Seminario del Departamento de
Ecuaciones Diferenciales y Análisis Numérico

Fecha : 30 de mayo de 2023
Hora  : 10:00
Lugar : Seminario del Departamento (Fac. de Matemáticas, 3a. planta, módulo 34)

Alberto Pérez Cervera

(U. Complutense de Madrid)

Resumen

The Kolmogorov forward and backwards equations are classical and well known equations describing the evolution of the conditional density of an associated SDE. During many decades, a usual approach to the solutions of these equations has been based on the spectral decomposition of the conditional density. Interestingly, in recent years, the eigenfunctions of the Kolmogorov backwards operator L* are being shown to store very important information if the SDE describes stochastic oscillatory motion. The aim of this talk is two fold: On the one hand, we want to introduce the audience to this framework, in particular showing - A meaningful stochastic phase function is encoded in the argument of the principal complex eigenfunction of L* - The principal purely real eigenfunction of L* corresponds with a meaningful stochastic amplitude function - A universal statistical description of the stochastic oscillator can be achieved if one transforms the system to the full principal complex eigenfunction of L* On the other hand, besides discussing contexts of application of this framework, we aim to raise important questions which still remain open in this promising framework: numerical and theoretical results suggest that the before mentioned stochastic phase and amplitude functions converge to their deterministic analogues. However, a rigorous proof is still missing. In addition, there might also be room for a dramatic improvement of the numerical procedure leading to the obtention of the L* eigenfunctions. Many open questions I look forward to discuss with Sevilla colleagues