• Fokker-Planck equation
• generalized stochastic flow
• isotropic Brownian flow
• regularization by noise
• rough coefficients
• stochastic continuity equation
• stochastic differential equation
• Wiener chaos

This thesis deals with the study of the stochastic continuity equation (SCE)
on R^d
under low regularity assumptions on the coefficients. This is the equations for the mass associated to an SDE on R^d.

In the first chapter Wiener pathwise uniqueness (i.e. uniqueness for solutions adapted to Brownian filtration) is proved for the SCE,. We use Wiener chaos to reduce Wiener uniqueness for the SCE to uniqueness for the Fokker-Planck
equation (FPE). The method consists of projecting the
equation on the Wiener chaos spaces and using the shift effect of the projectors in order to discard the Ito integral.

The second chapter deals with the SCE for flows, following Le Jan-
Raimond’s approach. Here we need that the FPE admits a particular semigroup as solution, which is guaranteed by the theory of Dirichlet forms under
mild assumptions on the coefficients. Wiener chaos gives Wiener uniqueness

(as before) and also existence. Another method of existence is based on
filtering a weak solution X of the associated SDE with respect to a certain
cylindrical Brownian motion W.


In the third chapter, we consider the case of a rough drift . Here a
phenomenon of regularization by noise can be observed: the results in the first
chapter give immediately Wiener uniqueness for the SCE, while uniqueness
does not hold in the deterministic case without additional
hypotheses on the drift. We cite an example of this phenomenon.


We prove also that, in many cases, strong uniqueness (i.e. Uniqueness
with respect to every filtration, not only Brownian filtration) holds for the
SCE. This is not surprising since a strong uniqueness result (due to Krylov-
Röckner) holds for the SDE. First, extending Ambrosio’s approach, we associate to every measure-valued solution of the SCE a superposition solution
N. Then, starting
from N, we build a weak solution of the SDE. This correspondence and
Krylov-Röckner’s result imply strong uniqueness for the SCE.


The last chapter is about a particular class of generalized flows, the
isotropic Brownian flows (IBFs). An IBF is a family of Brownian motions, indexed by their starting points in Rd , which are invariant in law
for translation and rotation; it can be found as (possibly generalized) solution S of an SCE with isotropic infinitesimal covariance function
K. Here we consider K’s driven by two parameters
α and η, related respectively to the correlation of the two-point motion and
to the compressibility of the flow. Studying the distance between the motions
of two points (which is a 1-dimensional diffusion), we find that coalescence
and/or splitting occur, depending of the values of α, η and d. This analysis makes rigorous the results for a simple model of turbulence
and shows that this situation cannot be described classically, thus motivating
the theory of generalized flows.


Finally two appendices recall preliminaries and technical results.