A Large Deviations Approach to the Homogenization of Nonlocal PDEs

05.11.2018, 10:15  –  Haus 9, Raum 2.22
Forschungsseminar Wahrscheinlichkeitstheorie

Dr. André de Oliveira Gomes

Homogenization theory studies, roughly speaking, the effects of high frequency oscillations on the coefficients of the solutions of partial differential equations (PDEs for short). In the simplest setting we are given a PDE with two natural scales, a macroscopic scale of order 1 and a microscopic scale of order \delta, the later measuring the period of oscillations. The interplay between Probability theory and PDEs is well known by the celebrated Feynman-Kac formula that expresses the solutions of parabolic PDEs as functionals of microscopic particles governed by stochastic differential equations (SDEs for short). By the use of Forward-Backward Stochastic Differential Equations (FBSDEs for short) this link is extended to quasilinear and semilinear parabolic PDEs. This is the core of the probabilistic approach we propose for the study of the homogenization problem that arises within the interplay between the scaling of the two parameters that get naturally involved: the homogenization (oscillation frequency intensity) parameter of the PDE itself and the parameter that tunes the source of the noise term in the underlying SDE.

It is our goal to study, when both parameters scale in some sense to be defined, the behavior of the solutions of the nonlocal PDEs that are associated to a certain class of FBSDEs with jumps. In order to be self-sufficient we intend to survey briefly how the connections between nonlocal PDEs and FBSDEs with jumps emerge. Secondly, before showing the results on the homogenization problem that we seek to expose, we explore how to build the large deviations principle for the system of FBSDEs involved in this work and we comment some specificities of that construction. With this in mind we hope to give a double flavor to this talk that comes with the combination of probabilistic techniques with analytical ones.