arXiv subjects

Szymon Sobczak

Publications and source records attributed to Szymon Sobczak.

3 recordsLinked to original sources

arXiv2026

Ekman Boundary Layers Under Transport Noise

We consider a 3D rotating Navier-Stokes equation with transport-stretching noise, posed between two horizontal plates, and study the joint limit of vanishing vertical viscosity and rapid rotation. Our driving noise depends only on the horizontal coordinates and its vertical component vanishes with the viscosity. Provided that the initial data is purely horizontal, we construct martingale weak solutions which converge in $L^2_ωL^\infty_tL^2_x$ to the layered strong solution of a 2D stochastic Navier-Stokes equation with damping. The damping coefficient is dependent on the limit of the ratio between vertical viscosity and inverse rotation rate.

math.AP↗

arXiv2026

Navier-Stokes with a fractional transport noise as a limit of multi-scale dynamics

We define a bona fide rough path solution for the Navier-Stokes equation with an additional rough transport term, and show that the SPDE on the three-dimensional torus driven by a fractional Brownian motion on $H^σ$ has solutions characterised as the effective limits of a slow/fast system. We further show that this rough path solution is equivalent to the widely used incremental notion of solution (the unbounded rough driver formulation), demonstrating broader applicability to other nonlinear SPDEs.

math.PR↗

arXiv2025

Fluctuations from a random fractional averaging limit

We consider a system of multiscale stochastic differential equations whose slow component is drivenby a fractional Brownian motion with Hurst parameter H greater than 1/2. Under ergodic assumptions ensuring the applicability of the fractional averaging and fractional homogenization theorems of Hairer and Li (arXiv:1902.11251, arXiv:2109.06948), we establish a fluctuation result. The deviation of the slow motion, scaled by epsilon^{1/2-H}, from its effective, time-dependent random limit converges, as the time-separation scale epsilon tends to zero, to the solution of a stochastic differential equation driven by a fractional Brownian motion and influenced by an additional space--time Gaussian field. Since the averaging principle and the fractional homogenization hold in different modes of convergence, obtaining the required joint convergence is a delicate matter. Moreover, neither the continuity of the Ito--Lyons solution map nor the martingale method is directly applicable for our purposes, so the proof requires several innovations. To establish the fluctuation theorem, we combine cumulant methods with a residue lemma and formulate the enlarged system as a rough differential equation in a suitable space.

math.PR↗