Stochastic Analysis, Mathematical Physics and Applications

The work of the group concentrates on a variety of problems and methods in geometric and stochastic analysis, very relevant to problems arising in Mathematical Physics. We study evolution equations in kinetic theory and interface problems that often involve the development of functional inequalities, calculus of variations, stochastic processes and differential geometry. The group works on the following problems.

Optimal estimates in Statistical Mechanics and convergence to equilibrium: The Boltzmann equation has been a paradigm for nonlinear dissipative evolutions just as the heat equation has for linear dissipative evolution. Many powerful methods frequently used in PDE such as the entropy-entropy dissipation method have their roots in the Boltzmann equation. Convergence to equilibrium for solutions of the spatially homogeneous Boltzmann equation hás been studied by many authors. A new method we introduced to study convergence for soft potentials was a way to use entropy production bounds without having pointwise lower bounds on the solution.While one has such bounds in the hard potential case, one does not in the soft potential case, arising from very long interactions. This and other new methods introduced in previous work will be developed to be applied to other dissipative evolution equations and to refine the results obtained.

Another strategy is of stochastic nature to study the rate of aproach to equilibrium for Boltzmann equation.In 1956 Kac introduced the notion of propagation of chaos and showed how this enabled to relate the rate of approach to equilibrium for a stochastic process describing random binary collisions in an N-body system.The Kac master equation for this process is linear.Kac's proposal was to estimate the rate of approach for Boltzmann eq in terms of that for the Master eq. Recent years have seen much progress in Kac program;strong information has been obtained on the rate of approach to equilibrium for the Master eq,which motivates further development of the rest of Kac program and in particular the refinement of his notion of propagation of chaos into a form that allows to deduce entropic convergence for Boltzman eq. from entropic convergence for the Master eq.It is also a goal to continue the work developed for

one-component to multicomponet interacting particle systems in the continuum towards new analytic methods and applications.

Exact solutions and nonlinear estimates for charged kinetic and fluid equations: The study of charged kinetic and fluid equations is of great interest for many technological applications.In addition to control and turbulence problems, the generation of large scale structures heavily contributes to the nature of particle and energy transport as well as disruption of steady equilibrium states.Present in experiments and in simulations of numerical codes, an understanding of the nature and control of these coherent structures can only by obtained by the construction of exact solutions and nonlinear growth estimates. The group focuses on the full Maxwell-Vlasov eqns and on reduced fluid eqns. Harmonic maps: Some mappings between Riemannian manifolds are critical points of a natural energy functional, generalizing the Dirichlet integral to the setting of Riemannian manifolds. In the late 70's, this area obtained a new

income from mathematical physics, in the guise of the non-linear sigma model or chiral model. As a consequence harmonic maps have attracted a much wider audience even beyond the mathematical community.They also have applications to the theory of liquid crystals, robotics, and stochastic processes. A recent key point about harmonic maps is the idea that the equation is an integrable system; the harmonic map equation admits a zero-curvature reformulation

and so corresponds to loops of flat connections. In this setting we deal with problems that are related with the geometry of harmonic maps into symmetric spaces.