Title
Non-Selection of Lagrangian Trajectories in the Zero-Noise Limit for a Class of Stochastic Regularizations
Abstract
Stochastic differential equations with irregular drift can exhibit striking non-uniqueness phenomena. A natural question is whether adding a small amount of noise selects a distinguished solution in the zero-noise limit, or whether the underlying non-uniqueness persists.
In this talk, I will discuss a negative answer to this selection problem for SDEs driven by divergence-free, Hölder-continuous vector fields. More precisely, we prove lack of selection in the zero-noise limit for vector fields with Hölder exponent \alpha \in (0,1), arbitrarily close to 1 but fixed. The result applies to a broad class of regularizing additive noises, including fractional Brownian motion and stable Lévy processes.
The proof combines deterministic and probabilistic ideas. On the deterministic side, we use pathwise Lagrangian arguments based on the analysis of flows associated with mixing velocity fields. On the stochastic side, we rely on estimates obtained through the stochastic sewing lemma. This allows us to prove that lack of selection occurs simultaneously for a large set of initial data, whose complement can be made arbitrarily small in Lebesgue measure.
This is joint work with Lucio Galeati and Massimo Sorella.
Please note that the seminar will take place in person in room 140 of Huxley Building.