My current work concerns the hydrodynamics of particles immersed in a thermally fluctuating, viscous, incompressible solvent. The governing equations stipulate conservation of momentum in the fluid, conservation of linear and angular momentum of the particle, and no-slip boundary conditions on the boundary of the particle. Is there existence and uniqueness for the solution? What are the limit theorems when time goes to infinity? These problems not only provide more detailed study of physical Brownian motions but also give a testing ground for the techniques in stochastic partial differential equations.
This thesis is a first step to answer these questions. We analyze parts of the system: stochastic Stokes equations in the whole space, passive point particle, passive particle with finite size. We characterize the regularity properties and statistical behaviors of the solution $u(t,x)$ and $p(t,x)$ to the stochastic Stokes equations in the whole space. We give existence and uniqueness results for passive particles (a point particles as well as a finite size particle), and we give limit theorems for a point particle when the time goes to infinity.