SCAPDE · Semi-Classical Analysis and Partial Differential Equations

Semi-classical analysis started to be developed about 50 years ago by the works of Sato and Hormander on micro-local analysis.Nowadays, it has reached great achievement with many applications to different topics in analysis including spectral theory, scattering theory, control theory, and some aspects in non linear equations, by the use of dispersive estimates and paraproduct techniques .The objective of our proposal is to develop new tools and applications in two directions : boundary value problems and connections between probability and semi-classical analysis. We expect to solve basic remaining open problems in the analysis of boundary problems, and to make contributions to develop new links between probability and analysis of partial differential equations.We will focus on four topics :- 1) Dispersive and Strichartz estimates for wave or Schrödinger equations in domains. Applications to the Cauchy problem for non linear waves in domains.- 2) Theoretical analysis of the optimal control operator in control theory.- 3) Analysis of Markov Chain Monte Carlo algorithm of Metropolis type via PDE's tools.- 4) Applications of probabilistic tools to the analysis of PDE.Topics 1) and 2) are strongly connected to progress in the analysis of boundary value problems.Topic 3) involves a generalization of the classical pseudo-differential calculus. The purpose of topic 4) is to develop a new field of research for deterministic PDE's (and therefore is not in the area of stochastic PDE's).All topics involve geometric analysis in the phase space.""