Soutenance de thèse : Coupling of stochastic processes in subRiemannian manifolds

In this thesis we study couplings of subelliptic Brownian motions in several subRiemannian manifolds: the free, step $2$ Carnot groups, including the Heisenberg group, as well as the groups of matrices $SU(2)$ et $SL(2,\mathbb{R})$.

Taking inspiration from previous works on the Heisenberg group we obtain successful non co-adapted couplings on $SU(2)$, $SL(2,\mathbb{R})$ (under strong hypothesis) and also on the free step $2$ Carnot groups with rank $n\geq 3$. In particular we obtain estimates of the coupling rate, leading to gradient inequalities for the heat semi-group and for harmonic functions. We also describe the explicit construction of a co-adapted successful coupling on $SU(2)$.

Finally, we develop a new coupling model "in one sweep" for any free, step $2$ Carnot groups. In particular, this method allows us to obtain relations similar to the Bismut-Elworthy-Li formula for the gradient of the semi-group by studying a change of probability on the Gaussian space.