Diffusion of regularity and propagation of isotropic singularities induced by quadratic differential operators

In this thesis, we study the diffusion of regularity and the propagation of isotropic microsingularities induced by quadratic forms, or by the corresponding quadratic differential operators, from two different perspectives. The first part is devoted to constructing a subelliptic diffusion on a closed manifold associated with a second order subelliptic operator and to establish convergence to equilibrium for such a diffusion. The construction we carry out is based on the notion of subunit curves, namely curves for which the velocity vector is controlled from above by the principal symbol of the operator under consideration, which is locally a quadratic form with respect to the frequency variables. To construct the random walk and prove the convergence result, we reduce the problem to the study of a constant coefficient operator that is locally equivalent to our second order subelliptic operator, in the sense that the diffusion generated by it induces a local diffusion for our operator. By using the compactness property this local diffusion can be lifted to a global diffusion, and the convergence result is then obtained via the spectral theory of the associated Markov operator. In the second part, we investigate the propagation of isotropic singularities for evolution equations whose generator is given by a Weyl-quantization of a global (that is, on both the position and frequency variables) complex quadratic form. In order to establish the propagation results, we develop the isotropic microlocal theory and we use it to relate the Shubin-Sobolev microsingularities of the solution of the evolution equation to those of the initial datum and the singular space associated with the quadratic form.