Seminário dos alunos - Formal Perturbation Theory for Stochastic Paths

Short annotation for mathematicians:

We discuss methods of perturbative QFT applied to certain stochastic differential equations (SDE), perturbations of the Ornstein-Uhlenbeck process. No knowledge of SDE or QFT is assumed.
Perturbative QFT amounts to some sort of generating function for formal power series expansions, and these methods have found many applications in pure and applied mathematics. This is the main motivation for this talk. The lack of mathematical proofs often comes together with the magical formulas derived from these methods, and this will also be examplified in this talk.
The application we discuss was the subject of joint work with Bruno Melo (ETH Zurich) and Thiago Guerreiro (PUC-RIO), published (PhysRevA) in January 2021.

Fine-print for physicists:

The motivation comes from experiments with optical tweezers, which roughly are laser beams able to trap certain particles around a point of mechanical equilibrium. The particle does not sit in the mechanical equilibrium, but jiggles stochastically around it; this stochasticity is known as Brownian motion and is the reason one uses SDE instead of ODE in the mathematical models. Often the working assumption is that the optical forces are a linear function of the displacement from mechanical equilibrium, which as we know is simply a first order Taylor approximation. We provided a way to deal with optical forces modelled by a linear function perturbed by small non-linearities, so potentially this method would be useful for sufficiently precise experiments. The validity of our method was probed by some numerical experiments, and for mysterious reasons it seems to work.