BCAM Scientific Seminar: Nonlinear differential equations with a quadratic nonlinearity and their stochastic duals

Location: Maryam Mirzakhani Seminar Room at BCAM
Time: 
- 15:30: coffee break
- 16:00: Seminar


Abstract:
In this talk, I would like to discuss the existence of some duality relations between (either ordinary or partial) nonlinear differential equations with a quadratic nonlinearity and certain types of stochastic processes with multiplicative noise. Such dualities can be first guessed by performing some (formal, although non-rigorous) manipulations with the path-integral representation of the corresponding stochastic processes. A more rigorous derivation relies on Ito calculus and the backward Kolmogorov equation. I discuss these connections by focusing on two paradigmatic examples: (i) the statistics of avalanches at the depinning transition of a driven elastic manifold in the presence of a disordered landscape and (ii) the Kardar-Parisi-Zhang (KPZ) equation. Throughout the talk, I will emphasize how such dualities open the door to some interesting (and to the best of my knowledge, unknown) mappings between (classical) depinning system and a Yang-Lee non-hermitian quantum field theory from one side, and between the KPZ equation and the stochastic density functional theory of the ideal gas from the other side. 

I will take a few minutes at the beginning of my talk to briefly mention some of my other lines of research, either in disordered systems, active matter, and stochastic thermodynamics, hoping it will spark some interest and motivate discussions with different teams.