Title
Periodic homogenization for singular SDEs
Abstract
In this talk we introduce singular stochastic differential equations (SDEs) with additive Brownian, as well as stable Lévy noise, and explain the phenomenon of regularization by noise, which enables to solve them. We then generalize the theory of periodic homogenization for SDEs with smooth enough drift (cf. Bensoussan, Lions, and Papanicolaou ’78 and Franke ’07) to the setting of periodic Besov drifts with negative regularity (in the so-called rough regime). For the solution of the martingale problem associated to the singular SDE, we prove existence and uniqueness of an invariant, ergodic probability measure with strictly positive Lebesgue density via proving a strict maximum principle for the singular Fokker-Planck equation. Furthermore, we prove a spectral gap on the semigroup of the diffusion and solve the singular Poisson equation with singular right-hand-side being equal to the drift itself. In the CLT scaling, we prove that the diffusion converges in law to a Brownian motion with diffusivity involving the solution to the Poisson equation. In the pure stable noise case, we rescale space accordingly and show convergence to the stable process itself (without diffusivity enhancement). This is ongoint work together with Nicolas Perkowski.
Please note that the seminar will take place in person in room 642 of Huxley Building.