solutions of stochastic ordinary differential equations in the sense of Itô (SODEs) ¹ such that solutions r (t,γ̃, q) depend “nicely” on some measure-valued process γ̃ and on the initial condition q. To proceed from SODEs to stochastic partial differential equations (SPDEs) for the mass distribution of particles, governed by the flow q ↦ r(t, γ̃, q), we proceed in a second step as follows: Suppose r(t, γ̃, q) is measurable in q with respect to the Borel σ-algebra B ^d of R ^d and integrable with respect to some initial measure μο.