Stochastic differential equations with rough coefficients

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

Non-explosion by Stratonovich noise for ODEs

We show that the addition of a suitable Stratonovich noise prevents the explosion for ODEs with drifts of super-linear growth, in dimension $d\ge 2$. We also show the existence of an invariant measure and the geometric ergodicity for the corresponding SDE.Comment: 16 page

Existence and uniqueness by Kraichnan noise for 2D Euler equations with unbounded vorticity

We consider the 2D Euler equations on $\mathbb{R}^2$ in vorticity form, with unbounded initial vorticity, perturbed by a suitable non-smooth Kraichnan transport noise, with regularity index $\alpha\in (0,1)$. We show weak existence for every $\dot{H}^{-1}$ initial vorticity. Thanks to the noise, the solutions that we construct are limits in law of a regularized stochastic Euler equation and enjoy an additional $L^2([0,T];H^{-\alpha})$ regularity. For every $p>3/2$ and for certain regularity indices $\alpha \in (0,1/2)$ of the Kraichnan noise, we show also pathwise uniqueness for every $L^p$ initial vorticity. This result is not known without noise

The enhanced Sanov theorem and propagation of chaos

We establish a Sanov type large deviation principle for an ensemble of interacting Brownian rough paths. As application a large deviations for the ($k$-layer, enhanced) empirical measure of weakly interacting diffusions is obtained. This in turn implies a propagation of chaos result in rough path spaces and allows for a robust subsequent analysis of the particle system and its McKean-Vlasov type limit, as shown in two corollaries.Comment: 42 page

Uniform Approximation of 2D Navier-Stokes Equations with Vorticity Creation by Stochastic Interacting Particle Systems

We consider a stochastic interacting particle system in a bounded domain with reflecting boundary, including creation of new particles on the boundary prescribed by a given source term. We show that such particle system approximates 2d Navier-Stokes equations in vorticity form and impermeable boundary, the creation of particles modeling vorticity creation at the boundary. Kernel smoothing, more specifically smoothing by means of the Neumann heat semigroup on the space domain, allows to establish uniform convergence of regularized empirical measures to (weak solutions of) Navier-Stokes equations.Comment: 37 page

Well-posedness of a reaction-diffusion model with stochastic dynamical boundary conditions

We study the well-posedness of a nonlinear reaction diffusion partial differential equation system on the half-line coupled with a stochastic dynamical boundary condition, a random system arising in the description of the evolution of the chemical reaction of sulphur dioxide with the surface of calcium carbonate stones. The boundary condition is given by a Jacobi process, solution to a Brownian motion-driven stochastic differential equation with a mean reverting drift and a bounded diffusion coefficient. The main result is the global existence and the pathwise uniqueness of mild solutions. The proof relies on a splitting strategy, which allows to deal with the low regularity of the dynamical boundary condition