We consider the Schrödinger equation on the one dimensional torus with a general odd-power nonlinearity $p \geq 5$, which is known to be globally well-posed in the Sobolev space $H^\sigma(\mathbb{T})$, for every $\sigma \geq 1$, thanks to the conservation and finiteness of the energy. For regularities σ < 1, where this energy is infinite, we explore a globalization argument adapted to random initial data distributed according to the Gaussian measures µs, with covariance operator $(1-\Delta)^s$, for s in a range $(s_p,\frac{3}{2}]$. We combine a deterministic local Cauchy theory with the quasi-invariance of Gaussian measures µs, with additional Lq-bounds on the Radon-Nikodym derivatives, to prove that the Gaussian initial data generate almost surely global solutions. These Lq-bounds are obtained with respect to Gaussian measures accompanied by a cutoff on a renormalization of the energy; the main tools to prove them are the Boué-Dupuis variational formula and a Poincaré-Dulac normal form reduction. This approach is similar in spirit to Bourgain’s invariant argument [7] and to arecent work by Forlano-Tolomeo in [18].

This paper contains a review of the results of the martingale approach to point processes and its applications to the theory of dynamical systems (in the engineering sense), mainly estimation problems.

Some of the topics reviewed are: stochastic intensity, absolutely continuous changes of measure, martingale representations, changes of times, filtering, queues, etc.