Parabolic equations and diffusion processes with divergence-free vector fields

The study of this thesis is motivated by the stochastic Lagrangian representations of solutions to the Navier-Stokes equations. The stochastic Lagrangian formulation to the Navier-Stokes equations is described by stochastic differential equations, which essentially represent the diffusions under divergence-free velocity fields. The associated stochastic differential equations are closely related to a class of parabolic equations and these two types of equations are the central objects of this thesis. The difficulty of the problem mainly comes from the low regularity of the velocity field. The key point is that we use the divergence-free condition to relax the regularity assumptions. The thesis is divided into two parts. The first part is the Aronson-type estimate which is an a priori estimate on the fundamental solutions (transition probability) independent of the smoothness of the coefficients. In the critical case, we obtain the Aronson estimate in its classical form, while in supercritical cases we obtain a weaker Aronson-type estimate. In the second part, we use approximation arguments to apply the Aronson estimate to the construction of solutions to the parabolic equations and the stochastic differential equations, and further regularity theory of the solutions is obtained for the critical case. Under the supercritical conditions, we will focus on the uniqueness of solutions to the parabolic equations and their relation to the construction of the diffusion processes.